Investigating Children’s Analogical Reasoning in Solving Mathematics And Science Problems

 

 

Hsin-Mei, E. Huang Department of Elementary Education, Taipei Municipal Teachers College, Taiwan

 

Abstract

The benefit of using analogy reasoning is that learners are able to apply a familiar structure to solve analogous problems in the target domain. Two studies examined the effect of problem content (i.e., its surface cover story) in children’s solving mathematics and science problems. Both studies provided the evidences that most children were able to identify the fields of problems based on their previous learning experience, however, such acquisition seems insufficient to help children activate the relevant knowledge about the deep structure in solving intradomain problems. Reminding did contribute certain advantages for children’s analogical problem solving and inferential thinking, especially when the source problems and target problems come from the same field. These results suggest that enforcing children’s ability in applying related knowledge to intradomain problem solving is expected.

Theoretical Framework and Objectives of the Research

The use of earlier examples in reminding appears to be a common method for solving problems. If textbooks are available, it’s frequently to observe students looking and skimming back through the pages for worked-out examples that look like the current problems. The core benefit of using analogy reasoning is that learners are able to apply a familiar structure (activated from the source) to solve analogous problems in the target domain (Holyoak, 1985; Novick & Hmelo, 1994). Furthermore, analogical reasoning effectively promote inferential learning (Donnelly & McDaniel, 1993; McDaniel & Donnelly, 1996).

A particular word problem type has a typical content. Problem solvers familiar with algebra are able to categorize algebra problems into certain types and after starting to read the problem they attempt to place the problem into a category as well as retrieve the set of previous knowledge about the problem solving task in which they are engaged, supposing it helps the problem-solving process. The superficial features of a word problem include the objects, settings, and surface cover story mentioned in the problem, and these superficial feature facilitate solvers to categorizing problems ( Blessing & Ross, 1996). Although superficial features of a problem presumably can affect a solver in accessing the problem deep structure, however it also affect solvers in application of problem skema less flexibility to other category problems that with similar deep structure. A problem’s superficial feature is distinct from its structural feature, which is the set of principles, potential relevant knowledge or equations important for solving the problem (Blessing & Ross, 1996). Some problems are with different content, but the deep structure of these problems are similar. If solvers could take intensive detection to the problems’ deep structure they may take advantage of their understanding of potential knowledge and apply them to solve novel problems.

In elementary school mathematics learning emphasizes objects manipulating, understanding and computing, whereas observation and experimental manipulating is the core of science learning. However, certain domain between mathematics and science are related. For example, consider the arithmetic-progression problem in algebra and constant-acceleration problem in science ( Bassok & Holyoak, 1989). Students learn various problem situations in the fields of mathematics and science respectively, and some knowledge can be flexibly applied to solve certain related problems between mathematics and science. For example, the basic ratio conceptions learned in mathematics can be applied to solve some spring-balance problems and air-volume problems which were included in the elementary school science curricula.

Mathematics are regarded as tools that are content free and domain independent of application. Such domain-independent concepts and representations of formal procedures, matched with appropriately abstract consideration of applicability, might be transferred to novel content areas (Bassok & Holyoak, 1989). Many studies have confirmed that primary school children can develop an abstract schema and make transfer. Still, it might be argued that problems with different contents have little impact on children’s analogical reasoning (e.g., Pierce, Duncan, Gholson, Ray & Kamhi, 1993; Zook & Di Vesta, 1991). Previous research has demonstrated that college or high-school students can produce transfer to structurally parallel physics problems with novel content (Bassok & Holyoak, 1989; Holyoak & Thagard, 1989). However, our understanding of the primary school children undertake their analogical reasoning in solving mathematical and science analogous problems has remained limited. The primary goal of this research was to fill this gap.

This research comprised of two related studies. The first study examined how children transfer mathematical knowledge to solve interdomain problems. The second study investigated how children transfer mathematical knowledge to solve structurally parallel science problems with novel contents. Both studies also examined the relationships between schema acquisition and inferential thinking.

