The Math Problem
Robert MacDuff, Ph.D.
President, InfoDynamics Applications Ltd.
macduff@mac.com 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
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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
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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.
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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.
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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), 487506, 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 visuospatial 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
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