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.     

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.     

An analytical comparison of students’ reasoning in the context of Inquiry-Oriented Instruction: The case of span and linear independence

In this report we analyze differences in reasoning about span and linear independence by comparing written work of 126 linear algebra students whose instructors received support to implement a particular inquiry-oriented (IO) instructional approach compared to 129 students whose instructors did not receive that support. Our analysis of students’ responses to open-ended questions indicated that IO students’ concept images of span and linear independence were more aligned with the formal concept definition than the concept images of Non-IO students. Additionally, IO students exhibited more coordinated conceptual understandings and used deductive reasoning at higher rates than Non-IO students. We provide illustrative examples of systematic differences in how students from the two groups reasoned about span and linear independence.     

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.     

Emergent modeling: discrete graphs to support the understanding of change and velocity

In this paper we focus on an instructional sequence that aims at supporting students in their learning of the basic principles of rate of change and velocity. The conjectured process of teaching and learning is supposed to ensure that the mathematical and physical concepts will be rooted in students’ understanding of everyday-life situations. Students’ inventions are supported by carefully planned activities and tools that fit their reasoning. The central design heuristic of the instructional sequence is emergent modeling. We created an educational setting in three tenth grade classrooms to investigate students’ learning with this sequence. The design research is carried out in order to contribute to a local instruction theory on calculus. Classroom events and computer activities are video-taped, group work is audio-taped and student materials are collected. Qualitative analyses show that with the emergent modeling approach, the basic principles of calculus can be developed from students’ reasoning on motion, when they are supported by discrete graphs.     

Toward a situation model in a cognitive architecture

The ability to coherently represent information that is situationally relevant is vitally important to perform any complex task, especially when that task involves coordinating with team members. This paper introduces an approach to dynamically represent situation information within the ACT-R cognitive architecture in the context of a synthetic teammate project. The situation model represents the synthetic teammate’s mental model of the objects, events, actions, and relationships encountered in a complex task simulation. The situation model grounds textual information from the language analysis component into knowledge usable by the agent-environment interaction component. The situation model is a key component of the synthetic teammate as it provides the primary interface between arguably distinct cognitive processes modeled within the synthetic teammate (e.g., language processing and interactions with the task environment). This work has provided some evidence that reasoning about complex situations requires more than simple mental representations and requires mental processes involving multiple steps. Additionally, the work has revealed an initial method for reasoning across the various dimensions of situations. One purpose of the research is to demonstrate that this approach to implementing a situation model provides a robust capability to handle tasks in which an agent must construct a mental model from textual information, reason about complex relationships between objects, events, and actions in its environment, and appropriately communicate with task participants using natural language. In this paper we describe an approach for modeling situationally relevant information, provide a detailed example, discuss challenges faced, and present research plans for the situation model.     

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.     

An Exploration of a Quantitative Reasoning Instructional Approach to Linear Equations in Two Variables With Community College Students

In this exploratory study, we examined the effects of a quantitative reasoning instructional approach to linear equations in two variables on community college students’ conceptual understanding, procedural fluency, and reasoning ability. This was done in comparison to the use of a traditional procedural approach for instruction on the same topic. Data were gathered from a common unit assessment that included procedural and conceptual questions. Results demonstrate that small changes in instruction focused on quantitative reasoning can lead to significant differences in students’ ability to demonstrate conceptual understanding compared to a procedural approach. The results also indicate that a quantitative reasoning approach does not appear to diminish students’ procedural skills, but that additional work is needed to understand how to best support students’ understanding of linear relationships.     

The use of cross multiplication and other mal–rules in fraction operations by pre-service teachers

Many studies show that prospective teachers often have misconceptions about fractions. In this case study, we report on some of the mal–rules used by a group of 60 prospective South African primary school teachers. The students’ written responses to two items focusing on addition and multiplication of fractions which formed part of an assessment, were analyzed. Semi-structured interviews were also used to elicit the reasoning used in the students’ calculations. Less than half of the participants completed both items correctly, and many of the other students displayed various mal–rules. To interpret the pre–service teachers’ misconceptions, we studied the rules used by the participants, and expressed them as theorems–in–action. An interesting mal–rule governing the multiplication of fractions was the widespread ‘cross multiplication’ rule which after some mutations led to other mal–rules, illustrating how students’ misconceptions can persist many years after their initial learning.     

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.     

Educative experiences in a games context: Supporting emerging reasoning in elementary school mathematics

Reasoning as a process supports students’ success in mathematics, yet reports on its development in elementary school are scarce. An action research project with grade 5 and 6 students investigated how growth in reasoning occurred within abstract strategy games. Reasoning within the board game context was framed by Dewey’s conceptualization of experience which emphasizes the importance of students’ active participation and reflection. Through characteristics of interaction and continuity, students analyzed moves, generalized toward strategies, and convincingly justified effective approaches through accepted structures of reasoning. Elaborating on reasoning as a process, results show that students can grow in their capability to reason through multiple experiences of developing convincing arguments in an authentic context.     

