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Questions used in Watson et al. (2003)
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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 abou...
Contexts in source publication
Context 1
... measure performance on statistical vocabulary, the current study used variations on two of the items described earlier, on random and sample, and replaced the item on average with one on variation, as reported in Watson, Kelly, Callingham & Shaughnessy (2003). The questions are shown in Fig. 1. In each case, a direct approach of asking for the meaning of the term was used. Examples then were requested and for variation, students were asked to put the word in a sentence. This approach was used to give the greatest opportunity for students to display their personal concept understanding and reduce the non-response ...
Context 2
... Level 4, Critical Aspects of Variation, items required employing of complex justification or critical reasoning. Although this analysis was based on a total of 44 individual partial credit items, some as parts of larger tasks, the items used in the current study contributed 9 items, 3 based on the codes 1 to 3, for each of the three questions in Fig. 1 (see Table I). In the case of these three questions, the analysis and determination of thresholds for the levels placed the Code 1 category of response for each question at Level 2, the Code 2 category at Level 3 for each question, and the Code 3 category at Level 4 for each ...
Context 3
... 2A, Sample 2B, and Sample 3. The students in Sample 1 were in grades 3, 5, 7, and 9 from 10 government schools in the Australian state of Tasmania. A subset of the students from Sample 1 who had completed specialized lessons focusing on variation in the chance and data curriculum and who had also completed a post-test including the questions in Fig. 1 made up Sample 2A. A further subset, Sample 2B was again re-tested 2 years later with the same survey. Sample 3 was a disjoint subset of students from Sample 1, matched by school for socio-economic status. Students from Sample 3 were re-tested 2 years later with the same survey but did not complete the specialized lessons. Students in ...
Context 4
... research team and classroom teacher administered the questions in Fig. 1 as part of a larger survey ( Watson et al., 2003). Surveys were competed by the students in class time and the entire survey took Sample 1 176 183 186 193 738 Sample 2A 72 82 91 90 335 Sample 2B 47 57 67 28 132 Sample 3 67 44 67 31 209 approximately 45 minutes. For the students in Sample 2A who experienced specialized lessons on chance ...
Context 5
... questions in Fig. 1 were analyzed using the four-step coding scheme in Table I. The responses from Sample 1 are used to discuss the distribution of levels of responses across grades for the three questions in Fig. 1. F-and t-tests were performed for the cohorts in different grades in Sample 1. Paired t-tests were used to indicate whether there was an ...
Context 6
... questions in Fig. 1 were analyzed using the four-step coding scheme in Table I. The responses from Sample 1 are used to discuss the distribution of levels of responses across grades for the three questions in Fig. 1. F-and t-tests were performed for the cohorts in different grades in Sample 1. Paired t-tests were used to indicate whether there was an increase in performance for Sample 2A after the series of specialized lessons on chance and data and for Sample 2B after a further 2 years. Paired t-tests were also used to gauge the rate of ...
Context 7
... sample of students who participated in this study is described as Sample 1, Sample 2A, Sample 2B, and Sample 3. The students in Sample 1 were in grades 3, 5, 7, and 9 from 10 government schools in the Australian state of Tasmania. A subset of the students from Sample 1 who had completed specialized lessons focusing on variation in the chance and data curriculum and who had also completed a post-test including the questions in Fig. 1 made up Sample 2A. A further subset, Sample 2B was again re-tested 2 years later with the same survey. Sample 3 was a disjoint subset of students from Sample 1, matched by school for socio-economic status. Students from Sample 3 were re-tested 2 years later with the same survey but did not complete the specialized lessons. Students in Samples 2B and Sample 3 were in grades 5, 7, 9, and 11 when they were re-tested after 2 years. The numbers of students in each sample and grade are shown in Table ...
Context 8
... research team and classroom teacher administered the questions in Fig. 1 as part of a larger survey ( Watson et al., 2003). Surveys were competed by the students in class time and the entire survey took Sample 1 176 183 186 193 738 Sample 2A 72 82 91 90 335 Sample 2B 47 57 67 28 132 Sample 3 67 44 67 31 209 approximately 45 minutes. For the students in Sample 2A who experienced specialized lessons on chance and data, students in grades 3 and 5 were taught ten lessons of work focusing on variation by an experienced teacher supplied by the research team. Details of these lessons are outlined in Watson & Kelly (2002a). Students in grades 7 and 9 were taught a unit of work that focused on variation by their usual mathematics teacher. The unit of work was devised and supplied by the research team. Details are given in Watson & Kelly (2002b). For the longitudinal follow-up that occurred 2 years later, there was some difficulty in locating students who were originally in grade 9 and had then changed schools for grade 11. As a result, only 26% of grade 9 students were ...
