Poster Children of Math Education

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By Barry Garelick

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. I have talked about this at length here, and here, so won’t go into further detail on it except to say that when I (and many others I know) was educated in the 50’s and 60’s math was taught with understanding and connections, not merely as 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)

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

We were also shown with diagrams what happens when a 5-inch piece of ribbon cut into [latex] \frac{2}{3} [/latex] inch lengths was used to show how [latex] 5 \div \frac{2}{3} [/latex] 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 numbers divided by a fraction, or the division of two proper fractions like [latex] \frac{2}{3} \div \frac{3}{4} [/latex]—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.  (The math behind it is explained in this short video put together by John Mighton, the Canadian mathematician who founded the JUMP Math program that is gaining popularity in Canada.)

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

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: [latex] 5 \div \frac{2}{3} [/latex] answers the question “How many [latex] \frac{2}{3}[/latex]’s are contained in 5?” and [latex] \frac{5}{6} \div \frac{3}{4}[/latex] tells how many [latex] \frac{3}{4}[/latex]’s are contained in [latex] \frac{5}{6}[/latex].  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.

For sixth graders, such procedural understanding is a good start into what fractional division is.  For fractions such as  [latex] \frac{5}{6} \div \frac{3}{4}[/latex], 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 its opposite”; also inverse operations.

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

[latex] \frac{4}{3} \times \frac{3}{4}x = \frac{5}{7} \times \frac{4}{3} [/latex], and [latex] x = \frac{5}{7} \times \frac{4}{3}[/latex]

… which shows that the invert and multiply procedure is equivalent to dividing [latex] \frac{5}{7}[/latex] by [latex] \frac{3}{4}[/latex] in the original problem.

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 just 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 know have?” and [latex]2 + 3 = 5[/latex].  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 [latex]3 \times 4[/latex], and it is not difficult for the student to make the connection.

Unlike whole number operations, however, 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.  While students undergoing instruction under the prevalent interpretations of Common Core may be able to recite an explanation they have been told, that is not the test of effectiveness of a math program. Their understanding is “rote understanding”.  What matters to me is whether a student knows what fractional division represents. If a student can solve the problem, “How many [latex] \frac{2}{3}[/latex] oz servings of yogurt are in a [latex] \frac{3}{4}[/latex] oz container” by dividing [latex] \frac{3}{4}[/latex] by [latex] \frac{2}{3}[/latex], and that this tells us how many [latex] \frac{2}{3}[/latex] are in [latex] \frac{3}{4}[/latex], 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 the memorization of basic computational skills.  Such skills are mistakenly viewed as rote learning and totally devoid of meaning.  This is a gross mischaracterization.  Liping Ma, author of “Knowing and Teaching Elementary Mathematics” and who taught elementary math in China, states that in the U.S. math in the lower grades is considered to be solely “‘basic computational skills’ … equivalent to an inferior cognitive activity such as rote learning.” (Ma, 2013)

In many US classrooms today, students must demonstrate an “understanding” of computational procedures before they are allowed to use standard algorithms.  Such topsy-turvy approaches to math education have been around for more than two decades, but the interpretation and implementation of Common Core have made them more popular.  To compensate for what reformers believe is a lack of understanding, the teaching of mathematics has been structured to drag work out far longer than necessary with multiple procedures, diagrams, and awkward, bulky explanations.

In so doing, 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. But this “stop and explain” approach to understanding undermines what the reformers want to achieve in the first place. It is “rote understanding”: an out-loud articulation of meaning in every stage that is the arithmetic equivalent of forcing a reader to keep a finger on the page, sounding out every word, every time, with no progression of reading skill.

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. Their views and philosophies are taken as faith by school administrations, school districts and many teachers – teachers who have been indoctrinated in schools of education that teach these methods.

The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way.  In so doing, the math reformers have unwittingly created a poster child in which “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)

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Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long-term substitute in a high school and middle school in California: “Confessions of a 21st Century Math Teacher.”