For those who have followed my writing, my apologies if you’ve heard this before: I majored in math, but didn’t know why the invert and multiply rule for fractional division worked until about twenty years ago. Not knowing the exact reason for the rule did not deter me from applying it, nor did it hold me back in my studies in math. Also, once I saw the derivation it was very obvious, since I had what young students lack: expertise coming from experience with algebraic manipulations and proofs over the years. I had more tools with which to work than students for whom it is all new.

In one of my telling’s of the story above, a well-known figure in reform math circles equated what I said to “I didn’t graduate from high school and I turned out just fine.” In the case of high school drop-outs, I would agree that they very likely incur serious opportunity costs. I was therefore curious as to what the commenter and math reformers in general believe the opportunity cost is for those who do not know the derivation of the invert and multiply rule.

The opportunity cost question goes beyond just the derivation of the fractional division algorithm. Math reformers tend to put a lot of stock in conceptual understanding in math, in the belief that traditional math 1) does not teach it, and 2) places too much emphasis on algorithms, procedures, and memorization.

The reformers’ view of the importance of conceptual understanding are exemplified in two comments I saw recently on Twitter:

*Math Learning is a 3-legged stool: 1) Procedural knowledge- Processes & shortcuts; 2) Conceptual knowledge-the WHY behind them; 3) Application knowledge- HOW they are used. We've focused on the first one for too long.**One of those legs is way more important than the others. I argue that the stool should sit upon a strong base of understanding. Most of those procedures and skills are no longer needed (and actually hinder understanding). I definitely do NOT place procedures or skills at the same level as conceptual understanding or applications. And, you can have strong conceptual understanding that will get you to a point where you can engage with interesting mathematics even if the skills are lacking.*

The first statement presents the view that traditionally taught math does not teach concepts enough nor adequately. The second statement is a more extreme version of the first and questions the extent to which procedures should be taught. It posits that an “understanding is all you need to succeed in math.”

In fact, procedures and understanding occur in tandem. For some procedures, the conceptual understanding may come prior to learning the procedure, and for others it comes after using it – sometimes many years later (as it did for me regarding the derivation of the algorithm for fractional division). And for still other procedures, the understanding may never come. The notion that all students must have 100% conceptual understanding for 100% of procedures 100% of the time, is 100% mistaken. Some students, for example, may not understand how the quadratic formula is derived. Still others, even PhD mathematicians, may be hard pressed to recall the proof of a particular theorem that they may use on a daily basis.

Traditionally taught math does in fact teach the concepts that inform the procedures. We just don’t obsess over them. Students in K-8 are presented with a slew of new information with which adults have been working for many years. There is an expectation among math reformers that students who are novices will be able to see the world through the same lens as adults who have long forgotten how it was when they were novices.

It is often unclear what math reformers mean by “conceptual understanding”. From what I’ve seen, the meaning is closer to “contextual understanding”. That is, what is it that we are doing when we multiply fractions, say, or divide them? What are the types of problems that such procedures are used in solving? What is it that the mathematics is doing?

What is important is that students see the connection between the concept and the procedure. They may or may not understand everything about the concept. For example, my co-author of “Traditional Math” has said that he is not able to derive the quadratic formula. He remembers, however, seeing the derivation of the formula, and saw the connection between the general form of a quadratic and how it leads to the formula, and finally how the formula is used to solve quadratic equations. This is a form of conceptual understanding.

A case-in-point of the reformer school of thought on conceptual understanding is a conversation I once had with past NCTM president Linda Gojak in the comments section of an article about math education in USA Today. She was bemoaning that students learn the procedure of moving the decimal point to the right or left a certain number of places when multiplying or dividing by a power of ten, but they do not understand why that works.

There are of course explanations as to why this works, and some books (particularly older texts from the ‘60’s and before) provide them and there are teachers who teach the explanations. It isn’t that hard to show that if we divide the number 400 by 10, using long division, we get 40; it is then a simple matter of showing that the 4 in the hundreds place is ten times the amount of the 4 in the tens place. By shifting the position via moving the decimal point one place to the left, we have changed the value of the 4 from 4 hundreds to 4 tens, or 40.

The example can be extended to multiplication, to the fractional parts of numbers and so on. You will get no disagreement from me that this is a good way to introduce the procedure of multiplying and dividing by powers of ten. (My co-author of “Traditional Math” shows how he explains it on pp 126-127 of the book. We also discuss the invert and multiply rule on p.128, and p. 185 (under “For Curious Students”) )

That said, most students seeing this explanation typically do not retain it in sufficient detail to allow them to explain it in their own words. It is more the case that they have seen a connection between place value and the procedure, just as my co-author saw the connection linking quadratic equations to the formula. Once having learned the procedure for multiplying or dividing by powers of ten, students go with that from then on. Ms Gojak and others of her ilk (like the two whose tweets I quoted earlier) have a problem with this and assume that traditional math does not focus enough attention on the concept part, and that procedure has been emphasized for far too long.

Those who use traditional methods do not find it necessary for students to show their understanding by using convoluted and inefficient methods that rely on first principles every time they wish to solve a problem. Such methods result in what Robert Craigen, math professor at University of Manitoba characterizes as “An out-loud articulation of ‘meaning’; 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.”

Emphasizing understanding at the expense of learning standard procedures holds students up when they’re ready to move on. Furthermore, knowing why a procedure works the way it does will not make a student more proficient at solving problems requiring such operations. Such practices and beliefs have opportunity costs. Many do not “turn out just fine”; rather they are often left not being able to solve problems they cannot understand with procedures they cannot perform.

Great article Barry! I find this quite disturbing "One of those legs is way more important than the others. I argue that the stool should sit upon a strong base of understanding. Most of those procedures and skills are no longer needed (and actually hinder understanding). I definitely do NOT place procedures or skills at the same level as conceptual understanding or applications. And, you can have strong conceptual understanding that will get you to a point where you can engage with interesting mathematics even if the skills are lacking."

Particularly disturbing given the fact that conceptual understanding seems to have no clear definition. I fear that such an attitude likely produces students who neither understand nor have skill. Perhaps it's an excuse to skimp on the practice that it takes to produce students who are fluent with procedures and skills - a way to get off the hook so to speak. After all, skill takes lots of practice and it's hard work.

This is a perfectly articulated treatment of your arguments as you’ve laid them out over the last few months. The statement “procedures and understanding occur in tandem” feels like a math education motto deserving of a t-shirt.

Our year is winding down over the next few weeks, but I would love to introduce a discussion about this topic in summer PD or before the fall session begins. Could I use this post as a starting point for discussion? I feel like you hit all your best points really effectively and in one piece here.

Regardless of that answer, thank you for continuing your advocacy and your posting here.