Recent Conference Publications and Presentations
2007
Bill, A., & Watson, J.M. (2007). Three student tasks in a study of
distribution in a “Best Practice” statistics classroom. In J. Watson & K.
Beswick (Eds.), Essential research, essential practice (Proceedings of the 30th
annual conference of the Mathematics Education Research Group of Australasia,
Hobart, pp. 123-132). Adelaide, SA: MERGA.
Three selected student tasks from a 2-week study of the statistical concept of
distribution in Year 9 class are examined. The tasks considered the exclusion of
outliers, analysis of data using a semi-formal framework (GICS) developed for
this study, and comparing two distributions. The pedagogy was modelled on
current statistics education research best practice, with an emphasis on the
cultivation of classroom dialogue where students explain and justify their
positions. Fathom™ software was used by the students in a computer laboratory,
and as a teaching aid in the classroom to support learning.
Watson, J.M., & Moritz, J.B. (2007). Developing aspects of distribution in
response to a media-based statistical literacy task. In Bulletin of the
International Statistical Institute 56th Session Proceedings Lisbon (Invited
Papers, Topic 37). [CD rom]. Lisbon: ISI.
This paper considers school students’ intuitive appreciation of distribution
when asked to represent a verbal description of an association of two variables.
The context is that of the interpretation of a claim in a newspaper article
rather than that of a course in formal statistics where students have
experienced instruction in correlation and regression. Hence the expectation is
for a range of responses in the realm of statistical literacy understanding
displayed by the time students reach the end of the middle years of schooling.
The task asked students to draw and label a graph to represent an “almost
perfect relationship between the increase in heart deaths and the increase in
the use of motor vehicles.” To express the relationship in a conventional
graphical form requires a visualisation of the distribution of paired values
across a range of possible single values for each variable (car usage and heart
deaths). Because no details are provided about variation in the distribution
except for the overall trend, the task provides the opportunity for students to
display their appreciation for what a realistic representation might look like.
A total of 1285 students were presented with this task in a survey format: 369
in Grade 6, 312 in Grade 8, and 604 in Grade 9. Responses are analysed within a
framework that acknowledges structural complexity in terms of the number of
elements of the tasks employed in the solution and the statistical
appropriateness of the response. The structural complexity is adapted from the
model of cognitive development devised by Biggs and Collis (1982). Responses of
the students are allocated to levels across grades and speculation about
development is presented. Development does not appear to be uniform and
suggestions are made about students’ previous experiences and about potentially
useful classroom experiences to improve performance.
2006
Watson, J.M. (2006). Issues for statistical literacy in the middle school. In
A. Rossman & B. Chance (Eds.), Proceedings of the Seventh International
Conference on Teaching Statistics: Working cooperatively in statistics
education, Salvador, Brazil. [CDRom]. Voorburg, The Netherlands: International
Association for Statistical Education and the International Statistical
Institute.
Focusing on the word “literacy” in the phrase “statistical literacy,” the
present study explored what happened to the non-numerically based aspects of
statistical literacy when students in Grades 7 and 9 were exposed to a unit of
work in chance and data that emphasized variation. To test the suggestion of
transfer of thinking skills to the literacy side of statistical literacy, 20
items from a larger survey were selected, upon which changes in literacy skills
could be measured. Ninety students in each of Grade 7 and Grade 9 were asked the
questions in a longer survey before and six weeks after taking part in a unit on
chance and data devised by their usual classroom mathematics teacher as part of
their schools’ mathematics programs.
2005
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.
This paper considers the evolution of research in statistics education since the
introduction of chance and data into the Australian mathematics curriculum in
1991 and presents selected outcomes of research into students’ understanding of
the content in the chance and data curriculum, using them to argue for a change
in emphasis in the classroom in the teaching of chance and data. These
suggestions might also influence current curriculum revisions taking place
within Australia and New Zealand. Building on the history of the discipline of
statistics and its introduction into the school curriculum, it is argued that
topics in the curriculum associated with expectation, such as the mean,
generally have preceded those associated with variation, such as the standard
deviation. Research however, suggests that children develop an appreciation of
variation before expectation, and this knowledge should influence the order of
the introduction of associated topics and their juxtaposition in the curriculum
and the classroom.
Watson, J.M. (2005). Developing an awareness of distribution. In K. Makar (Ed.),
Reasoning about distribution: A collection of current research studies
(Proceedings of the Fourth International Research Forum on Statistical
Reasoning, Thinking, and Literacy (~SRTL-4), University of Auckland, New
Zealand, 2-7 July). [~CD-ROM] Brisbane: University of Queensland.
