Professional Work
Teaching for and Understanding of Variation
Growing interest in students’ understanding of variation, for example by
David Green and Mike Shaughnessy in the 1990s, led to a recent project intended
both to document students’ understanding and to carry out a teaching
intervention designed to emphasize variation in relation to the teaching of the
chance and data curriculum. This project involved interviews with 73 students in
~Pre-Grade-1 and Grades 3, 5, 7,and 9 to profile understanding. It also involved
pre test, post test, and two-year longitudinal tests (the first and last with
control groups) with over 700 students in order to monitor change associated
with the teaching units. For Grades 3 and 5, a teacher was provided for a
10-lesson unit on chance and data. For Grades 7 and 9, units were provided for
teachers to choose from in their usual mathematics planning. The
pretest-posttest results for the experimental classes are summarised by grade in
the Table where the levels of improvement for four subscales – basic chance and
data, variation in chance, variation in data, and variation in sampling – plus
the total score are given. These are based on paired t-tests. For high school
grades there was variation among classes, with one class in each grade not
improving. These are encouraging outcomes but the comparison for some items and
subscales with the control groups after two years, indicates little difference.
Table. Improvement following teaching intervention emphasizing variation
Consideration of individual items and survey scales, as well as in depth
analysis of the 73 interviews has cast considerable light on the development of
understanding of variation in the context of topics specific to the chance and
data curriculum, for example vocabulary, pictographs, dice tossing, spinning
spinners, discussing the weather, and drawing objects from containers. As well
the juxtaposition of expectation and variation in students’ decision-making has
been found to be important to the development of understanding.
References
1. Watson, J.M., & Kelly, B.A. (2002). Can grade 3 students learn about
variation? In B. Phillips (Ed.), Proceedings of the Sixth International
Conference on Teaching Statistics: Developing a statistically literate society,
Cape Town, South Africa. Voorburg, The Netherlands: International Statistical
Institute.
2. Watson, J.M., & Kelly, B.A. (2002). Grade 5 students’ appreciation of
variation. In A. Cockburn & E. Nardi (Eds), Proceedings of the 26th Conference
of the International Group for the Psychology of Mathematics Education (Vol. 4,
pp. 385-392). Norwich,UK: University of East Anglia.
3. Watson, J.M., & Kelly, B.A. (2002). Variation as part of chance and data in
grades 7 and 9. In B. Barton, K.C. Irwin, M. Pfannkuch, & M.O.J. Thomas (Eds.),
Mathematics education in the South Pacific (Proceedings of the 25th annual
conference of the Mathematics Education Research Group of Australasia, Vol. 2,
pp. 682-289). Sydney, NSW: MERGA.
4. Watson, J.M., & Kelly, B.A. (2004). A two-year study of students’
appreciation of variation in the chance and data curriculum. In I. Putt, R.
Faragher, & M. McLean (Eds.), Mathematics education for the third millennium
:Towards 2010 (Proceedings of the 27th Annual Conference of the Mathematics
Education Research Group of Australasia, Townsville, Vol. 2, pp. 573-580).
Sydney, NSW: MERGA.
5. Kelly, B.A., & Watson, J.M. (2002). Variation in a chance sampling setting:
The lollies task. In B. Barton, K.C. Irwin, M. Pfannkuch, & M.O.J. Thomas
(Eds.), Mathematics education in the South Pacific (Proceedings of the 25th
annual conference of the Mathematics Education Research Group of Australasia,
Vol. 2, pp. 366-373). Sydney, NSW: MERGA.
6. Watson, J.M., & Kelly, B.A. (2002, December). School students’ understanding
of stacked dot (line) plots. Refereed paper presented at the Australian
Association for Research in Education conference, Brisbane. Available at:
http://www.aare.edu.au/02pap/index.htm
7. Watson, J.M. (2002). Lessons from variation research I: Student
understanding. In M.Goos & T. Spencer (Eds.), Mathematics – making waves.
(Proceedings of the Nineteenth Biennial Conference of the Australian Association
of Mathematics Teachers Inc., Brisbane, pp.261-268). Adelaide, SA: AAMT, Inc.
[Refereed paper]
8. Watson,J.M., & Kelly, B.A. (2003). Inference from a pictograph: Statistical
literacy in action. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley (Eds.),
Mathematics education research: Innovation, networking, opportunity (Proceedings
of the 26th annual conference of the Mathematics Education Research Group of
Australasia, Geelong, pp. 720-727). Sydney, NSW: MERGA.
