Research in Mathematics Education: New Directions for Two-Year Colleges

Research in Mathematics Research in Mathematics Education: New Directions for Education: New Directions for

Two-Year CollegesTwo-Year Colleges

April Strom, Scottsdale Community College (AZ)April Strom, Scottsdale Community College (AZ)Ann Sitomer, Portland Community College (OR)Ann Sitomer, Portland Community College (OR)

Mark Yannotta, Clackamas Community College (OR)Mark Yannotta, Clackamas Community College (OR)Amy Volpe, Glendale Community College (AZ)Amy Volpe, Glendale Community College (AZ)

RMETYC

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Purpose: To encourage and support quality research in mathematics education in two-year colleges, conducted by two-year college faculty.

New AMATYC Committee

Guiding Questions Why conduct research in mathematics education

at community colleges? Theory without practice is empty; practice without theory is

blind (Kwame Nkrumah, 1966).

Schoenfeld (2000) says “Research in mathematics education has two main purposes, one pure and one applied:

Pure (Basic Science): To understand the nature of mathematical thinking, teaching, and learning;

Applied (Engineering): To use such understandings to improve mathematics instruction.

What are some examples of research studies and findings conducted by two-year college faculty?

Current Trends in Math Education

Cognitive Research: Focus on student reasoning Quantitative and Proportional Reasoning (Thompson,

1994; Smith III & Thompson, 2008)

Covariational Reasoning (Carlson et al., 2002)

Advanced Mathematical Thinking (Rasmussen & Zandieh, 2005); Realistic Mathematics Education (Freudenthal, 1991)

Research-based curricula: Rational Reasoning Group (Arizona State); Abstract Algebra Group (Portland State)

Presentations Talk 1: Experience as a researcher (Ann

Sitomer)

Talk 2: Experience as a subject (Mark Yannotta)

Talk 3: Experience with research design (April Strom)

Talk 4: Experience with data snooping (Amy Volpe)

Exploring the Exploring the Mathematical Knowledge Mathematical Knowledge Embedded in Adults’ Life Embedded in Adults’ Life ExperiencesExperiencesAnn SitomerPortland State UniversityPortland Community College

Exploring the Mathematical Knowledge Embedded in Adults’ Life Exploring the Mathematical Knowledge Embedded in Adults’ Life ExperiencesExperiences

Adult returning students in Adult returning students in developmental mathematics developmental mathematics coursescoursesPlacement tests measure what adult returning students recall – or do not recall – about school mathematics.

Students who have been away from school often place into the most elementary mathematics classes offered by mathematics departments at community colleges.

But to what extent do adult students return to school with mathematical competencies not measured by placement tests?


To what extent do adult students return to school with mathematical competencies not measured by placement tests?

• Contexts for the questionTeachingResearch

• The study Setting

Proportional reasoningData collection

•Sample of student work on The Wage Problem

•Refining the research questions and the design of the study

An exploratory study An exploratory study


Contexts for the questionContexts for the question

To what extent do adult students return to school with mathematical

competencies not measured by placement tests?

PracticeMy work as a

teacher

ResearchAdults

learning mathematics

ResearchOut-of-school

mathematical practices


Contexts for the question: Contexts for the question: ResearchResearch

Out-of-school mathematical practices

Over the last 30 years, researchers have studied both adults’ and children’s mathematics in contexts outside of school:

•Liberian tailors (Lave, 1977)•Dairy workers (Scribner, 1984)•Young street vendors in Brazil (Carraher, Carraher, & Schliemann, 1985)•Female shoppers in the US (Capon & Kuhn, 1979; Lave, 1988)•Odds makers at the horse track (Ceci & Liker, 1986)•Bookies for an unofficial lottery game (Schliemann & Acioly, 1989)•Carpet layers (Masingila, Davidenko, & Prus-Wisniowska, 1996)•Apprentice iron workers (Martin, LaCroix, & Fownes, 2006; Martin & Towers, 2007)•Structural engineers (Gainsburg, 2007)