This paper is an informal account of observations about students’ developing
awareness of distribution as exhibited in responses to tasks used in Tasmanian
research over the past decade. The paper attempts a synthesis of individual
studies, most of which have been published task by task to illustrate detailed
student performance. Themes are drawn from the collection of tasks to build an
understanding of how intuitions develop before formal ideas of distribution are
introduced in the school curriculum. Graphical representations produced by
students are the basis of exploring the development over the years of schooling.
Watson, J.M., Kelly, B.A., & Izard, J.F. (2005). Statistical literacy over a
decade. 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. 775-782). Sydney: MERGA.
This study uses Rasch modelling to link student outcomes over the decade since
the introduction of chance and data into the curriculum of an Australian state
in 1993. Although improvement is observed over time for intact groups of
students, and between grade levels in a given year, improvement across cohorts
for given grades over time is not observed. The distribution of the items used
in the 2003 survey across the statistical literacy variable supports earlier
models of the hierarchical nature of statistical thinking obtained from a larger
pool of items.
Watson, J.M., & Callingham, R.A. (2005). Statistical literacy: From
idiosyncratic to critical thinking. In G. Burrill & M. Camden (Eds.), Curricular
Development in Statistics Education. International Association for Statistical
Education (IASE) Roundtable, Lund, Sweden, 2004 (pp. 116-162). Voorburg, The
Netherlands: International Statistical Institute.
This paper follows earlier research using a survey instrument devised to measure
statistical literacy understanding at the school level. Based on partial credit
Rasch analysis, the performance of 673 students in Grades 5 to 10 is reported
both overall and for three subgroups of items reflecting strands within
statistical understanding. The three strands are the basic measurement of
average and chance, the related ideas of sampling and inference, and the
representation of data and variation. A hierarchy of six levels of understanding
is presented, with differing trends across the grades discussed and an example
of individual student performance at each level given. Some of these examples
illustrate student differences in understanding for the different strands.
Implications for the school curriculum are considered with respect to potential
development across the years of schooling and realistic expectations for
students at various grade levels. Issues for further consideration and research
are raised in the final section.
Watson, J.M. (2005). Lessons from research: Students’ understanding of
statistical literacy. In M. Coupland, J. Anderson, & T. Spencer (Eds.), Making
mathematics vital (Proceedings of the 20th biennial conference of the Australian
Association of Mathematics Teachers, Sydney, pp. 253-260). Adelaide, SA: AAMT,
Inc.
This paper reports on the findings of research into the development of students’
understanding of statistical literacy over the years of schooling from grade 3.
Statistical literacy at the school level is associated with the application of
concepts from the chance and data part of the school curriculum in contexts that
expand with students’ experiences over the years and that ultimately require
critical questioning skills. Various tasks and examples of students’ levels of
performance are presented to illustrate the development of student
understanding. These focus on language, sampling in context, and questioning
claims.
2004
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.
Data collected from students involved in a project examining change in
understanding of statistical variation in relation to the chance and data
curriculum after instruction and after two years, are the basis for the analysis
reported in this study. Comparisons are made, using partial credit Rasch
analysis, between successive grades (3, 5, 7, and 9), within students after
instruction, within students after two years, and between students in the four
grades after two years depending on whether they were involved in the
instructional intervention or not. Results show varying magnitudes of
differences among grades, differing improvements after instruction, but little
difference between Intervention and Non-Intervention groups after a two year
period.
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.
This report considers the difference in performance of students in Grades 3, 5,
7, and 9 over a two-year interval, 2000 to 2002, on a survey of concepts in the
chance and data curriculum. Two types of comparisons occur. First, longitudinal
change within students is measured and this is compared for students in schools
where an instructional intervention occurred and those in schools with no
intervention. Second, in Grades 5, 7, and 9 in 2002 performance is compared with
that in the same grades in the same schools in 2000. Reasons for differences are
suggested based on the educational experiences that the 2002 students
experienced in the intervening time. Students answered questions concerning
basic chance and data understandings, and variation in chance, data and
graphing, and sampling. Scores on these four subscales as well as the overall
survey are used to make comparisons.
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.
This report considers beliefs about chance in relation to understanding of
random processes and luck. Changes in chance beliefs over two- and four-year
periods were measured for 265 students initially in Grades 3 and 6. Students
were asked three questions on repeated occasions, based on the meaning of the
word random, their beliefs about luck, and their beliefs about how luck affects
winning a lottery. Change was also measured for performance on three questions
related to chance measurement. The association of chance belief and chance
measurement was found to be weak. Educational implications are considered.
