The Language of Mathematical Meaning

My wonderful mother-in-law knows that I like quirky math things so when I received this image and joke from her recently I knew she knew how I see maths. The image appeared at the end of these lines.

A husband seeing his wife is busy cooking a meal asks if he can help. His wife asks him to peel half the potatoes and put them in a large pot to cook. The picture shows what he did, much to his wife's amusement.

The wonderful thing about this mathematically is that it is a mathematical play on the word "half". When we teach fractions in school we seldom give much time to the identification of the one or the whole to which the fraction refers. When I start a class on fractions I always ask the students if they would like half the money in my right hand or a quarter of the  money in my left hand. The value of a fraction is a mystery until you know the size of the one to which it refers.

This is why operations with fractions can be so difficult. Think about 1/2 ÷ 1/4. The answer is clearly 2, but 2 what? If you learned this procedurally and instrumentally then it's difficult to know. If, on the other hand, you learned it relationally then you can see that it refers to the 1/4. There are two quarters in a half. You can put it in context by thinking about the number of quarters in the first half of the Superbowl.

The reason why the husband did what he did was that he used the potatoes as the ones and not the whole group. He should really have peeled 4 potatoes and not 8 halves. The result is the same in term of the amount of peeled potato but pretty useless in terms of meal preparation.