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Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Editorial With this issue of the Mathematics Teaching-Research Journal, we begin the theme of democratization of mathematics and science education. Over the next several issues, our articles, your thoughts and the discussion will create the climate to arrive at a common shared understanding of what it means to have a democratic access to mathematics education. The word democratic includes equal access and existence of a level playing field. Do these two phrases also then include, creating the conditions necessary to address the learning needs of the students? Section 1 of the current issue addresses different aspects of the process of democratization. In Section 2 we bring mathematics classroom innovations from around the world and in Section 3 we present two reflective pieces upon our teaching work. Editorial News Our journal is now funded jointly by the National Science Foundation and CUNY's Office of Undergraduate Education. A new project undertaken by the teaching-research methodology - VISUALIZE- A TR-NYC project is aimed at creating a base, where students combine their tactile and visual sense to guide themselves in visualizing the geometry they create out of making paper models of polyhedra. The project is conducted jointly with the Siedlce group of teachers and mathematicians in Poland. The nets for the polyhedra, largely a creation of Waclaw Zawadowski and Krystoff Mostowski, are entertaining students of all ages in mathematics classes and outside.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
1
Acres of Diamonds in New York’s High Stakes Testing Environment;
Empowering Educators via the Internet Steve Watsoni
The Jefferson Math Project as a Prototype
The Jefferson Math Project (JMAP) is grounded in the New York commencement level
high school mathematics curriculum. It is a non-profit initiative of two New York City Teaching
Fellows, who both began their teaching careers at Thomas Jefferson High School in Brooklyn in
2003. As we initially worked to develop our own subject matter and teaching expertise in
mathematics, we began developing and refining a database and other resources for exploring and
analyzing historical assessment practices in the Regents Math A/B curriculum. We soon found
that we were looking at high stakes testing through a pragmatic new lens that facilitated the
alignment of instruction with state assessment practices, and we also found that other teachers
wanted the resources we were developing. We began giving away these resources through
teacher training programs at Thomas Jefferson High School, Brooklyn College and City College.
In March 2005, we presented JMAP at a symposium celebrating the fifth anniversary of the New
York City Teaching Fellows, after which 2,500 copies of the JMAP 611 CD-ROM were created
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
2
by the New York City Department of Education (NYCDOE) and distributed to mathematics
teachers throughout the City in September 2005.
JMAP also established a website (www.jmap.org) for mathematics teachers in March
2005, and provides lesson plans and teaching resources on all aspects of the New York Math
A/B curriculum, as well as the new curriculum currently being implemented. These materials
include over three gigabytes of JMAP books, workbooks, worksheets, solutions, grids, graphs,
exams, etc., which are available in MS Word, Adobe pdf, and other electronic formats. JMAP
materials are intended for the use of teachers in the classroom, but students and researchers are
welcome to use JMAP’s resources as well. Several thousand visitors, mostly from the City and
State of New York, regularly download lesson plans and teaching resources from the JMAP
website. During Regents week in June 2007, the website averaged over 5,000 visitors each day.
JMAP is listed as a resource for teachers by the Association of Mathematics Teachers of
New York State, Math for America, the Drexel Math Forum, the New York Math Exchange, the
Home School Math Network, NYLearns, and many other groups interested in mathematics
education in New York State. JMAP is not affiliated with any publisher, school, or other
corporate entity, though its co-founders are both high school math teachers employed by the
NYCDOE and both have personal affiliations with either universities or publishers. As such,
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
3
JMAP provides a prototype for a relatively low-cost lesson plan and resource management
system with high potential for influencing teaching praxis.
A Vision of Technology Empowered Educators
The following diagram provides a high level overview of the major elements of a lesson
plan and resource management system based on the JMAP model.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
4
This vision for a new system involves user friendly, readily available software, which seamlessly
integrates the following features:
database management of static and dynamic resources;
word processing; and
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
5
desktop publishing.
Commercially available, integrated suites of software, such as Microsoft’s Office Suite and
Adobe Acrobat/Reader, provide all required functionality in single integrated packages, and can
serve as the central core of such a lesson plan and resource management system. A fundamental
assumption behind this design specification is that teachers are more likely to invest the time and
intellectual effort necessary for systems mastery when the lesson plan and resource management
system is free of additional costs and grounded in pre-existing, user-friendly software with which
the teacher is already familiar. If such a public domain lesson plan management system is
created, curriculum specialists/website coordinators could then focus on the development and
dissemination of databases and teaching resources that are consistent with a sponsoring
organization’s topical and/or pedagogical agenda. Wikipedia style, refereed websites for the
development and dissemination of model lesson plans in editable formats could be used to
identify and disseminate best teaching practices.
The incremental costs of such an initiative would be limited to salaries or stipends for the
curriculum specialists/website coordinator(s) plus the relatively low costs of maintaining a
website. These curriculum specialists/website coordinators could be inside or outside the official
bureaucracy of public education. (JMAP is an example of curriculum specialists/website
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
6
coordinators who are outside the official bureaucracy.) These databases and resources, once
developed, are easily shared with large numbers of teachers (and students), as evidenced by the
JMAP statistics. To the extent that the lesson plans and resources thus developed and delivered
via Internet are value adding, they will be sought out and used by teachers, thereby constituting a
powerful means of communicating and influencing the implemented, as opposed to the stated,
curriculum.
Existing Prototypes of Lesson Plan Management Systems and Their Weaknesses
Many textbook publishers produce some form of technology-based lesson plan and
resource management systems on CD-ROMs, which typically accompany the teacher’s edition of
the textbook. Examples include:
Prentice Hall’s Resource Pro and Worksheet Builder;
Saxon’s Test Generator; and
McGraw Hill/Glencoe’s Teacher Works.
The above referenced programs are examples of proprietary programs belonging to “for-profit”
publishers. Their contents are typically copyrighted, non-editable, and relatively unfriendly to
users who wish to modify or adapt them to fit specific individual and/or classroom needs.
Almost all resources in these programs are presented in difficult or impossible to manipulate
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
7
formats, presumably and understandably to protect the proprietary interests of the publishers.
Any teacher seeking to adapt or change such fixed-format resources for the benefit of a specific
individual or classroom faces intentional technological impediments imposed by the publishers
as well as potential copyright infringement issues.
Teachers frequently draw resources for a single lesson from several sources. Proprietary
lesson plan management systems are typically not designed to facilitate the development of
lesson plans from more than a single source, and are limited to that which is provided by the
textbook publisher. This severely restricts teacher creativity in meeting the needs of students.
Although these resource and lesson plan management systems are promoted by textbook
publishers as being user friendly, this writer can attest to numerous obstacles encountered while
adding over 1500 Regents Math A and Math B questions to the databases, and in formatting and
printing hundreds of resources from the enhanced databases. Such obstacles are simply
overwhelming to the majority of teachers, thus preventing the realization of the potential of these
database management systems. These obstacles are overcome through the JMAP prototype, in
which responsibility for systems mastery, database management, and desktop publishing of
ready-to-use, high quality teaching resources is concentrated in individuals who are motivated
and competent in the associated technologies and academic content areas.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
8
JMAP as a Prototype of a Lesson Plan Management System in the Public Domain
As a co-founder of JMAP, this writer attributes JMAP’s success to the integration of: 1)
highly relevant databases of public domain resources (i.e. previously administered Regents
questions); 2) highly relevant databases of dynamic proprietary resources that have, in essence,
been allowed entry into the public domain through the generosity of their proprietary owners; 3)
associated lesson plans and teaching resources in editable electronic formats; 4) Adobe portable
document format (pdf) supplements; and 5) desktop publishing capabilities. All of the above
resources are intended to provide individual teachers with the ability to quickly and easily access
and adapt teaching resources for specific classrooms and individuals.
The major emphasis of JMAP is teacher empowerment through free resources with
minimal emphasis on securing private property rights or serving economic self-interests. This
orientation toward the public good without corresponding economic interest promotes open
sharing of educational resources and is essential to JMAP’s vision of a viable lesson plan and
resource management system.
The Need for An Improved Public Domain Technology Platform
While JMAP provides an effective prototype for the empowerment of teachers and the
creation and distribution of high quality teaching resources, an improved public domain
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
9
technology platform would overcome current obstacles and present additional educational
opportunities. JMAP currently uses proprietary database management and desktop publishing
software that is no longer being updated and improved. As time passes, the limitations of the
JMAP technology platform will become more obvious. A new, public domain technology
platform could facilitate improvements in JMAP as well as new initiatives based on the JMAP
prototype in other academic areas. Furthermore, a public domain technology platform, which
might conceivably take the form of enhanced templates and “wrap-around” programs for MS
Access, could be developed at relatively low cost and widely disseminated via the Internet.
Facilitating the Attainment of Educational Goals in Several Areas
As mentioned previously, the envisioned lesson plan and resource management system
will facilitate the attainment of educational goals in several areas, including: 1) teacher
preparation and subject matter awareness; 2) aligning classroom instruction with state mandated
assessments; 3) adapting lesson plans and learning materials to differentiated student and
classroom needs; and 4) reducing new teacher stress and turnover. These advantages are further
discussed in the following paragraphs.
Improving the Subject Matter Expertise of Teachers
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
10
The development and distribution of databases of Regents and related teaching resources
promotes awareness and understanding of the intended curriculum. This is important because
awareness and understanding of the intended curriculum is useful in the training and professional
development of teachers. Research has shown that other factors being equal, people who know
more math make better mathematics teachers than people who know less math (Darling-
Hammond, Holtzman, & Gatlin, 2005) (Decker, Mayer, & Glazerman, 2004) (Hill, Rowan, &
Ball, 2005). These findings are arguably applicable for all academic subject matter areas.
Aligning Instruction with Assessment
Historical assessment practices are important for understanding current and future
assessment practices, even when assessment practices are changing. Despite appearances to the
contrary, the curriculum changes slowly and incrementally. Accordingly, historical assessment
practices will continue to inform teachers and students with respect to future assessment
practices. With respect to mathematics, the new high school curriculum currently being
implemented throughout New York shares much of its topical content with the current Math A/B
curriculum. The differences are not so much in the area of what is taught, but rather in when a
particular topic is taught and when it is assessed. This is not surprising, as there is evidence that
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
11
the topics taught in mathematics have converged into a single common curriculum throughout
the developed countries of the world (Baker & Letendre, 2005).
There exists an inevitable dualism between the achievement of goals as measured by high
stakes testing and the roles of teachers in preparing students for examinations. This dualism
between Regents examinations and instructional practices was noted by Dr. John E. Bradley,
principal of Albany High School, approximately 130 years ago when he commented that "The
salutary influence of the primary examinations in stimulating both teachers and pupils to
thoroughness in the acquisition of the elementary branches suggested the extension of the system
to academic studies.” The occasion of Dr. Bradley’s remarks was the extension of Regents
examinations from the grade schools into the high schools of New York. (New York State
Education Department).
The NCTM assessment principle argues the need for aligning assessment and
instructional practices and asserts that
Good assessment can enhance students' learning in several ways. First, the tasks
used in an assessment can convey a message to students about what kinds of
mathematical knowledge and performance are valued. That message can in turn
influence the decisions students make—for example, whether or where to apply
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
12
effort in studying. Thus, it is important that assessment tasks be worthy of
students' time and attention. Activities that are consistent with (and sometimes the
same as) the activities used in instruction should be included….Assessment
should reflect the mathematics that all students need to know and be able to do,
and it should focus on students' understanding as well as their procedural skills.
Teachers need to have a clear sense of what is to be taught and learned, and
assessment should be aligned with their instructional goals (NCTM, 2000).
Numerous other studies suggest that alignment of curriculum and assessment practices is a
practical concern of educators (Firestone, Schorr, & Monfils, 2004)( (Hamilton, 2003). JMAP’s
position is that teacher awareness and understanding of historical patterns in assessment practices
is an effective means through which awareness and understanding of state learning standards is
acquired. Accordingly, curriculum alignment is good for both students and teachers.
Adapting Instructional Resources for Diverse Student Populations
Good teachers adapt their lessons and teaching resources to the needs and skill levels of
their students (Dewey, 1915/2001) (Fosnot, 2005). Inherent within this idea is the premise that
teachers add value to the generalized curriculum associated with textbooks and commercially
available teaching resources. Teachers have long used scissors and glue, mimeograph machines
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
13
and photocopiers, and pencil and paper to create and adapt teaching resources for their classes
and individual students. As computers and technology have increasingly asserted a presence in
our daily lives, a new generation of teachers has emerged that is willing and able to use the
Internet and, to a lesser extent, database programs and the desktop publishing capabilities of
personal computers, to replace scissors, glue, pencils and mimeograph machines. This new
generation of teachers will arguably be more effective in adapting lesson plans and teaching
resources to the diverse needs of their students if they are provided with Internet resources and
technology that integrates rich collections of resources in user-friendly formats. The New York
City Department of Education is among the most diversified public school systems in the world
and there is no single lesson plan or textbook that can ever meet the needs of all students and all
teachers. Thus, it seems reasonable that efforts be made to empower classroom teachers with
high quality, manipulable resources accompanied by desktop publishing capabilities. JMAP is a
prototype for such a system, and the JMAP prototype for teacher empowerment can easily be
extended to other academic subjects.
Reducing Stress Levels and Burnout Rates in New Teachers
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
14
New teachers face challenges in many critical areas, including but not limited to: 1)
developing subject matter expertise; 2) developing pedagogical expertise and classroom
management skills; and 3) developing several hundred lesson plans during their first years of
teaching, when they are arguably least capable of producing good lesson plans. Providing
adequate resources and facilities to support teachers in instructional practice is a recommended
practice for preventing and/or reducing teacher burnout (Wood & McCarthy, 2002). Although
textbooks and teaching resources are readily available from numerous sources, most of these
resources have serious limitations for New York’s teachers. These limitations include: 1) they
are not grounded in the New York State learning standards, assessment practices, or urban school
environments; 2) they are in formats that are not easily manipulable; and 3) they are disjointed
and non-integrated from both topical and pedagogical perspectives. A comprehensive lesson
plan and resource management system for each academic subject area could reduce the workload
and stress levels of new teachers by giving them starter lesson plans in manipulable formats that
can be adapted to a wide variety of classroom needs and pedagogical beliefs. Each lesson plan
would be accompanied by a complete history of how the topic has previously been assessed on
Regents examinations, and by additional teaching resources suitable for the topic of the lesson.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
15
By “lessening the load” on new teachers, such a lesson plan management system could
conceivably influence both teacher retention and quality of instruction.
An Opportunity Within Our Reach
Russell Conwell, founder of Temple University in Philadelphia, was renowned for a
series of lectures, sermons, and a book, all entitled “Acres of Diamonds,” in which he told the
story of a Persian farmer named Ali Hafed (Conwell, 1915). Ali searched the world over for
diamonds, never to learn that the farm he left as he began his quest would become the site of one
of the world’s great diamond mines. The morals of the story are: 1) sometimes the best
opportunities are in your own backyard; and 2) one should always look at the present situation
when looking for a better opportunity. Such is the case with education in New York City.
All Regents examinations pass into the public domain immediately upon publication and
administration, and this fact represents a diamond in our own back yard. These historical
documents provide an excellent teaching resource that teachers can own and do with as they
please. In this time of high stakes testing for all in New York, Regents questions constitute the
best available representation of the intended curriculum embodied in the state’s learning
standards for any given subject area, and as such, they have added significance for both students
and educators. They are the gold standard for understanding what the state of New York thinks
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
16
a student should know and understand, and they constitute a body of knowledge that can and
should be at or near the center of any curriculum in our state.
Summary
JMAP has demonstrated the need for and efficacy of technology based lesson plan and
resource management systems that empower teachers in the areas of subject matter expertise and
adapting lesson plans to diverse student populations. The JMAP approach offers potential for
reduced stress and turnover amongst new teachers as well as better alignment of instructions with
learning standards and assessment practices. Furthermore, the JMAP approach is a low cost
form of professional development with significant buy-in from teachers, and it can be replicated
and expanded beyond its current focus in mathematics. An opportunity exists for a sea-change in
teacher empowerment and educational publishing at a grass-roots level with minimal costs.
Additional Information and Questions
Feedback from educators, researchers, and others interested in the use of Internet-based
technologies for the empowerment of teachers and the improvement of teaching practice in
public education is encouraged. The opinions and vision expressed in this article are those of the
writer. Questions and comments concerning this article and its contents may be directed to its
author: The Jefferson Math Project is accessible at www.jmap.org.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
17
REFERENCES
Baker, D.P., & Letendre, G.K. (2005). National Differences, Global Similarities: World Culture
and the Future of Schooling (Chapter 10). Palo Alto, CA: Stanford University Press.
Conwell, R.H. (1915). Acres of Diamonds. United States: Harper and Brothers.
Darling-Hammond, L., Holtzman, D.J., & Gatlin, S.J. (2005). Does teacher preparation matter?
Evidence about teacher certification, Teach for America, and teacher effectiveness (13,
42). Education Policy Analysis Archives.
Decker, P., Mayer, D.P., & Glazerman, S. (2004). The effects of Teach for America on students;
Evidence from a national evaluation. Princeton, N.J.: Mathematica.
Dewey, J. (2001). The School and Society & The Child and the Curriculum. Mineola, NY: Dover
Publications, Inc. (Original work published 1915)
Firestone, W.A., Schorr, R.Y., & Monfils, L.A. (Eds.). (2004). The Ambiguity of Teaching to the
Test: Standards, Assessment, and Educational Reform. Chapters 2, 3, 4. Hillsdale, N.J.:
Lawrence Erlbaum.
Fosnot, C.T.,. (Ed.). (2005). Constructivism: Theories, Perspectives, and Practice. New York
and London: Teachers College Press.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
18
Hamilton, L. (2003). Assessment as a Policy Tool. In R.E. Floden (Ed.), Review of Research in
Education (Vol. 27, pp. 25-68). Washington, D.C.: American Educational Research
Association.
Hill, H.C., Rowan, B., & Ball, D.L. (Summer 2005). Effects of teachers’ mathematical
knowledge for teaching on student achievement. American Educational Research
Journal, 42, no. 2, 371-406.
NCTM (Ed.). (2000). Principles and Standards for School Mathematics. Reston, VA: National
Council of Teachers of Mathematics.
New York State Education Department. (November 24, 1987). History of Regents Examinations
1865 to 1987. In University of the State of New York State Education Department
Website. Retrieved May 20, 2006, from
http://www.emsc.nysed.gov/osa/hsinfogen/hsinfogenarch/rehistory.htm:
Wood, T., & McCarthy, C. (2002, Dec). Understanding and Preventing Teacher Burnout. ERIC
Digest. [Data File]. Washington DC: ERIC Clearinghouse on Teaching and Teacher
Education. Available from http://www.ericdigests.org/2004-1/burnout.htm
i STEVE WATSON is a doctoral student in Urban Education at the Graduate Center of the City University of New York. He also teaches high school mathematics at the International High School @ Prospect Heights in Brooklyn, and he is an adjunct faculty member of the mathematics
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
19
education department at Brooklyn College, where most of his students are New York City Teaching Fellows. Steve's general research interests are in mathematics education and the use of technology for teacher empowerment and new teacher orientation programs. After receiving an M.S. Ed. from Purdue University in 1973, Steve spent over twenty years as an executive with American General Corporation, a subsidiary of AIG, where he held positions including Senior Vice President for Government and Industry Relations, Senior Vice President and Chief Administrative Officer, and board member of two insurance companies. He joined the New York City Teaching Fellows in 2003 and co-founded the Jefferson Math Project with Steve Sibol during his first year of teaching. E-mail author.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007 Gender Differences
1
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Running head: GENDER DIFFERENCES
Gender Differences in Mathematics Departments at Colleges and Universities Across the
United States: Towards an Inclusive Environment
by
Alexander Rolón
Assistant Professor
Northampton Community College
3835 Green Pond Road
Bethlehem, PA 18020
610.861.4163
Gender Differences 2
Gender Differences in Mathematics Departments at Colleges and Universities Across the
United States: Towards an Inclusive Environment
Introduction
When one thinks of great mathematicians, women do not come to mind.
Perhaps it is because the development of mathematics, as we know it today, was
controlled by men. For example, Sir Isaac Newton, Carl Friedrich Gauss, and Gottfried
Leibniz were pioneers in the development of calculus. Women were not permitted to
become intellectuals in this sophisticated and complex subject. Mathematics departments
across the United States are predominantly male dominated. This disproportionate
interest is ingrained back in the elementary (Pérez, 2000) and middle school grades
(Hanson, 1992). Research (Eccles, 1986; Nichols, 1989; Zaslavasky, 1994) suggests that
females are not encouraged to enroll in higher level mathematics and science curricula
due to the lack of self identification, family discouragement and societal prejudices.
There is a need for more female representation in the field of mathematics and science.
One way to explain this discrepancy is rooted in Social Identity Theory. Women,
like men, challenge themselves but academically there exists a gap in the challenges they
Gender Differences 3
take as it relates to mathematics and science. Research further supports gender
differentiation when it comes to making mistakes in mathematics performance. Males
tend to find a scapegoat for their failures, while females often blame themselves for not
preparing adequately (Beal, 2000). When males perform well they attribute it to their
proficiency in the subject matter as if it were expected; by contrast women’s success is
highly praised to the extent of the effort or time they spent studying the material. The
lack of attribution to ability for success and the tendency to attribute failure to a lack of
ability promotes low mathematical self-esteem; hence creating a negative association or
bias toward other mathematics courses, allowing males to enroll in higher level
mathematics curricula. Males have a sense of superiority over females in mathematics.
Women who continue in mathematics courses usually do so in the presence of a
group with which they can identify. In addition to these groups, there are national and
local associations like The Association for Women in Mathematics and Women in Math,
among others. The creation of groups or cohorts helps facilitate and overcome some of
the sexist behaviors that exist in the mathematics community. Often times, the women in
these groups support each other and become successful. For this, and many other reasons,
group identification is essential to the success in future enrollment of women in
mathematics curricula.
