Understanding mathematics: Some key factors

Abstract

Mathematics is well known as a subject area where there can be problems in terms of understanding as well as retaining positive attitudes. In a large study involving 813 school students (ages approximately 10-12) drawn from two different school systems in Pakistan, the effect of limited working memory capacity on performance in mathematics was explored along with a survey of areas of difficulty and student attitudes. This involved looking at student perceptions of their experiences, the nature of the difficulties they have with mathematics and possible reasons for these difficulties. The overall aim is to explore the extent of the effect of working memory and to gain insights so that practical ways forward to enhance mathematics education can be identified. It was found that limited working memory capacity has a very strong influence on performance, confirming other studies. Indeed, if the cognitive load exceeds the capacity of working memory, understanding becomes a casualty, with consequent attitude deterioration. Students need to be able to see that mathematics has a purpose in being able to be applied to real-life situations. However, attempts to develop applications may often generate further working memory overload. Curricula devised by those outside the classroom can sometimes be inappropriate while topics causing the greatest problems at these ages and include areas of geometry, statistics and the applications of mathematics.

Keywords

• working memory capacity, attitudes to mathematics, difficulties in mathematics, cognitive overload

Kaynakça

1. Al-Ahmadi, F and Oraif, F. (2009). Working memory capacity, confidence and scientific thinking. Research in Science and Technological Education, 27(2), 225-243.
2. Alenezi, D.F. (2004). Difficulties Associated with Teaching and Learning Mathematics: A Study of Psychological Factors Affecting Pupils' Performance. MSc Thesis, Glasgow: University of Glasgow.
3. Alenezi, D.F. (2008). A Study of Learning Mathematics Related to some Cognitive Factors and to Attitudes. PhD Thesis, Glasgow: University of Glasgow. Retrieved from http://theses.gla.ac.uk/333/
4. Alhmali, R.J. (2007). Student Attitudes in the Context of the Curriculum in Libyan Education in Middle and High Schools, PhD Thesis, Glasgow: University of Glasgow.
5. Ausubel, D. P. (1968). Educational psychology a cognitive view. New York: Holt, Rinehart and Winston.
6. Brown, M., Brown, P.& Bibby, T. (2008). I would rather die: Reasons given by 16-years-olds for not continuing their study of mathematics. Research in Mathematics Education, 10(1), 3-18.
7. Cramer, K.A., Post, T.R. & delMas, R.C. (2002). Initial Fraction Learning by Fourth and Fifth Grade Students: A Comparison of the Effects of Using Commercial Curricula With the Effects of Using the Rational Number Project Curriculum. Journal for Research in Mathematics Education, 33(2), 111-144.
8. Duffin, J. & Simpson, T. (2000). Understanding their Thinking: the tension between the Cognitive and the Affective, In: Diana Coben (2002). Perspectives on Adults Learning Mathematics. Mathematics Education Library, 21, Section I, 83-99.

