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close this bookImproving Student Achievement in Mathematics (IAE - IBE - UNESCO, 2000, 48 p.)
View the document(introduction...)
View the documentThe International Academy of Education
View the documentPreface
View the documentIntroduction
View the document1. Opportunity to learn
View the document2. Focus on meaning
View the document3. Learning new concepts and skills while solving problems
View the document4. Opportunities for both invention and practice
View the document5. Openness to student solution methods and student interaction
View the document6. Small-group learning
View the document7. Whole-class discussion
View the document8. Number sense
View the document9. Concrete materials
View the document10. Students’ use of calculators
View the documentConclusions
View the documentAdditional resources
View the documentReferences
View the documentThe International Bureau of Education-IBE

4. Opportunities for both invention and practice

Giving students both an opportunity to discover and invent new knowledge and an opportunity to practise what they have learned improves student achievement

Research findings

Data from the TIMSS video study show that over 90% of mathematics class time in United States eighth-grade classrooms is spent practising routine procedures, with the remainder of the time generally spent applying procedures in new situations. Virtually no time is spent inventing new procedures and analysing unfamiliar situations. In contrast, students at the same grade level in typical Japanese classrooms spend approximately 40% of instructional time practising routine procedures, 15% applying procedures in new situations, and 45% inventing new procedures and analysing new situations.

Research evidence suggests that students need opportunities for both practice and invention. The findings from a number of research studies show that when students discover mathematical ideas and invent mathematical procedures, they have a stronger conceptual understanding of connections between mathematical ideas.

Many successful reform-oriented programmes include time for students to practise what they have learned and discovered. Students need opportunities to practise what they are learning and to experience performing the kinds of tasks in which they are expected to demonstrate competence. For example, if teachers want students to be proficient in problem solving, students must be given opportunities to practise problem solving. If strong deductive reasoning is a goal, student work must include tasks that require such reasoning. And, of course, if competence in procedures is an objective, the curriculum must include attention to such procedures.

In the classroom

Clearly, a balance is needed between the time students spend practising routine procedures and the time which they devote to inventing and discovering new ideas. Teachers need not choose between these activities; indeed, they must not make a choice if students are to develop the mathematical power they need. Teachers must strive to ensure that both activities are included in appropriate proportions and in appropriate ways. The research cited above suggests that attention to them is currently out of balance and that too frequently there is an overemphasis on skill work, with few opportunities for students to engage in sense-making and discovery-oriented activities.

To increase opportunities for invention, teachers should frequently use non-routine problems, periodically introduce a lesson involving a new skill by posing it as a problem to be solved, and regularly allow students to build new knowledge based on their intuitive knowledge and informal procedures.


Boaler, 1998; Brownell, 1945, 1947; Carpenter et al., 1998; Cobb et al., 1991; Cognition and Technology Group, 1997; Resnick, 1980; Stigler & Hiebert, 1997; Wood & Sellers, 1996, 1997.