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Proportional Reasoning: Lessons from Research in Data and Chance
Abstract
PRINCIPLES AND STANDARDS FOR SCHOOL Mathematics (NCTM 2000) places proportionality among the major concepts connecting different topics in the mathematics curriculum at the middle school level (p. 217). What concerns us about many of the problems presented to students, however, is that they are often posed purely as a ratio or proportion from the start. Often the statement of a problem is a giveaway that a proportion is involved. For example, the question “If 15 students out of 20 get a problem correct, how many students in a class of 28 would we expect to get the problem correct?” does not tap the depth of proportional reasoning that is required for meaningful problem solving.
... Variability is not only present in the data, but occurs from one entire sample to another. Research on reasoning in repeated sampling situations mainly developed around tasks where repeated samples were taken from a known mix of differently coloured candies (sweets) in a bowl (Kelly & Watson, 2002;Reading & Shaughnessy, 2000;Shaughnessy, Watson, Moritz & Reading, 1999;Watson & Shaughnessy, 2004). An example of a task used to explore the way learners reason about the variability of data in a sampling context, is the Lollie Task. ...
... More than half of the teachers (57%) however answered that they expected six red candies to be drawn (Table 1). This result concurs with other research findings on learner and pre-service teachers' reasoning about variability in the Candy Bowl Task Watson & Shaughnessy, 2004). A probable reason for this focus on centre and not possible variability could be that the teachers' experiences with and understanding of theoretical probability in their mathematics and statistics education or in professional development fostered limiting constructions and hindered their understanding of variability in a sampling context . ...
... Conversely, all teachers using proportional and distributional reasoning suggested a reasonable range for the number of reds pulled from the mix. This result concurs with the research of Watson and Shaughnessy (2004) who point out that in their research participants using explicit proportional reasoning 'were more likely to suggest a reasonable amount of variation around the expected mean of the samples' (p. 108). ...
Copyright: © 2013. The Authors. Licensee: AOSIS OpenJournals. This work is licensed under the Creative Commons Attribution License. The concepts of variability and uncertainty are regarded as cornerstones in statistics. Proportional reasoning plays an important connecting role in reasoning about variability and therefore teachers need to develop students' statistical reasoning skills about variability, including intuitions for the outcomes of repeated sampling situations. Many teachers however lack the necessary knowledge and skills themselves and need to be exposed to hands-on activities to develop their reasoning skills about variability in a sampling environment. The research reported in this article aimed to determine and develop teachers' understanding of variability in a repeated sampling context. The research forms part of a larger project that profiled Grade 8–12 teachers' statistical content and pedagogical content knowledge. As part of this larger research project 14 high school teachers from eight culturally diverse urban schools attended a series of professional development workshops in statistics and completed a number of tasks to determine and develop their understanding of variability in a repeated sampling context. The Candy Bowl Task was used to probe teachers' notions of variability in such a context. Teachers' reasoning mainly revealed different types of thinking based on absolute frequencies, relative frequencies and on expectations of proportion and spread. Only one response showed distributional reasoning involving reasoning about centres as well as the variation around the centres. The conclusion was that a greater emphasis on variability and repeated sampling is necessary in statistics education in South African schools. To this end teachers should be supported to develop their own and learners' statistical reasoning skills in order to help prepare them adequately for citizenship in a knowledge-driven society.
... Children participating in these studies had not previously studied probability and revealed a strong relationship between success in probability comparison and proportional reasoning. Moreover, proportional reasoning is considered to be linked to the acquisition of probability reasoning (Begolli et al., 2021;Bryant & Nunez, 2012;Watson & Shaughnessy, 2004). ...
