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 knowapart 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.

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).
Arrange the following numbers from the least to the greatest:
2.46        2.254        2.3       2.052          2.32
Many of your students have written:
2.052     2.3         2.32        2.46     2.254
An others have written:
2.052     2.254     2.32        2.46        2.3
Complete the following steps:
1. Describe and make sense of the error(s) committed by students;
2. Find a similar task in which the students’ reasoning would lead to the same error, confirming their strategy;
4. How would you intervene in these difficulties
This is the third in the series of posts on mathematical knowledge for teaching. The first is about Tangents to Curves and the second one is about Counting Cubes.
You may use the comment section below to answer the questions or share your thoughts about mathematics teaching.  I hope you find time to discuss this with your co-teachers.

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I'm a math teacher, researcher, writer, and facilitator of professional development for teachers. Email me at mathforteaching@gmail.com.