Recent
Research Journal Publications
Watson, J.M. (in press). The development of statistical understanding at
the elementary school level. Mediterranean Journal of Mathematics Education.
The massive increase in information and data available in the 21st century is
putting increasing pressure on school curricula around the world to prepare
students to meet the demands of using the information and data for the good of
society. Finding the place in an already crowded curriculum for the
technological and statistical literacy skills is not easy. The needs are present
across the curriculum but thus far statistical thinking has usually been placed
within the mathematics curriculum under such headings as “data and chance” or
“data handling.” Various suggestions have been made concerning what content
should be introduced at different year levels. This paper does not make specific
suggestions for students at different years of school or ages because of
differences across countries but presents a developmental picture of statistical
understanding from early childhood across the elementary into the middle school
years. Knowing where a child currently is in the developmental sequences can
indicate what activities can assist movement to higher levels of understanding.
It is hoped that greater appreciation of the potential for developing
statistical understanding will influence curriculum developers to find a place,
integrated across the traditional disciplines, to build the foundations required
for critical statistical literacy in the adult world.
Watson, J.M., & Kelly, B.A. (2007). Sample, random, and variation: The
vocabulary of statistical literacy. International Journal of Science and
Mathematics Education. [Electronic Version]
This paper considers the development of school students’ ability to define 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 emphasizing
variation for 335 students. After two years, 132 of these students and a further
209 students who were surveyed originally but did not take part in specialized
lessons, were surveyed again. The difference after two-years between the
performance of students who experienced the specialized lessons and those who
did not was considered, revealing no differences in performance longitudinally.
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. (2007). Inference as prediction. Australian Mathematics Teacher,
63(1), 6-11.
Inference, or decision making, is seen in curriculum documents as the final step
in a statistical investigation (Australian Education Council, 1991). For a
formal statistical enquiry this may be associated with sophisticated tests
involving probability distributions. For young students without the mathematical
background to perform such tests, it is still possible to draw informal
inferences based on data of various sorts, for example by comparing two
graphical representations (e.g., Watson & Moritz, 1999). In doing so it is
important to be able to state the assumptions that are the foundation for the
decision made (Whitin, 2006). This article considers a straightforward context
where students are asked to make predictions. These predictions are informal
inferences that can be based on aspects of the scenario, the students’
appreciation of the context, and their cognisance of the data presented.
Watson, J.M. (2007). The role of cognitive conflict in developing students’
understanding of average. Educational Studies in Mathematics, 65, 21-47.
Two strands of research motivate this study. One is the interest in school
students’ development of understanding of the concept of average, historically
part of the mathematics curriculum and prominent in the statistics curriculum
introduced in the early 1990s. The other is the belief of some educators that
students learn meaningfully when presented with cognitive conflict that
challenges incorrect or incomplete understandings. This study presented 58
students in Grades 3, 6, and 9 with a series of questions about the concept of
average. After initial levels of response were observed, students were presented
with alternative responses on video from other school students and asked to
choose which best resolved the task at hand. Initial responses confirmed the
levels of understanding in an earlier study based on the same questions.
Responses after the presentation of cognitive conflict were either at the same
level as before or higher, with no student finally agreeing with a lower level
response. The results are compared with longitudinal change in relation to
average and change resulting from cognitive conflict in relation to other areas
of chance and data. Implications of the research are considered.
Watson, J.M., & Kelly, B.A. (2007). Assessment of students’ understanding of
variation. Teaching Statistics, 29(3), 80-88.
Several tasks used in research studies are presented with assessment rubrics and
examples of the development of student understanding. The tasks focus on
students’ appreciation of variation in several contexts and illustrate the need
to discuss variation in the classroom and to ask students specifically about it
during assessment. [This paper won the C. Oswald Prize of the Royal Statistical
Society in the UK for the best paper of the year in the journal Teaching
Statistics.]
Watson, J.M., & Kelly, B.A. (2007). Development of student understanding of
outcomes involving two or more dice. International Journal of Science and
Mathematics Education. [Electronic version]
Data from 154 interviews with students in Grades 3 to 13 are analyzed to suggest
a developmental progression of conceptual understanding associated with the
sample space for two ordinary six-sided dice tossed simultaneously. The model is
then considered in the light of responses to an extension task involving three
six-sided dice with four sides painted black and two white. Forty-five of the
interviews are longitudinal interviews of students three or four years after the
original interviews, allowing for analysis of change in levels of understanding
across time. This study suggests some of the intuitions and understandings that
form the intermediate steps to a complete understanding of outcomes for two
dice, as well as documenting the conceptual regression likely to occur when a
more difficult task is encountered. Links to previous research, including
well-known misconceptions, and educational implications of the model are among
the discussion points.