This research aimed to relate Costa Rican students (11-16-year-olds) competence to compare probabilities in spinners and proportional reasoning in the comparison of ratios. We gave one of two questionnaires to a sample of 292 students (grade 6 to grade 10) with three probability comparison and three ratio comparison problems each. Globally both questionnaires cover six different proportional reasoning levels for each type of problem. Additionally, each questionnaire contains two comparison probabilities items intended to discover a specific bias. We analyze the percentages of correct responses to the items, strategies used to compare probabilities per school grade, and students’ probabilistic reasoning level. The results confirm more difficulty in comparing ratio than in comparing probability and suggest that the reasoning level achieved is lower than established in previous research. The main bias in the students’ responses was to consider the physical distribution of colored sectors in the spinners. Equiprobability and outcome approach were very scarce.
... , Pulos et Stage, 1983; Lamon, 1995; Lo et Watanabe, 1997; Norton, 2005). Certains auteurs soulignent également l'importance du raisonnement proportionnel pour l'apprentissage de la statistique (Cobb, 1999; Hammerman et Rubin, 2004; Mary et Theis, 2007; Shaughnessy, 2006; Watson et Shaughnessy, 2004). ...
Cet article rapporte quelques résultats d’une recherche qui consiste à décrire et à comprendre la manière de raisonner des élèves du troisième cycle du primaire lors de la résolution de problèmes statistiques et de proportionnalité. À travers l’analyse des cas de deux équipes d’élèves, nous verrons comment le contexte statistique peut influencer leur façon de raisonner en comparaison avec des contextes non statistiques. Les résultats montrent entre autres que les élèves ne fournissent pas une même interprétation au même contexte statistique d’un problème mathématique au primaire bien que ces élèves comprennent le problème et les contenus mathématiques impliqués.
Mots-clés : didactique des mathématiques, statistique au primaire, raisonnement proportionnel, contexte statistique, résolution de problèmes mathématiques au primaire
... The second attribute consists of comparing and ordering fractions (A2). Following missing-value problems, comparison problems are considered the second most widely researched and most commonly instructed proportional reasoning problem (Clark et al. 2003;Karplus et al. 1983;Noelting 1980a, b;Watson et al. 2008;Watson and Shaughnessy 2004). Hence, it is crucial to integrate A2 into the list of attributes. ...
In this paper, we discuss the process of identifying and validating students' abilities to think proportionally. More specifically, we describe the methodology we used to identify these proportional reasoning attributes, beginning with the selection and review of relevant literature on proportional reasoning. We then continue with the deliberation and resolution of differing views by mathematics researchers, mathematics educators, and middle school mathematics teachers of what should be learned theoretically and what can be taught practically in everyday classroom settings. We also present the initial development of proportional reasoning items as part of the two-phase validation process of the previously identified attributes. In particular, we detail in the first phase of the validation process our collaboration with middle school mathematics teachers in the creation of prototype items and the verification of each item-attribute specification in consideration of the most common ways (among many different ways) in which middle school students would have solved these prototype items themselves. In the second phase of the validation process, we elaborate our think-aloud interview procedure in the search for evidence of whether students generally solved the prototype items in the way they were expected to.
... Surveys conducted outside the school context often result in students seeing more clearly the reason to use samples due to the larger population and the inability to survey everyone. Carrying out sampling activities in the classroom as suggested by Watson and Shaughnessy (2004) in the context of drawing handfuls of lollies from a container with a given percentage of a certain colour, can also be useful. Students' discussion of their own methods of drawing handfuls is likely to bring out accusations of cheating or bias on the part of other students. ...
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 post-test 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.
