# Teaching primary maths

What is it that we want our learners to be able to do at the end of a lesson, or a term, or a year after our mathematics teaching?

Learners should be able to demonstrate the three actions of proficiency after a year of mathematics teaching: fluency, reasoning and problem solving. (Supplied)

We want learners to be able to answer the questions that we set them and to be able to answer the questions that appear on the Annual National Assessment they will do again later this year. However, correctly answering a range of questions from different sources requires teachers to pay attention to more than simply guiding learners towards correct answers in classrooms.

Over the last few weeks, Professor Mike Askew, an international expert in primary mathematics teaching and learning, has been working with primary teachers at Wits School of Education.

He suggests three “actions of proficiency” that teachers should build into their mathematics -lessons: -fluency, reasoning and problem—solving. (See diagram The three actions below.)

Fluency
Fluency refers to recall of basic number bonds, core skills such as adding 10 or 100 to any number, and definitions such as “odd” and “even” numbers. In foundation phase, this includes helping children to practice and then memorise facts related to number bonds to 10 and later to 20.

Reasoning
Reasoning involves thinking about relationships between ideas as well as explaining this thinking. For example, inviting children to explain why adding two odd -numbers will always give an even number and allowing children to use words and diagrams to show their thinking can provide opportunities for practising reasoning.

Children can also be encouraged to use known facts to work out new problems. For example, using the fact that 6 + 6 = 12 to work out the answer to 6 + 7, without having to add 6 objects to 7 objects, also supports the development of reasoning.  As  teachers, we might want to encourage a class to compare 6 + 6 with  6 + 7. We can question learners about what has changed and invite them to visualise and explain this change and its impact on the answer to the second problem.

By reasoning their way to the answer, rather than reverting to counting, children can build a network of connected information about number and calculating.

This network can then support  the building of a bigger bank of recalled facts. This is vital for developing e-fficiency, because we do not want children to always go back to concrete- counting to work out answers, especially when the number range they are dealing with starts growing  bigger.

Problem-solving
The final aspect that Askew mentions- is problem-solving. Problem-solving involves being able to represent problems in words, through acting them out and in diagrams.

It also involves choosing the correct- operations to solve a problem and being able to communicate this process.

A demonstration
Askew demonstrates the opportunity for all three actions of proficiency through this problem:
Shalati has 8 sweets. She is given some more sweets and ends up with 12 sweets. How many sweets was she given?

He encourages teachers to help children act out the problem. In other words, one child acts out being Shalati. She picks up 8 counters to represent her sweets. Another child acts out giving her some more sweets (in the form of counters in a different colour) and, meanwhile, the class is asked to keep track of how many sweets Shalati has in total.

Representing these actions can begin with a diagram and then move on to increasingly compressed representations, as shown above.

Fluency increases across these -different diagrams and eventually sums like 8 + 4 can be answered by recall without the need for any diagram.

However, fluency cannot develop without the support of the reasoning and problem-solving that are built into the actions and represented in the pictures above.

Also, these representations provide children with resources that allow them to reason and solve problems as the number range gets higher and the problems become more complex.

Reasoning and problem-solving
The communication skills that are built into reasoning and problem-solving allow teachers to hear children- express their current thinking and therefore to make decisions about the kinds of representations they are ready to grasp.

In this way, teachers can help children- to progress- towards more complex ideas.

Research shows that children enjoy the challenges associated with fluency, reasoning and problem-solving.

As mathematics teachers, we need to provide opportunities for children to engage in all three aspects if they are going to become mathematically proficient.

We have an email discussion group for primary maths teachers, where -teachers can share information about events, activities that work in class, questions and answers about primary maths teaching and learning.

If you would like to be part of the -discussion group, please email us on [email protected] or visit our website www.wits.ac.za/academic/humanities/education/14097/-primary_maths.html

Originally published in: The Teacher