Individuals get much of their self-esteem from interactions with their peers. The
case for females in mathematics is no exception. Without group identification their
psychic immunity decreases, and so does their ability to perform well. It is imperative
that we as professionals in the field think more inclusively about female capabilities to
Gender Differences 4
perform well as it relates to mathematics; hence recognizing that women are also part of
the realm of mathematics.
Enhancing Women in Mathematics Through History
Although the history of mathematics is male-centered, there were a few women
who were influential and contributed a great deal to the development of mathematics.
Maria Agnesi, often times referred to as the first woman professor of mathematics on a
university faculty. Sonya Kovalevsky impressed the French Académie des Sciences
judges, and unanimously voted her concealed paper, On the Rotation of a Solid Body
about a Fixed Point, the winner of the famous Prix Bordin. Because of the exceptional
merit of the work, the monetary value of the prize was raised from 3,000 to 5,000 francs,
a considerably sum of money at the time. Amelie Emmy Noether, who was a professor at
Bryn Mawr College in Philadelphia, PA and known for her work on the Theory of
Invariants. These women, as well as others who are not mentioned, were prodigies in
their field of expertise. A history of mathematics course could emphasize the
contributions of such prolific females, although it is not guaranteed to increase
awareness. Mathematics, as a field, is male dominated. Incorporating some biographical
vignettes of females into mathematics curricula will help shift the center and help to
communicate the message that women are also “good at math.”
Many women feel that graduating with a mathematics degree is equivalent to
winning a gold medal in the Olympics. They feel a great deal of accomplishment because
of their limited support from their male counterparts. Sexism governs the unwritten,
nonmathematical rules of this field. Women often times are told they are not good at
Gender Differences 5
math, that they are better at nurturing or in careers where creativity and intuition prevail
(Eccles, 1986). Therefore they begin to believe such prejudices and act on them.
Counselors, teachers, peers, or even parents who are not “good at math” discourage
women to explore other than mathematical opportunities. But once they have
accomplished success in mathematics, they want to share with other women. They feel it
is important to break the stereotypes and hence educate others and reduce the sexism that
exists in mathematics and science. Universities across the country need to have a more
mathematics diverse faculty. By having such diversity within the department, perhaps
they would attract more females in graduate programs. Furthermore, recognizing this
deficiency would create for a more inviting, less intimidating atmosphere for women.
Social Identity Theory suggests that there are some positive intergroup biases
when it comes to women succeeding in mathematics. Women value their educational
experience in fields like mathematics and science more so than males. Their success is
attributed to their preparation and perseverance. They feel as if they are equal to men in
the field. Elva Treviño Hart, a Chicano author, who holds a B.S. in Mathematics and an
M.S. in Computer Engineering from Stanford University, once said that being a math
major gave her the opportunity to express herself without being prejudiced about her
gender or her ethnicity in a field where she was “twice a minority”. She and similar
women were able to excel because they looked past the sexism and prejudices that this
field brings, and because they were determined to succeed regardless of the myriad of
obstacles presented to them. Many women don’t want to deal with the pressures and
often change careers or go into related fields where mathematics is needed, yet not
essential. Moreover, they view themselves as low achievers and feel as if they were more
Gender Differences 6
inferior to males. This feeling develops a negative association and belief that males are
better than females as it relates to mathematics competence and achievement.
Improving Intergroup Bias
How then do we go about creating an atmosphere of tolerance, respect, and
equality? The exploration of the following three activities incorporates methods to
improve intergroup bias: 1) recategorization, 2) decategorization and 3) mutual
differentiation. Each of these methods is equally important and they bear no order of
significance.
Recategorization
Many universities hold lecture series or a mathematics symposia sponsored by
mathematics departments, where distinguished mathematicians are invited to present their
latest research. A suggestion is that the faculty work together to feature more women in
this series or symposium. There are many women mathematicians to choose. For
instance Elva Treviño Hart, previously mentioned, is a good example of
multiculturalism as well gender issues in mathematics. Below you will read other
contemporary female mathematicians who are qualified for such lectures.
Margaret A. M. Murray who wrote a book in 2001 titled: Women Becoming
Mathematicians: Creating a Professional Identity in Post-World War II America, that
looks at the lives and careers of thirty-six of the approximately two hundred women who
earned Ph.D.s in mathematics in the United States from 1940 to 1959 an era when
American mathematical research enjoyed an unprecedented expansion, fueled by the
technological successes of World War II and the postwar boom in federal funding for
education in the basic sciences. Nevertheless women's share of doctorates earned in
Gender Differences 7
mathematics in the United States reached an all-time low. Murray explains: "…the book
examines the development of mathematical identity across the life span, from childhood
through adulthood and into retirement. It focuses on the process by which women, who
are actively involved in the mathematical community, come to ‘know themselves’ as
mathematicians. The women's stories are instructive precisely because they do not
conform to a set pattern; compelled to improvise, the women mathematicians of the
1940s and 1950s followed diverse paths in their struggle to construct a professional
identity in postwar America.”
Lai-Sang Young a pioneer in the field of topology. Her research is on continuous
flows on compact 2-manifolds. She has been the keynote speaker at different
mathematical association throughout the United States and Europe as well as her home
country of China.
Linda Goldway Keen who was born and raised in Bronx NY. Her research
involves studying the interplay between the analytic and geometric aspects of classifying
Riemann surfaces. Dr. Goldway Keen is one of the few mathematicians to have studied
this branch of mathematics.
Lenore Blum has a story of perseverance. She always dreamed on attending
Massachusetts Institute of Technology both because it was an excellent place for her to
study and also because her husband was there, but she was not accepted. She was
discouraged and was not set on taking a degree in mathematics but had other interests so
she enrolled in the Department of Architecture at Carnegie Institute of Technology in
Pittsburgh. She still wanted to attend MIT and had made a number of unsuccessful
applications to there but at last she made a successful one and began to study there while
Gender Differences 8
completing her first degree at Simmons College. She was awarded her B.S. from
Simmons in 1963 and continued working towards her doctorate at MIT.
These are great examples of contemporary female mathematicians who would
make excellent speakers at the lecture series or symposium throughout mathematics
departments at colleges and universities in the United States, with the end goal of
reducing biases in this field. The experience of both groups working together would be
sufficient to agree upon the fact that they are all professional mathematicians, regardless
of gender differences.
Decategorization
The male professors, after attending the lecture series or symposium, will discuss
with the lecturer how her research could be incorporated into their classes or their own
research. The purpose of this activity is to encourage more research and to have the
female mathematician as the focal point of the lecture or discussion. It is important to
have male mathematicians praise their female counterpart, with the objective of de-
emphasizing that mathematics is a male dominated field. Making these females
mathematicians the key person and expert in the field will confirm their role in the
mathematics world; hence, breaking the stereotypes that are associated with their
performance in mathematics.
Mutual Differentiation
Explaining how history has shaped the way we learn mathematics today would be
interesting to research to put into practice. The faculty will explore ways in which
females learn mathematics as opposed to men. Pedagogical practices will also be
explored to enhance the ways females perceive mathematics. Two of the key questions to
Gender Differences 9
answer are: Is a hands-on curriculum embedded in their programs and/or classes? Are
there different delivery methods rather than all lecture classes? The latter will explain
spatial ability and its influence in learning mathematics. Understanding how females
think is fundamental to the creation of a less biased environment. They would then
discuss how these approaches can benefit men as well. As a suggestion, a lesson attached
in Appendix A can be used as a collaborative group work where students will be aware of
female mathematicians. Mathematics reform calls for a more inclusive curricula where
creative minds are generated.
Summary
Inclusion of women in mathematics programs as well as an increase of the
number of female faculty at colleges and universities will yield an environment that is
tolerant and conducive to learning. However, there are many obstacles that this change
may possibly bring. Resistance on the faculty due to the lack of women input on this
issue is a possible downfall. Bringing females into the scheme of things may spark some
negativism creating tension among the male faculty. Yet this can be a constructive,
learning experience for all. Social Identity Theory explains how clusters among women
will be encouraged and expected. However, conveying the reality of sexist behaviors in
mathematics may put this issue into perspective. It is important to understand women’s
role and thoughts about sexism in the mathematics field in general as well as their
feelings about being a minority in the department.
Initiating discussion about gender differentiation and sexism at mathematics
departments among American institutions of higher learning, will make certain of one
thing: communication of ideas that otherwise would have been overlooked will be
Gender Differences 10
occurring. I am not an expert in sexism in the mathematics field; however, I am aware
that it exists. This is a step closer towards a more inclusive way of thinking. Being aware
of the biases should be the foremost, intricate step in understanding our own biases and
prejudices. Do we as teachers call on males more often than females? And when we call
on females, do we give the same wait-time for them to respond as we would for males? If
we can go beyond the basics and think about our own personal experiences, we can be at
a much better place in our professional lives.
We need to take initiative, be a pioneer, and be active participants in reducing
gender biases in the mathematics field. We must understand that we are all
mathematicians, regardless of sex or any other characteristics that come with each and
everyone of us; by doing so we build a better rapport within the mathematics community
and a more inclusive atmosphere.
In conclusion, in order to create an atmosphere of respect and tolerance one must
have knowledge, awareness, and skills when it comes to female competence in
mathematics. Knowing what to do when females are in our classes becomes an intricate
part of our professional growth and their interest in mathematics as a career; awareness of
the fact that sexism exists in today’s society will help with our own peace of mind to
accept; and we gain skills to be sensitive to everyone, especially women who have been
underrepresented in mathematics for a long time.
Gender Differences 11
References
Beal, C. (2000). Gaining confidence in mathematics: Instructional technology for girls.
Paper presented at the International Conference on Mathematics/Science
Education & Technology, Sand Diego, CA.
Eccles, J. (1986). Gender-roles and women’s achievement. Educational Research, 15(6),
15-19.
Hanson, K. (1992). Teaching mathematics effectively and equitably to females.
NY: Eric Clearinghouse on Urban Education.
Nichols, R. (1989). Gender and mathematics contests. Arithmetic Teacher, 41(5), 238-
243.
Pérez, C. (2000). Equity in standard-based elementary mathematics classrooms.
Retrieved July 10, 2007, from http://wge.terc.edu/equity.html.
Zaslavsky, C. (1994). A mind is a terrible thing to waste!: Gender, race, ethnicity, and
class. In Fear of math: How to get over it and get on with your life (pp. 69-98).
New Brunswick, NJ: Rutgers University Press.
Gender Differences 12
Appendix A
Lesson Plan #1: A Historical Perspective
Objective: To enhance student awareness of women mathematicians in education.
Grade Level: High School
Activity: Students will be paired and they will research a female mathematician of their
choice, with instructor approval. They will then create an article for a newspaper with a
short biography as well as one of her mathematical innovation. The article must be era
appropriate. The students will then have to put together a live interview of this person to
be presented in the class. The purpose is to emphasize the mathematical achievement(s)
of this person. (In the case where two male students are working together, they will need
to come up with a creative way to convey her contributions to mathematics.) The
instructor will help the students understand the mathematical concepts of these
mathematicians.
Assessment: A rubric will be developed for both the written article and the interview.
The students will have access to this rubric while they are researching and preparing their
presentations.
Technology: The students are encouraged to use power point presentations or other
technology to enhance their presentations.
Looking back: Individually, the students will write a journal entry reflecting on this
activity. In it they should explain what they learned, how valuable it was as well as their
reactions to the activity.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
1
Collaborative research relationships in Urban Science Classrooms1
Rowhea Elmesky and Anita Abraham Washington University in St. Louis & Philadelphia School District
INTRODUCTION
Anita (Teacher Researcher): As a science teacher at City High School, a school where a nearby university
conducted ongoing research in science classrooms, I had seen university researchers walking down the
halls, in classrooms and also in the principal’s office. Most of the teachers were suspicious about the
university researchers. They tried to avoid them, were apprehensive about being interviewed by them, and
afraid that they might accidentally say something that might put them in ‘trouble.’ In those days, I wasn’t
sure what the ongoing research was about, and I didn’t make any effort to know either.
Things started to change when our vice principal, Dr. Al, a former science department head, asked me to
join the Masters in Chemistry Education (MCE) program offered at the same nearby university. At the
same time, Dr. Kenneth Tobin, the main university researcher from the Graduate School of Education,
asked me if I would be interested in joining the research group already working at City High School. He
further explained to me that, as a part of the research team, university researchers would have access to my
classroom and I also would be participating in the research as a teacher-researcher. As a regular classroom
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
2
teacher, I didn’t consider myself a researcher and didn’t know what qualifications were expected for a
researcher. Moreover I wasn’t comfortable letting a university researcher into my classroom. I was worried
that, if things went out of control, those events would become the focus of their research findings. Sensing
my uneasiness, Dr. Tobin told me about the university researcher whom he had in mind for my classroom,
and assured me that I would be comfortable working with her.
When I shared this information with one of my coworkers, Ms. Cloud, a 30-year veteran teacher, her
reactions were negative, mainly because in her opinion educational researchers always concluded their
findings without any input from the classroom teacher or students. However, I anticipated that my situation
would be different because I would act as a teacher-researcher and my students would also become a part
of the research team as student-researchers. Hence, I agreed to be a part of the university team, excited that
my voice and my students’ voices would also be heard during the research process.
Rowhea (University Researcher): When I began working with Anita, I sensed her nervousness. Perhaps
she thought that I was there to criticize her as a teacher. For the purposes of the research study, I brought in
a video camera to record the class events several times a week, and the camera seemed to reinforce the
apprehension. Early on, I worked to assure Anita that I was present in her class to assist her in conducting
HER research. I attempted to position myself as an assistant – someone who would be a co-researcher on a
study that she would design – around questions that were of interest to her and would help her to improve
1 The research in this paper was supported in part by the National Science Foundation under Grant No. REC-0107022. Any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
3
self-selected aspects of her teaching. It took a while for us to establish a comfortable rhythm of trust and
for Anita to feel confident that I wanted to collaborate rather than dominate.
Nisha (student participant): I learned a lot from research. We sit in groups and talk about class an[d] stuff, I
never thought about the other kids and how they feel. I learned how Ms. A [Anita] cares about us. She
taught us to help other people in class. I get good grades. Class is just a big group of helpers for everybody.
(Journal entry, June 2002)
Maria (student researcher): We told Ms. Morris [the English teacher] about the research in your [Anita’s]
class and how we talk about what we like and what we don’t and all. She liked it. She said that she might
try it.
In this article, we describe some aspects of the research process in which a teacher researcher
(Anita), a university researcher (Rowhea), student researchers, and student research participants
collaborated on a study during 2002 in Anita’s 11th grade Chemistry class and her supplementary
laboratory at City High School. Conducted within a critical ethnographical framework, the study
focused on the teaching and learning of science in an urban high school where the majority of
students are African Americans from low socioeconomic backgrounds. The authors met through
our mutual involvement with an NSF-funded grant that invoked a model of collaborative
research (utilizing a ‘research with’ rather than ‘research on’ methodology) where teams were
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
4
created that consisted of two teacher-researchers from each participating urban school, at least
two student-researchers from each focal class, and university researchers. Thus we chose to
begin this paper with a series of reflections through multiple voices, to illuminate some of the
challenges and benefits associated with engaging in this collaborative model of research.
CONTROL OVER OR COLLABORATION WITH?
Anita: I have learned that model of ‘control over’ does not afford respect in urban school settings.
‘Collaboration with’ has allowed me to communicate care and build the understandings of my students. In
order to be a successful science teacher, I encouraged my students to give me feedback on my teaching
practices. Based on their suggestions and through negotiations, I tried to develop a learning community that
is built on respect and trust. As a teacher-researcher, I encouraged my students to make connections
between their experiences from their lifeworlds and science. Students brought new insights to common
concepts and raised questions that I had never considered. Now, three years following the completion of
this study, I am a teacher who is constantly evaluating and transforming my classroom practices in an
ongoing effort to create a learning community based on trust, collaboration, and shared responsibilities.
(Reflections, 2007)
Urban schools, such as City High where Anita taught, are marked by inequalities – visible in
school staffing, funding, courses offered, and the resources available. The schools are often
oppressive and hence become grounds for struggle – the teachers and/or administration against
students who are labeled as “resistant” or “unmotivated,” for example. This paper provides a
model for incorporating a critical ethnographic methodology and cogenerative dialogues as tools
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
5
for shifting school dynamics from control over to collaboration with – where participatory
critique is encouraged such that ongoing structural transformation in the classroom occurs and
schooling becomes a less oppressive experience and a more rewarding experience for both the
students and their teachers.
CRITICAL RESEARCH AS A TOOL FOR DAILY CLASSROOM CHANGE
When Barton (2001) discusses critical ethnography, she describes the research process as a
“dialectical theory- and practice-building process in which practice and research shape each
other in an endless cycle” (p. 907). Thus, critical ethnography calls for identifying the problems
and asks for transformation by connecting theory and practice. This dialectical relationship
between practice, theory, and research triggers local transformation of the structure by providing
tools for all participants to act in new ways as the findings from the research constantly inform
participants of their practices and vice versa. Critical ethnography also asks to increase the
agency of the participants who can draw strength from the research findings. Thus the research
process and associated findings can become a catalyst for growth and transformation (Seiler &
Elmesky, 2005).
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
6
Students as Researchers
Educational research that involves students as researchers “provides a way to obtain their
perspectives on what is salient in terms of school, teaching, learning, and myriad other issues”
(Tobin, 2006, p. 27). Student-researchers should be included in salient ways in the research so
that their perspectives on what is occurring in the school or neighborhood fields and ‘why’ can
emerge (Elmesky & Tobin, 2005). Student-researchers are empowered as they contribute
significantly to identifying patterns of coherence (as well as contradictions) within their
classrooms, in relation to the teaching and learning they experience.
In our study, student researchers engaged in activities such as the review and analysis of
videotapes, interviewing each other and fellow classmates, transcribing such interviews, writing
reflective journal entries, and developing video ethnographies that captured salient aspects of
their lifeworlds outside of school. Weekly, the researchers ate lunch together, during which time
they watched videotapes from class time and from within the laboratory. They were asked to
identify video vignettes of salient events that were taking place, and these video vignettes then
became central focal points for discussion. In addition, a selection of video vignettes was shared
with students who were participants within a captured video clip, in order to obtain their
perspectives and preserve and privilege their voices.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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7
Cogenerative Dialogues
Out of the many research methods that were engaged in this study, we highlight the role that
cogenerative dialogue can play in catalyzing deeper understandings and ongoing change in a
classroom. Cogenerative dialogue (LaVan, 2004; LaVan & Beers, 2005; Roth & Tobin, 2001;
Wassel, 2004) can be understood as a conversion between co-participants about shared events
and experiences. As Tobin, Zurbano, Ford, and Carambo describe (2003), “such conversions
might focus on participation, access and appropriation of resources and the co-occurrence of
given patterns of coherence and associated contradictions” (p. 55). During cogenerative
dialogues, every participant should have the opportunity to speak and be heard while also
listening with interest and respect to the perspectives of the others who are involved. Moreover,
participants can discuss collective responsibility, participants’ responsibility, curriculum,
experiences, and power relationships. A critical characteristic of cogenerative dialogues is that
the outcomes of these interactions can and should be immediately applied to improve the
teaching and learning of science in a science classroom.
Cogenerative dialogues can take many different forms and involve different participants at
different times. For instance, some are held after class time while others occur during class time.
Martin (2004) classifies cogenerative dialogues as formal, informal and classroom huddles.
Classroom huddles occurred between us (the authors) on a regular basis in between, during, and
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
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8
after each chemistry class. In these ‘huddles,’ we briefly discussed that day’s lesson, classroom
activities, how the materials would be distributed during the laboratory activity, and logistical
research details, such as where the video camera should be placed to best capture salient events
in the classroom. These classroom huddles were short, focused and occurred by moving to
physically stand in close proximity to each other; they enabled to evaluate and revaluate the
lesson before, during and after implementing it and to develop immediate suggestions and
implement ongoing modifications with the help of the university researcher. When a greater
amount of time was available right after the chemistry laboratory activity, informal cogenerative
dialogues occurred between us where we were able to sit down and discuss an individual
student’s performance or progress, laboratory group dynamics, and other salient events that
occurred during a particular laboratory activity.
During this research study, cogenerative dialogues occurred in different fields with different
participants. Formal cogenerative dialogues occurred about once a week between Rowhea and
student-researchers during their lunch time. Anita tried to attend most of these sessions but, at
times, her other responsibilities as a teacher prevented her from doing so. In these meetings,
Rowhea introduced the student-researchers to some basic concepts in sociocultural theory, and
we discussed classroom practices, especially Anita’s practices as a teacher, curriculum, and
laboratory activities, and came up with suggestions for improving our learning environment. In
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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9
order to help dissipate the power hierarchies existing, we sat wherever we felt comfortable, and
we constantly reminded the student-researchers that their interpretations and suggestions were
not going to affect their grades. We also watched video vignettes from previous classes, and
discussed concerns and suggestions. It is evident in the following entry from one student
research’s (Deidre’s) journal that she felt comfortable to be critical of Anita’s practices as a
teacher. She urged Anita to allow the students to access resources on their own to light the
Bunsen Burner.
I think Mrs. Abraham should trust us and plus the burner, She gotta go to group to group, lightning it and its gonna take a long time and we wanna do our lab real quick and by her keep goin to group to group she just need to give us like some matches or a lighter so we can [light] burner our own? Burner is easy to use. (Journal entry, February 2002)
These types of reflections were useful in helping us to identify practices that afforded and
truncated students’ performance within the laboratory setting. However, providing feedback in
writing lacks the interactive components of cogenerative dialogue and the possibilities to ask for
clarification. For example, in the following cogenerative dialogue, Rowhea was able to seek
clarification when a student researcher (Maria) advised that Anita needed to be more patient with
the students in the laboratory.
Maria: She is a nice teacher. She is all right. She explains how her culture is and everything. We ask her stuff. She explains why. She teaches but she still needs to be a little more patient with us also.
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10
Row: Can you give me an example of it? Do you remember a time in lab that she got frustrated with you?
Maria: Um, I think our group was asking for something. She was doing something else and she got like real mad like “I WILL BE THERE IN ONE SECOND” and I understand that you [Anita] are only one person but we need help also.
