The core of mathematics is problem solving: “Without a problem, there is no mathematics” (Holton et al., 1999).

A mathematical problem is any situation that must be resolved using mathematical tools but for which there is no immediately obvious strategy. If the way forward is obvious, it’s not a problem – it’s a straightforward application.

All mathematical programmes, including those for ALiM students, need to include complex, rich, and/or problematic tasks. The task for the teacher is to set the difficulty level high enough to challenge students, but not so high that they can’t succeed. Teachers who get this right create resilient problem solvers who know that with perseverance they can succeed.

Whaia te iti kahuranga ki te tuahu koe me he maunga teitei.

Aim for the highest cloud so that if you miss it, you will hit a lofty mountain.

### Why is this important?

Students only learn to handle complex tasks by being exposed to them.

Providing students with the opportunity to work on complex tasks – as opposed to a series of simple tasks devolved from a complex task – was crucial for stimulating [the students’] mathematical reasoning and building durable mathematical knowledge.

Francisco and Maher, 2005, quoted in Anthony and Walshaw, 2007, page 118

… if the teacher feels that students in a low-achieving group cannot solve multi-step problems and so does not pose them, the students will not learn how to solve them.

Sullivan, 2011, page 42

To become powerful learners of mathematics, students need opportunities to:

… sort, classify, structure, abstract, generalise, specialise, represent and interpret symbolically and graphically, justify and prove, encode and decode, formulate, communicate, compare, relate, recognise familiar structures, apply and evaluate applications, and automatise.

Anthony and Walshaw, 2007, page 120

It is the problem-solving potential of mathematics that makes it useful and interesting. If students are not exposed to problematic tasks, they are unlikely to see the point of mathematics, show interest in mathematics, or gain satisfaction from mathematics.

Tolerance of difficulty is essential in a problem-solving disposition because being stuck is an inevitable stage in resolving just about any problem. Getting unstuck typically takes time and involves trying a variety of approaches. Students need to learn this experientially.

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