# 2 WAYS TO TEACH AND LEARN MATHS

Mathematics can be taught and learned relationally or instrumentally.

Instrumental understanding means learning a set of procedures and could be described as ‘renting your maths just long enough for a test’!

Relational understanding means investigating concepts along a continuum, integrating related concepts, and could be described as ‘owning your maths’!

Relational vs Instrumental Video 1

Relational vs Instrumental Video 2

Richard Skemp believed that children could learn intelligently from a young age.

He defined two ways of teaching and learning Mathematics which he called Instrumental and Relational. The Research paper by Skemp explains this much better than we can, but to summarise Research Summary

**Instrumental** means a child knows a rule or procedure, and has the ability to use it.

Examples of rules and procedures include, but are not limited to:

• finding area by multiplying length by width,

• algorithms for addition, subtraction, multiplication and division,

• changing the sign on numbers moved from one side of the equals sign to other.

Other examples of instrumental explanations can be easily found on other websites. Unfortunately, the instrumental approach to teaching and learning maths is widespread. The learning theory was that children practise the rule or procedure, and understanding will follow. Current learning research tells us that in many cases,

• understanding never followed, or

• incorrect understanding followed, or

• if correct understanding did follow, the child did not know if it was important or related to any other parts of maths.

** Relational** means a child knows what to do and why. The child develops deep understanding of concepts, and the relationships between concepts.

Examples of relational understanding include, but are not limited to:

• explaining area is the amount of space a shape takes up in 2 of the 3 dimensions, relating this to the 3 dimensions being – up and down, left to right, front to back,

• thinking additively about place value to add and subtract, thinking multiplicatively about place value, and the distributive property to multiply and divide, and relating division to fractions,

• explaining the equals sign means equality (both sides are equal) so to remove a negative number from 1 side, we add the number to both sides.

Maths abounds with related concepts.

Skemp identified the short- and long-term effects of teaching and learning instrumentally and relationally:

REASONING IS MISSING FROM CLASSROOMS.

The Australian National Numeracy Review found that

‘Students need to learn mathematics in ways that enable them to recognise when mathematics might help to interpret information or solve practical problems, apply their knowledge appropriately in contexts where they will have to use mathematical reasoning processes, choose mathematics that makes sense in the circumstances, make assumptions, resolve ambiguity and judge what is reasonable.’

‘in Australia there was very little evidence of lessons involving mathematical reasoning’.

There exists ‘a syndrome of shallow teaching, where students are asked to follow procedures without reasons’ and ‘students’ strategies and reasoning could well challenge the teacher’s mathematical ‘comfort zone.’

While Instrumental teaching and learning involves following rules and procedures without reason, relational teaching and learning allows and encourages children to apply reasoning to their learning.

Relational teaching and learning develops BOTH the students’ and the teachers’ deep understanding of mathematical concepts, and BOTH the students’ and the teachers’ meta-language and capacity to explain!

21^{st} century learning tells us that when a child asks a mathematical question, it is no longer the teacher’s role to answer it! The teacher is not the ‘keeper of all knowledge’! With the amount of knowledge in the world doubling every 18 months, it would be impossible for the teacher to be the keeper of knowledge!!!

Children have all the knowledge in the world available to them. Children need to know some knowledge, but also how to locate more knowledge, assess and adapt to new knowledge, communicate knowledge, and to use knowledge to create more knowledge.

When a child asks a mathematical question, they are ready to investigate to find the answer! They use the knowledge that they already have to investigate to locate more knowledge, to assess the new knowledge, to communicate their knowledge – they have used their knowledge to create more knowledge!

When a child asks a mathematical question, they will not just be asking the teacher! They will be asking other students! There are 30 brains in the classroom – we can use all of them to learn!

When a child asks a mathematical question, Vygotsky’s research into the zone of proximal development tells us that a teacher cannot answer it! Our level understanding may be so far from the child’s level of understanding that the child cannot learn from us! The child needs to be learning with others within their zone of proximal development – other children!

This means we, as teachers, may not now teach the way we were taught! We may have been taught 20^{th} century learning where knowledge was delivered in a sequential, logical, controlled way.

But we need not fear being taken out of our comfort zone! Because children will be in their comfort zone! And as they question and investigate with others within their zone of proximal development, we teachers will find our own knowledge and understanding increases.

21^{st} century learning is relational.

What exactly does relational teaching and learning look like? It looks like

questioning, investigating and explaining!

thinking mathematically.

problem solving!

formative embedded assessment!

21st Century learning!

All Teaching Resources- every Teaching Plan, Video, Investigation, Reflection and Problem at A Learning Place A Teaching Place involve questioning, investigating and explaining to allow children (and teachers) to develop relational understanding of concepts.

The concept pages, provide information and guidance to related concepts to ensure children are explicitly investigating and explaining the relationships between concepts.

And as we read the Teaching Plans, watch the Videos, plan the Investigations, Reflections and Problems that your students will engage in, we teachers will find our own knowledge and understanding increases.

Research:

* Richard R. Skemp* Relational Understanding and Instrumental Understanding Department of Education, University of Warwick

* Lev Vygotsky* Interaction between Learning and Development

* National Numeracy Review *Report May 2008 Commissioned by the Human Capital Working Group, Council of Australian Government pages xi, 10, 30, 63, 99