# Developing Mathematical Thinking - Exploring and noticing

*Exploring and Noticing is part of our Developing Mathematical Thinking Primary and Secondary collections.This page for teachers accompanies the Primary and Secondary Exploring and Noticing resources. *

*You may wish to watch the recording of the webinars, which draw on the resources below to discuss how teachers can offer students opportunities to explore and build on their discoveries.*

During this webinar, we had a go at the following tasks:

The Number Jumbler

Add to 200

Cyclic Quadrilaterals

Tilted Squares

Mathematicians take great pleasure in being challenged by a problem they don't immediately know how to solve, and are excited by exploring a mathematical idea or question for the first time. They will often play with ideas, try some examples and test different approaches. Along the way, they look out for patterns and structure that might move their thinking forwards, and try to
make connections with what they already know.

If students are to work as mathematicians, willing to be 'playful' while being uncertain about how to proceed, then we may need to think about the following:

**Values and ethos**

- Believing that students can learn when presented with a problem or challenge to be explored, and then given an opportunity to share ideas
- Intending to build on students' ideas, questions and insights
- Valuing the 'messy' stages of working on a problem, and recognising that wrong turns and mistakes are a springboard to learning
- Recognising that there is never just one way to approach a problem, so students are encouraged to take responsibility and make choices
- Placing importance on students feeling a sense of ownership for the mathematics they are learning

**Structural considerations**

- In his 14 Practices for Building Thinking Classrooms in Mathematics, Peter Liljedahl makes suggestions about where students work (Practice 3) and ways to arrange the furniture (Practice 4) to ensure classrooms are most conducive to thinking
- Alan Wigley's 'Challenging Model' offers a useful framework for structuring lessons, in which students are presented with a task at the start of a lesson and are then given opportunities to collaborate and share ideas
- Making use of rich, and accessible low threshold high ceiling (LTHC) tasks offers students the opportunity to carry out some exploratory work before any teacher input
- There is a clear intention that the ideas arising in the exploration phase are built upon - see Ruthven's codification stage
- In Train Spotter's Paradise, Dave Hewitt alerts us to the richness that can be gained by looking at a particular situation in some depth

**Facilitating**

- Peter Liljedahl draws attention to the importance of considering 'When, where, and how tasks are given in the thinking classroom', see 14 Practices for Building Thinking Classrooms in Mathematics, Practice 6
- The teacher gives very clear messages that they are confident that students can contribute valuable ideas. Support offered by teachers will still expect students to do the important mathematical thinking.

"What have you tried so far?"

"What do you notice?"

"Is this linked to...?"

"What's the same and what's different?"

"What could you do next?"

"Can you find other examples?"

"Will this always work?"

"What might a mathematician ask next?"

As a result, students learn to believe in their own mathematical powers ("Can I have a bit more time to think about this?").

- Teachers only respond to 'keep-thinking questions', the questions students ask so that they can keep working, keep trying, and keep thinking - see Liljedahl's 14 Practices for Building Thinking Classrooms in Mathematics, Practice 5
- Messy work, use of manipulatives, jottings, trial and improvement, are all recognised as valuable, prior to organising ideas.
- Giving students time to think on their own, and an opportunity to discuss their thinking with one or two other people, as well as sharing their ideas with the whole class, highlights the value of exploration.
- The teacher models the questions and comments a mathematician might make:

"Ooh that's interesting!"

"That's surprising!"

"I think this might be linked to..."

"Ooh I wonder whether that always works."

**You may be interested in this collection of follow-up resources:**

Peter Liljedahl's book, Building Thinking Classrooms, in particular Practice 3 (Where students work in a thinking classroom), Practice 4 (How we arrange the furniture in a thinking classroom), Practice 5 (How we answer questions in a thinking classroom) and Practice 6 (When, where, and how tasks are given in the thinking classroom)

Mitch Resnick includes Play as one of his 'Four Ps of Creative Learning', and talks about all four in this recording of his Richard Noss lecture 'Sowing the seeds for a more creative society'

In their book Adapting and Extending Secondary Mathematics Activities, Stephanie Prestage and Pat Perks offer ways to create alternative versions of tasks, which will offer students a richer mathematical experience