When Asking For 3/4 + 6 – 1/4 is Racist

Oh, Canada.

We’re very late to this one and it is way outside our territory, but it feels necessary to post something. In December the Ontario Superior Court, in a 3-0 decision, ruled that requiring prospective teachers to pass Ontario’s Mathematics Proficiency Test (MPT) violated the Canadian Charter of Rights and Freedoms. The reason? According to the Court, the requirement unjustifiably

… has a disproportionate adverse impact on entry to the teaching profession for racialized teacher candidates …

Really.

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WitCH 77: Road to Nowhere

If nothing else, people can enjoy a great song.

This one puzzled us when we gave the 2021 Further Mathematics Exams a three-minute scan, but we didn’t bother then to think, um, further. Now, however, since graph theory is likely a mandatory part of Specialist 1&2, and thus also, at least technically, a mandatory part of Specialist 3&4, we have thought a little harder. So, after discussions with John Friend and with a colleague who we shall refer to as Professor Combo, here we are.

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Specialist Mathematics 1&2 Note-Sharing and Idea-Sharing

Perhaps people are done, but given the previous discussion, here and (weirdly) here, and a couple requests, it seems worth a shot.

Given VCAA’s pigheaded ramming through of their still-non-existent and now invisible new study design, some teachers have expressed concern about what and how to teach in Specialist Mathematics 1&2 in 2022. One teacher, we’ll call them Mr. Puzzled, emailed us:

I am a 11 & 12 specialist maths teacher and have been spending countless hours trying to prepare some sort of course outline for the upcoming year without clear advice from VCAA on what we need to cover. Continue reading “Specialist Mathematics 1&2 Note-Sharing and Idea-Sharing”

Another Fraction Question

OK, following up on our previous post, we have another fraction question. This one is not new, has appeared in the comments of a previous post, and many readers will have heard us bang on about it. Nonetheless, given the discussion on the previous question, and given the possibility that some new readers might not have yet read Marty’s Collected Sermons, it seems worthwhile giving the question its own post. Here it is:

    \[\color{Mulberry}\boldsymbol{(-1)^\frac26 \overset{?}{=} (-1)^\frac13}\]

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A Fraction Question

We have a question. One of the simplest possible fraction multiplications is

1/3 x 1/5 = 1/15

The question is, why? What is your simplest explanation of this equation?

To make the task absolutely clear, the explanation is to be suitable for an imagined average primary school student who is just being introduced to the notions of fraction multiplication. Explanations targeted at Peano or a Child Gauss or whomever are way, way off the point.

 

UPDATE (05/01/22)

Thank you to everyone for your comments and suggestions. They were very interesting to read. I’ll now give my own answer to the question, at which, of course, commenters are welcome and encouraged to hurl their objections. Just to be clear, I am, for once,* not pretending to be an expert. As with most commenters here, the teaching of such notions at the (upper?) primary level is well outside of my normal world. The question arose from my general concern with the undeniable awfulness of most primary level education, from specific and very enlightening discussions with Simon the Likeable, and from pseudo-teaching my nine year old daughter.

To be absolutely clear, my question was in no way intended as a trick, or as a surrogate for some other question, even if other questions naturally arose by implication. My question was intended exactly as stated:

What is the simplest explanation of 1/3 x 1/5 = 1/15 for a primary student just being introduced to fraction multiplication?

For this purpose, a deep-end definition of a/b x c/d is obviously inappropriate. At the other pole, the explanation of 1/3 x 1/5 of a pizza does not answer my question, even if it might be a valuable component of such an answer. I’m far from convinced even of that.

I agree with many of the commenters that the natural starting point is to interpret – in effect, define – “1/3 x” as “a third of”. All young children are introduced to fractions as “fractions of”: a third of a cake or whatever. We then use “1/3 x” to capture the notion of “a third of” in a precisely mathematical and symbolically manipulable way. This interpretation/definition is equivalent, as it must be, to defining “1/3 x” as “÷ 3”, but I don’t think that need concern us here; the “third of” expression is more intuitive for the intended audience.

Most easily, of course, children can take a third of suitable integer multiples: three cats or six apples or one Djokovic. That is, quantities that we are happy to divide into three equal groups. This easily extends to fractions of suitable natural numbers: a third of 6, and so on. As such, having students get used to 1/3 x 6 and the like is relatively straight-forward. (1/3 x 7, for example, is an important side story, but it can be left for later.)

But how, then, finally, does one explain 1/3 x 1/5? I’ll take it as agreed that it means “a third of 1/5”, but how do you take a third of 1/5? I think there is a simple and compelling answer.

The problem is that 1/5 is not “three things” (or a multiple of three things), as was the case with the cat clowder above. That leads many to replace 1/5 by a fifth of a pizza, and then out come the knives.

But wait a minute: 1/5 is three things. 1/5 is, of course, exactly three fifteenths. That is, the fraction 1/5 equals the fraction 3/15.** And, a third of 3/15 is obviously 1/15. In symbolic but easily decipherable and easily motivatable form,

1/3 x 1/5 = 1/3 x 3/15 = 1/15

This seems to me, by far, the most natural and simplest introduction to and explanation of non-trivial fraction multiplication. All that is required, which I agree is not a gimme, is a proper understanding of equal fractions and genuine automaticity of multiplication. (On the other hand, if a student does not have this background, the idea of their understanding fraction multiplication, or performing it well, is farcical.) But, except in conversation with Simon the Likeable, I have never seen such an approach clearly expressed.***

That’s it. Have at it.

 

*) Do not expect this to be repeated.

**) People who write that the two fractions are “equivalent” will be summarily shot. 

***) Definitely some commenters were suggesting similar approaches, but these approaches seemed to me clouded with extraneous ideas or activities. My apologies if I short-changed anyone, which is very possible.