Ultimately, basic math is a performance skill, just like playing an instrument or competing in athletics. And any accomplished violinist or basketball player will tell you that they spend a lot of time doing repetitive drills. Sure, you want to develop conceptual flexibility and help students apply basic math facts, but the foundation has to be built by doing the activity over and over again.
I don't love the comparison with performance skills because I think the grain size is a bit different. If I want to get better at shooting free throws I need to practice to refine my form and build muscle memory. A good start is just to shoot a ton of free throws. But in math there are a few hundred basic facts that we want students to learn. That performance skill logic suggests to teachers that we should focus on repetition, rather than using the principles of retrieval practice. Part of my point is that it's not just repetition, it's designing practice in a way that maximizes retrieval -- and too many teachers focus on repetition at the expense of retrieval.
But in practice "repetition" really is "retrieval"! This is especially true for low-level skills. We're both in agreement that shooting lots of free throws is essential to developing free throw skills. But each time that you shoot a free throw, you are "retrieving" the neural connections for that skill. So-called "muscle memory" doesn't really exist, because your muscles don't store information about how to do things. Instead, when we talk about "muscle memory", it's a metaphor for the complex process of developing the neural pathways to control a physical action.
Let's take your example, that a student needs to have automatic recall of the math fact that 4 + 6 = 10 (and also the math fact that 6 + 4 = 10 as well). There are many ways to help a student understand this, but one of the best ways is to ask that person what the sum of 4 and 6 is, and see if they can get the right answer. That's the most basic form of "retrieval" practice there is -- ask the student to retrieve what they know about 4 + 6. But in practice in order to help the student internalize this concept you have to perform the retrieval process a number of times, and that's repetition.
What would be another way to help students quickly recall the hundreds of math facts that they need to know that doesn't involve repetition and drill?
I think we largely agree, but I do want to split hairs a little bit because some of these small distinctions matter. I definitely agree with you in terms of math facts, because there are so many distinct facts students are doing more retrieval. But repetition isn't as effective for something like fraction division. If I just give students 30 identical fraction division problems in a row they aren't retrieving, they're just rehearsing the same process over and over again. Even a bit of variety -- whole number divided by fraction, fraction divided by whole number -- makes a big difference in what students are thinking about and retrieving. As students become more skilled I can mix in a wider variety -- fraction multiplication, whole number multiplied by a unit fraction, etc. If teachers just take the message to be "repetition and drill are important" they end up structuring practice with too much repetition for many skills. That's why I think emphasizing retrieval rather than repetition is a more useful framing here. Returning to math facts, I think the retrieval framing is helpful as well. Too much math fact practice involves kids retrieving the facts they already know well, and skip-counting or otherwise not retrieving the facts they don't know. Same thing -- I want to ask myself, "what are students retrieving from memory?" every time in order to design effective practice.
What do you think? Is that a distinction worth making?
I agree with Theodore. Repetition and drill is always needed. We should avoid redundant work however. When a student masters a topic, there is no need to do 40 exercises on the same thing
See my comment above for related thoughts, but I would also say that "mastering" a topic feels vague to me. Over and over in my teaching career I have been convinced that students understand something. I don't provide additional practice, and a few days later they've forgotten. The framing of repetition and drill tends to lean towards a lot of practice early on in the learning progression, and put less emphasis on sustained regular practice for days and weeks after first learning it.
Great post DK! This makes me think about executive function -- and that basic skills in self regulation and organization (akin to retrieval) can free up bandwidth for higher order self-management.
Yea I think that captures the distinction I'm talking about. I avoid the phrase "rote memorization" because it is only ever used pejoratively. If I asked my 7th graders to memorize "the derivative of sin x is cos x" that would be rote, the sentence has no meaning to them. In real math teaching there are degrees of understanding, and we are always trying to move up the ladder of more understanding and more complexity.
This is a nice discussion. There is no inquiry, exploration and transfer of learning without memorizing facts first.
There's nothing wrong with repetition and drill!
