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What is Changing in Math Education?
by Mathematically Correct
8 November 2004
[Originally published Feb. 13, 1996]
Mathematically Correct
PO Box 22083, San Diego, CA 92192-2083
The impending changes in mathematics education are not based on any change
in the mathematics that has been developed over thousands of years. Rather,
they are based on a cluster of notions from teaching philosophy and a desire
to implement them all at once. The driving force behind these changes is
dissatisfaction with the continued declines in the achievement of American
students, coupled with the idea that a set of goals should be developed that
all students can attain. The position taken is that poor math achievement is
the result of the traditional curriculum and the way it has been implemented
by teachers. The fact that math education in countries with high levels of
achievement does not look like these new programs, but rather like
intensified versions of our own traditional programs, is never addressed.
One of the philosophical components is the idea of Constructivism or
discovery learning. This notion holds that students will learn math better
if they are left to discover the rules and methods of mathematics for
themselves, rather than being taught by teachers or textbooks. This is not
unlike the Socratic method, minus Socrates. One of the problems with this
approach is that teachers must be extremely skilled in these methods.
Another is that "discovery" takes so long that considerably less material
can be covered. A third problem is that the children sometimes "discover"
the wrong "rules" and teachers don't always catch the error.
Another philosophical notion is the idea of Complete Math, which is a
replacement term for Whole Math. Just as Whole Language attempted to skip
the basics of phonics and go directly to reading literature, Whole Math
attempts to cast aside computational basics and go to final productions that
rely on math at some level. This view holds that math is not used only in
equations, but in writing and discussion as well. The implication is that
students should write essays and have group discussions about math. The
major problem with this method is that students end up spending hours
working on essays, again detracting from their chance to practice basic
skills. This has the substantial risk that potentially controversial moral
lessons make their way into assignments. Another problem with the emphasis
on language skills, which many feel is misplaced in math class, is the
disadvantage to students with speech and hearing problems or non-native
English speakers.
Proponents of Complete Math like to talk about "communicating
mathematically." However, the students do not learn the language of
mathematical exposition -- the terminology, symbols, and syntax needed for
communicating mathematically. Instead, their products are better
characterized as "communicating about math," -- written and spoken words and
pictures that have something to do with math but are a far cry from
"communicating mathematically." The process leaves many wondering what
English-math or art-math is.
The math changes are also based on the idea of Integrated Content. This view
argues that math applications do not neatly divide into content areas such
as algebra, geometry and trigonometry, and that math education should
therefore not be packaged in the traditional way. The implementation based
on this view is an admixture of the topics from various traditional areas
across multiple new courses of instruction. There are several serious
problems with this approach, all of which are the result of blurring
traditional milestones of development in mathematics abilities. If the
traditional developmental model is rejected, there is no clear model or
order to replace it. Therefore, inconsistency among various textbook
materials following this approach results, since there is no clear
prescribed content or sequence. This also leads to great difficulties in the
assessment of either individuals or whole programs.
A basic idea in learning theory is that complex skills can be analyzed to
see what the component parts are, so that these components can be taught
individually to develop component skills. In reading, for example, a student
learns how to decode words (phonics) and that the page is organized from
left to right and top to bottom. As these skills become automatic the
teacher and student pay less attention to them and a shift is made to more
complex reading tasks. Constructivism, Complete Math and Integrated Content
all work against the idea that component skills in math can be identified
and taught, and that these form building blocks for subsequent learning.
In support of their position, proponents of these approaches often note that
there may be more than one way to solve a problem, and one method may be
better for some while another method is better for others. One could join
two boards with nails, screws, or pegs and glue. However, it would take
knowledge and skill in all three to know which is the best approach in a
given application, as any carpenter would attest. The carpentry school
should not neglect any of these approaches just because some students favor
one while others prefer another.
Another idea behind the new programs is that of Cooperative Learning and the
related idea of Cooperative Assessment. This view is based on the notion
that real-world jobs involve cooperative efforts in groups, and that
competition among individual students is neither a good model of the real
world nor good for learning. This idea contradicts real-world experience
showing that those who do math in organizations rarely do so in a group
setting. In any case, the problems with this approach occur long before the
students reach the job market. When children work in groups in school, the
distribution of work, and of learning, is not equal. Teachers are supposed
to prevent this, but it happens anyway. Problems often occur from unequal
ability levels within a group. In such cases, the most advanced students do
the bulk of the work, with the others copying from them. In groups of equal
ability levels, students have been known to split up the work and then copy
answers. Group assessments are frequently objected to as well. They tend to
pull down the evaluations of the top students, while allowing weaker
students to pass without learning the material.
Despite admonitions to the contrary, having goals to be achieved by all
students gets interpreted as a need for a one-size-fits-all curriculum,
discouraging ability or achievement grouping and encouraging the use of
mixed ability groups. This amplifies the problems associated with the
cooperative approach. It also imposes unnecessary limits on individual
students, keeping them from progressing as far or as quickly as they might.
Yet another point emphasized by these new programs is the use of calculators
and computers. Based on the view that we live in an increasingly
technological society, these programs introduce the use of calculators as
early as kindergarten, and usually require students to have them available
at all times. The idea is that students shouldn't have to be bogged down
with mundane things like addition and subtraction, since calculators can do
these things for them. At higher levels, calculators that do fraction
problems or graphs are required. Opponents argue that the use of calculators
in the new programs is excessive and leads to a deficit of basic skills.
Algebra students have been know to reach for a calculator when faced with
the multiplication of two single-digit numbers or needing to divide 300 by
3.
Perhaps the most viable criticism of traditional programs offered by the
proponents of the new programs is that traditional students do not do as
well on problem solving (meaning word problems) as they do on straight
computation. An inspection of traditional texts will show that there are
plenty of word problems. There is some evidence to suggest that teachers
assign a smaller proportion of word problems than computation problems,
since the word problems are traditionally more difficult. In any case, the
new programs seem to ignore the fact that basic computation skills are
necessary in problem solving. If students lack the basic tools to yield
correct results, concepts will not help. Consequently, the new programs do
not appear to produce better problem solving skills as claimed.
Much criticism of the traditional programs made by the proponents of these
changes seems to be entirely misplaced. The new programs are said to
emphasize real-world problems more than traditional programs do. However,
inspection of course materials shows that the same "real" topics appear in
both, and some very "unreal" problems appear in the new textbooks. Another
claim is that traditional teachers are nothing but drill masters and unable
to relate math to applied problems. This notion is necessary to complete the
argument that teaching methods are to blame for inadequate achievement.
Finally, the proponents argue that students need better math self-esteem and
that math appreciation should be an assessment criterion. Building
self-esteem in math by decreasing computational skills seems seriously
misguided. As far as math appreciation is concerned, it may well be an
appropriate area for program evaluation, but it is difficult to see how the
assessment of individual student abilities should include this area.
Some of the features of this new approach, given occasionally in small
doses, may be beneficial. Unfortunately, all of these characteristics are
infused into the curriculum simultaneously in the new programs, and
traditional instruction is cast aside. One might reasonably expect that such
radical departures from traditional methods would be based on clear,
well-documented, overwhelmingly compelling, quantitative evidence of their
superiority. Sadly, this is not the case at all. In fact, the lack of
research support is striking. Perhaps the most unifying feature of these new
programs is that they are all experimental. This is not to say that
traditional mathematics education is perfect. Nor is there reason to believe
that we cannot find ways to improve traditional instruction. However,
prudence dictates that careful study, not just a hopeful philosophy, lay the
groundwork for these improvements.
MATHEMATICALLY CORRECT
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