... The terms proportional and distributional are particularly important for statistics and can have specialized meanings. Examples of the importance of proportional reasoning for handling statistical tasks are presented in Watson and Shaughnessy (2004). Ways of characterizing distributional reasoning are being developed by several groups of researchers, for example, Bakker and Gravemeijer (2004), Shaughnessy et al. (2004a), Shaughnessy, Ciancetta, Best, & Noll (2005). ...
provide an overview, interpretation, and focus of the research in learning and teaching stochastics / provide a framework to researchers who are new to this area / provide some reconceptualization of the literature for those who have already been involved in research in stochastics / point out several concerns . . . about the research in stochastics / suggest some research directions / [builds] a micro model of stochastic conceptual development (PsycINFO Database Record (c) 2012 APA, all rights reserved)
... Traditional links between proportional thinking and other parts of the mathematics curriculum have included measurement, similarity in geometry, trigonometry, and basic probability. More recently the links to chance and data have been highlighted in terms of the relationship of samples to populations (Saldanha & Thompson, 2002) and in the context of probability sampling and comparing data sets of different sizes (Watson & Shaughnessy, 2004). Further, the focus on quantitative literacy across a more eclectic school curriculum with higher level numeracy requirements (e.g., Madison & Steen, 2003) leads to specific statements of the need to apply proportional reasoning and calculations, often undertaken mentally, based on fractions, decimals and percents in many social and life-skills contexts (Steen, 2001; Watson, 2004). ...
In this article, data from a study of the mental computation competence of students in grades 3 to 10 are presented. Students
responded to mental computation items, presented orally, that included operations applied to fractions, decimals and percents.
The data were analysed using Rasch modelling techniques, and a six-level hierarchy of part-whole computation was identified.
This hierarchy is described in terms of the three different representations of part-whole reasoning — fraction, decimal, and
percent — and is elaborated by a consideration of the likely cognitive demands of the items. Discussion includes reasons for
the relative difficulties of the items, performance across grades and directions for future research.
... Merely repeating and reviewing tasks is unlikely to lead to improved skills or deeper understanding (e.g., Pfannkuch, 2005;Watson, 2004;Watson & Shaughnessy, 2004). ...
This paper provides an overview of current research on teaching and learning statistics, summarizing studies that have been conducted by researchers from different disciplines and focused on students at all levels. The review is organized by general research questions addressed, and suggests what can be learned from the results of each of these questions. The implications of the research are described in terms of eight principles for learning statistics from Garfield (1995) which are revisited in the light of results from current studies.
Cet article présente une vue d'ensemble de la recherche actuelle sur l'enseignement et l'étude de la statistique. Il résume les analyses qui ont été menées par des chercheurs de disciplines différentes. Le travail s'organise autour de questions de recherche générales qui ont été adressées, et suggère ce que l'on peut apprendre à partir de ces résultats au sujet de chacune de ces questions. Les implications de la recherche sont décrites en termes de huit principes sur l'étude de la statistique de Garfield (1995) qui sont repris selon les résultats d'études actuelles.
Tomar decisiones es un acto cotidiano en el ser humano, a mayor incertidumbre más difícil es decidir. A partir de una situación de aprendizaje, consistente en decidir entre dos juegos aleatorios con dados, se estudia la relación entre el pensamiento proporcional y el pensamiento probabilístico, considerando tres estados para el pensamiento proporcional y tres tipos de pensamiento probabilístico. bajo el enfoque de un estudio de casos instrumental, se analizan las decisiones y argumentos de estudiantes de secundaria chilenos. los resultados indican que existen relaciones tanto beneficiosas como perjudiciales entre el pensamiento proporcional y el probabilístico, y que las dificultades en la determinación de probabilidades no necesariamente obedecen a la ausencia del uso de proporciones. Se recomienda una enseñanza que considere la argumentación y el aprendizaje del espacio muestral para encauzar el uso de recursos intuitivos.