Context 9
... questions in Fig. 1 were analyzed using the four-step coding scheme in Table I. The responses from Sample 1 are used to discuss the distribution of levels of responses across grades for the three questions in Fig. 1. F-and t-tests were performed for the cohorts in different grades in Sample 1. Paired t-tests were used to indicate whether there was an increase in performance for Sample 2A after the series of specialized lessons on chance and data and for Sample 2B after a further 2 years. Paired t-tests were also used to gauge the rate of improvement for students in Sample 3 who did not receive specialized lessons. Difference scores were compared for individual grades across Samples 2B and 3 with t-tests to determine if there were differences depending on exposure to instruction. Using Cohen_s (1969) methodology and descriptors, the effect sizes of the differences for each test were determined. Table III shows the percent of students responding at each code for each grade for each of the questions. As can be seen there was an increase in performance with grade for BSample.^ More students in grades 7 and 9 responded at Code 3 than did students in grades 3 and 5. With the exception of students in grade 3, there was not very much change in performance across grades at Code 1 but there was an improvement from grade 7 to 9 at Code 2. For BRandom^ and BVariation^ there was little change in performance between grade 7 and 9. Grade 9 students performed marginally better on BVariation^ than grade 7 students. Table IV shows the mean code and standard error for each grade for each of the questions. There was a significant difference among grades on BSample^ (F = 2.62, p G .001), however t-tests showed a difference between grades 3 and 5, and 7 and 9, but no significant difference between grades 5 and 7. Of the significant differences, the greatest difference was between grades 3 and 5 with a medium effect size (0.70), with the difference between grades 7 and 9 representing a small effect (0.17). There was no significant difference between grades 7 and 9 on BRandom^ or BVariation.^ Results of t-tests between grades are shown in Table ...
Context 10
... questions in Fig. 1 were analyzed using the four-step coding scheme in Table I. The responses from Sample 1 are used to discuss the distribution of levels of responses across grades for the three questions in Fig. 1. F-and t-tests were performed for the cohorts in different grades in Sample 1. Paired t-tests were used to indicate whether there was an increase in performance for Sample 2A after the series of specialized lessons on chance and data and for Sample 2B after a further 2 years. Paired t-tests were also used to gauge the rate of improvement for students in Sample 3 who did not receive specialized lessons. Difference scores were compared for individual grades across Samples 2B and 3 with t-tests to determine if there were differences depending on exposure to instruction. Using Cohen_s (1969) methodology and descriptors, the effect sizes of the differences for each test were determined. Table III shows the percent of students responding at each code for each grade for each of the questions. As can be seen there was an increase in performance with grade for BSample.^ More students in grades 7 and 9 responded at Code 3 than did students in grades 3 and 5. With the exception of students in grade 3, there was not very much change in performance across grades at Code 1 but there was an improvement from grade 7 to 9 at Code 2. For BRandom^ and BVariation^ there was little change in performance between grade 7 and 9. Grade 9 students performed marginally better on BVariation^ than grade 7 students. Table IV shows the mean code and standard error for each grade for each of the questions. There was a significant difference among grades on BSample^ (F = 2.62, p G .001), however t-tests showed a difference between grades 3 and 5, and 7 and 9, but no significant difference between grades 5 and 7. Of the significant differences, the greatest difference was between grades 3 and 5 with a medium effect size (0.70), with the difference between grades 7 and 9 representing a small effect (0.17). There was no significant difference between grades 7 and 9 on BRandom^ or BVariation.^ Results of t-tests between grades are shown in Table ...
Context 11
... measure performance on statistical vocabulary, the current study used variations on two of the items described earlier, on random and sample, and replaced the item on average with one on variation, as reported in Watson, Kelly, Callingham & Shaughnessy (2003). The questions are shown in Fig. 1. In each case, a direct approach of asking for the meaning of the term was used. Examples then were requested and for variation, students were asked to put the word in a sentence. This approach was used to give the greatest opportunity for students to display their personal concept understanding and reduce the non-response ...
