Statistical modeling to promote students’ aggregate reasoning with sample and sampling

While aggregate reasoning is a core aspect of statistical reasoning, its development is a key challenge in statistics education. In this study we examine how students’ aggregate reasoning with samples and sampling (ARWSS) can emerge in the context of statistical modeling activities of real phenomena. We present a case study on the emergent ARWSS of two pairs of sixth graders (age 11–12) involved in statistical data analysis and informal inference utilizing TinkerPlots. The students’ growing understandings of various statistical concepts is described and five perceptions the students expressed are identified. We discuss the contribution of modeling to these progressions followed by conclusions and limitations of these results. While idiosyncratic, the insights contribute to the understanding of students’ aggregate reasoning with data and models, with regards to samples and sampling.     

Elementary preservice teachers’ reasoning about statistical modeling in a civic statistics context

Elements of statistical modeling can be implemented already in primary school. A prerequisite for this approach is that teachers are well-educated in this domain. Content knowledge, pedagogical content knowledge and (pedagogical) content related technological knowledge are core components of teacher education. We designed a course for elementary preservice teachers with regard to developing statistical thinking including the mentioned knowledge facets. The course includes exploring data and modeling and simulating chance experiments with TinkerPlots. We use the ‘data factory metaphor’ in fictive contexts and in contexts stemming from civic statistics for supporting the idea of modeling. We interviewed four participants of the course to assess and analyze their reasoning. We analyze how they model a given civic statistics contextual problem using the TinkerPlots sampler and how they evaluate their model with regard to a civic statistics context (the situation of hospitals in Germany).     

Developing prospective teachers’ covariational reasoning through a model development sequence

Forming part of a wider research study, the current study investigated prospective middle school mathematics teachers’ ways of covariational reasoning on tasks involving simultaneously changing quantities. As the introductory theme of a larger unit on derivative, a model development sequence on covariational reasoning was designed and experimented with 20 participants in a mathematical modeling course offered to prospective teachers. The participants’ developing abilities of covariational reasoning were documented under three categories: (i) identifying the variables, (ii) ways of coordinating the variables, and (iii) ways of quantifying the rate of change. The results revealed significant improvement in the prospective teachers’ ways of identifying and coordinating the variables, and in quantifying the rate of change. Moreover, the results indicated that preference for a particular way of thinking in identifying and coordinating the variables determined the prospective teachers’ way of quantifying the rate of change and thereby their level of covariational reasoning.     

The role of model comparison in young learners’ reasoning with statistical models and modeling

The goal of this study is to explore the role of model comparison, which is a key activity of young learners’ informal reasoning, with statistical models and modeling in the context of informal statistical inference. We suggest a framework to describe this reasoning (the RISM framework), and offer an illustrative case study of two-sixth graders showcasing its utility. In particular, we illustrate the benefit of untangling the informal modeling process into three separate, though not independent, modeling processes: modeling a conjecture, modeling data, and comparing them by means of a comparison model. This case study shows the possible progression of a comparison model, and its potential role as a catalyst for the development of the other two modeling processes. Finally, an expansion of our initial framework is discussed, highlighting the centrality of model comparisons.     

Examining students’ variational reasoning in differential equations

In this study, we explored how a sample of eight students used variational reasoning while discussing ordinary differential equations (DEs). Our analysis of variational reasoning draws on the literature with regard to student thinking about derivatives and rate, students’ covariational reasoning, and different multivariational structures that can exist between multiple variables. First, we found that while students can think of “derivative” as a variable in and of itself and also unpack derivative as a rate of change between two variables, the students were often able to think of “derivative” in these two ways simultaneously in the same explanation. Second, we found that students made significant usage of covariational reasoning to imagine relationships between pairs of variables in a DE, and that mental actions pertaining to recognizing dependence/independence were especially important. Third, the students also conceptualized relationships between multiple variables in a DE that matched different multivariational structures. Fourth, importantly, we identified a type of variational reasoning, which we call “feedback variation”, that may be unique to DEs because of the recursive relationship between a function’s value and its own rate of change.     

Validating a concept inventory for measuring students’ probabilistic reasoning: The case of reasoning within the context of a raffle

Assessing students’ conceptions related to independence of events and determining probabilities from a sample space has been the focus of research in probability education for over 40 years. While we know a lot from past studies about predictable ways students may reason with well-known tasks, developing a diagnostic assessment that can be used by teachers to inform instruction demands the use of familiar and unfamiliar contexts. This paper presents the current work of a research team whose aim is to create a formative concept inventory with strong evidence of validity that uses a psychometric model to confidently predict whether a student exhibits one or more misconception across many items. We illustrate this process in this paper using a particular item with a context of a raffle aimed to measure whether a student reasons with misconceptions related to independence or equiprobability. The results of two aspects of the validity process: cognitive interviews to assess response processes on individual items, and a large-scale administration to examine internal structure of the concept inventory revealed difficulties in assessing students’ reasoning about these key probability concepts and trends in the prevalence of misconceptions across grades. Results can provide guidance for others aiming to develop assessments in mathematics education and also support further possibilities for research into understanding students’ reasoning about independence and sample space.     

