Professional Work
Longitudinal
Development of Student Understanding in Chance and Data
The introduction of Chance and Data as a part of the suggested Australian
mathematics curriculum in 1991 brought probability and statistics to the serious
attention of mathematics educators for the first time. Although there had been
isolated study of tertiary students’ understanding of average and one
longitudinal study of probability in England, the decisions made in relation to
the content for the mathematics curriculum were basically made without the
benefit of research into students' abilities to handle the topics at various
stages of development. The Tasmanian research began in 1993 just before the
introduction of Chance and Data in Tasmania and has continued through the
collection of longitudinal data in 2003.
The objectives of this longitudinal project are (i) to follow the implementation
of the curriculum in Tasmania, (ii) to document student understanding using a
developmental model, and (iii) to make suggestions for classrooms and curriculum
developers. To this end two survey instruments, a 20-item short answer -
multiple choice questionnaire and an11-item media survey, and a nine-part
45-minute interview protocol were developed. The two survey instruments were
administered to 1027 students in grades 3, 6 and 9 in 13 schools in the seven
Tasmanian government school districts in 1993. From five of these schools 64
students were elected for individual interview. In 1995, 2107 students in grades
3, 5, 6, 8, 9 and 11,including 542 students from 1993, were surveyed again. A
similar follow up took place in 1997 with 1661 students surveyed. As well
interviews took place in1997 with about 25 of the students interviewed in 1993
and 22 students from South Australia who were interviewed in 1994.
The analysis of the data collected has taken place from several perspectives.
The SOLO Taxonomy with multimodal functioning of Biggs and Collis has assisted
in benchmarking levels of cognition in student responses which illustrate
unistructural, multistructural and relational functioning, and in some cases two
cycles of such functioning in problem solving contexts. A diagrammatic mapping
procedure of Collis and Watson has been used to further elucidate the processes
involved in solving complex chance and data problems. When considering
statistical ideas in social contexts, a hierarchy of statistical literacy
devised by Watson is being used to explain existing and desired student
behaviours. As well, changes in student cognitive functioning are being traced
with the intact longitudinal data set and changes brought about the
implementation of the curriculum are being monitored by considering the same
grades in succeeding years.
The areas of probability and statistics under investigation include chance
language, luck, fairness of dice, sample spaces, basic probability outcomes,
odds, average, sampling, randomisation, weighted means, conditional probability,
graphing (pictographs, bar graphs, and graphs of association), cause and effect
relationships, comparing data sets, and detecting errant claims in relation to
samples and populations.
The interim results of this project have informed several professional
development projects related to the chance and data part of the curriculum.
Suggestions have been made for the middle school curriculum. Digitised video
clips of students responding to questions have provided examples of what
teachers can expect in the classroom on many topics and expository writing has
taken place to assist teachers in teaching the topics. This research and
associated professional development work will continue for the next few years.
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References
1. Watson, J.M. (1992). What research is needed in probability and statistics
education in Australia in the 1990s? In B. Southwell, B. Perry, & K. Owens
(Eds.), Proceedings of the Fifteenth Annual Conference of the Mathematics
Education Research Group of Australasia (pp. 556-567). Kingswood, NSW: MERGA.
2. Watson, J.M. & Collis, K.F. (1993). Initial considerations concerning the
understanding of probabilistic and statistical concepts in Australian students.
In B. Atweh, C. Kanes, M. Carss, & G. Booker (Eds.), Contexts in Mathematics
Education (pp. 575-580). Brisbane: Mathematics Education Research Group of
Australasia.
3. Watson, J.M., Collis, K.F., & Moritz, J.B. (1994). Assessing statistical
understanding in Grades 3, 6 and 9 using a short answer questionnaire. In G.
Bell, B. Wright, N. Leeson, & G. Geake (Eds.), Challenges in Mathematics
Education: Constraints on Construction (pp. 675-682). Lismore, NSW: Mathematics
Education Research Group of Australasia.
4. Watson, J.M. (1994). Instruments to assess statistical concepts in the school
curriculum. In National Organizing Committee (Ed.), Proceedings of the Fourth
International Conference on Teaching Statistics. Volume 1 (pp. 73-80). Rabat,
Morocco: National Institute of Statistics and Applied Economics.
5. Watson, J.M., & Collis, K.F. (1994). Multimodal functioning in understanding
chance and data concepts. In J.P. da Ponte & J.F. Matos (Eds.), Proceedings of
the Eighteenth International Conference for the Psychology of Mathematics
Education. Volume 4 (pp. 369-376). Lisbon: PME.
