ENTER THE “DOTS”
RICHARD HEWKO
also known as Cognitive Instruction in Mathematical Modeling or
CIMM
Math Anxieties
It has been estimated that 80% of the high school graduates in the U.S. have some form of math
anxiety (anonymous grant reviewer with NSF). With about 12% of North Americans taking
careers in math oriented fields (engineering, physics, accounting, chemistry, statistics, or
teaching math), that leaves about 8% that are okay with math. Closer examination by personal
interview has shown that some of this latter group have high degrees of math anxiety when
asked, even though they have always scored well in math (sometimes never lower than an A+ in
math). Further questions reveal that they perform reasonably well; they just do not understand
math. Most say they have made their way through the courses by memorizing algorithms and
where to use what. Let me repeat: competency at math is NOT sufficient to reduce math anxiety
in many individuals. Only understanding can relieve the anxiety for these individuals.
Math anxieties are many and varied; some are difficult to sort from performance anxieties, but
there are many other kinds. It may be a mild case of math avoidance, a severe case of fight or
flight syndrome, or even a display of real physical symptoms such as flushing, stuttering,
freezing, and other characteristics of extreme stress. Math anxiety is extremely subjective and
can profoundly affect self esteem. It really does not depend on performance in math generally,
but rather on “feelings of not understanding”. They just feel they do not “get” math.
In the U.S., where many studies have been made recently, most students (essentially 100%) in
grade four love math and often it is their favorite subject. By grade eight only 20% still like
math, and the rest (80%) hate it. The curriculum in grade 5 through 8 is essentially composed of
the same topics, but some grade level appropriate questions are asked. Yet each year more and
more students start to hate math. What is going wrong?
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Cognitive Science
Recent studies in cognitive science such as reported in Fundamentals of Human Memory and
Cognition by Ellis and Hunt (1983) state these findings. More material is remembered when
1) there are relationships between various ideas.
2) there is grouping of material into groups of similarity.
3) there is semantic (meaningful) orientation than when there are abstract symbols only.
4) distinctions are made between groups and there is a minimum of overlap, blurriness,
or ambiguity.
5) we are able to visualize what it is about. For example pictures are more easily
recognized than words.
6) the words, objects, or concepts are concrete rather than abstract.
7) there is a consistent context, or consistent set of categories.
8) we do something with or to the material than when we just hear about it.
More Recent Cognitive Science
Recently scientists have been able to locate exact sites of various types of brain activity.
Mathematics seems to need the cooperation of many different locations in the brain. For
example the site for subitism (the area of the brain that allows infants and us to recognize small
numbers of objects instantly) is in a completely different part of the brain than the place where
symbolic manipulation takes place. Numerical calculations and language facility (and math is a
language) are in different places. To teach math successfully, these areas of the brain must work
together. Also geometry and statistical thought are at different sites. Many teachers do not
realize that proper understanding and problem solving involve using many different parts of the
brain working sequentially and together.
Linguistic Analysis
The scholar-philosopher Ernst Cassirer has written that a language is learned in three stages:
mimicry (imitation), analogical reasoning (metaphorical), and symbolic manipulation (syntax).
That is, at first we repeat certain phrases in a new language and concentrate on their rhythm and
phonemes. Then we begin to learn the syntax and grammar by analogy (oh, “this” is like “that”
in English). And finally, the syntax and grammar become automatic, and we start to think in that
language. We must gain enough fluency in math to be able to think in it to be successful.
However, in the standard sequence of teaching math we immediately jump from mimicry (one
horse and another horse makes two horses) to symbolic representations such as 1 + 1 = 2. For
many students this jump is too big, and they eventually begin to fall behind, hating math.
Not only is the symbolic representation abstract, it is highly ambiguous. For example 2x3 = 6
may mean many things -- from two groups of three objects, to three groups of two objects, to an
area 2 m by 3 m, to 2 groups of three subgroups, etc.
