**REASONING ABOUT SHAPE AS A PATTERN IN
VARIABILITY**

ARTHUR BAKKER

Freudenthal Institute

Utrecht University

The Netherlands

a.bakker@ioe.ac.uk

SUMMARY

This
paper examines ways in which coherent reasoning about key concepts such as
variability, sampling, data, and distribution can be developed as part of
statistics education. Instructional activities that could support such
reasoning were developed through design research conducted with students in
grades 7 and 8. Results are reported from a teaching experiment with grade 8
students that employed two instructional activities in order to learn more
about their conceptual development. A “growing a sample” activity had students
think about what happens to the graph when bigger samples are taken, followed
by an activity requiring reasoning about shape of data. The results suggest
that the instructional activities enable conceptual growth. Last, implications
for teaching, assessment and research are discussed.

Keywords: Design
research; Distribution; Instructional activities; Middle school level; Sampling

**__________________________**

*Statistics Education Research Journal,
3(2), 64-83, http://www.stat.auckland.ac.nz/serj*

*Ó International
Association for Statistical Education (IASE/ISI), November, 2004*

References

Bakker, A.
(2003). The early history of statistics and implications for education. *Journal of Statistics Education, 11*(1).
[Online: www.amstat.org/publications/jse/v11n1/bakker.html]

Bakker, A. (2004). *Design research in statistics education: On symbolizing and computer
tools.* Utrecht, The Netherlands: CD-Beta Press. [Online: http://www.library.uu.nl/digiarchief/dip/diss/2004-0513-153943/inhoud.htm]

Bakker, A., &
Gravemeijer, K. P. E. (2004). Learning to reason about distribution. In D.
Ben-Zvi & J. Garfield (Eds.), *The
challenge of developing statistical literacy, reasoning, and thinking *(pp.
147–167). Dordrecht, the Netherlands: Kluwer Academic Publishers.

Bakker, A., &
Hoffmann, M.H.G. (in press). Diagrammatic reasoning as the basis for developing
concepts: A semiotic analysis of students’ learning about statistical
distribution. *Educational Studies in Mathematics*.

Ben-Zvi, D.,
& Arcavi, A. (2001). Junior high school students’ construction of global
views of data and data representations. *Educational
Studies in Mathematics, 45*, 35–65.

Cobb, G. (1993).
Considering statistics education: A National Science Foundation Conference. *Journal of Statistics Education*, *1*(1). [Online: www.amstat.org/publications/jse/v1n1/cobb.html]

Cobb, P. (1999).
Individual and collective mathematical development: The case of statistical
data analysis. *Mathematical Thinking and
Learning, 1*(1), 5–43.

Cobb, P., Confrey, J.,
diSessa, A. A., Lehrer, R., & Schauble, L. (2003). Design experiments in
educational research. *Educational
Researcher, 32*(1), 9–13.

Cobb, P., Gravemeijer, K. P. E., Bowers, J., &
Doorman, M. (1997). *Statistical Minitools*
[applets and applications]. Nashville, TN & Utrecht: Vanderbilt University
& Freudenthal Institute, Utrecht University. [Translated and revised in
2001 by Bakker, A.] [Online: www.wisweb.nl/en]

Cobb, P., McClain, K.,
& Gravemeijer, K. P. E. (2003). Learning about statistical covariation. *Cognition and Instruction* *21*, 1–78.

Frege, F. L. G. (1976). *Wissenschaftlichter Briefwechsel* *[Scientific correspondence]* (Gabriel,
G., Ed.) (First ed. Vol. 2). Hamburg, Germany: Meiner.

Freudenthal, H. (1991). *Revisiting mathematics education: China
lectures*. Dordrecht, the Netherlands: Kluwer Academic Publishers.

Garfield, J., &
Ben-Zvi, D. (2004). Research on statistical literacy, reasoning, and thinking:
Issues, challenges, and implications. In D. Ben-Zvi & J. Garfield (Eds.), *The challenge of developing statistical
literacy, reasoning, and thinking *(pp. 397–409). Dordrecht, the
Netherlands: Kluwer Academic Publishers.

Glaser, B. G., &
Strauss, A. L. (1967). *The discovery of
grounded theory; Strategies for qualitative research*. Chicago: Aldine
Publishing Company.

Gravemeijer, K. P. E.
(1994). Educational development and developmental research. *Journal for Research in Mathematics
Education, 25*, 443–471.

Gravemeijer, K. P. E.
(1999, April). An instructional sequence of analysing univariate data sets*. *Paper presented at the Annual Meeting
of the American Education Research Association, Montréal, Canada.