Watson, J.M., & Chick, H.L. (2004). What is unusual? The case of a media graph.
In M. Johnsen-HØines & A. B. Fuglestad (Eds.), Proceedings of the 28th annual
conference of the International Study Group for the Psychology of Mathematics
Education (Vol. 2, pp. 207-214). Bergen, Norway: PME.
Three hundred and twenty-four middle school students considered a group of three
graphs in a newspaper article about boating deaths. The graphs contained
discrepancies and the students were asked to “comment on unusual features.” This
form of questioning produced a distribution of responses surprising to the
authors and perhaps challenging to current goals for statistical literacy. Of
these students, 201 answered the same question two years later and although
overall performance improved to some extent there were still very few high level
responses. The outcomes point to specific suggestions that can be made for
middle school classrooms in line with the goals of statistical literacy.
2003
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.
Although statistical variation does not receive detailed attention in
mathematics curriculum documents, students actually experience variation every
day of their lives. Among other varying phenomena, the weather provides a topic
of discussion for young and old. From early childhood, teachers are known to put
up weather calendar charts recording the weather for weeks at a time. This study
uses the weather context to explore students’ development of intuitive ideas of
variation from the third to the ninth grade. Three aspects of understanding
these intuitions associated with variation are explored in individual video
taped interviews with 66 students: explanations, suggestions of data, and
graphing. The development of these three aspects across grades is explored, as
well as the associations among them. Fifty-eight of the students also answered a
general question on the definitions of variation and variable and these
responses are discussed and compared with responses to the weather task. The
interview protocol may prove useful for teachers, particularly with younger
children, to appreciate students’ developing understanding of variation and
provide starting points for classroom work of a more specific nature, either
with respect to weather or other contextual topics.
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)
This paper considers the development of school students’ understanding of three
terms that are fundamental to statistical literacy: sample, random, and
variation. A total of 738 students in grades 3, 5, 7, and 9 were asked in a
survey to define and give an example for the word sample. Of these, 379 students
in grades 7 and 9 were also asked about the words random and variation.
Responses were used to describe developmental levels overall and to document
differences across grades on the understanding of these terms. Changes in
performance were also monitored after lessons on chance and data emphasising
variation for 335 students. After two-years, 199 of these students and a further
209 students who were surveyed originally but did not take part in specialised
lessons, were surveyed again. The difference after two-years between the
performance of students who experienced the specialised lessons and those who
did not was considered, revealing no differences in longitudinal performance.
For students in grades 7 and 9 the association of performance on the three terms
was explored. Implications for mathematics and literacy educators are discussed.
Watson, J.M. (2003). Statistical literacy at the school level: What should
students know and do? In Bulletin of the International Statistical Institute
54th Session Proceedings Berlin (Volume LX, Book 2, Invited Papers, Topic 49,
pp. 68-71). Berlin: ISI.
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 26th annual conference of the Mathematics Education Research
Group of Australasia, Geelong, pp. 728-735). Sydney, NSW: MERGA.
This study considers students’ predictions and explanations for outcomes when a
normal six-sided die is tossed 60 times. Changes are monitored after lessons on
chance and data and/or after two years. The study is motivated by two approaches
to probability advocated in the mathematics curriculum: one stressing
expectation based on theory and the other acknowledging variation during
experiments. The outcomes are discussed in light of other research and the
dilemmas created for students by these two approaches to probability.
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.
Pictographs are often used in the media to draw attention to data that would
likely be ignored in a table. In school, however, pictographs disappear from the
curriculum by the middle primary years. The outcomes of the research reported
here indicate that pictographs can provide a basis for rich tasks displaying not
only students’ counting skills but also their appreciation of variation and
uncertainty in prediction. The range of responses is discussed in relation to
other research and classroom implications.
2002
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]
Although the chance and data curriculum would not exist without variation –
variation in random processes and variation in data collection – the topic of
variation itself has not had a high profile in the chance and data curriculum.
If variation is the foundation of chance and data, what do school students
understand about the concept and how do they imagine variation to occur in
random processes? This session will present a summary of research findings for
students in grades 3 to 9, as well as the outcomes from interviews with some
6-year-olds that indicate children can discuss variation from an early age.
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
This study considers students’ understanding of a relatively new form of graph
in the data-handling curriculum: the stacked dot plot, or as it is more commonly
known, the line plot or dot plot. Students in grades 5 to 10 were presented with
various forms of a task asking for comparison of two stacked dot plots, one with
usual scaling of the base line and the other with gaps for zero-values omitted.