The parametric nature of two students’ covariational reasoning

Researchers have argued that covariational reasoning is foundational for learning a variety of mathematics topics. We extend prior research by examining two students’ covariational reasoning with attention to the extent they became consciously aware of the parametric nature of their reasoning. We first describe our theoretical background including different conceptions of covariation researchers have found useful when characterizing student reasoning. We then present two students’ activities during a teaching experiment in which they constructed and reasoned about covarying quantities. We highlight aspects of the students’ reasoning that we conjectured created an intellectual need that resulted in their constructing a parameter quantity or attribute, a need we explored in closing teaching episodes. We discuss implications of these results for perspectives on covariational reasoning, students’ understandings of graphs and parametric functions, and areas of future research.     

A framework for characterizing students’ cognitive processes related to informal best fit lines

Informal best fit lines frequently appear in school curricula. Previous research collectively illustrates that the adjective informal does not translate to cognitive simplicity. Using existing literature, we create a hypothetical framework of cognitive processes associated with studying informal best fit lines. We refine the framework using data from a cycle of design-based research about building students’ understanding of covariation. The refined framework includes student thinking processes for signifying observations as data, signifying data with scatterplots, perceiving aggregates in scatterplots, perceiving trends in aggregates, signifying trends with straight lines, and using straight lines as estimation tools. We explain how students’ perceptions of aggregates can proceed from the inside-out as well as from the outside-in. We also demonstrate how the amounts of variation encountered at different points in time and the extent to which students perceive straight lines to be abbreviations of linear covariation are important considerations for teaching and research.     

Didactical contract and responsiveness to didactical contract: a theoretical framework for enquiry into students’ creativity in mathematics

One of the manifestations of learning is the student’s ability to come up with original solutions to new problems. This ability is one of the criteria by which the teacher may assess whether the student has grasped the taught mathematics. Obviously, a teacher can never teach the ability to invent new solutions (at least not directly): he/she can ask for it, expect it, encourage it, but cannot require it. This is one of the fundamental paradoxes of the whole didactical relationship, which Guy Brousseau modelled in one of the best-known concepts in didactics of mathematics: the didactical contract. The teacher cannot be confident that the student will learn exactly what the teacher intended to teach and hence the student must re-create it on the basis of what he/she already knows. In this paper, the importance and the role of situations affording mathematical creativity (in the sense of production of original solutions to unusual situations) are demonstrated. The authors present an experiment (with 9–10-year-old children) that makes it possible to show how certain situations are more favourable (for all children) to express some characteristics of mathematical creativity.     

Every rose has its thorn: secondary teachers’ reasoning about statistical models

Statistical modeling is a core component of statistical thinking and has been identified by several countries as a curricular goal for secondary education. However, many secondary teachers have minimal preparation for teaching this topic. The goal of this research study is to learn about teachers’ perceptions of the role statistical models play in statistical inference and how these perceived purposes affect their reasoning about statistical models and inference. Problem-solving interviews were conducted with four in-service teachers who had recently taught a modeling and simulation-based introductory statistics course. Teachers’ responses suggest they may not see modeling variation as the primary purpose of statistical modeling and instead substitute two other purposes: making a decision and replicating the data collection process. Suggestions for how to build on teachers’ transitional conceptions and refocus attention on modeling variation are discussed.     

Broadening the sense of ‘dynamic’: a microworld to support students’ mathematical generalisation

In this paper, we seek to broaden the sense in which the word ‘dynamic’ is applied to computational media. Focussing exclusively on the problem of design, the paper describes work in progress, which aims to build a computational system that supports students’ engagement with mathematical generalisation in a collaborative classroom environment by helping them to begin to see its power and to express it for themselves and for others. We present students’ strengths and challenges in appreciating structure and expressing generalities that inform our overall system design. We then describe the main features of the microworld that lies at the core of our system. In conclusion, we point to further steps in the design process to develop a system that is more adaptive to students’ and teachers’ actions and needs.     

Coupon collector’s problems with statistical applications to rankings

Some new exact distributions on coupon collector’s waiting time problems are given based on a generalized Pólya urn sampling. In particular, usual Pólya urn sampling generates an exchangeable random sequence. In this case, an alternative derivation of the distribution is also obtained from de Finetti’s theorem. In coupon collector’s waiting time problems with $m$ kinds of coupons, the observed order of $m$ kinds of coupons corresponds to a permutation of $m$ letters uniquely. Using the property of coupon collector’s problems, a statistical model on the permutation group of $m$ letters is proposed for analyzing ranked data. In the model, as the parameters mean the proportion of the $m$ kinds of coupons, the observed ranking can be intuitively understood. Some examples of statistical inference are also given.     

