Posted on March 13, 2023


First day of Spring Break and I’m at the office; yes, that non-student has said good morning and availed themself of a snack from the snack bar, and I REALLY DO WANT TO SPEND AT LEAST 5 OF THE 28 HOURS (one of ’em is gone already) on the MOODLE course and 6 on the D2L one, but

Michael Pershan — the winner of the MTT2K first prize about Doing Better Than K*** Academy — has gathered research on … learning multiplication. Which means that a: I’m going to go through it and respond and b: I’ve got a model for something to do with my “annotated research on developmental math.” YES


“I think timed memorization activities that accidentally prompt students to use computational strategies can actually cause math anxiety, and it’s best to either use flexible timing or figure out ways to make sure students aren’t using strategies. “

He notes that there is hardly ***anything *** about subtraction or division in the research and I agree that this is worth of much more notice and … it inspires me to use some of those 5 hours to INCLUDE DIVISION in the same way we include subtraction.

He notes research showing that students *and* adults know a bunch of facts, but calculate some of them.

This reminds me of the whole-language proponents who claim that because adults don’t sound out words, we don’t need to teach it. People, this is like musicians using different strategies — but lots of practice with the skills when learning is still rather viable, eh?

He says yes, people should learn the facts by heart, especially because: FRACTIONS and everything else really depend on it, tho’ the research notes that mult. fluency was “uniquely predictive fraction procedures” (need an of in there?) …

Then: Would a cheat sheet help? Apparently no research on this.

Sue’s observations about using times tables charts:

In our PRe-Algebra course, a sheet w/ times tables to 25 and addition to the same (I think) is included in the excellent and downright affordable “notes” packet students purchas (the ALEKS subscription is not so affordable…).

Students are allowed very limited use of calculators. Many exams have a “no calculator” part you have to finish first before you can do the “calculator” part. My filtered sample of students who come for tutoring use that thing. I *know* that other students — many of whom fail the course — just use a calculator whenever they can. The ones who hang in there tend to learn a lot of basic math that they missed in K-12 because… they just used a calculator.

It *does* reduce the cognitive load, but not as much as knowing the facts does, because … you have to go look it up and find those other numbers. I would love to think that the “looking up” process facilitates making connections but, to be honest, I think they scan for the number, period. They don’t see the earlier factors and ‘skip count.’ Sometimes they do start to remember some of them — and two of ’em reveled in the novelty of “it just came into my brain!!!!” because in their previous years that had never happened. (I’m pretty sure “learning the times tables” was not part of the curriculum.)

Okay, here’s a thing: “multiplication is typically understood at an early age.”

Unless it’s not. I HOPE this gets addressed: the research cited states that “students already possess the fundamental conceptual capabilites for conceptualizing multiplication.” THAT IS NOT THE SAME AS DOING IT. Just memorizing the facts doesn’t do it, either, for too many people, and there must be a whole mess of “atypical” people out there who end up in college not understanding multiplication.

Okay,then the “should they learn strategies or go straight to memorizing? and apparently there is ONe Whole Study about that. NOTE: That article was in “Learning Disabilities Quarterly.”

Now we get to the idea that using strategies doesn’t get you to automaticity and while he doesn’t have citations, his logic is consistent with my experience with both strategies and using charts: you’re finding the answer, not making connections. Interesting that there doesn’t seem to be *research* about using some facts to get to others. I know there are ample materials out there suggesting “learn the 1,2,5 and 10 and you can get the rest.” I know Steve Chinn’s materials include it.

Michael thinks students should memorize the little ones, *teach more strategies using the facts,* then learn big ones.

I think this is an awesome way to get from the rote to the conceptual, *especially if you teach division in there, too.*

I appreciate that he simply states ways to memorize (assorted repeated retrieval practice), and states that yes, timing it means you’re retrieving, not computing, and that’s a good thing. He notes that yes, when people reflect back, timed tests were sources of anxiety and taking time limits away helps alleviate math anxiety — and cites an article by Steve Chinn 😉

Okay, goin’ to the top of the page because this sentence is particularly pithy.

Then he talks about incremental rehearsal — the “do first fact, then 1 and 2, then 1 and 2 and 3” — alas, without suggesting that this can be made a lot better and adapted in different ways.

Then he notes that there’s not much research on the kinds of routines (like flashcard drills and other incremental rehearsal) used in small group, when they’re tried in big groups.

And he finishes with “not really liking mnemonics.”

So, my thoughts right this second? What about the Tzur et all amazing and fun “build the multiplication concept” idea?

… and now, there’s a student here and/but I’m going to pull up Moodle.