Methods, Techniques and Data Sources

In the first study, 122 fifth grade children were selected from two public primary schools in Taipei county, Taiwan. Three kinds of source materials provided to children were as follows: Mathematical problems Reminding, Science problems Reminding, and No Reminding. Children were divided into three groups according to what kind of source material they’re learned. Children in each group were first presented with a source material and then they were requested to solve target problems consisting of two mathematical word problems. As Table 1 shows, for the MR group, the source included two Mathematical word problems, solution Reminding and line graphs of solutions. For the SR group, the source included two Science problems, solution Reminding and line graphs of solutions. The problems in source and target were with similar deep structure of mathematical basic ratio knowledge in both MR and SR. Children can apply the basic ratio knowledge learned from source to solve target problems. Meanwhile, after learning the source problems subjects were requested to identify the fields of source problems that they have learned before and then induce schema from source materials. The inferential thinking question is a multiple-choice test. Subjects have to generalize a correct statement that was induced from target problems solving. In the NR group, a story of magic leaves was given as a source material.

Table1: Source Problems and Target Problems In Study One and Study Two

	Study One	Study Two
Source: Two mathematical basic ratio word problems, solutions reminding , line graphs	MR	MR
Two spring-balance problems, solutions reminding , line graphs	SR	SR
Two spring-balance problems, solution reminding and line graphs		SPR
A story of magic leaves	NR	NR
Target: Two mathematical basic ratio word problems	MR, SR, NR
A spring-balance problem & an air volume problem		MR, SR, SPR, NR

In the second study, 162 fifth grade students were selected from two public primary schools in Taipei county. Subjects were divided into four groups according to what kind of source materials they learned. Four kinds of source materials given to children were MR, SR, NR, as well as Science problems and Pictures Reminding, and then they were requested to solve target problems which included two science problems. The SPR group was provided with pictures that represented the science problems contents, solution reminding and line graphs of solutions. As Table 1 shows, the target problems used in the four groups were a spring-balance problem and an air-volume problem in the science field, and the source materials in the MR, SR and NR groups were the same as those used in the first study. Subjects were also requested to identify the fields of source problems they have learned before and then induce schema from source materials in the MR, SR, and SPR groups.

Results and Conclusions

The results of the first study, in respect of the field identification of source problems, 91% of the MR group children identified source problems having been learned in mathematics field, whereas 7% identified the science field, and 2% identified the verbal field. In the SR group, 73% children identified source problems having been learned in science field, whereas 23% identified the mathematics field, and 2% identified the verbal and social study, respectively. The results show that most children were able to use previous experience to categorized problems. In the respect of comparison analogical reasoning problem solving and inferential performance among the MR, SR and NR groups, subjects’ previous academic performance (verbal and math achievement) were used as the covariate factors. The results of MANCOVA showed that reminding significantly affect children’s analogical and inference performance, wilks l =.81, p<.000. There are significant differences among the MR, SR and NR groups, F(2,117)=8.12, p<.000. Children in the MR group performed better in analogical problem solving than those in the NR group, but the difference was not significant between the MR and SR groups. Reminding also facilitated the children’s inferential thinking, F(2,117)=8.69, p<.000. The inferential performance of the MR and SR groups’ children outperformed those in the NR group. As the homogeneity of within-class regression coefficient is significant, wilks l =.83, p< .01, it is supposed that the type I error is existent and the possibility would be below .01 if we accept those forementioned results of analogical reasoning and inference performance.

Findings of the first study indicated that children performed the interdomain transfer of mathematical concepts and problem-solving procedures better than no reminding. When making transfer in problem solving as well as inferential thinking, children get certain benefits from reminding, even if the reminding drawn from domain specific of science facilitates children to inferential thinking. In addition, results showed that children in the MR and SR groups who could induce schema correctly from source problems performed better inferential thinking than those who induced incorrectly, t(81)=-3.92, p<.001.

From the second study, in respect of the field identification of source problems, 98% of the MR group children identified source problems having been learned in mathematics field before, whereas 2% identified the science field. In the SR group, 61% children identified source problems having been learned in science field, whereas 37 % identified the mathematics field. In the SPR group, 65% children identified source problems having been learned in science field whereas 28 % identified the mathematics field. The results are consistent with that in the first study, most children were able to categorized the problems’ fields according to their previous learning experience. As comparisons of analogical reasoning problem solving and inference performance among the MR, SR, SPR and NR groups, the subjects’ previous academic performance (verbal and math achievement) were used as covariate factors. The results of MANCOVA revealed that source material of reminding significantly affected children’s analogical performance and inferential thinking, wilks l =.77, p<.000. There were significant differences among the MR, SR, SPR and NR, F(3,156)=10.89, p<.000. Both the MR and SPR groups demonstrated better analogical ability than the NR group, however there was no significant difference among the MR, SR and SPR groups. Thus, reminding did promote children’s inferential thinking, F(3,156)=5.75, p<.001. As the homogeneity of within-class regression coefficient is significant, wilks l =.68, p< .000, it is supposed that the type I error is existent and the possibility of the type I error would be below .01 if we accept these forementioned results of analogical reasoning and inferential performance. The findings indicated that the interdomain transfer of science knowledge is possible for children solving science problems. Although children in the MR group performed better in analogical problem solving and inferential thinking than the NR group did, however, the mathematical procedures and solution reminding would not have helped the children sufficiently to make transfers to solve science problems. That is, a non significant trend was observed in the different content condition. In addition, the comparison of in reference performance, children in the MR, SR, SPR groups who induced schema correctly from source problems outperformed those who induced incorrectly, t(119)=-4.69, p<.001.

