The appearance of systematic errors in computation suggests relatively unlinked computational knowledge to conceptual knowledge, and hence difficulties with forward learning of mathematics. The provision of programs of good teaching, where concrete materials are used to exemplify and thus legitimize algorithmic processes, frequently are not effective for use with upper primary students: systematic errors often resurface. A novel and quite alternate approach to intervention is the Old Way/New Way (O/N) strategy (Lyndon, 1989) based on psychological principles of memory, forgetting and interference. In this article, issues associated with intervention, systematic errors and upper primary students are addressed through a discussion of results of previous research into seventh graders' subtraction knowledge development by overcoming error patterns in subtraction computation. By comparing re-teaching strategies and O/N, it is proposed that both good teaching and effective intervention strategies should be integral to the craft of teaching, particulary in the middle school.
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References
1.
Adelman, H. S. and Taylor, L. (1986) An Introduction to Learning Disabilities. Glenview, IL: Scott Foresman.
2.
Anderson, J. R. (1985) Cognitive Psychology and its Implications. New York: W. H. Freeman and Company.
3.
Ashlock, R. B. (1994) Error Patterns in Computation A Semi-Programmed Approach (6th edn) New York: Macmillan.
4.
Ashlock, R. B. , Johnson, M. L., Wilson, J. W. and Jones, W. L. (1983) Guiding each Child's Learning of Mathematics: A Diagnostic Approach to Instruction. Columbus, OH: Charles E. Merrill.
5.
Ausubel, D. (1968) Educational Psychology: A Cognitive View. New York: Holt, Rinehart and Winston.
6.
Baddeley, A. (1990) Human Memory: Theory and Practice. Hillsdale, NJ: Lawrence Erlbaum.
7.
Baxter, E. P. and Dole, S. (1990) 'Working with the Brain, not Against It: Correction of Systematic Errors in Subtraction', British Journal of Special Education17(1): 19-22.
Booker, G. , Irons, C. and Jones, G. (1980) Fostering Arithmetic in the Primary School. Canberra: Curriculum Development Centre.
10.
Borassi, R. (1994) 'Capitalising on Errors as "Springboards for Inquiry": A Teaching Experiment', Journal for Research in Mathematics Education25(2): 166-208.
11.
Bourke, S. F. (1980) Numeracy in School:An Integrated Approach to Community Expectation, Student Assessment, and Error Analysis. Unpublished master's thesis, Monash University, Melbourne.
12.
Brown, J. S. and Van Lehn, K. (1982) 'Towards a Generative Theory of "Bugs"', in T. P. Carpenter, J. M. Moser and T. A. Romberg (eds), Addition and Subtraction -A Cognitive Perspective, pp. 117-135. Hillsdale, NJ: Lawrence Erlbaum Associates.
13.
Brumfield, R. and Moore, B. (1985) 'Problems with the Basic Facts may not be the Problem', Arithmetic Teacher33(3): 17-18.
14.
Cole, P. and Chan, L. (1990) Methods and Strategies for Special Education. New York: Prentice Hall.
15.
Confrey, J. (1990a) 'What Constructivism Implies for Teaching', in R. B. Davis, C. A. Maher and N. Noddings (eds) Constructivist Views on Teaching and Learning of Mathematics, pp. 107-122. Reston, VA: NCTM.
16.
Confrey, J. (1990b) 'Inconsistencies and Cognitive Conflict: A Constructivist's View', Focus on Learning Problems in Mathematics12(3 & 4): 99-109.
17.
Connell, M. and Peck, D. (1993) 'Report of a Conceptual Change Intervention in Elementary Mathematics', Journal of Mathematical Behaviour12: 329-350.
18.
Cox, L. S. (1975) 'Diagnosing and Remediating Systematic Errors in Addition and Subtraction Computations', Arithmetic Teacher22(4): 202-220.
19.
Dole, S. (1999) Percent Knowledge: Effective Teaching for Learning, Relearning and Uniearning. Unpublished doctoral dissertation, Queensland University of Technology, Australia.
20.
