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Mathland’s Best Route via Arithmetic Canyon into Algebra Valley: A Virtual HikeGuide: Dr. June GastónBorough of Manhattan Community College, CUNYAMATYC Annual Conference, November 12, 2009

This virtual journey will include a survey of perils that must be overcome with resourcefulness in making mandatory algebraic connections. Recent research in teaching and learning that includes effective habits and strategies supporting Mathland survival will be discussed. Use of current and rising technologies will also be explored. This is a 21st century research-based exploration of k-14 math ed issues.

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Why do this exploration?

RESEARCH: American Mathematical Association of Two-Year Colleges

Professionals should continue to search for strategies toaddress various mathematics education issues such as:

• choice of appropriate mathematics content andeffective instructional strategies

• the use of technology in instruction

• teacher preparation

• professional development for full-time andadjunct faculty, and instructional support staff.

Source: American Mathematical Association of Two-Year Colleges (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College, p.16-17.

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What mathematics must 21st century k-12 students know to be prepared for college-level mathematics?

RESEARCH: National Mathematics Advisory Panel

Benchmarks for the Critical Foundations of Algebra Major Topics of School Algebra

Fluency With Whole Numbers1) By the end of Grade 3, students should be proficient with the addition and

subtraction of whole numbers.2) By the end of Grade 5, students should be proficient with multiplication and

division of whole numbers.

Fluency With Fractions1) By the end of Grade 4, students should be able to identify and represent

fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.

2) By the end of Grade 5, students should be proficient with comparing fractions and decimals and common percent, and with the addition and subtraction of fractions and decimals.

3) By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimals.

4) By the end of Grade 6, students should be proficient with all operations involving positive and negative integers.

5) By the end of Grade 7, students should be proficient with all operations involving positive and negative fractions.

6) By the end of Grade 7, students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality.

Geometry and Measurement1) By the end of Grade 5, students should be able to solve problems involving

perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).

2) By the end of Grade 6, students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three dimensional

shapes and solve problems involving surface area and volume.3) By the end of Grade 7, students should be familiar with the relationship

between similar triangles and the concept of the slope of a line.

Symbols and Expressions1) Polynomial expressions2) Rational expressions3) Arithmetic and finite geometric series

Linear Equations1) Real numbers as points on the number line2) Linear equations and their graphs3) Solving problems with linear equations4) Linear inequalities and their graphs5) Graphing and solving systems of simultaneous linear equations

Quadratic Equations1) Factors and factoring of quadratic polynomials with integer coefficients2) Completing the square in quadratic expressions3) Quadratic formula and factoring of general quadratic polynomials4) Using the quadratic formula to solve equations

Functions1) Linear functions2) Quadratic functions—word problems involving quadratic functions3) Graphs of quadratic functions and completing the square4) Polynomial functions (including graphs of basic functions)5) Simple nonlinear functions (e.g., square and cube root functions;

absolute value; rational functions; step functions)6) Rational exponents, radical expressions, and exponential functions7) Logarithmic functions8) Trigonometric functions9) Fitting simple mathematical models to data

Algebra of Polynomials1) Roots and factorization of polynomials2) Complex numbers and operations3) Fundamental theorem of algebra4) Binomial coefficients (and Pascal’s Triangle)5) Mathematical induction and the binomial theorem

Combinatorics and Finite ProbabilityCombinations and permutations, as applications of the binomial theorem and Pascal’s Triangle

Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p.15-16, 20.


• To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem solving skills.

• Computational proficiency with whole number operations is dependent on practice to develop automatic recall of addition and related subtraction facts, and of multiplication and related division facts. It also requires fluency with the standard algorithms for addition, subtraction, multiplication, and division – and a solid understanding of core concepts, such as the commutative, associative and distributive properties.

• The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percent, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.

Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p. xix, p.18-19.

RESEARCH: Commission on Mathematics and Science Education

Instruction should emphasize inquiry, relevance, and a multilayeredvision of proficiency as indicated by the National Research Council in Adding It Up and the National Mathematics Advisory Panel inFoundations for Success.

Proficiencies are:• Conceptual understanding

(comprehension of mathematical concepts, operations, and relations)

• Procedural fluency (skills in carrying out procedures flexibly, fluently, and appropriately)

• Strategic competence (ability to formulate, represent, and solve mathematical problems)

• Adaptive reasoning (capacity for logical thought, reflection, explanation, and justification)

• Productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy)

Source: Carnegie Corporation of New York, Institute for Advanced Study, Commission of Mathematics and Science Education (2009).The Opportunity Equation, p.24-25.

