What Are Some of the Potential Problems of the Various “New Math” Techniques Being Taught in Some American Schools?
by R. James Milgram - July 23, 2007
I give a brief history of the new math, and then discuss two of the topics introduced into the K-12 curriculum from the new math, problem solving as a separate part of instruction, and the introduction of foundational rules for arithmetic, the associative, commutative, and distributive laws, early in instruction. I indicate the serious problems with their implementation currently.
New Math dates back 50 years, starting with the New Math project (SMSG and other groups) developed in the mid- to late 1950s largely in response to Sputnik.
Unfortunately, the material was disseminated to the schools before it was ready and tended to be taught to all students rather than to the intended audience of top students. Many mathematicians also felt some of the material was inappropriate in the lower grades regardless of the audience.
There was a correction in the 1970s, but quite a few of the New Math perspectives remained in the curriculum. Among the more interesting and problematic were a focus on problem solving as a separate topic and the introduction of foundational ideas much earlier in the curriculum. These could have been extremely positive additions, but the way they are presented in K-12 today is a disaster.
There are a small number of basic processes - many isolated and studied by George Polya - that can help with some of the beginning steps in problem solving. However, to advance in this area, students have to know the underlying mathematics that relates to the problems they are trying to solve. As Polya (1944) pointed out,
We know, of course, that it is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge. Mere remembering is not enough for a good idea, but we cannot have any good idea without recollecting some pertinent facts; materials alone are not enough for constructing a house but we cannot construct a house without collecting the necessary materials (p. 9).
When we look at what is called problem solving in most current U.S. mathematics texts, it consists almost entirely of long lists of very special procedures from Polya's work that are usually taken out of context. Additionally, many current texts, particularly for so-called reform programs, make extravagant claims that are not supported by student outcomes. For example, they tend to claim that students can learn to solve problems without learning anything about the contexts in which the problems arise or the known mathematics that is related to them. As a result, after more than fifteen years of ever growing numbers of students taught reform style mathematics in K-12, my impression is that most mathematicians believe students entering college today are less prepared for college mathematics than ever before (Wilson, in press; Wilson, 2005). Moreover, key people in industry make the same assertion about the U.S. high school graduates applying for entry-level jobs.
The professional mathematics community agrees that having material in the school mathematics curriculum that is focused on applying and understanding the basic rules of arithmetic is potentially very helpful (Wilson, 2005). Key among these rules are the commutative, associative and distributive laws for addition and multiplication. This material provides a concrete foundation for algebra and helps provide a solid base for even more advanced mathematicscontent that has real value in college and the workplace. Unfortunately this material is almost never presented in a mathematically valid and coherent way in the early grades.
Most self-proclaimed reform texts advertise that they teach underlying concepts so students don't have to do endless drills practicing addition, subtraction, multiplication and division of numbers written in base-ten place value notation. By underlying concepts one might hope what is meant are the rules mentioned above. But this is not exactly the case, and even if it were, there is a crucial balance that has to be maintained. Mathematics is the most hierarchical of subjects. As is pointed out in a joint document whose development was supported by both the AMS and the MAA, the two major professional mathematics organizations in the United States,
The value of a mathematical education and the power of mathematics in the modern world arise from the cumulative nature of mathematics knowledge. A small collection of simple facts combined with appropriate theory is used to build layer upon layer upon layer of ever more sophisticated mathematical knowledge. The essence of mathematical learning is the process of understanding each new layer of knowledge and thoroughly mastering that knowledge in order to be able to understand the next layer (Mathematics Standards Study Group (n.d.)).
The document lists five basic principles that must be present in school mathematics instruction. Here are the first two:
Whole number arithmetic and the place value system are the foundation for school mathematics with most other mathematical strands evolving from this foundation. This foundation should be the subject of most instruction in early grades.
In every grade, the mathematics curriculum needs to be carefully focused on a small number of topics. Most mathematics instruction should be devoted to developing deepening mastery of core topics through computation, problem-solving and logical reasoning (Mathematics Standards Study Group (n.d.)).
Both an understanding of why the standard algorithms work and fluency with reasonably efficient implementations of them are essential prerequisites to the more advanced mathematics needed later. There are no shortcuts here.
Everything starts with place value. A whole number written in place value notation is given as a sequence of digits, a1a2 ... an, with 0 ≤ ai ≤ 9 for each ai. For example, in the number 3711, a1 = 3, a2=7, a3 = 1 and a4 = 1. The key thing students need to understand is that this is really shorthand for the expanded form
a1×10n-1 + a2×10n-2 + ... + an-1×10 + an = ∑i= 1 n ai×10n-i,
so that 3711 really means 3×103 + 7×102 + 1×10 + 1. Of course, this is not what our textbooks focus on. Instead students are taught how to say these numbers, so 3711 is thirty-seven hundred eleven, useful in reading but completely irrelevant in mathematics.
If place value had been taught properly, then students could easily understand the long-multiplication algorithm from the commutative, associative, and distributive laws. In a more sophisticated notation than would be used in actual instruction we have
(∑i=1n ai ×10n-i)×B = ∑i=1n ai×B×10n-i,
where B is the multiplicand (assumed written in base 10 place value notation). It follows that the product is written as a sum of powers of 10, 10n-i, multiplied by ai multiplied by the multiplicand. Noting that multiplication by 10n-i just puts n-i zeros on the right, and that ai multiplied by B is relatively direct - we can either expand B and multiply by ai using the distributive rule
ai×B = ai ×∑j=1m bj×10m-j = ∑ai ×bj×10m-j,
or just add an appropriate number of copies of B to itself - we get the standard stair-step algorithm. Of course, to my knowledge there is no standard textbook in this country that justifies the algorithm in this way. All the texts that teach this algorithm seem to do it in a purely procedural way - multiply single digits, handle carries, move the next line one step to the left..., and don't worry about why it works.
A few reform programs do not teach the stair-step algorithm. Instead they teach something called the lattice method. We can write
A×B = ( ∑i=1n ai ×10n-i )×(∑j=1m bj ×10m-j ) = ∑i=1n ∑j=1 m ai ×bj ×10 n+ m - i -j
and then do a very involved reordering of this last sum by gathering terms multiplied by the same power of 10,
∑k = 2 n+m (∑s =1k-1 as × bk-s ) × 10 n+m-k,
with the understanding that as = 0, and bk-s = 0 if s ≥ n or k-s ≥ m. Evaluating the sums of products
∑s = 1k-1 as × b k-s ,
then multiplying each by the appropriate 10n+m-k we obtain the ingredients for the lattice algorithm. Unfortunately, these texts make absolutely no mention of why the lattice method works. As with long-multiplication, the lattice method is taught purely procedurally.1
The discussion above only scratches the surface of the difficulties with implementing New Math in the K-12 curriculum. For example, we have not touched on definitions. However, problem solving and foundational concepts are among the most critical of these issues in terms of how they affect student outcomes.
1. When things are set out in this way, one sees that the lattice method is much more complex than the stair-step algorithm. It is not surprising that the texts using it do not explain the underlying concepts.
Mathematics Standards Study Group (n.d.) What is important in school mathematics Available at: http://www.maa.org/pmet/resources/MSSG_important.html
Polya, G. (1944). How to solve it. Princeton: Princeton University Press.
Wilson, W.S. (in press). Are our students better now? Available at: http://www.math.jhu.edu/~wsw/89/study89.pdf.
Wilson, W.S. (2005). Short response to Tunis's letter to the editor on technology in college, Educational Studies in Mathematics, 58, 415-420, Available at: http://www.math.jhu.edu/~wsw/ED/EDUCTunis.pdf, particularly see page 4.