Context 12
... analysis of Watson et al. (2003) produced a four-step coding scheme for each item based on two aspects of the overall responses to the parts of each question: the structural complexity and the statistical appropriateness. The complexity was judged in relation to a develop- mental model (Biggs & Collis, 1982) where elements related to the task were noted as they were employed. A code of 0 was given for tautological responses or idiosyncratic responses that did not show any appreciation of meaning. A code of 1 was given for responses reflecting a single idea related to the term or an example. For sample it was likely to reflect Ba test^ or Ba bit,^ whereas for random and variation it was more likely to be reflected in a valid example. Code 2 responses 1a) What does "sample" mean? 1b) Give an example of a "sample". 2a) What does "random" mean? 2b) Give an example of something that happens in a "random" way. 3a) What does "variation" mean? 3b) Use the word "variation" in a sentence. 3c) Give an example of something that "varies". Watson et al. (2003) reflected a straightforward but partial explanation of the term and an example. For sample this included Ba part of a whole,^ whereas for random and variation there was an acknowledgment of a process or change with a valid example. Code 3 responses gave more complete explanations, with nuances related to the examples given. Illustrations of responses for the codes for each term are given in Table I. Table I provides the overall percent of responses in each code for each question for the Watson et al. (2003) study. Watson et al. (2003) also performed a partial credit Rasch analysis on the overall scale, which measured students_ understanding of chance and data with an emphasis on variation. This analysis produced a variable map that estimated student abilities and item difficulties on the same scale. The underlying construct, representing understanding of variation within a chance and data context, was then considered in a developmental sense based on the expectations of the items and their placement along the scale. Four levels of increasing understanding were identified. At Level 1, Prerequisites for Variation, items upon which students were likely to be successful were associated with working out the environment of the question, reading simple graphs and tables, and reasoning intuitively about chance. At Level 2, Partial Recognition of Variation, items were associated with putting ideas in context, focusing on some single aspects of tasks while neglecting others. At Level 3, Applications of Variation, items were associated with consolidating and using ideas in context but being inconsistent in picking the most salient features. At Level 4, Critical Aspects of Variation, items required employing of complex justification or critical reasoning. Although this analysis was based on a total of 44 individual partial credit items, some as parts of larger tasks, the items used in the current study contributed 9 items, 3 based on the codes 1 to 3, for each of the three questions in Fig. 1 (see Table I). In the case of these three questions, the analysis and determination of thresholds for the levels placed the Code 1 category of response for each question at Level 2, the Code 2 category at Level 3 for each question, and the Code 3 category at Level 4 for each ...
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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 t...
Citations
... These key concepts, including the understanding of the effect of sample size and the idea that a sample characteristic-such as mean or median-can be used to compare distributions, are essential for understanding inferences (Bakker, 2004;Chance et al., 2004;Konold & Pollatsek, 2002;Saldanha & Thompson, 2002;Watson & Kelly, 2008). There is a strong relationship between these concepts: understanding the sampling distribution relies on understanding the key concept of a sample, in particular on understanding the balance between sample representativeness and sample variability (Batanero et al., 1994). ...
... This transition was reflected in students' worksheets when they referred to similar previous technical actions and in students' terminology that evolved from concrete terms to more abstract statistical terminology, for example from the term "viewing window" to "sample." The development of a statistical vocabulary is essential for students' understanding of concepts (Watson & Kelly, 2008). ...
This paper comprises the results of a design study that aims at developing a theoretically and empirically based learning trajectory on statistical inference for 9th-grade students. Based on theories of informal statistical inference, an 8-step learning trajectory was designed. The trajectory consisted of two similar four step sequences: (1) experimenting with a physical black box, (2) visualizing distributions, (3) examining sampling distributions using simulation software, and (4) interpreting sampling distributions to make inferences in real -life contexts. Sequence I included only categorical data and Sequence II regarded numerical data. The learning trajectory was implemented in an intervention among 267 students. To examine the effects of the trajectory on students’ understanding of statistical inference, we analyzed their posttest results after the intervention. To investigate how the stepwise trajectory fostered the learning process, students’ worksheets during each learning step were analyzed. The posttest results showed that students who followed the learning trajectory scored significantly higher on statistical inference and on concepts related to each step than students of a comparison group ( n = 217) who followed the regular curriculum. Worksheet analysis demonstrated that the 8-step trajectory was beneficial to students’ learning processes. We conclude that ideas of repeated sampling with a black box and statistical modeling seem fruitful for introducing statistical inference. Both ideas invite more advanced follow-up activities, such as hypothesis testing and comparing groups. This suggests that statistics curricula with a descriptive focus can be transformed to a more inferential focus, to anticipate on subsequent steps in students’ statistics education.
