Developing a statistical modeling framework to characterize Year 7 students’ reasoning

Some researchers advocate a statistical modeling approach to inference that draws on students’ intuitions about factors influencing phenomena and that requires students to build models. Such a modeling approach to inference became possible with the creation of TinkerPlots Sampler technology. However, little is known about what statistical modeling reasoning students need to acquire. Drawing and building on previous research, this study aims to uncover the statistical modeling reasoning students need to develop. A design-based research methodology employing Model Eliciting Activities was used. The focus of this paper is on two 11-year-old students as they engaged with a bag weight task using TinkerPlots. Findings indicate that these students seem to be developing the ability to build models, investigate and posit factors, consider variation and make decisions based on simulated data. From the analysis an initial statistical modeling framework is proposed. Implications of the findings are discussed.     

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.     

Evaluating students’ conceptual and procedural understanding of sampling distributions

Introductory statistics courses, which are important in preparing students for their daily lives, generally derive inferential statistics from informal knowledge. In this transition process, sampling distributions have an important place, yet research has shown that students often have difficulties with this concept. In order to increase their understanding of sampling distributions, students should have a strong conceptual foundation that is balanced with procedural knowledge. To address this issue, this study was designed to examine the relationship between college students’ procedural and conceptual knowledge of sampling distributions. With this aim in mind, an achievement test consisting of two sections – procedural and conceptual knowledge – was prepared. In answering the questions related to procedural knowledge, the participants were more successful in identifying the relationship between standard deviation of a population and sample means. However, they lacked theoretical knowledge about statements that they had heard or knew intuitively. Simulation activities provided in statistics courses may support students in developing their conceptual understanding in this regard.     

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.     

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.     

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.     

On the use of computational tools to promote students’ mathematical thinking

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. In this snapshot students employ dynamic geometry software to find great mathematical richness around a seemingly simple question about rectangles.

Editor: Uri Wilensky

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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.     

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...     

Third- to fourth-grade students’ conceptions of multiplication and area measurement

In this study, 34 children were evaluated in order to elucidate their multiplicative thinking and interpretation of the area formula of a rectangle, and to determine what roles these factors play in solving area measurement problems. One-on-one interviews and problem-solving tasks were employed to explore the problem-solving skills of the children regarding these two concepts. This study also explored how the associations changed throughout two consecutive phases, from the third to the fourth grades. The results indicated that in the third grade, multiplicative thinking was associated with the solving of area measurement problems. Third-grade children who understood the meaning of the multiplication symbol “p × q” in models (e.g., the set model and arrays) outperformed children who understood only partial multiplicative concepts or additive thinking; however, the association between multiplicative thinking and solving area measurement problems was not significant in the fourth grade. In contrast, children’s ability to interpret the area formula of a rectangle was associated with their performance at solving area measurement problems throughout the third and fourth grades. The way of interpreting the area formula was associated with the extent to which the children understood multiplication, area measurement, and the spatial concepts embedded in rectangular figures. The instructional implications of the study are discussed in terms of developing child abilities to solve area measurement problems by connecting multiplication and area measurement.     

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).     

Middle school students’ proportional reasoning in real life contexts in the domain of geometry and measurement

The purpose of this study was to examine middle school students’ proportional reasoning, solution strategies and difficulties in real life contexts in the domain of geometry and measurement. The underlying reasons of the difficulties were investigated as well. Mixed research design was adopted for the aims of the study by collecting data through an achievement test from 935 sixth, seventh and eighth grade students. The achievement test included real life problems that required proportional reasoning, and were related to the measurement of length, perimeter, area and volume concepts. In addition, task-based interviews were conducted on 12 of these students to collect more comprehensive data and to support the findings of the achievement test. Findings revealed that although students were mostly successful in giving correct answers, their reasoning lacked a clear argument of the direct and indirect proportional relationships between the variables and that they approached the problems by superficial characteristics of the problems.     

