The Myth about Traditional Math Education, or Who Believes Test Scores Anyway?
By Barry Garelick
Most discussions about mathematics and how best to teach it in the K-12 arena break down to the inevitable bromides about how math was traditionally taught and that such methods were ineffective. The conventional wisdom on the “traditional method” of teaching math is often heard as an opening statement at school board meetings during which parents are protesting the adoption of a questionable math program: “The traditional method of teaching math has failed thousands of students.” A recent criticism I read expanded on this notion and said that it wasn’t so much the content or the textbooks (though he states that they were indeed limited) but the teaching was “too rigid, too inflexible, too limited, and thus failed to adequately address the realities of educating a large, diverse, and rapidly changing population during decades of technological innovation and social upheaval.”
There is some confusion when talking about “traditional methods” since traditional methods vary over time. Textbooks considered traditional for the last ten years, for example, are quite different than textbooks in earlier eras. For purposes of this discussion, I would like to confine “traditional” to methods and textbooks in use in the 40′s, 50′s and 60′s. And before we get to the question about teaching methods, I want to first talk about the textbooks in use during this time period. A glance at the textbooks that were in use over these years shows that mathematical algorithms and procedures were not taught in isolation in a rote manner as is frequently alleged. In fact, concepts and understanding were an important part of the texts. Below is an excerpt from a fifth grade text of the “Study Arithmetic” series (Knight, et. al. 1940):
The above excerpt is a thoughtful discussion designed with the goal of having students understand that a fraction is really a representation of division–and why this is so . The equal division of three cupcakes among four people, or the equal division of a 3 inch line into four parts, is an extension of the idea of division of a whole number by a lesser whole number, which students have already mastered. Students already know that if 12 cupcakes are equally divided among four people, then each person gets 12/4 cupcakes. This idea is extended by starting with the problem of 3 divided by 4, and expressing 3 as 12 fourths. The problem is now stated as 12 fourths divided among 3 people, so that each person receives 3 fourths. This idea is then applied to a line three inches long, so that fractions are ultimately related to a number line, and the final point made that fractions are a representation of division. This is a key concept and ultimately underscores an idea of representing a fractional part as a unit unto itself. That is, 3/4 of an inch can be thought of as a unit (i.e., there are four such units in a three inch line) which is a cornerstone idea when fractional division is studied later.
Absolutely no one has argued or is arguing for memorization without understanding, and that caricature of traditional methods one of the bigger stumbling blocks in the debate. In light of the tracking of students which prevailed in the past, the traditional method could be said to have failed thousands of students because those students who were sorted into general and vocational tracks weren’t given the chance to take the higher level math classes in the first place—the instructional method had nothing to do with it. The excerpt above is not atypical of texts at that time. I believe that any poor instruction that was practiced was not inherent to such books–it was incidental to it.
Which now leads to the question of how prevalent was bad teaching? Some evidence in the form of test data exists to shed light on the effectiveness of traditional methods–both textbooks and instructional quality. Specifically, test scores from the Iowa Tests of Basic Skills ( for grades 3 through 8 ) and the ITED (high school grades) have been documented from the early 40’s through the 80’s for the State of Iowa.
The scores (in all subject areas, not just math) show a steady increase from the 40’s to about 1965, and then a dramatic decline from 1965 to the mid-70’s. The same pattern of ITBS scores through the 80’s was noted by Bishop (1989) for Indiana and Minnesota.
One conclusion that can be drawn from these test scores is that the method of education in effect during that period appeared to be working. And by definition, whatever was working during that time period was not failing. That the math could have been made more challenging and covered more topics in the early grades does not negate the fact that the method was effective. While some may argue that standardized tests scores do not measure true knowledge or “authentic” problem solving skills, the rise of the ITBS scores during this period has been of considerable interest to various researchers for some time (including Dan Koretz who wrote about it extensively in a study he wrote for the Congressional Budget Office (1986), and Bishop (1989) ). The interest of researchers is not confined to what caused the meteoric rise in test scores, but also what caused the sudden and dramatic drop starting in 1965. Also of interest is that SAT scores followed the same pattern nationwide as the ITBS/ITED scores.
Among the possible explanations offered for the decline are increased drug use in the mid-60′s, permissiveness, increase in divorces and single family homes, as well as the progressivist trends in education resulting in student-centered and needs-based courses. (See http://www.educationnews.org/commentaries/156298.html for a more extensive discussion of this last item.) Another explanation offered is that the population of test takers starting around that time began to include more minority students, resulting in a dilution effect. That argument fails to explain, however, why the same pattern of declining test scores for the SATs exists for the ITBS and ITED test scores which were not limited only to college bound students. Also significant is the fact that the population of test takers in Iowa, Minnesota and Indiana remained primarily white which has been noted by Bishop (1989) and Murray (1992). Specifically, the U.S. Census of 1950 shows that the population in Iowa was 99.2 percent white, declining by 0.7 percentage points to 98.5 percent white by 1980. Similarly, the populations of Minnesota and Indiana were 99 and 95.5 percent white in 1950, dropping respectively to 98.2 and 92.8 by 1970. (Hobbes, 2002).
In light of such data, it is disingenuous to say that traditional math ” failed to adequately address the realities of educating a large, diverse, and rapidly changing rapidly changing population during decades of technological innovation and social upheaval,” per the criticism I had quoted earlier. The education establishment continues to advance faddish techniques such as group and collaborative learning, inquiry-based and problem-based learning, while it pays lip service to traditional approaches, calling it a “balanced approach”. While there are aspects of teaching and texts of the past that could definitely be improved, the question remains why the educational establishment remains intent on throwing the baby out with the bath water.
Bishop, John. 1989. Is the Test Score Decline Responsible for the Productivity Growth Decline? The American Economic Review (Vol. 79, No. 1)
Congressional Budget Office. 1986. Trends in Educational Achievement. Prepared by Daniel Koretz of Congressional Budget Office’s Human Resources and Community Development Division. Congress of the United States. Available at: http://www.cbo.gov/ftpdocs/59xx/doc5965/doc11b-Entire.pdf
Hobbes, Frank and Stoops, N. 2002. “Demographic Trends of the 20th Century”. U.S. Census Bureau. Washington DC. November.
Knight, F.B., Studebaker, J.W., and Ruch, G.M. 1948. “Study Arithmetics; Book 5″. Scott, Foresman and Company.
Murray, Charles. 1992. What’s Really Behind the SAT-Score Decline? , Public
Interest, 106 (1992:Winter) p.32