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How should 21st century k-12 mathematics be taught?

According to research, what approaches and strategies work? What are the best practices?

What strategies can teachers use to help elementary school students achieve computational fluency?

What instructional strategies can be used to keep adolescents motivated to learn abstract pre-algebraic and algebraic concepts?

Are k-12 teachers prepared to teach these concepts?

RESEARCH: How the Brain Learns K-12 Math

Title: How the Brain Learns Mathematics (Paperback)Author: David A. Sousa Publisher: Corwin Press, CA Year: 2008ISBN-10: 1412953065 ISBN-13: 978-1412953061

Product DescriptionDiscusses cognitive mechanisms for learning mathematics and factors that contribute to mathematics difficulties,examines how the brain develops an understanding of number relationships, and connects to NCTM curriculum focalpoints.

http://www.corwinpress.com/booksProdDesc.nav?prodId=Book230967

RESEARCH: How the Brain Learns K-12 Math

Criteria for Long-Term Storage of Information

In everyday life, working memory quickly and/or permanently

stores information that has:• survival value and/or• an emotional connection

In the classroom, working memory saves information if it:• makes sense when connected to past experiences and prior

knowledge• has meaning or relevance for the learner

Sousa, David A. How the Brain Learns Mathematics, Thousand Oaks, CA: Corwin Press, 2008, p. 54-56.


Instructional Practices

• Team Assisted Individualization (TAI) improves students’ computation skills, but not conceptual understanding and problem solving skills. This pedagogical strategy involves heterogeneous groups of students helping each other, individualized problems based on student diagnostic test results, specific teacher guidance, and rewards based on both group and individual performance.

• Teachers’ regular use of formative assessment improves their students’ learning, especially if teachers use the assessment to design and to individualize instruction.

• Mathematically gifted and motivated students appear to be able to learn mathematics faster than students proceeding through the curriculum at a normal pace, with no harm to their learning, and should be allowed to do so.

Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p. p xxii-xxv .


Instructional Practices involving Computation and Calculators

• Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computation. Results are consistent for students with learning disabilities and students who perform in the lowest third of a typical class.

• The Panel cautions that to the degree that calculators impede the development of automaticity, fluency in computation will be adversely affected. The Panel recommends that high-quality research on particular uses of calculators be pursued, including both their short- and long-term effects on computation, problem solving, and conceptual understanding.

• Calculators should not be used on test items designed to assess computational facility.

Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p. p xxii-xxv .


Instructional Practices using Computer Assisted Instruction (CAI)

• CAI drill and practice, is a useful tool in developing students’ automaticity (i.e., fast, accurate, and effortless performance on computation), freeing working memory so that attention can be directed to more complex tasks.

• CAI tutorials are useful in introducing and teaching specific subject-matter content to specific populations. Additional research is needed to identify which goals and which populations are served well by tutorials, and features of effective tutorials and of their classroom implementation.

Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p.51 .


Teacher education programs and licensure tests for• early childhood and early childhood special education teachers

should fully address the topics in the Critical Foundations of Algebra, as well as the concepts and skills leading to them

• elementary and elementary special education teachers should fully address all topics in the Critical Foundations of Algebra and topics typically covered in introductory Algebra

• middle school and middle school special education teachers should fully address all of the topics in the Critical Foundations of Algebra and all of the Major Topics of School Algebra.

...teachers must know in detail and from a more advanced perspective the mathematical content they are responsible for teaching and the connections of that content to other important mathematics, both prior to and beyond the level they are assigned to teach.

Source: National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education, p. 19, p.38.

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What must 21st century developmental math students know to be prepared for college-level mathematics?

RESEARCH: Report of the 100% Math InitiativeRecommended Curriculum and Instructional Objectives for Developmental Mathematics Courses

Foundations of Mathematics Foundations of Algebra I Foundations of Algebra II

1. Read, write and round whole numbers and decimals. 2. Perform basic operations on whole numbers. 3. Evaluate expressions containing whole number

exponents and square roots. 4. Apply the order of operations to whole number

expressions. 5. Factor whole numbers, define prime numbers and use

prime factorization to determine LCM’s and GCD’s.

6. Perform basic operations on rational numbers (fractions).

7. Perform the basic operations on decimals. 8. Express the meaning of percent as it relates to

fractions and decimals. 9. Solve basic verbal applications of percents.10. Use the basic tools of arithmetic to solve applications

problems involving whole numbers, fractions and decimals.

