Posted in Elementary School Math, Number Sense

Bob is learning calculation

Bob is an elementary school student. He is learning to calculate. He just learned about addition and multiplication but there are some things that he doesn’t understand. For example, how come 1+3 = 3 + 1? How can it be the same thought Bob? Every morning I have 1 piece of bread for breakfast while Dad has 3 pieces. If I have 3 pieces while Dad has 1 piece, I will be too full and Dad will be hungry?

When they added three numbers, Bob did not understand (1+2) + 1 = 1 + (2+1). Usually I like to drink 1 cup of coffee with 2 spoons of milk then afterwards have a piece of bread. I would not feel well if I first drink a cup of coffee then afterwards drink 2 spoons of milk while having 1 piece of bread. How come they are the same, thought Bob.

The most confusing part was after the lesson on fraction. Bob learned that 1/2 = 2/4. So when he got back home he tried to share 6 apples with his sister Linda. He divided the 6 apples into two groups – 2 apples in one group and 4 apples in another group.

apples, dividing apples

From the set of two apples he gave 1 to Linda. That’s 1/2. From the set of four apples, he took 2, that’s 2/4. It is equal he said. But Linda did not agree with him because she got 1 apple less that he. Bob thought, how can this be? Why would 1/2 = 2/4 not work for apples!

The next day, the teacher asked Bob to add 1/2 and 2/4? Bob wrote 1/2 + 2/4 = 3/6 because taking 1 apple from 2 apples then 2 apples from 4 apples, he must have taken a total of 3 apples from 6 apples!

This story is adapted from A Framework of Mathematical Knowledge for Teaching by J. Li, X. Fan, and Y Zhui at the EARCOME5 2010 conference.

Point for reflection:

What has Bob missed about the meaning of addition of natural numbers? the meaning of fraction?

You may want to read the following posts about math knowledge for teaching:

Posted in Algebra

Math knowledge for teaching fractions and decimals

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).
Task 2
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;
  3. Find a similar task in which the students’ reasoning would lead to a right answer;
  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.