Dole, S. (1995) 'Gaining Access to Mathematical Literacy: Application of Old Way/New Way to Mathematics Instruction for Adult Learners', Proceedings of the 3rd Annual International Conference on Post-Compulsory Education and Training, Brisbane.
21.
Dole, S. (1993) 'Error Patterns and Subtraction Knowledge Development A Comparison of Methods', in J. Novak (ed.) Proceedings of the Third International Seminar on Misconceptions and Educational Strategies in Science and Mathematics. Ithaca, NY: Correll University. (Electronic publishing of seminar proceedings)
22.
Cole, S. and Cooper, T. (1995) 'Errors as Springboards for Remediation of Year 7 Students' Subtraction Knowledge', in B. Atweh and S. Flavell (eds) Eighteenth Annual Conference of the Mathematics Education Research Group of Australasia (MERGA), pp. 235-240, Northern Territory University.
23.
Dyer, F. (1973) 'The Stroop Phenomenon and its Use in the Study of Percentual, Congitive and Response Processes', Memory and Cognition1(2): 106-120.
24.
Erlwanger, S. H. (1975) 'Case Studies of Children's Conceptions of Mathematics', Journal of Mathematical Behaviour1(3): 157-283.
25.
Fernald, G. M. (1971) Remedial Techniques and Basic School Subjects (Edited by Lorna Idol) Austin, TX: ProEd.
26.
Fischbein, E. (1987) Intuition in Science and Mathematics: An Educational Approach. AH Dordrecht, Holland: D. Reidel.
27.
Fuson, K. and Briars, D. J. (1990) 'Using a Base-Ten Blocks Learning/Teaching Approach for First and Second Grade Place Value and Multidigit Addition and Subtraction', Journal forResearch in MathematicsEducation21(3): 180-206.
28.
Gable, R. A. , Enright, B. E. and Hendrickson, J. M. (1991) 'A Practical Model for Curriculum-Based Assessment and Instruction in Arithmetic', Teaching Exceptional Children24(1): 6-9.
29.
Gagne, R. (1983) 'Some Issues in the Psychology of Mathematics Instruction', Journal for Research in Mathematics Education14(1): 7-18.
30.
Ginsburg, H. (1977) Children's Arithmetic: The Learning Process. New York: Van Nostrond.
31.
Graeber, A. and Baker, K. M. (1991) 'Curriculum Materials and Misconceptions Concerning Multiplication and Division', Focus on Learning Problems in Mathematics13(3): 25-38.
32.
HMSO (1975) A Language for Life. Report of the Committee of Inquiry Appointed by the Secretary of State for Education and Science under the Chairmanship of Sir Alan Bullock.
33.
Houston, J. P. (1991) Fundamentals of Learning and Memory, 4th edn.Orlando, FL: Harcourt Brace Jovanovich.
34.
James, W. (1890) Principles of Psychology. New York: Holt, Rinehart and Winston.
35.
Kephart, N. C. (1960) The Slow Learner in the Classroom. Columbus, OH: Charles E. Merrill.
36.
Lampert, M. (1986) 'Knowing, Doing and Teaching Multiplication', Cognition and Instruction3(4): 305-342.
37.
Leinhardt, G. (1988) 'Getting to Know: Tracing Student's Mathematical Knowledge from Intuition to Competence', Educational Psychologist23(2):119-144.
38.
Lyndon, H. (1995) 'Conceptual Mediation A New Approach to an Old Problem', Voice A Journal for Educators3(2):00-00
39.
Lyndon, H. (1989) 'I Did it My Way! An Introduction to the Old Way/New Way Methodology', Australasian Journal of Special Education3(1): 32-37.
40.
Lyndon, E. H. and Dawson, C. J. (1995) The Conceptual Mediation Program: A Practical Outcome from a Novel Perspective on Conceptual Change, paper presented at the Australian Association of Science Education Annual Conference, Adelaide, July 1995.
41.
Mansfield, H. and Happs, J. (1992) 'Using Grade-Eight Students Existing Knowledge to Teach about Parallel Lines', School Science and Mathematics92(8): 450-454.
