Learning experience

Motivation and purpose

Learning experiences should be motivating and their purpose clear to the student.

Mathematics is often promoted to students as an investment in the future and for some students this is sufficient motivation to keep them working at it. For others, however, this is not persuasive and the mathematics provided in school must provide its own motivation if such students are to continue to participate actively. All students, however, should have opportunities to experience the satisfaction and pleasure that mathematics can bring. Students should use mathematics in decision making and problem solving about situations that are interesting in their own right and not simply because they demonstrate some mathematical idea.

Effective learning requires that students feel able to risk making mistakes without fear of the consequences. This means that the purpose of activities and hence expectations must be clear to students so that they know which risks are reasonable. Learning about families of functions may involve students in predicting what a set of related graphs will look like. Students should not be inhibited from conjecturing or making quick sketches for fear that they will be judged negatively if their early tries don’t work or are messy. Later, however, they would be expected to correctly predict graph shapes and sketch them with care.

Inclusivity and difference

Learning experiences should respect and accommodate differences between learners.

Linguistic, cultural, gender and class differences between students are often regarded as adequate explanations for differences in mathematical achievement. This Framework starts from the premise, however, that a common cause of many students’ failure to learn mathematics in a sustainable and robust way is an inadequate match between the curriculum and the experiences and understandings of students. For example, many children come to school able to count collections of 6 or 7 by pointing and saying the number names in order, but they do not have the visual memory to recognize 6 or 7 at a glance. Others (and this may be more common in some Aboriginal and Torres Strait Islander communities), may recognize 6 or 7 objects at a glance without being able to say the number names in order. In each case, the students’ existing knowledge should be recognized and used as the starting point for further learning. In each case, it should be extended to include the complementary knowledge, with the new knowledge being linked to, building on and challenging the students’ existing ideas and strategies, so that over time they develop mathematical understandings which are both commonly accepted and over which they feel some ownership.

Independence and collaboration

Learning experiences should encourage students to learn both from, and with, others as well as independently.

Collaborative learning can enhance mathematical learning in a number of ways. Firstly, by working together and pooling ideas, students can develop ideas and solve problems which may be inaccessible to them individually. Secondly, students’ command of mathematical ideas and mathematical language is likely to improve when they try to describe, explain or justify. Thirdly, discussion is one of the ways students come to understand that others may not interpret things in the same way or share their point of view. Finding that a friend is not convinced that all quadrilaterals will tessellate may motivate the student to rethink, clarify and refine his or her ideas and ways of talking about them, and to develop better arguments to justify the claim. The skeptical friend may be provoked to think about what she or he knows in fresh ways or to work on what she or he doesn't know, perhaps coming to see that quadrilaterals can be thought of as two triangles with a common edge and that this provides a way of showing how all quadrilaterals tile.

Working individually is also important in mathematics. It should enable students to ensure a personal grasp of concepts, processes and procedures. In turn, they should develop confidence in their capacity to do mathematics for themselves. Students will need help to develop strategies for getting started and persevering in mathematical situations.

Supportive environment

The school and classroom setting should be safe and conducive to effective learning.

High levels of unproductive anxiety about school mathematics are common, even among students who achieve well. This anxiety is associated with certain beliefs about mathematics. Firstly, there is a widespread and deep-seated view that you either have ‘a mathematical mind’ or you don’t and that those who do are quick thinkers, can respond instantly to tasks and recognize an appropriate solution strategy immediately.

Secondly, mathematics is seen to be either right or wrong and the feeling of exposure associated with being wrong is quite debilitating for many students. In order to cope and avoid feelings of failure, some students ‘don’t try’ and, as a result, progress very little as they proceed through school.

The belief that students’ confidence in mathematics will increase if they have continued success sometimes tempts us to explain exactly what to do, reducing the risk of error, and expecting and accepting less high-level thinking of certain students. Many learners become debilitated by continued success on personally easy tasks, however, and increasingly avoid situations in which they might make mistakes or be found out, and become less able to take the risks needed for higher-level learning. It is important that students learn to flounder in a constructive way rather than to avoid all stress and struggle, and that the mathematics classroom be typified by challenge within a supportive learning climate.