We’re really, really trying to avoid new WitCHes right now, but people keep throwing them at us. This one comes courtesy of Simon the Likeable, who suggested we watch the following, and who will pay dearly for it.

**UPDATE (11/09/20)**

Commenter has SRK has flagged a related video (and will pay dearly for it):

I’ll be interested to read the comments on this one, because in one sense I have some sympathy for the point that the video is trying to make: Here’s a bunch of mental models for multiplication, but if you push on them, lots of problems start appearing, so perhaps we should not try to reduce our understanding of all cases of multiplication to just one of these mental models. Multiplication is as multiplication does.

My charitable interpretation of the video is something like: (i) let’s take for granted how multiplication works with natural numbers, and write down some of the general properties. (ii) how does multiplication work for integers? let’s assume that it follows those properties we liked from multiplication of natural numbers, and see what the consequences are. (iii) how does multiplication work for rationals? Same thing. A lot of the other discussion is to point out that, to the extent that those other mental models work, (repeated addition, area, scaling, “of” etc.) it’s because those mental models follow those general properties, not because they are the essence of multiplication.

Thanks, SRK. Your summary of the video seems clear and accurate, and doesn’t need to be characterised as charity.

SRK,

What is the essence of multiplication?

(Is that not a good question to ask?)

How should we think about multiplication, if we are a mathematics teacher?

How should we think about multiplication, if we are a mathematician?

Good questions.

Storyteller, I’m flattered that you might think I have something worthwhile to say about this.

1. My answer to your first question is the one you have anticipated. It may be that the question is well-formed, in the sense that there is a true statement of the form “the essence of multiplication is …” But I’m sceptical that the answer would be informative or useful to a teacher of mathematics (perhaps even a mathematician, but I’m not qualified on that).

2. As to your second question, I’m inclined to have a fairly ecumenical, lightweight approach. For the purposes of teaching high school students, multiplication is as multiplication does. So assuming that they’re au fait with multiplication of the naturals, then we can justify the rules for multiplication of all integers, then the rules for multiplication of rationals. “Justification” should here be understood in the sense of making the rules seem like a reasonable way for multiplication to work, which helps students reliably remember and apply them. Real numbers are tricky, because we really only deal with rules for “simplifying” products of irrationals that can be written as roots, but the same approach seems reasonable. Beyond this point, things diverge depending on what VCE subject students are taking:

a) Students taking general / further mathematics need to understand multiplication of matrices. For this cohort, an approach using linear transformations is out of the question, so it’s essentially just teach the rule (row x column).

b) Students taking specialist encounter complex numbers. Here I just state that (a + bi)(c + di) is defined to follow the same multiplication axioms as for reals, with i^2 = -1.

So how should high school teachers think of multiplication? I guess in summary I would say that it’s best to not get too attached to any one mental model or analogy. It’s most important that students can DO multiplication (be it of integers, rationals, reals, complex, matrices, etc.) accurately and efficiently. Of course there is the issue of giving students strategies to recognise WHEN multiplication is an appropriate thing to do, and that is where mental models and analogies can be useful. But I’m sceptical that the best way to do this is to reduce all of multiplication to one analogy, because that would involve trying to explain one analogy in terms of another, and confusion is the likely result.

I appreciate your response SRK and enjoyed reading and thinking about it.

It is clear you had something worthwhile to add.

It struck me while reading it, and reflecting on other things I have read about this issue, that how we are situated, now and in the past, effects (for better or worse) how we think about things and thus what we pass on.

How we are situated can make it hard to listen to the wisdom of others situated differently (especially more highly).

Re your point that it is “most important that students can DO multiplication”.

It seems fairly widespread now that, in the formative years, no matter what, understanding is #1 and can DO means different things to different people.

A discussion centring on the relationship between doing and understanding (in mathematics) would make for a good conversation.

On the point about “understanding” and “doing” multiplication in the formative years (which I understand to be primary school, or perhaps early high school).

I have a lot of sympathy for one aspect of the view given in the video: if you haven’t got 7 x 8 at your fingertips, then you should at least have some useful strategies for working it out. If that’s what we mean by “understanding”, then I’m all in favour. But I’d be surprised if anyone really disagrees with that. I think the disagreement is with people who think that being able to “work out” 7 x 8 is a *substitute* for automatic recall of basic facts, rather than a *complement* to it.

Talk of “understanding” maths is notoriously slippery. I think everyone wants students to “understand” maths, but we need to be clear and specific on what that understanding looks like. I find that often the word “understanding’ is used in pernicious ways to denigrate what students have learned because there is something else that the student hasn’t learned. For instance, we might say that the student who has rote-learned 7 x 8 = 56, but wouldn’t think of using 7^2 + 7 to “work out” 7 x 8, does not “understand” multiplication (or perhaps their 7 times tables, or whatever). Maybe, but so what? Knowing your times-tables is useful. Having, in addition to that, more flexible strategies is great, and no doubt is better than just having rote-learned the times-tables. But describing this in terms of the presence or absence of “understanding” seems unhelpful. I would prefer to think of this as: what do we want students to be able to do, can they do it, how does it help them learn more maths in the future?

Thanks, Storyteller and SRK. I think you are discussing about as intelligently as one can what amounts to a pointless, and thus very damaging, question.

