Nurturing students' resourcefulness

This guidance is part of our Primary and Secondary 'Developing mathematical mindsets' collections.

To complement this guidance, we hosted two teacher webinars. You may wish to watch the recording/s before or after taking a look at the guidance below.

We'd like our students to believe they have the capacity to tackle new problems by drawing on their knowledge and skills, and if necessary, making connections in new and creative ways.

What can we do in our mathematics classrooms to prepare our students, so that they approach new challenges with confidence? 

It may be helpful to think about the following:

Values and ethos in our classrooms

• We appreciate that mathematics comprises the arbitrary and the necessary, and we teach accordingly, emphasising connections whenever possible
• Maths is more than just getting the right answer, it is about being able to explain and justify how one reached it; different methods are valid, if they can be justified
• We try to avoid doing anything for students that they can do for themselves - the teacher's role is to draw on students' capabilities and build on their ideas
• We don't expect students' first attempts to always be successful
• Mathematical ability is not fixed; everyone can learn new approaches

Structural considerations

• Alan Wigley's 'challenging model' encourages us to start lessons with a problem, build on students' ideas (think-pair-share), and set aside time for plenary sessions to draw together key ideas
• Oracy skills are given high priority so students have opportunities to articulate their ideas and engage with others to develop their understanding
• Inviting students to look for a way, rather than the way, means they are less reliant on memory or guessing what's in the teacher's head (which can lead to anxiety), and are more likely to be successful
• If students are to be flexible and creative, giving them thinking time, and encouraging them to pause and choose between alternative approaches before they start on a problem, can be helpful
• Feedback and assessment focuses on students' willingness to work mathematically (experimenting, trying different approaches, collaborating, questioning)

Facilitating

The teacher models the questions and comments a resourceful mathematician might ask/make and, over time, encourages students to ask these questions themselves:

• What could you/I do? Which method might you/I use? Why?
• This reminds me of...
• This hasn't worked, what could you/I try now?
• How did you do it? Why?
• Did anyone do it in a different way?
• Why do both strategies work?
• Which strategy is better? Why?
• What else could you do?

Some teachers use the published students' solutions to NRICH problems to encourage their students to consider the merits of different approaches.

Below we have selected a few tasks that exemplify our whole collection of problems which challenge students to be resourceful. We hope these make good starting points for you.

Can be tackled in multiple ways, and some can be used to illustrate the value of multiple representations:

Shape times shape (Age 7-11)

Coded hundred square (Age 7-11)

Through the window (Age 7-11)

Consecutive seven (Age 11-14)

Mixing lemonade (Age 11-14)

Always a multiple? (Age 11-14)

Special numbers (Age 11-14)

Odds, evens and more evens (Age 11-14)

Warmsnug double glazing (Age 14-16)

Which is bigger? (Age 14-16)
 

Can help students appreciate that working systematically is a valuable resource at their disposal:

Heads and feet (Age 5-7)

Sealed solution (Age 7-14)

Two and two (Age 7-16)

What's possible? (Age 14-17)
 

Can be used to challenge students to overcome barriers:

Make 37 (Age 5-11)

Cryptarithms (Age 11-14) 

Unequal averages (Age 11-14)

Product sudoku (Age 11-16)

These collections of Primary and Secondary problems, organised by year group, are designed to encourage students to see the value of becoming more resourceful. 

References

Malcolm Swan's article Improving reasoning: analysing alternative approaches

Three linked articles by Dave Hewitt: 

"If I'm having to remember..., then I'm not working on mathematics"

Arbitrary and Necessary Part 1: A Way of Viewing the Mathematics Curriculum

Arbitrary and Necessary Part 2: Assisting Memory

Arbitrary and Necessary Part 3: Educating Awareness

Colin Foster's article Problem solving in the mathematics curriculum: From domain-general strategies to domain-specific tactics

I Can't Do Maths series of three podcasts on the NCETM website