The findings of the two studies exhibit the evidence that most children have knowledge of problem field identification. Some children categorized science problems in the mathematics field. It’s probably the science problems were presented as word problems and look like mathematical word problems. Furthermore, a clear effect of reminding on children’s problem solving and inferential thinking, but the reminding was in effect of interdomain problem solving than intradomain problem solving. Children identified the problem fields according to their previous learning, but with the intradomain transfer which lacked one-to-one correspondent surface similarities, it seemed uneasy for children to take an intensive detection to problems’ deep structure.

Implications for Instruction and Future Research

The overall findings in this research indicate children’s analogical problem solving is influenced by the representation constructed. Reminding plays an important role in promoting children’s reasoning and inference thinking. Inducing schema from source correctly would facilitate later inferential thinking on target problems. Though the theoretical importance of reminding in learning is stressed, researchers still should pay attention to understand how children transfer across domains. A high level of transfer from mathematics to science, or from science to mathematics, require a relatively demanding analogical remind-and-map process (Holyoak & Thagard,1989; Bassok & Holyoak, 1989) as well as the amount of problem-space exploration (Pierce, et al., 1993)to relate domain-specific representation to analogical reasoning in other content domains. Effective instruction should encourage children to completely explore the various problem states and move operators during base acquisition (Pierce, et al, 1993) and then facilitate and transfer the particular knowledge to solve novel problems. In addition, we frequently find gaps between disciplines and what is actually taught, and these gaps can have a lasting impact on children’s learning and problem solving. When appropriate, merging concepts from two or more disciplines can make for powerful learning experience (Jacobs, 1989). It’s important for teachers to discover possibilities for potential linkages and natural connections among subjects matters that will expand students’ learning and analogical transfer.

Acknowledgement

This research was supported by a grant from the National Science Council under grant NSC 87-2511- S-081B-001. The opinions do not reflect the views of the foundation.

Reference

Bassok, M., & Holyoak, K. J.,(1989). Interdomain transfer between isomorphic topics in algebra and physics. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15(1), 153-166.

Blessing, S. B. & Ross, B. H. (1996). Content effects in problem categorization and problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition, 22(3), 792-810.

Donnelly, C. M. & McDaniel, M. A. (1993). Use of anaology in learning scientific concepts. Journal of Experimental Psychology: Learning, Memory, and Cognition, 19, 975-987.

Holyoak, K. J.,(1985). The pragmatics of analogical transfer. In G. H. Bower (Ed.). The psychology of learning and instruction, (V.19, p.59-87.). San Diego, CA: Academic Press.

Holyoak, K. J., & Thagard, P. R.,(1989). A comptational model of analogical problem solving. In S. Vosniadou, & A. Ortony (1989). Similarity and analogical reasoning. p.242-266. Cambridge University Press.

Jacobs, H.H.（1989）. Mapping the big picture. Integrating curriculum & Assessment K-12. Association for supervision and curriculum development, USA.

McDaniel, M. A. & Donnelly, C. M. (1996). Learning with analogy and elaborative interrogation. Journal of Educational Psychology, 88(3), 508-519.

Novick, L. K., & Hmelo, C. E.,(1994). Transferring symbolic representations across nonisomorphic problem. Journal of Experimental Psychology: Learning, Memory, and Cognition, 20(6), 1296-1321.

Pierce, K. A., Duncan, M. K., Gholson, B., Ray, G. E. & Kamhi, A. G. (1993). Cognitive load, schema acquisition, and procedural adaptation in nonisomorphic analogical transfer. Journal of Educational Psychology, 85(1), 66-74.

Zook, K. B. & Di Vesta, F. J. (1991). Instructional analogies and conceptual misrepresentations. Journal of Educational Psychology, 83(2), 246-252.