by Barry Garelick
The New York Times recently published a piece called "The Faulty Logic of the Math Wars" by Alice Crary (philosophy professor at Eugene Lang College) and W. Stephen Wilson (a math professor at Johns Hopkins University). It focuses on the beliefs and practices of those known as math reformers. For over twenty years, there has been a battle between two philosophies of how best to teach math in the K-12 arena. The differences of opinion have resulted in what has come to be known as the "math wars".
While the article itself is worth reading, I found the reaction of the readers to be equally fascinating. They revealed the ideological divide that defines this "war". I was reminded of Tom Wolfe's famous description of the reaction of the New Yorker literati to his 1965 article in the New York Herald Tribune that criticized the culture of The New Yorker magazine: "They screamed like weenies over a wood fire."
Many of those who commented on the Times article about math agreed with the premise of the article and expressed their appreciation for a viewpoint that supported the teaching of standard algorithms such as adding and multiplying multi-digit numbers. Others accused the authors of casting the situation as one of either/or, and of exaggerating that the teaching of standard algorithms in the early grades is avoided. Several commenters went on to conclude that the math war is a fiction. One comment exclaimed: "The "Faulty Logic" here is that there should be a Math War at all." Other commenters asserted that there is in fact a "balance" of techniques going on in classrooms.
Balance and Understanding
The "balance" argument has been used for years. Math reformers argue that they don't advocate leaving out the teaching of standard algorithms, but believe that both alternative strategies and the standard algorithm should be taught. Reformers believe that the alternative strategies provide a better means of ultimately understanding why the standard algorithms work. "Balance" largely remains undefined in these discussions, however. In addition, there are arguments about the pedagogical approach to be used; is problem-based and inquiry-based learning superior to direct, whole-class instruction? Both reformers and traditionalists will argue that they use both, but the degree to which reformers rely on inquiry-based approaches (and the methods used) differ from traditional approaches.
Among the many arguments about "balance", in the end it all comes down to "understanding" . The reform camp argues that they do in fact teach procedures–they just teach them with meaning, so that students can understand what they are doing. The implication is that those on the traditional/classical side of teaching math do not teach procedures with meaning–i.e., "understanding". Furthermore, the reformers contend that the standard algorithms are 1) difficult to teach and 2) they do not lend themselves to understanding how they work. I am reminded of a dialogue between a friend of mine—a math professor—and a public school administrator. My friend was making the point that students need basic foundational skills in order to succeed in math. The administrator responded with "You teach skills. But we teach understanding."
Although several comments to the Times article addressed the issue of "understanding", one comment that stands out appeared elsewhere. It was written by Keith Devlin, a mathematician at Stanford (also known as "that math guy" who gives talks on NPR) and appeared in a post at his blog in which he wrote a lengthy criticism of the article. In moderating the many comments that responded to his piece, he left this comment at his blog:
"The fact that [the standard algorithms] do not work well educationally is made abundantly clear by the fact that, though they were used as the standard educational procedure for arithmetic for hundreds of years, the majority of people, even today, are not proficient in arithmetic and exhibit little real understanding of the place-ten number system. They may be "easy to teach", â¦ but the overwhelming evidence is that they are difficult to learn, and indeed, most students do not learn them! (They certainly did not when I was in elementary school, and I am now in my mid-sixties! I remember being one of the few students in my class who "got them", and only after a long, hard struggle.)" (Bolding in Keith's original comment).
I have a difficult time understanding how this argument can be made. Some time ago, I wrote an article published here in Education News that addressed the myths about traditional math teaching. In it, I showed test scores (in all subject areas, not just math) from the Iowa Tests of Basic Skills (for grades 3 through 8 ) and the ITED (high school grades) from the early 40's through the 80's for the State of Iowa. The scores 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. (See "The Myth About Traditional Math Education.")
Once again, I offer these test scores as evidence that the method of education in effect during a period that relied on the teaching and mastery of standard algorithms (and has been criticized for failing thousands of students) appeared to be working. And by definition, whatever was working 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 methods were effective. Nevertheless, there will undoubtedly be those who argue that standardized tests such as ITBS do not measure true knowledge or "authentic" problem solving skills.
