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Effects of heuristics instruction on pupils' achievement in solving non-routine problems
Author
Wong, Swee Oi
Supervisor
Lim-Teo, Suat Khoh
Abstract
The National Council of Teachers of Mathematics declared problem solving to be the focus of school mathematics. It was supported by the Cockcroft Report (1982) in the United Kingdom, and the Curriculum Development Centre (1982) in Australia. Moving in the same direction, the Ministry of Education in Singapore revised the syllabus for mathematics education in schools in 1992 (Ministry of Education, 2001), emphasising the central role of problem solving in the recent mathematics curriculum in the belief that problem solving in the essence of mathematics learning. In the current mathematics classrooms in Singapore, relevant non-routine mathematical problems are employed. Pupils are required to use heuristics in mathematical problem solving and thinking, and not mere application of algorithmic skills or simple recall.
This research sought to study the effects of heuristic instruction on pupils' achievement in solving non-routine problems. The intention was to use the results to provide the basis for a course of action to improve pupils' achievement through innovative problem solving instructional programs.
The research was conducted in a government-aided primary school in Singapore. Thirty-nine male and female pupils in one entire primary six class from the EM2 stream participated in this study.
The study comprises three phases. Phase 1 involved a paper-and-pencil test, the Pre-Test, consisting of five non-routine mathematical problems requiring multiple heuristics, inferential, deductive or inductive reasoning, with varying difficulty level. In Phase 2, pupils were introduced to heuristics through twelve lessons spread over 4 weeks. The major part of the programme is the teaching of heuristics, designed for this study, to equip pupils with sufficient knowledge and skill so as to enable them to use heuristics like look for a pattern or patterns, make supposition(s), draw a model, tabulation and systematic listing to solve non-routine problems. It consisted of three 1-hour lessons per week for four weeks. Normal mathematics lessons took up the other mathematics periods of the week. The first two sessions in each week were more directed as only one or two heuristics were taught in each session. In the last session of each week, and whenever possible during the normal mathematics lesson in class, collections of assorted problems testing all the heuristics that were taught were given to the students out of context, to allow for practice in heuristics selection and heuristics implementation. In Phase 3, another paper-and-pencil test, the Post-Test, with five parallel questions to that in Phase 1 was carried out in the week immediately following the end of the intervention. There was no time limit for both the tests. They were scored by a modified version of the Analytic Scale for Problem-solving developed by Charles, Lester and O'Daffer (1987). This scale consisted of 0, 1 or 2 points assigned to the two components; Appropriate Choice of Heuristics and Getting an Answer. The total maximum score for each problem was 4 marks.
The results indicated that pupils were more proficient in selecting and implementing the appropriate heuristics learnt in the Heuristics Instruction Programme to solve the non-routine problems in the Post-Test as compared to the Pre-Test. The pupils' mean scores were 9.49, (SD= 3.39) for the Pre-Test and 16.03, (SD= 4.26) for the Post-Test. After the programme, they could effectively choose and coordinate the use of multiple heuristics to produce solutions that were more organised and elegant, requiring more reasoning and fewer steps. They showed more flexibility and willingness to think about and use heuristics to get diverse solutions. Generally, there was also a decline in the inefficient and inappropriate use of pre-instructional heuristics like guess-and-check, number manipulation, logical reasoning, try simple cases and unsystematic listing as the pupils were able to apply some specific heuristics in the problem solving process.
This research sought to study the effects of heuristic instruction on pupils' achievement in solving non-routine problems. The intention was to use the results to provide the basis for a course of action to improve pupils' achievement through innovative problem solving instructional programs.
The research was conducted in a government-aided primary school in Singapore. Thirty-nine male and female pupils in one entire primary six class from the EM2 stream participated in this study.
The study comprises three phases. Phase 1 involved a paper-and-pencil test, the Pre-Test, consisting of five non-routine mathematical problems requiring multiple heuristics, inferential, deductive or inductive reasoning, with varying difficulty level. In Phase 2, pupils were introduced to heuristics through twelve lessons spread over 4 weeks. The major part of the programme is the teaching of heuristics, designed for this study, to equip pupils with sufficient knowledge and skill so as to enable them to use heuristics like look for a pattern or patterns, make supposition(s), draw a model, tabulation and systematic listing to solve non-routine problems. It consisted of three 1-hour lessons per week for four weeks. Normal mathematics lessons took up the other mathematics periods of the week. The first two sessions in each week were more directed as only one or two heuristics were taught in each session. In the last session of each week, and whenever possible during the normal mathematics lesson in class, collections of assorted problems testing all the heuristics that were taught were given to the students out of context, to allow for practice in heuristics selection and heuristics implementation. In Phase 3, another paper-and-pencil test, the Post-Test, with five parallel questions to that in Phase 1 was carried out in the week immediately following the end of the intervention. There was no time limit for both the tests. They were scored by a modified version of the Analytic Scale for Problem-solving developed by Charles, Lester and O'Daffer (1987). This scale consisted of 0, 1 or 2 points assigned to the two components; Appropriate Choice of Heuristics and Getting an Answer. The total maximum score for each problem was 4 marks.
The results indicated that pupils were more proficient in selecting and implementing the appropriate heuristics learnt in the Heuristics Instruction Programme to solve the non-routine problems in the Post-Test as compared to the Pre-Test. The pupils' mean scores were 9.49, (SD= 3.39) for the Pre-Test and 16.03, (SD= 4.26) for the Post-Test. After the programme, they could effectively choose and coordinate the use of multiple heuristics to produce solutions that were more organised and elegant, requiring more reasoning and fewer steps. They showed more flexibility and willingness to think about and use heuristics to get diverse solutions. Generally, there was also a decline in the inefficient and inappropriate use of pre-instructional heuristics like guess-and-check, number manipulation, logical reasoning, try simple cases and unsystematic listing as the pupils were able to apply some specific heuristics in the problem solving process.
Date Issued
2002
Call Number
QA63 Won
Date Submitted
2002