Ultimately, basic math is a performance skill, just like playing an instrument or competing in athletics. And any accomplished violinist or basketball player will tell you that they spend a lot of time doing repetitive drills. Sure, you want to develop conceptual flexibility and help students apply basic math facts, but the foundation has to be built by doing the activity over and over again.
I don't love the comparison with performance skills because I think the grain size is a bit different. If I want to get better at shooting free throws I need to practice to refine my form and build muscle memory. A good start is just to shoot a ton of free throws. But in math there are a few hundred basic facts that we want students to learn. That performance skill logic suggests to teachers that we should focus on repetition, rather than using the principles of retrieval practice. Part of my point is that it's not just repetition, it's designing practice in a way that maximizes retrieval -- and too many teachers focus on repetition at the expense of retrieval.
But in practice "repetition" really is "retrieval"! This is especially true for low-level skills. We're both in agreement that shooting lots of free throws is essential to developing free throw skills. But each time that you shoot a free throw, you are "retrieving" the neural connections for that skill. So-called "muscle memory" doesn't really exist, because your muscles don't store information about how to do things. Instead, when we talk about "muscle memory", it's a metaphor for the complex process of developing the neural pathways to control a physical action.
Let's take your example, that a student needs to have automatic recall of the math fact that 4 + 6 = 10 (and also the math fact that 6 + 4 = 10 as well). There are many ways to help a student understand this, but one of the best ways is to ask that person what the sum of 4 and 6 is, and see if they can get the right answer. That's the most basic form of "retrieval" practice there is -- ask the student to retrieve what they know about 4 + 6. But in practice in order to help the student internalize this concept you have to perform the retrieval process a number of times, and that's repetition.
What would be another way to help students quickly recall the hundreds of math facts that they need to know that doesn't involve repetition and drill?
I think we largely agree, but I do want to split hairs a little bit because some of these small distinctions matter. I definitely agree with you in terms of math facts, because there are so many distinct facts students are doing more retrieval. But repetition isn't as effective for something like fraction division. If I just give students 30 identical fraction division problems in a row they aren't retrieving, they're just rehearsing the same process over and over again. Even a bit of variety -- whole number divided by fraction, fraction divided by whole number -- makes a big difference in what students are thinking about and retrieving. As students become more skilled I can mix in a wider variety -- fraction multiplication, whole number multiplied by a unit fraction, etc. If teachers just take the message to be "repetition and drill are important" they end up structuring practice with too much repetition for many skills. That's why I think emphasizing retrieval rather than repetition is a more useful framing here. Returning to math facts, I think the retrieval framing is helpful as well. Too much math fact practice involves kids retrieving the facts they already know well, and skip-counting or otherwise not retrieving the facts they don't know. Same thing -- I want to ask myself, "what are students retrieving from memory?" every time in order to design effective practice.
What do you think? Is that a distinction worth making?
I agree with Theodore. Repetition and drill is always needed. We should avoid redundant work however. When a student masters a topic, there is no need to do 40 exercises on the same thing
See my comment above for related thoughts, but I would also say that "mastering" a topic feels vague to me. Over and over in my teaching career I have been convinced that students understand something. I don't provide additional practice, and a few days later they've forgotten. The framing of repetition and drill tends to lean towards a lot of practice early on in the learning progression, and put less emphasis on sustained regular practice for days and weeks after first learning it.
Great post DK! This makes me think about executive function -- and that basic skills in self regulation and organization (akin to retrieval) can free up bandwidth for higher order self-management.
That makes sense -- and in the same way, those skills are often invisible to us once we have them.
That explains the difference between memorization and rote memorization.
Yea I think that captures the distinction I'm talking about. I avoid the phrase "rote memorization" because it is only ever used pejoratively. If I asked my 7th graders to memorize "the derivative of sin x is cos x" that would be rote, the sentence has no meaning to them. In real math teaching there are degrees of understanding, and we are always trying to move up the ladder of more understanding and more complexity.