9. Watson, J.M., & Kelly, B.A. (2003). Predicting dice outcomes: The dilemma of
expectation versus variation. In L. Bragg, C. Campbell, G. Herbert, & J. Mousley
(Eds.), Mathematics education research: Innovation, networking, opportunity
(Proceedings of the 26thannual conference of the Mathematics Education Research
Group of Australasia, Geelong, pp. 728-735). Sydney, NSW: MERGA.
10. Watson, J.M., & Kelly, B.A. (2003). Statistical variation in a chance
setting. In N.A. Pateman, B.J. Dougherty, & J.T. Zilliox (Eds.), Proceedings of
the 27th conference of the International Group for the Psychology of Mathematics
Education held jointly with the 25th conference of ~PME-NA (Vol. 4, pp.
387-394). Honolulu, HI:Center for Research and Development Group, University of
Hawaii.
11. Watson, J.M., & Kelly, B.A. (2003). The vocabulary of statistical literacy.
In Educational Research, Risks, & Dilemmas: Proceedings of the joint conferences
of the New Zealand Association for Research in Education and the Australian
Association for Research in Education [~CD-ROM]. Auckland, New Zealand,
December, 2003. (Refereed paper) Available at:
http://www.aare.edu.au/03pap/alpha.htm
12. Watson, J.M., & Kelly, B.A. (2003). Developing intuitions about variation:
The weather. In Lee, C. (Ed.), Reasoning about variability: Proceedings of the
Third International Research Forum on Statistical Reasoning, Literacy, and
Thinking. [CD-ROM] Mt. Pleasant, MI: Central Michigan University.
13. Watson, J.M., Kelly, B.A., & Izard, J.F. (2004, December). Student change in
understanding of statistical variation after instruction and after two years : An
application of Rasch analysis. Refereed paper presented at the annual conference
of the Australian Association for Research in Education, Melbourne, December,
2004.
14. Watson, J.M., & Kelly, B.A. (2002). Emerging concepts in chance and data.
Australian Journal of Early Childhood, 27(4), 24-28.
15. Watson, J.M., Kelly, B.A., Callingham, R.A., & Shaughnessy, J.M. (2003). The
measurement of school students’ understanding of statistical variation.
International Journal of Mathematical Education in Science and Technology,
34,1-29.
16. Watson, J.M., & Kelly, B.A. (2004). Expectation versus variation: Students’
decision making in a chance environment. Canadian Journal of Science,
Mathematics and Technology Education, 4, 371-396.
17. Watson, J.M., & Kelly, B.A. (2004). Statistical variation in a chance
setting: A two-year study. Educational Studies in Mathematics, 57,121-144.
18. Watson, J.M., & Kelly, B.A. (2005). The winds are variable: Student
intuitions about variation. School Science and Mathematics, 105, 252-269.
19. Watson, J.M., & Kelly, B.A. (2005). Cognition and instruction: Reasoning
about bias in sampling. Mathematics Education Research Journal, 17(1), 24-57.
20. Watson, J.M., & Kelly, B.A. (2004). A two-year study of students’
appreciation of variation in the chance and data curriculum. In I. Putt, R.
Faragher, & M. McLean (Eds.), Mathematics education for the third millennium:
Towards 2010 (Proceedings of the 27th Annual Conference of the Mathematics
Education Research Group of Australasia, Townsville, Vol. 2, pp. 573-580).
Sydney, NSW: MERGA.
21. Watson, J.M., & Kelly, B.A. (2006). Expectation versus variation: Students’
decision making in a sampling environment. Canadian Journal of Science,
Mathematics and Technology Education, 6, 145-166.
22. Watson, J.M., Callingham, R.A., & Kelly, B.A. (in press). Students’
appreciation of expectation and variation as a foundation for statistical
understanding. Mathematical Thinking and Learning.
23. Watson, J.M. (2005). Variation and expectation as foundations for the chance
and data curriculum. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A.
McDonough, R. Pierce, & A. Roche (Eds.), Building connections: Theory, research
and practice (Proceedings of the 28th annual conference of the Mathematics
Education Research Group of Australasia, Melbourne, pp. 35-42). Sydney: MERGA.
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