Providing a Foundation for Deductive Reasoning: Students' Interpretations when Using Dynamic Geometry Software and Their Evolving Mathematical Explanations

A key issue for mathematics education is howchildren can be supported in shifting from `because it looks right' or`because it works in these cases' to convincing arguments which work ingeneral. In geometry, forms of software usually known as dynamicgeometry environments may be useful as they can enable students tointeract with geometrical theory. Yet the meanings that students gain ofdeductive reasoning through experience with such software is likely to beshaped, not only by the tasks they tackle and their interactions with theirteacher and with other students, but also by features of the softwareenvironment. In order to try to illuminate this latter phenomenon, and todetermine the longer-term influence of using such software, this paperreports on data from a longitudinal study of 12-year-old students'interpretations of geometrical objects and relationships when using dynamicgeometry software. The focus of the paper is the progressivemathematisation of the student's sense of the software, examining theirinterpretations and using the explanations that students give of thegeometrical properties of various quadrilaterals that they construct as oneindicator of this. The research suggests that the students' explanations canevolve from imprecise, `everyday' expressions, through reasoning that isovertly mediated by the software environment, to mathematicalexplanations of the geometric situation that transcend the particular toolbeing used. This latter stage, it is suggested, should help to provide afoundation on which to build further notions of deductive reasoning inmathematics.     

Written justifications to multiple-choice concept questions during active learning in class

Increasingly, instructors of large, introductory STEM courses are having students actively engage during class by answering multiple-choice concept questions individually and in groups. This study investigates the use of a technology-based tool that allows students to answer such questions during class. The tool also allows the instructor to prompt students to provide written responses to justify the selection of the multiple-choice answer that they have chosen. We hypothesize that prompting students to explain and elaborate on their answer choices leads to greater focus and use of normative scientific reasoning processes, and will allow them to answer questions correctly more often. The study contains two parts. First, a crossover quasi-experimental design is employed to determine the influence of asking students to individually provide written explanations (treatment condition) of their answer choices to 39 concept questions as compared to students who do not. Second, we analyze a subset of the questions to see whether students identify the salient concepts and use appropriate reasoning in their explanations. Results show that soliciting written explanations can have a significant influence on answer choice and, when it does, that influence is usually positive. However, students are not always able to articulate the correct reason for their answer.     

The interplay among gestures, discourse, and diagrams in students’ geometrical reasoning

This study explores interactions with diagrams that are involved in geometrical reasoning; more specifically, how students publicly make and justify conjectures through multimodal representations of diagrams. We describe how students interact with diagrams using both gestural and verbal modalities, and examine how such multimodal interactions with diagrams reveal their reasoning. We argue that when limited information is given in a diagram, students make use of gestural and verbal expressions to compensate for those limitations as they engage in making and proving conjectures. The constraints of a diagram, gestures and linguistic systems are semiotic resources that students may use to engage in geometrical reasoning.     

Proof, Explanation and Exploration: An Overview

This paper explores the role of proof in mathematics education and providesjustification for its importance in the curriculum. It also discusses threeapplications of dynamic geometry software – heuristics, exploration andvisualization – as valuable tools in the teaching of proof and as potentialchallenges to the importance of proof. Finally, it introduces the four papers in this issue that present empirical research on the use of dynamicgeometry software.This revised version was published online in September 2005 with corrections to the Cover Date.     

Proofs produced by secondary school students learning geometry in a dynamic computer environment

As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Géeomèetre to solve geometry problems structured in a teaching unit. The teaching unit had theaims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their conception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamicgeometry computer environments to improve students' proof skills.This revised version was published online in September 2005 with corrections to the Cover Date.     

‘Because Christian was mean and killed innocent people!’: Swedish primary school students’ causal reasoning about the origins of the nation

The topic of this article is how Swedish primary school students aged 12–13 use causal reasoning when they explain a historical event that is usually considered the ‘origin of the nation’. The study is based on student texts about the rise to power of Gustav Vasa, who is traditionally portrayed as the ‘founding father’ of Sweden. The analysis of the students’ causal reasoning takes into account how many, and what kinds of, causal factors the students use. The main finding of the study is that one category of students give causal explanations that adhere very close to the traditional image of the event, with Vasa as an important and heroic agent pitted against an antagonist, king Kristian II. Another category of students instead give generic explanations with very little historical context. Of these, the former category shows greater causal complexity than the latter. In both categories, there are instances of students failing to causally connect agents to the event, suggesting that teaching practices may need to address this issue.     

