And the problem with many existing times tables practice systems.
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A discussion I had with Jason a couple weeks ago:
Once we have quick-retrieval “math facts” practice integrated into our system (times tables, etc.) we’ll want to extend that to higher subjects for similar practice with trig identities, derivative/integral rules, probability distributions, etc.
This discussion was spurred by a report from a student who said they were having trouble remembering basic derivatives like $\dfrac{\textrm d}{\textrm dx} \left[ \sqrt{x} \right]$ and they would always end up explicitly working out every step, even though they had accumulated tons of practice doing that.
It struck me as being is exactly like what happens when a student doesn’t memorize their times tables and instead recalculates the multiplication each time instead of explicitly practicing their recall.
But the first step is going to be setting up this kind of practice environment for math facts in arithmetic. It’ll be rapid retrieval practice, starting with small sets of facts and gradually combining them into larger groups and focusing on the hardest facts.
We’re going to make sure to avoid the failure mode that plagues so many times tables practice systems out there: when your practice is randomized across the entire times table, or even if you limit it to a longitudinal or lateral subset (e.g., “just facts involving 7 and 8”), you end up serving up way too many easy facts.
Think about it like this: in a 12x12 times table including 0, over a third of the facts involve 0, 1, or 2. Over 70% of the facts involve a small number (0, 1, 2, 3, 4, 5).
So if you say that the student needs to answer some number of facts correctly in some amount of time, maybe even subject to a loose accuracy threshold, and you choose these facts randomly across the table, or even if you limit to a longitudinal or lateral subset later in the table… you’re still going to be serving up a crap-ton of easy facts and the student could quite easily “pass” your success criterion by nailing the easy facts while struggling on facts that are even just moderately challenging.
The way around this is to be very careful about what subsets of the table you’re practicing on, and very careful about how you select questions to test the student on. In particular:
- After the student learns the easy facts, they need to quizzed on subsets of the table containing harder facts and leaving out easy facts.
- Each individual question must be timed. If a student doesn’t respond quickly enough, they miss the question. (Of course, this timed retrieval practice should come AFTER the student is able to compute results untimed.)
- And if they ever miss a question, it needs to come up again (spaced out with some other questions in between) and they have to get it right.
There should be no way to get through the tasks without actually knowing the facts. The system has to be like the Terminator and hunt down what facts you don’t know, serve them to you, and force you to learn them. There should be no way for a student to “get by” despite not knowing some facts. No place to hide.
It’s not good enough for a student to know the easiest 80% of facts in the table. They need to know 100% of the facts and their practice always needs to be targeted to the facts on which they are the shakiest (while, of course, scaffolding up to harder facts by first covering any easier “prerequisite” facts that the student hasn’t yet learned).
Last summer, I started mapping out our math facts curriculum, generating a bunch of questions/content, hooking this up in our knowledge graph, thinking about the question selection algo… of course, as it often goes, I got pulled onto other higher-priority opportunities that arose before we could close the loop. (Looking to get back on this in the spring, hopefully.)
But one really interesting thing I found while looking through the research on multiplication instruction was that… well, to put it bluntly, a lot of it seemed to lack common sense. And I’m not just talking about modern stuff (e.g., Boaler) – the lack of common sense stretched back at least 40 years ago.
For instance, consider this quote in the widely cited paper Knowing, Doing, and Teaching Multiplication back from 1986:
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*“The possession of principled understanding is thought to enable the knower to invent procedures that are mathematically appropriate and to recognize that what he or she knows can be applied in a variety of different contexts.
Someone who has a principled understanding of multidigit multiplication might figure out 76 x 8 in the following way: ‘76 is 1 more that 75, which is 3 times 25; to get 8 x 75, then 8 times 25 is 200, 3 times 200 is 600; now we need 8 x 1, which is 8. So 76 x 8 equals 608.’”*
If you want to do 76 x 8, why wouldn’t you just do 560 + 48 = 608? That seems way, way easier, and it’s still easy when the problem turns into 86 x 8 (whereas the strategy above breaks down). If you have developed instant recall on your fundamental facts, I just don’t get the sense that any of those strategies are useful.
Here’s my take on arithmetic strategies:
- The point of a basic arithmetic “strategy” is really just to get the kid to understand the “why” behind a fact before they memorize it (and ideally to also make it easier to memorize).
- It’s good to start with a computation strategy so that the kid knows what the heck the fact actually means, but once a kid is able to compute a fact using a strategy, I think they need to immediately be thrown into instant recall practice so that they don’t start using the strategy as a crutch. The longer they practice with the strategy beyond the point of baseline conceptual understanding, the harder it is going to be to get them to stop using the strategy and perform instant recall instead.
- Other educational services sometimes go crazy with tons of alternative strategies for math facts, and I think that’s a mistake because not only does it delay the transition to retrieval practice, but having too many strategies to choose from can also “paralyze” a kid.
- For instance, many resources teach “+8” as “add 10, then subtract 2.” But in my experience kids do just fine counting upwards as long as they’re using benchmarks along the way: for instance, to do 9 + 8, you can just think “starting at 9, go up 1 to 10, then we need to go up 7 more to 17.” And if they learn “add 10, then subtract 2” then some kids will get confused about when they should do that and when they should just count upwards.
- Likewise, many resources teach “x8” as “double it, and then double it again, then double it again.” But who actually remembers their “x8” facts that way? Only the kids who are slow at arithmetic. The kids who are fast just recall the x8 fact without thinking. They know that they could compute 4 x 8 as 8+8+8+8 and that’s sufficient for understanding what 4 x 8 means. Sure, some students are naturally going to think of 8+8+8+8 as two groups of 2x8 and compute it that way, but if a student isn’t thinking of it like that, it seems like a waste of time to force them to learn a “derived fact” strategy that doesn’t come naturally to them. In the end this is all going to be memorization anyway.
- For instance, many resources teach “+8” as “add 10, then subtract 2.” But in my experience kids do just fine counting upwards as long as they’re using benchmarks along the way: for instance, to do 9 + 8, you can just think “starting at 9, go up 1 to 10, then we need to go up 7 more to 17.” And if they learn “add 10, then subtract 2” then some kids will get confused about when they should do that and when they should just count upwards.
- That said, I think it would also be a mistake to ask kids to memorize facts before they can even compute the facts from scratch using some strategy – otherwise, they probably won’t understand what the facts even mean, which will make it really difficult to memorize the facts. (For example: if someone doesn’t know how to compute multiplication using repeated addition, they’re not going to understand why 3 x 5 is bigger than 3 x 4, they’re not going to understand why you can find 15 / 3 by recalling the associated multiplication fact 3 x? = 15.)
Anyway, it’s 1:37am and I could sit here and keep ranting away into the night but I should probably go to sleep.
TLDR: math facts practice is on the roadmap and when it comes it’s gonna be the best you’ve ever seen. It’ll be peak efficiency and nobody is going to get through it without actually knowing their facts;)
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