Of interest were students’ abilities to interpret the information as presented
in the graphs, to put the information in context, and to distinguish the
statistically appropriate form of graph with an explanation. Performances across
grades and forms of the task are compared. For a subset of students in grades 5,
7, and 9 performance is compared for one form of the task before and after a
unit of study on chance and data. Discussion includes implications for the
classroom in relation to current interest in statistical literacy across the
curriculum.
Callingham, R.A., & Watson, J.M. (2002, December). Implications of differential
item function in statistical literacy: Is gender still an issue? Refereed paper
presented at the Measurement Special Interest Group of the Australian
Association for Research in Education conference, Brisbane. Available at:
http://www.aare.edu.au/02pap/index.htm
Statistical literacy is a complex developmental construct requiring both
mathematical skills and contextual understanding. The development of statistical
literacy is an important objective of classrooms where the curriculum is
approached through considering problems that require the active engagement of
learners with relevant social material. Such approaches are often advocated for
the middle years of schooling. Little attention has been paid, however, to the
effects of these approaches on male and female students. This paper reports on a
study that considers Differential Item Functioning (DIF) with respect to gender
of questions on a statistical literacy scale derived from archived data.
Multi-faceted Rasch models were applied to polytomous data to determine the
interactions between gender and item. Three criteria were applied to the
results: statistical significance, replicability and substantive explanation of
DIF. The results suggested that although there was no overall difference in the
average performance of male and female students, items requiring numerical
responses or calculations were less difficult for male students and, conversely,
items demanding written explanations were less difficult for female students.
The implications of these findings for both assessment and teaching are
discussed.
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.
This report focuses on one aspect of a larger study of school students’
understanding of statistical variation. Although the study included students in
grades 3, 5, 7, and 9, this paper will focus on grade 5 students only. Students
experienced a unit of 10 lessons on the chance and data part of the mathematics
curriculum conducted over an eight-week period. Lessons included a particular
emphasis on variation and its role in statistical understanding. Pre- and
post-tests were administered and improvements were found in overall performance
and for variables reflecting appreciation of variation in chance, variation in
data, and variation in sampling. Some comparisons are made with grade 3
students’ performance.
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.
Responses of 73 students to an interview protocol based on selecting 10 lollies
from a container with 50 red, 20 yellow, and 30 green are categorised with
respect to centre and spread of numerical answers and to reasoning expressed in
justification of the answers. Results are compared to earlier survey research
and small-scale interview studies.
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.
As part of a larger project studying school students’ understanding of
statistical variation, 92 students in grade 7 and 90 students in grade 9
participated in a unit of work related to the chance and data curriculum,
emphasising variation with respect to the topics covered. Students completed
pre- and post-tests devised to assess understanding before and after the lessons
in the unit. This paper reports on the teaching arrangements for the classes
taking part and the change in performance after the unit.
Watson, J.M. (2002). Creating cognitive conflict in a controlled research
setting: Sampling. 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.
This paper reports on research that created a controlled environment for
interviewing individual students on the topic of sampling, allowing for
cognitive conflict from other students. At various points in the interview the
student was shown video extracts with contrasting views to those expressed and
ask for a reaction. Outcomes are discussed with respect to (a) the outcomes for
37 students, in terms of their reaction to the cognitive conflict presented, and
(b) the methodology, in terms of modeling cognitive aspects of a classroom
environment in a controlled setting.
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.
This paper will report on outcomes observed in an investigation that involved
teaching chance and data with an emphasis on understanding the part that
variation plays in processes associated with chance measurement and data
collection/analysis. Classes of students in grades 3, 5, 7, and 9 took part in
the study but this report will focus on children in grade 3. They were taught a
unit of 10 lessons over eight weeks and given pre and post tests in association
with the teaching of the unit. Of interest was not only their learning about
basic probability and data handling but also their developing understanding of
the influence that variation has on outcomes in relation to the observation of
pattern. The question of the age at which children can start appreciating the
influence of variation creates special interest in this group of students.
Reprinted in B. Phillips (Ed.), ICOTS6 Papers for School Teachers (pp. 83-88).
Voorburg, The Netherlands: International Statistical Institute.
2001
Watson, J. M., & Chick, H. L. (2001). A matter of perspective: Views of
collaborative work in data handling. In M. van den Heuvel-Panhuizen (Ed.),
Proceedings of the 25th conference of the International Group for the Psychology
of Mathematics Education (Vol. 4, pp. 407-414). Utrecht: Freudenthal Institute.