I wouldn't be quite so much against the strategies that are taught in lieu of the standard algorithms, if they were used to help explain how the standard algorithms work, thus effecting the "understanding" the reformers claim to be so concerned about. In actual practice, however, the alternatives are left by themselves. Students are left to work with partial sums and partial products (and other methods such as the lattice method for multiplication) in the early grades, and do not learn the standard algorithms until 4th and 5th grades, in programs such as Everyday Math, and Investigations.
Partial sums and partial products are nothing new and were taught even in earlier eras. Below are two figures taken from an arithmetic book published in 1948 (Study Arithmetics, Grade 5), showing these two methods. It was not uncommon to provide such instruction. Mastery of the standard algorithms were already achieved when such strategies were introduced, and provided interesting and helpful insights. As a side dish, it complimented the main course effectively. But now, in the interest of understanding, the side dishes have become the main course–and the standard algorithms are now side dishes. Judging by the comment from Keith Devlin and others, reformers consider these algorithms to be an inferior way to teach math.
Which brings us to what is actually meant by "understanding". What the reform camp means by understanding is different than from what many mathematicians and those in the more traditional camp mean. The reform approach to "understanding" is teaching small children never to trust the math, unless you can visualize why it works. If you can't "visualize" it, you can't explain it. And if you can't explain it, then you don't "understand" it. According to Robert Craigen, math professor at University of Manitoba, "Forcing students to use inefficient procedures that require ham-handed handling of place value so that they articulate "meaning" out loud in every stage is the arithmetic equivalent of forcing a reader to keep his finger on the page and to sound out every word, every time, with no progression of reading skill."
The power of math, however, is allowing for exploration of concepts that cannot be visualized. Math is what takes over when our intuition begins to fail us. It gives our mind new powers by extending principles–often beyond what can be visualized. A student who learns how to add and multiply two and three digit numbers, has the understanding to apply the procedure to addition and multiplication of numbers with any number of digits. A student who understands the procedures for long division can use that procedural understanding to learn what a repeating decimal is—and ultimately why certain numbers repeat. As far as understanding why the algorithms work, sometimes that is learned before the procedure, and other times as a consequence of doing the procedure and seeing how it works.
Such procedural understanding applies also to fractions. Much ado is made over the necessity of students learning why the invert and multiply rule for fractional division works. I agree it should be taught, but as I said some students will get it then, others later. These concepts are easier to convey—and understand—once the student has the algebraic tools by which to do so. But there is an expectation that students in 5th or 6th grade be able to "explain and understand" why such rule works. Such a student may be able to solve the problem: "How many 3/4 ounce servings of yogurt are in a cup that contains 7/8 ounces?" But if the student cannot explain why the invert and multiply rule works, or be able to represent the problem pictorially, he or she may be judged to lack "understanding".
What Are the Two Sides Saying?
Those seeking to prove that the math wars are only a fiction typically argue that "we're saying the same things; we're in agreement" Reformers will agree that students need to learn math facts and procedures, but in the next breath will say that they also need to learn how to think critically and use higher order thinking skills." This side is keen on understanding with a begrudging nod to the procedures that make understanding possible. Understanding is their predominant goal without basing it on the careful accretion of the foundational skills that make that happen.
Reform math in various forms has pervaded math teaching for over twenty years. Could it be that the failures of understanding are due to the beliefs and techniques of the reformers rather than the other way around? If it were true that we were all saying the same things, would we have the large number of students counting on their fingers in algebra classes, or being unable to work with fractions?
Why don't those arguing for better math education look at what those students are doing who are succeeding in pursuing majors in science, engineering or math? It is likely that you will see students learning standard algorithms and practicing many drills and problems (deemed dull, tedious and "mind numbing") and other techniques viewed by reformers as not resulting in true, deep understanding. But such an outcome based investigation is not occurring. Some parents whose children are not doing well in math believe what they hear from school administrators that "Maybe your child just isn't good at math." The parents who recognize the inferior math programs in K-6 for what they are, get their children the help they need.
And in answer to the statement that we're all saying the same thing: No. We're not saying the same thing at all.
Knight, F.B., Studebaker, J.W., and Ruch, G.M. 1948. "Study Arithmetics; Book 5. Scott, Foresman and Company.