Think about that question… do you know the answer?

Along with English (or as it is known: “multiliteracies“) unit this semester, I’m also doing a math subject (or as it is known: “mathematics”).

While mathematics still mathematics, there’s still been several shake ups in pedagogy and terminology since I went to school: For example “times tables” are now “number facts”.

The weird thing in doing research for last weeks assignment (which is my excuse of the month for lack of updates), is finding out a lot of these “new fangled” concepts are not that new at all. I found “number facts” in some books almost as old as me (late 80’s).

Another concept I think is quite relevant is the “Three Facets of Mathematics” from Payne and Rathmell, which appears to date back to 1975 (some NCTM publication).

The basic idea is that what people do with math needs to make sense. Being able to punch in symbols and get an answer might get you an A+ on the test – but real world situations are not frequently symbolic (symbols are possibly more prevalent in a digital society) – understanding concepts is much more important than interpreting symbols.

Payne and Rathmell add two other ways of illustrating concepts that students need to be familiar with – a physical or visual representation (e.g. two circles joining three circles), a verbal representation (e.g. two more than three, add two and three), in addition to the symbolic (2 + 3 = ?). The symbolic needs to be recognised as the most abstract and complex of these illustrations, so should really be introduced *after* students understand the concept (or at least near the end of reaching that understanding). The other point of using three facets of math is that the language used in each illustration needs to match up.

This “math has to make sense in three ways” really came clearer when we were dealing with dividing by a fraction. When dividing we often think of it as partitioning or sharing into equal groups. [20 ÷ 4] becomes “I have twenty apples, and four people: how many apples can I give each person?” This will not work with fractions. [20 ÷ ¾] will not become “I have twenty apples, and three quarters of a person, how many apples can I give this twisted remnant of a human?” It’s either nonsensical (you don’t get three-quarters of a person) or not the question you are trying to solve (the midget gets all the apples [or maybe ¾], he’s one, right?).

In dividing by fractions we need to use another interpretation of dividing – quotitive, or rather than asking “how many do each get”, ask “how many can I give out”. [20 ÷ 4] becomes “I have twenty apples, and I am giving out baskets of four: how many baskets can I give out?”. This **will** work with fractions. [20 ÷ ¾] becomes “Each student needs three quarter pieces of fruit, how many can I feed with 20 apples?” This can then be transformed visually as shown below.

And this language can be used for situations beyond slicing (fruit and pizza connect well with schoolkids). You could also use volumes. Below is “I have a 20L of wine in a barrel. How many bottles can I fill (each bottle contains 750mL, or ¾L)? Although, I’m not sure if wine is a responsible illustration to use for 8th graders.

I still like the slicing representation better. To me it will explain the algorithm step-by-step. First, multiply by the denominator 4 (to get the number of slices), and then divide by the numerator 3 (to get how many groups of three quarters there are all together) to reach your final answer. You can then move the students onto to pizzas (eighths) to check that the algorithm holds for different denominators. Then perhaps move them to non-slicing situations and see that the algorithm still holds (or does it?)

Really lame pictures by zayzayem.

Answer: “STUDENTS”. (Smart ass, yes. But it should be true for all teachers of all subjects.)

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