The instinct of many students to cancel these fractions kicked in and there was some uncertainty as to whether the 2's in the first fraction or the g's in the second would cancel. No, I dutifully informed them, you can only cancel factors that are common to all terms of the fraction. Done and dusted, I hoped, and we could get back on with the main thrust of the lesson - rearranging those formulae.

But students were not convinced by this. With the silent sound of 'I didn't really get that' hanging heavy in the air, I realised my errors: (i) I had whisked quickly over an important learning point in my efforts to drive the lesson towards the intended goal, refusing to be led down this tangential avenue, and (ii) I had presented this

*necessary**must-be*maths fact to students as though it were an*arbitrary*fact, instead of designing a more powerful experience where they could see it for themselves.
Following these realisations, I wrote the following on the board:

...and an impromptu point of inquiry was born. My students seem to really enjoy getting to the bottom of problems posed this way and certainly came away with a clear understanding that it

*must be*that cancelling the b's in this fraction was not a 'safe' algebraic operation. Their findings? The two fractions are only equivalent*sometimes*- when one (or both) of two particular conditions is met. (I'll leave finding those two conditions to the reader.)
Ideally, I believe the spirit of mathematical inquiry should permeate every maths lesson - certainly whenever students are encountering new

*must-be*maths facts. As to what constitutes a 'mathematical inquiry', well I believe the answer to this is very broad indeed - from points of inquiry like these that may last just a few minutes to those that last an hour lesson or beyond (for example, those that can be found at www.inquirymaths.co.uk.)
I think it is also important when manipulating algebra that students do not lose sight of the fact that the letters represent

*numbers*and*operations on numbers*- that it is generalised arithmetic. If forgotten, algebra can come to seem like an*arbitrary*language with strict and potentially confusing grammatical laws. If students have a good awareness that algebra always represents the way numbers work, they can fall back on 'substituting a few different numbers' as a check for whether manipulating an algebraic expression in a certain way is valid. (This is not foolproof though, of course)
Finally, an important distinction emerges when students are told the rule compared to when they 'see it' for themselves - the question of

*on whose authority is this fact true?*If simply told the fact, the students have no choice but to accept the authority of the teacher. If students see the fact for themselves, the fact clearly stands as true*on its own*- or indeed, on the*students'**own*authority.*Must-be*maths facts are not true simply because I, or any other teacher, says they are, they are true because they simply*must-be!*That is the nature of mathematics, and that is what students in our classes should experience.
I like the inquiry that they used to check whether or not it made sense. I think it's really important to teach our students checking mechanisms that they can use for themselves to help them convince themselves that they are right (or wrong!).

ReplyDeleteYou wrote: "No, I dutifully informed them, you can only cancel factors that are common to all terms of the fraction."

When discussing this with pupils I've found it useful to remind students about what they are doing when simplifying/cancelling down a fraction such as 4/8. When asked the students very quickly reply that they are "dividing the numerator and denominator by the same amount". That's when you can point out that simplifying algebraic fractions is just the same - you're simply dividing the numerator and the denominator by the same amount. If you then ask them to divide (a+2b) by 2 they'll soon realise they can't unless they introduce extra fractions or decimals.

Yes, I like that idea - that's a good question to prompt thinking, and it takes the algebraic process back to it's 'concrete roots' in numbers. I remember doing this 'numerical checking for myself' of algebraic rules back when I was in school and it giving me confidence that a rule I was using was correct (or otherwise!)

ReplyDeleteHi James ... love the focus of the blog on conceptual learning! On what constitutes an inquiry, I think the main distinction inquiry teachers make is between inquiries that are rooted in the students' everyday “real life” and interests or those, like yours and mine, that start with intriguing mathematical prompts. For me, there are two key features of an inquiry: (1) students initiate inquiry by asking questions and making observations; and (2) students have a role in structuring and regulating the inquiry (using the cards). The first feature helps to generate the energy and excitement of classroom inquiry, while the second one helps to channel that energy and excitement into mathematically-valid inquiry pathways. I hope we get to discuss mathematical inquiries some time. In the meantime, I look forward with great interest to following your blog. Regards, Andrew

ReplyDeleteHi Andrew, I agree that there's a distinction to be made. There does seem to be a lot of emphasis on forming an inquiry from a 'real-life' prompt doing the rounds in the maths education world, but I think it's equally important to realise that the process of inquiry can form a powerful learning process when dealing with purely abstract mathematical ideas. Hope to catch you at the next La Salle conference!

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