President, Infodynamics Applications Ltd

The Math Problem Robert MacDuff, Ph.D. President, InfoDynamics Applications Ltd. [email protected] 480-205-6135 One thing is perfectly clear to all concerned: there is a math problem. It is not a local problem; it is national in scope. It is not something new. Evidence suggests that this problem has been with us for a long time. There are various ways in which to classify the math problem. Many educators focus on the dismal performance of American students on international tests. Others see it in terms of the small number of engineering students at local universities, where the demand for trained engineers is constantly increasing. The problem is apparent from the simple fact that 80% or more of the students, by the time they leave school, suffer from math phobia1. This includes many who go on to teach in elementary schools. These individuals are curriculum casualties, collateral damage of a curriculum that only successfully educates a select few. An important point to emphasize is that the problem is not an intelligence problem, nor is it genetic. The immensity of this problem is so huge it is hard to comprehend. Consider the problem in this light: “If the goal of mathematics education were to engender math phobia, would choosing our current mathematics programs be a good choice?” Even successful students and teachers perceive mathematics as the execution of algorithms where symbols have little or no meaning. The public is rightly concerned that their investment in math education, as measured by testing, is not producing satisfactory gains. If trying harder could have solved the math problem, we would have started to see significant results by now. It’s time to try something different. Smarter, or Harder, Longer and Louder

To solve a problem, the first requirement is to understand what the problem is. The reason for lack of progress is that the math problem is poorly defined. Most mathematics education researchers have targeted teacher knowledge, both content and pedagogical, as the problem. Others assume that it is caused by a lack of parental support, by effects of change or breakdown in society or unwillingness of students to learn. All math approaches point to one or more of these issues as the crux of the problem.

An alternative possible source of the math problem, simply stated, is that it lies in the mathematics itself. Could mathematics itself be flawed? Frege2, Russell3, Cassirer4, 5, Kline6, and Hart7 and others have pointed out difficulties in the foundations of mathematics.

If the teachers, despite their considerable exposure to current mathematical content, cannot master it, then we must consider the possibility that the problem is with the content itself, rather than with the teachers. Certainly, if teachers have not been able to

1 master this content, we cannot expect that any redesign of teaching methods, re-ordering of topic sequences, raising of standards or programs of high-stakes testing will result in their students being able to do so!

Our research suggests that the problem lies in the difficulties imposed on the students by asking them to learn a flawed mathematics content. Their difficulties in turn, lead to a drastic underestimation of their capabilities and thence to shortchanging them in their education. The solution therefore requires a profound re-thinking of the foundations and assumptions of mathematics itself. An Alternative Approach

The system of ideas that is emerging from this process can be described as a “mathematics of quantity”, which takes as its starting point the consideration of collections of objects. This approach separates the learning of mathematics into four major subsections: conceptual understanding (grouping structure), symbol construction (algorithmic manipulations), problem solving and mathematical reasoning. As defined here, conceptual understanding and mathematical reasoning are not to be found in standard approaches to mathematics education. And yet these are the critical components. Conceptual understanding lies in both the relationships between objects and groups of objects and the objects themselves. In other words, the mathematics involves both numbers and objects. (Frege8, in the latter years of his life, wrote on February 3, 1924: “My efforts to become clear about what is meant by number have resulted in failure.”) Number, as a component of the mathematics of quantity, emerges as the symbolization of a quotient relationship between groups of objects. This approach is the only one that defines number directly rather than tacitly through a set of axioms. The mathematics is rooted in observations of collections of objects and results of manipulation of those collections. Mathematics of quantity is the science of objects without internal structure. In this sense it is similar to Euclidean geometry, which is the science of objects with internal structure. Dots act as diagrammatic objects and play a role similar to that played by lines and points in geometry. Just as geometry takes as its starting point a set of postulates, or “self-evident” statements about the nature of space, our approach takes as postulates, statements about the nature of collections of objects. Foremost among those is the principle – or postulate - of invariance of quantity under regrouping.

