This article describes an advanced learning technology used to investigate hypotheses about learning by teaching. The proposed technology is an instance of a teachable agent, called SimStudent, that ...learns skills (e.g., for solving linear equations) from examples and from feedback on performance. SimStudent has been integrated into an online, gamelike environment in which students act as "tutors" and can interactively teach SimStudent by providing it with examples and feedback. We conducted 3 classroom "in vivo" studies to better understand how and when students learn (or fail to learn) by teaching. One of the strengths of interactive technologies is their ability to collect detailed process data on the nature and timing of student activities. The primary purpose of this article is to provide an in-depth analysis across 3 studies to understand the underlying cognitive and social factors that contribute to tutor learning by making connections between outcome and process data. The results show several key cognitive and social factors that are correlated with tutor learning. The accuracy of students' responses (i.e., feedback and hints), the quality of students' explanations during tutoring, and the appropriateness of tutoring strategy (i.e., problem selection) all positively affected SimStudent's learning, which further positively affected students' learning. The results suggest that implementing adaptive help for students on how to tutor and solve problems is a crucial component for successful learning by teaching.
Although the notion of proof is important for all learners’ mathematical experiences, there has been limited attention to what it might involve and look like to introduce students and prospective ...teachers to proof. In this paper we argue for the importance of having a coherent approach to introducing students and prospective teachers to proof, and we discuss the theoretical basis of a learning trajectory relevant to both groups. We also discuss an instructional sequence that aimed to promote the learning trajectory among English secondary students and U.S. prospective elementary teachers, drawing on data from two multi-year design experiments. The learning trajectory comprises two milestones: (1) seeing a need to learn about proof and (2) developing an operationally functional conceptualization of proof. The “need” in milestone 1 entails an aspect of epistemological justification applicable to both students and prospective teachers, and a further aspect of pedagogical justification applicable to prospective teachers.
•It is unclear what it might involve and look like to introduce learners to proof.•A learning trajectory is proposed and its theoretical basis discussed.•The learning trajectory is applicable to both students and prospective teachers.•An instructional sequence was designed to promote the learning trajectory.•Data from two design experiments support the promise of the instructional sequence.
•Reasoning-and-proving (RP) is important for all students’ learning of mathematics.•RP receives little attention in classrooms and students face serious difficulties with it.•Textbooks can offer an ...important leverage point for supporting classroom work on RP.•Little is known about how RP is treated in existing textbooks.•Significant methodological challenges surround textbook analyses on RP.
The activity of ‘reasoning-and-proving’ can serve as a vehicle to mathematical sense making and is thus important for students’ learning of mathematics at all levels of education. Yet, reasoning-and-proving does not receive appropriate attention in typical classroom practice, and many students face serious difficulties with it. One important, albeit underexplored and insufficiently exploited, leverage point for supporting classroom work on reasoning-and-proving is textbooks. As a first step towards a possible longer-term goal to develop textbooks that can appropriately support classroom work on reasoning-and-proving, it is necessary to understand how reasoning-and-proving is treated in existing textbooks and to grapple with the many methodological challenges that surround textbook analyses on reasoning-and-proving.
Proof Constructions and Evaluations Stylianides, Andreas J.; Stylianides, Gabriel J.
Educational studies in mathematics,
11/2009, Letnik:
72, Številka:
2
Journal Article
Recenzirano
In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof, and we examine their ability to construct proofs and evaluate their own ...constructions. We claim that the combined "construction-evaluation" activity helps illuminate certain aspects of prospective teachers' and presumably other individuals' understanding of proof that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid. Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof, the additional consideration of prospective teachers' evaluations of their own constructions overruled this interpretation and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account of our findings, and we discuss implications for research and instruction.
The activity of
reasoning-and-proving
is at the heart of mathematical sense making and is important for all students’ learning as early as the elementary grades. Yet, reasoning-and-proving tends to ...have a marginal place in elementary school classrooms. This situation can be partly attributed to the fact that many (prospective) elementary teachers have (1) weak mathematical (subject matter) knowledge about reasoning-and-proving and (2) counterproductive beliefs about its teaching. Following up on an intervention study that helped a group of prospective elementary teachers make significant progress in overcoming these two major obstacles to teaching reasoning-and-proving, we examined the challenges that three of them identified that they faced as they planned and taught lessons related to reasoning-and-proving in their mentor teachers’ classrooms. Our findings contribute to research knowledge about major factors (other than the well-known factors related to teachers’ mathematical knowledge and beliefs) that deserve attention by teacher education programs in preparing prospective teachers to teach reasoning-and-proving.