Cogenerative dialogue provided a space for hearing the students’ voices and enabled Anita to
reflect on her practices and transform the structures to allow for greater student and teacher
empowe. Conversations like this one helped Anita perceive emotion as a central part of
instruction. Maria brought to our attention that the generation of positive emotional energy
(Collins, 2004) in the classroom (through patience) would encourage a positive atmosphere for
learning. Some of the other actions that Anita took to build positive emotional energy were
positive reinforcement (instead of being loud), avoiding argument, and complimenting students
for being successful in class and laboratory. More importantly, through the research process,
Anita was increasingly conscious of regularly reflecting upon her practices.
Anita: Everyday I tried to spend a couple of minutes reflecting on my actions, and at times asking the following question to myself, “If I were a student, would I want me as a teacher?”
In addition to providing critical remarks, the students also shared their perceptions of
teaching practices that they considered successful, during cogenerative dialogues. The following
cogenerative dialogue excerpt was in relation to Anita’s practice of allowing the students to
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
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11
select their own laboratory groups, contrary to other teachers. The students enjoyed this
autonomy. Evident in her comments, Deidre felt that working with familiar peers assisted the
process to proceed smoothly and in an enjoyable manner:
When we are in the lab and we have to pick who we are in the group with, and you work with people you are already familiar with. Some teachers just put you with anybody if you don’t like that person and you are not familiar with that person you are not going to work because you don’t know anything about them. So you work with your friends and like we have the lab [Rate of Reaction], and we had to mix the chemicals, look at the color change and time it for one second or two second. It was fun.
Cogenerative dialogues as windows into student practices. On some occasions, the video
being watched during cogenerative dialogues spurred conversations that encouraged the study
practices of different students’ practices and related aspects of the learning environment. For
example, Maria made the following comment as we watched a videotape of the students engaged
in a lab (Flame Test Laboratory Activity).
But at 11:07 [AM] we seem like we all were writing down our observation and getting along well. Look at Earl. Earl the type of person that doesn’t do any work. He the one that copy and stuff like that. But he not dumb! Earl ain't dumb! He smart he just don’t wanna do it, He don’t wanna comprehend. He don’t wanna seem like he smart.
Earl was considered to be a disruptive student by other teachers. As his seat was near the
door, he preferred to look outside the classroom than to focus inside. During classroom
instruction, instead of paying attention and writing notes, he usually put his head down. Anita
tended to ignore him because he was not threatening her classroom environment. When Maria
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12
brought Earl to our attention, Anita went to the counselor’s office to learn about Earl’s living
situation. She was told that Earl and his sister were living in a foster home. Even though efforts
to reunite them with their mother were being made by the child protection services, they were
constantly being moved from placement to placement. With no permanent home, the situation
was very stressful. Every day Earl brought his toothbrush and other personal belongings to
school because he was not sure whether he would be moved to a new foster home by the end of
the day. Through cogenerative dialogues around video footage of the classroom, Anita obtained
new insights into why particular students may act in particular ways.
Cogenerative dialogues and collective responsibility. When the students were in the
laboratory, there were constant requests for Anita’s assistance. The students expected her to
assist groups and she did – continuously circling throughout the duration of the laboratory
activity from group to group. While Rowhea also assisted students who were in close proximity
of the video camera, the most responsibility, initially, seemed to be placed on Anita. This issue
of the lack of human resources in the lab was raised by a student researcher during one
cogenerative dialogue. As Maria explained, the historical experiences of the students (since they
had never participated in a science laboratory activity previously in ninth and tenth grades)
invoked a higher demand for help then Anita and I had expected.
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13
Row Yea, I know what you mean. When I am in that lab, it seems like there are a million people [calling out], “Ms Abraham, Ms Abraham.”
Maria This is our first time for doing something. This is our first time being in the lab. It is our first time all this stuff. It is the first time. But I think she can get more help somewhere else too. She needs to find some more help.
Interestingly, the students began to take responsibility for their own and each other’s
practices. Students kept an eye on their group members and on other groups to make sure that
they were following procedures correctly. They often provided information by answering
questions, sharing procedures, talking through the process and modeling for each other. For
example, during laboratory activity on physical and chemical changes, one group wanted to
finish the laboratory activity quickly and decided to put the baking powder directly into the
vinegar without first wrapping the powder up inside a paper towel, as the procedure required
them to do. However, surprisingly, this didn’t go unnoticed by another group’s member who
reacted quickly by shouting, “Stevenson you wrong! Don’t take it out! You wrong.” Such
interactions indicate that the students themselves were acting as resources for others within the
laboratory, illustrating an emergent spirit of collective responsibility.
CONCLUSIONS
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14
The potential to transform the classroom lies in knowing oneself as a teacher and/or learner, and that only by collectively seeking to expose and examine the structures associated with the process of teaching and learning can contradictions be resolved to afford greater agency for all classroom participants. (Martin, 2004, p. 203)
This critical research on the teaching and learning of science in Anita’s chemistry classroom
demonstrates the possibilities for improving the teaching and learning of science. Cogenerative
dialogues between teachers and students can act as a tool to fill the gap between a teacher and
her students. However, due to their busy roster, teachers can barely find time to engage in
cogenerative dialogues with their students. We suggest that administrators provide support to
teachers who would like to engage in cogenerative dialogues with their students.
School and classroom structures and available resources impact the teaching and learning of
science in a classroom. Because structures can be changed to afford the learning of students in
the classroom and because teachers’ practices are also a part of the structure, we propose that
teachers should regularly reflect with students, through cogenerative dialogues, on their practices
to identify those that are successful in the classroom setting. When teachers identify successful
practices, they can reinforce them to achieve their own as well as their students’ goals.
Although educational research findings are always utilized to improve teaching and learning
in a classroom, the reality is that traditional research dynamics do not afford the immediate
participants of a study with opportunities to reap the benefits; rather the implications of the
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15
research findings are for future classrooms. A research ‘with’ methodology empowers students
and teachers during the research process. That is, the model of critical research we discuss in this
article introduces a view of educational research that utilizes research as a tool that is
immediately effective and designed to encourage a sense of empowerment. In this manner, teams
of teacher and student researchers become integrated and natural parts of a classroom routine
where the learning environment is characterized by an openness to examining practices and
taking responsibility for one’s own actions.
REFERENCES
Elmesky, R., & Tobin, K. (2005). Expanding our understandings of urban science education by
expanding the roles of students as researchers. Journal of Research on Science Teaching, 42,
807-828.
Barton, A. (2001). Science education in urban settings: Seeking new ways of praxis through
critical ethnography. Journal of Research in Science Teaching, 38, 899-917.
Collins, R. (2004). Interaction ritual chains. Princeton, NJ: Princeton University Press.
LaVan, S. K. (2004). Cogenerating fluency in urban science classrooms. Unpublished doctoral
thesis, University of Pennsylvania, Philadelphia.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
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LaVan, S.K., & Beers, J. (2005). The role of cogenerative dialogue in learning to teach and
transforming learning environments. In K. Tobin, R. Elmesky, & G. Seiler (Eds.), Improving
urban science education: New roles for teachers, students, and researchers (pp. 147-163).
New York: Rowman & Littlefield Publishers, Inc.
Martin, S. (2004). The cultural and social dimensions of successful teaching and learning in an
urban classroom. Unpublished doctoral thesis, Curtin University of Technology, Perth, WA.
Roth, W.-M., & Tobin, K. (2001). The implications of coteaching/cogenerative dialogue for
teacher evaluation: Learning from multiple perspectives of everyday practice. Journal of
Personnel Evaluation in Education, 15, 7-29.
Seiler, G., & Elmesky, R. (2005). The who, what, where, and how of our urban ethnographic
research. In K. Tobin, R. Elmesky, & G. Seiler (Eds.), Improving urban science education:
New roles for teachers, students, and researchers (pp. 1-19). New York: Rowman &
Littlefield Publishers, Inc.
Tobin, K. (2006). Qualitative research in classrooms: Pushing the boundaries of theory and
methodology. In K. Tobin and J. Kincheloe (Eds.), Doing educational research – A
handbook (pp. 15-58). Rotterdam, The Netherlands: Sense Publishers.
Tobin, K., Zurbano, R., Ford, A., & Carambo, C. (2003). Learning to teach through coteaching
and cogenerative dialogue. Cybernetics & Human Knowing, 10, 51-73.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
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Wassell, B. (2004). On becoming an urban teacher: Exploring agency through the journey from
student to first year practitioner. Unpublished doctoral thesis, University of Pennsylvania,
Philadelphia.
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A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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A Pragmatic Mathematics: A new skills-for-life mathematics course addressing the
NSACS’s revelation of the dismal quantitative literacy of America’s college graduates
Dr. Elie Feder Kingsborough Community College - CUNY
Department of Mathematics and Computer Science [email protected]; [email protected]
1. INTRODUCTION
U.S. colleges are failing in their responsibility of training students in the practical
mathematical skills necessary to successfully enter society. This is the conclusion that was reached by
The American Institutes for Research’s new study examining the literacy of U.S. college students
(American Institutes for Research [AIR], 2006). “The National Survey of American College Students
[NSACS],” is based upon a sample of 1,827 graduating students from randomly selected 2-year and 4-
year, public and private, universities and colleges across the United States. According to Stephane
Baldi, the NSACS’s director at the American Institutes for Research, the study is intended to be used as
a tool to help college and university administrators identify specific academic areas where students
have literacy gaps that need to be rectified. The study reveals that students struggle most with
quantitative literacy, which the NSASC defines as follows:
The knowledge and skills required to perform quantitative literacy tasks, that is, to identify
and perform computations, either alone or sequentially, using numbers embedded in printed
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materials. Quantitative examples include balancing a checkbook, figuring out a tip,
completing an order form, or determining the amount of interest on a loan from an
advertisement.
The study concludes that approximately twenty percent of U.S. college graduates completing
four year degrees – and thirty percent earning two year degrees – have only basic or below basic
quantitative literacy skills. This means they are unable to estimate if their car has enough gas to get to
the next gas station, or calculate the total cost of ordering office supplies. The results indicate
shortcomings in the educational system’s preparation of students to meet the mathematical challenges
of the real world.
As these shortcomings have considerable effects upon American society, the results of this
study indicate a crisis which must be addressed. “Many situations bring people into contact with
mathematics, including buying products, conducting business, producing products, managing people
and technology, using science and technology” (Arney, 1999). The rate of growth of mathematically
based occupations is about twice that for all other occupations (National Research Council, 1990).
Almost 40 percent of the workforce does not have sufficient quantitative literacy for jobs that pay more
than $26,900, on average. Additionally, close to two-thirds of new jobs will require quantitative skills
typical of those who currently have some college or bachelor’s degree. America cannot remain a first-
rate economic power with a population that has second-rate mathematical literacy. Additionally, if
educators cannot fulfill their economic responsibility to help our youth and adults achieve quantitative
literacy, they will also fail in their cultural and political missions to create good neighbors and good
citizens (Carnevale and Desrochers, 2003).
In response to the findings of the study, a new problem-based method of teaching basic
mathematical skills to those students who have difficulty in applying mathematics to their everyday
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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lives is presented. A course is designed which is not truly a mathematics course, but can be construed
as a skills-for-life course which involves mathematics. Mathematics is not taught as a subject, but as a
tool to be used on a day-to-day basis. This change in attitude impacts what is taught, what is not taught,
and most significantly, how it is taught. These changes signify an appropriate response to the challenge
presented by the findings of the NSACS.
The paper is organized as follows. Section 2 discusses the notion of quantitative literacy, and
surveys the literature and studies regarding the emphasis on quantitative literacy in the American
mathematical educational system. Section 3 relates the successful Israeli model of improving the
quantitative literacy of its weaker students while simultaneously maintaining strong standards for its
advanced students. Section 4 elucidates the new pragmatic mathematic1 course designed to improve
America’s quantitative literacy, illustrates its teaching method and content, and compares it to
traditional mathematics courses. Section 5 is devoted to addressing various oppositions towards the
implementation of this new course. Section 6 provides some concluding remarks.
2. BACKGROUND ON QUANTITATIVE LITERACY AND ITS OPPOSITION
2.1. The notion of quantitative literacy. The concept of numeracy, or quantitative literacy, emerged
in the Crowther report (1959), where “numerate” is defined as “a word to represent the mirror image of
literacy … an understanding of the scientific approach to the study of phenomena - observation,
hypothesis, experiment, verification [- and] the need in the modern world to think quantitatively.” The
landmark report, A Nation at Risk (U.S. Department of Education, 1983), calls for higher standards for
all students in mathematics, as well as curricula that would teach students to “apply mathematics in
1 Throughout this work, the term pragmatic mathematic will be used specifically to refer to the new course developed in this paper.
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everyday situations”. Carnevale and Desrochers (2003) underscore that “most Americans seem to have
taken too little, too much, or the wrong kind of mathematics”. They recommend that “to fully exploit
mathematics as a practical tool for daily work and living, mathematics needs to be taught in a more
applied fashion.” In Mathematics and Democracy: The Case for Quantitative Literacy (Steen, 2001),
the Quantitative Literacy Design Team remarks: “Typical numeracy challenges involve real data and
uncertain procedures, but require primarily elementary mathematics. In contrast, typical school
mathematics problems involve simplified numbers and straightforward procedures, but require
sophisticated abstract concepts.”
In their insightful discussion of quantitative (in their terms, mathematical) literacy, Amit and
Fried (2002) comment that most educators struggle to supply the term with a precise definition which
enables one to determine its applicability to particular students. Despite this difficulty, they contend
that “at the heart of this notion lies students’ openness to mathematics”, rather than their mastery of
particular skills. Since it is unavoidable that students will confront mathematics in their lives, educators
must ensure that these encounters do not cripple them with fear. They conclude that “the
mathematically literate society is, thus, one characterized by a sense of ease, of feeling at home, with
mathematical ideas and mathematically presented information.” A similar attitude is presented by
Briggs, Sullivan and Handelsman (2004) who comment as follows:
Providing liberal arts students with a worthwhile experience in a quantitative literacy course
requires overcoming significant psychological obstacles. Students who take such courses often
are victims of previous mathematics courses and instructors. As a result, they harbor genuine
fears of mathematics, they have lost confidence in their quantitative skills, and they have little
belief that mathematics might be of use in their future. A successful quantitative literacy
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course cannot subject students to more of the same experiences they have had in previous
mathematics courses.
Although some of the characterizations of quantitative literacy found in the
literature seem to differ, the difference is largely one of perspective more than of
substance. Many authors focus on the practical manifestations of quantitative
literacy– the possession of practical mathematical skills; while others define the
underlying cause of quantitative literacy – openness to and understanding of
mathematics. However one precisely defines quantitative literacy, almost all agree
about its necessity in today’s world. The urgent need for effective quantitative literacy
courses and programs in American colleges is expressed in recent reports
commissioned by the Mathematical Association of America [MAA] (Sons, 1995),
The American Mathematical Association of Two-Year Colleges [AMATYC] (Cohen,
1995), and the College Board (Steen, 1997). In 2001, the case for quantitative literacy
reappeared in a report that has inspired a new dialogue on the subject (Steen, 2001). A
recently released report, Beyond Crossroads II, extends the dialogue on quantitative
literacy by underscoring the interdisciplinary nature of quantitative literacy and
making a “call to coordinate across the disciplines to create a curriculum that
effectively supports quantitative literacy in our colleges (Blair, 2006).”
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2.2. Related studies. The findings of the NSACS provide evidence for the shortcomings of American
college’s mathematical education. This problem is not new to the American mathematical educational
system. It simply provides additional evidence for the prevalence of a problem which was previously
recognized as a serious flaw in the American educational system. As evidence of this problem, The
Program for International Student Assessment [PISA] indicates that while U.S. high school students
match their peers in other nations when it comes to mathematical skills, this is not the case regarding
practical mathematical skills in which they ranked 24th out of 29 industrialized nations (NCES, 2004a).
Furthermore, The Trends in International Mathematics and Science Study [TIMSS] finds that
American eighth-grade students rank 15th internationally in mathematical achievement (NCES, 2004b).
In a similar vein, The National Assessment of Educational Progress [NAEP], billed as “the nation’s
report card”, reveals that only 36% of fourth graders, and 30% of eighth graders, have reached a level
of proficiency in mathematics (NCES, 2004c). These findings, which were reported prior to the
NSASC, indicated that the nation’s educational techniques must be improved to better prepare students,
throughout K-12, for the mathematical challenges that life presents. One would naturally assume that
this problem which is prevalent throughout K-12 would not vanish in American colleges. NSACS
confirms this assumption and demands that mathematics education reform be extended to American
colleges as well.
In response to the findings of weak mathematical performance of American
students throughout K-12, many educators have supported the implementation of a
Standards-Based Mathematics Curriculum in American schools. “The NCTM
Standards”, developed by the National Council of Teachers of Mathematics (NCTM,
1989, 1991, 1995, 2000), shift the focus of mathematics education from
memorization, rote learning, and application of facts and procedures, to the
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development of conceptual understanding and reasoning. The NCTM Standards are
based on a set of core beliefs about mathematics as a body of knowledge and about
the learning processes that effectively promote mathematical understanding and
literacy (Goldsmith and Mark, 1999).
The need for reform in collegiate mathematics education has also been
documented in several national reports, and change in this area has begun. Specific
recommendations for curriculum change in two-year colleges are made in Curriculum
in Flux (Davis, 1989). A new undergraduate curriculum is put forth in Reshaping
College Mathematics (Steen, 1989). Everybody Counts (National Research Council,
1989) calls for detailed changes in mathematics education starting from kindergarten
all the way to graduate school. Additionally, Moving beyond Myths (National
Research Council, 1991) proposes major changes in undergraduate mathematics
education. The MAA’s Guidelines for Programs and Departments in Undergraduate
Mathematical Sciences (1993) recommends that every college graduate should be able
“to analyze, discuss, and use quantitative information; to develop a reasonable level of
facility in mathematical problem solving; to understand connections between
mathematics and other disciplines; and to use these skills as an adequate base of life-
long learning.” Significant work has also been done regarding calculus educational
reform in American colleges [Crocker (1990), Ross (1994), Tucker and Leitzel
(1994)].
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The shortcomings revealed by the NSACS are specifically regarding students of two-year
colleges and the lower division of four-year colleges. Mathematics education of these students is
referred to as “introductory college mathematics” by the AMATYC’s Crossroads in Mathematics:
Standards for Introductory College Mathematics before Calculus (Cohen, 1995). Crossroads develops
standards for introductory college mathematics education with the following two goals in mind: “to
improve mathematics education at two-year colleges and at the lower division of four-year colleges and
universities and to encourage more students to study mathematics.” One theme in these standards is
that “the mathematics that students study should be meaningful and relevant” and that the problems
presented should “provide a context as well as a purpose for learning new skills, concepts and theories.
A similar sentiment is described by Haver and Turbeville (1995) in their formulation of the goals of a
mathematics course designed for nonscience majors. They explain:
The goals of the course are to develop, as fully as possible, the mathematical and quantitative
capabilities of the students; to enable them to understand a variety of applications of
mathematics; to prepare them to think logically in subsequent courses and situations in which
mathematics occurs; and to increase their confidence in their ability to reason mathematically.
In their recent report Beyond Crossroads (Blair, 2006), the AMATYC presents a renewed
vision for introductory college mathematics education by providing new Implementation Standards,
which “focus on student learning and the learning environment, assessment of student learning,
curriculum and program development, instruction, and professionalism.” These standards are designed
to “clarify issues, interpret, and translate research to bring standards-based mathematics instruction into
practice” in an attempt to reform American introductory mathematics education.
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Many other conferences and articles have echoed the sentiments that the American
educational system must reform the way it teaches mathematics, and refocus on practical applications.
Crossroads suggests that “introductory mathematics courses hold the promise of opening new paths to
future learning and fulfilling careers to an often neglected segment of the student population” (Cohen,
1995). Despite numerous suggestions and implementations of reform in introductory college
mathematics education, the recent findings of NSACS reveal that these changes have not been as
widespread or as effective as would be desired. A large number of American college graduates are still
failing in their quantitative literacy and are not properly prepared for the mathematical challenges
which life presents. Hopefully, the adoption of the new Implementation Standards suggested in Beyond
Crossroads will help address these shortcomings.
2.3. Opposition to an educational system focused on quantitative literacy. Despite the strong and
widespread support for a shift of the American educational system towards quantitative literacy, there
is some tough opposition. Kaiser (1999) reveals that the educational method in Germany is more
focused on theoretical aspects of mathematics, while that of England is more focused on the pragmatic
side of mathematics. Thus, the impetus towards quantitative literacy can be seen as a push that America
follows England’s lead. However, Gardiner (2004) strongly cautions against such an approach and
furnishes evidence of the failings of the English educational system in equipping its students with basic
mathematical skills. He attributes this failing to the paradigm shift which occurred in England after the
publication of the Cockcroft report (Cockcroft, 1982), and advises educators to heed the warning of
Hyman Bass (quoted in (Steen, 2004)), who describes with uncanny accuracy what occurred in
England in the late 1980’s and 1990’s:
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The main danger … is the impulse to convert a major part of the curriculum to this form of
instruction. The resulting loss of learning of general (abstract) principles may then deprive the
learner of the foundation necessary for recognizing how the same mathematics witnessed in
one context, in fact applies to many others.
Gardiner concludes that “mathematics and mathematics teaching are simply hard, and that
there is no “cheap alternative” to facing the fact that abstraction is a crucial part of elementary
mathematics - almost from the outset.” He cautions against making England’s mistake, and suggests
“that current abysmal levels of achievement indicate the need for hard work and incremental
improvement, rather than the launch of yet another bandwagon.” Gardiner’s points are insightful and
must be considered in any attempt at improving the American educational system. America must learn
from England’s mistakes and cannot afford to deprive its students of the true essence and beauty of
mathematics, and at the same time rob them of learning its basic skills. Simultaneously, the abysmal
levels of mathematical proficiency of America’s students must not be ignored, and its educational
methods must be improved.