11. Apply the concepts of ratio and proportion.12. Apply the concept of percent to applications such as

percent increase/decrease, discount, interest, commission and sales tax.13. Write and solve equations for applications of percents.14. Apply the concepts of perimeter, area and volume to

basic geometric figures.15. Use the U.S. and Metric systems in applications of measurements of length, weight and mass and

volume.16. Perform conversions between the U.S. and Metric

systems and between units in the same system.17. Express the meaning of variable and use symbols to

represent unknowns.18. Solve single step linear equations.19. Evaluate the mean of a distribution of numbers and

interpret simple bar graphs and pie charts.

1. Develop a conceptual understanding of different uses of variables.

2. Understand and apply the concepts of expression and equation.

3. Operate on expressions, and recognize and generate equivalent forms.

4. Know the laws of exponents for multiplication, division, and taking of powers.

5. Know the meaning of a negative exponent. 6. Know the distributive property and be able to add like terms. 7. Be able to multiply binomials. 8. Factor out a monomial factor. 9. Factor simple trinomials.10. Factor difference of two squares.11. Understand and apply absolute value in terms of distance and

square roots.12. Understand and apply the principles for solving linear

equations and inequalities.13. Understand and apply the concept of slope as rate of change.14. Understand and apply the Pythagorean Theorem.15. Apply algebraic methods to solve a variety of real-world and

mathematical problems.16. Reduce simple algebraic fractions.17. Solve ratio and proportion problems.18. Use tables and graphs as tools to interpret expressions,

equations, and inequalities.19. Create and identify a linear relationship based on patterns in

collected data.20. Use symbolic algebra to represent situations and to solve

problems, especially those that involve linear relationships.

21. Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

22. Generalize patterns and relationships using tables, verbal rules, equations, and graphs.

23. Perform basic operations on simple algebraic fractions.24. Translate among numerical, symbolic, and graphical

representations of relationships.25. Analyze linear relationships by investigating rates of change,

intercepts, and zeros.26. Model problem situations using systems of equations and

solve these systems.27. Understand and apply the concept of independent and

dependent variable in real world contexts.

1. Rewrite expressions by understanding and applying the concepts of combining like terms, the distributive property, and factoring.

2. Know the difference between simplifying an expression and solving an equation.

3. Understand and apply literal equations to real world problems. 4. Understand and apply the Pythagorean Theorem and

properties of similar triangles. 5. Apply algebraic methods to solve a variety of real world and

mathematical problems. 6. Perform operations on simple rational expressions and

equations. 7. Explore relationships between symbolic expressions and

graphs of lines, paying particular attention to the meaning of intercept and slope.

8. Find the slope of a line from a graph or two points. 9. Find an equation of a line given slope and a point or two

points.10. Understand and apply systems of equations in two variables

including their graphs.11. Understand and apply direct and inverse proportional

reasoning.

12. Write equivalent forms of equations and inequalities and solve them with fluency --mentally, with paper and pencil, and using technology.

13. Understand relations and functions and use various representations for them.

14. Understand and apply the concept of domains and ranges in real world contexts.

15. Find zeros of quadratic functions by factoring techniques, tables, and graphs.

16. Model and solve contextualized problems using various representations, such as graphs tables, and equations.

17. Analyze the effects of parameter changes on the graphs of functions.

18. Analyze linear and nonlinear functions of one variable by investigating rates of change, intercepts, zeros, and local and global behavior.

19. Identify linear and nonlinear functions when represented by tables, graphs, and equations.

20. Draw reasonable conclusions about a problem situation being modeled.

21. Model real-world phenomena with a variety of nonlinear functions such as exponential (growth and decay) and quadratic.

Source: “Building a Foundation for Student Success in Developmental Math,” Massachusetts Community College Executive Office. (2006).

100% Math Initiative: Building a foundation for student success in developmental math. Boston: Author, p. 40-41.

RESEARCH: American Mathematical Association of Two-Year Colleges

• The desired student outcomes for developmental mathematics courses should be developed in cooperation with the partner disciplines. The content for these courses also should address mathematics anxiety, develop study and workplace skills, promote basic quantitative literacy, and create active problem solvers.

• Topics in algebra, geometry, statistics, problem solving and experience using technology should be integrated throughout developmental courses. However, students should still be expected to perform single digit arithmetic, without the use of a calculator.

Source: American Mathematical Association of Two-Year Colleges (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College, p.41-42 .

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What strategies can instructors use to help developmental students achieve computational proficiency?

How should developmental mathematics be taught? According to research, what approaches and strategies work? What are the best practices?

What causes the high failure rates? What strategies can help curb them?