Although the notion of assumptions is important in mathematical activity as early as the elementary school, there is limited research on how to help elementary teachers develop mathematical knowledge ...for teaching related to assumptions. In this paper, we discuss the theoretical foundation and implementation of an intervention that aimed to promote three key elements of this knowledge among prospective elementary teachers. We developed the intervention in a 4-year design experiment that we conducted in an undergraduate mathematics course for prospective elementary teachers. The intervention's design utilized the notion of productive ambiguity in the context of a deliberately ambiguous task where the role of assumptions surfaced and was reflected upon in purposefully organized ways. We focus on the implementation of the intervention in the last of 5 research cycles of our design experiment to exemplify our theoretical framework and to discuss the promise of the intervention to promote the three targeted elements of knowledge. The approach to promoting mathematical knowledge for teaching that we discuss in the paper offers a paradigmatic case of how teacher educators can use productive ambiguity to design learning opportunities for prospective teachers to intertwine mathematical learning with pedagogical awareness thus developing pedagogically functional mathematical knowledge.
In this article we elaborate a conceptualisation of
mathematics for teaching as a form of applied mathematics (using Bass's idea of characterising mathematics education as a form of applied ...mathematics) and we examine implications of this conceptualisation for the mathematical preparation of teachers. Specifically, we focus on issues of design and implementation of a special kind of mathematics tasks whose use in mathematics teacher education can support the development of knowledge of mathematics for teaching. Also, we discuss broader implications of the article for mathematics teacher education, including implications for mathematics teacher educators' knowledge for promoting mathematics for teaching.
In a study involving 66 masters and 60 sixth-grade students, we conducted Principal Component Analysis to identify more-and-less competent problem posers based on performance criteria rather than, as ...in prior research, relying on participants’ mathematical experience or background. Also, to cast light on characteristics of competent posers, we explored possible patterns in the problem-posing process based on the two identified groups’ self-reports and eye-movements. The results showed that: masters students had a significantly lower proportion of the more-competent group and a higher proportion of the less-competent group than sixth graders; more-competent posers perceived a better understanding of the problem-posing tasks than less-competent posers; more-competent posers exhibited significantly more fixation time on completing the entire problem-posing activities than less-competent posers, though this pattern disappeared across particular stages of the problem-posing process; and more-competent posers appeared to engage in a more purposeful search and processing to construct their problems than less-competent posers.
•A data-driven approach, on the basis of certain performance criteria, was used to identify competent problem posers.•More-competent problem posers do not necessarily have higher mathematical experience or knowledge base than their less-competent counterparts.•More-competent problem posers perceived a better understanding of problem posing tasks than less-competent problem posers.•More-competent problem posers showed more willingness to engage in problem posing tasks than less-competent posers.•More-competent problem posers appeared to engage in a more purposeful viewing pattern to construct problems than less-competent posers.
This systematic review aims to provide a complementary to existing synopses of the state-of-the-art of mathematics education research on
proof
and
proving
in both school and university mathematics. ...As an organizing framework, we used Cohen et al.’s triadic conceptualization of instruction, which draws attention not only to the main actors of the didactical process (i.e., the
Teacher
and
Students
) and the
Content
around which the actors’ work is organized (herein, content related to proof and proving), but also to the relationships among the actors and the content. Out of the 103 papers we reviewed, almost half fell in the Student-Content category, which is consistent with the existence of a substantial number of frameworks, methods, and research findings related to students’ engagement with proof and proving. About a quarter of the papers fell in the Student–Teacher-Content category, which reflects an emphasis on viewing instructional practice in proof and proving in a holistic, systemic way. Only few papers fell in the categories that did not include Content in them, namely, the categories of Student, Teacher, and Student–Teacher; this suggests mathematics education research on proof and proving has a strong disciplinary identity, which potentially differentiates it from other mathematics education research strands. About a fifth of the papers were oriented towards ‘breaking ground’ through making an explicit theoretical and/or methodological contribution (Student–Teacher-Content and Content were the main categories where such contributions appeared), whilst the majority of the papers were focused on ‘building ground’ through elaborating or employing existing methodological and/or theoretical approaches.
Following Star (2005, 2007), we continue to problematize the entangling of type and quality in the use of conceptual knowledge and procedural knowledge. Although those whose work is guided by types ...of knowledge and those whose work is guided by qualities of knowledge seem to be referring to the same phenomena, actually they are not. This lack of mutual understanding of both the nature of the questions being asked and the results being generated causes difficulties for the continued exploration of questions of interest in mathematics teaching and learning, such as issues of teachers’ knowledge.
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