3. A SUCCESSFUL MODEL FOR IMPROVING QUANTITATIVE LITERACY
In the search for a middle ground between the democratization of mathematics with its risk of
“watering down” mathematics on the one hand, and the maintenance of high mathematical standards on
the other, Amit and Fried (2002) present the successful model of Israel’s reformulation of their
National Completion Examinations in Mathematics (NCEM) administered at the end of high school. In
the early 1990’s, educators realized that the high level of mathematics demanded by their NCEM’s
intimidated weaker students, and caused many to terminate their mathematical studies as early as ninth
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grade. This presented a serious impediment to the nation’s aspirations towards high quantitative
literacy for its citizens. At the same time, however, they did not want to lower their standards and
history of high achievement for their advanced students. To solve this dilemma, in 1996 they created an
alternative for weaker students by dividing their basic NCEM into two parts. The basic part is designed
with the reasonable expectation that all students can pass. One of the foci of this part of the test is
mathematical questions involving common sense and everyday experience. The hope was that the
weaker students would be able to face their mathematical inadequacies and strive for a modicum of
success in their mathematical endeavors. Instead of insisting on a standard which these students could
not achieve, Israel devised a bifurcated system which enabled its weaker students to pursue a more
reasonable goal, and thereby continue their mathematical training. In order to ensure that the standards
for the higher level students were not compromised, the completion of the exam demanded passing the
more advanced section as well. Although this is the case, Amit and Fried comment that:
The hope is that the option of taking this part of the examination independently of the rest of
the examination will encourage students to continue to study mathematics earnestly until the
end of high school, that they will work through the Basic Questionnaire with success, and that
this sense of success will push them eventually to complete the whole NCEM. There is
already evidence that this hope is not futile.
The data collected in the years following this shift indicate the success of the model. The number of
students who took the basic exam increased - an indication that once the students were given the
opportunity to take an exam more suited to their level, they became less intimidated and rose to the
occasion. Interviews with teachers reflected that these students felt more motivated to continue
studying mathematics, and gained a sense of competency through passing the basic part of the exam.
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Although the evidence of Amit and Fried is limited to high school educational reform, the
concept should hold true in college educational reform as well. Assuming this to be the case, the Israeli
model indicates a successful method of maintaining America’s high mathematical achievement, while
pursuing reforms aimed at increasing its quantitative literacy. Namely, educators must continue their
advanced, more theoretical methods of mathematics instruction for stronger students, but
simultaneously reach out to weaker students and provide instruction more suited to their level and
interests. With this goal in mind, Section 4 introduces a new mathematics course, designed specifically
for weaker students. This pragmatic mathematic course should not be offered to stronger students who
are capable of higher mathematical achievement. By restricting the focus on practical mathematics to
its weaker students, America will follow Israel’s successful model and avoid the catastrophe which
occurred when England “watered down” its mathematics education for strong and weak students alike.
4. THE PRAGMATIC MATHEMATIC APPROACH
This section focuses on how the lesson learned from the Israeli model should be implemented in
teaching weaker students in American colleges. It elucidates the particulars of the pragmatic
mathematic approach and differentiates it from the traditional approach.
4.1. The traditional approach. It is common wisdom that to prepare students for life, the applicability
of mathematics to everyday situations must be demonstrated (see Senge (2000), for instance).
However, it is often overlooked that the role of practical examples must be different when teaching
strong, as opposed to weak students. In a traditional mathematics course, taken by both strong and
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weak students, a rigorous approach is taken2. The focus is on content knowledge; to teach the student
certain pieces of subject matter (Cohen, 1995).This subject matter is presented in a logical sequence,
determined by the intrinsic mathematical development of the ideas. After a brief introduction regarding
motivation for a given topic, the mathematical theory is taught in an abstract setting. The methods of
computation are derived from this theory and extended to real life applications, wherever possible. The
applications are essentially introduced as an afterthought to the fundamental ideas. This approach is
effective in teaching advanced students who comprehend the underlying mathematical theory, follow
the computational methods derived from this theory, and appreciate the practical applications enabled
by these methods. They are engaged each step of the way, are truly involved in the mathematical
process of gaining knowledge, and reap the full benefits of the education provided. They are not part of
the troubling statistic regarding quantitative illiteracy.
This method, however, is inappropriate for students lacking in quantitative literacy. The
traditional college algebra or precalculus courses, which are primarily designed to prepare students for
calculus, do not provide the breadth and applicability of mathematics needed by liberal arts students
(Sons, 1995). They are not interested in the theory, find the computational methods difficult, and are
consequently not prepared to comprehend the real life applications. Their apathy, coupled with the
intricacies of the material, obfuscates them. Due to their weakness in comprehending advanced
mathematics, they are robbed of the opportunity of acquiring basic mathematical skills well within their
capabilities. Unfortunately they proceed to fail a quantitative literacy test (such as NSACS) and, more
importantly, are not properly equipped with the fundamental degree of mathematical ability necessary
for life’s challenges. It must be remembered that “making mathematics relevant and meaningful is the
collective responsibility of faculty” (Cohen, 1995) and the results of NSACS reveal a failure in this
responsibility which must be addressed.
2 For a more thorough discussion of the traditional mathematics education approach, see Quirk (n.d.).
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4.2. A direction revealed by NSACS. One of the five Implementation Standards of
Beyond Crossroads (Blair, 2006) is “Curriculum and Program Development”. It
suggests that:
Mathematics departments will develop, implement, evaluate, assess, and revise courses,
course sequences, and programs to help students attain a higher level of quantitative literacy
and achieve their academic and career goals.
In attempting to discover an appropriate method for implementing this standard and reaching out to
college students with poor quantitative literacy, one aspect of the NSACS stands out and provides
direction to educators. Namely, the study reveals that students who take classes which stress analytic
thinking and applying theories to practical problems, have a higher degree of quantitative literacy
(AIR, 2006). This correlation suggests the introduction of a pragmatic mathematic course which
reaches out to students who have no intrinsic interest in mathematics, but realize its necessity in the
modern world3. This course addresses the frequent question posed by students, “Why do we need to
learn this stuff?” The pragmatic approach better motivates the student to study mathematics. Instead of
immersing students in pure mathematics and its multifarious abstractions, this course only teaches the
mathematical skills necessary in the modern society, and therefore succeeds in conveying these skills to
the students4. By giving the students a level of comfort with mathematics and allowing them to realize
the power and usefulness of mathematics, they are assisted in overcoming their fear of mathematics
and improving their quantitative literacy. 3 Though the correlation does not prove that the cause of the increased quantitative literacy is these courses, it is nonetheless a correlation which suggests a direction for needed reform. 4 Being that modern society is rapidly changing, the mathematical skills necessary in modern society are also rapidly changing. Thus, while the approach of this course is fixed, the particulars must be adjusted by the instructor to the changing demands of society. This being said, for a sample of some mathematical skills necessary in today’s society, see Section 5.4.
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4.3. The pragmatic mathematic approach: What is taught? The new method that the pragmatic
mathematic utilizes in teaching these students is exactly the opposite of the traditional sink-or-swim
approach. The traditional approach unrealistically attempts to prepare all students to become
mathematical experts. On the other hand, the pragmatic mathematic aspires to train weaker students in
practical mathematical areas. The practical examples do not complement the theory, but are the focus
of this course. It is not a pure mathematics class, but a preparatory class for life’s challenges. As such,
each lesson focuses on a problem which the students encounter in their daily lives. The syllabus is
designed to have approximately 40 practical problems which will serve as springboards to introduce
mathematical skills and methods. Instead of concocting unrealistic word problems to illustrate remote
applications of an abstract subject, this course focuses on real life problems which every student relates
to and appreciates. Rather than the example of Phil, the farmer, using the quadratic formula to help
plant his field, this course considers the example of Steve, the student, using fractions to determine if
he will make it to the next gas station. A teacher of this course will not be concerned about skipping
abstract topics whose mathematical significance may be great, but whose practical significance is
small, or nonexistent. These topics simply do not belong in a pragmatic mathematic course, but in a
true mathematics course. Just as Shakespeare is not taught to beginners in reading, mathematical
abstraction should not be taught to beginners in mathematics. Will these students comprehend and
appreciate the full picture of mathematics? Absolutely not! They will not understand the theory, nor
will they have a solid grasp on the rigor of the computational methods. An honest analysis leads to the
realization that this is not appropriate for these students, as is evidenced by their failure to grasp real
mathematics in the traditional educational system. In a pragmatic mathematic course, these students
derive from mathematics the practical tools they truly need. As was the case with Israel’s high school
students, these positive experiences allow the students to become comfortable with mathematics. They
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will enjoy the pragmatic mathematic as its beneficial function is immediately apparent. This method
addresses the American educational system’s shortcomings in educating its weaker students, and will
find success in increasing the nation’s quantitative literacy through teaching these students only what is
truly necessary.
4.4. The pragmatic mathematic approach: How is it taught? Besides for impacting what material is
taught, the pragmatic mathematic approach provides a new method as to how the material should be
taught. This method is elucidated by explaining the basic approach towards each lesson in a pragmatic
mathematic course, and then by illustration through an example. Each lesson begins by engaging the
students with a practically motivated exercise. Once the students gain interest in the problem, the
instructor demonstrates that its solution demands mathematics, and introduces the skills necessary to
solve the problem. Instead of insisting on a rigorous mathematical solution, the most efficient method
of solving the problem is illustrated. Whenever possible, tricks or shortcuts are introduced to simplify
the solution. The method can be illustrated with a simple example. On a regular basis, students are
faced with the task of computing a tip. Assume that their restaurant bill totals $51.07, and they want to
give a 15% tip. The NSACS reveals that many college graduates are perplexed by such a task. Why is
this so? Are American students incapable of such a simple procedure? Certainly not! The explanation is
usually that they were never properly taught how to calculate it. They were instructed how to take
2.47% of 0.0456, and other complex percent problems. They have a vague recollection of moving the
decimal two spots to the left and multiplying, but have forgotten long ago how to carry out this multi-
step process. They were never shown how straightforward it is to take 15% of a number, especially
when precision is unnecessary. It is in this context that percents are introduced - with the simplest, most
practical problems, which can be solved by a shortcut. The instructor emphasizes the significance of
rounding in real-life problems. Students are taught to quickly compute 10%, half it to get 5%, and add
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these to get the desired 15%. No student should have trouble applying this method, especially with
practice. How many mathematicians evaluate a 15% tip by doing long multiplication by 0.15? Most
employ some shortcut which they have discovered on their own. So why not teach this method to
students who cannot figure it out by themselves? Why burden them with multiplying $51.07 by 0.15
when there is a simpler and quicker alternative? Just as in a regular mathematics course tricks cannot
be allowed to substitute for real mathematics, so too in a pragmatic mathematic course, real
mathematics cannot be allowed to substitute for tricks, and cloud the path towards a practical solution.
The goal of this course is not to make its students into mathematicians, but to give them the tools and
confidence needed to apply mathematics to their everyday lives. Although not every lesson lends itself
to the simplicity involved in computing tips, this example illustrates the approach of this course and
should, therefore, serve as a model for other lessons.
4.5. Education regarding real-life institutions. The pragmatic mathematic course possesses another
feature which addresses the failings of college graduates in the NSACS, but is unrelated to their
mathematical education per se. Often times, students are relatively well equipped with mathematical
skills, but are ignorant regarding the real world institutions which invoke these skills. For instance,
computing interest payments on a mortgage or credit card involve basic mathematics. Yet, even the
greatest “math genius” would be unable to perform these tasks without knowing the concept of a
mortgage, or the meaning of APR. This is specifically applicable to students with poor quantitative
literacy who fear anything involving mathematics and, therefore, never acquaint themselves with these
basic financial phenomena. With this realization in mind, this course attempts to overcome the
students’ fear of mathematics and teaches mathematical skills together with their accompanying real
world knowledge. The concepts of interest, insurance, credit card fees, deposit slips, checkbooks,
investments, odds, and other such notions are elucidated. By familiarizing students with sufficient
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applications of mathematics, they are assisted in overcoming their dread of mathematics, and become
more suited to handle future situations involving mathematical skills.
The success of the course will be increased by designing hands-on lessons. This can be
accomplished by distributing: checkbooks, deposit slips, credit card offers, cell phone plans, lottery
rules, odds for sporting events, food labels containing nutritional information, and any other material
which invokes mathematics. This arouses the interest of the students and allows them to realize, in a
concrete manner, the value of improved quantitative literacy.
4.6. Comparable courses. A similar, but more sophisticated, pragmatic course design is suggested by
Bernard L. Madison (2004). The University of Arkansas course, developed by Madison, is centered on
newspaper and magazine articles which can only be understood or critiqued by applying mathematical
skills. A comparison between his course and the pragmatic mathematic course indicates that the subject
matter and mathematics involved in Madison’s university course are more complex and are suited to
higher level students.
Another similar liberal arts mathematics course is offered at University of Colorado at Denver
by William L. Briggs. The course has three stated goals: (1) to strengthen and broaden students’
quantitative skills; (2) to restore students’ confidence in using those skills; and (3) to demonstrate the
immediate relevance and applicability of mathematics to students’ lives and careers. (Briggs et al.,
2004). From their experiences, the authors conclude that “if student engagement is secured early in the
course, it can change student attitudes favorably and lead to an effective learning experience.” It would
be a fruitful study to thoroughly compare the three courses (Madison’s, Briggs’s and the pragmatic
mathematic) in their content and degree of success.
5. ADDRESSING OBJECTIONS TO THE PRAGMATIC MATHEMATIC COURSE
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This section discusses a number of objections which can be raised against the implementation
of the pragmatic mathematic course in American colleges.
5.1. The level of the course. One might raise the following objection to the pragmatic mathematic
course: “Is this truly a college mathematics course? After all, the mathematical topics covered are of an
elementary nature and should have been mastered before entering college. Students who have difficulty
with these skills should take remedial classes to rise up to college standards!” To this objection, two
responses are offered. Firstly, the NSACS revealed the sobering fact that thirty percent of graduates of
American two-year colleges are severely lacking in their practical mathematical skills. This is
compared to the zero percent of students who graduate without remediation or placing out of remedial
courses. Colleges are simply not succeeding in preparing their students for life. What good is a
mathematics education which prepares students to pass a test requiring numerous calculations, if they
stumble as soon as they encounter a real life mathematical problem? Is this truly a college mathematics
course? Although it is not a college course in the current educational system, that is precisely the
problem. It should be, as indicated by the results of the NSACS. Educators cannot ignore the findings
of this national study and blindly assume that the current approach is flawless. They must rise to the
challenges presented by the directors of the study and change the manner in which weaker students are
educated. The pragmatic mathematic course suggested in this paper rises to this new challenge.
Additionally, one cannot judge the level of a mathematics course merely by the technical
skills it involves. Any mathematics teacher is well aware that many students who have mastered the
requisite technical skills become dumbfounded by challenging word problems. This explains why
many students performed poorly in the NSACS test of practical mathematics skills, despite the fact that
all community college graduates have either passed or placed out of remediation. Mathematics cannot
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be limited to the application of step-by-step algorithms, but must include understanding, thinking and
applying mathematics to new situations. This is arguably the most difficult and important part of
mathematics, and is precisely what the current educational system is failing to teach its students. Thus,
although the pragmatic mathematic course covers mathematical skills which are considered elementary
in their technical level, it teaches its students how to understand and apply these skills to new problems.
It therefore truly merits the status of a college-level mathematics course.
5.2. The scope of the course. Another objection which might be raised is that by restricting its lessons
to truly practical examples, this course unnecessarily limits the scope of its students’ mathematical
education. After all, by the end of the semester its students have only learned how to solve
approximately 40 practical problems. A more traditional approach, however, would provide students
with tools to handle a larger variety of problems. There are two responses to this objection. First,
although the lessons are centered about practical problems, these problems are invariably solved by
mathematical methods which can be generalized to other problems. These 40 problems train the
students to think mathematically and prepare them to solve other mathematical challenges. More
importantly, Amit and Klein (2002) note that the major hindrance to students’ advancement in
mathematics is not their weak intellectual faculties, but is their frightful attitude towards mathematics.
Because of their early failings in mathematics, they shy away from anything involving mathematics.
An effective method of overcoming this obstacle is through accustoming students to applying basic
mathematics, and helping them realize the usefulness of mathematics in making important decisions. A
course meeting the Crossroads standards is one in which “the students will have the opportunities to be
successful in doing meaningful mathematics that fosters self-confidence and persistence” (Cohen,
1995). When mathematics is presented as a concrete tool instead of an abstract pursuit, it becomes
demystified in their minds. By breaking through their inner resistances, educators can open up a world
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of mathematics which was previously closed to these students. Once students become comfortable with
mathematics, they readily learn the basic skills they need. Thus, while the initial approach of this
course is limited to the practical, its objective extends much further. Hopefully, this course will provide
an effective method of reaching those students which the educational system has failed thus far, and
will enjoy the success of Israel’s model which reached out to its weaker students in the reformulation
of its NCEM’s.
5.3 Avoiding the “England Disaster”. Beside addressing the troubling results revealed in the recent
studies, this new course stands up to Gardiner’s objections as well. In order to avoid the “England
disaster” which resulted from shifting the focus of all mathematics education towards the practical, this
syllabus should only be used as a substitute for a “liberal arts mathematics” syllabus. In general,
students who take this course are those who have no plans of advancement in mathematics, but need to
satisfy a college requirement. These classes attract students who struggle with quantitative literacy.
Since the studies underscore the failings of the traditional methods for the lower twenty to thirty
percent of students, it is only these weaker students who must be targeted by a new approach.
However, this syllabus must not be implemented for stronger students who can, and must learn the true
rigor and theory involved in mathematics. The Israeli model, which found success through clearly
differentiating between its standards for stronger and weaker students, must be followed. Advanced
classes must continue in the traditional educational approach, keeping in mind that “numeracy and
mathematical literacy are desirable byproducts of school mathematics” (Gardiner, 2004). America will
thereby maintain the strengths of its stronger students, while simultaneously alleviating the weaknesses
of its weaker students.
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5.4. The quantity of material. One may suggest that there are not enough practical mathematical
problems to provide material for an entire semester. In order to address this concern, numerous
examples are listed by topic. These examples are merely a fraction of the numerous problems which
students with poor quantitative literacy are confounded by on a daily basis. Paying attention to every
day experiences furnishes many similar examples. Teachers are encouraged to elicit examples
involving mathematical reasoning from the students’ daily routines. As the objective is to help the
students gain the mathematical skills which they require in their lives, they will direct the instructor to
the areas they find difficult. Additionally, such an approach is an effective means of fostering the
interest of the students.
5.4.1 Financial Topics. Comparing credit card offers; computing interest on a credit card
based upon APR; computing simple and compound interest on mortgages, loans and
investments; balancing checkbooks; deposit slips; comparing investments: stocks, bonds,
cash, mutual funds; determining profits/losses on investments; retirement plans; income taxes.
5.4.2. Consumer Topics. Comparing the value of two products in the grocery store;
comparing nutritional information on food products; comparing cell phone offers; comparing
prices for different types of gasoline: full vs. self and premium vs. regular; computing miles
per gallon of a car; determining if a car has enough gas to reach a gas station; metric system
conversions; computing tips; determining square yardage of a room to buy carpet; analysis of
insurance premiums using mathematical expectation; comparing insurance plans based upon
deductibles, percent coverage and out-of-pocket expenses; determining price of an item with a
given percent off sale; estimating and determining sales tax.
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5.4.3. Recreational Topics. Understanding probability; methods of counting; analysis of
lotteries; odds for sporting events; examination of card and dice games; interpreting statistics,
bar graphs, line graphs, and circle graphs.
5.4.4. Miscellaneous Topics. Mixture problems; work problems; averages and grades;
greatest common divisor and least common multiple; speed/miles per hour; and scientific
notation.
6. CONCLUSION
Many studies have underscored the failing of America’s mathematics educational system in
imparting quantitative literacy to its students at all levels. The NSACS has provided new evidence of
the severity of this problem in the nation’s colleges. The pragmatic mathematic course is a new
suggestion to help remedy this problem. Educators at both two-year and four-year colleges are
encouraged to offer this course as a liberal arts mathematics course. The author of this paper requests to
be kept informed of any progress. Hopefully, this course will build a foundation for raising the
quantitative literacy of American college graduates and adequately preparing them for the challenges of
the modern world.
ACKNOWLEDGEMENTS
The author would like to thank Aron Zimmer and Rachel Sturm-Beiss for their help with this paper.
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REFERENCES
American Institutes for Research (AIR), (2006). The National Survey of America’s College Students:
The Literacy of America’s College Students. Washington, DC: Author. Amit, M., & Fried, M.N. (2002). High-Stakes Assessment as a Tool for Promoting Mathematical
Literacy and the Democratization of Mathematics Education. Journal of Mathematical Behavior, 21, 499-514.
Arney, D.C. (1999). Undergraduate Mathematics for the Future: Modeling and Solving Problems,
Understanding the New Sciences. Retrieved January 14, 2007 from http://www.dean.usma.edu/math/activities/ilap/workshops/1999/files/arney1.pdf.
Blair, R. (Ed.). (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two
Years of College. Memphis, TN: American Mathematical Association of Two-Year Colleges (AMATYC).
Briggs, W., Handelsman, M.M. & Sullivan, N. (2004). Student Engagement in a Quantitative Lieracy
Course. The AMATYC Review, 26(1). Carenvale, A.P., & Desrochers, D.M. (2003). The Democratization of Mathematics. In B.L. Madison,
& L.A. Steen (Eds.), Quantitative Literacy: Why Numeracy Matters for Schools and Colleges, (pp. 21-31). Princeton, NJ: National Council on Education and the Disciplines.
Cockcroft, W.H. (1982). Mathematics Counts. Report of the Committee of Inquiry into the
Teaching of Mathematics in Schools. London: Her Majesty’s Stationery Office.
Cohen, D. (Ed.). (1995). Crossroads in Mathematics: Standards for Introductory College Mathematics before Calculus. Memphis, TN: American Mathematical Association of Two-Year Colleges (AMATYC).
Crocker, D. A. (1990). What has happened to calculus reform? The AMATYC Review, 12(1), 62-66.
Crowther Report (1959). 15 to 18: Report of the Central Advisory Council for Education (England). London: Her Majesty’s Stationery Office.