Comments on “Statistical reasoning with set-valued information: Ontic vs. epistemic views”

In this comment, several paragraphs from the paper “Statistical reasoning with set-valued information: Ontic vs. epistemic views” have been selected and discussed. The selection has been based, on one side, on a personal view of what can be considered the most clarifying points in the paper and, on the other side, on the aspects I am more familiar with and interested in and being quite unequivocally ontic-oriented. For sure, it is a biased selection, but the aim of these comments is that of sharing what I have found to be more appealing within the discussion and I would like to point out in connection with my own expertise.     

Rejoinder on “Statistical reasoning with set-valued information: Ontic vs. epistemic views”

This note replies to comments made on our contribution to the Low Quality Data debate.     

Teachers’ metacognitive and heuristic approaches to word problem solving: analysis and impact on students’ beliefs and performance

We conducted a 7-month video-based study in two sixth-grade classrooms focusing on teachers’ metacognitive and heuristic approaches to problem solving. All problem-solving lessons were analysed regarding the extent to which teachers implemented a metacognitive model and addressed a set of eight heuristics. We observed clear differences between both teachers’ instructional approaches. Besides, we examined teachers’ and students’ beliefs about the degree to which metacognitive and heuristic skills were addressed in their classrooms and observed that participants’ beliefs were overall in line with our observations of teachers’ instructional approaches. In addition, we investigated how students’ problem-solving skills developed as a result of teachers’ instructional approaches. A positive relationship between students’ spontaneous application of heuristics to solve non-routine word problems and teachers’ references to these skills in their problem-solving lessons was found. However, this increase in the application of heuristics did not result in students’ better performance on these non-routine word problems.     

Fifth-grade students�� approaches to and beliefs of mathematics word problem solving: a large sample Hungarian study

Mathematical word problems used in Verschaffel et al.??s (Learning and Instruction 7:339?C359, 1994) study were applied in several follow-up studies. The goal of the present study was to replicate and extend the results of this line of research in a large sample of Hungarian students using an alternative set of data-gathering and data-analysis techniques. 4,037 students forming a nationwide representative sample of the Hungarian fifth-grade student population (aged 10?C11) completed the test. The test contained five word problems from the list of 10 P(??problematic)-items from Verschaffel et al.??s test. In contrast to all previous research in this domain, we used a multiple-choice format, where three options were given for each task: (a) routine-based, non-realistic answer, (b) numerical response that does take into account realistic considerations, (c) a realistic solution stating that the task cannot be solved. The hypotheses of this study were: (1) Students?? responses will confirm previous results, i.e. upper elementary school students prefer to respond to P-items by means of the routine-based answer; (2) Most students will demonstrate a more or less consistent preference for a given answer type (a, b or c) over problems; (3) Students?? school math marks will have low correlation indices with students?? achievement on these word problems. Our results confirm student??s overall tendency to follow non-realistic approaches when doing school word problem solving. The tendency even holds when confronting students with various kinds of realistic answers. Our results show that students demonstrate response patterns over problems, and that the correlation with math school performance is significant but small.     

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.     

Students’ challenges with polar functions: covariational reasoning and plotting in the polar coordinate system

Covariational reasoning has been the focus of many studies but only a few looked into this reasoning in the polar coordinate system. In fact, research on student's familiarity with polar coordinates and graphing in the polar coordinate system is scarce. This paper examines the challenges that students face when plotting polar curves using the corresponding plot in the Cartesian plane. In particular, it examines how students coordinate the covariation in the polar coordinate system with the covariation in the Cartesian one. The research, which was conducted in a sophomore level Calculus class at an American university operating in Lebanon, investigates in addition the challenges when students synchronize the reasoning between the two coordinate systems. For this, the mental actions that students engage in when performing covariational tasks are examined. Results show that coordinating the value of one polar variable with changes in the other was well achieved. Coordinating the direction of change of one variable with changes in the other variable was more challenging for students especially when the radial distance r is negative.