Fractions, ratios, proportions, percents and the rational number system are developed out of a process of reasoning about relationships between quantities. Students learn that they have freedom to take any quantity or segment of a line as their unit and other quantities or segments as multiples of it. In the second course of the program, real numbers are developed out of reasoning about line segments, much as was done by the ancient Greeks.

 It is interesting in this connection that while Euclid talks about number as an abstraction, his discussion of ratio and proportion in Book V show him reasoning in terms of quantity relationships, much as we do. That this is so is made more apparent by the stunning diagrams in Byrne’s 1847 translation.9

2 Changing the nature of mathematics is just the first step; the next is to base curriculum development on the latest results of investigations in neuroscience that have uncovered evidence that different structures in the brain handle different aspects of mathematical thinking.10, 11 For example, it has been shown that the numeral representation “10”, the word “ten”, and ten dots () activate very different parts of the brain.12 Surprisingly enough, evidence strongly supports the conclusion that the brain is already hardwired for numerosity, and possibly for proportional reasoning. However, most current approaches to mathematics education fail to stimulate areas of the brain that may be responsible for the ability to visualize and comprehend mathematical statements. The Cognitive Instruction in Mathematical Modeling (CIMM) curriculum is designed to coordinate activation of parts of the brain, in particular that part which is hardwired for numerosity. It is this coordinated activation, the connection of the physical to the symbolic, which results in conceptual understanding: the sense of knowing the meaning of what one is doing. The next step is to recognize that mathematics education has a larger context than just the study of quantitative or numerical relations. It needs to develop students’ power to think and reason about the world they live in, and to foster their ability to understand and use the scientific method that underlies modern technology and much of modern culture.

Cognitive Instruction in Mathematical Modeling (CIMM) is an approach that opens the door to a deeper understanding of this larger context. This broader context includes: 1/ Linguistics - learning to express ideas using multiple representational systems, including the written word to articulate the structure of diagrammatically represented contexts. Students are given representational tools to express their thinking in the form of models. 2/ Mathematics and science integration - Integrating mathematics and science is a necessary component in development of a deeper understanding. A deep understanding of science is difficult without math, and conversely, a deep understanding of math is difficult without science. 3/ Mathematical modeling - using mathematics to construct models. The Modeling Instruction program at Arizona State University (ASU)13 shows that the model-building approach14 rapidly develops students’ ability to think about and analyze real-world situations.

The last step involves changes in pedagogy necessary to instill understanding in the student. In the vast majority of mathematics classrooms, the teacher is engaged in doing mathematics 90% or more of the time, the student 10% or less. The modeling pedagogy developed at ASU for creating an active engagement environment has greatly increased the student’s percentage of time on task.

The CIMM Program

 I find this type of instruction painful and I often wonder as to how students endure it. It appears to be happening for control reasons rather than for learning.

3 The essential features of our program, Cognitive Instruction in Mathematical Modeling (CIMM), include: 1/ Training. The use of these materials is not obvious, and even the best-educated math teachers need additional training. For those teachers that need it, this training also raises their understanding of math to a grade eight level. 2/ Coaching. Trainers (often other teachers who have been using this program) work with teachers in a classroom environment on implementing the program. 3/ CIMM pedagogy, where the focus of instruction is on constructing mathematical models of different systems. 4/ Neuroscience. Teachers learn about the structure of the brain and about learning as coordinated activation of different parts of the brain. Teachers learn to lead students in constructing models using four different representational systems, each of which activates different areas of the brain. 5/ Integration of five intellectual domains of knowledge: linguistics, psychology, mathematics, philosophy and cognitive science. 6/ Engineered materials that develop conceptual understanding as well as the means of constructing symbols to represent mathematical thinking.