Influences on Students' Mathematical Reasoning and Patterns in its Development: Insights from a Longitudinal Study with Particular Reference to Geometry

We report some findings of the Longitudinal Proof Project, which investigated patterns in high-attaining students' mathematical reasoning in algebra and in geometry and development in their reasoning, by analyses of students' responses to three annual proof tests. The paper focuses on students' responses to one non-standard geometry item. It reports how the distribution of responses to this item changed over time with some moderate progress that suggests a cognitive shift from perceptual to geometrical reasoning. However, we also note that many students made little or no progress and some regressed. Extracts from student interviews indicate that the source of this variation from the overall trend stems from the shift over the three years of the study to a more formal approach in the school geometry curriculum for high-attaining students, and the effects of this shift on what students interpreted as the didactical demands of the item.     

The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.     

Some assembly required: How scientific explanations are constructed during clinical interviews

This article is concerned with commonsense science knowledge, the informally gained knowledge of the natural world that students possess prior to formal instruction in a scientific discipline. Although commonsense science has been the focus of substantial study for more than two decades, there are still profound disagreements about its nature and origin, and its role in science learning. What is the reason that it has been so difficult to reach consensus? We believe that the problems run deep; there are difficulties both with how the field has framed questions and the way that it has gone about seeking answers. In order to make progress, we believe it will be helpful to focus on one type of research instrument—the clinical interview—that is employed in the study of commonsense science. More specifically, we argue that we should seek to understand and model, on a moment‐by‐moment basis, student reasoning as it occurs in the interviews employed to study commonsense science. To illustrate and support this claim, we draw on a corpus of interviews with middle school students in which the students were asked questions pertaining to the seasons and climate phenomena. Our analysis of this corpus is based on what we call the mode‐node framework. In this framework, student reasoning is seen as drawing on a set of knowledge elements we call nodes, and this set produces temporary explanatory structures we call dynamic mental constructs. Furthermore, the analysis of our corpus seeks to highlight certain patterns of student reasoning that occur during interviews, patterns in what we call conceptual dynamics. These include patterns in which students can be seen to search through available knowledge (nodes), in which they assemble nodes into an explanation, and in which they converge on and shift among alternative explanations. © 2011 Wiley Periodicals, Inc. J Res Sci Teach 49: 166–198, 2012     

Developing Sixth Graders’ Inquiry Skills to Construct Explanations in Inquiry‐based Learning Environments

The purpose of this study is to investigate how sixth graders develop inquiry skills to construct explanations in an inquiry‐based learning environment. We designed a series of inquiry‐based learning activities and identified four inquiry skills that are relevant to students’ construction of explanation. These skills include skills to identify causal relationships, to describe the reasoning process, to use data as evidence, and to evaluate explanations. Multiple sources of data (e.g., video recordings of learning activities, interviews, students’ artifacts, and pre/post tests) were collected from two science classes with 58 sixth graders. The statistical results show that overall the students’ inquiry skills were significantly improved after they participated in the series of the learning activities. Yet the level of competency in these skills varied. While students made significant progress in identifying causal relationships, describing the reasoning process, and using data as evidence, they showed slight improvement in evaluating explanations. Additionally, the analyses suggest that phases of inquiry provide different kinds of learning opportunities and interact with students’ development of inquiry skills.     

Scientific explanations and piagetian operational levels

Explaining natural phenomena is an important goal in science teaching. A logical analysis reveals that causal explanations exhibit formal operational structures in that they consist of implication statements chained together through transitive reasoning. It was hypothesized in the present study that individuals who do not reason formally will have difficulty in learning explanations presented in instruction. To test this hypothesis, the effect of levels of operational thought on the explanations which ninth-grade (n = 26) and college (n = 40) physical science students reconstructed after instruction was investigated. Subjects in the study were classified through Piagetian tests as concrete or formal operational. Both concrete and formal subjects were successful in recalling explanations requiring the chaining of two implication statements. Formal operational subjects performed significantly better than concrete operational subjects in three of the four tests of the reconstruction of complex explanations requiring the chaining of six implication statements. In teaching complex causal explanations to students at the concrete operational level, it is suggested that teachers be prepared to furnish some external structuring which the students can rely on in logically relating the various propositions of the explanation to one another.     