RESEARCH: 100% Math Initiative

Instructors should:• vary their classroom methodology to actively engage

students in the learning process. (Most developmental education students have an attention span of approximately 15 minutes. Adjust the lesson appropriately. For example, switch from lecture to group discussion or student demonstration of a concept or procedure.)

• employ a broad range of pedagogical approaches that both match the range of material and actively engage students. Use the “rule of four” and present information four ways: graphically, numerically, symbolically, and verbally.

• orient their presentation to the real world application of the material.

• be aware of different learning styles among their students and adjust their instructional approach accordingly.

• identify and implement strategies for assisting students with study skills and integrate these skills directly into course and classroom activities.

Source: “Building a Foundation for Student Success in Developmental Math,” Massachusetts Community College Executive Office. (2006). 100% Math Initiative: Building a foundation for student success in developmental math. Boston: Author, p. 31-33

RESEARCH: 100% Math Initiative

Instructors should also:• clarify specific competency-based expectations that a

developmental math student must meet. These can include student portfolios and/or departmental final exams, which measure student proficiency.

• convey the value of homework (emphasize it, offer incentives for completed homework, check it regularly, and provide feedback to students).

• attend professional development workshops that address strategies for accommodating different learning styles, integrating study skills into instruction, using technology, creatively engaging students, and advising students.

• regularly collaborate with specialists and support staff who work with learning-disabled students.

• have access to a handbook for developmental mathematics instructors that includes materials concerning administrative and logistical issues, curriculum and syllabus information, recommended instructional approaches, and an inventory of academic support resources.

• collaboratively select textbooks (in print or online) that include varied instructional approaches, are contextually rich, incorporate numerous applications, are activity-based, hands-on and most effectively meet student needs.

Source: “Building a Foundation for Student Success in Developmental Math,” Massachusetts Community College Executive Office. (2006). 100% Math Initiative: Building a foundation for student success in developmental math. Boston: Author, p. 31-33

RESEARCH: Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices

Instructors should use technology:• to facilitate acceleration of coursework • to transform developmental courses by offering traditional

content in an intensified time frame• to target specific skills gaps

Ex: Community College of Denver’s FastStart@CCD allows students to complete two levels of remedial math within one semester. Though acceleration is a key component, the holistic program draws effectiveness from a mix that includes student support, a learning community format, interactive teaching, and career exploration delivered in a first-year experience class. Ex: Ivy Tech Community College, Evansville, replicated the CCD accelerated model in fall 2007, with similar initial outcomes…efforts have increased to identify and remediate students who may test close to the “cut-off score” as a way to reduce unnecessary time and money spent in developmental coursework. Self-paced, competency-based, and modular courses have proven successful for some students, as have “refresher courses” for returning students.

• for instruction Ex: Instructional software (e.g., tutorial) programs have been in use for many years in community college classrooms, yet less than 40% of two-year colleges reported use of a Learning Management System or Course Management System in 2007.

• to supplement rather than replace traditional delivery methods because studies show no clear consensus on the effectiveness of technology-based delivery methods.

Source: Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices , Rhonda M. Epper and Elaine DeLott Baker, January 2009, p.8-10,http://www.gatesfoundation.org/learning/Documents/technology-solutions-for-developmental-math-jan-2009.pdf

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What current technology trends are helping to bridge the gap between arithmetic and algebra?

RESEARCH: Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices

General Trends: Open Education Resources, Digital Game-Based Learning, Social Networking, Virtual Worlds

• Use of Web 2.0 technologies in developmental mathAdvocates of digital game-based learning believe math drill-and-practice games could be effective for higher math levels if appropriately designed. “Net Generation” attraction to games and research on effectiveness of Digital Game-Based Learning (DGBL) may prove such math games helpful to developmental math students.

• Reconciling “Net Generation” learner profiles with profiles of the developmental student population Penn State University researchers suggest that the game generation prefers doing many things simultaneously by using various paths toward the same goal, is less likely to become frustrated when facing a new situation, prefers being active, learning by trial and error, and figuring things out by themselves rather than by reading or listening.

• Use of open coursewareThe National Repository of Courseware (NROC) maintains and expands a courseware library. The content is distributed free-of-charge to students and teachers at public websites including www.HippoCampus.org.

• Web-base learning initiativesThe Open Learning Initiative (OLI) of Carnegie Mellon University uses intelligent tutoring systems, virtual laboratories, simulations, and frequent opportunities for assessment and feedback. OLI builds college level web-based courses that are intended to provide effective instruction that promotes learning.

