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.
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Thursday, September 10, 2015

What Things In Our Math Curricula Do You Think Have Become Outdated (...Or Even Obsolete?)

The always interesting Keith Devlin writes here about the future of math. As a mathematician, he has a good perspective on the ways in which math education has changed. For Devlin, our education systems are lagging behind the world of math. Math has changed, but has how we teach it?

 I will quote a whole paragraph here:
"To most people, mathematics means applying standard techniques to solve well defined problems with unique right answers. They have good reason to think that. Until the end of the 19th Century, that’s exactly what it did mean! But with the rise of the modern science and technology era, the need for mathematics started to change. By and large, most people outside mathematics did not experience the change until the rapid growth of the digital age in the last twenty years. With cheap, ubiquitous computing devices that can do all of the procedural mathematics faster and more accurate than any human, no one who wants – or wants to keep – a good job can now ignore that shift from the old “application of known procedures” to new emphasis on creative problem solving."

Jordan Ellenberg is also quite persuasive on this topic, and if you haven't read his book, How Not To Be Wrong, you must. Machines can do the laborious work of long calculations, so our brains are freed up to do what we do best-think.  Conrad Wolfram is another who sees computing as a large part of the future of math.

We may not need to do long laborious calculations by hand, but we do, of course, need basic number sense, combined with our intuitions and our operational skills. One example from my own schooling is calculating approximations of square roots by hand- I remember doing it in school, but I am not aware of anyone who teaches it now. Of course, if you are a certain age, you remember having to use slide rules and log tables. Probably nobody is nostalgic for that!

I got to thinking today, after an interesting conversation with @MathiesUnite. We talked about how, with the dawn of Desmos, drawing graphs by hand is a less useful skill than ever. Have you ever planned a lesson that involved drawing a graph, and looked around 20 minutes later, and some kids hadn't even put the scales on the axes yet?  I have. I have seen terrible hand drawn graphs. The worst. I have seen graphs so scary bad they  didn't even look like any more than a bunch of squiggles on a page. Why  should we not use the tools at hand to draw graphs more effectively?

But Desmos (and others) are so much more than that-you can compare two data sets, at a glance, within minutes, freeing up time for discussion, interpretation, and analysis. Nobody would argue that we don't need good data interpretation skills these days.  

 @BrianPenfound offers some pushback on the idea of digital tools replacing graphing by hand completely:

The discussion got me thinking, and I posed this question, later:
That's not to say our students will do FiveThirtyEight level data visualizations, nor that they don't need to know how to use conventional graphs. I just wondered, in the age of pictographs, incredible fan-made baseball data visualizations, and so on, if that might be another "new" area of math we should explore with our students.

 The definition of data visualization offered by Wikipedia makes me wonder if it might not be its own literacy, on its own.

So graphing by hand might be one, do you have any other ideas for items from math curricula that you suspect could be obsolete, or, at the least, outdated in their conception?




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