Students need to experience a wide range of problems so that they can develop skills and strategies that have realworld relevance. This principle is embedded in the stem to the achievement objectives in the mathematics curriculum.
In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to …
A good problem is “problematic” – it is centred around a genuine problem. Unlike “problems” that can be solved by applying a simple procedure, problematic tasks challenge students’ thinking and involve them in testing, proving, explaining, reflecting, and interpreting. Such tasks:
- are accessible and extendable
- allow individuals to make decisions
- promote discussion and communication
- encourage originality and invention
- encourage “what if?” and “what if not?” questions
- contain an element of surprise.
Adapted from Ahmed, 1987
A problematic task offers serious potential for discussion and learning. When students explain their ideas, respond to the ideas of others, and allow their thinking to be challenged, they are “doing maths”. For those who think maths is all about “getting it right” and who need the positive reinforcement of long lines of ticks next to “answers” in their books, this can be difficult to accept. But students need to be weaned from this counterproductive orientation.
… studies suggest that a classroom orientation that consistently defines task outcomes in terms of the answers rather than the thinking processes entailed in reaching the answers negatively affects the thinking processes and mathematical identities of learners.
Anthony and Walshaw, 2007, page 122
Where teachers value process over answers, the emphasis shifts from “Did you get it right?” to “Do you understand?” and “What did we learn?” Teachers need to make this emphasis explicit.
Word problems are usually attempts to put mathematics into contexts, but putting problems into words doesn’t automatically make them real. This can be an issue for students, who get confused by the fact that, simultaneously, they have to pay attention to things that they would normally ignore and ignore things that they would normally assume mattered.
Think about ways in which this word problem is unrealistic:
A caterer at a wedding supplies 5 pavlovas. At the end of the wedding, she sees three quarters of all the pavlovas were eaten. How many pavlovas were eaten?
Most obviously, why would a caterer ever bother with such a calculation? They could hardly assume that, next time around, they should only provide 3 ¾ pavlovas!
The challenge for teachers is to provide students with problems that draw on their experience of reality, rather than asking them to suspend it.
Such authentic contexts provide sense-making and experientially real situations for children, rather than simply serve as cover stories for proceduralised and frequently irrelevant tasks.
English, 2004, page 3
Mathematical problems do not, however, need to be set in reallife contexts. Imaginative contexts can provide engaging and engrossing opportunities for mathematical explorations. And playfulness and humour can increase student engagement and encourage creative thinking.
Watson (2004) argues that ‘realistic’ does not mean that tasks must necessarily involve real contexts … she advocates that tasks should be seen as ‘realistic’ not because they relate to any particular everyday context, but because they make students think in ‘real’ ways.
Anthony and Walshaw, 2007, page 114
For an example of a rich mathematical problem set in a mythical context, see the Icarus and Daedalus problem on the Math Pickle website at http://www.youtube.com/watch?v=R4oINmqHXVY&feature=player_embedded#!
Unpacking word problems
ALiM students are likely to need deliberate support in interpreting word problems. But it is important not to begin by unpacking the problem for them, assuming that they are incapable of getting meaning from the words. Instead of supporting them to learn, this will encourage dependency. Rather, expect the students to do their utmost (working individually or in twos or threes) to make sense of the problem, and then get them to explain to you what they believe it is about. You can then support them to complete the interpretation process.
Don’t be too tidy
The way word problems are constructed can limit student thinking. Real-life problems rarely have a single, straightforward solution, so students need exposure to problems that allow for different solutions. Such problems are important in shifting the focus from the answer to the mathematical thinking involved.
It is also important that students (including ALiM students) are not restricted to problems that have tidy, whole number solutions. If they are, they will be ill-equipped to deal with more realistic problems, which typically involve messier numbers, both as inputs (for example, measurement data) and as solutions. Real-life problems will provide students with the impetus and need to move beyond the set of counting numbers.