Watson, J.M., Callingham, R.A., & Kelly, B.A. (2007). Students’
appreciation of expectation and variation as a foundation for statistical
understanding. Mathematical Thinking and Learning, 9, 83-130.
This study presents the results of a partial credit Rasch analysis of in-depth
interview data exploring statistical understanding of 73 school students in six
contextual settings. The use of Rasch analysis allowed the exploration of a
single underlying variable across contexts, which included probability sampling,
representation of temperature change, beginning inference, independent events,
the relationship of sample and population, and description of variation.
Interpretation of the demands of increasing code levels for the resulting
variable revealed an increasing appreciation of and interaction between the
ideas of variation and expectation. Student progression in understanding is
illustrated with kidmaps and educational implications are considered.
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.
Although errors in reasoning about conditional probability have been the focus
of interest of psychologists for a long time, the development of conditional
reasoning in school students has received little attention. This paper considers
the responses of 69 students across grades 3 to 13 in an attempt to model the
development of appropriate reasoning skills. Although acknowledging the
misconceptions observed by earlier researchers, this study seeks a positivist
approach to describe developing understanding by combining mathematical
appropriateness with structural complexity. The results of the study are
intended to inform other researchers, teachers, and curriculum developers.
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.
A task adapted from one used by Tversky and Kahneman (1971) was used in an
interview or questionnaire context with 122 students from Grade 3 to Grade 13.
Two questions assessed student understanding of the relationship of a sample to
a population and of the expected value of the arithmetic mean, with and without
information on a single value from the sample. Combining responses to the two
questions, increasingly complex hierarchical sequences were identified in the
observed responses in relation to the expected value of the means, an
Expectation variable, and in relation to the degree that ideas about variation
were used to support judgments, a Variation variable. Using data from another
task answered by 68 of the students, responses were associated with observed
development of understanding of sampling more generally, a Basic Sampling
variable. Overall, the associations of levels of response among the variables
were not strong, suggesting more explicit discussion of sampling issues is
required in classrooms.
Watson, J.M., Kelly, B.A., & Izard, J.F. (2006). A longitudinal study of student
understanding of chance and data. Mathematics Education Research Journal, 18(2),
40-55.
This study uses Partial Credit Rasch analysis to study a complex data set of
student responses to survey items that were classroom-administered tests about
understanding of chance and data collected from 1993 to 2003 in the Australian
state of Tasmania. Data were collected from a total of 5514 individual students
across Grades 3 to 11 over the decade, with 896 of these students providing at
least one repeated measure. Students completed a core of common items that
allowed Rasch analysis to be performed and all students were subsequently placed
on the same scale for comparison. The purpose of the analysis is to consider
average cohort change over time and trends in performance during the first 10
years after the curriculum was introduced in Tasmania. Implications for the
education system and curriculum implementation are considered.
Chick, H.L., Pfannkuch, M., & Watson, J.M. (2005). Transnumerative thinking:
finding and telling stories with data. Curriculum Matters, 1, 87-108.
A critical component in the development of students’ statistical thinking and
reasoning is transnumerative thinking; that is, changing representations of data
to engender an understanding of observed phenomena. Examples from Years 6 to 9
New Zealand students’ and Australian students’ representations of data from a
given multivariate dataset are described. Their representations are discussed in
terms of their developing abilities to explore data and unlock the stories
contained therein. The implications of changing the focus of statistics
instruction and the curriculum from merely teaching students how to construct
graphs to exploring and representing patterns and relationships in data are
presented.
Watson, J.M., & Kelly, B.A. (2005). The winds are variable: Student intuitions
about variation. School Science and Mathematics, 105, 252-269.
This study uses the context of the weather to explore the development of
students’ intuitive ideas of variation from pre-grade one to the ninth grade.
Three aspects of understanding these intuitions associated with variation are
explored in individual video taped interviews with 73 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. (2005). Cognition and instruction: Reasoning about
bias in sampling. Mathematics Education Research Journal, 17(1), 24-57.
Although sampling has been mentioned as part of the chance and data component of
the mathematics curriculum since about 1990, little research attention has been
aimed specifically at school students’ understanding of this descriptive area.