9. Friel, S.N., Curcio, F.R., & Bright, G.W. (2001). Making Sense of Graphs: Critical Factors Influencing Comprehension and Instructional Implications. Journal for Research in Mathematics Education, 32(2), 124-158.
10. Harries, T and Barmby P (2007). Representing and Understanding Multiplication. Journal for Research in Mathematics Education, 9, 33-45.
11. Harries, T. & Suggate, J. (2006). Exploring links across representations of numbers with young children. International Journal for Technology in Mathematics Education, 13(2), 53-64.
12. Hindal, H, Reid, N., and Badgaish, M. (2008). Working Memory, Performance and Learner Characteristics. Research in Science and Technological Education, 27(2), 187-204.
13. Hussein, F. and Reid, N. (2009). Working memory and difficulties in school chemistry. Research in Science and Technological Education. 27(2), 161-185.
14. Johnstone, A. H., and El-Banna, H. (1986). Capacity, Demands and Processes - A Predictive Model for Science Education. Educational in Chemistry, 23, 80-84.
15. Johnstone, A. H., and El-Banna, H. (1989). Understanding learning difficulties - a predictive research model. Studies in Higher Education, 14,159-68.
16. Johnstone, A. H., & Reid, N. (1981). Toward a model of attitude change. European Journal of Science Education, 3(2), 205-212.
17. Jung, E-S., and Reid, N. (2009). Working memory and attitudes. Research in Science and Technological Education, 27(2), 205-223.
18. Kato, Y., Kamii, C., Ozaki, K. & Nagahiro, M. (2002). Young Children’s Representations of Groups of Objects: The Relationship between Abstraction and Representation. Journal for Research in Mathematics Education, 3(1), 30-45.
19. Kirschner, P.A., Sweller, J. & Clark, R.E. (2006). Why Minimal Guidance during Instruction Does not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-based, Experiential, and Inquiry-Based Learning. Educational Psychologist, 41(2), 75-86.
20. Kouba, V., Zawojewski, J. & Strutchens, M. (1997). What do students know about numbers and operations? In P.A. Kenny & E. A. Silver (Eds), Results from the sixth mathematics assessment of the National Assessment of Educational Progress. Reston, VA : Council of Teachers of Mathematics, 87-140.
21. Kyriacou, C. and Goulding, M. (2006). A systematic review of strategies to raise pupils' motivational effort in Key Stage 4 mathematics EPPI Centre. London : Institute of Education.
22. Likert, R. (1932). A technique for the measurement of attitudes. Archives of Psychology, 140, 5–55
23. Lo, J-J. & Watanabem T. (1997). Developing Ratio and Proportion Schemes: A Story of a Fifth Grader. Journal for Research in Mathematics Education, 28(2), 216-236.
24. Lucangeli, D., Tressoldi, P. E. & Cendron, M. (1998). Cognitive and Metacognitive Abilities Involved in the Solution of Mathematical Word Problems: Validation of a Comprehensive Model. Contemporary Educational Psychology, 23, 257-275.
25. Ma, X., Kishore, N. (1997). Assessing the Relationship between Attitude towards Mathematics and Acievement in Mathematics : A Meta Analysis. Journal for Research in Mathematics Education, 28(1), 26-47.
26. Matthews, A. and Pepper, D. (2005). Evaluation of participation in A level mathematics: Interim report. London: Qualifications and Curriculum Agency.
27. McLeod, D.B. (1994). Research on affect and mathematics Learning in the JRME: 1970 to the present. Journal for Research in Mathematics Education 24, 637-647.
28. Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and experimental curriculum. Journal for Research in Mathematics Education, 30,122-147.
29. Nardi, E. and Steward, S. (2003). Is mathematics T.I.R.E.D? A profile of quiet disaffection in the secondary mathematics classroom. British Educational Research Journal, 29(3), 345-367.
30. Osgood, C. E., Suci, C. J. & Tannenbaum, P. H. (1957). The measurement of meaning. Urbana, IL: University of Illinois Press.
31. Pascual-Leone, J. (1970). A mathematical model for the transition rule in Piaget's developmental stages. Acta Psychologica, 32, 301-345.
32. Piaget, J. (1963). The Child’s Conception of the world. Peterson, N.J.: Little Field, Adams.
33. Ponte, J.P., Matos, J.F., Guimares, H.M., Leal, L.C., Canavarro, A.P. (1991). Students' Views and Attitudes towards Mathematics teaching and Learning: A case of a Curriculum Experience. Educational Studies in Mathematics, 26(4), 347-365.
34. Reid, N. (2003). Getting Started in Pedagogical Research in Higher Education. LTSN Physical Science, Hill: Higher Education Academy. Retrieved from http://www.heacademy.ac.uk/resources/detail/subjects/physsci/Practice-guide-getting-started-ped-research
35. Reid, N. (2009). Working Memory and Science Education. Research in Science and Technological Education, 27(2), 245-250.
36. Reid, N., & Skryabina, E. (2002). Attitudes towards physics. Research in Science and Technological Education. 20(1), 67-81.
37. Reid, N. and Yang, M-J. (2002). Open-ended problem solving in school chemistry: a preliminary investigation. International Journal of Science Education, 24(12), 1313 – 1332
38. Reid, P. (2002). Problem solving by primary school children with particular reference to dyslexics. MSc Thesis, Glasgow: University of Glasgow.
39. Tall, D. (2004). Thinking through three worlds of mathematics. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 4, 281–288
40. Tirosh, D. (2000). Enhancing Prospective Teachers' Knowledge of Children's Conceptions: The Case of Division of Fractions. Journal for Research in Mathematics Education, 31 (1), 5-25.
41. Watson, J.M. & Moritz, J.B. (2000). Developing Concepts of Sampling, Journal for Research in Mathematics Education, 31(1), 44-70.