Exploring Repunits

This article describes a discovery activity which is organized and developed at two different levels of treatment corresponding to the different degree of mathematical maturity of students. Using an 8-digit calculator, students at the first level (7-9th graders) explore patterns which lead them to discover the properties of repunits and their relationships to many mathematical curiosities. At the second level (10-12th graders) students may be interested in further extension exploring why these patterns emerge and offering algebraic justification.     

How Do Students Know What They Learn in Middle School Mathematics Is True?

This article presents ways in which students ascertain that what they have learned in mathematics is true. Students in the middle school (and a few from other grades) were interviewed by prospective and in‐service teachers. Students were asked what they had learned recently in mathematics and how they knew it was true. The answers were grouped by the author according to the justification schemes used by the students in their explanations. Students interviewed used three kinds of justification schemes: externally based, empirical, and analytic. For each kind, examples are provided of students' justifications. Additional insights are included from the reflections of the interviewers. Some suggestions are offered regarding how teachers can help increase their students' ability to give convincing arguments in mathematics.     

Examining epistemic beliefs across conceptual and procedural knowledge in statistics

The purpose of this paper was to examine whether students’ epistemic beliefs differed as a function of variations in procedural versus conceptual knowledge in statistics. Students completed Hofer’s (Contem Edu Psychol 25:378–405, 2000) Discipline-Focused Epistemological Beliefs Questionnaire five times over the course of a semester. Differences were explored between students’ initial beliefs about statistics knowledge and their specific beliefs about conceptual knowledge and procedural knowledge in statistics. Results revealed differences across these contexts; students’ beliefs differed between procedural versus conceptual knowledge. Moreover, students’ initial beliefs about statistics knowledge were more similar to their beliefs about conceptual knowledge rather than procedural knowledge. Finally, regression analyses revealed that students’ beliefs about the justification of knowledge, attainability of truth and source of knowledge were significant predictors of examination performance, depending on the examination. These results have important theoretical, methodological and pedagogical implications.     

Middle school students’ patterning performance on semi-free generalization tasks

This longitudinal study empirically addresses the issue of structure construction and justification among a class of US seventh and eighth-grade Algebra 1 students (mean age of 12.5 years) in the context of novel semi-free pattern generalization (PG) tasks before and after a teaching experiment that emphasized a multiplicative thinking approach to patterns. We compared the students’ PG responses before and after the experiment and found that (1) one source of variability in their abduced structural processing was in part due to an initial conceptual preference toward thinking either in parts or in wholes and (2) a multiplicative understanding of structures significantly aided them in PG conversion (e.g., from the visual to the alphanumeric) and processing (e.g., from nonstandard to standard function-based formulas). Our findings provide both necessary and sufficient conditions for constructing, establishing, and justifying valid structures in the case of (semi-) free figural patterning tasks.     

Beliefs and engagement structures: behind the affective dimension of mathematical learning

Beliefs influencing students’ mathematical learning and problem solving are structured and intertwined with larger affective and cognitive structures. This theoretical article explores a psychological concept we term an engagement structure, with which beliefs are intertwined. Engagement structures are idealized, hypothetical constructs, analogous in many ways to cognitive structures. They describe complex “in the moment” affective and social interactions as students work on conceptually challenging mathematics. We present engagement structures in a self-contained way, paying special attention to their theoretical justification and relation to other constructs. We suggest how beliefs are characteristically woven into their fabric and influence their activation. The research is based on continuing studies of middle school students in inner-city classrooms in the USA.     

“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.     

Developing conceptions of statistics by designing measures of distribution

Students often learn procedures for measuring, but rarely do they grapple with the foundational conceptual problem of generating and validating coordination between a measure and the phenomenon being measured. Coordinating measures with phenomenon involves developing an appreciation of the objects and relations in each as well as establishing their mutual correspondence. We supported students?? developing conceptions of statistics by positioning them to design measures of center and of variability for distributions that they had generated through repeated measure of a length. After students invented and explored the viability of their measures individually, they participated in a public (whole-class conversation) forum featuring justification and reflection about the viability of their designed measures. We illustrate how individual invention enticed students to attend to, and to make explicit, characteristics of distribution not initially noticed or known only tacitly. Conceptions of statistics and of relevant characteristics of distribution were further expanded as students justified and argued about the utility and prospective generalization of particular inventions. Teachers supported student learning by highlighting prospective relations between characteristics of measures and characteristics of distribution as they emerged during the course of activity in each setting.     

Justifying and Proving in the Cabri Environment

This paper describes a long term teaching experiment carried out with students from the 9th–10th grades. Geometrical constructions in the Cabri environment were selected as a specific field of experience, within which the sense of theory may emerge. The idea of construction constitutes the key to accessing the idea of theorem, moving from a generic idea of justification towards the idea of validating within a geometrical system. The study aims at clarifying the role of the Cabri environment in this teaching-learning processes: analysis of protocols shows the possible evolution of a justification into a proof but at the same time indicates that this evolution is not expected to be simple and spontaneous.This revised version was published online in September 2005 with corrections to the Cover Date.     

Intertwining special education and mathematics education perspectives to design an intervention to improve student understanding of symbolic numerical magnitude

In our study we examined changes in student justifications over time with an intervention that drew from the best instructional practices in the fields of special education and mathematics education. These justifications were provided by teacher-identified struggling second-grade students while engaging in symbolic numerical magnitude comparisons. Following screening, we conducted 8 instructional sessions to promote conceptual understanding of fundamental ideas for numerical magnitude. Using data collected from 71 instructional tasks, we analyzed the types of justifications students provided and how these justifications changed over time. Prior to the intervention, most student justifications involved few components of a valid mathematical justification. Over the course of the study, students provided more valid and generalizable mathematical justifications.     

Coherentism,Truth, and Witness Agreement

Coherentists on epistemic justification claim that all justification is inferential, and that beliefs, when justified, get their justification together (not in isolation) as members of a coherent belief system. Some recent work in formal epistemology shows that “individual credibility” is needed for “witness agreement” to increase the probability of truth and generate a high probability of truth. It can seem that, from this result in formal epistemology, it follows that coherentist justification is not truth-conducive, that it is not the case that, under the requisite conditions, coherentist justification increases the probability of truth and generates a high probability of truth. I argue that this does not follow.