... A large body of the statistics literacy generally examined the development of students' statistical knowledge (Callingham and Watson, 2004), revealing statistical literacy level of the adults (DelMas, 2002;Schield, 2006) or defining how to use the statistical knowledge in real world (Ben-Zvi and Garfield, 2004;Martinez-Dowson, 2010;Rumsey, 2002;Schield, 2006;Watson and Kelly, 2008). Some studies also suggested some new ways for enhancing statistical knowledge in young ages. ...
This study attempts to determine whether gamification can be used as a pedagogical technique to overcome the challenges in teaching statistics. A post-test quasi-experimental design was carried out in gamified and non-gamified groups in order to reveal the effect of gamification elements in cultivating students’ statistical literacy skills. Students in gamified group were also interviewed to understand the function of gamification process. The results suggest that; although gamifying the instructional process had a positive impact on developing students’ statistical literacy in medium and high score students; surprisingly the influence of the gamification to the low- achieved scores were not positive. The positive impact was discussed in accordance with the gradual structure of statistical literacy and suggestions for successful gamification applications due to the context were included.
... The results of this study showed below average understanding of the university students about symbols and terminologies used in statistics for distinct purposes. Watson and Kelly, (2008) also probed low achievements of the students in this respect. Students exhibited low performance in central tendencies while the studies of (Koleza, & Kontogianni, 2013;Mokros & Russell, 1995) showed better performance of the students. ...
The main aim of universities is to conduct research studies on different aspects of human life for the purpose of development and prosperity. In this regard the importance of statistical literacy cannot be denied in every discipline of knowledge. Therefore, the main objective of this study was to investigate the statistical literacy of the university students at BS level. Quantitative survey research design was employed where sample of the study comprised of randomly selected 360 BS level students from 9 public sector universities of Khyber Pakhtunkhwa (Pakistan). The data were collected through self-developed instrument namely Instrument for Statistical Literacy (ISL). The study demonstrated that students had low statistical literacy at BS level; it was therefore, recommended that Higher Education Commission of Pakistan may include a significant portion of practical work in the approved curriculum at university level for the enhancement of statistical literacy.
... There are various studies in the related literature regarding the concepts of average and variation. Some of them revealed that students experience difficulties in defining the concepts of average and variation (Watson & Moritz, 2000;Watson & Kelly, 2008), and in using the concepts of average and variation when comparing two data sets (Mc Gatha et al., 2002;Watson & Moritz, 1998). Other studies focused on the role of different contexts or graphical representations in students' interpretation of the concepts of average and variation (Mc Gatha et al., 2002;Shaugnessy & Pfannkuch, 2002;Watson et al., 2014). ...
The aim of the current study was to investigate the statistical literacy of seventh grade students regarding the concepts of average and variation on bar and line graphs presented in alternative real-life contexts. To this end, the statistical literacy levels of the seventh grade students were initially determined. Subsequently, definitions, interpretations and evaluations generated by the students at varying levels of statistical literacy were examined in further depth. More specifically, a qualitative survey study was conducted with 164 seventh graders from two public middle schools in Ankara, Turkey. The data, collected by means of a-five-item instrument, were analyzed by utilizing the modified version of the six-hierarchical-level framework developed by Watson and Callingham (Statistics Education Research Journal, 2(2), 3–46, 2003). The findings revealed that most of the students’ statistical literacy levels were identified as either Level 3 or Level 4 in the framework with respect to the concept of average, and either Level 1 or Level 2 with respect to the concept of variation. Hence, it can be inferred that while many of the students could interpret average in the given contexts, they had difficulty in interpreting variation. Although definitions and evaluations related to both concepts were found difficult by seventh graders, the graphs, particularly bar graphs, which students were provided with, were observed to be helpful for students in their evaluation of variation. Additionally, students benefitted from some of the contexts they were provided while evaluating variation. Finally, the analysis revealed some discrepancies among students’ definitions and their interpretations or evaluations.
... Watson y Kelly [65] indican que la relación entre el lenguaje y la comprensión del sujeto (en este caso la estadística) es compleja. Como los autores mencionan, hasta hace relativamente poco tiempo, la medición de la comprensión estadística se ha centrado en procedimientos con respuestas numéricas y no en descripciones de terminología, debido principalmente a que los profesionales que analizan la alfabetización estadística, tienen tendencia a centrarse más en aspectos procedimentales, como encontrar probabilidades de eventos, dibujar gráficas o calcular promedios, que en la alfabetización. ...