6. Watson, J.M., Collis, K.F., & Moritz, J.B. (1995). Children’s understanding
of luck. In B. Atweh & S. Flavel (Eds.), Proceedings of the Eighteenth Annual
Conference of the Mathematics Education Research Group of Australasia (pp.
550-556). Darwin, NT: Mathematics Education Research Group of Australasia.
7. Watson, J.M. (1995). Statistical literacy: A link between mathematics and
society. In A. Richards, G. Gillman, K. Milton, & J. Oliver (Eds.), Flair:
Forging links and integrating resources (pp. 12-28). Adelaide, SA: Australian
Association of Mathematics Teachers. [1995 Hanna Neumann Memorial Lecture]
Reprinted in Reflections, 20(3), 36-45, August, 1995.
8. Pereira-Mendoza, L., Watson, J.M., & Moritz, J.B. (1995). What’s in a graph?
In A. Richards, G. Gillman, K. Milton, & J. Oliver (Eds.), Flair: Forging links
and integrating resources (pp. 301-307). Adelaide, SA: Australian Association of
Mathematics Teachers.
9. Moritz, J. B. (1995, December). Words’ worth. In R. Tytler (Chair),
Contemporary Approaches to Research in Mathematics, Science, Health and
Environmental Education. Symposium conducted at Deakin University, Melbourne.
10. Moritz, J.B., Watson, J.M., & Collis, K.F. (1996). Odds: Chance measurement
in three contexts. In P. C. Clarkson (Ed.), Technology in mathematics education
(pp. 390-397). Melbourne: Mathematics Education Research Group of Australasia.
11. Watson, J.M., & Pereira-Mendoza, L. (1996). Reading and predicting from bar
graphs. Australian Journal of Language and Literacy, 19, 244-258.
12. Watson, J. M., & Moritz, J. B. (1997). Student analysis of variables in a
media context. In Phillips, B. (Ed.), Papers on Statistical Education Presented
at ICME-8 (pp. 129-147). Hawthorn, Australia: Swinburne Press.
13. Watson, J.M., Collis, K.F., & Moritz, J.B. (1997). The development of chance
measurement. Mathematics Education Research Journal, 9, 60-82.
14. Watson, J.M. (1997). Assessing statistical literacy using the media. In I.
Gal & J.B. Garfield (Eds.), The Assessment Challenge in Statistics Education
(pp. 107-121). Amsterdam: IOS Press and The International Statistical Institute.
15. Moritz, J.B., & Watson, J.M. (1997). Pictograph representation: Telling the
story. In N. Scott & H. Hollingsworth (Eds.), Mathematics creating the future
(pp. 222-231). Adelaide: Australian Association of Mathematics Teachers, Inc.
16. Watson, J.M., & Pereira-Mendoza, L. (1996). Reading and predicting from bar
graphs. Australian Journal of Language and Literacy, 19, 244-258.
17. Watson, J.M., Collis, K.F., & Moritz, J.B. (1997). The development of chance
measurement. Mathematics Education Research Journal, 9, 60-82.
18. Moritz, J. B. (1998). Statistical literacy and adolescent risk. In L.
Pereira-Mendoza, L. S. Kea, T. W. Kee, & W. Wong (Eds.), Statistical education:
Expanding the network. (Proceedings of the Fifth International Conference on
Teaching of Statistics) Vol. 1, (pp. 451-457). Singapore: International
Association for Statistical Education.
19. Moritz, J. B. (1998). Long odds: Longitudinal development of student
understanding of odds. In C. Kanes, M. Goos, & E. Warren (Eds.), Teaching
mathematics in new times, Proceedings of the Twenty First Annual Conference of
the Mathematics Education Research Group of Australasia, Vol. 2, (pp. 373-380).
Gold Coast: Mathematics Education Research Group of Australasia.
20. Watson, J.M., & Moritz, J.B. (1998). Longitudinal development of chance
measurement. Mathematics Education Research Journal, 10(2), 103-127.
21. Watson, J.M., Moritz, J.B., & Pereira-Mendoza, L. (1998). Interpreting a
graph in a social context. The Mathematics Educator, 3 (1), 61-71.
22. Watson, J.M. (1998). Assessment of statistical understanding in a media
context. In L. Pereira-Mendoza (Ed.), Statistical education – Expanding the
network. Proceedings of the Fifth International Conference on Teaching
Statistics (pp. 793-799). Voorburg: International Statistical Institute.
23. Watson, J.M. (1998). Numeracy benchmarks for years 3 and 5: What about
chance and data? In C. Kanes, M. Goos, & E. Warren (Eds.), Teaching mathematics
in new times. Volume 2 (pp. 669-676). Brisbane: Mathematics Education Research
Group of Australasia.
