Teaching children fractions
Worldwide, most children learn the rules for adding, subtracting, multiplying and dividing fractions in Grade 5. These are reinforced by revision and extension through to Year 12.
They form the basis for ratio and proportion in Years 7 and 8. They are the same rules needed for algebra and calculus.
Without a sound working knowledge of fractions and how to manipulate them, a child will not progress in Mathematics.
But don't calculators make fraction work easy?
They certainly can. But what calculators can never do is to know when to add, subtract, multiply or divide. The student must tell the calculator what to do. That means that the student must know what to do before enlisting the help of the calculator. The student who has laboured with the rules and who can apply them easily in any context, can use a calculator with skill and purpose. A child without that knowledge has, in a calculator, a billion dollar brain which they cannot use.
But how do you develop the knowledge of when to do what?
- By working through hundreds of word problems.
- By learning how to read and symbolise questions.
- By applying rules without the help of a calculator.
It is more important for students to understand than to memorise by rote
This is nonsense! Why should memorising rules by rote stop a student from understanding what is going on? In the hands of a skilled teacher, the fact that an entire class knows the rules can serve as a perfect introduction to a Why Do the Rules Work? lesson.
And, in many cases, curious students may have already asked themselves why the rules work and come up with an answer.
Children must be able to visualise fractions
To start with perhaps. Showing what a fraction is by shading part of a rectangle may help. Showing that 3/5 is the same as 6/10 may be helped by a diagram. But trying to get a student to visualise 4/7 times 3/11 will confuse the vast majority of students.
If you teach them to "multiply the tops and multiply the bottoms" you can explain why it works later.
"Invert and Multiply" has become synonymous with lack of understanding
Yes it has. But generations of students have learned this rule and can apply it with no effort and get the correct answer every time and still don't understand it!
If you would like a book filled with explanations which teach "understanding", try the National Council of Teachers of Mathematics publication Principles and Standards for School Mathematics. The National Council is the supreme body of mathematics teachers in the US. Their insidious ideas have produced two obvious results:
- In the Third International Mathematics exam – taken by hundreds of countries – the US came third last.
- The US and Britain advertise daily for overseas maths teachers.
How can parents help?
Check your child's textbook for the following headings:
- What is a fraction?
- Equivalent fractions
- Proper and improper fractions – changing mixed numerals to improper fractions
- Even whole numbers can be thought of as fractions
- Cancelling fractions
- Expressing fractions with different denominators
- Adding and subtracting fractions using a common denominator
- Word problems
- Multiplying fractions
- Cancelling before multiplying
- Multiplying fractions by a whole number
- Multiplying fractions by mixed numerals
- Word problems
- Dividing one fraction by another
- Dividing a fraction by a whole number
- Dividing a whole number by a fraction
If the book seems to spend more time on drawing diagrams of fractions rather than exercises to test to see if students can work with fractions, then get a different textbook.
If you can't find a satisfactory text, make up 10 questions to test your child's knowledge of the rules. They shouldn't be difficult questions but they should test the ideas in the list above. Get your child to do these 10 questions and mark them. Repeat this process each week.
Don't think you can't do it? Of course you can. It's not so much a question of knowledge, it's a question of whether you think it's worth while.
But my children are too old to learn Year 5 work
Nonsense! This is a case of a Problem-Needing – To – Be – Fixed.
At Smith Mathematics Coaching we teach fractions to students in all years up to their HSC. We use the same list as above and we test students' knowledge by short tests. Once students have mastered fractions, their response is usually one of anger! Basically their comments are something like:
Why wasn't I taught these rules in Year 5!
Their comments should be compulsory reading for all students studying to become teachers.