Professional Work
Cognitive Development of School Students in Mathematics: Volume Measurement,
Fractions, Decimals and Problem Solving
Over a decade several related studies have taken place focusing on the
development of students’ understanding of various mathematical concepts using
the theoretical foundation of the SOLO Taxonomy with multimodal functioning of
Biggs and Collis. The overall educational purpose of these studies was to inform
classroom teachers and curriculum planners of the increasingly complex structure
exhibited by students in solving mathematical problems during the school years.
This can aid in setting reasonable expectations as well as in suggesting
activities to increase levels of performance. Theoretical outcomes of the work
included extensions to the SOLO model and confirmation of its usefulness.
The mathematical topics investigated as part of the research included area,
volume, fractions, decimals, multiplication, division and problem solving
generally. Besides the basic modal structure (ikonic and concrete symbolic) and
cyclic levels of sophistication within modes (unistructural, multistructural and
relational), a diagrammatic mapping procedure was developed to display the
processing involved during problem solving. Although devised within the SOLO
context, the procedure has been taken up more widely by researchers studying
mathematical problem solving. Further, evidence gleaned from the study of
volume, fractions and decimals suggested two cycles of development within the
concrete symbolic mode, one needed to establish a basic mathematical concept
(e.g., fraction) and the second to apply it in problem solving situations.
As well as documenting developmental change and problem solving strategies,
another interest of the research was the use of visual processing and other
imagery during problem solving. This grew out of the increasing evidence of
multimodal functioning, particularly with support from the ikonic mode apparent
in solutions to concrete symbolic problems. Using measures of visual imagery and
cognitive processing in a 2 x 2 design, it was found that success at problem
solving was related to logical operational ability, but not to vividness of
visual imagery. There also was no interaction of the two abilities. Further it
was possible to distinguish between students who do not draw diagrams because
they are able to generate and use appropriate visual imagery to solve a problem,
and students whose imagery and understanding of a problem are so poor that they
are unable to produce useful images or diagrams. Those in the former group had
either high operational skills or high vividness of visual imagery (or both),
while the latter group was low on both, suggesting that these students have
fewer resources available to them when solving problems.
References
Chick, H.L., Watson, J.M. & Collis, K.F. (1988). Using the SOLO Taxonomy for
error analysis in mathematics. Research in Mathematics Education in Australia,
June, 1988, 34-46.
Watson, J.M., Chick, H.L., & Collis, K.F. (1988). Applying the SOLO Taxonomy to
errors on area problems. In Pegg, J. (Ed.), Mathematical Interfaces (pp.
260-281). Armidale, N.S.W.: Australian Association of Mathematics Teachers.
Watson, J.M. & Mulligan, J. (1990). Mapping solutions to an early multiplication
word problem. Mathematics Education Research Journal, 2(2), 28-44.
Collis, K.F. & Watson, J.M. (1991). A mapping procedure for analysing the
structure of mathematics responses. Journal of Structural Learning, 11, 65-87.
Collis, K.F., Watson, J.M., & Campbell, K.J. (1992). Multimodal functioning in
novel mathematical problem solving. In B. Southwell, B. Perry, & K. Owens
(Eds.), Proceedings of the Fifteenth Annual Conference of the Mathematics
Education Research Group of Australasia (pp. 236-243). Kingswood, NSW: MERGA.
Campbell, K.J., Watson, J.M., & Collis, K.F. (1992). Volume measurement and
intellectual development. Journal of Structural Learning, 11, 279-298.
Watson, J.M., Campbell, K.J., & Collis, K.F. (1993). Multimodal functioning in
understanding fractions. Journal of Mathematical Behavior, 12, 45-62.
Campbell, K.J., Collis, K.F., & Watson, J.M. (1993). Multimodal functioning
during mathematical problem solving. In B. Atweh, C. Kanes, M. Carss, & G.
Booker (Eds.), Contexts in Mathematics Education (pp. 147-151). Brisbane:
Mathematics Education Research Group of Australasia.
Collis, K.F., Watson, J.M., & Campbell, K.J. (1993). Cognitive functioning in
mathematical problem solving during early adolescence. Mathematics Education
Research Journal, 5, 107-121.
Watson, J.M. (1994). A diagrammatic representation for studying problem solving
behavior. Journal of Mathematical Behavior, 13, 305-332.
Watson, J.M., Collis, K.F., & Campbell, K.J. (1995). Developmental structure in
the understanding of common and decimal fractions. Focus on Learning Problems in
Mathematics, 17(1), 1-24.
Campbell, K.J., Collis, K.F., & Watson, J.M. (1995). Visual processing during
mathematical problem solving. Educational Studies in Mathematics, 28, 177-194.
Watson, J.M., Campbell, K.J., & Collis, K.F. (1996). Fairness and fractions in
early childhood. In P. C. Clarkson (Ed.), Technology in mathematics education
(pp. 588-595). Melbourne: Mathematics Education Research Group of Australasia.
Watson, J.M. (1996). What's the point? Australian Mathematics Teacher, 52(2),
40-43.
Watson, J.M. (1997). Children’s construction of ‘fair’ representations of
one-third. Australian Journal of Early Childhood, 22(2), 34-38.
Mulligan, J.T., & Watson, J.M. (1998). A developmental multimodal model for
multiplication and division. Mathematics Education Research Journal, 10(2),
61-86.
Watson, J.M., Campbell, K.J., & Collis, K.F. (1999). The structural development
of the concept of fraction by young children. Journal of Structural Learning and
Intelligent Systems, 13(3-4), 171-193.
Contact me...
Faculty of Education
University of Tasmania
Private Bag 66 Hobart Tasmania Australia 7001
Phone: 61-3-6226-2570; Fax: 61-3-6226-2569
Jane.Watson@utas.edu.au