Davis, R.M. (Ed.). (1989). A curriculum in flux: Mathematics at two-year colleges (A report of the Joint Subcommittee on Mathematics Curriculum at Two-Year Colleges). Washington, DC: Mathematical Association of America (MAA).
Gardiner, T. (2004). What is Mathematical Literacy? UK: University of Birmingham. Retrieved February 9, 2006, from http://www.pims.math.ca/.
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Goldsmith, L.T. & Mark, J. (1999). What is Standards-Based Mathematics Curriculum? The Constructivist Classroom, 57 (3), 40-44.
Haver, W. & Turbeville, G. (1995). An Appropriate Culminating Mathematics Course. The AMATYC
Review, 16(2), 45-50. Kaiser, G. (1999). Comparative Studies on Teaching Mathematics in England and Germany. In Kaiser,
G. & Luna, E. & Huntley, I. (Eds.), International Comparisons in Mathematics Education (pp. 140–150). London: Falmer Press.
Madison, B.L. (2004, December). To Build a Better Mathematics Course. All Things Academic, 5 (4).
Mathematical Association of America (MAA). (1993). Guidelines for programs and departments in undergraduate mathematical sciences. Washington, DC: Author.
National Center for Education Statistics (NCES). (2004a). International Outcomes of Learning in Mathematics Literacy and Problem Solving: PISA 2003 Results from the U.S. Perspective (NCES 2005-003). Washington, DC: U.S. Department of Education: Author.
National Center for Education Statistics (NCES). (2004b). Highlights from the Trends in International
Mathematics and Science Study (TIMSS) 2003 (NCES 2005-005). Washington, DC: U.S. Department of Education: Author.
National Center for Education Statistics (NCES). (2004c). The Nation’s Report Card: Mathematics
Highlights 2003 (NCES 2004-451). Washington, DC: U.S. Department of Education: Author. National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and
evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (NCTM). (1991). Professional standards for teaching mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (NCTM). (1995). Assessment standards
for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards in school mathematics. Reston, VA: Author.
National Research Council. (1989). Everybody counts. Washington, DC: National Academy Press: Author.
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National Research Council. (1990). A Challenge of Numbers. Washington, DC: National Academy Press: Author.
National Research Council. (1991). Moving beyond myths: Revitalizing undergraduate mathematics. Washington, DC: National Academy Press: Author.
Quirk, B. (n.d.). Traditional K-12 Math Education: Chapter 1 of Understanding the Original NCTM Standards, retrieved on January 14, 2007 from http://www.wgquirk.com/Genmath.html.
Ross, S. C. (Ed.). (1994, January). NSF Beat: $2.1 million in NSF calculus grants. UME Trends.
Senge, P. (2000). Schools That Learn: A Fifth Discipline Fieldbook for Educators, Parents, and Everyone who cares about Education. New York: Doubleday Currency.
Sons, L. R. (Chair). (1995). Quantitative reasoning for college graduates: A complement to the Standards. (Report of the CUPM Subcommittee on Quantitative Literacy). Washington, DC: Mathematical Association of America.
Steen, L. A. (Ed.). (1989). Reshaping college mathematics. Washington, DC: Mathematical Association of America.
Steen, L. (Ed.). (1997). Why Numbers Count: Qualitative Literacy for Tomorrow’s America, The College Board, New York.
Steen, L.A. (Ed.). (2001). Mathematics and Democracy: The Case for Quantitative Literacy. Princeton,
NJ: National Council on Education and the Disciplines. Steen, L.A. (2004). Achieving Quantitative Literacy. MAA Notes, 62. Washington D.C.: Mathematics
Association of America.
Tucker, A. C., & Leitzel, J. R. C. (1994). Assessing calculus reform efforts. Washington, DC: Mathematical Association of America.
U.S. Department of Education. The National Commission on Excellence in Education (1983). A Nation at Risk: The Imperative for Educational Reform, Washington, DC: U.S. Government Printing Office.
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Using Alternative Numeral Systems in Teaching Mathematics
Farida Kachapova School of Computing
and Mathematical Sciences Auckland University of
Technology [email protected]
Murray Black School of Computing
and Mathematical Sciences Auckland University of
Technology [email protected]
Ilias Kachapov 13 Fairlands Ave
Waterview Auckland 1026, New
Zealand [email protected]
1. Introduction
The purpose of this article is to stimulate interest in number theory and history of
mathematics. Teachers can include parts of this article as additional topics in their
classroom teaching. On one hand, the origin of our decimal numbers and possible
numeral systems can make an exciting lesson and lead to quite advanced topics in
mathematics. On the other hand, this material does not require any specific knowledge
and can be explained even to primary students. So it can be used by high school
teachers to motivate students and expand their horizons.
A numeral system is a language where numbers are represented by symbols –
numerals. In the modern mathematics we use the decimal numeral system. It has base
10, which means that all numerals are made of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The
probable reason for using base 10 is that humans have 10 fingers and they used them
for counting. 10 is the base of the most common numeral system but it is not the only
possible one. Some native Americans used spaces between their fingers for counting,
so their numeral system had base 8.
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The binary system (with base 2) has originated in China, was studied by G.Leibniz
in the 17th century and had useful applications in computer science in the modern
time. The hexadecimal system (with base 16) also has applications in computer
science because its base can be written as a power of two: 16 = 2 4.
The numeral systems with base 12 and base 60 were also used in the past. We still
have 12 hours in the clock and 12 months in the year. When we measure time we use
60 minutes in an hour and 60 seconds in a minute; we have 60 seconds in a degree as
angular measure.
In practice we mostly use the decimal numeral system. The alternative numeral
systems can be used by mathematics teachers to stimulate students’ interest in
mathematics and research. The students will realise that the decimal system is not the
only possible one and even not the best one, so they will learn to think “outside the
square”. The binary, senary and other alternative numeral systems can be explained
in simple terms at different levels starting from primary school. At the same time they
lead to some interesting topics in number theory and general algebra at the tertiary
level, such as modular arithmetic and Mersenne primes.
2. Senary Numeral System
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Now we will take a closer look at the senary numeral system – the numeral system
with base 6.
Imagine aliens from another planet that have three fingers on each hand. They use 6
fingers for counting, so their numeral system has base 6 and all the numerals are
constructed from 6 digits: 0, 1, 2, 3, 4, 5. Besides each alien has 6 limbs: two arms,
two legs and two wings.
The aliens have the same arithmetic as we do but their arithmetic has a simpler
representation because it is expressed in the senary numeral system instead of our
decimal numeral system.
We will write numerals in the senary system with a subscript 6. So numbers 0, 1, 2,
3, 4, 5 are represented by numerals 06 , 16 , 26 , 36 , 46 , 56 respectively. Next number
6 is represented by 106 , 7 is represented by 116 , etc. Actually humans with 5 fingers
on each hand can show two-digit senary numerals if they use each hand to show a
digit from 0 (a fist with no fingers out) to 5 (all fingers out).
3. Arithmetic in the Senary System
Addition table:
16 + 16 = 26
26 + 16 = 36 26 + 26 = 46
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36 + 16 = 46 36 + 26 = 56 36 + 36 = 106
46 + 16 = 56 46 + 26 = 106 46 + 36 = 116 46 + 46 = 126
56 + 16 = 106 56 + 26 = 116 56 + 36 = 126 56 + 46 = 136 56 + 56 = 146
Multiplication table:
26 × 16 = 26 36 × 16 = 36 46 × 16 = 46 56 × 16 = 056
26 × 26 = 46 36 × 26 = 106 46 × 26 = 126 56 × 26 = 146
26 × 36 = 106 36 × 36 = 136 46 × 36 = 206 56 × 36 = 236
26 × 46 = 126 36 × 46 = 206 46 × 46 = 246 56 × 46 = 326
26 × 56 = 146 36 × 56 = 236 46 × 56 = 326 56 × 56 = 416
26 ×106 = 206 36 ×106 = 306 46 ×106 = 406 56 ×106 = 506
The senary addition table is shorter than the decimal one but otherwise is not very
different.
The senary multiplication table is much easier to learn for the aliens’ children than
the decimal one for the humans’ children because it has several patterns in it:
1. Last digits 2, 4 and 0 make a cycle in the first and third columns.
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2. Last digits 3 and 0 make a cycle in the second column.
3. In the fourth column the first digits are 0, 1, 2, 3, 4, 5 and the last digits are the
same in reverse order.
Notation: for digits an, …, a1, a0, the symbol an … a1 a0 will denote the numeral
in the senary system made of these digits (versus a product an ⋅ …⋅ a1 ⋅ a0).
As with any other base, there are algorithms for transferring numbers from senary
form to decimal form and vice versa.
Senary to Decimal. For a numeral an an-1… a1 a0 in the senary system its decimal
form is given by the formula an ⋅6n + an-1 ⋅6n-1 +…+ a1 ⋅ 61 + a0 ⋅ 60.
Examples:
12346 = 1×63 + 2×62 + 3×6 + 4 = 310,
543216 = 5×64 + 4×63 + 3×62 + 2×6 + 1 = 7465.
Decimal to Senary. To transform a natural number in decimal form to senary form
we keep dividing numbers by 6 and then write all remainders from right to left.
Example: 1244 ÷ 6 = 207 with remainder 2
207 ÷ 6 = 34 with remainder 3
34 ÷ 6 = 5 with remainder 4
5 ÷ 6 = 0 with remainder 5
So 1244 = 54326.
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Since the senary multiplication table has several patterns, divisibility criteria are
easier in the senary system. To check divisibility of a number by 2, 3 or 6 we use only
its last digit: if the last digit is a, then the number can be written as 6b + a for some
integer b.
Suppose a natural number n ends on a digit a in the senary system.
Divisibility by 2 (3, 6). n is divisible by 2 (3, 6) if and only if a is divisible by 2 (3,
6).
We can write this in different terms.
Divisibility by 2. n is divisible by 2 if and only if a equals 0, 2 or 4.
Divisibility by 3. n is divisible by 3 if and only if a equals 0 or 3.
Divisibility by 6 = 10 6. n is divisible by 6 = 106 if and only if a equals 0.
Divisibility by 4. Suppose a natural number n has digits a and b as its last two
digits in the senary system. Then n is divisible by 4 if and only if the number ab is
divisible by 4.
Proof. The numeral ab denotes the number 6a + b. For some integer c,
n = 36c + 6a + b = 4 ⋅ 9 + 6a + b.
So n is divisible by 4 if and only if 6a + b is divisible by 4, which is if and only if
ab is divisible by 4.
Next divisibility criterion is proven similarly.
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Divisibility by 9 = 13 6. Suppose a natural number n has digits a and b as its last
two digits in the senary system. Then n is divisible by 136 if and only if the number
ab is divisible by 136.
Divisibility by 5. A natural number n is divisible by 5 if and only if a sum of all its
digits in the senary system is divisible by 5.
Proof. The proof is easy if we use congruency modulo m. Two integers x and y
are said to be congruent modulo m if (x − y) is divisible by m; this is denoted by
x ≡ y (mod m).
Suppose n = ak … a1 a0. Then n = ak ⋅6k +…+ a1 ⋅ 6 + a0 .
Apparently 6 ≡ 1 (mod 5). For i = 1, 2,…, k by properties of the congruency,
6 i ≡ 1 (mod 5), ai 6 i ≡ ai (mod 5) and n ≡ ak +… + a1 + a0 (mod 5). Hence n is
divisible by 5 if and only if a0 + a1 +…+ ak is divisible by 5.
Next divisibility criterion is proven similarly.
Divisibility by 7 = 11 6. A natural number n = ak … a1 a0 is divisible by 116 if and
only if a sum of its signed digits a0 − a1 + a2 −… ak is divisible by 116.
4. Fractions in the Senary System
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A fraction 0.c1c2 c3 … in the senary system is transformed to the following
decimal numeral: ...ccc
+++3
3
2
21
666.
Examples: 0.346 = 18
11
6
4
6
3
2=+ = 0.6111… ;
0.111…6 = 5
1
6
11
1
6
1
6
1
6
1
6
1
32=
!
"=+++ ... = 0.2.
The following example demonstrates how to transform a decimal numeral to senary
form by subsequent multiplication by 6.
Transforming 16
1 to senary form. 6×16
1 = 8
3 = 8
30 ; 6×
8
3 = 4
12
4
9= ;
6×4
1 = 2
11
2
3= ; 6×
2
1 = 3. We collect the whole parts of all results to get a senary
numeral: 0.0213. So 16
1 = 0.02136.
In the decimal system some fractions can be written as finite decimals and others as
infinite (recurring) decimals. For example, 2
1 = 0.5 and 25
1 = 0.04 are finite;
3
1 = 0.333… and 6
1 = 0.1666… are infinite. In general, a fraction k
1 can be written
as a finite decimal if and only if any prime factor of k is either 2 or 5. The reason is
that 2 and 5 are the only prime factors of 10.
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For base 6 the only prime factors are 2 and 3. So a fraction k
1 has a finite
expression in the senary system if and only if any prime factor of k is either 2 or 3.
Since multiples of 3 occur more often than multiples of 5, the fractions with finite
expressions occur more often in the senary system than in the decimal system:
Senary Decimal
62
1 = 0.36 2
1 = 0.5
63
1 = 0.26 3
1 = 0.333…
64
1 = 0.136 4
1 = 0.25
65
1 = 0.111…6 5
1 = 0.2
610
1 = 0.16 6
1 = 0.1666…
611
1 = 0.0505…6 7
1 = 0.142857…
612
1 = 0.0436 8
1 = 0.125
613
1 = 0.046 9
1 = 0.111…
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Base 60 seems even more practical for expressing fractions, since it has three prime
factors: 2, 3 and 5.
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5. Prime Numbers and Perfect Numbers in the Senary System
If a number in the senary system ends on 0, 2 or 4, then it is divisible by 2. If a
number in the senary system ends on 0 or 3, then it is divisible by 3. So any prime
number in the senary system (except 2 and 3) ends on 1 or 5. These are the first
eleven prime numbers:
Decimal
5 7 11 13 17 19 23 29 31 37 41
Senary
56 116 156 216 256 316 356 456 516 1016 1056
We should emphasize that most properties of numbers (for example being a prime
number) are independent of the numeral system because numerals are only symbols
for expressing numbers. But some numeral systems express number properties in a
simpler form than others.
Perfect numbers are another class of numbers that have a simpler form in the senary
system. A perfect number is a natural number, which is a sum of its positive factors,
excluding itself. Here are a few first perfect numbers:
6 = 1+2+3,
28 = 1+2+4+7+14,
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496 = 1+2+4+8+16+31+62+124+248,
8128,
33550336.
These are their senary forms:
6 = 106,
28 = 446,
496 = 21446,
8128 = 1013446,
33550336 = 31550333446.
One can see that every numeral on the right of this list, except 6, ends on 44. It can
be proven that in the senary system every perfect number (except 6) has 44 as its last
two digits.
Thus, the senary system would make a better symbolic base for dealing with
numbers in mathematics but due to historical reasons we use the decimal system and
it is too late to change it now.
References
Ore, O. (1988). Number theory and its history. New York: Courier Dover Publications
Mathematics Teaching-Research Journal On-Line
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Volume 2 Issue 1 Date September 2007
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Lam, L.Y.& Ang, T.S. (2005). Fleeting footsteps: tracing the conception of arithmetic and
algebra in ancient China. River Edge, N.J.: World Scientific.
Kachapova, F. and Kachapov, I. (2005) Senary Numeral System. Workshop at the 9-th
conference of NZ Association of Mathematics Teachers. Retrieved October 1, 2005 from
NZAMT website http://www.nzamt.org.nz/nzamt9/ka/Senary%20System.ppt
Senary. (2007). In Wikipedia, The Free Encyclopedia. Retrieved April 22, 2007,
from http://en.wikipedia.org/w/index.php?title=Senary&oldid =121979068
Perfect number. (2007). In Wikipedia, The Free Encyclopedia. Retrieved April 22, 2007, from
http://en.wikipedia.org/w/index.php?title=Perfect_number&oldid=123382935
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Effective Procedures in Teaching Mathematics
Farida Kachapova Auckland University of Technology
Murray Black Auckland University of Technology
<[email protected]> Ilias Kachapov
.
Introduction and Framework In our practice of teaching tertiary mathematics we want students to gain both
conceptual and procedural knowledge and develop their problem solving abilities. In
mathematics, procedural knowledge is the knowledge of symbolic representations,
algorithms and rules; conceptual knowledge is the knowledge of core concepts,
principles and their interrelations (Byrnes & Wasik, 1991). On the tertiary level, the
dominance of procedural mathematics is a characteristic of mathematics itself.
Mathematical theories are based on axioms and derivation rules, thus this knowledge
is highly procedural by nature: it must be derived from the fundamental definitions
and axioms by a finite sequence of logical steps (Tossavainen, 2006). Also
understanding a mathematical concept often does not provide the relevant procedural
knowledge, for example the definitions of a limit, a derivative or an integral are not
linked to the methods of their evaluation.
One approach to learning mathematics labeled the ‘dynamic action view’ (Byrnes
& Wasik, 1991), justifies the ‘traditional’ emphasis on procedural knowledge.
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Another approach, the ‘simultaneous action view’, focuses on conceptual knowledge
using the assumption that student’s conceptual knowledge will necessarily increase
their procedural knowledge (Haapasalo & Kadijevich, 2000).
In constructivism, in particular in Piaget’s theory of cognitive development
(Piaget, 1985), conceptual knowledge and procedural knowledge are both integral
parts of the learning process. The ‘iterative model’ developed these views further
(Rittle-Johnson, Siegler, & Alibali, 2001). Their research shows the causal relations
between conceptual and procedural knowledge: concepts and procedures develop
iteratively reinforcing each other. Increased conceptual knowledge leads (through
training) to gains in procedural and problem solving abilities. Use of correct
procedures leads to improved conceptual understanding.
The authors of this paper share the iterative views. Therefore in our teaching
practice we look for effective methods to improve students’ procedural knowledge
that will in turn enhance their conceptual knowledge. Here we describe some non-
traditional methods that we used in courses in algebra, calculus and probability theory
at the Auckland University of Technology and the Moscow Technological University
for several years.
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Substitution Method
The authors used the substitution method in teaching some topics of secondary
and tertiary mathematics, where other methods were traditionally applied. They
noticed that substitution simplifies learning and mastering some mathematical
techniques, especially for weaker students. The students who use substitution do not
have to guess; they just apply simple formulas, get answers faster and make fewer
mistakes on the way.
Completing the Square
To complete the square in a quadratic y = cbxax ++2 we introduce a new
variable a
bxt
2+= . Then the linear term in the quadratic cancels and at the end we
substitute x back. This can be justified as follows:
=+!=+"#
$%&
'!+"
#
$%&
'!=
!=
+=
=++= ca
batc
a
btb
a
bta
a
btx
a
bxt
cbxaxy422
2
22
2
2
2
a
bc
a
bxa
42
22
!+"#
$%&
'+= .
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The new variable means the change to the set of coordinates where the
corresponding parabola has a simple form: symmetrical in the y-axis.
The following three examples illustrate this.
Example 1.
( ) ( ) =!=+!+!=!=
+==++ 27363
3
376
222ttt
tx
xtxx ( ) .x 23
2
!+
Example 2.
=!=+"#
$%&
'+!"
#
$%&
'+=
+=
!=
=+!8
123
4
55
4
52
4
5
4
5
3522
2
2ttt
tx
xt
xx
.x8
1
4
52
2
!"#
$%&
'!=
Solving Quadratic Equations
The same substitution a
bxt
2+= helps to solve quadratic equations in an easier
way than traditional factorising or quadratic formula.
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Example 3.
0158122
=!! xx .
3
1
3
1
+=
!=
tx
xt
. 0153
18
3
112
2
=!"#
$%&
'+!"
#
$%&
'+ tt ,
03
4912
2=!t ,
6
7
36
492±== t,t . 1)
6
5
3
1
6
7
3
1
1!=+!=+= x,tx . 2)
2
3
3
1
6
7
3
1
2=+=+= x,tx .
Example 4.
0142
=++ xx . 2
2
!=
+=
tx
xt. ( ) ( ) 01242
2
=+!+! tt , 032
=!t ,
3±=t . 1) 3221
!!=!= x,tx . 2) 3222
+!=!= x,tx .
The factorising method is not very useful in Example 3 because the roots are
fractional, and does not work at all in Example 4 because the roots are irrational. The
quadratic formula works in all cases but it is harder to apply than the substitution
formula a
bxt
2+= .
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Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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Evaluating Limits of the Form 0
0 .
When we have to evaluate a limit of the form ( )xflimax!
with 0!a , it is often
useful to do the substitution axt != ; then we get 0!t . It is often easier to study
the behaviour of a function near point 0 than near other points. Here are a few
examples.
Example 5.
( ) ( )=
!=
+=
!=="
#
$%&
'=
!
!(( t
tlnlim
tx
xt
x
xlnlim
tx
1
5
5
0
0
5
6
05
( )1
1
0
!=!
!!
" t
tlnlimt
.
(Using the fact that: ( ))1
1
0
=+
! z
zlnlimz
.
Example 6.
=!
"#
$%&
'+
=
+=
!=
="#
$%&
'=
! (( t
tcos
lim
tx
xt
x
xcoslim
tx 2
2
2
2
0
0
2 0
2
)
)
)
)) t
tsinlimt 20 !
!
"=
2
1
2
1
0
==! t
tsinlimt
. (Using the fact that: )10
=! z
zsinlimz
.
Example 7.
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( )( )
=+
+=
+=
!=="
#
$%&
'=
(( )
)
)
)
) 55
44
0
0
5
4
0 tsin
tsinlim
tx
xt
xsin
xsinlim
tx
=!" tsin
tsinlimt 5
4
0
5
4
5
5
4
4
5
4
00
!="!=## tsin
tlim
t
tsinlim
tt
. (Using the fact that: )10
=! z
zsinlimz
.
In the last example many students who apply the traditional technique give the
wrong answer 5
4 .
The substitution axt != always helps to find the limits of rational functions of
the indeterminate form 0
0 . With this substitution we can avoid factorising, which
involves trial and error process and can be sometimes quite difficult. Here are a few
examples.