Source: Technology Solutions for Developmental Math: An Overview of Current and Emerging Practices , Rhonda M. Epper and Elaine DeLott Baker, January 2009, p.15-16,

http://www.gatesfoundation.org/learning/Documents/technology-solutions-for-developmental-math-jan-2009.pdf

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What other issues need to be explored?

Ongoing explorations

• How can high school coursework and testing that permits calculator use, and college developmental coursework and testing that prohibits calculator use, be brought into clearer alignment?Many students in high school experience mathematics in context, using technology. A majority of high school exit examinations allow the use of graphing calculators. In contrast, some higher education mathematics placement exams test only basic arithmetic and algebraic computation without technology. These differences in the use of technology need to be addressed. Collaborative efforts to implement standards-based mathematics can be an initial step in minimizing the need for remediation in postsecondary mathematics education, address the critical need for students to complete algebra, and help to ease the transition from high school to college.

American Mathematical Association of Two-Year Colleges (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two Years of College, p.73-74.

Ongoing explorations

• How can high school testing be brought into better alignment with college testing so that high school students will be better prepared for college entry-level assessment exams in math?

70% of students entering CUNY’s community colleges failed their placement exams in reading, writing, or math and were required to take remedial courses.

Grading of NYS high school mathematics Regents examinations…about the new Integrated Algebra Exam administered for the first time in June 2008…the NYSED announced today that a raw score of 30 points out of 87 (just 34.5 percent) was all that students were required to earn to achieve a passing grade of 65. In the State’s headlong race to lead American students to the bottom rung of the industrialized world’s academic ladder, we’ve proudly declared a 35 to be our 65. . . . What now passes in New York State for high school level competency, represented by the new Integrated Algebra I Regents Exam, is by any measure an international laughingstock, an exam that a typical sixth grader in China could ace with hardly a second thought. (“NYS Algebra Regents” 2008)

Annenberg Institute for School Reform at Brown University (with John Garvey), Education Policy for Action Series, Education Challenges Facing New York City, (2009). Are New York City’s Public Schools Preparing Students for Success in College? p. vii, p.20.

Ongoing explorations• What can be done about poor k-14 attendance in New York

City? How does poor attendance impact math achievement? Are socioeconomic factors an issue?

Chronic Absence Starts Young and Grows With AgeCitywide, more than 20% of elementary school pupils missed more than a month of school in 2007-08; nearly 40% of high school students missed that much. Moreover, the rates of severe chronic absence increase with the grade level. While 4.5% of elementary pupils missed 38 days (nearly two months), 24% of high school students missed that much.

Why Attendance Matters – early chronic absenteeism sets the stage for failure....recent research by the National Center for Children in Poverty at Columbia University shows that children who have poor attendance in kindergarten tend to do poorly in first grade, and that children with a history of poor attendance in the early elementary grades have lower levels of academic achievement throughout their school years...Schools with high levels of absenteeism tend to have slower-paced instruction overall, harming the achievement levels of strong students as well as those who struggle, a report by the Open Society Institute suggests...good attendance is a prerequisite for academic achievement. However, under No Child Left Behind, schools are primarily judged on their students’ performance on standardized tests in math and reading—not their attendance rates.Nauer, Kim, White, Andrew and Yerneni, Rajeev. Strengthening Schools by Strengthening Families, Community Strategies to Reverse Chronic Absenteeism in the Early Grades and Improve Supports for Children and Families, Center for New York City Affairs, The New School, Milano the new school for Management and Urban Policy, October 2008, p.14, p.20.

Next explorations

• What gets college students to math class and gets them there on time? Rewards for attendance and punctuality? Penalties for absence and lateness? Quiz at the beginning of each session? Accelerated coursework? Team or group activities? Individualized instructional approaches?

An effective college online early intervention system?

The Community College of Allegheny County (CCAC) has Online Early Intervention enabling faculty members to refer struggling students for assistance via a secure website. The student support staff members who receive these referrals attempt to contact the students in order to connect them to college resources.

The goals of the CCAC Online Early Intervention System are to: - Reach struggling students as early as possible

- Connect students to college resources to resolve their issues- Help students to maintain or improve their GPA- Improve student retention rates

(CCAC defines struggling students as those who have poor attendance, are continually late for class or leave early, don’t take notes, are inattentive, don’t participate, or are unlikely to be successful in a given course.)

CCAC Online Early Intervention, http://www.ccac.edu/default.aspx?id=151239

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Questions or interest in collaborating on k-14 mathematics education research?

Contact:Dr. June GastónBMCC-CUNYMathematics DepartmentRoom [email protected]: (212) 220-1342

For information concerning the fractal landscapes in this presentation, use the link at the right corner under each image