... Es por ello que se plantea necesario una formación más específica para los profesores en este campo, ya que como demuestran algunas investigaciones (ver [41,42]), algunos 156 E. Molina-Portillo, J. M. Contreras, J. D. Godino, F. Ruz profesores de matemáticas no siempre usan una terminología correcta en su lenguaje de instrucción, y en su lugar utilizan el lenguaje cotidiano. Como inciden Watson y Kelly [65] algunos profesores pueden incluso sentirse incómodos con la terminología estadística y, por lo tanto, evitar el uso de estos términos en la instrucción para evitar tener que explicar a la clase el significado de los conceptos formales subyacentes. Esto puede provocar una problemática ya que algunos términos tienen acepciones diferentes en su uso diario y su uso especializado (ver [65]). ...
... Como inciden Watson y Kelly [65] algunos profesores pueden incluso sentirse incómodos con la terminología estadística y, por lo tanto, evitar el uso de estos términos en la instrucción para evitar tener que explicar a la clase el significado de los conceptos formales subyacentes. Esto puede provocar una problemática ya que algunos términos tienen acepciones diferentes en su uso diario y su uso especializado (ver [65]). ...
Opinion mining and sentiment analysis are two topics with growing interest in artificial intelligence. The latest research in these areas has evolved in increasing complexity and sophistication using text mining and natural language processing techniques. Twitter is a popular social networking platform where users post and read messages. These messages, known as “tweets”, can convey opinions about a range of topics and therefore may be subject to sentiment analysis tasks. Possible strategies go from concept-based approaches and data mining, to the crudest keyword spotting methods. This paper describes the building of an average general sentiment index using Spanish tweets. For this purpose, an existing emotional lexicon for Spanish words has been tailored to assign sentiment polarity to texts. Some results and methods to evaluate the quality of the resulting index are shown later.
... It indicates that the statistical literacy of students is low. A description of the students`with low statistical literacy can also be found in the study [4,5,6,7] which stated "children produced fewer idiosyncratic descriptions of data, possessed intuitive knowledge of center and spread and were constrained in analysis and interpretation by knowledge of data context". Taking into account the previous explanation, the purpose of this reseach is to describe the statistical literacy of high school students in reading, interpreting and presenting data as part of the basic ability that must be mastered by every student. ...
One of the foundations for high school students in statistics is to be able to read data; presents data in the form of tables and diagrams and its interpretation. The purpose of this study is to describe high school students' competencies in reading, interpreting and presenting data. Subjects were consisted of male and female students who had high levels of mathematical ability. Collecting data was done in form of task formulation which is analyzed by reducing, presenting and verifying data. Results showed that the students read the data based on explicit explanations on the diagram, such as explaining the points in the diagram as the relation between the x and y axis and determining the simple trend of a graph, including the maximum and minimum point. In interpreting and summarizing the data, both subjects pay attention to general data trends and use them to predict increases or decreases in data. The male estimates the value of the (n+1) of weight data by using the modus of the data, while the females estimate the weigth by using the average. The male tend to do not consider the characteristics of the data, while the female more carefully consider the characteristics of data.
... Several studies have been conducted with statistical vocabulary as main or side attention. Watson and Kelly (2008) conducted a study to students in grades 3, 5, 7, and 9 about their ability to define three fundamental statistical terms: sampe, random, and variation. (Pfannkuch, 2011) did a study on informal inferential reasoning in which the teacher will introduce the vocabulary in the beginning of the class, and then write it on the board every time it comes up in discussion. ...
... Statistical literacy can be thought as a combination of statistics and literacy. The statistics part involves understanding of statistical concepts, while the literacy part is about the ability to express statistical concepts in words, as well as to communicate, discuss, and argue about it (Watson & Kelly, 2008). It is evident that subject-specific vocabulary is an inherent part in supporting the 'literacy' aspect of statistical literacy. ...
Inferential statistics has been proven to be a hard concept for students to understand.
On the other hand, as the future citizen of modern society, statistics becomes
important for the students to deal with data-laden information they encounter on
daily basis. Informal Inferential Reasoning is a current topic in international
statistics education proposed to bridge these two problems. The present study is
focused on developing 7th grade students’ informal inferential reasoning in
Indonesia. It is based on Realistic Mathematics Education (RME) also known as
Pendidikan Matematika Realistik Indonesia (PMRI). The goal is to contribute to
instructional materials in developing IIR as well as accompanying theory on how
this material works. The research question of this study is, how can 7th grade
students be supported to develop IIR? We used design research as the methodology.
