Some cognitive factors relevant to mathematics instruction

Abstract

Our understanding of cognitive processes has progressed sufficiently in the last few years to enable us to generate novel instructional techniques that can enhance substantially learning of subjects such as mathematics. This paper will review briefly some research intended to contribute to this process. There are two relevant aspects. Firstly, recent work has thrown light on schema acquisition while learning mathematics, and on techniques for detecting schemas in mathematics learners. Secondly, other research has assessed the distribution of cognitive resources while learning mathematics and other related subjects leading to the design of instructional techniques to facilitate schema acquisition. The Relevance of Schemas to Mathematica) ExpertiseCognitive research findings have revealed the importance of schema acquisition in successful leaming and problem solving in mathematics (see Sweller, 1988). A problem schema can be defined as a cognitive construct that allows problem solvers to recognise a problem as belonging to a specific category that requires particular moves for solution. This definition includes problem states, problem-solving operators, and their relations.Consider a problem such as, "If a boat travelled downstream in 120 minuten with a current of 5 kilometres per hour and the return trip against the same current took 3 hours, what was the speed of the boat in stil) water?" Irrespective of the person's skill in computation and their basic linguistic and factual knowledge (e.g., what "stil) water" means, that rivers have currents that run only downstream), this problem is beyond the capability of someone who does not know the specific relations between speed of boat, rate of current, and time involved in river current contrast problems. In the absence of such schematic knowledge, there is no context for computation and calculation. The person needs to be able to identify the problem as belonging to a given type (e.g., river current contrast) in order to determine what information from the text should be used, in what sequence, and through what operations (see Mayer, 1987, pp. 345-373).

Word Problem-Solving Instruction in Inclusive Third-Grade Mathematics Classrooms

“…Their line of research has shown that explicit instruction in problem-solution and schema-broadening instruction as well as self-regulated learning strategies is necessary to enhance mathematical problem-solving performance. However, none of these studies examined the long-term effects of schema-mediated instruction or the benefits of visual representation techniques, such as those involving diagrams, that are helpful scaffolds in organizing information in the problem, thereby reducing the level of cognitive load associated with the problem-solving task (Pawley, Ayres, Cooper, & Sweller, 2005;Sweller & Low, 1992;Sweller, Van Merrienboer, & Paas, 1998).…”

Reducing cognitive load by mixing auditory and visual presentation modes.

“…This research has been generated by the following assumptions concerning our basic cognitive architecture: (a) People have a very limited working memory that is able to hold and process only a few items of information at a time; (b) People have a huge long-term memory that is effectively unlimited in size; (c) Schema acquisition is a primary learning mechanism. Schemata are defined as cognitive constructs that permit people to categorize information in the manner in which it will be used (see Low & Over, 1990, 1992; Low, Over, Doolan, & Michell, 1994; Sweller & Low, 1992). Schemata have the functions of storing information in long-term memory and of reducing working memory load by permitting people to treat multiple elements of information as a single element; (d) Automation of cognitive processes, including automatic use of schemata, is a learning mechanism that also reduces working memory load by effectively bypassing working memory.…”