Impact of simulator-based instruction on diagramming in geometrical optics by introductory physics students

We examine the conceptual development resulting from an instructional experiment with an interactive learning environment in geometrical optics for introductory high school physics. How did teaching-learning processes come to change the ways in which students depicted various everyday optical situations in paper and pencil graphical representations? We view conceptual development as a process resulting in part from increasingly aligning one's practices to a target community by means of participating in a community of practice that uses the target concepts. For formal science learning, this participation requires changes in concepts, epistemological attitude, and in the development and use of representational tools, including diagrams and technical language, as a means of communication. Results of our instructional experiment indicated that students went through major conceptual developments as reflected in the diagrams they constructed and supported by other representational tools and as judged in terms of several perspectives: in identifying the formation of shadows and images, in recognizing the eye as a participating factpr in the optical system, and in changing the types of justifications they provided in their optical reasoning from presuppositional to causal.     

The role of contradiction and uncertainty in promoting the need to prove in Dynamic Geometry environments

In many geometrical problems, students can feel that the universalityof a conjectured attribute of a figure is validated by their action in adynamic geometry environment. In contrast, students generally do not feelthat deductive explanations strengthen their conviction that a geometricalfigure has a given attribute. In order to cope with students' convictionbased on empirical experience only and to create a need for deductiveexplanations, we developed a collection of innovative activities intended tocause surprise and uncertainty. In this paper we describe two activities, thatled students to contradictions between conjectures and findings. We analyzethe conjectures, working methods, and explanations given by the studentswhen faced with the contradictions that arose.This revised version was published online in September 2005 with corrections to the Cover Date.     

Students’ reasons for preferring teleological explanations

The teleological bias, a major learning obstacle, involves explaining biological phenomena in terms of purposes and goals. To probe the teleological bias, researchers have used acceptance judgement tasks and preference judgement tasks. In the present study, such tasks were used with German high school students (N?=?353) for 10 phenomena from human biology, that were explained both teleologically and causally. A sub-sample (n?=?26) was interviewed about the reasons for their preferences. The results showed that the students favoured teleological explanations over causal explanations. Although the students explained their preference judgements etiologically (i.e. teleologically and causally), they also referred to a wide range of non-etiological criteria (i.e. familiarity, complexity, relevance and five more criteria). When elaborating on their preference for causal explanations, the students often focused not on the causality of the phenomenon, but on mechanisms whose complexity they found attractive. When explaining their preference for teleological explanations, they often focused not teleologically on purposes and goals, but rather on functions, which they found familiar and relevant. Generally, students’ preference judgements rarely allowed for making inferences about causal reasoning and teleological reasoning, an issue that is controversial in the literature. Given that students were largely unaware of causality and teleology, their attention must be directed towards distinguishing between etiological and non-etiological reasoning. Implications for educational practice as well as for future research are discussed.     

What Do Students’ Explanations Look Like When They Use Second-Hand Data?

Explanation studies underlined the importance of using evidence in support of claims. However, few studies have focused on students' use of others' data (second-hand data) in this process. In this study, students collected data from a local water source and then took all the data back to the classroom to create scientific explanations by using claim–evidence–reasoning model on a new mobile application. A middle school science teacher from a Midwest town participated with four sixth-grade classes. After collecting their own data from a local water source, students created explanations by analyzing the data they collected (first-hand data), and by analyzing existing data set collected by another school from another river (second-hand data). By analyzing the health of these two water sources, students created two scientific explanations. Students participating in this study created stronger explanations when analyzing the data they generated (first-hand data).     

Modelling Molecular Mechanisms: A Framework of Scientific Reasoning to Construct Molecular-Level Explanations for Cellular Behaviour

Although molecular-level details are part of the upper-secondary biology curriculum in most countries, many studies report that students fail to connect molecular knowledge to phenomena at the level of cells, organs and organisms. Recent studies suggest that students lack a framework to reason about complex systems to make this connection. In this paper, we present a framework that could help students to reason back and forth between cells and molecules. It represents both the general type of explanation in molecular biology and the research strategies scientists use to find these explanations. We base this framework on recent work in the philosophy of science that characterizes explanations in molecular biology as mechanistic explanations. Mechanistic explanations describe a phenomenon in terms of the entities involved, the activities displayed and the way these entities and activities are organized. We conclude that to describe cellular phenomena scientists use entities and activities at multiple levels between cells and molecules. In molecular biological research, scientists use heuristics based on these intermediate levels to construct mechanistic explanations. They subdivide a cellular activity into hypothetical lower-level activities (top-down approaches) and they predict and test the organization of macromolecules into functional modules that play a role in higher-level activities (bottom-up approaches). We suggest including molecular mechanistic reasoning in biology education and we identify criteria for designing such education. Education using molecular mechanistic reasoning can build on common intuitive reasoning about mechanisms. The heuristics that scientists use can help students to apply this intuitive notion to the levels in between molecules and cells.     