This paper reports on a study of students’ collaborative group work in a grade
5/6 classroom on an open-ended task from the chance and data part of the
mathematics curriculum. It considers (a) students’ beliefs about collaborative
group work compared to their actions; (b) observations of student knowledge,
learning, and outcomes during collaboration compared to understanding displayed
in individual interviews after work was completed; (c) students’ accounts of
events that took place in their groups compared with what was recorded on
videotape; and (d) students’ perceptions of the task and their beliefs about the
mathematics curriculum.
Watson, J. M., & Mortiz, J. B. (2001). The role of cognitive conflict in
developing students’ understanding of chance measurement. In J. Bobis, B. Perry,
& M. Mitchelmore (Eds.) Numeracy and beyond (Proceedings of the 24th Annual
Conference of the Mathematics Education Research Group of Australasia, Vol. 2,
pp. 523-530). Sydney, NSW: MERGA.
In individual interviews, twenty students in each of grades 3, 6, and 9
responded to a task involving chance measurement, then viewed video recordings
of other students’ conflicting responses and decided which response they
preferred. Seven students improved their levels of reasoning and seven agreed
with higher-level prompts without expressing reasoning. Only two students agreed
at some point with lower-level prompts, and both reverted to the level of their
original response in conclusion. Educational implications are noted.
2000
Moritz, J. B., & Watson, J. M. (2000). Reasoning and expressing probability
in students' judgements of coin tossing. In J. Bana & A. Chapman (Eds.),
Mathematics education beyond 2000 (Proceedings of the 23rd Annual Conference of
the Mathematics Education Research Group of Australasia, Vol. 2, pp. 448-455).
Perth, WA: MERGA.
A survey item based on a newspaper article about coin tosses was administered to
1256 students from grades 6 to 11. Few students determined the probability of
four successive tails. Most students considered that heads and tails were
equally likely for a subsequent fifth toss, often describing the probability as
“50-50”. Students in higher grades were more likely to respond appropriately.
Results are discussed with reference to equiprobability, independence, the
gambler’s fallacy, and the outcome approach to probability.
1999
Watson, J.M. (1999). Lessons from chance and data research I: Student
understanding. In K. Baldwin & J Roberts (Eds.), Mathematics - The next
millennium (pp. 100-110). Adelaide, SA: Australian Association of Mathematics
Teachers, Inc.
This paper will summarise some of the results of recent research into students'
understanding of chance and data concepts. Examples will be given to demonstrate
where students from grade 3 to grade 9 have difficulties, as well as where they
sometimes exceed expectations. Topics covered will include aspects of
probability, predicting from bar charts, average and comparing data sets, and
sampling. It is hoped that the lessons learned from the students' responses will
assist teachers in anticipating how their own students will perform.
Watson, J.M. (1999). Lessons from chance and data research II: For the
classroom. In K. Baldwin & J Roberts (Eds.), Mathematics - The next millennium
(pp. 311-320). Adelaide, SA: Australian Association of Mathematics Teachers,
Inc.
This workshop will present several ideas for classroom lessons arising from
recent research related to the chance and data part of the mathematics
curriculum. Potential experiences for students in the middle and secondary years
of schooling have arisen from interview protocols used with students in grades
3, 6 and 9 and a media survey used with grades 6 and 9. These are associated
with understanding issues of fairness and sample size in various settings, with
understanding average as used in different contexts, and with representing data
sets and relationships in graphical forms.
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.
Conjunction fallacy problems have been the focus of much previous research at
the tertiary level. This paper examines responses from over 3000 students in
grades 5 to 11 to two survey items. The items asked for P(A), P(B), and P(A and
B) of everyday events, expressed either as a probability or as a frequency.
Analysis revealed development in expressing probability, but only marginal
improvement in reducing incidence of conjunction errors. Longitudinal analysis
of 113 students asked the same questions over two- and four-year intervals was
consistent with the cross-sectional analysis.
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.
Two survey items asking for estimates of probability or frequency of conditional
events, (A|B) and (B|A), were completed by 2719 school students in grades 5 to
11. Cross-sectional and longitudinal analyses revealed improvement with grade in
expressing probability numerically and in distinguishing conditional events.
Conditional events were better distinguished for the frequency item than the
probability item. Comparisons with responses to other probability items
indicated understanding of conditional probability was related to development of
basic chance measurement.
Shaughnessy, J. M., Watson, J., Moritz, J., & Reading C. (1999, April). School
mathematics students' acknowledgment of statistical variation. In C. Maher
(Chair), There's more to life than centers. Presession Research Symposium
conducted at the 77th Annual National Council of Teachers of Mathematics
Conference, San Francisco, CA.
1998
Chick, H.L., & Watson, J.M. (1998). Showing and telling: Primary students'
outcomes in data representation and interpretation. In C. Kanes, M. Goos, & E.