CIMM has been pilot-tested extensively by trained teachers. This was done first with ninth grade remedial math students in an inner-city school, and is now being developed and tested with fifth and seventh-grade mainstream classes. We are finding that by changing the approach to mathematics, we can enable teachers, and subsequently their students, to master its content. The success of our program has proved to be dramatic, so much so that Paradise Valley Unified School District, third largest in the state of Arizona, has chosen to implement CIMM in their elementary schools as a means of eliminating the need for remedial math programs at their high schools. Their data center has been tasked with tracking and analyzing outcomes of the implementation. The teachers who have gone through the introductory programs we offer, which are typically of several weeks’ duration, generally emerge enthusiastic about the program, and often undergo a dramatic shift in their attitudes about mathematics and in their confidence in being able to understand and use it. It is the enthusiastic support and participation of the teachers who are using CIMM that has led to its rapid acceptance and growth. Many of these teachers in turn, have contributed to its further development.

Summary

Identifying mathematics itself as the primary cause of the math problem was the starting point in development of CIMM. Rethinking the nature of the problem to include the mathematics itself opened the way to apply recent research in cognitive science as a guide for curriculum change. The curriculum changes made it possible to utilize the modeling pedagogy developed at ASU. Only in the light of these changes is it possible to see that mathematics education had reduced itself to algorithmic manipulations of symbols.

4 The CIMM program, while still under construction, is transforming mathematics education to include conceptual understanding (grouping structure), symbol construction (algorithmic manipulations), problem solving and mathematical reasoning. The end result is that all of the notoriously difficult topics (fractions, negative numbers, place value, exponents, etc.) become trivial. Our solution is simple, as it bridges the gap between the concrete mathematics of elementary school and the abstract symbolism of high school.

References: 1/ Arem, Cynthia, Conquering Math Anxiety, Brooks/Cole, Canada, 2003. 2/ Frege, Gottlob, The Foundations of Arithmetic, Philosophy Library Inc., NY, 1953. 3/ Russell, Bertrand, The Principles of Mathematics, Bradford & Dickens, London 1951. 4/ Cassirer, Ernst, The Problem of Knowledge, Yale University Press, New Haven, 1950. 5/ Cassirer, Ernst, Substance and Function and Einstein’s Theory of Relativity, The Open Court Publishing Company, Chicago, 1923. 6/ Kline, Morris, Mathematics, The Loss of Certainty, Oxford University Press, NY, 1980. 7/ Hart, George W., Multidimensional Analysis; Algebras and Systems for Science and Engineering, Springer-Verlag, New York, 1995. 8/ Frege, Gottlob, Posthumous Writings, The University of Chicago Press, 1979. 9/ Byrne, Oliver, The First Six Books of the Elements of Euclid, in which Coloured Diagrams and Symbols are Used Instead of Letters for the Greater Ease of Learners, William Pickering, London, 1847. http://www.math.ubc.ca/~cass/euclid/book5/book5.html 10/ Dehaene, Stanislas; Piazza, Manuela; Philippe, Pinel; and Cohen, Laurent; “Three parietal circuits for number processing”, Cognitive Neuropsychology 20 (3,4,5,6), 487- 506, 2003. 11/ Ansari, Daniel; and Donlan, Chris; Thomas, Michael S.C.; Ewing, Sandra A.; Peen, Tiffany; and Karmiloff-Smith, Annette, “What makes counting count? Verbal and visuo- spatial contributions to typical and atypical number development”, Journal of Experimental Child Psychology 85, 50-62, 2003. 12/ Ansari, Daniel “Does the Parietal Cortex Distinguish between ‘10,’ ‘Ten,’ and Ten Dots?”, Neuron 53, (20), 165-167, 2007. 13/ Hestenes, David and Jackson, Jane, Findings of the ASU Summer Graduate Program for Physics Teachers (2002-2006), report submitted to the NSF, http://modeling.asu.edu/R&E/Findings-ASUgradPrg0206.pdf, 2006. 14/ Wells, Malcolm; Hestenes, David; and Swackhamer, Gregg, “A Modeling Method for High School Physics Instruction”, American Journal of Physics 63, 606-619, 1995, http://modeling.asu.edu/R&E/ModelingMethod-Physics_1995.pdf.

May 2008