Out-of-school mathematical practices

Over the last 30 years, researchers have studied both adults’ and children’s mathematics in contexts outside of school:

•Liberian tailors (Lave, 1977)•Dairy workers (Scribner, 1984)•Young street vendors in Brazil (Carraher, Carraher, & Schliemann, 1985)•Female shoppers in the US (Capon & Kuhn, 1979; Lave, 1988)•Odds makers at the horse track (Ceci & Liker, 1986)•Bookies for an unofficial lottery game (Schliemann & Acioly, 1989)•Carpet layers (Masingila, Davidenko, & Prus-Wisniowska, 1996)•Apprentice iron workers (Martin, LaCroix, & Fownes, 2006; Martin & Towers, 2007)•Structural engineers (Gainsburg, 2007)

A partial catalog of mathematical competencies developed outside of

school



Adults learning mathematics

• The role of affect (Evans, 2000; Wedege & Evans, 2006)

•Translation between worlds (Benn, 1997; Martin et al., 2006)










The study: SettingThe study: SettingThe students participating in the study are enrolled in Basic Mathematics at a community college in an urban setting.

Participating students were enrolled in four of the 12 sections of this course offered on campus Fall 2009. The participation rate in each section varied from about 20% to 75% of the students enrolled in a section.

Basic Math, along with a developmental reading and a

developmental writing course, are standard prerequisites for most lower division collegiate courses offered at

the college.


The study: SettingThe study: SettingBasic Mathematics

“Use fractions, decimals, percents, integer arithmetic, measurements, and geometric properties to write, manipulate, interpret and solve application and formula problems. Introduce concepts of basic statistics, charts and graphs.”

What are the core ideas of the course?

Problem solving

Proportional reasoning


The Study: Proportional The Study: Proportional reasoningreasoningWhat is meant by proportional reasoning?

“I propose that proportional reasoning means supplying reasons in support of claims made about the structural relationships among four quantities (say, a, b, c, d) in a context simultaneously involving covariance of quantities and invariance of ratios or products; this would consist of the ability to discern a multiplicative relationship between two quantities as well as the ability to extend the same relationship to other pairs of quantities” (Lamon, 2007, p. 638, emphasis added).


The Study: Proportional The Study: Proportional reasoningreasoningWhy examine proportional reasoning for this study?

“If people with little or no schooling really understand proportional relations in these contexts, or if highly schooled individuals who have difficulty understanding proportionality in school-type settings fail to exhibit such difficulty in informal learning contexts, then there is something important to be understood” (Schliemann & Carraher, 1993, p. 49).


The study: Data collectionThe study: Data collectionThree types of data are being collected:

•Participating student’s responses to a brief biographical survey

•Participating students’ work done on a problem solved collaboratively during the second class meeting of the term, as well as field notes taken while students collaborated on a solution to this problem.

•Selected participating students’ responses to interview questions about their work on The Wage Problem, about the mathematics they have used outside of school, and on a selected tasks from the research literature on proportional reasoning.










Students’ written work on The Wage Students’ written work on The Wage ProblemProblem

Two things to consider•What mathematics do the students know before they return to school?•Are there strategies or concepts that students are bringing to bear on The Wage Problem that suggest that a student is building on her/his out-of-school experiences?

A guiding principle“These researchers [ethnographers] have consistently tried to understand mathematical problem solving in the same way as their subjects” (Eisenhart, 1988, p. 110).


Students’ written work on The Wage Students’ written work on The Wage ProblemProblem

Page 8 of the submitted work with computational support on pages 9 and

10.


ObservationsObservations


RefinementsRefinementsGoals of data analysis from pilot study

•Is there evidence of mathematical competencies with which adult students are returning to school?

If so, how might these competencies be categorized and further studied?

If not, what another type of experimental design might uncover these competencies?