Warren (Eds.), Teaching mathematics in new times. Volume 1 (pp. 153-160).
Brisbane: Mathematics Education Research Group of Australasia.
Students, in triads in a near-classroom environment, were video-taped as they
worked on interpreting and representing supplied data. Their responses were
categorised using the SOLO taxonomy. Representation skills varied from copying
out some of the data to relating two variables graphically; interpretation
skills varied similarly. Moreover, there appeared to be connections between the
two skills. The study also considered the nature and effectiveness of the
collaboration which took place within groups.
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.
Since January 1997 there has been much debate within Australia on the type of
numeracy benchmarks which should apply to children in Years 3 and 5. As well as
debate on the breadth and depth of understanding, there has been difficulty in
some areas establishing what children actually do know and can do. Although the
data included in this report were not collected to answer questions related to
numeracy benchmarking, they may help inform the debate about what children in
Years 3 and 5 know and can do in the area of chance and data.
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.
This paper will report on the use of a three-tiered model to assess the
statistical understanding of school students in the context of a media report.
Data from 229 Australian students will be presented to show levels of
performance in relation to the model reflecting increasing complexity of
response. Four groups of students were tested at two-year intervals with one of
these groups also tested after four years. The media item used in this report
contained a misleading claim about a population based on an unrepresentative
sample. The assessment was designed to determine if students could question the
claim. Results indicate that educators have a long way to go in creating a
statistically literate citizenry.
Watson, J.M. (1998). Professional development of teachers using CD-ROM
technology. In L. ~Pereira-Mendoza (Ed.), Statistical education Expanding the
network. Proceedings of the Fifth International Conference on Teaching
Statistics (pp. 921-927). Voorburg: International Statistical Institute.
The need to provide professional development for teachers in statistics and
probability arises because many teachers did not cover the topics in their
preservice training. The need to use technology arises in countries like
Australia because of the low density of teachers separated by vast distances. A
government funded projected entitled "Learning the Unlikely at Distance
Delivered as an Information Technology Enterprise" (LUDDITE), explored various
technologies and developed a ~CD-ROM as part of a professional development
package. The ~CD-ROM included motivating curriculum material to complement a
commercial text and video. The package of materials was trialed in several
contexts and feedback from these trials will be discussed, as well as the
difficulties associated with turning a pilot project into a commercial product.
1997
Lidster, S. T., Watson, J. M., & Chick, H. L. (1997). Developing cognition in
representing and interpreting data. In N. Scott & H. Hollingsworth (Eds.),
Mathematics creating the future (pp. 202-209). Adelaide: Australian Association
of Mathematics Teachers.
This case study examines the development, in school students, of higher level
cognitive functioning in data organisation, representation and interpretation.
Such higher level functioning is evident when students use forms of data
representation for the purpose of illustrating relationships among variables
that have been identified or hypothesised from the raw data. The ability to
achieve this depends on the level of facility with skills such as data
representation and hypothesising relationships, and the extent to which these
can be synthesised.
Moritz, J. B., & Watson, J. M. (1997). Graphs: Communication lines to students?
In F. Biddulph & K. Carr (Eds.), People in mathematics education, Vol. 2 (pp.
344-351). Waikato: Mathematics Education Research Group of Australasia.
Graph comprehension is considered a basic skill in the curriculum, and essential
for statistical literacy in an information society. How do students interpret a
graph in an authentic context? Are misleading features apparent? Responses to
questions about a graph-based advertisement suggest that students commonly fail
to appreciate scaling difficulties, to relate a graph as relevant in the context
of a standard interpretation task, and to apply numeracy skills for calculations
based on data in graphical representations.
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.
Students from Grade 3 to 9 were asked to represent given information on numbers
of books children had read, using picture-cards of children and books. The
interview protocol included questions requiring interpretation and prediction,
which allowed for a wide range of responses, from imaginative story-telling to
the use of sophisticated statistical reasoning. This paper discusses some issues
for student understanding when representing data, interpreting the
representation, and using it to make predictions.
Watson, J.M. & Chick, H.L. (1997). Collaboration in mathematical problem
solving. A paper presented at the Annual Conference of the Australian
Association for Research in Education, Brisbane, Queensland.
