I wrote this tweet because I am very curious about what mental models or imagery others may have when presented with multiplication facts:
Is there anyone out there who *hasn't* memorized their times tables, and is willing to describe what they "see", or their workarounds?— Matthew Oldridge (@MatthewOldridge) September 19, 2015
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.
An amazing array of responses https://t.co/Uaoq69oagr generated by @Sue_Cowley 's question! #mathchat Constructivism at its best.— Cristina Milos (@surreallyno) July 6, 2014
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.@surreallyno @Sue_Cowley I can honestly say I have no imagery whatsoever when I think of a multiplication fact. I just hear the number.— Matthew Oldridge (@MatthewOldridge) July 6, 2014
So if you say, 8x8, I think "64". I "see" nothing. But I can break down this fact many ways.Learned rote, then made sense later? @Madame_JB— Matthew Oldridge (@MatthewOldridge) September 24, 2015
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.