This summary provides an overview of chapter 5 of Effective pedagogy in Mathematics/Pangarau: Best evidence synthesis iteration [BES] (Anthony and Walshaw, 2007). This chapter discusses the forms of mathematical tasks used by teachers to support their students in learning mathematics. The full document can be downloaded from this page.
It is important to choose appropriate mathematical tasks and activities for students.
Key Points to be understood:
- Tasks should be designed to actively engage students in mathematical thinking.
- Tasks should take into account students’ previous knowledge and experiences.
- A range of tools should be used to support students’ understanding of the mathematical concepts involved in tasks.
The primary purpose of tasks in a mathematics lesson should be to engage students with a mathematical idea.
- While there is a place in a mathematics class for practice and consolidation, “tasks that require students to engage in complex and non-algorithmic thinking promote exploration of connections across mathematical concepts” (p 97).
- Tasks that are open ended, rather than procedural, provide students with “something they need to think about, not simply a disguised way of practising already demonstrated algorithms.” (p.97). Some examples of open ended tasks:
- Instead of finding the average of a list of numbers ask the students to suggest a possible list of numbers for a given average. (For example, the average number of family members of students in the class is 4. What might this look like for a class of 20 students?)
- As an exercise to encourage addition and subtraction ask “If Jimmy and his 2 brothers took out a total of 10 books from the library, how many might they each have taken?”
- Instead of learning lists of rules to identify quadrilaterals such as diamond, rectangle, kite, parallelogram, and rhombus, you could ask students to “Draw as many different 4 sided shapes as you can and describe their features.”
- Implementation of tasks should focus on students’ strategies not just the answer to problems. For this reason, tasks with multiple possible solution strategies are to be encouraged. Asking students to explain their methods helps them develop their mathematical understandings. Students’ success on a task should be judged by their understanding rather than superficial indicators such as speed of completion or correctness of answers.
- Productive mathematical tasks encourage the development of learning dispositions: reflection, generalisation, curiosity, conjecture and exploration. Enquiry based tasks, exploring students’ questions (for example, what shapes can be in a tessellation?) are more appealing to students than prescriptive tasks. However, keeping mathematics interesting and fun should not be at the expense of content. Use of manipulatives and individual choices of activity can in some instances allow students to avoid ‘real maths’ instruction.
In selecting or developing mathematical tasks, consideration needs to be given to the ability of students and to their previous life experiences.
Tasks should be appropriate for students’ level of ability:
- Productive task engagement requires that tasks relate closely enough to current knowledge and skills to be understood but be different enough to extend students’ thinking. If tasks are too easy or too hard they are not motivating and are unlikely to engage students. “Tasks that are too easy or too hard have limited cognitive value” (p119).
- Students do not all progress along a common developmental path at the same rate. Use your ongoing evaluation of students’ performance to help pitch the tasks at the right level. For this to be possible you need to know what the students are capable of. This is one particular strength of the Numeracy Development Project.
Contexts used in tasks should relate to students’ life experiences where possible:
- Use of real life contexts that are appropriate to your students’ experiences may make the mathematics more meaningful, accessible and appealing for students. Deciding which contexts are familiar for students is challenging (for example addition with money is not necessarily a familiar context as children usually buy things one at a time).
- It is important that the context does not obscure the mathematics involved in the problem; overly complicated contexts can lead to a task being more about interpreting the question than the actual mathematics.
A range of tools should be used to support students’ understanding of the mathematical concepts involved in tasks.
- For the purposes of this discussion, tools include but are not restricted to: physical equipment, diagrams, graphs, examples, illustrations, problem contexts, equations, calculators and computers. Such resources can be used to provide a frame of reference to support the development of mathematical understanding.
- Materials need to provide a scaffold to think about mathematical ideas rather than the focus for the learning. Using a range of materials to illustrate the same key idea helps students build conceptual mathematical ideas.
- Getting students to record their own representations of learning can be valuable as it can help teachers make sense of students’ strategies and provide evidence of student learning. Students own representations are also often the most accessible to the student. When representations are chosen by the teacher the students may not see the link between the representation and the maths as clearly as when they use their own model.
- Using both informal and formal recording is important. Students should be given opportunities to create their own representations before being introduced to conventional ones; this may make the effectiveness of conventional forms more apparent to students.
- Artefacts, including pictures, diagrams and symbols, can be used to provide a ‘bridge’ between the context of the task and the mathematical ideas involved.