Discourse Generation for Instructional Applications: Identifying and Exploiting Relevant Prior Explanations

To reap the benefits of natural language interaction, tutorial systems must be endowed with the properties that make human natural-language interaction so effective. One striking feature of naturally occurring interactions is that human tutors and students freely refer to the context created by prior explanations. In contrast, computer-generated utterances that do not draw on the previous discourse often seem awkward and unnatural and may even be incoherent. The explanations produced by such systems are frustrating to students because they repeat the same information over and over again. Perhaps more critical is that, by not referring to prior explanations, computer-based tutors are not pointing out similarities between problem-solving situations and therefore may be missing out on opportunities to help students form generalizations. In this article, we discuss several observations from an analysis of human-human tutorial interactions and provide examples of the ways in which tutors and students refer to previous explanations. We describe how we have used a case-based reasoning algorithm to enable a computational system to identify prior explanations that may be relevant to the explanation currently being generated. We then describe two computational systems that can exploit this knowledge about relevant prior explanations in constructing their subsequent explanations.     

University Students Explaining Adiabatic Compression of an Ideal Gas—A New Phenomenon in Introductory Thermal Physics

This study focuses on second-year university students?? explanations and reasoning related to adiabatic compression of an ideal gas. The phenomenon was new to the students, but it was one which they should have been capable of explaining using their previous upper secondary school knowledge. The students?? explanations and reasoning were investigated with the aid of paper and pencil tests (n?=?86) and semi-structured interviews (n?=?5) at the start of a thermal physics course at the University of Eastern Finland. The paper and pencil test revealed that the students had difficulties in applying content taught during earlier education in a new context: only a few of them were able to produce a correct explanation for the phenomenon. A majority of the students used either explanations with invalid but physically correct models, such as the ideal gas law or a microscopic model, or erroneous dependencies between quantities. The results also indicated that students had problems in seeing deficiencies or inconsistencies in their reasoning, in both test and interview situations. We suggest in our conclusion that the contents of upper secondary school thermal physics courses should be carefully examined to locate the best emphases for different laws, principles, concepts, and models. In particular, the limitations of models should be made explicit in teaching and students should be guided towards critical scientific thinking, including metaconceptual awareness.     

From visual reasoning to logical necessity through argumentative design

Our main goal in this study is to exemplify that a meticulous design can lead pre-service teachers to engage in productive unguided peer argumentation. By productivity, we mean here a shift from reasoning based on intuitions to reasoning moved by logical necessity. As a subsidiary goal, we aimed at identifying the kinds of reasoning processes (visual, inquiry-based, and deductive) pre-service teacher's students adopt, and how these reasoning processes are interwoven in peer-unguided argumentation. We report on a case study in which one dyad participating in a pre-service teachers program solved a mathematical task. We relied on three principles to design an activity: (a) creating a situation of conflict, (b) creating a collaborative situation, and (c) providing a device for checking hypotheses/conjectures. We show how the design afforded productive argumentation. We show that the design of the task entailed argumentation which first relied on intuition, then intertwined the activities of conjecturing and checking conjectures by means of various hypotheses-testing devices (measurement, manipulations, and dynamic change of figures with Dynamic Geometry software), leading to a conflict between conjectures and the outcome of the manipulation of DG software. Peer argumentation then shifted to abductive and deductive considerations towards the solution of the mathematical task. These beneficial outcomes resulted from collaborative rather than adversarial interactions as the students tried to accommodate their divergent views through the co-elaboration of new explanations.     

Teachers’ pedagogical reasoning and action in the digital age

Beginning teachers are entering the profession with increasing confidence in their ability to use digital technologies which has the potential to change the way teachers of the future make pedagogical decisions. This paper explores how pedagogical reasoning and action might occur in the digital age, comparing Schulman’s 1987 model with the reality for a small sample of digitally able beginning teachers as part of the emerging generation of teachers. The latter were examined through a multiple case study during their first year of teaching as they made decisions about using digital technologies within their teaching practice which gave an insight into pedagogical reasoning and action through the use of open‐ended interviews and observation. The conclusion drawn is that while the pedagogical reasoning and action model remains relevant, it was based on an assumption that teaching involves knowledge being passed from a teacher to their students, which was found to restrict innovation by digitally able teachers. A broader interpretation of knowledge and teaching within this model building on emerging learning theory could help reform practice once again, providing a framework for teachers in the digital age.