24. Watson, J.M. (1998). The role of statistical literacy in decisions about
risk: Where to start. For the Learning of Mathematics, 18(3), 25-27.
25. Watson, J.M., & Moritz, J.B. (1999). Interpreting and predicting from bar
graphs. Australian Journal of Early Childhood, 24(2), 22-27.
26. Moritz, J.B. (1999). Graphing data: Relating representation and
interpretation. In K. Baldwin & J Roberts (Eds.), Mathematics – The next
millennium (pp. 90-99). Adelaide, SA: Australian Association of Mathematics
Teachers, Inc.
27. Watson, J.M. (1999). Lessons from chance and data research I: Student
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28. Watson, J.M. (1999). Lessons from chance and data research II: For the
classroom. In K. Baldwin & J Roberts (Eds.), Mathematics – The next millennium
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Inc.
29. Mortiz, J.B., & Watson, J.M. (1999). The conjunction fallacy and
longitudinal development of chance expression. In J.M. Truran & K.M. Truran
(Eds.), Making the difference. (pp. 380-387). Sydney, NSW: Mathematics Education
Research Group of Australasia Incorporated.
30. Watson, J.M., & Mortiz, J.B. (1999). Longitudinal understanding of
conditional probability by school students. In J.M. Truran & K.M. Truran (Eds.),
Making the difference. (pp. 522-529). Sydney, NSW: Mathematics Education
Research Group of Australasia Incorporated.
31. Shaughnessy, J.M., Watson, J., Moritz, J., & Reading C. (1999, April).
School mathematics students' acknowledgment of statistical variation. Paper
presented at the NCTM Research Presession Symposium: There's more to life than
centers, 77th Annual NCTM Conference, San Francisco, CA.
32. Watson, J.M., & Moritz, J.B. (1999). The development of the concept of
average. Focus on Learning Problems in Mathematics, 21(4), 15-39.
33. Watson, J.M., & Moritz, J.B. (1999). The beginning of statistical inference:
Comparing two data sets. Educational Studies in Mathematics, 37, 145-168.
34. Watson, J.M. (2000). Statistics in context. Mathematics Teacher, 93, 54-58.
35. Watson, J.M., & Moritz, J.B. (2000). Developing concepts of sampling.
Journal for Research in Mathematics Education, 31, 44-70.
36. Watson, J.M., & Moritz, J.B. (2000). The longitudinal development of
understanding of average. Journal of Mathematical Thinking and Learning, 2(1 &
2), 11-50.
37. Watson, J.M., & Moritz, J.B. (2000). Development of understanding of
sampling for statistical literacy. Journal of Mathematical Behavior, 19,
109-136.
38. Torok, R., & Watson, J. (2000). Development of the concept of statistical
variation: An exploratory study. Mathematics Education Research Journal, 12,
147-169.
39. Watson, J.M. (2001). Longitudinal development of inferential reasoning by
school students. Educational Studies in Mathematics, 47, 337-372.
40. Watson, J.M., & Moritz, J.B. (2001). Development of reasoning associated
with pictographs: representing, interpreting, and predicting. Educational
Studies in Mathematics, 48, 47-81.
41. Watson, J.M., & Moritz, J.B. (2002). School students’ reasoning about
conjunction and conditional events. International Journal of Mathematical
Education in Science and Technology, 33, 59-84.
42. Watson, J.M., & Kelly, B.A. (2002). Emerging concepts in chance and data.
Australian Journal of Early Childhood, 27(4), 24-28.
43. Watson, J.M., & Moritz, J.B. (2003). The development of comprehension of
chance language: evaluation and interpretation. School Science and Mathematics,
103, 65-80.
44. Watson, J.M., & Moritz, J.B. (2003). Fairness of dice: A longitudinal study
of students’ beliefs and strategies for making judgments. Journal for Research
in Mathematics Education, 34, 270-304.
45. Watson, J.M., Caney, A., & Kelly, B.A. (2004). Beliefs about chance in the
middle years: Longitudinal change. 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. 581-588). Sydney, NSW: MERGA.
46. Watson, J.M., & Caney, A. (2005). Development of reasoning about random
events. Focus on Learning Problems in Mathematics, 27(4), 1-42.
47. Watson, J.M., & Kelly, B.A. (2007). The development of conditional
probability reasoning. International Journal of Mathematical Education in
Science and Technology, 38, 213-235.
48. Watson, J.M., & Kelly, B.A. (in press). Development of student understanding
of outcomes involving two or more dice. International Journal of Science and
Mathematics Education.
49. 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.
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