We conducted two cycles of teaching experiments whose subjects were the 7th grade
the students and one teacher in Laboratory Secondary School of State University of
Surabaya. The first cycle consisted of 6 students, while the second cycle consisted
of 27 students and one teacher. The data were collected through video recordings
of the teaching experiment and students’ written works.
The data then analyzed by comparing and contrasting the Hypothetical Learning
Trajectory to the Actual Learning Trajectory. As the result of the analysis, six
lessons on developing IIR were designed; (1) Scout staff; (2) Comparing the Dots;
(3) Are girls taller than boys?; and (4) Social Media Addiction.
... The findings on sample manifest that the concept of sample is not easily understood and confused with other concepts, prejudiced sample choices are made, proper sample size cannot be determined, and there is no emphasis on this subject in the books. As a matter of fact, Watson and Kelly [18] specified that students associate the statistical terms to other concepts. Moreover, Watson [2] pointed out that it takes time to develop part-whole relationships and sample-population relationship in the early ages. ...
The purpose of this study is to define teacher views about the difficulties in learning
and teaching middle school statistics subjects. To serve this aim, a number of interviews
were conducted with 10 middle school maths teachers in 2011–2012 school year in the
province of Trabzon. Of the qualitative descriptive research methods, the semi-structured
interview technique was applied in the research. In accordance with the aim, teacher
opinions about the statistics subjects were examined and analysed. Similar responses
from the teachers were grouped and evaluated. The teachers stated that it was positive
that middle school statistics subjects were taught gradually in every grade but some
difficulties were experienced in the teaching of this subject. The findings are presented
in eight themes which are context, sample, data representation, central tendency and
dispersion measure, probability, variance, and other difficulties
... Cognitive development frameworks are useful to inform the sequencing and presentation of content. Changes to the school curriculum to facilitate the further development of statistical literacy were proposed by Watson and Kelly (2008) based on theoretical underpinnings from their longitudinal cross-grades application of the Watson et al. (2003) framework to describe school students' understanding of a number of fundamental statistical concepts including variation. They found that in Year 9, 50% of sampled students could not define variation. ...
There has been an increasing focus in recent years on theoretical frameworks to describe cognitive development of statistical concepts. There is now a need to encourage the use of these frameworks to inform practice in the teaching and learning of statistics. This paper focuses on frameworks that describe the levels of cognitive development of the concept of variation. Recent research proposing theoretical frameworks on, or referring to, reasoning about variation are synthesised. Discussion follows on the use of theoretical frameworks to inform the teaching and learning cycle for statistics courses, especially the design of curriculum, learning activities and assessment tasks. INTRODUCTION Variation is a broad construct with a critical role underpinning statistical thinking and during the past decade statistics education researchers have become more aware of the importance of variation. The use of structured models of cognitive development to develop coding schemes for analysing student responses was an early phenomenon in this focus on reasoning about variation (e.g., Watson, Kelly, Callingham & Shaughnessy, 2003). Detailed information about the cognitive structure of reasoning about variation is needed to better plan for teaching and learning in statistics courses. This paper first presents necessary components of the concept of variation and cognitive frameworks proposed to help articulate how reasoning about variation develops. Then, a variety of teaching and learning situations are described to demonstrate how these frameworks have informed practice. Finally, some implications for future teaching and learning practice are shared.
Previous research has shown that students form fragmented understandings about variability even after instruction and that these primitive notions function as obstacles to a robust understanding of variability. To investigate the effect of disruptions to students’ primitive notions of variability, we examined undergraduate students’ reasoning about variability when distributions of quantitative datasets to be compared (a) have equal ranges, (b) do not include extreme values, and (c) have approximately the same number of different values. We designed homework questions and collected student responses to these questions from a large number of students attending an introductory statistics course. We also reorganized these questions into interview tasks and conducted task-based interviews with students. The results showed that students either did not address variability in these narrowly framed situations or provided limited, ambiguous, or inconsistent responses. The findings suggest that students typically think about variability in terms of range’s colloquial meaning and how different the values are from each other. Students’ reasoning was often contingent upon the particular and more prevalent characteristics of the distributions on which they were asked to work. Future work should explicate ways to utilize students’ available conceptions of variability in teaching the concept’s normative meaning.