42.
Melton, A. and Von Lockum, W. (1941) 'Retroactive and Proactive Inhibition in Retention: Evidence for a Two-Factor Theory of Retroactive Inhibition', American Journal of Psychology54: 157-173.
43.
Posner, G. , Strike, K., Hewson, P. and Gertzog, W. (1982) 'Accommodation of a Scientific Computation: Toward a Theory of Conceptual Change', Science Education66: 211-227.
44.
Postman, L. and Underwood, B. (1973) 'Critical Issues in Interference Issues', Memory and Cognition1: 19-40.
45.
Putnam, R. T. , Lampert, M. and Peterson, P. C. (1990) 'Alternative Perspectives on Knowing Mathematics in Elementary Schools', Review of Research in Education16: 57-149.
46.
Rauff, J. V. (1994) 'Constructivism, Factoring and Beliefs', School Science and Mathematics94(8): 421-426.
47.
Read, G. (1987) 'Followingthe Progress of Childrenwho have Received "Remedial" Tuition', Support for Learning2(1): 52-59.
48.
Resnick, L. B. (1992) 'From Protoquantities to Operators: Building Mathematical Competence on a Foundation of Everyday Knowledge', in G. Leinhardt, R. Putnam and R. A. Hattrup (eds) Analysis of Arithmetic for Mathematics Teaching, pp. 373-429. Hillsdale, NJ: Lawrence Erlbaum.
49.
Resnick, L. B. (1982) 'Syntax and Semantics in Learning to Subtract', in T. P. Carpenter, J. M. Moser and T. A. Romberg (eds) Addition and Subtraction A Cognitive Perspective, pp. 136-155. Hillsdale, NJ: Lawrence Erlbaum Associates.
50.
Resnick, L. , Nesher, P., Leonard, F., Magone, M., Omanson, S. and Peled, R. (1989) 'Conceptual Bases ofArithmetic Errors: The Case ofDecimal Fractions', Journal for Research in Mathematics Education20(1): 8-27.
51.
Rowell, J. A. , Dawson, C. J. and Lyndon, E. H. (1990) 'Changing Misconceptions: A Challenge to Science Education', International Journal of Science Education12(2): 167-175.
52.
Stroop, J. R. (1935) 'Studies of Interference in Serial Verbal Reactives', Journal of Experimental Psychology18: 643-662.
53.
Swedosh, P. (1990) 'Reducing the Incidence of Mathematical Misconceptions in "Middle Band" Students', in J. Truran and K. Truran (eds) Making the Difference: Proceedings of the Twenty-Second Annual Conference of the Mathematics Education Research Group of Australasia. Adelaide: MERGA.
54.
Swedosh, P. and Clark, J. (1998) 'Mathematical Misconceptions We Have an Effective Method for Reducing their Incidence but will the Improvement Persist?', in C. Kanes, M. Goos. E. Warren (eds) Teaching Mathematics in New Times: Proceedings of the Twenty-FirstAnnual Conference of the Mathematics Education Research Group of Australasia (2), pp. 588-595. Broadbeach: MERGA.
55.
Tirosh, D. (1990) 'Inconsistencies in Students' Mathematical Constructs', Focus on Learning Problems in Mathematics12 (3 & 4): 111-129.
56.
Underwood, B. J. and Postman, L. (1969) 'Extra-Experimental Sources of Interference in Forgetting', Psychological Review67(2): 73-95.
57.
Underwood, B. J. (1966) Experimental Psychology, 2nd edn.New York: AppletonCentury Crofts.
58.
Underwood, B. J. (1957) 'Interference and Forgetting', Psychological Review64(1): 49-60.
59.
Underwood, B. J. (1948) 'Retroactive and Proactive Inhibition after Five and Forty-Eight Hours', Journal of Experimental Psychology38: 29-38.
60.
Vinner, S. (1990) 'Inconsistencies: Their Causes and Function in Learning Mathematics', Focus on Learning Problems in Mathematics2(3 & 4): 85-99.