I am currently introducing multiplication to my 6-year-old, earlier than planned, because she ran into a chanting tables song-video (in Mandarin). She loved the patterns, but had no sense what the patterns meant, what 2 x 4 or 4 x 2 meant, so I figured to show her. We sat down for about ten minutes with cuisenaire rods, and now she knows what multiplication *means*, at least until she gets to fractions or negatives. Now she can get on with learning what the multiples *are* and how multiplication *works*.

In brief, what’s the big fucking deal?

I’m not entirely sure, but my sense is that there are really two points of action here, which are related.

1. One debate is about what we want students to learn in their early mathematics education. I think some people genuinely do not believe that having automatic recall of timestables is all that important and valuable. These people would instead emphasise puzzle solving, playing with manipulatives, having multiple strategies for working out basic facts, etc. I think this debate involves a combination of empirical and ideological issues, and the fact that these aren’t clearly disentangled leads to a lot of talking past one another.

2. The second debate is about the very hard problem of how to build a student’s ability to transfer learnt skills / concepts to unfamiliar contexts. I think there’s a view that memorising timestables, or memorising a standard algorithm for multiplication, isn’t conducive to learning transfer. What these students are missing, so the thought goes, is a “deeper understanding” of what multiplication is. If these students possessed this understanding, they would then be in a better position to recognise when multiplication can be applied, and thus would be more successful at learning transfer.

Thanks, SRK.

Re 1. Yes, there are definitely people who don’t see any value in learning the tables by heart. These people are idiots.

Re 2. I totally agree that the automatic tables and memorised algorithms are not sufficient. Kids also need ten minutes of cuisenaire rods.

No BFD when 10 minutes with CRs etc. works.

When that does not work, BFD. And that seems to be how it all started.

A question I would like to know the answer too is, just what % of kids does what you have and are doing, not work?

No, it’s not a BFD. It’s an SFD.

Can you ask your question again (preferably with the spelling and grammar sorted)?

With what percentage of young learners would your method fail?

Thanks, Storyteller. (“For”, not “With”, but I’ll now leave you alone.)

It’s a fair question, and we both know you have much more idea than me what the answer might be. Not every kid has an attentive, mathematically trained and underemployed father to watch what is going. I also have the sense that Lillian has a strong innate number sense; she seems to pick up the ideas quickly. So, I was being flippant to suggest it would always, or typically, be that easy.

But I still think it’s fundamentally an SFD, unless *no one* pays attention for the whole seven years of primary school. THEN it’s definitely a BFD.

The first thing is, asking WHAT multiplication is, is trivial and distracting in the context we’re currently discussing, of lower primary kids getting going. Multiplication is repeated addition, and we’re done. (Maybe “multiple addition” is a more accurate term – adding this many of those at the same time.) 3 x 5 means three yellow rods in line. 5 x 3 means five green rods in line. That’s it.

No matho-philosophical discussions are going to alter that, or help kids one bit. Sure, there are “models” of multiplication, which amount to pictures of HOW it works, but the idea that the “models” offer more practical value than that, either to teachers or to the kids, is silly.

The second thing is, I’m really sceptical, pretty much always, of the idea that deep and prolonged worrying of meaning help kids in maths get it. To have no sense of meaning, to reduce the maths to symbol manipulation is of course crazy, and that definitely happens. But there’s usually not *that* much meaning to be had with a new concept or a given misunderstanding. And trying to indicate the same meaning ten different ways is an obvious recipe for futility.

What the kids need, almost always, is lots and lots of practice, on a clearly delimited set of facts and techniques, to see how the meaning plays out, to get a feel for the meaning.

I will try to do better with me grammar. 🙂

So a game of balance and common sense. ✅

How did we come to deviate from that?

Thanks, storyteller, and yes. “Common sense”, definitely. “Balance”, too, although that word suggests a 50-50 thing, which is wrong. Not that we have units, but 90-10 or 95-5 gives the better sense.

I address these points in my talk, at 44:00 and 46:20.

Re balance, it does.

However, the notion of a balanced life has never meant equal (at least to me). I need to find a new word/phrase.

I can recall three clear moments when change was born and subsequently gained momentum.

1. The claim from universities (well, some people in universities with power at that time) that too many high school graduates could do — but not understand (mainly calculus).

Greater understanding was demanded.

2. The desire that more/all students were successful.

3. Learning mathematics, in the primary and lower secondary school years, was a cruel experience for many students.

As a result, slowly, methods/… that did work, for some, were largely shelved — for all.

Also, what was once accepted as success — changed.

It is open to opinion whether or not what replaced those methods/… work for more/all and whether or not what success became is better/of use/…

Of course those three moments happened during the period in which … (things you referenced above, and much more).

Storyteller, I think 2 and 3 have probably been more powerful underminers than 1.

1. provided reason to change what happened in the lower years. The changes to assessment in the senior years played a significant role.

Prior to 1., people resisted change in the lower years, like that advocated by 3., because of the need to be prepared for the senior years.

I’m still not convinced. I’m not sure advocacy from the universities pushed anything. It seems to me that either power was taken from the universities to control school education, or the universities simply relinquished that power. In any case, I don’t see universities doing much now except whining about the consequences.

I guess this is a follow up video to the one above, and relates back to an early post on multiplication tables… https://www.youtube.com/watch?v=ylrFTLcRdIc

I found this one FAR worse than the one above.

Jesus. This guy is seriously getting up my nose.

I wonder how he would go about learning the capitals of the world.

Well the capital of Mexico is Mexico City, the capital of Panama is Panama City, so I guess he could thereby reduce the problem of learning the capitals to the problem of learning the nations.