This study considers the initial understanding of bias in sampling by 639
students in grades 3, 5, 7, and 9. Three hundred and forty-one of these students
then undertook a series of lessons on chance and data with an emphasis on
chance, data handling, sampling and variation. A posttest was administered to
285 of these students and two years later all available students from the
original group (328) were again tested. This study considers the initial level
of understanding of students, the nature of the lessons undertaken at each grade
level, the post-instruction performance of those who undertook lessons, and the
longitudinal performance after two years of all available students. Overall
instruction was associated with improved performance, which was retained over
two years but there was little difference between those who had or had not
experienced instruction. Results for specific grades, some of which went against
the overall trend are discussed, as well as educational implications for the
teaching of sampling across the years of schooling based on the classroom
observations and the changes observed.
Watson, J.M., & Chick, H.L. (2005). Collaborative statistical investigations in
diverse settings. International Journal of Mathematical Education in Science and
Technology, 36, 573-600.
This study presents a continuing investigation of influences on outcomes
achieved by students working in groups of three on tasks related to chance and
data. Earlier research described final mathematical outcomes and identified 17
factors influencing three types of short-term outcomes for groups working in an
isolated setting. The current report documents the 17 factors for groups working
in a classroom setting and 1654 events for all groups in both settings are
identified and each associated with a factor and a short-term outcome.
Consideration is then given to variables that have the potential to influence
the factors, the short-term outcomes, and their interaction. The overarching
variable is the setting within which the collaboration took place. Within each
setting, however, two other variables operated: the task carried out, the
age/grade of students, gender balance, or collaborative characteristics. The
influences of these variables are described within the two settings before
consideration is given to the overall influence of the settings.
Callingham, R.A., & Watson, J.M. (2005). Measuring statistical literacy. Journal
of Applied Measurement, 6(1), 19-47.
This study considers the measurement of Statistical Literacy understanding that
goes beyond the basic chance and data skills and knowledge in the mathematics
curriculum. This understanding requires application of mathematical skills in a
range of contextual situations and draws on aspects of statistics, such as
variation and inference, which may not be explicit in the school curriculum. The
study reports the outcomes from tests of Statistical Literacy given to 673
students from Grades 5 to 10. It confirms the nature and structure of a
previously identified previously identified construct of Statistical Literacy
and proposes three subgroups of items that address aspects of Statistical
Literacy that might usefully be measured by classroom teachers.
Watson, J.M., & Kelly, B.A. (2004). Statistical variation in a chance setting: A
two-year study. Educational Studies in Mathematics, 57, 121-144.
A series of 13 survey questions based on a 50-50 spinner is used to explore
school students’ understanding of statistical variation in a chance setting.
Five questions set the context by assessing understanding of theoretical
expectation and representation of repeated trials in a stacked dot (line) plot.
Four questions provide opportunity to display appreciation of variation from
point expectation and four questions address variation from distributional
expectation. Three hundred and seventy-five students in Grades 3 to 9 answered
some or all of these questions. These students then took part in a unit of study
on chance and data emphasizing variation. Of these students, 334 answered a
post-test including the same items and a further subset of 199 students
completed a longitudinal survey of the same items two years later. Analysis of
the initial data showed a progression of understanding across the years of
schooling, plateauing at Grade 7, improvement for all grades after instruction,
and generally sustained and continuing improved performance after two years.
Educational issues are considered.
Watson, J.M., & Caney, A. (2005). Development of reasoning about random events.
Focus on Learning Problems in Mathematics, 27(4), 1-42.
The development of school students’ understanding of luck and random events is
explored in three related studies based on tasks well known in the research
literature. In Study 1, 99 students in grades 3 to 9 were interviewed on three
tasks and surveyed on two tasks about luck and random behaviors in chance
settings. In Study 2, 23 of these students were interviewed on the same tasks,
three or four years later to monitor developmental change. In Study 3 a
different group of 15 students was interviewed with two of the tasks and
prompted with conflicting responses of other students on video. The aim of Study
3 was to monitor the influence of cognitive conflict in improving student levels
of response. Four levels of response are identified across tasks, reflecting
increasing structural complexity and statistical appropriateness. Implications
for teachers, educational planners, and researchers are discussed in the light
of other researchers’ findings.
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.
Sixty-six students in grades 3 to 9 were interviewed using a protocol that
involved both theoretical questions about and experimentation with two spinners
used in a carnival game. Of interest were students’ initial responses to
questions and the changes in response and reasoning that occurred after
experimentation. Because the experimentation took place in a probabilistic
setting, there was the possibility of variation occurring that could appear
contrary to expectation. Student responses were hence analysed based on a 2 x 2
matrix of initial estimate of chance outcome by actual experimental outcome, and
with subsequent reasoning judged as appropriate based on the cell occupied.
