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.
Posted in Misconceptions, Number Sense

Technically, Fractions are Not Numbers

It is misleading to put fractions alongside the sets of numbers – counting, whole, integers, rational, irrational and real. The diagram below which are in many Mathematics I (Year 7) textbooks is inviting misconceptions.

WRONG WAY

Fraction is a form for writing numbers just like the decimals, percents, and other notations that use exponents and radicals, etc.

The fraction form of numbers is used to describe quantities that is 1) part of a whole, 2) part of a set, 3) ratio, and 4) as an indicated operation. Yes, it can also represent all the rational numbers but it doesn’t make fractions another kind of number or as another way of describing the rational numbers. Decimals can represent both the rational and the irrational numbers (approximately) but it is not a separate set of numbers or used as another way of describing the real numbers! Note that I’m using the word number not in everyday sense but in mathematical sense. In Year 7, where learners are slowly introduced to the rigor of mathematics and to the real number system, I suggest you start calling the numbers in its proper name.

I prefer the Venn diagram to show the relationships among the different kinds of numbers like the one shown below:

The Real Number SYSTEM
The Real Numbers

The diagram shows that the set of real numbers is composed of the rational and the irrational numbers. The integers are part of the set of rational numbers just like the counting numbers are members of the set of whole numbers and the whole numbers are members of the set of integers. The properties of each of these set of numbers can be investigated. We do not investigate if fraction is closed or is commutative under a certain operation for example, but we do it for the rational numbers.

You may want to know why we invert the divisor when dividing fractions. Click the link.