1. I asked everyone to draw a rectangle on cm squared paper whose dimensions were 2. As part of this I asked them to explain what ‘dimensions’ meant so everyone used w and h for width and height of their rectangle.

2. Next I asked them to calculate the area and perimeter of their rectangles, referring these measures as A and P. This brought about misconceptions about how to calculate the perimeter of a rectangle when different children explained their methods. Having sorted out which of their three methods was correct we moved on.

3. Now each child had a rectangle with four pieces of information: w, h, A and P.

4. I asked them to make a sketch of their own rectangle on plain (non-squared) sugar paper showing just two of the four pieces of information, i.e. h and A or w and P or A and P, together with their name. So for example Lily drew her rectangle and wrote w = 6 and A = 30.

5. Everyone stuck their piece of sugar paper around the classroom walls and everyone then had to work out the two missing pieces of information from a selection of the data around the room.

Thus each child engaged with doing and undoing. A further key pedagogic issue was how their collective pieces of sugar paper formed a ‘worksheet’ which they collectively had ownership of. This was a far more powerful aspect of their learning by contrast to any worksheet I might have given them to do. ]]>

Cheers, Sam

]]>Hope your well,

Sam

]]>I think the initial approach is a good idea, and hopefully will give students the opportunity to become aware of the equivalence of e.g. 3x and x+x+x… and also perhaps reveal an awareness of some of the x’s being ‘duplicated’ on both sides, which might later suggest a method for solving equations (subtracting x’s from both sides).

Perhaps some subtraction or multiplication between the boxes might be interesting here too as an opportunity for generalisation? [i.e. does the method you are using to fill in the boxes still hold for these other operations?] But perhaps this would over-complicate, or take the focus away from the awarenesses you wish to ‘force’.

I can see you have tried to introduce a ‘reason’ why we (students) might want to solve such equations in lesson 4, but they are still sort of ‘arbitrary’ questions. I wonder if there is some kind of investigation (perhaps starting with one of the geometric examples and extending/generalising it) that would achieve this whilst at the same time require the use of some other mathematical skills such as conjecturing. generalising, and so on?

D

]]>The how I decide who gets what question and why is so important to teaching mathematics, so I wish I had a nice answer. Hopefully this example will give some insight into that;

With year 7’s the other my question to the class was “what pairs of numbers sum to 10?”, as I spoke to students and observed their workings a particular student had written down all the whole number pairs for this, my question was then “how many pairs are there?” to which she answered “11 or 6 if they are the same (referring to 8 + 2 as 2 + 8)”, at this stage it was clear to me that the student was happy that they had all the possible solutions. As we had recently studied negative numbers I wrote on her work 17 + ____ = 10. The student comfortably answered this, so I asked the “how many pairs” question again. Comparing this to another student in the same lesson, that when I spoke to and observed his work had already split his workings into three parts whole, decimals and negatives. My question of “how many pairs?” was answered with infinity, so my question was “how can we represent all these solutions?” He first spent some time writing this algebraically, with success, so I asked if a graph could show this.

I do make an effort to get to know my students (how successful this is I don’t know), but it with this in mind that I wish we kept our students together with the same teacher for their secondary education.

I’ll finish this on something I can answer confidently, 47 is odd and has an even number of factors, 48 is even and has an even number of factors, and 49 is odd and has an odd number of factors.

Thanks again,

Sam

]]>I think there is a good argument to say we should just teach squaring as numerical operation. I think when teaching powers squaring is a terrible example to use anyway. 2 + 2 = 2 x 2 = 2^2 instead I favour what does 57^24 mean. It is also easy for a student to miss “three squared is three times three” in discussion, compared with “eighteen to the power of twenty seven is eighteen times eighteen times eighteen times … eighteen times”. What are your thoughts about power of 2.

Thanks again,

Sam

]]>Yeah Mike some examples of what students do would add greatly to this and like you I am wondering about how Sam decides who gets which question and why he decides that, but I AM happy with tasks having some kind of lurking mathematical idea that Sam would like students to get a glimpse of. And a big question for me is how a teacher coordinates all the ideas and approaches in the class so that everyone gets some glimpse of that idea.

Hurrah and hurrah again for wanting numbers to be seen as products.

Tell me something interesting about 47, and 48, and 49 ….

And what would be the equivalent ‘trick’ with addition to the double and half trick for multiplication?

Have a good year Sam!

]]>I like the bit about root(5) being an instruction, and then not, such is the multiplicity of signifiers in maths. I really like the follow-up: ‘… we should also be looking for opportunities to shift the relationship that students have with mathematical objects… ‘ more. I think this is embodied in your first task, where you are shifting from thoughts about rectangles and area, to factors and combinatorics.

Have you thought about other examples of shared signifiers that are like square root (instruction and… label?)? + springs to mind… Or that are shared in some other way? And how could we teach these different signs without just pointing and telling, do you think?

I am currently interested in the power 2.. we call it ‘squared’ (a geometric representation) of course it doesn’t have to be, it can just be an numerical operation… This is talked about by Goold in MT17, and of course by David Pimm. I wonder what might be the consequences of these types of metaphors?

Well, thanks for the article, I enjoyed it, please let me know when you do the next one ðŸ™‚

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