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.


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