These data were used to determine a developmental progression among the students
in the study. As well, data were available on these students’ responses to
survey questions involving a single spinner of the type used in the interview
protocol. Data for one and two spinners hence could be considered together. The
outcomes of the study lead to specific suggestions for the classroom in terms of
teaching expectation and variation in probabilistic settings.
Watson, J.M., & Callingham, R.A. (2003). Statistical literacy: A complex
hierarchical construct. Statistics Education Research Journal, 2(2), 3-46.
The aim of this study was, first, to provide evidence to support the notion of
statistical literacy as a hierarchical construct and, second, to identify levels
of this hierarchy across the construct. The study used archived data collected
from two large-scale research projects that studied aspects of statistical
understanding of over 3000 school students in grades 3 to 9, based on 80
questionnaire items. Rasch analysis was used to explore an hypothesised
underlying construct associated with statistical literacy. The analysis
supported the hypothesis of a unidimensional construct and suggested six levels
of understanding: Idiosyncratic, Informal, Inconsistent, Consistent
non-critical, Critical, and Critical mathematical. These levels could be used by
teachers and curriculum developers to incorporate appropriate aspects of
statistical literacy into the existing curriculum.
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.
One hundred-eight students in Grades 3, 5, 6, 7, and 9 were asked about their
beliefs concerning fairness of dice before being presented with a few dice (at
least one of which was “loaded”) and asked to determine whether each die was
fair. Four levels of beliefs about fairness and four levels of strategies for
determining fairness were identified. Although there were structural
similarities in the levels of response, the association between beliefs and
strategies was not strong. Three or four years later, we interviewed 44 of these
students again using the same protocol. Changes and consistencies in levels of
response were noted for beliefs and strategies. The association of beliefs and
strategies was similar after three or four years. We discuss future research and
educational implications in terms of assumptions that are often made about
students’ understanding of fairness of dice, both prior to and after
experimentation.
Watson, J.M., & Moritz, J.B. (2003). The development of comprehension of chance
language: evaluation and interpretation. School Science and Mathematics, 103,
65-80.
Comprehension of chance language, such as found in newspapers, is a fundamental
aspect of statistical literacy. Students' understandings of chance language were
explored through responses to two items in surveys administered to 2726 students
from grades 5 to 11. One item involved evaluating the chance expressed in
phrases from newspaper headlines using a number line, and responses were
described in four levels of chance language evaluation. The other item involved
interpreting, in context, an expression of percent chance, and responses were
described in four levels of chance language interpretation. Students in higher
grades were more likely to demonstrate higher levels of both evaluation and
interpretation. The association between levels of evaluation and interpretation
was further explored generally and in relation to one of the headlines involving
percent. Implications for mathematics educators in relation to chance language
in the curriculum across the years of schooling are discussed.
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.
This paper presents a questionnaire devised to assess school students’
understanding of statistical variation. The questionnaire is based on earlier
research of students’ understanding of the chance and data curriculum and recent
work, more specifically related to variation. It was devised, piloted, revised,
and administered to 746 students in grades 3, 5, 7, and 9 in ten Tasmanian
schools. The analysis of outcomes was carried out in three stages: a
hierarchical coding scheme was developed based on a structural model of
cognitive development; a Rasch analysis was carried out to produce a variable
map of student performance and item difficulty on a single scale; and an
holistic model of development was suggested for the questionnaire. Outcomes for
individual items are presented to illustrate the range of student responses, and
possible rubrics for use by teachers. For some items, comparisons are made with
results of other researchers.
Watson, J.M. (2002). Inferential reasoning and the influence of cognitive
conflict. Educational Studies in Mathematics, 51, 225-256.
This study follows two earlier studies of school students’ abilities to draw
inferences when comparing two data sets presented in graphical form (Watson and
Moritz, 1999; Watson, 2001). Using the same interview protocol with a new sample
of 60 students, 20 from each of grades 3, 6, and 9, cognitive conflict was
introduced in the form of video clips of reasoning expressed by students in the
earlier studies. This methodology was intended to mimic the type of
argumentation that might take place in the classroom but in a controlled setting
where identical arguments could be presented to different students. Interviews
were videotaped and analysed in a similar fashion to the earlier studies in
order to document change associated with the presentation of cognitive conflict.
