Friday, September 25, 2015

Connective Tissue and Bone: On Multiplication Facts

Jo Boaler is making the news again for her stance on teaching times tables. I don't wish to wade into that debate, per se, but to consider how personal and idiosyncractic our knowledge of multiplication facts can be.

I wrote this tweet because I am very curious about what mental models or imagery others may have when presented with multiplication facts:

The "see" part was inspired by an incredible Twitter conversation started a while back by @Sue_Cowley and @surreallyno.

The Storify of that discussion is well worth your time. I know it blew my mind.

I don't personally have access to any of the interesting mental models, visuals, strategies or schema that others report having. I just don't I can't wind back the clock to how I learned to multiply. I just have the schema I have. 7x8 ("fifty-six"). 8x9 ("seventy-two).

My working hypothesis is that I learned solely by rote (flashcards and worksheets).  I don't fully remember. If my hypothesis is true, my ability to break multiplication facts down, and find the connective tissue between them, came later, through repeated exposure to them, and lots of practice.

I am not willing to wade into the knowing vs. understanding debate here. I know this fact, and I understand what "64" means. But my 7 year old self didn't have the benefit of the more than 30 years of experience I was later to get with multiplication facts. Did my seven year old self "know", or "understand" that 8x8=64? I learned up to the 12 times tables, but I stumble on some of the 11 and 12 times table facts.

Children should have the opportunity to make sense of numbers, to play with them, and find what I call the connective tissue, the very fabric that makes up all numbers. I won't take a stand against automaticity, as an end goal, but I think the means are more important.* If my school aged self simply memorized the times tables, then didn't do any further work making sense of them, I probably wouldn't have the rich schema for numbers that I have. This is the worst case scenario described by Boaler and others. In my case, I had many more years of math to come, and I use a fair amount of math in my job.

Further, I would have liked things like playing dice or card games to practice facts, or listening to others share their thoughts and strategies during number talks. There's a lot of nice and balanced ideas in this Reddit thread about learning times tables.

I think that, especially in K-8, children should be encouraged to "play" with numbers-find patterns, break them apart, make connections between them. I used to do a lot of lessons with the hundreds square, do you know how many interesting tasks you can build from that?  Again, I am calling this the connective tissue of number (or lack of connective tissue, when it comes to primes, but that's another story). Maybe just learning multiplication facts on their own is all bone, no tissue.

I have since heard stories about professional mathematicians who stumble when presented with certain facts (like 7x8), or fellow teachers who use what they call "workarounds" with things like the 6 times table.  I'm not sure at this point though, what's a "workaround", and what's a "strategy". My mind is my mind, and yours is yours. We need tissue, and we need bone.

*I am also leaving aside the thorny, knotty issues of what constitutes "fluency", "automaticity", and "memorization". Others have covered that quite well.


  1. Hey Matt,

    I feel similarly to yourself, in that I didn't really formulate any work-arounds as I was first learning my multiplication tables. For me, it was a bit of a game with my teacher - how many tables could I learn by the end of the school year? I remember it being really intrinsically motivating to see my accomplishment over the year (we had a start system so that after each table we memorized, we received a start sticker). I also feel that a lot of my number sense came afterwards. For example, one of my favourites is 3^4 = (3x3)x(3x3)=9x9=81 rather than (3x3x3)x3=27x3=81.

    I thought it might be interesting to share my experiences as an adult with multiplication work-arounds. I used to be a table games dealer at a local casino. As such, we had to payout odds of 6:1, 8:1, 11:1, 17:1, and 35:1. The 6, 8, and 11 to 1 odds were OK since I already had those multiplication facts in long-term memory. The 17 and 35 odds were a bit more challenging to learn as an adult. I remember that I learned 17x1, 17x2, 17x3, 17x5 and 17x10 first (which is interesting in of itself). The facts 17x4 and 17x6 were a bit more challenging, and I recall using 17x5 - 17 and 17x5 + 17 for these at first. Eventually, I had to be faster and needed to commit 17x4 and 17x6 to memory - you DO NOT want to keep a busy table of gamblers waiting too long for their money!! It seems to me that the work-arounds did help me start to remember the more challenging facts, or at least gave me a "fall-back" in case I couldn't recall them quickly. Over time, with repeated practice, I did eventually get these facts quicker (much to the relief of my gamblers haha).

  2. Very perceptive and honest. As a youngster I had to use workarounds to allow me to master multiplication facts. I’m not always sure about where a particular workaround originated (Did I pick it up from a friend, or did I “invent” it myself?) Whatever, I definitely relied on them in order to master the “facts”. They were not something that developed after rote memorization of the times tables.

    I don’t think that my workarounds would count as connective tissue: For example, in grade three I learned the 9 times table by recognizing that the two digits always summed to 9 and the first digit of 9 times n is one less than n. At the time, I didn’t know why this was the case, just that it worked and saved me some grief.

    If some mathematicians stumble on the “multiplication facts” then the need for being able to recall them them instantaneously from memory is questionable. Somehow I got along fine without ever completely memorizing them.

  3. Research on cognitive arithmetic:

    Most items: full text available for download.

  4. What an interesting post, Matt! As I was reading it, I was thinking that playing more with these facts could almost provide a link between number sense and spatial sense. Like you, I just see the answers to these multiplication facts in my head (lots and lots of drill and kill types of activities when I was younger), but I needed to really force myself to see an image of what this answer really looks like. Would this image help some students -- particularly ones that benefit from more visuals -- have a great connection/understanding of multiplication? I wonder how this would connect with an overall understanding of number. Lots of food for thought here.