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 phobia^{1}. 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? Frege^{2}, Russell^{3},
Cassirer^{4, 5}, Kline^{6}, and Hart^{7} 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 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. (Frege^{8}, 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.

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 approach^{14} 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__

__ __

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.

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. h__ttp://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 Phys*ics 63,
606-619, 1995, __http://modeling.asu.edu/R&E/ModelingMethod-Physics_1995.pdf__.

May
2008

· 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}

·· 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.