Example 8.
=!=
+=="
#
$%&
'=
!+
!+
!( 1
1
0
0
12
23
2
2
1 tx
xt
xx
xxlimx
( ) ( )( ) ( )
=!!+!
!!+!
" 1112
2113
2
2
0 tt
ttlimt
3
5
32
53
32
53
02
2
0
=!
!=
!
!=
"" t
tlim
tt
ttlim
tt
.
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Example 9.
=+=
!=="
#
$%&
'=
!+!
!+!( 2
2
0
0
1021143
674
23
23
2 tx
xt
xxx
xxxlimx
( ) ( ) ( )( ) ( ) ( )
=++
++=
!+++!+
!+++!+=
"" ttt
tttlim
ttt
tttlim
tt
23
23
023
23
0 43
32
1022121423
627242
3143
32
2
2
0
=++
++=
! tt
ttlimt
.
Example 10.
=+=
!=="
#
$%&
'=
!!!
!!+
( 1
1
0
0
13
4352
234
24
1 tx
xt
xxx
xxxlimx
( ) ( ) ( )( ) ( ) ( )
=!+!+!+
!+!+++=
" 11113
4131512
234
24
0 ttt
tttlimt
7
15
714113
151782
714113
151782
23
23
0234
234
0
=+++
+++=
+++
+++=
!! ttt
tttlim
tttt
ttttlim
tt
.
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Obviously all these limits can be calculated by l’Hôpital’s rule. But the
substitution method gives students the chance to calculate these limits easily before
they have learned differentiation.
Integration
Substitution is widely used for integration. Here we will consider the integrals,
where substitution is not usually used but can be quite useful.
These integrals have the form ! ++ cbxax
dx
2 and dx
cbxax
BAx
! ++
+2
. So we do not
need partial fractions to integrate these particular types of rational functions. The idea
is the same as for the problem of completing the square: in each case the substitution
a
bxt
2+= is used.
The following three examples illustrate this.
Example 11.
=
+
=
+
=
=
!=
+=
=++ """
19
49
2
2
92
2
1
2
1
522 222
t
dt
t
dt
dtdx
tx
xt
xx
dx =
+!"
#$%
&'
13
29
2
2
t
dt
=+!!"
#$$%
&!"
#$%
&+=+!
"
#$%
&''=
((CxtanCttan
2
1
3
2
3
1
3
2
2
3
9
2 11C
xtan +!
"
#$%
& +'
3
12
3
1 1 .
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Example 12.
=
!
!=
+!
=
=
+=
!=
=! """
25
15
1
5
15
5
1
5
1
52 222
t
dt
t
dt
dtdx
tx
xt
xx
dx
=+!
+=+
+
!!=
""""
#
$
%%%%
&
'
+
!!
(!= ) Ct
tlnC
t
t
lndt
tt15
15
2
1
5
1
5
1
2
1
5
1
1
5
1
1
2
5
5
1
Cx
xln +
!=
25
5
2
1 .
Example 13.
=+
!=
=
!=
+=
=++
!"" dtt
t
dtdx
tx
xt
dxxx
x
13
741
1
463
34
22
( )=
+!
+ "" dt
t
dtt
t
13
7
13
6
6
4
22
( ) ( ) =+!+= !Cttantln 3
3
713
3
2 12
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( ) ( )( ) Cxtanxxln ++!++= !13
3
7463
3
2 12 .
The authors noticed that many students had difficulties with applying standard
techniques to the problems described above; but they can easily master and apply the
substitution technique. As the reader can see, the advantage of this method is that it is
algorithmic. So the students only have to remember the formula a
bxt
2+= and do
routine operations like expanding brackets and simplifying polynomials. The method
does not involve trial and error or guessing, unlike the ordinary techniques of
factorising and completing the square.
We invite readers to teach students to apply the substitution method to the classes
of mathematical problems described above, and see how this will enhance the
students’ learning.
Using Probability Trees: Making Procedures Meaningful
In basic courses in probability theory many secondary and tertiary teachers use
probability trees as a tool for teaching conditional probabilities, (e.g. Lipschutz &
Lipson, 2000, pp. 87-89). We put some thought and investigation into this practice. It
is easy for students to apply probability trees to simple problems. But often this
technique is not justified, so the students apply it without a clear understanding of its
meaning and limitations. Let us consider the following common example.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Example 14.
A test for a certain disease shows positive with probability 0.8 on an ill person and
with probability 0.05 on a healthy person. 10% of the population are affected by this
disease. If the test on a person shows positive what is the probability that this person
is ill?
In our lessons we explain that every person in this example represents an
outcome, or an elementary event. The sample space Ω (the set of all outcomes) can be
represented as a union of two events H 1 and H 2 (hypotheses):
H 1 = {a person is ill} and H 2 = {a person is healthy}.
Consider event A = {the test shows positive}. Construct a probability tree, where
the first level of branches represents hypotheses and the second level shows
conditional probabilities:
H1
H2
A∩H1
A∩H2
Ac∩H2
Ac∩H1 0.1
0.9
0.8
0.05
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
The students usually quickly learn to multiply probabilities along each branch and
then add some of the products:
P(A) = 0.1×0.8 + 0.9×0.05 = 0.125; ( )( )( )
6401250
80101
1.
.
..
AP
HAPA|HP =
!=
"= .
Common mistakes here are choosing wrong hypotheses and applying probability
trees to the problems, where this method is not appropriate. For example, some
students came up with such a probability tree for Example 14:
To avoid these mistakes, we justify the method of probability trees in the
following three steps.
1. The hypotheses H1, H2 ,…, Hn are disjoint events, which union contains all
outcomes.
2. Multiplication theorem: if P(H k) > 0, then P(A! H k) = P(H k) ⋅ P(A | H k),
k = 1, 2,…, n. It justifies multiplying probabilities along each branch.
positive result result
ill
ill
healthy
healthy
negative result result
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
3. Partition theorem: ( ) ( )!=
"=n
k
kHAPAP
1
. It justifies adding the products for
the branches corresponding to event A.
With these underlying principles the students will choose the correct hypotheses
and use the correct conditional probabilities in their probability tree. We call this
‘meaningful procedural knowledge’ because the procedure – probability trees – is
applied with understanding of the underlying mathematical concepts and theorems. It
is also useful to encourage the students to think about proofs of the multiplication and
partition theorems. One can be explained by the teacher and the other offered as an
exercise.
In past the authors taught probability trees as an algorithm. Now we teach them
together with the three justifying steps. We notice that our students more often get
correct solutions for this kind of problems than they used to in the past.
Conclusion
The described procedures were used for several years at the Auckland University
of Technology and the Moscow Technological University. Verbal responses from
students and their assessment results show effectiveness of these procedures. The
students who applied the suggested procedures are more successful than the ones
using traditional procedures, in technical manipulations as well as in learning the
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
relevant concepts. Applied to certain types of mathematical problems, these
procedures involve fewer technical details, are logical, relatively universal and
eliminate most of memorising. We would like to hear from the readers who used the
described procedures and we will appreciate their feedback on how this affected their
students’ learning.
References
Byrnes, J.P. & Wasik, B.A. (1991). Role of conceptual knowledge in mathematical procedural
learning. Developmental Psychology, 27(5), 777-786.
Haapasalo, L. & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation.
Journal für Mathematikdidaktik, 21(2), 139-157.
Lipschutz, S. & Lipson, M.L. (2000). Probability. (2nd ed.). New York: McGraw-Hill.
Piaget, J. (1985). The equilibrium of cognitive structures. Cambridge, MA: Harvard University Press.
Rittle-Johnson, B., Siegler, R.S., & Alibali, M.W. (2001). Developing conceptual understanding and
procedural skill in mathematics: an iterative process. Journal of Educational Psychology, 93(2),
346-362.
Tossavainen, T. (2006). Conceptualising procedural knowledge of mathematics – or the other way
around. Retrieved February 26, 2007, from website:
http://www.distans.hkr.se/rikskonf/Grupp%206/
Tossavainen.pdf
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Counter-Examples and Paradoxes in Teaching Mathematical Statistics: A Case Study
Farida Kachapova Auckland University of Technology
Murray Black Auckland University of Technology
Sergiy Klymchuk Auckland University of Technology
Ilias Kachapov < [email protected]>
Introduction and Framework Counter-examples are a powerful and effective tool for scientists, researchers and
practitioners. They are good indicators showing that a suggested hypothesis or a chosen
direction of research is wrong. Before trying to prove a conjecture or a hypothesis it is
often worth to look for a possible counter-example. It can save lots of time and effort.
Creating examples and counter-examples is neither algorithmic nor procedural and requires
advanced thinking which is not often taught at school (Selden & Selden, 1998; Tall, 1991;
Tall, et al. 2001). Many students are used to concentrate on techniques, manipulations,
familiar procedures and do not pay much attention to concepts, conditions of theorems and
rules, reasoning and justifications. As Seldens argue, coming up with examples requires
different cognitive skills from carrying out algorithms – one needs to look at mathematical
objects in terms of their properties. To be asked for an example can be disconcerting.
Students have no prelearned algorithms to show the ‘correct way’ (Selden & Selden,
1998).
2
There are several publications on using counter-examples in teaching/learning of
mathematics, in particular calculus (Gelbaum and Olmstead, 1964; Peled & Zaslavski,
1997; Zaslavski & Ron, 1998; Bermudez, 2004; Gruenwald & Klymchuk, 2003;
Klymchuk, 2004 & 2005). There are three well-known books on counter-examples in
statistics at an advanced level (Stoyanov, 1997; Romano, 1986; Wise & Hall, 1993). But
we could not find any publication on using counter-examples in teaching/learning of first-
year probability and statistics. So we decided to apply counter-examples along with
paradoxes as a pedagogical strategy in our first-year probability course.
The main objective of the study was to check our assumptions on how effective the
usage of counter-examples is for deeper conceptual understanding, eliminating students’
misconceptions and developing creative learning environment in teaching/learning of first-
year probability course.
In this study, practice was selected as the basis for the research framework and, it was
decided ‘to follow conventional wisdom as understood by the people who are stakeholders
in the practice’ (Zevenbergen & Begg, 1999). The theoretical framework was based on
Piaget’s notion of cognitive conflict (Piaget, 1985). Some studies in mathematics education
at secondary level (Swan, 1993; Irwin, 1997) found conflict to be more effective than
direct instruction. ‘Provoking cognitive conflict to help students understand areas of
mathematics is often recommended’ (Irwin, 1997). Swedosh and Clark (1997) used
conflict in their intervention method to help undergraduate students to eliminate their
misconceptions. ‘The method essentially involved showing examples for which the
misconception could be seen to lead to a ridiculous conclusion, and, having established a
conflict in the minds of the students, the correct concept was taught’ (Swedosh and Clark,
1997). Another study by (Horiguchi & Hirashima, 2001) used a similar approach in
3
creating discovery learning environment in their mechanics classes. They showed counter-
examples to their students and considered them as a chance to learn from mistakes. They
claim that for counter-examples to be effective they ‘must be recognized to be meaningful
and acceptable and must be suggestive, to lead a learner to correct understanding’
(Horiguchi & Hirashima, 2001). Mason and Watson (2001) used a method of so-called
boundary examples, which suggested creating by students examples to correct statements,
theorems, techniques, and questions that satisfied their conditions. ‘When students come to
apply a theorem or technique, they often fail to check that the conditions for applying it are
satisfied. We conjecture that this is usually because they simply do not think of it, and this
is because they are not fluent in using appropriate terms, notations, properties, or do not
recognise the role of such conditions’ (Mason and Watson, 2001). In our study, often not
the lecturer but the students were asked to create and show counter-examples to incorrect
statements, so the students themselves established a conflict in their minds. The students
were actively involved in creative discovery learning that stimulated development of their
advanced statistical thinking.
The Study
The students from a first-year course ‘Probability Theory and Applications’ were given
mathematical statements and asked to create counter-examples to disprove these
statements. They had enough knowledge to do that. However, for most of the students this
kind of activity was absolutely new, very challenging and even created psychological
discomfort and conflict for a number of reasons. In the beginning some of the students
could not see the difference between “proving” that the statement is correct by an example
and disproving it by an example. It agrees with the following observation from Selden &
Selden (1998): ‘Students quite often fail to see a single counter-example as disproving a
4
conjecture. This can happen when a counter-example is perceived as “the only” one that
exists, rather than being seen as generic’. To illustrate the idea of disproving by a counter-
example it might be helpful to use non-mathematical examples first. For instance, it might
be discussed with students: What does it take to disprove the statements like ‘all
Scandinavians are blond’ or ‘there are no numbers such that when they are spelled they
contain the letter "a". Apart from the activity on using counter-examples the students were
also given some paradoxes and were asked to explain them.
In our study we did not use ‘pathological’ cases. All exercises given to the students
were within their knowledge and often were related to their common misconceptions.
Below are examples of the incorrect statements to be disproved by counter-examples
and the paradoxes to be explained that were discussed with the students.
Counter-Examples
Use counter-examples to disprove the following incorrect statements.
1) Pairwise independence of events implies their independence.
2) a) If events A and B are independent, then they are conditionally independent.
b) If events A and B are conditionally independent, then they are independent.
3) Uncorrelated random variables are independent.
a) Consider the case of discrete random variables.
b) Consider the case of continuous random variables.
4) Pairwise independence of random variables implies their mutual independence.
Paradoxes
We consider the following problems as paradoxes because the correct answer to each
of them contradicts intuition.
5
Galton’s paradox. (Grimmett & Stirzaker, 2004, p. 14).
You flip three fair coins. At least two results are alike (the same). There is 50-50
chance that the third one is a head or a tail. Therefore the probability that all three results
are alike equals 0.5. Do you agree?
Simpson’s paradox. (Grimmett & Stirzaker, 2004, p. 19).
A doctor has performed clinical trials to determine the relative efficacies of two drugs,
with the following results:
Table 1
Results of Drug Treatment
Women Men
Drug 1 Drug 2 Drug 1 Drug 2
Success 200 10 19 1000
Failure 1800 190 1 1000
Total 2000 200 20 2000
The success rate of Drug 1 is 219/2020 ≈ 0.108 and of Drug 2 is 1010/2200 ≈ 0.459, so
the overall success rate is greater for Drug 2.
Among women the success rates are:
200/2000 = 0.1 for Drug 1 and 10/200 = 0.05 for Drug 2.
Among men the success rates are:
19/20 = 0.95 for Drug 1 and 1000/2000 = 0.5 for Drug 2.
So the success rates are greater for Drug 1 when the proportions are calculated for men
and women separately.
Which drug is better?
6
Monty Hall paradox. (Grimmett & Stirzaker, 2004, p. 12).
Suppose you are in a game show, and you are given the choice of three doors of which
one contains a prize. The other two contain gag gifts like a goat or a donkey. You pick a
door, say, No. 1. The host (who knows what behind the doors) opens door 3, which has a
donkey. He then says to you, “Do you want to pick door 2?”. Is it to your advantage to
switch your choice? That is, will your probability of winning increase if you switch to door
2?
The intuition of many students tells them that switching the door does not change the
probability of winning. Actually this probability increases from 3
1 to 3
2 .
Prisoners’ paradox. (Grimmett & Stirzaker, 2004, p. 11). There are three prisoners, A,
B, and C. The warden tells them that two of them will be released and one will be
executed. But he is not permitted to reveal to any prisoner the fate of that prisoner.
A asks the warden to tell him the name of one of his cohort who will be released. The
warden obliges and says, “B will be released.” Assume that the warden tells the truth.
a) What are A’s and C’s respective probabilities of dying now?
b) If A could switch fates with C now, should he?
This paradox is similar to Monty Hall paradox. Contrary to what intuition tells us, the
conditional probabilities of dying are different for A and C (3
1 and 3
2 respectively).
St Petersburg’s paradox. (Grimmett & Stirzaker, 2004, p. 55).
In a game of chance, a player pays a fixed fee to enter, and then a fair coin is tossed
repeatedly until a head appears ending the game. If the first head appears after n tosses,
then the player gets $2 n.
a) What is the expected win of a player?
7
b) What is the “fair” entrance fee?
The answer for both questions is ∞. It seems that any high fee is worth paying to enter
this game, which contradicts common sense.
The Questionnaire
After several weeks of using such exercises in class the students were given the
following questionnaire to investigate their attitudes towards the usage of counter-
examples and paradoxes in learning/teaching.
Question 1.
Do you find counter-examples and paradoxes useful for understanding this course?
a) Yes Please give the reasons:
b) No Please give the reasons:
Question 2.
Do you feel confident using counter-examples and paradoxes?
a) Yes Please give the reasons:
b) No Please give the reasons:
Question 3.
Do you find this strategy effective?
a) Yes Please give the reasons:
b) No Please give the reasons:
Question 4.
Would you like this kind of activity to be a part of assessment?
a) Yes Please give the reasons:
b) No Please give the reasons:
8
Findings from the Questionnaire
The statistics from the questionnaire are presented in the following table.
Table 2
Summary of Findings from the Questionnaire
Number of Students
Question 1
Useful?
Question 2
Confident?
Question 3
Effective?
Question 4
Assessment?
Yes No Yes No Yes No Yes No
11
100%
10 1
91% 9%
7 4
64% 36%
11 0
100% 0%
4 7
36% 64%
The majority of the students found counter-examples and paradoxes useful for
understanding the course. The typical comments from those students were as follows:
- they are both entertaining and informative;
- they are helpful because we can look back at them when we do assignments;
- we go though reasoning of counter-examples and paradoxes that helps
understanding the course.
About 2/3 of the students (64%) felt confident using counter-examples and 36% did
not. The ones who answered ‘no’ to the question about confidence provided the following
comments:
- counter-examples are difficult;
- sometimes they are confusing;
- I need more practice with them.
9
All surveyed students found the method of counter-examples effective and provided
the following typical comments:
- it improves my understanding of probability and random variables;
- it builds my logical skills;
- it strengthens my thinking ability.
About 2/3 of the students (64%) did not like the idea of counter-examples being a part
of assessment. In their comments they wrote that they could cope only with simple
counter-examples in assessment or with counter-example problems only in home
assignments but not in class tests. To some extent this last result contradicts the responses
to questions 2 and 3, where many students indicated that they felt confident using counter-
examples and that they considered the method effective.
Conclusions and Recommendations
The statistical results of this study show positive attitudes of the students towards using
paradoxes and counter-examples as a pedagogical strategy in a first-year course in
probability and random variables. All students surveyed stated that the pedagogical
strategy was effective. The majority of the students (91%) stated that the strategy was
useful for understanding the course. Many students commented that this method helped
them improve their logical skills and critical thinking and made the learning environment
more creative and entertaining. Though most of the surveyed students did not encounter
counter-examples in past and often found them challenging, they also found them useful
and effective and wanted to practise more with such problems.
As with any other case study the question is: to which extent can the results of the
study be generalised? The question remains regardless of the number of students surveyed
in the study. It doesn’t matter whether there are 11 students or 20 students or 50 students in
10
a class – there is only one lecturer and one learning environment and the number of
students surveyed is drop in the ocean compared to the number of students in the world
studying the first-year university probability course. In addition in this particular class
many students were in year 2 and 3 of their studies, therefore they had a better
mathematical background than typical year 1 students. It makes the study a bit biased. So
the results of the study can be treated as an invitation for colleagues to try the suggested
strategy with their own students and see how it works with them. It definitely worked for
our students.
As the first step in introducing counter-examples the authors recommend that a lecturer
provides a paradox or a counter-example and asks the students to explain or justify it. Next
the students can be asked to create their own counter-examples for a given incorrect
statement. And finally, the lecturer can ask the students to decide whether a given
mathematical statement is correct, so the students have to come up with a proof to show
that the statement is true, or with a counter-example to show that the statement is wrong. In
a one-semester course we tried to lead the students through these three steps with a certain
amount of success. But we observed that many students needed a lot more practice to
succeed and feel more confident in this area.
Further Study
We would like to extend the study to measure the effectiveness of this pedagogical
strategy on the students’ exam performance on the questions that require good
understanding of concepts, not just manipulations and techniques. We plan to compare the
performance of two groups of students with similar backgrounds. In one group we will
extensively use counter-examples and paradoxes, with the other group being the control
11
group. Then we will use statistical methods to establish whether the difference is
significant or not.
References
Bermudez, C.G. (2004). Counterexamples in calculus teaching. Paper presented at the 10th International
Congress on Mathematics Education (ICME-10). Copenhagen, Denmark.
Gelbaum, B.R. & Olmstead, J.M.H. (1964). Counterexamples in Analysis. San Francisco: Holden-Day.
Grimmett, G. & Stirzaker, D. (2004). Probability and Random Processes. (3rd ed.). New York: Oxford
University Press.
Gruenwald, N. & Klymchuk, S. (2003). Using counter-examples in teaching calculus. The New Zealand
Mathematics Magazine. 40(2), 33-41.
Horiguchi, T. & Hirashima, T. (2001). The role of counterexamples in discovery learning environment:
Awareness of the chance for learning. Proceedings of the 1st International Workshop on Chance
Discovery (pp. 5-10). Matsue, Japan.
Irwin, K. (1997). What conflicts help students learn about decimals? Proceedings of the International
Conference of Mathematics Education Research Group of Australasia (pp. 247-254). Rotorua, New
Zealand.
Klymchuk, S. (2005). Counter-examples in teaching/learning of calculus: Students’ performance. The New
Zealand Mathematics Magazine. 42(1), 31-38.
Klymchuk, S. (2004). Counter-examples in calculus. New Zealand: Maths Press.
Mason, J. & Watson, A. (2001). Getting students to create boundary examples. MSOR Connections, 1(1),
9-11.
Peled, I. & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that (also)
explain. Focus on Learning Problems in Mathematics. 19(3), 49-61.
Piaget, J. (1985). The Equilibrium of Cognitive Structures. Cambridge, MA: Harvard University Press.
Romano, J.P. (1993). Counterexamples in probability and statistics. Chapman & Hall/CRC.
Selden, A. & Selden, J. (1998). The role of examples in learning mathematics. The Mathematical Association
of America Online. Retrieved February 2, 2007, from website: www.maa.org/t_and l/sampler/rs_5.html
Stoyanov, J.M. (1997). Counterexamples in probability. (2nd ed.) England: Wiley.