AARE Web site: http://www.swin.edu.au/aare/
The research literature on collaborative group work in problem solving contains
generally positive results in terms of both the processes involved and the
cognitive outcomes achieved. The project reported in this paper sought to look
at several aspects of collaborative problem solving in relation to higher order
functioning in the chance and data part of the mathematics curriculum. Previous
work with students in Grades 6 and 9 suggested a hypothesis that students
working in groups may produce higher level outcomes in the process of finding
and justifying associations in data sets than those working individually. In the
current study, students in Grades 3, 6 and 9 worked in groups on two chance and
data tasks: one related to fair dice and the other related to associations among
variables presented on data cards. Whereas other reports from this project have
focussed on the mathematical achievement of students completing tasks, this one
will look specifically at students' levels and types of collaboration in
relation to issues raised in the literature. The particular focus will be on
determining what changes in cognitive functioning take place and what
collaborative behaviours contribute to this change.
Watson, J. M., & Moritz, J. B. (1997). The C&D PD CD: Professional development
in chance and data in the technological age. In N. Scott & H. Hollingsworth
(Eds.), Mathematics creating the future (pp. 442-450). Adelaide: Australian
Association of Mathematics Teachers.
The Chance and Data Professional Development (PD) CD-ROM is described along
with a textbook and video used as part of a trial PD package for teachers of
Chance and Data. The multimedia CD-ROM includes curriculum documents, classroom
activities, and examples of student responses to a variety of questions from
research. Evaluation of the PD package and its effectiveness is based on
responses from teachers and other evaluators who trialed the package.
Watson, J. M., & Moritz, J. B. (1997). Teachers' views of sampling. In N. Scott
& H. Hollingsworth (Eds.), Mathematics creating the future (pp. 345-353).
Adelaide: Australian Association of Mathematics Teachers.
The importance of sampling in the school curriculum is discussed in the context
of changes to the mathematics curriculum in the 1990s. Over 100 teachers
responded to questions related to what sampling means, where it fits in the
curriculum and approaches to teaching the topic. These responses are discussed
in relation to professional development initiatives in statistics education.
Watson, J. M., & Moritz, J. B. (1997). Measuring teachers' reactions to new
areas of the curriculum: A case study from chance and data. A paper presented at
the Annual Conference of the Australian Association for Research in Education,
Brisbane, Queensland
The evaluation of teachers' reactions to a new area of the curriculum is
important in order to identify factors limiting effective curriculum
implementation in the classroom. Teachers may lack confidence because the
subject matter was not part of their preservice training. They may be
uncomfortable with the different teaching methodologies seen as appropriate for
the new content. They may perceive deficiencies in the professional development
provided to help then bridge the gap between old and new content. This report
will focus on a profiling instrument devised to characterise the factors which
are related to teachers' effectiveness in teaching topics in the chance and data
part of the mathematics curriculum. The profile measures teachers' knowledge of
topics, teaching methodology and planning, anticipation of common student
responses, confidence teaching particular topics, beliefs about the usefulness
of statistics in society, familiarity with resources in the area, and perceived
needs for professional development. The report is based on data from 115
teachers of primary and secondary grades around Australia.
1996
Moritz, J. B., Watson, J. M., & Pereira-Mendoza, L. (1996). The Language of
Statistical Understanding: An Investigation in Two Countries. Paper presented at
the Annual Conference of the Australian Association for Research in Education,
Singapore.
Statistical understanding of students from Grades 3 to 9 in Australia and
Singapore was assessed using a 20-item survey which involved items based on past
research into probability and statistics. Responses were analysed using the
language analysis software NUD·IST, and were classified according to the SOLO
Model with Multimodal Functioning developed by Biggs and Collis. Results from
the two countries are discussed in the light the curriculum and teaching
practices.
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.
What are students' views of odds? Students were asked to interpret a newspaper
headline, "North at 7-2". Three different perspectives were distinguished: (1) a
probability view often using traditional part-whole ratios, (2) a frequency view
involving scores and frequency of wins, and (3) a social view, usually involving
betting and money exchange in part-part ratios. Each view followed a
developmental sequence, with interaction between them.
Watson, J. M., & Moritz, J. B. (1996, July). The art of bar charts: Steps in
interpreting and predicting. Poster presented at the International Congress on
Mathematical Education, Seville, Spain.
This paper presents example responses from 30 students in Grades 3, 5, 7 and 9
to an interview protocol involving an adjustable bar chart. Of major interest
are (i) the interpretations given to the bar chart as presented, (ii) the
compensation which takes place when the chart is changed, and (iii) the criteria
which are used when predictions are made based on the chart. As the interviews
were videotaped, it is possible to provide both dialogue and still graphics of
the students' responses. A hierarchical development is evident in responses:
those which do not conserve the numbers present in the situation, those which
only use out-of-school experience to make predictions, those which base
predictions only on the frequencies presented in the chart, and those which
integrate the information provided and out-of-school experience to make
predictions. Implications of these results for teaching are discussed.
