**THIS WEEK**: We explore the phenomenon of

*Letter as Object*, something we met briefly in ALG 9C, ie we look at contexts where a letter represents a pure number (of objects) or a quantity (the price or length or mass or some other numerical

*quality*of an object), but where there might be a temptation to use the letter as a shorthand for the object itself (as in

*'a*stands for apple,

*b*stands for banana' - the classic

*fruit salad algebra*). We will see that sometimes one can slide harmlessly between letter as object and letter as quantity, but that sometimes it leads to a completely fake algebra.

**MONDAY**: In this pair of tasks

*m*stands for a numerical quality (mass) in the first task and for a pure number in the second. Though the answer is 5

*m*in both cases, we have found that the second kind of task is substantially more demanding than the first - does that hold for your students?

In the first task, it is easy to interpret

*m*as simply standing for

*mints*rather than the mass of a mint, but still come up with the correct expression, 5

*m*, albeit misinterpreted as 5

*mints*. In the second task students have to cope with the idea that

*m*is most definitely a number, but one whose value we don't know, so that it is not possible to arrive at a specific numerical answer.

**TUESDAY**: Careful! Needles and pins.

*Because of all my pride, the tears I gotta hide...*This turns out to be quite a Searching task, though let's hope it doesn't quite lead to tears!

The task involves an expression in which the letters stand for numbers but where the temptation to treat them as objects is very strong (for us as well as our students) and where this leads to a complete misinterpretation of the expression.

*Why can't I stop*

And tell myself I'm wrong

I'm wrong, so wrong

Why can't I stand up

And tell myself I'm strong

And tell myself I'm wrong

I'm wrong, so wrong

Why can't I stand up

And tell myself I'm strong

*....*

The extract below is from my article

*Object lessons in algebra?*that appeared in

*Mathematics Teaching*98 in 1982. Still worth reading!

**WEDNESDAY**: In this task, the misinterpretation that results from treating the letters as objects is not as jarring... The given expression stands for the

*cost*(in number of pence) of 5 bananas and 2 coconuts, so it is wrong to simply translate it as

*5 bananas and 2 coconuts*. Nevertheless this translation does make some kind of sense - the story is about 5 bananas and 2 coconuts, whereas the corresponding interpretation of the expression 2

*p*+ 5

*n*in ALG 11B, as

*2 pins and 5 needles*, does not fit that story at all.

It is interesting to consider the alternative version of task 11C shown below.

Here students are quite likely to come up with the right expression

*,*though not necessarily for the right reason - we can't be sure whether students who write 5

*b*+ 2

*c*fully realise that this represents a number of pence and that it is not simply telling us about the number of bananas and coconuts bought. So this alternative version is not as useful as the original for revealing students' thinking. The task below, which appeared recently on Twitter, is even less effective as a diagnostic tool: here the correct option (B, I assume) can be chosen by simply treating it as an abbreviation of the verbal statements, ie by reading 5

*t*+ 2

*c*= 3.70 as

*5 teas and 2 coffees cost 3.70*(pounds), rather than appreciating that 5

*t*+ 2

*c*actually represents the number of £s spent on the drinks.

**THURSDAY**: We present the first of two tasks involving sets of coloured rods, which we ask students to symbolise in different ways.

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