Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course

We present results from a classroom-based intervention designed to help a class of grade 10 students (14–15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students’ solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs. Grounded on a conception of proof that includes both empirical and deductive mathematical argumentations, we show the trajectories of some students progressing from developing basic empirical proofs towards developing deductive proofs and understanding the role of conjectures and proofs in mathematics. Our analysis of students’ solutions is based on networking Boero et al.’s construct of cognitive unity of theorems, Pedemonte’s structural and referential analysis of conjectures and proofs, and Balacheff and Margolinas’ cK¢ model, while using Toulmin schemes to represent students’ productions. This combination has allowed us to identify several emerging types of cognitive unity/rupture, corresponding to different ways of solving conjecture-and-proof problems. We also show that some types of cognitive unity/rupture seem to induce students to produce deductive proofs, whereas other types seem to induce them to produce empirical proofs.     

KNOWLEDGE USE IN THE CONSTRUCTION OF GEOMETRY PROOF BY SRI LANKAN STUDENTS

Within the domain of geometry, proof and proof development continues to be a problematic area for students. Battista (2007) suggested that the investigation of knowledge components that students bring to understanding and constructing geometry proofs could provide important insights into the above issue. This issue also features prominently in the deliberations of the 2009 International Commission on Mathematics Instruction Study on the learning and teaching of proofs in mathematics, in general, and geometry, in particular. In the study reported here, we consider knowledge use by a cohort of 166 Sri Lankan students during the construction of geometry proofs. Three knowledge components were hypothesised to influence the students’ attempts at proof development: geometry content knowledge, general problem-solving skills and geometry reasoning skills. Regression analyses supported our conjecture that all 3 knowledge components played important functions in developing proofs. We suggest that whilst students have to acquire a robust body of geometric content knowledge, the activation and the utilisation of this knowledge during the construction of proof need to be guided by general problem-solving and reasoning skills.     

Learning Through Constructing Representations in Science: A framework of representational construction affordances

Compared with research on the role of student engagement with expert representations in learning science, investigation of the use and theoretical justification of student-generated representations to learn science is less common. In this paper, we present a framework that aims to integrate three perspectives to explain how and why representational construction supports learning in science. The first or semiotic perspective focuses on student use of particular features of symbolic and material tools to make meanings in science. The second or epistemic perspective focuses on how this representational construction relates to the broader picture of knowledge-building practices of inquiry in this disciplinary field, and the third or epistemological perspective focuses on how and what students can know through engaging in the challenge of representing causal accounts through these semiotic tools. We argue that each perspective entails productive constraints on students’ meaning-making as they construct and interpret their own representations. Our framework seeks to take into account the interplay of diverse cultural and cognitive resources students use in these meaning-making processes. We outline the basis for this framework before illustrating its explanatory value through a sequence of lessons on the topic of evaporation.     

Values and norms of proof for mathematicians and students

In this theoretical paper, we present a framework for conceptualizing proof in terms of mathematical values, as well as the norms that uphold those values. In particular, proofs adhere to the values of establishing a priori truth, employing decontextualized reasoning, increasing mathematical understanding, and maintaining consistent standards for acceptable reasoning across domains. We further argue that students’ acceptance of these values may be integral to their apprenticeship into proving practice; students who do not perceive or accept these values will likely have difficulty adhering to the norms that uphold them and hence will find proof confusing and problematic. We discuss the implications of mathematical values and norms with respect to proof for investigating mathematical practice, conducting research in mathematics education, and teaching proof in mathematics classrooms.     

Students’ understanding of the structure of deductive proof

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.     

Fostering empirical examination after proof construction in secondary school geometry

In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term “empirical examination” refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers’ roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers’ actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement.     

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.     

Unpacking the logic of mathematical statements

This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61 students in six small sections of a bridge course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion ofproof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them, i.e., determine their correctness.     