Watson, J.M. & Moritz, J.B. (1996). Student Analysis of Variables in a Media
Context. In Brian Phillips (Ed.), Papers on Statistical Education presented at
International Association for Statistical Education, ICME-8, Seville, Spain.
(pp129-147), Swinburne Press, Hawthorn, Australia.
Judging statistical claims from the media is fundamental to being statistically
literate. In making a judgements it may be appropriate to graph, or at least
visualise, the claim being made. Such is often the case when cause and effect
assertions are made for two or more variables. The contexts in which claims are
made can be quite varied and can overlap areas of the high school curriculum
other than mathematics. It is important for mathematics teachers to be aware of
the contexts which can motivate an understanding of the statistical principles,
and it is important for teachers of other subject areas such as science, social
science, health and technology to be aware of the mathematics necessary to make
sense of statistical claims. A survey item based on a newspaper article which
claimed an "almost perfect relationship between the increase in heart deaths and
the increase in the use of motor vehicles" was administered to 1291 students to
explore their ability to produce a graphical representation of the claimed
relationship and their ability to question the claim itself. Implications of the
hierarchy of understanding for high school teachers of mathematics and other
subjects are discussed.
1995
Callingham, R. A., Watson, J. M., Collis, K. F., & Moritz, J. B. (1995).
Teacher attitudes towards chance and data. In B. Atweh & S. Flavel (Eds.),
Proceedings of the Eighteenth Annual Conference of the Mathematics Education
Research Group of Australasia (pp. 143-150). Darwin, NT: Mathematics Education
Research Group of Australasia.
Seventy-two teachers from Tasmanian government primary and secondary schools
were surveyed regarding (i) their agreement with statements relating to personal
confidence with chance and data, (ii) their views of the importance of
statistics in society, and (iii) their confidence in teaching chance and data.
Differences across gender and school type were found in measures of individual
items and also combined scales. These results help to specify needs for
professional development.
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.
What do students really think graphs mean? Based on interviews with students
from Canada, Australia and Singapore, students' interpretations of different
graphs are examined, and the implications of these differing interpretations for
teaching are considered. Some examples of the potential of newspapers for
developing statistical ideas for primary and secondary students are included.
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: MERGA.
This paper presents an analysis of two questionnaire items which explore
students' understanding of the concept of luck in relation to the development of
ideas of formal probability. The items were administered to 1014 students in
Grades 3, 6 and 9 in Tasmanian schools. The analysis was based on the multimodal
functioning SOLO model. The results lead to a hypothesised structure and
implications for curriculum and teaching practice.
Watson, J. M., Collis, K. F., & Moritz, J. B. (1995, November). The development
of concepts associated with sampling in grades 3, 5, 7 and 9. Paper presented at
the Annual Conference of the Australian Association for Research in Education,
Hobart.
AARE Web site: http://www.swin.edu.au/aare/
As part of a study of chance and data concepts covered in recent Australian
curriculum documents, a sample of 171 girls was chosen from grades 3, 5, 7 and 9
in a private school in an Australian capital city. On the basis of responses to
a 20-item questionnaire and a media survey, 30 students across the grades were
selected for 45-minute interviews on nine problem solving protocols related to
statistical thinking. For this report only items related to sampling will be
analysed: one interview protocol, one item from the 20-item questionnaire and
two items from the media survey. The purpose is to propose a developmental
sequence for learning of concepts associated with sampling over the years of
schooling. This model will then be tested when data from the complete study are
analysed. The theoretical basis for the analysis is the SOLO Taxonomy with
Multimodal Functioning developed by Biggs and Collis. Educational implications
of this developmental sequence will be addressed.
1994
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.
This paper presents some results of a pilot study which devised a 20-item
paper-and-pencil, short-answer/multiple-choice questionnaire to assess students'
understanding of statistics and probability in Grades 3, 6 and 9. The items are
presented, with discussion of response differences over the three grades, and
the level and type of cognitive functioning associated with responses
1993
Watson, J.M., Collis, K.F., & Moritz, J.B. (1993, Sept.) Assessment of
statistical understanding in Australian schools. Paper presented at the
Statistics '93 conference, Wollongong, NSW.
This paper will describe the initial phases of a three-year project whose
purpose is to devise assessment instruments and follow the implementation of the
Chance and Data content of A National Statement on Mathematics for Australian
Schools. A discussion of the rationale used in devising items will be followed
by descriptions of the four types of instruments developed. Finally, the results
of pilot trials carried out in recent months will be presented.
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