Toward a model for students’ combinatorial thinking

Combinatorics has many applications in different disciplines, however, only a few studies have explored students’ combinatorial thinking at the upper secondary and tertiary levels concurrently. The present research is a grounded theory study of eight Year 12 and five undergraduate students, who have participated in semi-structured interviews and responded to eight combinatorial tasks. Three types of combinatorial tasks were designed: combinatorial reasoning, evaluating, and problem-posing tasks. In the open coding phase of data analysis, seventy-one codes were identified which categorized into seven main categories at the axial coding phase. At the selective coding phase, five relationships between categories were identified that led to designing a model of students’ combinatorial thinking. The model consists of three movements: Horizontal, vertical downward, and vertical upward movement. It is asserted that this model could be used to improve the quality of teaching combinatorics, and also as a lens to explore students’ combinatorial thinking.     

The effect of project-based learning on students’ statistical literacy levels for data representation

The point of this study is to define the effect of project-based learning approach on 8th Grade secondary-school students’ statistical literacy levels for data representation. To achieve this goal, a test which consists of 12 open-ended questions in accordance with the views of experts was developed. Seventy 8th grade secondary-school students, 35 in the experimental group and 35 in the control group, took this test twice, one before the application and one after the application. All the raw scores were turned into linear points by using the Winsteps 3.72 modelling program that makes the Rasch analysis and t-tests, and an ANCOVA analysis was carried out with the linear points. Depending on the findings, it was concluded that the project-based learning approach increases students’ level of statistical literacy for data representation. Students’ levels of statistical literacy before and after the application were shown through the obtained person-item maps.     

Online assessment of students’ reasoning when solving example-eliciting tasks: using conjunction and disjunction to increase the power of examples

ZDM – Mathematics Education - We argue that examples can do more than serve the purpose of illustrating the truth of an existential statement or disconfirming the truth of a universal...     

Students’ reasoning about relationships between variables in a real-world problem

This paper reports on part of an investigation of fifteen second-semester calculus students’ understanding of the concept of parametric function. Employing APOS theory as our guiding theoretical perspective, we offer a genetic decomposition for the concept of parametric function, and we explore students’ reasoning about an invariant relationship between two quantities varying simultaneously with respect to a third quantity when described in a real-world problem, as such reasoning is important for the study of parametric functions. In particular, we investigate students’ reasoning about an adaptation of the popular bottle problem in which they were asked to graph relationships between (a) time and volume of the water, (b) time and height of the water, and (c) volume and height of the water. Our results illustrate that several issues make reasoning about relationships between variables a complex task. Furthermore, our findings indicate that conceiving an invariant relationship, as it relates to the concept of parametric function, is nontrivial, and various complimentary ways of reasoning are favorable for developing such a conception. We conclude by making connections between our results and our genetic decomposition.     

Middle school students’ reasoning about data and context through storytelling with repurposed local data

Publicly-available datasets, though useful for education, are often constructed for purposes that are quite different from students’ own. To investigate and model phenomena, then, students must learn how to repurpose the data. This paper reports on an emerging line of research that builds on work in data modeling, exploratory data analysis, and storytelling to examine and support students’ data repurposing. We ask: What opportunities emerge for students to reason about the relationship between data, context, and uncertainty when they repurpose public data to explore questions about their local communities? And, How can these opportunities be supported in classroom instruction and activity design? In two exploratory studies, students were asked to pose questions about their communities, use publicly-available data to investigate those questions, and create visual displays and written stories about their findings. Across both enactments, opportunities for reasoning emerged especially when students worked to reconcile (1) their own knowledge and experiences of the context from which data were collected with details of the data provided; and (2) their different emerging stories about the data with one another. We review how these opportunities unfolded within each enactment at the level of group and classroom, with attention to facilitator support.     

Assessing science students’ attitudes to mathematics: a case study on a modelling project with mathematical software

This article reports the attitudes of students towards mathematics after they had participated in an applied mathematical modelling project that was part of an Applied Mathematics course. The students were majoring in Earth Science at the National Taiwan Normal University. Twenty-six students took part in the project. It was the first time a mathematical modelling project had been incorporated into the Applied Mathematics course for such students at this University. This was also the first time the students experienced applied mathematical modelling and used the mathematical software. The main aim of this modelling project was to assess whether the students’ attitudes toward mathematics changed after participating in the project. We used two questionnaires and interviews to assess the students. The results were encouraging especially the attitude of enjoyment. Hence the approach of the modelling project seems to be an effective method for Earth Science students.     

Promoting uncertainty to support preservice teachers’ reasoning about the tangent relationship

Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers’ thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way of promoting preservice teachers’ reasoning about the tangent relationship.     

Capturing and characterizing students’ strategic algebraic reasoning through cognitively demanding tasks with focus on representations

In mathematics education, it is important to assess valued practices such as problem solving and communication. Yet, often we assess students based on correct solutions over their problem solving strategies—strategies that can uncover important mathematical understanding. In this article, we first present a framework of competencies required for strategic reasoning to solve cognitively demanding algebra tasks and assessment tools to capture evidence of these competencies. Then, we qualitatively describe characteristics of student reasoning for various performance levels (low, medium, and high) of eighth-grade students, focusing on generating and interpreting algebraic representations. We argue this analysis allows a more comprehensive and complex perspective of student understanding. Our findings lay groundwork to investigate the continuum of algebraic understanding, and may help educators identify specific areas of students’ strength and weakness when solving cognitively demanding tasks.