12
Swan, M. (1993). Becoming numerate: Developing conceptual structures. In Willis (Ed), Being numerate:
What counts? (pp. 44-71). Hawthorne VIC: Australian Council for Educational Research.
Swedosh, P. & Clark, J. (1997). Mathematical misconceptions – can we eliminate them? Proceedings of the
International Conference of Mathematics Education Research Group of Australasia (pp. 492-499).
Rotorua, New Zealand.
Tall, D. (1991). The psychology of advanced mathematical thinking. In Tall (Ed), Advanced Mathematical
Thinking (pp. 3-21). Kluwer: Dordrecht.
Tall, D., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., et al. (2001). Symbols and the
bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and
Technology Education, 1, 81–104.
Wise, J.L. & Hall, E.B. (1986). Counterexamples in probability and real analysis. New York: Oxford
University Press.
Zaslavsky, O. & Ron, G. (1998). Students’ understanding of the role of counter-examples. Proceedings of the
22nd Conference of the International Group for the Psychology of Mathematics Education. 1, 225-232.
Stellenbosch, South Africa.
Zevenbergen, R. & Begg, A. (1999). Theoretical framework in educational research. In Coll, R.K., et al
(Eds) SAME Papers (pp. 170-185). New Zealand.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Planning a route for a performing trip of a music team: A group project in 4th grade mathematics
Grażyna Pawłowska,
Szkoła Podstawowa Nr1 im. Gustawa Morcinka w Warszawie,Poland e-mail: [email protected]
Overview
I have used a project-based approach in two 4th grade classes at a grammar school. The project entailed planning the most efficient route for a music team. The group work method used was presented in NIM and Forum of Education (Pawlowska, 2001). This was the first experience of this kind for for my students, although they had already worked with the collaborative group method in the context of an integrated approach.
I wanted all pupils to understand precisely what is expected of them in similar situations, to be clear about the principles of the cooperative work and to find the task doable as a whole. I wanted the children – also those working slower or requiring help of a teacher or classmates – to experience satisfaction and success. In order not to discourage able pupils with too simple problems, each set of them contained separate, more difficult problems designed for the members of the team and its leader. In order to accommodate children, who after initial teaching still have problems with reading, the instruction was written in larger font and tasks were discussed in front of the blackboard. The aim of the class was to use counting by memory in the practical context of calculating the length of a car route on the basis of a map of Poland. The skill of using the map is very useful in life and I wanted children to also learn map skills. Observing them, I think that the problem posed in front of them was interesting, convincing and speaking to childrens‘ imagination.
As usual with group work, pupils had to take on certain roles and perform actvities connected with them. This simple approach engages pupils imagination and reinforces their emotional attitude toward their work. During the time of the proposed game, every group became a musical team preparing a performance tour. The route commenced in Frombork through Ostróda to Mława to Płock. There were three concerts on each day in each of the cities. In order to make the problem more real I suggested that each of the music teams played well but was not famous enought to have a manager. Consequently, each member of the team had to play and also plan the route of the travel. The work had to be divided: each member of the team was supposed to plan the route on a different interval of the trip and calculate its length.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
The computational part of the work consisted of adding lengths of individual pieces read off the map. After finishing its calculations, each group had to glue coordinated pieces of the map onto a poster board indicating each of the travel intervals. The groups also included tables with the intervals and their calculated length, as well as the list of villages and towns passed along the road. The presentation was designed to enhance and diversify the work of the teams. Goals and Objectives The general aims of a class period were to: --Develop the competency of applying mental calculus to daily life problem solving, and in particular to problems encountered while using car road maps. -- Develop the competency to create a simple outline showing the problem situation. -- Develop the capacity for collaborative work. The proposed design of the class will also impact: --Competency of reading with understanding. --Competency of using tables as means for organization and presentation of information. The proposed problem is also useful for the realization of wider, interdisciplinary educational aims such as: --Familiarization with geographic regions contained in the road map. --Familiarization with the history and development of the passing communities. --Familiarlization with the scale of distance in Poland. --Introduction to the art of writing ads and announcements. --Organization of the musical program of the team. Among the detailed outcomes, students will be able to: -Plan the route of the trip from given initial site to the final site using the car map. -Make a sketch of the route. -Read the distances between different cities. -Complete the form describing the route in accordance with the template; -Do simple mental calculations applied to practical problem. -notice the relationships between different details of the map. -Synthisize details and make a useful chart or table.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Materials The design involves the following didactical materials in addition to a large map of Poland:
a) three different fragment maps,limited by the horizontal lines passing through the passing and final destinations (Fig.1)
b) instructions for the leader of the team c) instructions for the team’s members d) three forms for each of the members of the
team e) one form to collect information for the
leader of the team e) a handout with information about the cities
passed along the route f) large piece of the poster board to integrate
all the elements of the work.
Method
The teacher informs the class that each of the teams is responsible for plotting the performance route for a music team, scheduled to play three concerts in three different cities on one day. In the morning the teams leave Frombork where they stayed overnight and have to go to Ostróda, where they have the first concert at 10:00am in the gymnasium. The trip to Mława follows, where they have to play at 14:00 (2 p.m.) in the Cultural Center. Finally, for the evening they have engagement for 19:00 (7 p.m.) in the discoteque in Płock.
While presenting the outline of the game/activity, the teacher shows the essential cities on the map of Poland and briefly describes their history, development, significance for the region. Next, she shows the set of materials to be received by each of the groups. She discusses the distribution of tasks for the team. The teacher reads the instructions aloud and coordinates the forms with the map. Finally, she explains the sequential activities using the demonstration map.
Each member of the team is responsible for one interval of the trip and must design the route and compute its length. The activities are:
Frombork , miasto w województwie warmi_sko -mazurskim, nad Zalewem Wi_lanym, na skraju Równiny Warmi_skiej. 2,7 tys. mieszka_ców (2000). Port rybacki, przysta_ _eglugi pasa_erskiej, regularne po__czenia z Krynic_ Morsk_. Spe_nia funkcje miejscowo_ci wypocz ynkowej. Dobrze rozwini_te zaplecze noclegowe.
Historia Prawa miejskie nadano w 1310 osadzie, która rozwin__a si_ wokó_ warowni biskupów warmi_skich. Od 1466 w granicach Polski. Miejsce pracy i _mierci M. Kopernika, który mieszka_ we Fromborku w lat ach 1512 -1516 i 1522 -1543. W XVI i XVII w. rozwój handlu, powstanie portu i huty szk_a. W okresie rozbiorów pod panowaniem pruskim. W 1945, znacznie zniszczony, powróci_ do Polski.
Zamek we Fromborku, w_a_ciwie zespó_ katedralny wraz z umocnieniami obronnymi, za_o_ony na miejscu staropruskiego grodu. Umiejscowiony na wysokim wzgórzu. Ok. 1278 kapitu_a warmi_ska przenios_a tu swoj_ siedzib_ ze zniszczonego przez Prusów Braniewa. Wówczas wybudowano tu pierwszy ko_ció_, którego wygl_d nie jest znany, przypuszczalnie by_ on otoczony umocnieniami z drewna i ziemi. Istniej_cy dzisiaj ko_ció_ katedralny zbudowano w latach 1329 -1388. Wtedy te_ prawdopodobnie zacz_to wznoszenie murowanych umo cnie_
wzgórza. W latach 1510 -1543 z czteroletni_ przerw_ pracowa_ tu i mieszka_ Miko_aj Kopernik, b_d_cy kanonikiem warmi_skim. W 1626 wojska szwedzkie pod wodz_ Gustawa Adolfa zdoby_y i ograbi_y miasto i katedr_ ze wszystkich skarbów, biblioteka katedraln a i zbiory Kopernika zosta_y wywiezione do Szwecji.
Rys. 1. Collection of team’s materials
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RAZEM:
Planowana godzina wyjazdu
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(godzina:minuty)
Czas przejazdu trasy przy
pr_dko_ci 60 km na godzin_
(godziny:minuty)
Planowana godzina
przyjazdu do Ostródy –
oblicz ! (godzina:minuty)
Potrzebna ilo__ paliwa przy
zu_yciu 10 litrów na 100 km
(litry)
Warto__ potrzebnego
paliwa przy cenie 3 z_ za litr
(z_)
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pr_dko_ci 60 km na godzin_
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Planowana godzina
przyjazdu do P_ocka –
oblicz ! (godzina:minuty)
Potrzebna ilo__ paliwa przy
zu_yciu 10 litrów na 100 km
(litry)
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przejazd Frombork -Ostróda 7:00 I.
koncert w Ostródzie 10:00 10:30
przejazd: Ostróda -M_awa 11:00 II.
koncert w M_awie 14:00 14:30
przejazd: M_awa -P_ock 16:00 III>
koncert w P_ocku 19:00 20:00
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Matematyka GRA -IV, ©Paw_owscy`2002 KAPELA: Praca w grupach (dodawanie)
Instrukcja dla zespo_u i jego lidera.
Jeste_cie zespo_em muzycznym, wybieraj_ cym si_ na tras_ koncertow_. W sobot_ macie
zagra_ a_ trzy koncerty, wi_c musicie dok_adnie to zaplanowa_. Z samego rana musicie
pojecha_ swoim busem z Fromborka do Ostródy, gdzie przed po_udniem , o godzinie 10:00
gracie w miejscowym gimnazjum. Potem czeka was podró_ do M_awy , gdzie o godzinie
14:00 gracie w Domu Kultury. Na wieczór macie umówiony koncert w dyskotece w
P_ocku o godzinie 19:00 .
Wasz zespó_ gra bardzo _adnie, ale jeszcze nie zdobyli_cie s_awy, wi_c nie macie
mena_era. Ka_dy z muzyków musi nie tylko gra_, ale i planowa_ tras_ – tak jak pilot
rajdowy w rajdzie samochodowym . Lider zespo_u nadzoruje prac_ wszystkich i pomaga
tym, którzy sobie nie radz_.
Zadania dla lidera (zadania na szarym tle nie s_ obowi_zkowe) :
1. Sprawd_ , czy koledzy prawid_ow o policzyli d_ugo_ci tras.
2. Wpis z d_ugo_ci tras do tabelki zbiorczej (pola I. II. i III.)
3. Policz ca_kowit_ d_ugo__ sobotniej trasy (pole |IV. RAZEM: ).
4. Pomó_ kolegom zrozumie_ i wykona_ zadania dodatkowe.
5. Wpis z wyniki zada_ dodatkowych do tabelki zbiorczej.
6. Oblicz potrzebn_ ilo__ paliwa i jego koszt.
7. Rozplan uj rozmieszczenie mapek i tabelek na kartonie, i pomó _ kolegom je naklei_.
8. Nakle jcie razem karteczki z plakatami informuj_cymi o koncertach.
Matematyka GRA -IV, ©Paw_owscy`2002 KAPELA: Praca w grupach (dodawanie)
Instrukcja dla pilota trasy (zadania na szarym tle nie s_ obowi_zkowe):
1. Odszukaj na mapce swoje miast o pocz_tkowe i ko_cowe , i zaznacz tras_ przejazdu.
2. Naszkicuj tras_ w tabelce .
3. Wpisz do tabelki nazwy co najmniej 4 kolejnych miast mijanych na Twoim
odcinku trasiy (pola nr 1a., 2a., 3a., ....) . Cz___ pól mo_e pozosta_ pusta.
4. W polu obok ka_dego miasta, wpisz d_ugo__ drogi od miasta poprzedniego (pola nr
1b., 2b., 3b., ...) . Nie zapom nij o mie_cie ko_cowym, które ju_ jest wpisane!
5. W polu „RAZEM” tabelki podsumuj d_ugo__ ca_ej trasy.
6. Je_li potrafisz i masz czas, to wype_nij kolejne pola „Tabel i zada_ dodatkowych”.
Podpowied_: pr_dko__ 60 km na godzin_ to to samo co 1 km na minut_
7. Wytnij i naklej na karton swoj_ mapk_, dopasowuj_c j_ do mapek kolegów.
8. Naklej na karton sw oj_ tabel_ .
9. Wykonaj projekt plakatu informuj_cego o koncercie waszego zespo_u w mie_cie, do
którego prowadzi Twoja trasa. Podaj miejsce, dat_ i godzin_ .
Figure 1. Fragment of a map obtained by one of the groups/
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
-find the initial and final town on the map. - draw the route on the map. - make a draft of the route on the appropriate form. -wirte in at least 4 names of passing towns. -write in the distances between them. -calculate the sum of the distances for the total trip.
Team leaders have their own tasks. They are responsible for the work of the members of
the teams: they should check whether the members do their work correctly; offer suggestions; and assist where needed. They organize common activities including: naming the team; planning the composition of maps and forms on the final poster presentation; and creating the announcements. The leader also supervises filling out the final form, plans the course for the day and sums up the length of the whole route. The forms for the particular intervals of the route contain additional tasks with instructions and needed information. Understanding these instructions is part of the overall task, hence it’s not discussed by the teacher.
When all the members of the team end their activities, the team aprpoaches the construction of the final product on the poster board. Each team member cuts out their map and glues them to the board making sure all individual maps fit together. The team leader supervises the gluing and adds the handout about one of the concerts. The team continues to work on the announcements of their concerts. (The teacher demonstrated different possible arrangements of the final product using the magnetic pieces.) The lesson and project can be accomplished in about 45 minutes. After checking the group submissions it is important that the teacher display each groups work (Figure 2.).
Figure 2. Example of the work design
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Outcomes The described design was realized in two 4th grades with19 and 18 pupils respectively. Pupils were informed about upcoming activity and about the car maps and their utilization one class in advance; the appointed leaders were good in mathematics. The class experienced real joy and satisfaction in both classes. I was available to help children but only after each question was directed back to the team leaders. I was reminded children about the necessity of mutual help, about the final product depending on the collaboration of every one in the team. I looked for students who needed help the most and was helping them in more difficult situations. The real difficulty was reading off real distances from the map. I had also encouraged children to organize their work better.
Figure 3. Working Group. In the background there is a blackboard with attachad materials..
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Assessment and evaluation of knowledge I used the following rubric to assess pupils‘ work - the member of the team could get 5 points for basic activities, which - gave 2 pts for starting all activities - gave 2 pts for correct reading off the distances from the map - gave 1 point for correct sum of all the distances. Captain could also get 5 points for basic activities: -2 points for starting -1 point for correct writing in the data into the table; - 1 point for finding sum -1 point for good leadership of the team (averaging 4 points by team members). Both captain and the team members could get max 3 points for additional activities.
Figure 4. One of the work results.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
The distribution of points for the tasks is presented Figure 5. The average was 4.05. More than 75% of students received 4 points which means that basic tasks were done correctly or at most with small corrections. The additionaL task was taken by 4 pupils in the class Ivd, including two team captains.
Histogram of teams‘ average results (Figure 6) shows that all teams, except one, averaged above 3.5 pts, and four out of nine teams averaged more than 4.5 pts. These data confirm the subjective
assessment that the lesson was able to motivate collaboration in the majority of the teams. Corrected and graded products were discussed a couple of days later. The classroom where the posters were produced was changed into a gallery of finished work. The exhibition evoked deep interest amongst the children. Every child received the grade for individual work in a team in agreement with the accepted criteria. Besides two best teams in each class received the special prize to be divided by the team members. Assessment of effectiveness
The value of the project based learning is in its effectiveness. At the same time it is difficult to assess objectively because of the individual characteristics of the pupils. One measure can be the sum of all activities undertaken by pupils. All activities are counted independently of the correctness of the result. There were 6 activities in the described lessons which can be objectively registered on the basis of the documentation. Those activities together with the percentage of activities undertaken is presented in Table 1. Activity 3 was
0.0
1.0
2.0
3.0
4.0
5.0
6.0
b1.1
b1.3
b2.1
b2.3
b3.1
b3.3
b4.1
b4.3
b5.1
b5.k
d1.2
d2.1
d2.3
d3.1
d3.3
d4.1
d4.3
Figure 5. Distributionof points
0.0
1.0
2.0
3.0
4.0
5.0
6.0
b1 b2 b3 b4 b5 d1 d2 d3 d4
Figure 6. Team Averages
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
not required accounting for the smaller percentage. The presented data suggest that all or almost all students intensively participated in the classroom work. This confirms my subjective assessment. Summary and discussion
The sequence of activities in grade 4 as well as the effectiveness demonstrates that the group work method worked in this case. The lesson achieved it’s outlined goals. The coefficient of difficulty was 0.8 which means that the problems were well chosen for the class. Even children from the class who usually work slower and are a less able team presented the final effect of their work.
An additional problem was done by few children so the coefficient of difficulty is not reliable as there was not much time for this problem. Pupils who had undertaken the problem demonstrated high levels of competency and work organization. More difficult problems are needed so that more able students experience have satisfaction with the work.
The class design can be easily adapted for use in mathematics, biology, language, history and music. Use of the music team performance task including characteristics of the geography, history, legend and curiosities of passing cities) worked in the case of mathematics. At the same time the children had more time for the actual task of mathematical computation of the best route. Expanding the range of subjects to be included would necessitate using additional class periods. Although for this project I didn’t plan collaboration with other teachers I think it would be interesting and useful to realize multidisciplinary project. Talking with my colleagues, I think such a collaboration is possible. References
1. G. Pawłowska. Zamawiamy sadzonki na klomby. Praca w grupach, klasa V. Nauczyciele i
Matematyka 35 (2000) 10. 2. G. Pawłowska. Płacimy podatki. Praca w grupach na temat procentów, klasa V. Nauczyciele
i Matematyka 36 (2000) 21. 3. G. Pawłowska. Witraż: Praca w grupach na przykładzie lekcji w geometrii w klasie piątej.
Nauczyciele i Matematyka 40 (2001) 21. 4. G. Pawłowska. Planujemy zakupy na wycieczkę: Praca w grupach, klasa IV. Część I. Forum
Edukacji 2/3 (2001) 70.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
5. G. Pawłowska. Planujemy zakupy na wycieczkę: Praca w grupach, klasa IV. Część II. Forum Edukacji 4 (2001) 72.
Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Mathematical Competencies in the nascent state“on the basis of the Educational Project „In the Middle of the Way…“ Elżbieta Jaworska, Bożena Makulska-Dąbkowska, Elżbieta Ostawiczuk, Andzrej Wawrzyniak, Andrzej Werner1 Mathematics Teacher-Coaches, Department of Education, Office of the City Warszawa, Poland. A person taken by the new idea is a weirdo
until the idea is succesful. Mark Twain Education is the main determinant of the cultural and economical development of a society. Hence the strategic role of education should be high on the list of government’s priorities. Already during the seventies of the twentieth century, countries of the European Economical Community (OPEC) and now EU started to attach increasing importance to the formulation of the correct educational policy2. The educational priorities of EU contain increasing the quality of education at every level, development of vocational education, mutual recognition professional qualification by the member states, and the development of higher education. European pedagogical categories contain multidisciplinary approaches, equalization of educational opportunities, raising the quality through the reform of curriculum, stimulation of the creative, innovative approaches by the teachers. The idea of the vision is to raise a European as a citizen of the World.
1 Originally the authors of the project „In the Midde of the Way…“ was created by the teacher-coaches of mathematics – Andrzej Wawrzyniak and Andrzej Werner and full of enthusiasm teachers of mathematics, Elżbieta Jaworska and Elżbieta Ostaficzuk. In the Fall 2004, the team was augmented by Bożenę Makulską-Dąbrowską. Since then all five teachers constituted the group of teacher-coaches. At present Elżbieta Ostaficzuk works as aa consultant to the Małopolska Centrum of Teacher Development. The project „In the Middle of the Way …“ is the object of our formative concerns and interventions. 2 Tadeusiewicz, G. – Education in Europe PWN, Warszawa-Łodź, 1977.
2
Polish society, although it continuously learns, is badly educated.3 Most probably it can’t be otherwise because independent Poland has been left with the results of the absence of concern for education from the previous years. Expenditures for education kept steadily below 4% of NGP, the level beneath which, according to UNESCO, one encounters the „educational death“ of the society, thus creating deep depression in Polish education. For the first time this low level was crossed in 1990. Unfortunately again, according to the World Bank. Poland had systematically lowered its educational spending towards 2.9% of NGP. The reasons for of the low conditions of teaching mathematics, the Queen of Science, are multidimensional. First of all since 1986, mathematics is not required for the final graduation exam from high school4. Many years of carelessness in the education of teachers and overloading the content of the curriculum causes the deepening of the crisis. This state of affairs requires a systemic approach of improvement. From the beginning of the nineties that is from the beginning of the deconstruction of previous reality, the difficult process of the transformation of the socio/economical system, the Polish school is in the state of continuous „tectonic movement“. The reform of education had entered the schools with big effort; its beginning are well taken by R. Nowakowski5: „Aims and expectations deduced from superficially understood meaning of life, understood as free movement around the garbage hill of the world had deformed the conception of education. As a result, teaching of mathematics not only doesn’t shape the rational man being able to distinguish the objective world from the illusion and being able to find a way of understanding phenomena in the broadest sense of word, but also is not able to reach assumed mistaken goals.“ These critcal words convey not only the factual state of affairs but also the emotions which are woken up by the issue of education reform. There are two worlds in the school millieu: the world of child/pupil and the world of the adult teacher. All have good intentions and enthusiasm. Children are convinced that they will be at least as succesful as their parents if not more. Adult teachers are convinced they will never reproduce the behavior of that „terrible mathematics teacher“ they had themselves. How to help them in
3 Kwieciński Z. – Education vis-à-vis hopes and dangers of the contemporary ssociety- presentation at the 3rd Polish Pedagogical Congress, Poznań, 21-13 Sept. 1998. 4 The exam might be restored by 2010, according to the last Minister of Education, Roman Giertych. 5 Nowakowski, R. – Dokad zmierzasz, edukacjo? O tendencjach I sytuacji nauczania matematyki (I nie tylko w niej). Wydawnictwo Naukowo Oswiatowe, Wrocław, 2005,p.5.
