An examination of middle school students’ representation practices in mathematical problem solving through the lens of expert work: towards an organizing scheme

Representation is viewed as central to mathematical problem solving. Yet, it is becoming obvious that students are having difficulty negotiating the various forms and functions of representations. This article examines the functions that representation has in students’ mathematical problem solving and how that compares to its function in the problem solving of experts and broadly in mathematics. Overall, this work highlights the close connections between the work of experts and students, showing how students use representations in ways that are inherently similar to those of experts. Both experts and students use representations as tools towards the understanding, exploration, recording, and monitoring of problem solving. In social contexts, experts and students use representations for the presentation of their work but also the negotiation and co-construction of shared understandings. However, this research also highlights where students’ work departs from experts’ representational practices, hence, providing some directions for pedagogy and further work.     

Facilitating Students’ Problem Solving in a Technological Context: Prospective Teachers’ Learning Trajectory

In the past decade, there has been an increased emphasis on the preparation of teachers who can effectively engage students in meaningful mathematics with technology tools. This study presents a closer look at how three prospective teachers interpreted and developed in their role of facilitating students’ mathematical problem solving with a technology tool. A cycle of planning–experience–reflection was repeated twice during an undergraduate course to allow the prospective teachers to change their strategies when working with two different groups of students. Case study methods were used to identify and analyze critical events that occurred throughout the different phases of the study and how these events may have influenced the prospective teachers’ work with students. Looking across the cases, several themes emerged. The prospective teachers (1) used their problem solving approaches to influence their pedagogical decisions; (2) desired to ask questions that would guide students in their solution strategies; (3) recognized their own struggle in facilitating students’ problem solving and focused on improving their interactions with students; (4) assumed the role of an explainer for some portion of their work with students; (5) used technological representations to promote students’ mathematical thinking or focus their attention; and (6) used the technology tools in ways consistent with the nature of their interactions and perceived role with students. The implications inform the development of an expanded learning trajectory for what we might expect as prospective teachers develop an understanding of how to teach mathematics in technology-rich environments.     

Modelling mathematical argumentation: the importance of qualification

In recent years several mathematics education researchers have attempted to analyse students’ arguments using a restricted form of Toulmin’s [The Uses of Argument, Cambridge University Press, UK, 1958] argumentation scheme. In this paper we report data from task-based interviews conducted with highly talented postgraduate mathematics students, and argue that a superior categorisation of genuine mathematical argumentation is provided by the use of Toulmin’s full scheme. In particular, we suggest that modal qualifiers play an important and previously unrecognised role in mathematical argumentation, and that one of the goals of instruction should be to develop students’ abilities to appropriately match up warrant-types with modal qualifiers.     

The role of abduction in proving processes

This paper offers a typology of forms and uses of abduction that can be exploited to better analyze abduction in proving processes. Based on the work of Peirce and Eco, we describe different kinds of abductions that occur in students’ mathematical activity and extend Toulmin’s model of an argument as a methodological tool to describe students’ reasoning and to classify the different kinds of abduction. We then use this tool to analyze case studies of students’ abductions and to identify cognitive difficulties students encounter. We conclude that some types of abduction may present obstacles, both in the argumentation when the abduction occurs and later when the proof is constructed.     

“Accepting Emotional Complexity”: A Socio-Constructivist Perspective on the Role of Emotions in the Mathematics Classroom

A socio-constructivist account of learning and emotions stresses the situatedness of every learning activity and points to the close interactions between cognitive, conative and affective factors in students’ learning and problem solving. Emotions are perceived as being constituted by the dynamic interplay of cognitive, physiological, and motivational processes in a specific context. Understanding the role of emotions in the mathematics classroom then implies understanding the nature of these situated processes and the way they relate to students’ problem-solving behaviour. We will present data from a multiple-case study of 16 students out of 4 different junior high classes that aimed to investigate students’ emotional processes when solving a mathematical problem in their classrooms. After identifying the different emotions and analyzing their relations to motivational and cognitive processes, the relation with students’ mathematics-related beliefs will be examined. We will specifically use Frank’s case to illustrate how the use of a thoughtful combination of a variety of different research instruments enabled us to gather insightful data on the role of emotions in mathematical problem solving.     

