The Poster Children of Math Education

(From "Math Education in the U.S.")

I had a discussion about math education recently with a principal of a middle school where I had once worked.  I think highly of him and enjoy our discussions, even though we may not see eye to eye.  In our last discussion he expressed the following view:

“I think we have spent too many years teaching kids to 'do math', and not nearly enough time teaching true number sense. In education, we often swing too far in the opposite direction. I recall learning to divide fractions by ‘invert and multiply’, but no one ever taught me why that works. Just one example of doing vs. understanding.”

The invert and multiply example has for years served as the poster child for the reform math movement. It is used as evidence that traditionally taught math is math taught wrong because it is presented as a bunch of tricks, relying on rote memorization with no conceptual understanding or connections to other concepts—students should see that math makes sense.

Before I get too far into this, let me say that I believe that students should be taught why the invert and multiply rule for fractional division works, and I have done so in classes that I have taught.  I will also say that the accusations about traditionally taught math are in large part based on mischaracterizations. When I and many others I know were educated in the 50’s and 60’s, math was taught with understanding, and connected with prior concepts, and was not taught as merely rote memorization. 

In the Interest of Full Disclosure and a Slight Digression

In the interest of full disclosure, let me say that like the principal, I was not taught why invert and multiply works. As I told the principal when he brought the subject up, I did not find out how it worked until about 15 years ago, despite my having majored in mathematics. 

We were first shown how to divide fractions using the common denominator method (see Figure 1).

Figure 1 (Source: Buswell, Brownell, Sauble, 1955)

We were also shown with diagrams what happens when a 5 in piece of ribbon cut into 2/3 in lengths was used to show how 5 ÷  2/3 works (See Figure 2). The method shown in my (and most) arithmetic books at that time, stopped short of explaining the math behind why the divisor is inverted and then multiplied.  We were only shown that in all cases—whole or mixed number divided by a fraction, or the division of two proper fractions like 2/3 ÷ 3/4—multiplying the dividend by the reciprocal of the divisor produced the same answer as the common denominator method, or counting intervals as in the picture above. 

Thus, the invert and multiply procedure was extended to apply to all fraction divisions by virtue of the pattern we were seeing, but without the mathematical explanation behind it.

Figure 2: (Source: Buswell, Brownell, Sauble; 1955)

Although I was not instructed as to the math behind why the method works, the explanations I have described above illustrate what the various fractional divisions represent: 5÷ 2/3 answers the question “How many 2/3’s are contained in 5?” and 5/6÷3/4  tells how many 3/4’s are contained in 5/6.  I might not have known why the invert and multiply rule worked, but I did know what the fractional division represented, and how it was used to solve problems.

Such procedural understanding is a good start for sixth graders into what fractional division is. For fractions such as 5/6÷3/4, the explanation for the invert and multiply rule is easier to convey—and understand—once the student has the algebraic tools by which to do so. Until that time, however, some teachers explain that the reason "invert and multiply" works is because "dividing by a number is the same as multiplying by its reciprocal" (inverse operations).  It is similar to "subtracting a number is the same as adding it's opposite"; also inverse operations.

Usually in seventh grade, students have learned the essentials for solving simple equations such as 6x=24, which can then be extended to fractions to explain why the rule works.  For example, an equation like 3/4 x=5/7 can be solved by dividing both sides by   Since dividing by 3/4 is done to leave  with a coefficient of 1, students are taught that this goal is also achieved by multiplying 3/4 by its reciprocal 4/3 since the product is 1.  Both sides are then multiplied by 4/3:

4/3 × 3/4 (x) = 5/7 × 4/3;  x = 5/7  × 4/3

 These steps show that the invert and multiply procedure is equivalent to the original problem of dividing 5/7 by 3/4.

When Understanding is Part and Parcel to Procedure—and When it is Not

The educational arena has been dominated by the fetish of understanding for more than 100 years. The prevailing group-think amongst the educational establishment and math reform movement is the fear that students will be “doing” math but not “knowing” math as the principal I know had expressed to me. .

Rote (i.e., non-understanding) learning is pretty hard to accomplish with elementary whole number math. The very learning of procedures is, itself, informative of meaning, and the repetitious use of them conveys understanding to the user.  When learning to add and subtract, students make the connection between “I have 2 apples and got 3 more; how many do I now have?” and 2 + 3 = 5.  Similarly, multiplication is understood so that “3 apples are in each bag, and there are 4 bags; how many apples in all” can readily be represented by 4 × 3, (i.e, 4 groups of 3) and it is not difficult for the student to make the connection.

But unlike whole number operations, the conceptual underpinnings of fractional division are not part and parcel to the procedure. Even with an algebraic explanation, some kids will get it, and some will not. Those who do get it may or may not remember why it works.   That is not the test of effectiveness of a math program. And while students may be able to recite an explanation they have been told, thus acquiring a rote understanding, what is more important is whether a student knows what fractional division represents.  If a student can solve the problem "How many 2/3 oz. servings of yogurt are in a 3/4 oz. container" by dividing 3/4 by 2/3, and knows that this tells us  how many 2/3’s are in 3/4, then I judge that student to have sufficient understanding.  A student who has that understanding but does not know why the invert and multiply rule works is not at any significant disadvantage in solving fractional division problems.

What the Poster Child Hath Wrought

The “invert and multiply” example has served as a poster child for the reform math movement, and in their minds constitutes proof that traditionally taught math is nothing more than memorization of basic computational skills.  Such skills are mistakenly viewed as rote learning and totally devoid of meaning.  This is a gross mischaracterization.  According to Liping Ma, author of “Knowing and Teaching Elementary Mathematics” and who taught elementary math in China, in the U.S., basic computational skills are viewed as an inferior cognitive activity such as rote learning. (Ma, 2013).

Today’s math teaching methods in lower grades often require students to demonstrate a conceptual understanding of computational procedures before they are taught and allowed to use standard algorithms. Such topsy turvy approaches to math education have been around for three decades, and in the last fifteen years have become de riguere because of the interpretation and implementation of Common Core.  When students are ready to move on, claiming that they “get it”, they are forced to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations to make sure they really get it. Students are forced to show what passes for understanding at every point of even the simplest computations. Instead, they should be learning procedures and working effectively with sufficient procedural understanding.

The approaches to math teaching in the lower grades in schools is a product of many years of mischaracterizing and maligning traditional teaching methods. The math reform movement touts many poster children of math education and has succeeded in foisting its beliefs upon ever growing populations of new teachers. These new teachers believe this is the only way.  The new poster child is now someone for whom “understanding” foundational math is not even “doing” math.

References

Buswell, Guy T., William A. Brownell,  Irene Saubel. (1955) “Arithmetic We Need; Grade 6”; Ginn and Company.

Ma, Liping. (2013) A Critique of the Structure of U.S. Elementary School Mathematics. AMS, Notices; Vol. 60., No. 10 (DOI: http://dx.doi.org/10.1090/noti1054