Ways in which engaging with someone else’s reasoning is productive

Classrooms which involve students in mathematical discourse are becoming ever more prominent for the simple reason that they have been shown to support student learning and affinity for content. While support for outcomes has been shown, less is known about how or why such strategies benefit students. In this paper, we report on one such finding: namely that when students engage with another’s reasoning, as necessitated by interactive conversation, it supports their own conceptual growth and change. This qualitative analysis of 10 university students provides insight into what engaging with another’s reasoning entails and suggests that higher levels of engagement support higher levels of conceptual growth. We conclude with implications for instructional practice and future research.     

Innovations in statistical modeling to connect data,chance and context

Statistical modeling is emerging as a fertile research environment in which to promote and learn about student statistical reasoning processes. We outline a paradigm shift toward a modeling perspective that is occurring in statistics education research and how statistical modeling processes involve connecting data, chance and context. The innovative task and software designs and theoretical frameworks that are under development for explicating student reasoning and pedagogy are discussed. In conclusion, we reflect on statistics education research.     

Inquisitive logic as an epistemic logic of knowing how

In this paper, we present an alternative interpretation of propositional inquisitive logic as an epistemic logic of knowing how. In our setting, an inquisitive logic formula α being supported by a state is formalized as knowing how to resolve α (more colloquially, knowing how α is true) holds on the S5 epistemic model corresponding to the state. Based on this epistemic interpretation, we use a dynamic epistemic logic with both know-how and know-that operators to capture the epistemic information behind the innocent-looking connectives in inquisitive logic. We show that the set of valid know-how formulas corresponds precisely to the inquisitive logic. The main result is a complete axiomatization with intuitive axioms using the full dynamic epistemic language. Moreover, we show that the know-how operator and the dynamic operator can both be eliminated without changing the expressivity over models, which is consistent with the modal translation of inquisitive logic existing in the literature. We hope our framework can give an intuitive alternative interpretation to various concepts and technical results in inquisitive logic, and also provide a powerful and flexible tool to handle both the inquisitive reasoning and declarative reasoning in an epistemic context.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

Representing scientific activity: Affordances and constraints of central design and enactment features of a model‐based inquiry unit

This research explores how explaining an anchoring phenomena and engaging students in investigations, as central designs of a model‐based inquiry (MBI) unit, afforded or constrained the representation of scientific activity in the science classroom. This research is considered timely as recent standards documents and scholars in the field have highlighted the significance of identifying what features of scientific activity are important and how these can be represented for students in classrooms. Through taking advantage of qualitative research methods to closely examine the enactment of an MBI unit, both affordances and constraints were identified for each design. More specifically, explaining an anchoring phenomenon provided a context for more authentically framing the work of students, while investigations afforded students insight into the role these play in the refinement of models. Further, the teacher's attempts to support student reasoning and, at times, reasoning for students when they were found struggling were the most salient constraints identified connected to explaining an anchoring phenomenon and engaging students in investigations.     

A hypothetical learning trajectory for conceptualizing matrices as linear transformations

In this paper, we present a hypothetical learning trajectory (HLT) aimed at supporting students in developing flexible ways of reasoning about matrices as linear transformations in the context of introductory linear algebra. In our HLT, we highlight the integral role of the instructor in this development. Our HLT is based on the ‘Italicizing N’ task sequence, in which students work to generate, compose, and invert matrices that correspond to geometric transformations specified within the problem context. In particular, we describe the ways in which the students develop local transformation views of matrix multiplication (focused on individual mappings of input vectors to output vectors) and extend these local views to more global views in which matrices are conceptualized in terms of how they transform a space in a coordinated way.     

How students use the ‘see similar example’ feature in online mathematics homework

Homework is one of students’ opportunities to learn mathematics, but we know little about what students learn from homework. This study employs the instructional triangle and didactic contract to explore how students used the ‘see similar example’ feature in an online homework platform and how that use reflected their learning goals. Findings indicate students used similar examples to troubleshoot, to check if they were on the right track, and to see the form of the answer. Students also sought to unpack the reasoning in solution steps, used solutions as templates for solving their own problems, and sometimes copied answers. One student did a ‘see similar example’ problem for more practice. Students’ goals included completing the homework, maximizing their score, and understanding the content. This research lays groundwork for future work characterizing what students learn from homework and how features that provide students with similar examples help or hinder their learning.     

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.