### Research shows that children can learn mathematics relationally from a young age.

But often they are taught instrumentally.

### Each way of teaching and learning has long and short term effects.

#### Teacher is

Teaching Instrumentally

#### Teacher is

Teaching RELATIONALLY

#### Teacher is

Teaching Instrumentally

All appears well because there is a match between teacher and child goals. Teacher and child ‘do’ mathematics. One sign the child is learning instrumentally is that they don’t know if they are right until the teacher tells them.

The child learns ever more rules, developing shallow understanding, causing problems when a question does not fit a ‘rule’. The child may continue to study mathematics to pass tests, but will drop mathematics as soon as they can.

The child tries to understand relationally what is being taught instrumentally. The child tries to develop deep understanding but is being taught shallowly.

The child feels that they are not smart enough to Understand mathematics. They disengage. They drop Mathematics as soon as they can. As adults, they tell their children that they couldn’t do mathematics either.

#### CHILD is

LEARNING

RELATIONALLY

Frustration for the teacher because the child doesn’t want to know why. Frustration for the child because they just want to know how to do it.

The child and the teacher develop relational understanding of mathematics. Child continues to study mathematics and works in a mathematics related career.

The child and the teacher both develop deep relational understanding

The child and the teacher both continue to develop deep relationalunderstanding of mathematics. Child continues to study mathematics and works in a mathematics related field and saves the world!

#### Teacher is

Teaching Instrumentally

#### CHILD is LEARNING

Instrumentally uttam

All appears well because there is a match between teacher and child goals. Teacher and child ‘do’ mathematics. One sign the child is learning instrumentally is that they don’t know if they are right until the teacher tells them.

The child learns ever more rules, developing shallow understanding, causing problems when a question does not fit a ‘rule’. The child may continue to study mathematics to pass tests, but will drop mathematics as soon as they can.

#### CHILD is LEARNING

RELATIONALLY

Frustration for the teacher because the child doesn’t want to know why. Frustration for the child because they just want to know how to do it.

The child and the teacher develop relational understanding of mathematics. Child continues to study mathematics and works in a mathematics related career.

#### Teacher is

Teaching RELATIONALLY

#### CHILD is LEARNING

Instrumentally

The child tries to understand relationally what is being taught instrumentally. The child tries to develop deep understanding but is being taught shallowly.

The child feels that they are not smart enough to Understand mathematics. They disengage. They drop Mathematics as soon as they can. As adults, they tell their children that they couldn’t do mathematics either.

#### CHILD is LEARNING

RELATIONALLY

The child and the teacher both develop deep relational understanding

The child and the teacher both continue to develop deep relationalunderstanding of mathematics. Child continues to study mathematics and works in a mathematics related field and saves the world!

### Examples of INSTRUMENTAL TEACHING AND LEARNINGinclude, 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 teaching and learning 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 mathematics.

### Examples of RELATIONAL TEACHING AND LEARNING include, but are not limited to:

- explaining area is the amount of space a shape or surface takes up in 2 of the 3 dimensions, explaining that the 3 dimensions are up and down, left to right, front to back,
- thinking additively about place value, addition and subtraction; thinking multiplicatively about place value, multiplication, division, fractions, and metric measurement, using the distributive property to multiply and divide, and relating division to fractions,
- explaining that 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.