Using educational software to support collective thinking and test hypotheses in the computer science curriculum

This study analyses the discourse among the teacher and the students, members of three (3) small groups, who learn in the environment of a stand-alone computer. Two educational environments are examined: the first one, a “virtual laboratory” (Virtual scale-DELYS) and the second one, a computer modeling environment (ModelsCreator). The ‘Virtual Scale’ environment provides users with curriculum focused feedback and in that sense it can be categorized as directive. The ModelsCreator environment provides users merely with a representation of their own conception of curriculum concepts, so it can be categorized as an open-ended environment. The goal of this research is to exemplify the way the two educational software environments support (a) the development of collective thinking in peer— and teacher-led discussion and (b) students’ autonomy. The software tools of the “Virtual scale” along with the resources provided for the problem solving created an educational framework of hypothesis testing. This framework did not limit the students’ contributions by directing them to give short answers. Moreover, it supported the students’ initiatives by providing tools, representations and procedures that offered educationally meaningful feedback. Based on the above results, we discuss a new educationally important structure of software mediation and describe the way the two software activities resourced collective thinking and students’ initiatives. Finally, for each type of software environment, we propose certain hypotheses for future research regarding the support of collaborative problem solving.     

Grounded blends and mathematical gesture spaces: developing mathematical understandings via gestures

This paper examines how a person’s gesture space can become endowed with mathematical meaning associated with mathematical spaces and how the resulting mathematical gesture space can be used to communicate and interpret mathematical features of gestures. We use the theory of grounded blends to analyse a case study of two teachers who used gestures to construct a graphical anti-derivative while working on a professional development task in a calculus modelling activity. Results indicate that mathematical gesture spaces can encourage mathematical experimentation, lighten the cognitive load for students and can be limited by a person’s physical constraints.     

Appropriating geometric series as a cultural tool: a study of student collaborative learning

The aim of this article is to illustrate how students, through collaborative small-group problem solving, appropriate the concept of geometric series. Student appropriation of cultural tools is dependent on five sociocultural aspects: involvement in joint activity, shared focus of attention, shared meanings for utterances, transforming actions and utterances and use of pre-existing cultural knowledge from the classroom in small-group problem solving. As an analytical point of departure, four mathematical theoretical components are identified when appropriating the cultural tool of geometric series: (1) estimating of parameters, (2) establishing of the general term, (3) composing of the sum and (4) deciding on convergence. Analyses of five excerpts focused on the students’ social processes of knowledge objectification and the corresponding semiotic means, i.e., lecture notes, linguistic devices, gestures, head movements and gaze, to obtain shared foci and meanings. The investigation of these processes unveils the manner in which the students established links to pre-existing mathematical knowledge in the classroom and how they simultaneously combined the various mathematical theoretical components that go into appropriating the cultural tool of geometric series. From the excerpts, it is evident that the students’ participation changes throughout their involvement in the problem-solving process. The students are gaining mathematical knowing through a process of transforming and by establishing shared meanings for the concept and its theoretical components.     

Teachers attending to students’ mathematical reasoning: lessons from an after-school research program

There is a documented need for more opportunities for teachers to learn about students’ mathematical reasoning. This article reports on the experiences of a group of elementary and middle school mathematics teachers who participated as interns in an after-school, classroom-based research project on the development of mathematical ideas involving middle-grade students from an urban, low-income, minority community in the United States. For 1 year, the teachers observed the students working on well-defined mathematical investigations that provided a context for the students’ formation of particular mathematical ideas and different forms of reasoning in several mathematical content strands. The article describes insights into students’ mathematical reasoning that the teachers were able to gain from their observations of the students’ mathematical activity. The purpose is to show that teachers’ observations of students’ mathematical activity in research sessions on students’ development of mathematical ideas can provide opportunities for teachers to learn about students’ mathematical reasoning.     

Geometrical representations in the learning of two-variable functions

This study is part of a project concerned with the analysis of how students work with two-variable functions. This is of fundamental importance given the role of multivariable functions in mathematics and its applications. The portion of the project we report here concentrates on investigating the relationship between students’ notion of subsets of Cartesian three-dimensional space and the understanding of graphs of two-variable functions. APOS theory and Duval’s theory of semiotic representations are used as theoretical framework. Nine students, who had taken a multivariable calculus course, were interviewed. Results show that students’ understanding can be related to the structure of their schema for R³ and to their flexibility in the use of different representations.     

