Using reasoning and logic to explain and justify solutions is the one of the main purposes of mathematics. In fact it is very difficult to think of any purposeful mathematics that does not employ reasoning and logic. It is therefore our responsibility as teachers to ensure that students see mathematics as a structure for logical thinking rather than a series of similar puzzles that are solved by applying familiar algorithms or using standard past paper solutions.
At GCSE, the National Curriculum details the expectation that pupils use and apply mathematics and the new National Curriculum at KS3 has put even more emphasis on the application of mathematics by describing Key Processes and Key Concepts. This area has been neglected at AS and A2 despite being explicitly included in the specifications:
For example, these aims are taken from the current Edexcel AS/A2 specification:
Students should:
develop abilities to reason logically and recognise incorrect reasoning, to generalise and to construct mathematical proofs extend their range of mathematical skills and techniques and use them in more difficult, unstructured problems
And this is one of the main assessment objectives:
Construct rigorous mathematical arguments and proofs through use of precise statements, logical deduction and inference and by the manipulation of mathematical expressions, including the construction of extended arguments for handling substantial problems presented in unstructured form.
Students therefore need to be encouraged to develop their powers of reasoning and understand and construct mathematical proofs.
Taking Action
If we look explicitly at exam style questions students can improve their reasoning by explaining each step of their solution. This can then be taken further by asking the students to explain how the solution would change if certain parameters within the question are varied
Example
Here is a solution to completing the square of a quadratic expression without comments:
x2 + 4x + 10
(x+2)2 - 4 + 10
(x+2)2 + 6
If the teacher asks the student to insert a comment to accompany each line and to explain what is useful about the method (these can be called revision comments as they are particularly useful when students are looking back at previous work) the solution could look like this:
x2 + 6x + 10
I have noticed that x2 + 6x is part of the perfect square x2 + 6x + 9 so I have replaced x2 + 6x with (x+3)2 - 9. 3 is half of 6 and 3 squared is 9.
(x+3)2 - 9 + 10
The -9 and the +10 simplify to make +1
(x+3)2 + 1
This form of the quadratic equation is useful because it tells me the coordinates of the turning point of the quadratic curve (-3, 1)
Depending on their ability students can now be asked to think further about the solution by being asked some or all of the following questions:
Explain what would happen if the coefficient of x is an odd number.
Explain what happen if the coefficient of x2 is 2.
Why does this make the problem more difficult?
Generalise your solution for the quadratic equation x2 +px + q
Generalise your solution for the quadratic equation ax2 +bx + c
The latter questions are now taking the student closer to the concept of proof, the last question being a nice lead in to a proof of the quadratic formula.
Students find mathematical proof very difficult to master and since they are part of the specification (proofs of the sums of series and the laws of logs for example) teachers have to develop these skills.
One way of developing standard proofs is to shuffle the proof and ask the student to reorder the mathematical expression in a logical order with their own comments written between the lines.
For example:
It is possible for students just to memorise proofs by rote but if they understand the key elements of the proof (in this case reversing the series and adding) they do not have to memorise large tracts of meaningless mathematics (Richard Skemp).
Case Study
An AS class were divided into pairs and given the jumbled proofs of the series above and the sum of an arithmetic series. After a brief introduction by the teacher on the theme of How do we know that a formula always works?, they were asked to complete the proofs, insert their comments and then present one of the proofs to the rest of the class.
During the construction of the proofs, most students were able to decide on the correct order of the algebraic expressions. However most were reluctant or unable to say why they had chosen that order and why one line (or a combination of two lines) led to the next. Two students finished the tasks early and were asked to investigate and prove the sum of the first n odd numbers.
To facilitate the process the teacher wrote on the whiteboard:
1 What do we definitely know to be true?/What can we assume?
2 What do we want to know?
How do we get from statement 1 to statement 2?
The first statement generated a class discussion about where we can start a proof (in this case it required the class to understand and know the nth term of an arithmetic sequence).
After completing the proofs, the teacher asked pairs to present their solutions, choosing less confident pairs for the easier proofs. By listening to students' explanations she was able to pinpoint areas that needed development as well as the concepts which the students found difficult.
Further examples
Find proofs that depend on GCSE mathematics but which require AS standard thinking skills:
For example:
Prove that a prime number is either one more or one less than a multiple of 6.
Using a triangle on a circle and a diameter, prove the sine rule.
Clue: You will need to use the following circle theorems:
Angles subtended by the same arc are equal at the circumference.
The angle at the circumference is a right angle when subtended by the diameter.
This is an interesting challenge for the teacher and is an excellent task for using a shuffled series of diagrams and/or dynamic geometry.
The Mathsnet site has many proofs of Pythagoras' theorem and their subscription site for A level contains interactive proofs of the sums of series and the rules of logarithms.
Reading
The Psychology of Learning Mathematics (Richard Skemp) Penguin Books ISBN 0-14-022668-0