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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

3. Learning new concepts and skills while solving problems

Students can learn both concepts and skills by solving problems.

Research findings

Research suggests that students who develop conceptual understanding early perform best on procedural knowledge later. Students with good conceptual understanding are able to perform successfully on near-transfer tasks and to develop procedures and skills they have not been taught. Students without conceptual understanding are able to acquire procedural knowledge when the skill is taught, but research suggests that students with low levels of conceptual understanding need more practice in order to acquire procedural knowledge.

Research by Heid suggests that students are able to understand concepts without prior or concurrent skill development. In her research with calculus students, instruction was focused almost entirely on conceptual understanding. Skills were taught briefly at the end of the course. On procedural skills, the students in the conceptual-understanding approach performed as well as those taught with a traditional approach. Furthermore, these students significantly outperformed traditional students on conceptual understanding.

Mack demonstrated that students’ rote (and frequently faulty) knowledge often interferes with their informal (and usually correct) knowledge about fractions. She successfully used students’ informal knowledge to help them understand symbols for fractions and develop algorithms for operations. Fawcett’s research with geometry students suggests that students can learn basic concepts, skills and the structure of geometry through problem solving.

In the classroom

There is evidence that students can learn new skills and concepts while they are working out solutions to problems. For example, armed with only a knowledge of basic addition, students can extend their learning by developing informal algorithms for addition of larger numbers. Similarly by solving carefully chosen non-routine problems, students can develop an understanding of many important mathematical ideas, such as prime numbers and perimeter/area relations.

Development of more sophisticated mathematical skills can also be approached by treating their development as a problem for students to solve. Teachers can use students’ informal and intuitive knowledge in other areas to develop other useful procedures. Instruction can begin with an example for which students intuitively know the answer. From there, students are allowed to explore and develop their own algorithm. For instance, most students understand that starting with four pizzas and then eating a half of one pizza will leave three and a half pizzas. Teachers can use this knowledge to help students develop an understanding of subtraction of fractions.

Research suggests that it is not necessary for teachers to focus first on skill development and then move on to problem solving. Both can be done together. Skills can be developed on an as-needed basis, or their development can be supplemented through the use of technology. In fact, there is evidence that if students are initially drilled too much on isolated skills, they have a harder time making sense of them later.

References:

Cognition and Technology Group, 1997; Fawcett, 1938; Heid, 1988; Hiebert & Wearne, 1996; Mack, 1990; Resnick & Omanson, 1987; Wearne & Hiebert, 1988.