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.

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.

Thank you for your comment and being a loyal reader. Of course you may use this as a starting point. (You can even make a T-shirt with the motto on it if you'd like! If you do, let me know, and I'll buy one.)

Thank you for your permission. ETA will be either June or August for a discussion, depending on what agenda flexibility we’re allotted.

I teach high school math in the Sacramento area of California. I mostly teach Calculus, so I have extremely driven 11th and 12th graders but many of whom show a diverse array of mathematical holes.

I’m eager to approach my curriculum planning for ‘24-‘25 with your arguments in mind.

Great article! Thank you for describing the situation so well.

If I'm trying to solve a complex math equation that involves applying the invert-and-multiply rule, I want to be able to apply the invert-and-multiply procedure quickly and almost mindlessly, using as little of my working memory as possible. Understanding WHY invert-and-multiply works is of no value to me while I'm in the midst of problem-solving.

Understanding WHY invert-and-multiply works can be of some value when it's initially taught to the learner. At this stage, understanding why it works is helpful because a) it convinces the learner of the procedure's correctness; and b) it helps the learner appreciate why the invert-and-multiply procedure is such a useful one to memorize. However, it's not critical that the learner forever retain the explanation of why it works, since they can always look up the explanation, or reconstruct it, should they feel a need to do so.

Thanks for your comment. I would add that in lower grades, the demonstration is done by a visual graphic using whole number divided by fraction. The fraction divided by fraction proof is best explained after they've had some algebra. In either case, most students do not retain the explanation as I alluded to in the article and I do not believe that it needs to be "obsessed over" so that students must demonstrate and "show understanding" by using inefficient strategies in lieu of the algorithm; i.e., being held up when ready to move on. Teaching the derivation provides both an introduction to presenting the algorithm as well as providing a connection; i.e., they know it comes from somewhere, like the example I used for the quadratic formula. My co-author cannot derive it, but he knows where it comes from; it's not some formula handed to him in isolation. Same with place value explanation of why we move the decimal point when multiplying or dividing by powers of ten.

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.

Thank you for your comment and being a loyal reader. Of course you may use this as a starting point. (You can even make a T-shirt with the motto on it if you'd like! If you do, let me know, and I'll buy one.)

Curious about where you teach and what grades?

Thank you for your permission. ETA will be either June or August for a discussion, depending on what agenda flexibility we’re allotted.

I teach high school math in the Sacramento area of California. I mostly teach Calculus, so I have extremely driven 11th and 12th graders but many of whom show a diverse array of mathematical holes.

I’m eager to approach my curriculum planning for ‘24-‘25 with your arguments in mind.

Great article! Thank you for describing the situation so well.

If I'm trying to solve a complex math equation that involves applying the invert-and-multiply rule, I want to be able to apply the invert-and-multiply procedure quickly and almost mindlessly, using as little of my working memory as possible. Understanding WHY invert-and-multiply works is of no value to me while I'm in the midst of problem-solving.

Understanding WHY invert-and-multiply works can be of some value when it's initially taught to the learner. At this stage, understanding why it works is helpful because a) it convinces the learner of the procedure's correctness; and b) it helps the learner appreciate why the invert-and-multiply procedure is such a useful one to memorize. However, it's not critical that the learner forever retain the explanation of why it works, since they can always look up the explanation, or reconstruct it, should they feel a need to do so.

edited Apr 21Thanks for your comment. I would add that in lower grades, the demonstration is done by a visual graphic using whole number divided by fraction. The fraction divided by fraction proof is best explained after they've had some algebra. In either case, most students do not retain the explanation as I alluded to in the article and I do not believe that it needs to be "obsessed over" so that students must demonstrate and "show understanding" by using inefficient strategies in lieu of the algorithm; i.e., being held up when ready to move on. Teaching the derivation provides both an introduction to presenting the algorithm as well as providing a connection; i.e., they know it comes from somewhere, like the example I used for the quadratic formula. My co-author cannot derive it, but he knows where it comes from; it's not some formula handed to him in isolation. Same with place value explanation of why we move the decimal point when multiplying or dividing by powers of ten.