Approach to mathematical problem solving and students’ belief systems: two case studies

The goal of the study reported here is to gain a better understanding of the role of belief systems in the approach phase to mathematical problem solving. Two students of high academic performance were selected based on a previous exploratory study of 61 students 12–13 years old. In this study we identified different types of approaches to problems that determine the behavior of students in the problem-solving process. The research found two aspects that explain the students’ approaches to problem solving: (1) the presence of a dualistic belief system originating in the student’s school experience; and (2) motivation linked to beliefs regarding the difficulty of the task. Our results indicate that there is a complex relationship between students’ belief systems and approaches to problem solving, if we consider a wide variety of beliefs about the nature of mathematics and problem solving and motivational beliefs, but that it is not possible to establish relationships of causality between specific beliefs and problem-solving activity (or vice versa).     

Assessment Practices: Empowering Mathematics and Science Teachers in Rural Secondary Schools to Enhance Student Learning

Aligned with recent changes to syllabuses in Australia is an assessment regime requiring teachers to identify what their students ‘know’ and ‘can do’ in terms of the quality of understanding demonstrated. This paper describes the experiences of 25 secondary science and mathematics teachers in rural schools in New South Wales as they explore the changing nature of assessment and its implications on their classroom practice. To help reconceptualise these changes, teachers were introduced to a cognitive structural model as a theoretical framework. Throughout the 2-year study, teachers attended a series of professional development sessions and received ongoing consultative support. Each session was taped and transcribed while interviews were conducted with each teacher at the end of both years. Analysis of these data using a grounded theory approach identified seven major components of teacher practice impacted by the study. The core component was questioning while the six contributing components were teachers’ pedagogical practices, attention to cognition, teaching strategies, assessment linked to pedagogy, classroom advantages for students, and classroom advantages for teachers. These findings represent a major shift in teachers’ perceptions of assessment from a focus on the accumulation of students’ marks to one of diagnosis as a means of directing teaching to enhance students’ scientific and mathematical understandings.     

Teacher-education students’ views about knowledge building theory and practice

This study investigated the effects of engaging students to collectively learn and work with knowledge in a computer-supported collaborative learning environment called Knowledge Forum on their views about knowledge building theory and practice. Participants were 24 teacher-education students who took a required course titled “Integrating Theory and Practice in Teaching.” Data mainly came from (1) student discourse recorded in a Knowledge Forum database, (2) a survey that examined students’ views about knowledge building, and (3) interviews with regard to students’ perceived barriers to implementing knowledge building theory in teaching. Findings suggest that with sustained discourse to construct their collective understanding of the relationships between theory and practice in teaching for a semester, the participants were able to attain more informed and practical views about knowledge building theory. In addition, students’ perceived barriers to implementing knowledge building in teaching were identified and strategies to help overcome these barriers discussed.     

Students�� Experiences of Autonomy, Competence, Social Relatedness and Interest Within a CSCL Environment in Vocational Education: The Case of Commerce and Business Administration

To prepare students for effective workplace learning, it is necessary to have insight into the contextual characteristics that affect students’ developing interest. Aiming at students to become selfregulated learners, teachers should act as mindful coaches, encouraging their students to monitor the quality of collaborative group work. A field study was conducted within the context of a Computer Supported Collaborative Learning (CSCL) project fostering self-regulated learning. Students and teachers made use of an electronic instrument that assessed and visualised students’ experiences of the quality of group learning over time. 137 vocational students in commerce and business administration participated in a 6 months project requiring to work in small learning groups. A SEM model, based on self-determination theory, fitted the data quite well. Perceived autonomy, competence, and social relatedness seemed to be good predictors of students’ situational interest. Qualitative interview data revealed not only the added value of process-oriented reflection and within-group discussion, but also some shortcomings concerning the effectiveness of (implementing) CSCL.     

Using the SEE-SEP Model to Analyze Upper Secondary Students’ Use of Supporting Reasons in Arguing Socioscientific Issues

To achieve the goal of scientific literacy, the skills of argumentation have been emphasized in science education during the past decades. But the extent to which students can apply scientific knowledge to their argumentation is still unclear. The purpose of this study was to analyse 80 Swedish upper secondary students’ informal argumentation on four socioscientific issues (SSIs) to explore students’ use of supporting reasons and to what extent students used scientific knowledge in their arguments. Eighty upper secondary students were asked to express their opinions on one SSI topic they chose through written reports. The four SSIs in this study include global warming, genetically modified organisms (GMO), nuclear power, and consumption. To analyse students’ supporting reasons from a holistic view, we used the SEE-SEP model, which links the six subject areas of sociology/culture (So), environment (En), economy (Ec), science (Sc), ethics/morality (Et) and policy (Po) connecting with three aspects, knowledge, value and personal experience (KVP). The results showed that students used value to a greater extent (67%) than they did scientific knowledge (27%) for all four SSI topics. According to the SEE-SEP model, the distribution of supporting reasons generated by students differed among the SSI topics. Also, some alternative concepts were disclosed in students’ arguments. The implications for research and education are discussed.     

