Critical Thinking

Primary Mathematics

Views of Mathematics

The way in which we teach mathematics is likely to be influenced by what we think mathematics is.

What do you think mathematics is?

Indicate how closely you agree with the statements in ‘Critical Thinking Primary Mathematics: Task 1’ by ringing the appropriate number

Group task• In groups of about 4 people:• Discuss which of these statements you agree with and

disagree with and why;• Brainstorm how the teaching of mathematics would be

influenced by strongly agreeing that ‘Mathematics is a set of facts, concepts and procedures which must be learned’ (Platonist view) or by strongly agreeing that ‘Mathematics is made up of ideas that are created or discovered through human endeavour’ (discovery view);

• Divide a large sheet of paper in half. On one side indicate the characteristics of mathematics teaching influenced by a ‘Platonist view’ and on the other the characteristics of mathematics teaching influenced by a ‘discovery view’.

Instrumental and Relational understanding

• Richard Skemp famously made a distinction between understanding of mathematics in an instrumental way or in a relational way.

• Instrumental understanding involves understanding

mathematics as lists of unconnected rules so that mathematical knowledge involves having a set of procedures for performing specific mathematical tasks.

• Relational understanding involves being able to draw on a network of interrelated conceptual structures to construct meaning in relation to any specific mathematical problem.

Example of calculation strategies demonstrating instrumental and relational

understanding:

The problem: How much marking?

• A teacher has 25 pupils in her class.

• Each pupil has completed 16 mathematics problems for a homework task.

• How many problems does the teacher have to mark?

Demonstration of instrumental understanding

2 5

x 16

1 5 0

2 5 0

4 0 0

Demonstration of relational understanding

4 x 25 = 100

16 = 4 x 4

Therefore

16 x 25 = 400

Both solution strategies achieved the correct answer

Which strategy do you think demonstrates a more profound understanding of mathematics? Why?

• Divide the strategies on the sheet ‘Critical Thinking: Primary Mathematics Task 2’ into those which appear to demonstrate instrumental understanding and those that appear to demonstrate relational understanding of mathematics.

• Think of examples of calculation strategies used by pupils in your class that could be put into each group.

Discussion of efficient calculation

strategies • Since the introduction of the Numeracy

Strategy in 1998 it has been common in English primary school classrooms for teachers to ask pupils to explain their strategies for working out calculations.

• For example, strategies given by pupils in a class of 8 year olds who were asked to calculate 3 x 14 mentally:

• Pupil 1 “I doubled 14 to get 28 then I added ten which gave 38 then I added the remaining 4 which gave 42”

• Pupil 2 “I said 3 x 10 = 30 and 3 x 4 = 12, 30 + 12 = 42”

• Pupil 3 “I knew 3 x 15 was 45, then I subtracted 3”

• Pupil 4 “I added 14 and 14 and 14”

• Which of these strategies do you think children in your class might use?

• Are any of these strategies more efficient than others? If so, why?

• Which strategies might you want to encourage?

• What questions might you ask to help pupils who used less efficient strategies move to using the more efficient strategies?

Thinking about which strategies are the most efficient for any particular calculation involves

thinking critically.

In small groups brainstorm:• Teaching and learning strategies that will

encourage pupils to use flexible calculation strategies (i.e. strategies that are most appropriate for particular calculations)

• Teaching and learning strategies that will encourage pupils to think about and discuss the efficiency of the calculation strategies they use

On a large sheet of paper list the questions that you might ask in order to encourage the consideration and use of flexible calculation strategies.

Critical thinking and word problems

• Consider the following solution to the problem: How many buses holding 32 passengers are needed to take 174 pupils on a school visit?

Solution: 0 0 5 remainder 14

32) 1 7 4

1 6 0

14• Has the pupil correctly solved the problem?

Mathematics involves more than being able to carry out calculations correctly and efficiently.

Mathematics involves being able to use calculations to solve problems.This involves thinking critically.

In their solution of the buses problem, the pupil demonstrated much mathematical knowledge:

• They correctly decided which operation (division) they needed to use.

• They knew a procedure for carrying out this calculation. • They seemed to know that multiplication is the inverse of division as

to find how many lots of 32 there are in 174 they presumably used trial and error to find that 5 lots of 32 = 160.

However, this pupil still got an incorrect solution. Six buses were needed, not 5 buses remainder 14.

What did the remainder 14 mean? If the pupil had thought critically about this they might have arrived at the correct answer.

Processes involved in word problems

In small groups discuss and then list the processes involved in solving word problems such as the bus problem, e.g.

1) Read, or listen to, the problem and pick out the relevant information

2) Decide which operation is needed

3)

Etc.

Discuss the sort of mathematical problems you give (or might give) to your pupils that

involve them in critical thinking. Divide a large sheet of paper in half:

• On one half list the characteristics of problems that encourage critical thinking

• On the other half list suggested teaching strategies to encourage critical thinking when engaged in problem solving.