3
their endeavor? Professor Zawadowski6 usually says:“Teach in agreement with medical principles – do not harm!“ In the contemporary reality of the school one needs to notice a pupil who doesn’t want to learn. It seems correct to say that the responsible for the education of teachers should accept that at present, pedagogical studies prepare the army (of teachers) for the war „that was“. That means that the teacher – constantly trained professionally- is in effect helpless in the classroom. Then he/she more often than not uses the freedom to do bad in the name of doing good7. One needs to notice the pupil who doesn’t want to learn in the present school reality; one needs to develop means and strategies of facilitating pupil’s interest, to impact the motivation. Convince the pupil that the basic goal of every individual is independence and self- assertion. The community of teachers is standing at present in front of such issues and attempts to solve them. The realization of the Project „In the Middle of the Way…“ became Warsaw’s laboratory of developing those two qualities of independence and self-assertion amongst teachers of mathematics, and therefore amongst their pupils. The idea of the test was born on the background of teachers‘ concern about the development of mathematical competencies by students of liceum and technicum of Warsaw. Mathematics teachers of Warsaw were carried by the creative, professional impulse - the authors of the test proposed the cycle of workshops to Warsaw teachers of mathematics whose aims were: - presenting the teachers with the advantages of didactic measurement in teachers systems of teaching; - preparing the content of the large scale test for students of 2nd grade of Warsaw high schools. The workshops discussed the issues of -three dimensions of the teaching content; -typology of written assignments; -statistical analysis of written assignments and results of testing; -graphical representation of the results, and percentiles; -interpretation and communication of the results. Before the problems for the test were chosen, a questionnaire was conducted 6 Prof. W. Zawadowski, known mathematician, founder of Stowarzyszenie Nauczycieli Polskich. 7 The teacher generally asseses the pupil negatively. Kwiatkowska „„Chaos i autonomia, czyli o wolności i przymusie w wychowaniu”, w listopadzie 2004, w Warszawie
4
amongst the Warsaw teachers of mathematics concerning the conduct of mathematics education in their schools. The average number of hours of mathematics teaching per week was 8 hours in licea and 6 hours in technica. Using this information, the scope of material to be presented on the test was decided: numbers and sets, functions and their properties, linear, quadratic and trigonometric functions of acute angle, polynomials of higher degree, plane geometry. The tests „In the Middle of the Way…“ 2004, 2005, 2006, 2007 were designed to meet the Programatic Principles as well as the Examination Standards. The tasks with this content were grouped in three categories: FiW-functions and their properties, RiN – equations and Inequalities; GiT – geometry and Trigonometry. Problems are designed in the Basic level as well as in the Advanced level. In 2005, in order to enrich educational diagnosis, a multidimensional description of problem solving had been estanlished, where: A – is the analysis of the problem, that is using the mathematical language, understanding and creation of symbols, drawings, familiairty with mathematical terminology; M – choice of the method, that is meritoric correctness based on the familiarity with mathematical theory U-independence of learning; R – correctness of computations and transformations of equations. The level of mathematical competency was investigated with the help of the difficulty coefficient for the given competency in a diagnosed cohort of students: - larger than 0.75 – mastery of the competence by the cohort; - between 0.30 – 0.75 – the whole cohort needs to review given competence - less than 0.30 – the cohort needs to learn the competency anew. Several cycles of the test had shown that important element of teachers‘ knowledge about student competencies is the information about student errors and the frequency of their occurrence. The errors were coded in the following manner: - d – absence of familiarity with the definition of the concept; - w - absence of knowledge of properties of the object, rules of operation, a theorem (that is the sentence in the form If…then… - p - errors in computations, graphs, rounding off; - k - errors of the type d,w or p. Succesive editions of the Test In the Middle of the Way… were conducted to obtain precise measurements and independence of the results, Each of the editions of the test were piloted with cohorts of 100 students. While checking
5
the tests, teachers were giving points: 1 – the mathematical activity done correctly 0 – the mathematical activity done wrongly x – mathematical activity not undertaken. Correct non-standard solutions get the maximal number of points. In 2004 there was 9,000 students taking the test, whose analysis allowed the teachers to formulate extensive diagnosis of their classes. The value of the difficulty coefficient for different categories:
Taxonomic category B Understanding of the concept C Application of knowledge in similar situations
D Application of knowledge in Problem
situations.
Difficulty Coefficient 0.64
0.31
0.37
It turned out that our pupils manifest certain type of functional limitation. They can apply their knowledge in situations directly described like tests, but can’t if the instruction is more general and does not contain suggestions about the mathematical domain involved or the method of solving. Equally troubling are results in geometry and trigonometry. Independently of the school, student knowledge in this areas is sporadic. The coefficient of difficulty of plane geometry problems was p=0.38, what translates itself into: -50% pupils doesn’t know how to solve the problem; -25% can solve it partially and 25% - completely. The dsitribution of points obtained by students in this problem suggests that they don’t know basic geometrical facts, don’t know how to solve problems which do not have any numerical data. We suggest the cause to be the absence of coherence in the program of geometry and small number of hours devoted to plane geometry. The coefficient of difficulty of trigonometric problems was p=0.49. That translates into -34% of students could not react at all to the problem -33% solved partially; -33% students solved it correctly. We think that these results are due to the fact that trigonometrical content appears late, and therefore students can retain
6
the acquired knowledge. There was 7,000 students participating in the 2005 edition of the test In the Middle of the Way…The difficulty coefficient reached by the test had following numbers
Taxonomic category B Understanding of the concept C Application of knowledge in similar situations
D Application of knowledge in Problem
situations.
Difficulty Coefficient 0.47
0.22
0.57
The analysis of results allows the teacher to reflect upon teacher’s system of teaching. Of special interest is the analysis of pupils‘ work from the point of view of new competencies being introduced in schools at present, called anchoring competencies. The analysis of the change in the mastery of A M U R competencies was made throughout different ediitions of the test. The comparison of difficulty coefficients:
Fig. 2 Comparison of difficulty coefficients of the test In the Middle of the Way…
7
0,44
0,45
0,38
0,49
0,37
0,22
0,32
0,39
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
F i W R i N G i T Zestaw
podtesty
wspó?czynnik
?atw
oœ
ci
2004
2005
The analysis of the comparisons suggest that there is a development in pupils‘ mastery of anchoring competencies. The questionnaire given to the teachers whose students participated in the test inform that 61% of them have adapted their methods of teaching to needs of the classrooms. Work on the project In the Middle of the Way… initiated the process of activization of professional development of mathematics teachers in Warsaw. More than 85% of teachers expressed interest in further development of professional knowledge conencted with the assessment of student mastery of mathematics. The Project In the Middle of the Way… is recognized by those Warsaw teachers who recognize the need to improve one’s own quality of work; who treat teaching profession as a necessary profession in contemporary society, who give high priority to the issue of learning assessment. May 2007
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The Process of Doing Mathematics By
Rony Gouraige I have been actively involved for the last year as a mentor to several students under the New
York City Louis Stokes Alliance for Minority Participation program. This experience forced me
to think about the best approach to teaching undergraduates with a limited knowledge of
mathematics how to actually do mathematics. I hope that my ideas on this matter may prove
useful to someone who is acting as a mentor. I also hope to encourage teachers to become
mentors by highlighting one of the less obvious benefits of mentoring.
I began with four students in the summer of 2006. By the spring of 2007, two of my students
had moved on to schools outside of New York City. My students started with a background that
included three semesters of calculus and a course in linear algebra. From the outset, I had two
goals in mind. First, I wanted to teach my students some mathematics that would be useful no
matter what careers they decided to pursue. Second, I wanted my students to gain the experience
of actually doing mathematics, which in my mind meant that they should do some original
research.
Mathematics Teaching-Research Journal On-Line
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Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
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With these aims in mind, I had to first of all decide what subject to study. My background is in
non-commutative algebra. For a variety of reasons, I decided that this area was not appropriate
for an undergraduate research project. There seems to me to be at least two approaches that one
could adopt. Choose a subject that requires no specialized knowledge, but which is rich in open
and challenging problems that could be solved without resorting to any complex machinery. On
the other hand, one could spend some time teaching the students some specialized knowledge,
and then give them problems that could be solved with the techniques that they had learned,
perhaps supplemented by further theory according to the circumstances. I rejected the first
approach because in my mind it was inconsistent with the first goal I stated above. To pursue the
second approach, I finally decided to teach my students some number theory. Number theory
has a rich history, is an active research area, and, most importantly for my immediate aims, has
motivated the development of much of the abstract machinery which pervades modern
mathematics
I spent the summer of 2006 teaching my students elementary number theory, roughly the
equivalent of a junior-senior level undergraduate course in number theory. However, I adopted
the point of view that in teaching number theory, I would make maximal use of the basic
structures of abstract algebra: Groups, rings, fields, and vector spaces. This was consistent with
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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my first goal. After the initial period of teaching my students elementary arithmetic, I had to
now focus on my second goal, which was to get my students to do original research. Not being a
professional number theorist, I was not conversant with the open problems, even the elementary
ones. Frankly, I did not want to simply assign problems that were already well-known, on the
assumption that such problems were already receiving plenty of attention. I decided to bring to
bear my experience in non-commutative algebra, and look for problems in arithmetic that had
analogues in algebra. Having done some work in the theory of central simple algebras, it was
natural that I would look for the matrix analogues of results in elementary number theory. This
is hardly a novel approach, but it turned out to be remarkably fruitful. For example, one of my
students is currently investigating the analogies between Euler’s totient (also called phi) function
and row equivalence of matrices while another is looking at the analogues of Fermat’s sum of
two squares theorem over modular rings of integers.
How does one initiate a student into the process of doing research? The first thing that I had to
overcome was the natural tendency of my students to turn to a textbook for the answer to a
problem that they could not solve. More fundamentally, I wanted to encourage my students to
embrace the unknown as an opportunity for discovery. My constant refrain is that at some point,
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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no book, or paper, or person will have the answer to the question that one is asking. This fear of
the unknown is primarily a psychological barrier, and it takes some time to overcome.
My next goal was to inculcate the notion that the most difficult part of doing mathematics is not
in proving theorems, but rather in coming up with theorems that are worthy of being proved. For
the novice in mathematics, the notion of proof and the techniques of proof seem to be of
paramount importance. This is not entirely wrong, but it risks a fundamental misunderstanding
of the nature of mathematics. To focus on proofs is to take the theorems to be proved as given.
But in the practice of doing research, discovering the theorem to be proved comes before one can
even begin to undertake the search for a proof.
How does one make new discoveries? I have spoken to other mathematicians about this topic,
and it seems that the process of discovery is as individualized as one’s fingerprints. So whatever
I write about this topic is necessarily biased and reflective of my own temperament and abilities.
I advise my students to pursue analogies and to examine many examples. The analogies one
pursues will change as one’s knowledge increases. However, I have been influenced by my
advisor, Ravindra Kulkarni, to think in terms of three fundamental categories in mathematics:
Number, space, and symmetry. The key is to appreciate the subtle and symbiotic relationship
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Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
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between these categories. This is not the place to elaborate on these themes, but the idea is to
pursue analogies that follow from these basic categories.
The analysis of examples is of fundamental importance, and underappreciated by students. How
often have you observed that a student can state a theorem and repeat its proof verbatim, but is
completely lost when asked what the theorem implies about a basic example? A good example
gives meaning to a theorem. Better yet, the examination of many examples is a fruitful way to
make new discoveries. Of course, the discoveries that one makes will reflect one’s talents. But
the examination of many examples is a critical step in the process of discovery, and ought not to
be bypassed. This is inductive reasoning. Students find it very difficult at first because it takes a
lot of hard work. I encourage my students to engage in this process, and to persevere even if it
doesn’t yield immediate results. The process of doing research involves hard work and tenacity
as well as creativity, and I want my students to understand and accept this.
In closing, I want to comment about some of the benefits of mentoring for the mentor. I teach at
a community college, and as a result, have a heavy teaching load. The free time that I have for
research is precious. Why should I devote any of that free time to mentoring students? Each
teacher will have to find their own answer to this question, but my experience has taught me that
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Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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mentoring can be personally rewarding and at the same time can further one’s own research.
The personal rewards are obvious, and need no elaboration from me. I think that it is the benefit
to one’s research that tends to be overlooked when one is considering whether or not to become a
mentor. If one is actively engaged in research, why take the time to mentor someone who is
unprepared to understand or contribute to the work that you are doing? I would argue that
mentoring such a student forces you to look very carefully at basic questions. These questions
have been the stimulus for original research for millennia. The time that one spends
contemplating such questions can only benefit one’s research program. This benefit may seem
tenuous at best, but I am convinced that if a researcher brings to bear his or her creativity in the
examination of such problems, then they will discover new avenues of investigation that they
might not otherwise pursue or even be aware of.
R. Gouraige
Bronx Community College
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Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
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Reflections of a Teacher Educator.
Harrie Broekman, Freudenthal Institute for Math and Science Education, University of Utrecht, The Netherlands.
If we expect a lot of effect of teachers researching their own practice, we as teacher trainers
have not only ‘to teach as we preach’, but also ‘research our own practice’.
Shortly after writing about my reading of an interesting article and writing about it 1 for some
colleagues, I realised that as a participant in a project for Teacher Researchers2, I had to look
more carefully at my own teaching. Not only what I’m doing in reality, but also the main ideas
behind my activities had to be ‘researched’, or at least reflected on in a critical way.
The following is part of the ongoing reflection about “what tasks to use in my teacher training
courses, why to use them and how to use them”.
The main background ideas for selection of and work on tasks in teacher education.
1 See appendix 2 A EU sponsored project ‘Professional Development of Teacher Researchers, called the Krygowska project.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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1. Mathematics is a human activity; a lot of doing and reflecting on the doing as well on the
background of this doing. The strategy ‘think, share and compare’ supports this.
2. In Realistic Mathematics Education both the horizontal mathematisation (real contexts
used3) as well as vertical mathematisation (processing within the mathematical system4)
are important.
3. Teachers’ stories are an important tool for building a strong fundament for making
teachers’ tacit (hidden and not conscious) knowledge explicit.
4. Reflection – private, but also with the support of others – is useful and maybe even
necessary for learning.
5. Implicitly is the importance of a questioning style, about which Vrunda Prabhu5 wrote:
“The questioning style also has the hope that the enquiring attitude m the regular
classroom discourse become a part of students way of learning and creating their own
mathematics”
6. As a mathematics teacher educator it is my task to investigate the most effective methods
of improving learning in my pre-service and in-service courses.
7. As a trainer/educator of teacher educators it is also my task to investigate teaching and
learning processes in general; the selection of classroom tasks included. 3 Transforming a problem field into a mathematical problem 4 Going ‘deeper’ in the mathematical structures etc 5 Vrunda Prabhu, Independence of learning, MTRJ Online. V2;N1.
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Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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An example used as a starter to ‘open the minds of participating pre-service and in-service
teachers for aspects of teaching, worthwhile to write about’.
A short article, written by a teacher, was given to a group of in-service teachers to comment, and
– more specific - to ‘look for’ the aspects of her experience the teacher wrote about in this article
and what aspects she didn’t mention. I expected a lively discussion about what is interesting to
write about and what is interesting to read about, and that happened and made us all become
aware of our privileged focus points.
I named the activity: “learning from a human activity described in a teachers’ story”.
The teachers’ story
A primary teacher – Lonneke Boels – described in a Dutch teachers’ magazine how she added to
the textbook (paragraph about tables and graphs) the question: “bring to the class newspapers,
journals, telephone books, bus- and train time schedules, or other sources with tables and graphs
in them. The children had to select from their materials at least one graph, one table and one
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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diagram, and glue that on a A4 page. Next they had to circle one point in each figure and write
down what the meaning was of that specific point in the given context.”
The results were shared, compared. The teacher describes in the article what she observed as
learning outcomes for the children and – according to her reflection, important for her own
learning – what she didn’t expect, what surprised here and what she intended to do as a result of
that.
[Lonneke didn’t mention that she was working in a classroom culture that is supportive,
conversational and respectful; a community where ideas are valued (regardless of their
mathematical validity), where there is trust (no ridiculisation) and where risk-taking is
rewarded.]
Two illustrations of some important aspects of mathematics education.
Horizontal and vertical mathematising illustrated by two short stories.
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A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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Story A. My daughter had to go to the railway station every working day of the week. She had
the possibility to travel with bus 3 or bus 4, for both directions. Since she is not a routine person
she wanted to have different routes on each day. Is that possible?
After a group of teachers worked on this many of them came with the (correct) answer NO.
The next question I gave them was: can you show this with some kind of a visualisation? And
next: can you prove it?
These next questions were given because I wanted to involve the teachers in a discussion about
‘modes of communication’ and about the triple ‘ being sure’, ‘being able to convince a friend’,
and ‘ being able to convince a math teacher ‘ (proof?) The given ‘ realistic situation’ was used to
provoke different solutions. This time I did not expect a discussion about that aspect.
I was surprised by the results, both by the creative solutions as well as by the inability of some
participants to visualise or explain their thoughts.
This made us – the participants agree with my decision to use a second, more inner-mathematical
context, for further exploration.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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Story B. Each participant of my in-service group is asked to choose 5 lattice points on a ‘square-
dotted’ paper with a coordinate system on it, and to find the midpoints of the connecting lines
between each pair of dots. After my question if anybody found a midpoint without whole
coordinates the answer was YES.
So, the next question that came up (asked by a participant) was: is it possible to find 5 points in
such a way that all of the midpoints have whole coordinates?
And the whole group started to think, share, compare and reflect on the process.
What didn’t happen was the spontaneous emerging of the question that could have made the
work on these two problems into a real “mathematics learning by horizontal and vertical
mathematising”. This question seemed to be in the Zone of Proximal Development of the
participants. So, they needed a teacher/coach to ask some question to help them to ‘learn’.
The best question I could think of was: ‘What is similar in the problems A and B?’
A bit later in the conversation I also was the person who asked the question ‘What presentation
can you use to show that?’
Reflection on the work.
Mathematics Teaching-Research Journal On-Line
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Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
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Afterwards I realised that we missed in the conversation attention for the question: ‘how can we
help our students to start asking these kind of questions themselves and become independent of a
teacher?’ Next sessions with this group we have to find a moment to reflect on this question.
For me the most important part of the experience with these tasks (story 1 and 2) and the work
on it was the reflection of one of the participants.
“At the beginning I didn’t understand why you asked the second question, the one about
‘presentation’. But then I listened to the reactions of my colleagues and thought that is a bit crazy
that all of us tried to use ‘language’ (words) to ‘show’ the similarity in the problems A and B.
Only when you pushed us to think deeper about the second question we started to look for other
‘modes’ of communication. Maybe that is the reason we don’t stress enough the ‘visual’ ways of
describing in our own classrooms. We don’t use them easily ourselves”
These different modes of communication are – for sure – much more salient in contemporary
society (Kress&Jewitt6, 2003, p.1) and are for that reason stressed in the PISA tests as can be
seen in the main processes (8 in 2000, or 7 in 2003), called representation (number 6) and
mathematical communication (number 3).
6 Kress, G. and Jewitt, C. , 2003, Íntroduction’, in: Kress, G. and Jewitt, C. (eds) Multimodial literacy, New York, NY, Peter Lang, pp.1-18.
Mathematics Teaching-Research Journal On-Line
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Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
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Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
My personal learning.
As a result of the described experience - and some other experiences - I’m much more focused at
finding and using opportunities /tasks that can act as a challenge for participants in mathematics
teaching courses. Challenges to work on, to reflect on their own work in mathematics, but also to
reflect on the work of their students and to investigate their own role in the (mathematics)
learning of their students.
Appendix.
Harrie Broekman, Freudenthal Institute for Math and Science Education, Utrecht University, was
reading for you:
Ana Maria Lo Cicero, Yolanda De La Cruz, Karen C. Fuson.
Teaching and Learning Creatively: Using Children’s Narratives.
Teaching Children Mathematics, May 1999, pp.544-547.
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
Children’s Math Worlds project seeks to integrate student’s social, emotional and cultural
experiences into classroom mathematics.
“We build on the individual experiences, interests, and practical mathematics knowledge that
diverse children bring to our classrooms.”
There needs to be a balance between building on children’s knowledge and teaching within
the zone of proximal development (also called the ‘learning zone’).
The project uses a Vygotskian model for unfolding, formulating, and solving mathematics
problems from children’s experience. This model describes one way in which teachers build on
children’s prior knowledge about various situations to facilitate student’s construction of
understandings of formal mathematical concepts, symbolism, and problems. The unfolding
multiple narratives of different children’s experiences provide a framework that is co-constructed
by the teacher and children and within which teachers relate new mathematical ideas to children'’
lives. The ZPD -–learning zone – is what children can accomplish with assistance. The teacher
leads the children from a starting point to more advanced mathematical knowledge. This
knowledge includes being better at listening, explaining and helping one another understand;
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
learning more advanced, efficient and accurate solution methods; and learning mathematical
symbolism, language, and new ideas.
- Getting started: eliciting and using children’s stories.
- Understanding, listening and describing.
- Putting a story in mathematical terms.
- Problem solving, reflecting and explaining.
- The co-constructing process. The classroom conversation is co-constructed by all those
involved. The active participants in a conversation each direct the conversation in certain
ways. Each contribution stimulates thinking. Throughout the conversation, personal
meanings are continually being constructed and reconstructed in ways that are influenced by
the classroom process.
Conclusion
Listening to children, putting their stories in a mathematical context, using children’s labeled
mathematics drawings and number drawings, and eliciting explanations from children about how
they solved problems are powerful approaches. But these approaches need constant leadership by
Mathematics Teaching-Research Journal On-Line
A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)
Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York
Volume 2 Issue 1 Date September 2007
Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.
the teacher so that children can progress in their knowledge of mathematical methods,
vocabulary, and understanding.
In Utrecht we say that the teacher is needed to foster and coach the horizontal
mathematising as well as the vertical mathematising. In classroom settings interaction plays
an important role.
Part of the teachers’ role is: giving students opportunities to share their ideas, opinions, and
questions (creating a classroom environment in which mathematical thinking is encouraged and
valued) The selection of tasks or learning situations/contexts as well as the teachers’ own
questions are an important ingredient in this teachers’ role.