Change was documented with respect to the levels of observed response for two
parts of the protocol, and for the use of displayed variation in the graphs.
Implications of the methodology for future research and teaching are discussed.
Watson, J.M., & Kelly, B.A. (2002). Emerging concepts in chance and data.
Australian Journal of Early Childhood, 27(4), 24-28.
Interviews with seven six-year-old children shed light on the emerging ideas
related to four themes within the chance and data part of the mathematics
curriculum: appreciation of uncertainty and variation, observing and creating
representations, appreciating the need to count and conserve quantity, and using
data for interpreting and predicting.
Watson, J.M., & Moritz, J.B. (2002). Quantitative literacy for pre-service
teachers via the internet. Mathematics Teacher Education and Development, 4,
43-56.
Quantitative literacy involves not just basic number skills, often called
numeracy, but also the ability to integrate basic skills in contexts that
require high levels of literacy to interpret situations and make judgments. This
paper discusses the development, implementation, and evaluation of a unit
designed to assist pre-service teachers of mathematics to see the relevance of
quantitative literacy to daily life, to understand and discuss aspects of
quantitative literacy themselves, and to prepare and deliver lessons in schools
using an internet site as a teaching resource. Forty pre-service teachers (a)
selected a newspaper article and developed student questions and brief comments
for discussion with teachers, identifying issues for mathematics teaching; (b)
developed a teaching unit for one or more class lessons and implemented it in
the classroom; (c) prepared an evaluation of the experience; and (d) discussed
their colleagues’ work in relation to these tasks online. Responses indicated
most of the pre-service teachers could engage with these tasks and came to
believe that they supported effective teaching in the classroom. Some
difficulties are identified, however, related to the implementation of the
project, and suggestions are made for future variations.
Chick, H.L., & Watson, J.M. (2002). Collaborative influences on emergent
statistical thinking—A case study. Journal of Mathematical Behavior, 21,
371-400.
The purpose of this case study is to examine how collaboration affects the
emergent statistical thinking of a group of three Grade 6 boys. Results of
previous studies of students in Grades 3, 6, and 9 suggested that (a) when
finding and justifying associations in data sets students working in groups may
produce higher level outcomes than those working individually, and (b) there are
numerous factors that influence the success or otherwise of collaborative
activity. The current study, based on detailed analysis of video tape and
transcripts of a group working collaboratively on a data handling task,
documents various factors that affect collaboration and how these contribute to
the attainment of desirable cognitive outcomes in terms of the task set. These
outcomes are classified by emergent statistical themes and insight is gained
into how naïve statistical thinking begins to develop during the collaborative
process. Implications for educators and researchers are considered.
Watson, J.M., & Chick, H.L. (2002). A case study of graduate professional
development in the TAFE sector. Australian Educational Researcher, 29(2), 73-91.
There has been very little provision of professional development for teachers of
mathematics in the technical and further education (TAFE) sector and even less
evaluation and reporting on such programs. In 1993, the University of Tasmania
developed an extensive professional development program in mathematics that was
designed for TAFE teachers as well as those from the K–12 sector. This case
study describes the project, considering such issues as the professional
development needs of the TAFE teachers and what components should be included in
a professional development program. Evidence is provided for the short- and
long-term effectiveness of the program. The flexible structure of the program
could be used as a model for professional development in other TAFE curriculum
areas.
Watson, J.M. (2002). Discussion: Statistical literacy before adulthood.
International Statistical Review, 70, 26-30.
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.
The objective of this study was to provide baseline data in an area of the
mathematics curriculum that is beginning to receive greater attention than
previously. Four survey items were completed by 2615 students in grades 5 to 11.
Two survey items asked for estimates of probability or frequency for everyday
events (A), (B), and their conjunction (A and B). Two survey items asked for
estimates of probability or frequency for conditional events, (X|Y) and (Y|X).
Cross-sectional and longitudinal analyses revealed improvement with grade in
expressing probability numerically and in distinguishing conditional events, but
no change in incidence of conjunction errors. The relationships of responses to
conjunction items with those to conditional items, and of both with responses to
other items of basic chance measurement were considered. Implications were
related to interpretation of the results in terms of previous research and
suggestions for educators.
Watson, J.M., & Moritz, J.B. (2001). Development of reasoning associated with
pictographs: representing, interpreting, and predicting. Educational Studies in
Mathematics, 48, 47-81.