•What themes emerge from the data? How might these themes refine future research questions aimed at uncovering the mathematical knowledge embedded in adult’s life experiences? Do the emerging themes suggest questions that might help us understand how adults students’ mathematical knowledge interacts with the mathematics they are learning in their mathematics classes?


Thank youThank you

Questions?

Math 299: A Bridge to Math 299: A Bridge to University University MathematicsMathematicsMark YannottaPortland State UniversityClackamas Community College

OverviewOverviewBackground on bridge courses and Math 299

The Math 299 Class of 2009Some preliminary results from the data

Participant activity

My dissertation area:My dissertation area:What are the challenges and opportunities associated with developing and sustaining a mathematics bridge course in a community college setting?

What do we know about What do we know about mathematics bridge mathematics bridge courses?courses?About 40% of colleges and universities

in the US offer a dedicated courseNo consensus on content (although

many people argue that proof should be integral)

Research supports that “bridging” does not occur in a single course

Community colleges might provide some new direction for these courses

PORTLAND STATE UNIVERSITYPORTLAND STATE UNIVERSITYDEPARTMENT OF MATHEMATICS AND STATISTICSDEPARTMENT OF MATHEMATICS AND STATISTICSBA/BS DEGREE REQUIREMENTSBA/BS DEGREE REQUIREMENTS

Mth 251, 252, 253, 254: Calculus I-IV (16)Mth 256 or Mth 421: Differential Equations

(4/3)Mth 261: Introduction to Linear Algebra (4)

Mth 311, 312: Advanced Calculus (8)

Mth 344: Group Theory (4)

Additional 21 - 28 credits of elective courses

A GAP IN THE CURRICULUM

The Evolution of Math 299The Evolution of Math 299

Year Students Instructor(s) Curriculum

2005 10 3 proof, topology & group theory

2007 6 1 group theory, proof & math history

2008 6 2 group theory (g.r.) & topology

2009 9 1 group theory (g.r.)

2010 ?? 1 group theory (g.r.)

How did I get involved with How did I get involved with this abstract algebra this abstract algebra curriculum?curriculum?

2006: I took a topics course with Sean Larsen. I started thinking about ways to incorporate more research-based ideas into Math 299 when I taught it again.

2007: Sean contacted me about being Co-Pi on a collaborative NSF grant.

2008: The grant was funded and I began incorporating some of the materials into the class.

2009: In conjunction with the grant, we used a modified version of the group theory curriculum in Math 299 and collected data at CCC.

Benefits of guided Benefits of guided reinventionreinventionFreudenthal (1991) argues that,

“knowledge and ability, when acquired by one’s own activity, stick better and are more readily available than when imposed by others” (p. 47).

◦ The students actively participate in developing symbols, notation systems, definitions, and theorems.

Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Norwell, MA: Kluwer Academic Publishers.

The students in my 2009 The students in my 2009 class:class:

Consisted of 9 community college students aged 17 - 35 years (6 in the 18 - 24 age category)

4 males, 5 females*4 math majors, 2 engineering majors, 1 music

major, 2 undecided Various math backgrounds and experience

◦4 completed differential equations (2 had taken a different version of the transition course in 2008)

◦1 completed calculus III◦3 completed calculus II◦1 completed college algebra

The classroom The classroom environment:environment:

Low risk◦ An elective course◦ Little homework was required◦ Inexpensive (<$200 to take the course)◦ Graduate student grading (A, B, or Pass)

High level of participation◦ Participation protocols were established

early in the term◦ Students were expected to be active

participants in the class every day

Some preliminary results Some preliminary results from the datafrom the dataMy students were VERY active in class

discussionsThey used sophisticated geometric

reasoning and were resistant to moving toward more axiomatic reasoning

The were several moments throughout the term when students seemed empowered and took ownership of the mathematics

The exit interview data suggests that most of the students thought they had changed the way they thought about mathematics

Activity 1: Defining the Activity 1: Defining the symmetries of the Equilateral symmetries of the Equilateral Triangle Triangle

Let F stand for a flip across the vertical axis and R stand for a 120-degree clockwise rotation.

