# Explicit modelling of reasoning and processes behind actions

In teaching mathematics, the challenge often lies in making abstract concepts and problem-solving strategies explicit to students. Students often struggle to grasp the underlying reasoning and processes that drive mathematics. Explicit Teaching in Maths provides strategies for the teacher which help them to design and present instruction and learning to students in a meaningful way.

Teacher modelling is a way we can intentionally make clear the reasoning and processes behind mathematical ideas and concepts. Listen to our podcast, Explicit modelling of reasoning and processes behind actions, for more, where Allan Dougan (AAMT) and Dr Kristen Trippet (Australian Academy of Science) discuss how we model the reasoning and processes behind actions.

This article answers the questions:

- What do we mean by modelling in maths?
- How do we model the reasoning and processes behind actions in maths?
- How do we make the most of modelling in classroom teaching?

## What is Teacher modelling in mathematics?

Mathematics is a subject where concepts can easily remain invisible to students. Teacher modelling allows teachers to showcase and explain the reasoning and processes behind mathematical concepts and practices. It often involves breaking down complex problems, strategies and skills into manageable steps that students can understand and replicate.

Teacher modelling is about thinking aloud and making visible those practices that are often concealed within a mathematician's mind.

## What is mathematical reasoning?

According to the Australian Curriculum V9.0: Understand this learning area, students are reasoning mathematically when they can:

- explain their thinking
- deduce and justify strategies used and conclusions reached
- adapt the known to the unknown
- transfer learning from one context to another
- prove that something is true or false.

Explicit modelling is a powerful tool for developing students’ mathematical reasoning abilities. Model mathematical reasoning by analysing, generalising and justifying mathematical situations. By thinking aloud, teachers can model this process effectively.

**Start by analysing a problem**: 7+8, I can do that with 5+2+8!

**Then generalise the approach**: It doesn’t matter what order I add these numbers up in, I will get the same result.

**Justify the learning:** I can use this approach to solve other problems too.

Using mathematical reasoning helps build patterns of thinking and mathematical fluency.

## Beyond worked examples

Worked examples are the prime example of modelling and mathematical reasoning in action, but they are just the tip of the iceberg. While they provide a helpful starting point for students, relying solely on worked examples will not lead to a comprehensive understanding of mathematical concepts.

## Fade in/Fade out

The idea of Fade in/Fade out support refers to **when** and **why** support is provided. This concept is useful when thinking about modelling.

Teachers will often start a lesson with a worked example and then **fade out** their modelling while students proceed with their learning. This approach doesn’t allow for the flexibility and responsiveness required for the most effective modelling. Modelling should not be a one-time occurrence. Instead, it should be scaffolded throughout a lesson. Build your modelling up through your lesson and **fade in** with instruction and modelling as students need it.

## What does explicit modelling and mathematical reasoning look like?

Let’s unpack it with an example.

**Example 1:**

**Problem:** A student is asked to solve the equation 7+8 and to explain how they did it.

**Student:** I added 2 to 8, which made 10, and then added 5.

This strategy is called ‘bridging to 10’ and involves children using their knowledge of addition up to 10 as a base to then work out sums with totals over 10. A modelled response to this student’s work might look like this:

**Teacher:** Hang on a second, you said you added 2 to the 8, let’s take a look at that. So, 7+8 is the same as 5+2+8, where did the 7 go in that? Oh yes! (5+2)+8 also equals 15! What happens if we change the order of the numbers?

**Student**: 8+5+2, that also equals 15.

In this example, the student has modelled their own mathematical awareness, and this has then been extended with the teacher modelling both equivalence and the associative property in the working out of the problem.

## Planning for modelling

Modelling is most effective when it is intentional and focused. Before teaching a lesson, ask yourself:

- What is the mathematical concept I want students to understand?
- What is the important mathematical skill I want them to see?
- What are the most effective ways I can model these strategies to students?
- How can I be responsive in my modelling (for example,
**fading in**)?

There is so much more to maths than just getting the right answer! Explicit modelling is a powerful tool for making mathematical concepts and practices accessible to students. By incorporating modelling, educators empower students to not only find the right answers, but also to develop their mathematical reasoning skills, as well as their understanding of the underlying processes and strategies that mathematics is built upon. Modelling helps students to develop their thinking and come up with efficient strategies for tackling unknown mathematical problems. And *this* is the work of a true mathematician!