A developmental model involving four response levels is proposed concerning how
students arrange pictures to represent data in a pictograph, how they interpret
these pictographs, and how they make predictions based on these pictographs. The
model is exemplified by responses from three related interview-based studies. In
Study 1, examples of each response level are provided from 48 preparatory- to
tenth-grade students. Students from higher grades were more likely to respond at
higher levels. In Study 2, 22 students were interviewed longitudinally after a
three-year interval; many improved in response level over time, although a few
responded at lower levels. In Study 3, 20 third-grade students were interviewed
and then prompted with conflicting responses of other students on video; many
improved their initial responses to higher levels after exposure to the
conflicting prompts. Association among levels of representing, interpreting, and
predicting were explored. Educational implications are discussed concerning
reasonable expectations of students and suggestions to develop these skills in
students at different grades.
Watson, J.M. (2001). Longitudinal development of inferential reasoning by school
students. Educational Studies in Mathematics, 47, 337-372.
This study follows an earlier study of school students’ abilities to draw
inferences when comparing two data sets presented in graphical form (Watson &
Moritz, 1999). Forty-two students who were originally interviewed in grades 3 to
9, were subsequently interviewed either three or four years later. The results
for individual student development add to the credibility of the cross-age
observations, as well as support the hierarchical framework suggested by the
original study. Changes in levels of performance and strategies for drawing
conclusions are documented. A further step from the original study is the
consideration of how students used the variation displayed in the graphical
presentation of the data sets as a basis for decision-making. Implications for
teaching and for further research are discussed.
Chick, H.L., & Watson, J.M. (2001). Data representation and interpretation by
primary school students working in groups. Mathematics Education Research
Journal, 13, 91-111.
Twenty-seven grade 5/6 students, working in triads in a near-classroom
environment, were video-taped as they considered a supplied data set over three
45-minute sessions. They were asked to hypothesise about associations in the
data and represent these on a group poster. Each student was assigned to three
categories: one for the observed level of interpreting the information provided
in the data set, one for the observed level of representing the chosen data, and
one for the type of collaboration observed in the group. In addition, students
were asked their views on the group work. Levels of interpretation and
representation skills were related and there was some indication of a possible
association with the type of group collaboration. There was no association of
type of collaboration and students’ views on group work. Descriptive aspects of
the three characteristics—interpretation, representation, and collaboration—are
considered, as are implications for future research and for the classroom.
Watson, J.M., & Chick, H.L. (2001). Factors influencing the outcomes of
collaborative mathematics problem solving—An introduction. Mathematical Thinking
and Learning, 3(2&3), 125-173.
This study is an investigation of the factors that influence the effectiveness
of collaboration on open-ended mathematical tasks. Students in grades 3, 6 and 9
worked in groups of three on two chance and data tasks: one related to fair dice
and the other related to associations among variables presented on data cards.
The groups’ outcomes and the types of collaboration observed are investigated in
relation to issues raised in the literature. Various phenomena are identified
that influence cognitive “lifting,” “hovering,” and “falling,” that is,
improvement, no change, and reduction in levels of functioning, respectively.
These phenomena include cognitive factors, social or interpersonal factors, and
external factors.
Watson, J.M. (2001). Profiling teachers’ competence and confidence to teach
particular mathematics topics: The case of chance and data. Journal of
Mathematics Teacher Education, 4, 305-337.
This paper presents an instrument for assessing teacher achievement and teacher
need in relation to the chance and data part of the mathematics curriculum. The
development of a profiling instrument, to be used with teachers, had two main
objectives. The first was to assist in assessing teacher achievement in the
context of proposals for the adoption of professional standards for mathematics
teachers. The second was to assess professional development needs for teachers
in the light of changes to the mathematics curriculum. The background for the
development of the instrument is presented, followed by a description of the
instrument and the results of responses to it from 43 Australian teachers. Uses
for the instrument and further development possibilities are also discussed.
Watson, J.M., & Chick, H.L. (2001). Does help help?: Collaboration during
mathematical problem solving. Hiroshima Journal of Mathematics Education, 9,
33-73.
This paper considers the circumstances surrounding instances where help is
sought and/or provided in a collaborative problem-solving situation. Video-taped
observations of nine groups of three grade 5/6 students working over three
45-minute periods on an open-ended task allowed for the documentation of
questions asked, answers provided, and outcomes achieved. Help associated with
questioning was provided in two contexts: by students or teachers in response to
student questions and by teachers through questions. Hierarchical levels were
defined for questions, answers, and outcomes. It was then possible to compare
the levels of each component of this sequence for student-initiated and
teacher-initiated questions. As well, unsolicited help provided by students was
documented. The lower level of questioning by students in comparison to teachers
was influential throughout the question-answer-outcome sequence, indicating that
student soliciting of help through questioning was not as effective as
unsolicited help offered through the questioning of teachers. Some differences
associated with gender or gender-composition of groups were observed and these
are discussed in relation to the context of the study and other findings in the
literature.