In the right-hand column of the table, express each of your symmetries in terms of combinations of F and R.

Activity 2: Completing the Activity 2: Completing the Group TableGroup Table

I R R2 F FR FR2

I

R

R2

F

FR

FR2

A few ideas for a sustainable A few ideas for a sustainable transition course model at transition course model at CCC CCC Teach in way that combines the acquisition

of knowledge with opportunities for students to participate in meaningful and contextualized mathematical activity

Run a transition (bridge) course once a year

Alternate curriculum every year:◦ Y1: Group Theory Curriculum ◦ Y2: Advanced Calculus Curriculum

Widen the net to include high school and reverse transfer students

A Case Study of a Secondary Mathematics Teacher’s

Understanding of Exponential Function: A Theoretical

Framework

April D. Strom

“The greatest shortcoming of the human race is our inability to understand the exponential function.”

Albert Bartlett (1976)

Literature Review

Multiplicative Conceptual Field Theory– Multiplicative Reasoning– Exponential function (splitting, multiplicative unit)– Recursion– Rate of Change

Covariational Reasoning

Research Questions What conceptions does a secondary

mathematics teacher hold about exponential growth and decay?

How effective are the current attributes of the developed exponential function framework in explaining the ways of thinking about the concept of exponential function?

How does an instructional unit - focused on exponential function - facilitate the development of a secondary mathematics teacher’s understanding of exponential function?

Theoretical Perspective

Cognitive constructivism:– the philosophical belief that learning occurs through

experiences and action, rather than through knowledge passed on by others (Steffe & Thompson, 2000).

… focus on describing the mental images of an individual to learn one’s ways of thinking and approaches to problem situations (Piaget, 1970).

Exponential Function Framework

In groups, discuss the following:

What are the critical elements of knowledge that students should know/build for understanding exponential functions?

What are the critical reasoning abilities that students should use for understanding exponential functions?

Methodology: Data Collection

1 secondary mathematics teacher: “Ben”

All 16 teachers

4 teachers (Pilot) + Ben

Functions course2 teachers (report on 1) All 16

teachers

Teaching Experiment Methodology

Middleton, Carlson, Flores, Baek, & Atkinson (2004)

Steffe & Thompson (2000)

Exponential Function Framework:

Notation and Language

ENLN Use notation to represent

€

bx as

€

x factors of

€

b for rational values of

€

x.

ENLL Use language to represent

€

bx as

€

x factors of

€

b for rational values of

€

x.

ENLM Use multiple representations to represent the multiplicative processes of exponential functions.

ENLP Use parametric changes to alter representations of an exponential function.

ENLI Use implicit definition of exponential functions to represent exponential situations recursively.

ENLE

Use explicit definition of exponential functions to represent exponential situations explicitly.

Exponential Function Framework:

Reasoning AbilitiesERCR Use covariational reasoning to describe exponential behavior by

attending to incremental changes in the independent variable. ERCR1: Output Changes ERCR2: Amounts of Change ERCR3: Constant Percent Change ERCR4: Multiplicative Rate of Change

ERFR Use exponential function reasoning to compare and contrast exponential functions with other functions (i.e., linear functions).

ERFR0: Inappropriate Use of Additive Reasoning ERFR1: Compare Amount of Change with Percent Change ERFR2: Compare Constant Rate with Changing Rate ERFR3: Compare Constant Rate with Multiplicative Rate

ERMR Use multiplicative reasoning to describe exponential behavior. ERMR1: Proportional Parameters ERMR2: Recursive Change by Constant Factor ERMR3: Constant Proportionality of Outputs

ERPIR Use partial-interval reasoning to describe exponential behavior. ERPIR1: Partial Change ERPIR2: Partial Factors

Results: Thinking about Exponents

When simplifying expressions with integer exponents, Ben said:

“the exponent tells you the number of base numbers to be multiplied together”

Ex: 32 = 3 · 3

So why not use this thinking for fractional exponents?