Enhancing students’ mathematical reasoning in the classroom: teacher actions facilitating generalization and justification

A proof is a connected sequence of assertions that includes a set of accepted statements, forms of reasoning and modes of representing arguments. Assuming reasoning to be central to proving and aiming to develop knowledge about how teacher actions may promote students’ mathematical reasoning, we conduct design research where whole-class mathematical discussions triggered by exploratory tasks play a key role. We take mathematical reasoning as making justified inferences and we consider generalizing and justifying central reasoning processes. Regarding teacher actions, we consider inviting, informing/suggesting, supporting/guiding and challenging actions can be identified in whole-class discussions. This paper presents design principles for an intervention geared to tackle such reasoning processes and focuses on a whole-class discussion on a grade 7 lesson about linear equations and functions. Data analysis concerns teacher actions in relation to design principles and to the sought mathematical reasoning processes. The conclusions highlight teacher actions that lead students to generalize and justify. Generalizations may arise from a central challenging action or from several guiding actions. Regarding justifications, a main challenging action seems to be essential, while follow-up guiding actions may promote a further development of this reasoning process. Thus, this paper provides a set of design principles and a characterization of teacher actions which enhance students’ mathematical reasoning processes such as generalization and justification.     

Multimedia resources designed to support learning from written proofs: an eye-movement study

This paper presents two studies of an intervention designed to help undergraduates comprehend mathematical proofs. The intervention used multimedia resources that presented proofs with audio commentary and visual animations designed to focus attention on logical relationships. In study 1, students studied an e-Proof or a standard written proof and their comprehension was assessed in both immediate and delayed tests; the groups performed similarly at immediate test, but the e-Proof group exhibited poorer retention. Study 2 accounted for this unexpected result by using eye-movement analyses to demonstrate that participants who studied an e-Proof exhibited less processing effort when not listening to the audio commentary. We suggest that the extra support offered by e-Proofs disrupts the processes by which students organise information, and thus restricts the extent to which their new understanding is integrated with existing knowledge. We discuss the implications of these results for evaluating teaching innovations and for supporting proof comprehension.     

Justification enlightenment and combining constructions of knowledge

This case study deals with a solitary learner’s process of mathematical justification during her investigation of bifurcation points in dynamic systems. Her motivation to justify the bifurcation points drove the learning process. Methodologically, our analysis used the nested epistemic actions model for abstraction in context. In previous work, we have shown that the learner’s attempts at justification gave rise to several processes of knowledge construction, which develop in parallel and interact. In this paper, we analyze the interaction pattern of combining constructions and show that combining constructions indicate an enlightenment of the learner. This adds an analytic dimension to the nested epistemic actions model of abstraction in context.     

Relationship between inductive arithmetic argumentation and deductive algebraic proof

In this paper, we present a cognitive analysis of the relationship between the argumentation process leading to the construction of a conjecture and its algebraic proof in solving Calendar Algebra problems. To solve this kind of problem, students encounter two sources of potential difficulties: the shift from using arithmetic in the argumentation to using algebra in the proof and the shift from an inductive argument towards a deductive proof. Thus, the aims of this article are to describe these cognitive difficulties and to show how students overcome them. Methodologically, we compare students’ problem solving process corresponding to three problems presented in the first four lessons of a teaching experiment. The analysis and comparison between these three resolution processes is performed using Toulmin’s model.     

Latinx* popular culture imaginaries: examining Puerto Rican children’s social discourses in interpreting telenovelas

Abstract

In this paper, we present a collaboration project within one urban Puerto Rican classroom, focused on constructing a critical literacy inquiry curriculum grounded in the students’ out-of-school literacy practices in their communities, including their experiences with media and popular culture. We focused on a critical literacy and media inquiry unit centered on the students’ self-selected subject of the telenovela. Here, we examine one student’s work to highlight two overarching findings: (1) the visibility of the students’ complex understanding of the media landscapes in telenovelas, particularly the construction of dominant social discourses across telenovela worlds, and (2) the ways that bringing children’s mediatized cultural imaginaries in their creative work supports an approach to literacy in classrooms, where explorations of discourses of power emerge from the students’ knowledge. In order to articulate how children actively examine and construct discourses across multiple social worlds, we examine these findings using the Four Resource Model, and elements of discourse analysis, as theoretical and analytical frameworks, focusing on the construction of identities, worlds and meanings in relation to the social discourses of telenovelas.     