Watson, J.M. (2000). Preservice mathematics teachers’ understanding of sampling:
Intuition or mathematics. Mathematics Teacher Education and Development, 2,
121-135.
This paper considers 33 preservice secondary mathematics teachers’ solutions to
a famous sampling problem, well known for confounding educated adults. Of
particular interest is the use of intuition and/or formal mathematics in
reaching a conclusion. The relationships of solution strategy to students’
background in formal mathematics and to gender are also considered. Implications
for teaching statistics at both the secondary and preservice teacher education
levels are discussed briefly.
Torok, R., & Watson, J. (2000). Development of the concept of statistical
variation: An exploratory study. Mathematics Education Research Journal, 12,
147-169.
An appreciation of variation is central to statistical thinking, but very little
research has focused directly on students’ understandings about variation. In
this exploratory study, four students from each of grades 4, 6, 8, and 10 were
interviewed individually on aspects of variation present in three settings. The
first setting was an isolated random sampling situation, whereas the other two
settings were realistic sampling situations. Four levels of responding were
identified and described in relation to developing concepts of variation.
Implications for teaching and future research on variation are considered.
Watson, J. M., & Moritz, J. B. (2000). Developing concepts of sampling. Journal
for Research in Mathematics Education, 31, 44-70.
In developing ideas associated with statistical inference, a key element
involves developing concepts of sampling. The objective of this research is to
understand the characteristics of students' constructions of the concept of
sample. Sixty-two students in grades 3, 6, and 9 were interviewed using
open-ended questions related to sampling; written responses to a questionnaire
were also analysed. Responses were characterized in relation to the content,
structure and objectives of statistical literacy. Six categories of construction
were identified and described in relation to the sophistication of developing
concepts of sampling. These categories illustrate helpful and unhelpful
foundations for an appropriate understanding of representativeness and hence
will help curriculum developers and teachers plan interventions.
Watson, J. M., & Moritz, J. B. (2000). Development of understanding of sampling
for statistical literacy. Journal of Mathematical Behavior, 19(1), 109-136.
The development of understanding sampling is explored through responses to four
items in a longitudinal survey administered to over 3000 students from grades 3
to 11. Responses are described with reference to a three-tiered framework for
statistical literacy, including defining terminology, applying concepts in
context, and questioning claims made without proper justification. Within each
tier increasing complexity is observed as students respond with single,
multiple, and integrated ideas to four different tasks. Implications for
mathematics educators of the development of sampling concepts across the years
of schooling are discussed.
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.
The development of the understanding of average was explored through interviews
with 94 students from grades 3 to 9, follow-up interviews with 22 of these
students after three years, and follow-up interviews with 21 others after four
years. Six levels of response were observed based on a hierarchical model of
cognitive functioning. The first four levels described the development of the
concept of average from colloquial ideas into procedural or conceptual
descriptions to derive a central measure of a data set. The highest two levels
represented transferring this understanding to one or more applications in
problem-solving tasks to reverse the averaging process and to evaluate a
weighted mean. Usage of ideas associated with the three standard measures of
central tendency and with representation are documented, as well as strategies
for problem solving. Implications for mathematics educators are discussed.
Watson, J. M. & Moritz, J. B. (1999). The beginning of statistical inference:
Comparing two data sets. Educational Studies in Mathematics, 37, 145-168.
The development of school students' understanding of comparing two data sets is
explored through responses of students in individual interview settings.
Eighty-eight students in Grades 3 to 9 were presented with data sets in
graphical form for comparison. Student responses were analysed according to a
developmental cycle which was repeated in two contexts: one where the numbers of
values in the data sets were the same and the other where they were different.
Strategies observed within the developmental cycles were visual, numerical, or a
combination of the two. The correctness of outcomes associated with using and
combining these strategies varied depending upon the task and the developmental
level of the response. Implications for teachers, educational planners and
researchers are discussed in relation to the beginning of statistical inference
during the school years.
Watson, J. M., & Moritz, J. B. (1999). The development of concepts of average.
Focus on Learning Problems in Mathematics, 21(4), 15-39.