Thoughts about Fractional Exponents

“I know I’ve explained this before, but I don’t even remember what I said.”

“I’d just tell students that they should go back to the rules for whole number powers because the rules are the same for any type of power.”

Fractional Exponents

Thinking about bx as representing x factors of b for fractional values of x

Ex: 31/2 represents 1/2 the number of factors of 3.

What is 1/2 the number of factors?

Partial Factors Emerged

If then

Ben: “31/2 is a number that when I multiply itself by another 31/2 gives me just 3”

€

32 = 3⋅3

€

31 = 3? ⋅3?

Partial Factors Emerge“…it takes me five 3 to the 1/5ths to give me just 3”

€

31

5 ⎛

⎝ ⎜

⎞

⎠ ⎟

5

Filling In the Missing Pieces

+1+0.5

+0.5

Ben: “Two factors of because

that’s a half factor. Here I am multiplying by just a factor of one .

Here’s two .”

€

3

3

€

3

3

€

3

3

×3

3

×3

3×13

Research Question #1 Teacher’s Conceptions of Exponential Function

– Partial-Interval Reasoning & Partial Factors• Building strong conceptions of partial factors facilitates in building

mature understanding of multiplicative change & exponential function.

– Multiplicative Reasoning• Ben relied on recursion to make sense of tasks; recursive action to

recursive process

– Exponential Function Reasoning• Used his thinking about linear behavior to facilitate his thinking about

exponential behavior• Difficulty distinguishing between quadratic functions and exponential

functions -- “anything that has a power” was exponential

– Covariational Reasoning• Used amounts of change as a dominant way of thinking, especially in

the beginning of study; at times he used constant percent change to describe exponential behavior -- both were meaningful to Ben

Research Question #3 Effectiveness of Instructional Unit

– Exponential Decay: More difficult to describe• A paralleled emphasis on exponential decay provided opportunities for Ben to

think about exponential decay in light of exponential growth

– Contextualization facilitated Ben’s ability to make sense of the task• Condensed version of the framework (elimination of Conceptual

Understandings -- merged with other components)

– Comparing and contrasting exponential and linear behavior facilitated Ben’s thinking

• Ben thought about constant percent change relative to constant amount change

Research Implications

Research-based materials

– Exponential Functions developed with the conceptual framework in mind

– Rational Reasoning, Arizona State University• Materials focused on college algebra and precalculus

Questions?

Developmental Math Students’ Beliefs and Self-Efficacy

Amy VolpeGlendale Community College

What Motivated My Research?

You Tube

Facebook

Researchers See Data Everywhere

• Everyone knows its Greg Episode 1.mp4

Question• What insights did you gain from this video about the

students’ perceptions of mathematics and their math teacher?

• Title: “Greg in mathematical misadvantage”• Sign: “A mindless worker is a happy worker so shut up and

do your job!!! ”• Student: “This math problem is too hard should we use

subtraction or addition?” Teacher, “I think we should use both”.

• Teacher just states what operations will or won’t be used without justification. Students accept this, but attempt to generate creative representations of mathematical operators.

• Operation Symbols: “We shouldn’t fight. We all have one common enemy. Pre-algebra.”

• Teacher: “That’s it. I can’t take it anymore. You kids make the easiest problems so difficult to teach. I quit!”

Student Mathematical Beliefs• Studies have shown that student beliefs about their

capability in mathematics, the usefulness of mathematics and their enjoyment of mathematics are related to learning and how students perform in the classroom (Schoenfeld, 1989, Nickson 1992).