Making decisions is a daily act in the human being, to greater uncertainty more difficult it is to decide. From a learning situation, consisting of deciding between two random games with dices, the relationship between proportional thinking and probabilistic thinking is studied, considering three states for proportional thinking and three types of probabilistic thinking. under the focus of an instrumental case study, the decisions and arguments of chilean high school students are analyzed. the results indicate that there are both beneficial and harmful relationships between proportional and probabilistic thinking, and that the difficulties in determining probabilities are not necessarily due to the absence of the use of proportions. A teaching that considers argumentation and learning of the sample space is recommended to channel the use of intuitive resources. PAlAbRAS C
CLAVE:-Pensamiento probabilístico-Pensamiento proporcional-Incertidumbre-Heurística-Toma de decisiones
Many decisions in politics, economics, and society are based on data and statistics. In order to participate as a responsible citizen, it is essential to have a solid grounding in reasoning about data. Reasoning about data is a fundamental human activity; its components can be found in nearly every profession and in most school curricula in the world. This chapter reviews past and recent research on reasoning about data across all ages of learners from primary school to adults. Specifically in this chapter, the term reasoning about data is defined, the implementation of reasoning about data in the curricula of different countries is investigated, and research studies of learner reasoning about distribution, variation, comparing groups, and association, which are fundamental concepts when reasoning about data, are reviewed. The research review presented includes references to existing frameworks and taxonomies that can assess learner reasoning in regard to these concepts and discusses the influence of digital tools to enhance learner statistical reasoning. Finally, some insights for future directions in research about reasoning about data are provided.
Risk is everywhere yet the concept of risk is seldom investigated in high school mathematics. After presenting arguments for teaching risk in the context of high school mathematics, the article describes a case study of teaching risk in two grade 11 classes in Canada-an all-boy independent school (23 boys) and a publicly funded religious school (19 girls and 4 boys). The findings suggest that the students possessed intuitive knowledge that risk of an event should be assessed by both its likelihood and its impact. Following and amending pedagogic model of risk (Levinson, 2012), the study suggests that pedagogy of risk should include five components: 1) knowledge, beliefs, and values, 2) judgment of impact, 3) judgment of probability, 4) representations, and 5) estimation of risk. These components do not necessarily appear in the instruction or students' decision making in chronological order,; furthermore, they influence each other. The implication for mathematics education is that a meaningful instruction about risk should go beyond mathematical representations and reasoning and include other components of the pedagogy of risk. The article also illustrates the importance of reasoning about rational numbers (rates, ratios, and fractions) and their critical interpretation in the pedagogy of risk. Keywords: probability teaching and learning, risk literacy, teaching and learning of risk, inquiry-based learning, understanding of rational numbers; risk estimation.
Matematika merupakan salah satu mata pelajaran yang menantang pemahaman siswa; pokok bahasan peluang dalam topik statistik merupakan hal yang bermanfaat dan mempunyai aplikasi ke berbagai disiplin ilmu. Namun, pokok bahasan ini termasuk yang kurang dipahami oleh siswa. Penelitian ini bertujuan untuk mengkaji pemahaman siswa dalam menyelesaikan masalah peluang (probabilitas) dan untuk mengetahui masalah yang dihadapi oleh siswa dalam menyelesaikan masalah peluang. Untuk tujuan tersebut, satu studi kualitatif dengan pendekatan studi kasus telah dilakukan dengan jumlah sampel secara purposif sebanyak delapan orang pada siswa kelas sepuluh di sekolah menengah negeri di daerah Johor Bahru, Malaysia. Data penelitian pada tahap awal didapat dengan memberikan ujian diagnostik kepada sampel, kemudian empat orang diantaranya dipilih untuk di wawancara secara mendalam mengenai jawaban yang diberikan. Wawancara difokuskan kepada pola pemahaman siswa dalam menyelesaikan masalah peluang dan masalah yang di hadapi oleh mereka dalam menyelesaikan masalah peluang tersebut. Data kualitatif kemudian dianalisis menggunakan kaedah analisis tematik. Hasil studi mendapati beberapa tema dan pola yang menjelaskan tentang pemahaman siswa dalam menyelesaikan masalah peluang yaitu miskonsepsi, penguasaan konsep yang lemah dan tidak memahami kalimat dan istilah.