Software that assists learning within a complex abstract domain: the use of constraint and consequentiality as learning mechanisms

This paper describes a research project into undergraduates’ use of a software tool to learn symbolic logic—a complex abstract domain that has been shown to be intimidating for students. The software allows the students to manipulate proofs in certain ways and then calculates the consequences of their actions. A research method has been developed that allowed students’ use of this tool to be modelled, and this model was then used to identify, refine and create visual cues that provide support for students’ reasoning. The focus of this paper is the role of the software as an artefact to aid students’ visualisation of reasoning processes rather than the logic itself. The main mechanisms by which this visualisation is supported are the imposition of constraints on the actions available and the demonstration to students of the consequences of their actions. The study shows that the software encouraged experimentation with different routes to a proof, and constituted a challenge to fixated reasoning.     

Beliefs about justification for knowing when ethnic majority and ethnic minority students read multiple conflicting documents

We examined the role of justification for knowing beliefs in learning and comprehension when ethnic majority and ethnic minority students from the same school classes read five conflicting documents on the scientific issue of sun exposure and health. Results showed that the more ethnic minority students trusted scientific authorities and the less they relied on personal opinion when validating knowledge claims in the domain of science, the more they learned from and the better they comprehended the documents. In contrast, justification for knowing beliefs did not seem to play a role in learning and comprehension among ethnic majority students. These results may reflect that the documents represented more of a challenge to the ethnic minority students, with justification beliefs affecting learning and comprehension processes to a greater extent when the task is perceived as an ill-structured problem. This study is probably the first to indicate different relationships between various justification beliefs and performance in different language and cultural groups, having theoretical as well as educational implications.     

Actions speak louder with words: The roles of action and pedagogical language for grounding mathematical proof

Theories of grounded and embodied cognition posit that situated actions are central constituents in cognitive processes. We investigate whether grounding actions influence reasoning, and how pedagogical language influences the action–cognition relationship. Undergraduate students (N = 120) generated proofs for two mathematical tasks after performing either grounding or non-grounding actions. Grounding actions facilitated key mathematical insights for both tasks, but did not lead to superior proofs. Pedagogical language in the form of prompts (prospective statements) and hints (retrospective statements) accompanying grounding actions enhanced proof performance on one task but not the other. Results from transfer tasks suggested that participants learned to apply their mathematical insights to new contexts. The findings suggest that relations between action and cognition are reciprocal: actions facilitate insight, while pedagogical language strengthens the influence of task-relevant actions for proof production. Pedagogically supported grounding actions offer alternative ways of fostering mathematical reasoning.     

Making the transition to formal proof

This study examined the cognitive difficulties that university students experience in learning to do formal mathematical proofs. Two preliminary studies and the main study were conducted in undergraduate mathematics courses at the University of Georgia in 1989. The students in these courses were majoring in mathematics or mathematics education. The data were collected primarily through daily nonparticipant observation of class, tutorial sessions with the students, and interviews with the professor and the students. An inductive analysis of the data revealed three major sources of the students' difficulties: (a) concept understanding, (b) mathematical language and notation, and (c) getting started on a proof. Also, the students' perceptions of mathematics and proof influenced their proof writing. Their difficulties with concept understanding are discussed in terms of a concept-understanding scheme involving concept definitions, concept images, and concept usage. The other major sources of difficulty are discussed in relation to this scheme.This article is based on the author's doctoral dissertation completed in 1990 at the University of Georgia under the direction of Jeremy Kilpatrick.     

Learner agency and the curriculum: a critical realist perspective

Abstract

Agency, understood as the capacity to act independently and to make one’s own choices, is considered central to children’s development. Thus, education, and hence education curricula, have a role in the development of learner agency. While curriculum development is a key focus for educational theory, research, policy, and classroom practice, the potential implications of curriculum content selections for learner agency remain underexplored. Theoretically, this paper engages with critical realism, explaining how it can provide theoretical foundation for a more comprehensive view of learner agency and, by implication, more balanced curricula. Empirically, the paper draws on the findings from a content analysis of the national curriculum documents of four countries with relatively high scores in international comparative tables, England, Australia, Hong-Kong, and Canada, to develop a new typology of primary curricula. Based on the extent of emphasis placed on knowledge versus skills, values, and attitudes, three types of curricula were identified: knowledge-based, skills-oriented, and learner-centred. Due to its significant theoretical and practical influence globally, we focus on the knowledge-based model and its likely impact on students’ agency. We conclude by highlighting the importance of making learner agency a key orientation of the curriculum and suggesting directions for future research.