Hancock, C., Kaput, J.
J., & Goldsmith, L. T. (1992). Authentic enquiry with data: critical
barriers to classroom implementation. *Educational
Psychologist 27*(3), 337–364.

Konold, C., &
Higgins, T. (2003). Reasoning about data. In J. Kilpatrick, W. G. Martin &
D. Schifter (Eds.), *A research companion
to principles and standards for school mathematics* (pp. 193–215). Reston, VA:
National Council of Teachers of Mathematics.

Konold, C., &
Pollatsek, A. (2002). Data analysis as the search for signals in noisy
processes. *Journal for Research in
Mathematics Education, 33*, 259–289.

Konold, C., Robinson,
A., Khalil, K., Pollatsek, A., Well, A. D., Wing, R., & Mayr, S. (2002).
Students' use of modal clumps to summarize data. In B. Phillips (Ed.), *Proceedings of the International Conference
on Teaching Statistics*, *Cape Town,
South Africa*. [CD-ROM] Voorburg, The Netherlands: International Statistics
Institute.

Makar, K., &
Confrey, J. (2004). Secondary teachers’ reasoning about comparing two groups.
In D. Ben-Zvi & J. Garfield (Eds.), *The
challenges of developing statistical literacy, reasoning, and thinking *(pp.
353–373). Dordrecht, the Netherlands: Kluwer Academic Publishers.

McGatha, M., Cobb, P.,
& McClain, K. (2002). An analysis of students’ initial statistical
understanding: Developing a conjectured learning trajectory. *Journal of Mathematical Behavior, 21*,
339–355.

Mokros, J., & Russell,
S. J. (1995). Children’s concepts of average and representativeness. *Journal for Research in Mathematics
Education, 26*, 20–39.

Peirce, C. S. (1976). *The new elements of mathematics* (Eisele,
C., Ed.) (Vol. I-IV). The Hague-Paris/Atlantic Highlands, N.J.:
Mouton/Humanities Press.

Petrosino, A. J.,
Lehrer, R., & Schauble, L. (2003). Structuring error and experimental
variation as distribution in the fourth grade. *Mathematical Thinking and Learning, 5*(2&3), 131–156.

Russell, S. J., &
Corwin, R. B. (1989). Statistics: The shape of the data*. Used numbers: Real data in the classroom. Grades 4-6*. Washington,
DC: National Science Foundation.

Schwartz, D. L.,
Goldman, S. R., Vye, N. J., Barron, B. J., & Cognition and Technology Group
at Vanderbilt (1998). Aligning everyday and mathematical reasoning: The case of
sampling assumptions. In S. P. Lajoie (Ed.), *Reflections on statistics: Learning, teaching, and assessment in grades
K-12* (First ed., pp. 233–273). Mahwah, NJ: Lawrence Erlbaum Associates.

Simon, M. A. (1995).
Reconstructing mathematics pedagogy from a constructivist perspective. *Journal for Research in Mathematics
Education, 26*, 114–145.

Steinbring, H. (1980). *Zur Entwicklung des
Wahrscheinlichkeitsbegriffs - Das Anwendungsproblem in der Wahrscheinlichkeitstheorie
aus didaktischer sicht.* [*On the
development of the probability concept - The applicability problem in
probability theory from a didactical perspective*]. Bielefeld: Institut für
Didaktik der Mathematik der Universität Bielefeld.

Van den Boer, C. (2003).
*Als je begrijpt wat ik bedoel. Een
zoektocht naar verklaringen voor achterblijvende prestaties van allochtone
leerlingen in het wiskundeonderwijs* [If you know what I mean. A search for
an explanation of lagging results of mathematics education among ethnic
minority students]. [CDROM] Utrecht, The Netherlands: Beta Press.

Van Oers, B. (2000). The
appropriation of mathematics symbols: A psychosemiotic approach to mathematics
learning. In P. Cobb, E. Yackel & K. McClain (Eds.), *Symbolizing and communicating in mathematics classrooms; Perspectivies
on discourse, tools, and instructional design* (pp. 133–176). Mahwah, NJ:
Lawrence Erlbaum Associates.

Wilensky, U. (1997).
What is normal anyway? Therapy for epistemological anxiety. *Educational Studies in Mathematics, 33*,
171–202.

ARTHUR BAKKER

Since
September 1, 2004:

Institute of
Education

University of
London

23-29 Emerald
Street

London WC1N
3QS

United
Kingdom