No one can teach mathematics without knowing mathematics but not everyone who knows mathematics can teach it well. Below are two tasks about teaching fractions and decimals that would give us a sense of the kind of mathematical knowledge we teachers need to know apart from knowledge of the content of mathematics. As teachers it is expected of us to have knowledge of students difficulties and misconceptions in specific domains of mathematics. We are also expected to know the different representations or models of concepts to design an effective instruction. The two tasks were used in a study about mathematical knowledge for teaching of pre-service teachers.

Task 1

You are teaching in 7th grade. You want to work on multiplication of fractions, using the following numbers:

a) 10 x 3 b) 10 x 3/4 c. 10 x 1 1/5 d. 10/11 x 1 1/5

- Create a problem using an everyday context, accessible to students and easily visualized, that uses the repeated addition sense for multiplication;
- Prepare an illustration that works and that you could use for all numbers to help students visualize the operation;
- Show, for each case, with the illustration and specific explanations, how one can make sense of c) from the answer obtained in a).

- Describe and make sense of the error(s) committed by students;
- Find a similar task in which the students’ reasoning would lead to the same error, confirming their strategy;
- Find a similar task in which the students’ reasoning would lead to a right answer;
- How would you intervene in these difficulties

I like the questions. I am confounded by the examples.

I don’t think my 6th grade students, who would err in Task 2, who do so in the way you suggest.

And for Task 1 d. 10/11 x 11/5 … seriously? – no, wait – oh, I see – take 10 of the 11 parts in 11/5. I was boggled enough by c. 10 x 11/5 (I have to draw that – really?) that I then pictured an array model for d. which would be quite a mess.

I try to give examples that students can visualize and draw easily enough – and then generalize to ‘bigger’ numbers. What I don’t want them to do is jump to the algorithm – which they know, but do not understand.