Dec 23, 2022Liked by Michael Pershan

This makes me wonder if the Montessori (as expressed by AMI, Montessori being a broadly and weirdly applied label) approach to addition fits the research best. Briefly, in an AMI classroom students have the "addition strip board" that lets them quickly build a mathematical model and see the answer without always counting.

Expand full comment
Dec 25, 2022Liked by Michael Pershan

Karen Wilding, an early maths specialist in the UK, advocates a similar approach. She gets children to model a calculation using ten frames and to find the answer by subitizing instead of counting. Combined with a lot of work on composition of number, this leads to children being able to calculate using memorised facts once the ten frames / visual supports are removed.

Expand full comment

I'm not really familiar with it, but I'd want to just be sure that at some point kids are remembering the fact, not just using a visual or counting.

Expand full comment

Michael --

In his blog post “Rehearsal first; retrieval practice later - an important distinction,” Tom Sherwood points out that there are many ways to use flashcards that do not work well. One is failure to do enough rehearsal first.


Cognitive experts say we need to give students one way to solve problems (like standard algorithms – or recall based on rehearsal) before teaching them multiple strategies (such as how to calculate the facts). If students need to choose between methods, some of which they cannot manage cognitively in the more complex cases -- it quickly results in overload of their exceptionally limited working memory – leading to problem solving failure. Kirschner, Sweller, and Clark discuss this in their seminal 2006 paper. This is just another case of where what does make sense mathematically is the wrong strategy to teach children because it overloads working memory.

Cognitive science also says learning multiple strategies with similar steps also results in mistakes because of “interference:” students mis-remember which similar steps go with which procedure.

What the science is suggesting is, for math facts, start with substantial rehearsal, then do retrieval. When they have over-learned the facts so they have an automated strategy to apply that works every time (retrieval), then get into variations that may indeed help with conceptual understanding, and at an older age they may more often have the WM capacity and factual recall needed to use the strategies successfully.

The problem with the NCTM "teach strategies" approach is that it denies what scientists have found is true about how the brain works. Science denial hurts kids.

Expand full comment

You are absolutely right that if working memory is a static capacity that can't be improved by mental exercise then trying to teach struggling kids to reason rather than memorize is a losing proposition, since reasoning just requires too much cognitive load for that individual. I hadn't thought about that till I read your comment, it sent me on a bit of rabbit hole dive.

Looks like the scientific consensus on whether working memory capacity can be improved is divided. For example, Klingberg, T. (2010). Training and plasticity of working memory. Trends in cognitive sciences, 14(7), 317-324, suggests improving cognitive capacity is possible, whereas Redick, T. S., Shipstead, Z., Harrison, T. L., Hicks, K. L., Fried, D. E., Hambrick, D. Z., ... & Engle, R. W. (2013). No evidence of intelligence improvement after working memory training: a randomized, placebo-controlled study. Journal of experimental psychology: General, 142(2), 359, claims there is no evidence supporting such a conclusion. I couldn't figure out a way to read more than the abstract for these papers, so I can't give credence to one over the other. Do you have any further insights?

Expand full comment

Cameron -- Working memory is a complex topic and there is extensive more recent research in the field. But the consensus among experts is that in general, working memory capacity for the types of visual and sound data involved in academic learning, most adults have a maximum capacity of 3-5 chunks that is biologically determined. The good news is that no matter what your capacity is, WM limits can be worked around by practice that automates quick recall of facts and procedures from LTM in a variety of distinctive contexts. The evidence is "that educational achievement gaps come from differences in learning opportunities and that better access to such opportunities can help close those gaps." See https://doi.org/10.1073/pnas.2221311120

Expand full comment

If you want to teach children multiplication facts and prevent them trying to derive, there is a very simple way: songs!

Look up 'Children Love to Sing'. Their times table songs are on YouTube and Spotify. First the children sing the facts in order, then the song mixes them up for a "random test".

Something this easy should not work this well but it really does. I was introduced to this approach as an inexperienced teacher several years back and within a couple of months, my entire class of 8-year olds had memorised every multiplication fact up to 12 X 12.

Try it!

Expand full comment

Michael, I like your deep dive into why derivable facts aren't remembered as well if students can keep deriving, thanks.

I wanted to note that in my reading of some cognitive research (summarized in Make It Stick & elsewhere), the act of *writing* is better for memory & consolidation than *reading*. So I've found that making the flash cards is more important than using the flash cards [yes, using them properly is a retrieval self-test, and that more beneficial for learning than reading, of course].

So I've had my high school students write out the things they need to memorize, such as trigonometry facts or derivative formulas; write them every time you need them until they are automatic. That seems to #MakeItStick for them; what do you think?

Expand full comment