Balancing Perspectives on Mathematics Instruction

Focus on Exceptional Children, May 2003 by Jones, Eric D, Southern, W Thomas

Generally, teachers first model the correct response for students who are in the initial phases of acquisition. Once teachers have shown students the correct response, they lead or prompt the students in the performance. As students' performances indicate that scaffolding is unnecessary, prompts are less explicit and eventually are withdrawn.

Finally, students are expected to respond independently. Teachers can use the student's performances during independent responding to test or to evaluate student learning and the effectiveness of instruction. When the tasks have been partially or previously learned, scaffolding may consist of fewer prompts and less precise prompts.

With the conclusion of instruction, the direct instruction model calls for opportunities for review and practice. A basic tenet is to provide much practice for acquisition, which is then distributed to maintain knowledge and skills (Englemann & Carnine, 1991). Carnine (1989) outlined a set of guidelines for providing effective practice and review. Practice activities should:

* Involve the practice of skills that the student has acquired already and can perform independently. If the student requires assistance, the skill will require additional teaching.

* Introduce information to review activities cumulatively.

* Distribute practice to build retention.

* Emphasize relationships to make learning more meaningful.

* Pre-teach components of strategies and algorithms.

* Require quicker response times.

* Use varied examples for review.

Although practice is deemed necessary for effective instruction, the efficiency of instruction is determined in part by the extent to which well designed instructional interactions can reduce the amount of practice necessary to acquire and maintain knowledge and skills. For example, Darch, Carnine, and Gersten (1984) demonstrated that well selected instructional content and well designed explicit instruction in generalizable strategy instruction resulted in more rapid acquisition of mathematical problem-solving strategies and required less practice than instruction that would be available from typical commercial mathematics curricula.

Constructivism

The constructivist perspective on appropriate approaches to instruction has been harder to define than direct instruction. Thus, the risk of misrepresenting it, or at least not representing it to the satisfaction of all concerned, is greater. The constructivist ideology emerged from the earlier discovery learning approach to instruction, which gained some popularity-as well as resounding criticism-in the 1960s. Advocates of constructivism (e.g., Cobb, 1994a, 1994b; Driver, Asoko, Leach, Mortimer, & Scott, 1994; Gadanidis, 1994; Palincsar, 1998; Poplin, 1988a, 1988b; Pressley, Harris, & Marks, 1992; Woodward & Montague, 2002) assert, as did the advocates of discovery learning (e.g., Bruner, 1961; Hawkins, 1966, Rogers, 1968) that students acquire the most meaningful understandings and appreciations of their learning and problem-solving experiences if they are engaged in learning activities that allow them to discover relationships and solutions for themselves.