• Kloostermann et al (1996) performed a three year longitudinal study of twenty-nine students in the first through fourth grade to observe how their beliefs evolved over time. The data from the study suggested that:

1. Student beliefs about their ability in mathematics were relatively homogeneous over time.

2. Student beliefs about their ability were correlated with their performance.

3. Student beliefs about collaborative work often mirrored the beliefs expressed (sometimes unintentionally) by their teacher and varied over time with respect to their teacher.

Self-Efficacy• Albert Bandura: Social Cognitive Theory• Self-Efficacy Defined: “…people's judgments of their

capabilities to organize and execute courses of action required to attain designated types of performances" (1986, p. 391).

“Unless people believe that they can produce desired effects by their actions, they have little incentive to act…Such beliefs influence aspirations and strength of goal commitments, level of perseverance in the face of difficulties and setbacks, resilience to adversity, quality of analytic thinking, causal attributions for successes and failures and vulnerability to stress and depression.” (1996, p. 1206)

The TIMMS Study: An example of a study that surveyed student beliefs

• The TIMMS study is an international study aimed at assessing fourth and eighth grade students’ mathematics and science achievement.

• The study implements probability sampling in an effort to draw inferences within and between countries.

• In addition, similar studies were executed in 1995, 1999 and 2003 and efforts have been made so the data can also be analyzed across time.

• The use of probability sampling allows the possibility of drawing inferences about the US population of 4th and 8th grade student beliefs about mathematics and their correlations with achievement.

TIMMS Questions on Mathematical Self-efficacy

a) I usually do well in mathematicsb) I would like to take more mathematics in schoolc) Mathematics is more difficult for me than for many of

my classmatesd) I enjoy learning mathematicse) Sometimes, when I do not initially understand a new

topic in mathematics, I know that I will never really understand it

f) Mathematics is not one of my strengthsg) I learn things quickly in mathematics

Example Output from TIMMS

These distributions are significantly different with p < .001, but note that the large sample size, n = 8672, can effect this.

My Pilot StudyResearch Question: What are my Basic Arithmetic students

beliefs about the study of mathematics and their mathematical ability? Do their beliefs change by the end of the semester after experiencing an inquiry based approach to mathematics?

Population: My four Basic Arithmetic Classes from Glendale Community College (2 day and 2 night classes.)

Method: Administer a survey to students at the beginning and end of the semester on their math beliefs and self efficacy beliefs.

Data Analysis: Descriptive statistics of aggregate survey responses. Comparison analysis of pre and post surveys. Cluster analysis of students based on survey responses.

The Pilot SurveyThe following survey was given as a take home assignment due the 2nd

class period. I accepted late assignments for up to one month into the semester. The possible responses were :

Strongly Disagree Disagree Neutral Agree Strongly Agree 1 2 3 4 5

1. I enjoy mathematics.2. Mathematics is about following the rules.3. People are born good or bad at mathematics.4. Mathematics makes sense to me.5. I think I will be successful in this class.6. I am good at math.7. Mathematics is useful in my everyday life.8. I need to know mathematics to be successful in life.9. A strong understanding of mathematics is necessary for my career

choice.10.I would pursue a career that involves a lot of mathematics skills.

Response rate: 75 out of 109 students completed the survey.

Some Preliminary ResultsQuestion 1: I enjoy Mathematics

Question 2: Mathematics is about following the rules.

3. People are born good or bad at mathematics.

4. Mathematics makes sense to me.

5. I think I will be successful in this class.

6. I am good at math.

7. Mathematics is useful in my everyday life.

8. I need to know mathematics to be successful in life.

9. A strong understanding of mathematics is necessary for my career choice.

10. I would pursue a career that involves a lot of mathematics skills.

What next?

• Revise survey questions• Use significant results to peruse research for

other measurement instruments• Scale up in future; include random sample• Research is an iterative process!

Questions?Ideas?

Comments?

Thank you!

Contact Information

• April Strom [email protected]

• Mark [email protected]

• Ann [email protected]

• Amy [email protected]

mailto:[email protected]




Documents

Research in Mathematics Education: New Directions for Two-Year Colleges