What do students learn while solving mathematics problems?

Abstract

Evidence is accumulating that the means-ends problem-solving strategies used conventionally by novice problem solvers are relatively ineffective as vehicles for the acquisition of schemata characteristic of experts. We suggest that a means-ends strategy places a heavy load on cognitive processing capacity and that this load retards knowledge acquisition. A series of three experiments using trigonometry problems was carried out in which the problem goal was modified with the intention of disrupting the strategy used by novices. It was hypothesized that development of adequate cognitive representations of the sine, cosine, and tangent ratios would be enhanced as a consequence. Our results indicated that preventing novice problem solvers from using meansends analysis resulted in fewer mathematical errors both during acquisition and on subsequent problems including transfer problems. This provided some evidence for our contention that a means-ends strategy places a heavy load on cognitive processing capacity, which retards knowledge acquisition.

Positive transfer and negative transfer/antilearning of problem-solving skills.

“…The second research history concerns the effects of two related features of goal setting in CDCTs: goal-specificity (e.g., Burns & Vollmeyer, 2002;Owen & Sweller, 1985;Sweller & Levine, 1982;Vollmeyer, Burns, & Holyoak, 1996), and goal-difficulty (e.g., Kanfer & Ackerman, 1989;Kanfer, Ackerman, Murtha, & Nelson, 1994). In studies of the goal specificity effect, participants are either instructed to first learn about a complex control system by controlling it to a criterion (SG: Specific Goal), or are simply told to explore the system (NSG: Non-Specific Goal).…”

“…The effect of varying the specificity and difficulty of reaching a goal on control performance was first examined by Sweller (Sweller & Levine, 1982;Sweller, Mawer, & Ward, 1983), and has since been replicated (e.g., Berry & Broadbent, 1984;Burns & Vollmeyer, 2002;Geddes & Stevenson, 1997;Miller, Lehman, & Koedinger, 1999;Owen & Sweller, 1985;Trumpower, Goldsmith, & Guynn, 2004;Vollmeyer et al, 1996). Burns and Vollmeyer's (2002) recent account of the phenomena develops on Dual-Space theory (Klahr & Dunbar, 1988;Simon & Lea, 1974).…”

Instructional Design Consequences of an Analogy between Evolution by Natural Selection and Human Cognitive Architecture

“…Such non-specific or goal-free problems reduce working memory load because the complex procedure described above for solving a conventional problem is substituted by a procedure that simply involves finding a set of givens that will allow something to be calculated. The goal-free effect occurs when goal-free problems result in more learning than conventional problems and has been demonstrated on many occasions (Ayres 1993;Bobis, Sweller and Cooper 1994;Burns and Vollmeyer 2002;Geddes and Stevenson 1997;Miller, Lehman and Koedinger 1999;Owen and Sweller 1985;Paas, Camp and Rikers 2001;Sweller 1988;Sweller and Levine 1982;Sweller, Mawer and Ward 1983;Tarmizi and Sweller 1988;Vollmeyer, Burns and Holyoak 1996). It needs to be noted that because they provide no central executive guidance to problem solvers, goal-free problems should only be constructed using problems with a limited number of possible moves and where all moves have educational significance.…”

Thinking with a Theory: Theory-prediction Consistency and Young Children’s Identification of Causality

“…Owen & Sweller (1985) have demonstrated that high school students given a ''non-specific goal'' of finding out how to solve problems in geometry showed greater understanding of the underlying mathematical Table 2. Means and standard deviations of preliminary assessment scores for children in each of the four groups (n = 12 for each group)…”

“…Such a goal might encourage other approaches, such as pattern matching strategies. A procedural or, in the terms of Owen & Sweller (1985), a more non-specific goal, with an emphasis on finding out how, might be more successful at encouraging a greater depth of thought. Thus, it was predicted that this goal would give rise to higher theory scores, greater theory-prediction consistency and more reflection upon unexpected outcomes.…”