Although these forms of multiplication are not so bad, when we begin to use division, what is
6/3 = 2? Is this partitive ( 6 dots divided by 3 dots per group gives 2 groups) or quotative (6 dots
divided by three groups is two dots per group), or geometric (6 square feet divided by 3 feet
gives a width of two feet)? Does it mean a rate (6 km in 3 hr is 2 km/hr), a fraction (6 pieces of
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pizza with 3 pieces making one pizza forms two pizzas), or a slope (a quotient relationship
showing motion in the y compared to motion in the x directions)? I have noticed over ten
completely different meanings for this simple operation (including ratio and proportion,
statistics, probability, partition into subgroups, etc).
Students who are otherwise very intelligent have trouble knowing which operations of division
we are modeling. Linguistically these ambiguities lead to difficulties in processing material
learned in math class and translating it into practical applications. That is, the student can only
operate in the symbolic region of the brain at best, and can not move among various
representations such as words, numbers, symbols, geometric shapes, etc. The student shuts
down.
History of Math
The Greeks taught that we should elevate math as a pure and undiluted subject that proposed
symbolic and abstract relationships rather than concrete ones. There should be a minimum of
contact between math and its applications. Math must be kept pure so that it remains ambiguous,
and may therefore be used to model many systems and ideas.
In the Renaissance, we rediscovered the Greek ideals, and a golden age of math proceeded into
the present age. But math has always been taught to the elite. There does seem to be a small
cadre of very bright people who are drawn to this like moths to a flame.
But what about our needs in the 21st century, where we need 20% -30% of the population to be
able to do the occupations that need to apply math? Where are the new applied mathematicians
going to come from? Do we need to teach math in different ways if we expect more people to be
able to learn it?
What is Math, Really?
Most of us have been raised thinking that mathematics is the symbolic manipulation involved in
algebraic equivalencies like 1 + 2 = 3. Mathematics actually is the study of the relationships
involved in reality. There are many mathematical languages that can express these relationships
well, such as algebra, geometry, graphing, topology, statistics, etc, each with its own special
syntax and grammar, and each with its own advantages and disadvantages.
Most of us understand math only as this syntax and grammar (especially of algebra) and miss the
story. Although algebra is a very powerful (expressive) language, it is not intuitive for most
people. That is, it is very symbolic and abstract with little real meaning. We need a way to build
a semantic step (filled with meaning) between the story we want to tell and the abstract
symbolism of algebra – an analogical reasoning step.
Enter the Dots
Thus my colleague Rob MacDuff decided to build a new language that could explain the story in
a way that was easier to understand. Together we decided to build course materials that are
visual, meaningful, full of relationships, concrete, and that use contexts involving systems and
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groups. We also wanted student skills learned in the math course to be easily transferred into
science contexts.
Instead of 1 + 2 = 3 we use
.
This new language using dots is highly intuitive and meaningful. Then in a series of small steps
we take this very simple but flexible language so that it looks very much like algebra. In other
words, our method helps students to move from simple subitism and counting, from icons to
labels and symbols, to variables by introducing intermediate steps of analogical reasoning.
Labels are simple names for parts --- they can not be manipulated or operated on like symbols.
The courses “Cognition Ignition” and “Catch a Wave” are results of that goal.
We also explicitly help students by giving them or guiding them to find suitable conceptual tools
(tools to think with) and representational tools (tools to help them visualize relationships).
A Brief Bibliography
Math Anxiety
1. Overcoming Math Anxiety, Sheila Tobias, W.W. Norton & Co, 1994, New York
2. They’re Not Dumb, They’re Different, Sheila Tobias, Research Corporation, 1990,
Tucson, AZ
3. Succeed with Math: Every Student’s Guide, Sheila Tobias
4. Mind Over Math, Kogelman and Warren, McGraw Hill, 1978, New York
5. Conquering Math Anxiety, Cynthia Arem, Brooks/Cole, 2003, Canada
Cognitive Science
6. Fundamentals of Human Memory and Cognition, Ellis and Hunt, W.C. Brown, 1983,
Dubuque, Iowa
7. Women’s Ways of Knowing, Belenky, Clinchy, Godberger, Tarule; Basic Books, 1986,
New York
Philosophy of Mathematics
8. Where mathematics Comes From, Lakoff and Nunez, Basic Books, 2000, New York
9. Mathematics and the Physical World, Morris Kline, Dover Publications, 1981, New York
10. Philosophy of Symbolic Forms, Vol. 1, Ernst Cassirer. Yale University Press, 1955, New
Haven, CT.
2007
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