For students to become effective learners of mathematics, they must be actively engaged, and want and be able to take on the challenge, persistent effort and risks involved. This is most likely to occur when the student personally experiences an environment that is supportive but mathematically challenging and when processes that enhance sustained and robust learning are promoted.
Opportunity to learn
Learning experiences should enable students to engage with, observe and practice the actual ideas, processes, products and values which are expected of them.
Students should practice, (that is, ‘do’) mathematics, but doing mathematics involves much more than the repetition of facts and procedures; it also involves working mathematically across all the strands. If students are to learn to deal flexibly with fractions in both routine and non-routine ways, select the most suitable approach for a situation, try alternatives, or adapt procedures to different situations, they will need to
see these processes modeled by others, and engage in them for themselves. This will require that they understand the concepts about fractions which underpin procedures and processes. Thus, they will need to engage in activities which focus on the meaning of fractions, rather than simply procedures for dealing with them.
A limited number of skills should be automated and some repetition will help with this. Students should, however, practice the skill which they actually need to develop: for example, to develop mental addition skills, students need spaced and varied practice with a repertoire of alternative addition strategies and an emphasis on choosing strategies suited to the particular numbers. Repetitive exercises on a written vertical addition algorithm are unlikely to improve mental addition, indeed, they are more likely to interfere with it.
Connection and challenge
Learning experiences should connect with students’ existing knowledge, skills and values while extending and challenging their current ways of thinking and acting.
Learners’ interpretations of new mathematical experiences depend on what they already know and understand: for example, often the first experience children have of the decimal point is in the context of money and measure. As a result, many develop the idea that the decimal point is a separator of two whole numbers (the dollars and the cents; the meters and the centimeters), a reasonable ‘first estimate’ of the meaning of the decimal point, even though inadequate in the longer term. Good teaching will help students to clarify or bring to the surface their understanding of decimals in order to extend, refine or discard unhelpful ideas and construct a more sophisticated understanding of the meaning of the decimal point.
If additional ideas about decimals are introduced without connecting to and challenging their existing ideas, students may continue to hold on to their earlier understanding, even when they are apparently successful with the new material. Thus, students may have learned that the first column to the right of the decimal point is the tenths and the second position is the hundredths and yet, underneath this, still think of the decimal point as separating two whole numbers. This can lead them to a number of different errors. For example, they may expect a book with a Dewey Decimal Number 3.125 to come after one with Number 36.65, continue the sequence 1.2, 1.4, 1.6, 1.8 … with 1.10, and, round an answer of $3.125 to $4.25. Dealing with each of these errors separately is likely to be unproductive and inefficient – the underlying misconception should be addressed.
A mathematical challenge to this understanding could occur when the student finds that the book is not where he expects it to be, punches the sequence into a calculator by repeatedly adding 0.2 and find the calculator goes from 1.8 to 2.0, or estimates that $25 ÷ 8 is very close to $3 and cannot be more than $4. To learn from the challenge or conflict, the student must recognise it, see errors as a useful source of feedback,
believe that mathematics is supposed to make sense and that inconsistent answers need to be thought about, and respond by trying to find some way of dealing with the inconsistency.
Action and reflection
Learning experiences should be meaningful and encourage both action and reflection on the part of the learner.
Mathematical learning is most successful when students actively engage in making sense of new information and ideas. If students face mathematical situations that are not inherently meaningful, then they are forced to conclude either that mathematics does not make sense or that they themselves are incapable of making that sense. Providing students with isolated facts and procedures which they are expected simply to imitate and remember, or with partial explanations of concepts disconnected from their other mathematical ideas forces them to resort to learning strategies based on the passive reception of mathematical concepts and processes and the role imitation of procedures. The result is likely to be short term storage that needs to be topped up
regularly, rather than effective long-term learning.
How students respond to a task and what they learn from it depends upon their conception of the task. Students should be helped to distinguish between activities which provide drill to increase accuracy and efficiency on already-learned procedure, and activities which develop understanding of mathematical concepts and processes or require them to apply mathematics in new ways. For the former, they should expect to be able to get started almost immediately and, otherwise seek help.
For the latter, persistence, thoughtfulness, struggle and reflection are expected as they work out what to do for themselves. Students should be taught to reflect upon what did and did not work and why, and how it connects to other mathematics.
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 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. 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.
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