Aware that the use of concrete math materials has met with mixed results, Patrick W. Thompson, co-director, Quantitative Reasoning Project, San Diego State University, has been studying ways to use concrete materials more effectively to increase mathematical understanding. Effective instruction using concrete materials begins, Thompson stresses, with recognizing that the materials themselves have no inherent ability to convey a mathematical concept. Only teachers can help students understand concepts by the way they use materials and the questions they pose for students.
Research shows that concrete materials are interpreted by different students in different ways. Because of this, Thompson believes that teachers need to be aware of all the possible interpretations students may make using a specific material, and to be accepting of unconventional approaches to problem solving. Encouraging students to discover solutions in their own creative ways enables them to gain confidence in their math ability.
Unfortunately, Thompson reports, teachers often reject creative problem solving when trying to teach a standard algorithm. For example, when base-ten blocks are used to teach addition and subtraction of whole numbers, teachers frequently correct students who begin working from the left with the largest blocks. It is unconventional but not incorrect, he insists, to work left to right.
When teachers discourage unconventional but legitimate approaches, students may learn the desired algorithm but miss out on opportunities for deeper understanding. For this reason, Thompson reminds teachers to ask themselves “What should students understand?” rather than “What should students learn to do?”
Thompson believes that concrete materials can be a very effective teaching tool, but, he concedes that they are not easy to use well. Concrete materials provide a prop, an opportunity to discuss abstract mathematical concepts using tangible objects in ways that help students discover solutions empirically.
“Concrete Materials and Teaching for Mathematical Understanding”, Arithmetic Teacher, Volume 78, Number 8, June 1994, pp.556-558.
Published in ERN, September/October 1994, Volume 7, Number 4.