Kata kunci: miskonsepsi siswa, pendidikan matematika, pembelajaran peluang, sekolah menengah Malaysia
Teachers of grades Pre-K-8 are charged with the responsibility of developing children's statistical thinking. Hence, strategies are needed to foster statistical knowledge for teaching (SKT). This report describes how writing prompts were used as an integral part of a semester-long undergraduate course focused on building SKT. Writing prompts were designed to help assess and develop the subject matter knowledge and pedagogical content knowledge of prospective teachers. The methods used to design the prompts are described. Responses to a sample prompt are provided to illustrate how the writing prompts served as tools for formative assessment. Pretests and posttests indicated that prospective teachers developed both SKT and knowledge of introductory college-level statistics during the course. It is suggested that teacher educators employ and refine the prompts in their own courses, as the method used for writing and assessing the prompts is applicable to a broad range of statistics and mathematics courses for teachers.
The Oregon Mathematics Leadership Institute (OMLI) project served 180 Oregon teachers, and 90 administrators, across the K-12 grades from ten partner districts. OMLI offered a residential, three-week summer institute. Over the course of three consecutive summers, teachers were immersed in a total of six mathematics content classes– Algebra, Data & Chance, Discrete Mathematics, Geometry, Measurement & Change, and Number & Operations—along with an annual collegial leadership course. Each content class was designed and taught by a team of expert faculty from universities, community colleges, and K-12 districts. Each team chose a few “big ideas” on which to focus the course. For example, the Algebra team focused on algebraic structure and properties of the concept of a group, while the Data & Chance team centered their activities on the exploration of ideas of central tendency and variation using statistics and data analysis software packages. The content in all of the courses was addressed through deep investigation of the mathematics of tasks that had been selected and adapted from resources for K-12 mathematics classrooms. In addition to mathematics content, the courses were designed with specific attention to socio-mathematical norms, issues of status differences among learners, and the selection and implementation of group-worthy tasks for group work. The faculty attended sessions grounded in the work of Elizabeth Cohen on strategies for working with heterogeneous groups of learners (Cohen, 1994; Cohen et al, 1999) which was central to the OMLI design and implementation. Institute faculty modeled these strategies in the Institute classrooms and made their moves as transparent as possible, so that the teachers would be able to grapple with these strategies during the Institute and plan for implementation in their own classrooms. The Data & Chance course also modeled uses of technology in instruction using Tinkerplots. Generalization and justification were emphasized as mathematical ways of learning and knowing, and institute faculty conducted classroom discussions that intentionally modeled pushing for generalization and justification.
This study presents the results of a partial credit Rasch analysis of in-depth interview data exploring statistical understanding of 73 school students in 6 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.
An appreciation of variation is central to statistical thinking, but very little research has focused directly on students’
understanding of 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 real world 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.
This study follows an earlier study of school students' abilities to draw inferences when comparing two data sets presented
in graphical form (Watson and 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.
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.
School Mathematics Students' Acknowledgment of Statistical Variation
- Michael Shaughnessy
- Jane Watson
- Jonathan Moritz
- Chris Reading
Shaughnessy, Michael, Jane Watson, Jonathan Moritz,
and Chris Reading. "School Mathematics Students'
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of Teachers of Mathematics, San Francisco, California,
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Data and Chance In Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress
- Judy Zawojewski
- Michael Shaughnessy
Zawojewski, Judy, and Michael Shaughnessy. " Data and
Chance. " In Results from the Seventh Mathematics Assessment of the National Assessment of Educational
Progress, edited by Edward Silver and Pat Kenney, pp.
235–68. Reston, Va.: National Council of Teachers of
Mathematics, 2000.
In Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress
- Judy Zawojewski
- Michael Shaughnessy
Zawojewski, Judy, and Michael Shaughnessy. "Data and
Chance." In Results from the Seventh Mathematics Assessment of the National Assessment of Educational
Progress, edited by Edward Silver and Pat Kenney, pp.
235-68. Reston, Va.: National Council of Teachers of
Mathematics, 2000.