Sam Otten
EDUC 782
Fall 2009
Word Count: 1086
Essay on Mathematical Understanding
Why Understanding?
The National Council of Teachers of Mathematics (NCTM) developed their
Principles and Standards (2000) on a foundation of mathematical learning with
understanding. What justifies such a foundation? Instead of relying solely on
philosophical arguments that correspond with a conception of mathematics as a sensemaking discipline, NCTM cited a small collection, and in so doing alluded to a larger
research base (Kilpatrick, Martin, & Schifter, 2003), of empirical studies pointing to the
value of learning mathematics with understanding. For example, Bransford, Brown, and
Cocking (1999), in their tome on learning, found that understanding is a key factor with
respect to learners’ proficiency in complex domains. In particular, learners who
memorize facts or procedures without the associated understandings may not know which
facts are relevant under certain circumstances or what procedure is appropriate in certain
situations. In this way, learning with understanding is valuable not only for its own sake
but in support of the concurrent learning of facts and procedures.
Learning with understanding, according to NCTM, also supports future learning
of mathematical skills and concepts. The work of Schoenfeld (1988) and Skemp (1976)
points to the usefulness of understanding when attempting to make sense of new
situations and building new meanings. Moreover, NCTM posits that learners who
understand mathematics will be more self-motivated to continue learning and more
1
confident than learners who do not understand. (It is interesting, however, that the
paragraph stating this, at the top of page 21 in PSSM, does not contain any references.)
A third leg of NCTM’s argument for learning with understanding rests on context.
In our modern, computer-based society, we no longer need to rely on humans to perform
routine mathematical computations—computers are usually faster and less likely to err.
Thus, an individual who only learns procedures is no better than a computer. An
individual who understands, however, can be an intelligent user of such technologies and
can become adaptively prepared to meet the new challenges of the future. As Jeremy
Kilpatrick wrote, “the quality of the lives our citizens lead depends on whether they are
equipped with mathematical tools for thinking about problems that confront them” (1983,
p. 306). The “tools for thinking” that he references are certainly characterized by
understanding, not divorced from it.
Whence Misunderstanding?
The above is evidence for some of the values of learning with understanding.
Though some may disagree with the role that understanding should play in school
mathematics, and may present countervailing evidence to this effect, everyone can
usually agree on the detrimental quality of persistent misunderstanding. Why is it, then,
that such misunderstanding exists? For example, many students believe that
a  b2  a 2  b 2
or
2
2
2


(Matz, 1980). These misunderstandings seem to
3a  5b 3a 5b
be related to valid algebraic identities, such as 2a  b  2a  2b and
3a  5b 3a 5b


,
2
2
2
which may have been overgeneralized into the errors above. The difference between the
valid identities and the student errors has to do with the concept of linear transformations.
2
Multiplication by 2 and division by 2 (or multiplication by 1/2) are linear transformations
and so, thinking of them as a function, it is true that f a  b  f a  f b . Squaring and
taking the reciprocal of division by 2, however, are not linear transformations, so the
same property does not apply. Errors of the type mentioned above may come from an
expectation that all transformations behave as nicely as linear transformations.
Another misunderstanding that has been documented in the research literature is
confusion with heights of triangles (Gutiérrez & Jaime, 1999). Many learners believe that
the height of a triangle should be a side of the triangle, and others believe that the height
should always run vertically with respect to their paper. It seems plausible that the heightas-side-of-a-triangle misunderstanding is related to the similar (and valid) notion of baseas-side-of-a-triangle. Also, the height-runs-vertically-on-the-page misunderstanding is
likely rooted in the typical practice of drawing triangles so that the base runs horizontally,
forcing the height to commonly run vertically. (A similar finding with young children is
that they do not recognize triangles that sit askew on their paper.)
One possible broad explanation for these misunderstandings lies in Hewitt’s
distinction between the necessary and arbitrary in mathematics (Hewitt, 1999).
Necessary truths are grounded in the axioms of mathematics which themselves are
grounded in the reality of our universe and so, in this sense, are determined. Examples of
necessary truths are the common properties of the whole numbers and the Pythagorean
Theorem. Arbitrary aspects of mathematics, on the other hand, could be otherwise—we
have simply agreed on them as conventions. For example, we use the “+” symbol for
addition and we tend to draw triangles with their base at the bottom. Both of these things
could be different without affecting the underlying mathematical truths, the necessary.
3
With respect to the misunderstandings identified above, perhaps learners have
conflated the necessary and the arbitrary. They may believe that rules such as the
distributive property were arbitrarily decided upon (indeed, the term “rule” may suggest
this) rather than determined by the behavior of the number system, and so it would make
sense to also apply this rule to similar situations such as squaring a binomial. This may
also be reinforced by the fact that this arbitrary approach is easier than the necessary
approach, so if mathematics is arbitrary, why not make it easy? In the case of triangle
altitudes, it is possible that learners have interpreted the common but arbitrary orientation
of triangles with base on bottom as a necessary aspect of the situation, thus concluding
that heights should always run vertically.
What role does instruction play? If teachers are themselves unaware of the
distinction between the mathematically necessary and the mathematically arbitrary, then
it will be nearly impossible for them to guide students in making the distinction.
Additionally, focusing on rules and procedures may send the message that the discipline
is completely arbitrary, undercutting the possibility of students seeing the need to search
for necessary truth (e.g., Erlwanger’s Benny (1973)).
Where from here?
I believe that there is promise, some explanatory power, in the notion of the
necessary and the arbitrary in mathematics. Future work may develop the framework
further and use it to make sense of some student misunderstandings. Such work would
certainly need to incorporate interviews with students for the purpose of uncovering their
conception of mathematics. It may be worthwhile to examine teacher’s conceptions of
mathematics along these dimensions, as well, to investigate my claims above about the
4
role of instruction. In particular, the domain of algebraic properties and
overgeneralizations seems particularly ripe for applying these ideas.
References
Bransford, J. D., Brown, A. L., & Cocking, R. R. (1999). How people learn: Brain, mind,
experience, and school. Washington, DC: National Academy Press.
Erlwanger, S. H. (1973). Benny's conception of rules and answers in IPI mathematics.
Journal of Children's Mathematical Behavior, 1, 7-26.
Gutiérrez, A., & Jaime, A. (1999). Preservice teachers' understanding of the concept of
altitude of a triangle. Journal of Mathematics Teacher Education, 2, 253-275.
Hewitt, D. (1999). Arbitrary and necessary: A way of viewing the mathematics
curriculum. For the Learning of Mathematics, 19(3), 2-9.
Kilpatrick, J. (1983). Editorial. Journal for Research in Mathematics Education, 14, 306.
Kilpatrick, J., Martin, W. G., & Schifter, D. (Eds.). (2003). A research companion to
Principles and Standards for School Mathematics. Reston, VA: NCTM.
Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of
Mathematical Behavior, 3, 96-166.
National Council of Teachers of Mathematics. (2000). Principles and standards for
school mathematics. Reston, VA: Author.
Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of
'well-taught' mathematics courses. Educational Psychologist, 23, 145-166.
Skemp, R. R. (1976). Relational understanding and instrumental understanding.
Mathematics Teaching, 77, 20-26.
5