Students’ experiences with contrasting learning environments: The added value of students’ perceptions

This study investigated the effects of two contrasting learning environments on students’ course experiences: a lecture-based setting to a student-activating teaching environment. In addition, the evaluative treatment involved five research conditions that went together with one of four assessment modes, namely, portfolio, case-based, peer assessment, and multiple-choice testing. Data (N = 608) were collected using the Course Experience Questionnaire. Results showed that the instructional intervention (i.e. lectures versus student-activating treatment) influenced students’ course experiences, but in the opposite direction to that expected. In declining order, the following scales (5 out of 7) revealed statistically significant differences: Clear Goals and Standards; the General scale; Appropriate Workload; Good Teaching; and Independence. Moreover, when the assessment mode was considered, also the Appropriate Assessment scale demonstrated significant differences between the five research conditions. Moreover, the same teaching/learning environments led to diverse students’ perceptions. While the perceptions of lecture-taught students were focused and concordantly positive, students’ course experiences with student-activating methods were widely varied and both extremely positive and negative opinions were present. Students’ arguments in favour of the activating setting were the variety of teaching methods, the challenging and active nature of the assignments and the joys of collaborative work in teams, whereas students expressed dissatisfaction with the perceived lack of learning gains, the associated time pressure and workloads, and the (exclusive) use of collaborative assignments and related group difficulties.     

Portraits of middle school students constructing evidence-based arguments during problem-based learning: the impact of computer-based scaffolds

A critical step in problem-based learning (PBL) units occurs when groups present their solution to the central problem. This is challenging for middle school students because it involves the creation of an evidence-based argument (Krajcik et al., Journal of the Learning Sciences 7:313–350, 1998). Using a mixed method design, this study investigated (a) the impact of computer-based argumentation scaffolds on middle school students’ argumentation ability, and (b) what middle school students used for support and why during a PBL unit. Data sources included persuasive presentation rating scores, argument evaluation ability test, videotaped class sessions, and prompted interviews. Results included a significant impact on average-achieving students’ argument evaluation ability, and use of the scaffolds by the small groups to plan their research and keep organized.     

Students’ conceptual practices in science education

Recent research has to a limited extent explored the characteristics of students’ conceptual practices as sociocultural phenomena in general and in science education in particular. I approach this issue by studying a group of students while solving a particular scientific problem from A to Z, and as part of this analyse how different cultural means (the knowledge domain and the tools in use) structure the students’ interactions and how their interpersonal relations change over this period of time. The aim is to illustrate how these cultural means intersect in productive and less productive ways during the students’ conceptual practices. The study has its point of departure in a design experiment where a group of four students, together with their teacher, solve different problems related to the biological phenomenon of sequencing a DNA molecule (the insulin gene). Video-recordings of the students’ interactions constitute the basis for this analysis. The cultural means strongly structure the students’ conceptual practices during their problem solving processes. Whereas the knowledge domain structured the whole process, the significant roles of the website and the computer-based 3D model of the insulin gene were especially apparent during the second part of the trajectory. The intersection of these cultural means appear productive in terms of disciplinary knowledge when the students’ became aware of how to handle this relationship. The interpersonal relations between the students and their teacher altered slightly in the beginning and became increasingly more fixed during the students’ progression.     

Revealing educationally critical aspects of rate

Rate (of change) is an important but complicated mathematical concept describing a ratio comparing two different numeric, measurable quantities. Research referring to students’ difficulties with this concept spans more than 20 years. It suggests that problems experienced by some calculus students are likely a result of pre-existing limited or incorrect conceptions of rate. This study investigated 20 Australian Year 10 students’ understanding of rate as revealed by phenomenographic analysis of interviews. Eight conceptions of rate emerged, leading to the identification of four educationally critical aspects of the concept which address gaps in students’ thinking. In addition, the employment of phenomenography, to reveal conceptions of rate, is described in detail.