The development of concepts of average is explored for students from grades 3 to
11 through four survey items administered to 2250 students. A cognitive model
describes the structural development of the concepts from single ideas into more
complex descriptions to derive a measure of average from a data set. In
association with this development, the presence of ideas related to the three
common measures of average - mean, median and mode - is monitored in different
contexts. Implications for mathematics educators of the structure of student
outcomes and the choice among the three measures are discussed.
Watson, J. M., & Moritz, J. B. (1999). Interpreting and predicting from bar
graphs. Australian Journal of Early Childhood, 24(2), 22-27.
Bar graphs are commonly used throughout society as a simple tool for
representing data. How do students in early childhood interpret bar graphs in a
familiar context? Thirty Grade 3 students were interviewed to explore their
interpretations of a bar graph of How Children Travel To School and their
predictions for various changed circumstances. Student responses to the task of
interpreting the graph reflected three levels of graph comprehension: reading
the data, reading between the data, and reading beyond the data.
Watson, J.M. (1998). Professional development for teachers of probability and
statistics: Into an era of technology. International Statistical Review., 66,
271-289
The focus of this paper is the professional development of teachers of
probability and statistics at the school level. Within a world where the
statistics curriculum is changing at the school level, the professional
development needs of teachers of statistics are changing and the technology to
meet these needs is changing. This paper reviews the work in the field,
describes the development of a multimedia package for professional development
of statistics teachers and looks to the future.
Watson, J. M., & Moritz, J. B. (1998). Longitudinal development of chance
measurement. Mathematics Education Research Journal, 10(2), 103-127
Understanding of chance measurement and how this develops over time is explored
by analysis of response data collected in 1993, 1995, and 1997. The data are
analysed to document changes in levels of observed outcomes for students in
relation to the SOLO developmental model, extending the previous work of Watson,
Collis and Moritz (1997). The chance and data curriculum was introduced in
Tasmania in 1993 and comparisons over the years indicate no improvement in
outcomes by 1997 with respect to chance measurement. Gender differences,
favouring males, are observed for selected secondary school grade levels.
Watson, J. M., Moritz, J. B., & L. Pereira-Mendoza (1998). Interpreting a graph
in a social context. The Mathematics Educator, 3(1), 61-71.
Mulligan, J. T., & Watson, J. M. (1998). A developmental multimodal model for
multiplication and division. Mathematics Education Research Journal, 10(2),
61-86
This paper presents an analysis of young students' development of multiplication
and division concepts based on a multimodal functioning SOLO model. The analysis
is drawn from two sources of data: a two-year longitudinal study of 70 grade 2
through to 3 students' solutions to 24 multiplicative word problems and
exemplars from a problem-centred teaching project with grade 3 students. An
increasingly complex range of counting, additive and multiplicative strategies
based on an equal-grouping structure demonstrated conceptual growth through
ikonic and concrete symbolic modes. The solutions employed by students to solve
any particular problem reflected the mathematical structure they imposed on it.
A SOLO developmental model for multiplication and division is described in terms
of developing "structure" and associated counting and calculation strategies.
Watson, J. M., Collis, K. F., & Moritz, J. B. (1997). The development of chance
measurement. Mathematics Education Research Journal, 9, 60-82.
This paper presents an analysis of three questionnaire items which explore
students' understanding of chance measurement 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, using the NUD·IST text
analysis software, was based on the multimodal functioning SOLO model. An
analysis of the results and a developmental model for understanding chance
measurement are presented, along with implications for curriculum and teaching
practice.
Watson, J. M., Collis, K. F., Callingham, R. A., & Moritz, J. B. (1995). A model
for assessing higher order thinking in statistics. Educational Research and
Evaluation, 1(3), 247-275.
As in other areas of the school curriculum, the teaching, learning and
assessment of higher order thinking in statistics has become an issue for
educators following the appearance of recent curriculum documents in many
countries. These documents have included probability and statistics across all
years of schooling and have stressed the importance of higher order thinking
across all areas of the mathematics curriculum. This paper reports on a pilot
project which applied the theoretical framework for cognitive development
devised by Biggs and Collis to a higher order task in data handling in order to
provide a model of student levels of response. The model will assist teachers,
curriculum planners and other researchers interested in increasing levels of
performance on more complex tasks. An interview protocol based on a set of 16
data cards was developed, trialed with Grade 6 and 9 students, and adapted for
group work with two classes of Grade 6 students. The levels and types of
cognitive functioning associated with the outcomes achieved by students
completing the task in the two contexts will be discussed, as will the
implications for classroom teaching and for further research.
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