Gabriel J. Stylianides

Procedural and Conceptual Knowledge: Exploring the Gap Between Knowledge Type and Knowledge Quality

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.RésuméDans la lignée de Star (2005, 2007), nous continuons de problématiser l’absence de distinction entre type et qualité lorsqu’il est question de connaissance des concepts et des procédures. Bien que ceux dont les travaux se fondent sur les types of connaissances et ceux dont les travaux se fondent sur les qualités des connaissances semblent faire référence aux mêmes phénomènes, ce n’est pas le cas en réalité. Le manque de compréhension réciproque, aussi bien de la nature des questions posées que des résultats obtenus, cause des difficultés pour l’exploration de questions importantes en enseignement et en apprentissage des mathématiques, par exemple la question des connaissances des enseignants.

Learning by teaching simstudent: technical accomplishments and an initial use with students

The purpose of the current study is to test whether we could create a system where students can learn by teaching a live machine-learning agent, called SimStudent. SimStudent is a computer agent that interactively learns cognitive skills through its own tutored-problem solving experience. We have developed a game-like learning environment where students learn algebra equations by tutoring SimStudent. While Simulated Students, Teachable Agents and Learning Companion systems have been created, our study is unique that it genuinely learns skills from student input. This paper describes the overview of the learning environment and some results from an evaluation study. The study showed that after tutoring SimStudent, the students improved their performance on equation solving. The number of correct answers on the error detection items was also significantly improved. On average students spent 70.0 minutes on tutoring SimStudent and used an average of 15 problems for tutoring.

Studying the Effect of Tutor Learning Using a Teachable Agent that Asks the Student Tutor for Explanations

We have built Sim Student, a computational model of learning, and applied it as a peer learner that allows students to learn by teaching. Using Sim Student, we study the effect of tutor learning. In this paper, we discuss an empirical classroom study where we evaluated whether asking students to provide explanations for their tutoring activities facilitates tutor learning - the self-explanation effect for tutor learning. The results showed that students in the self-explanation condition displayed the same amount of learning gain as students in the non-self-explanation condition, but with a significantly smaller number of problems tutored (during the same time). The study also showed an apparent increase in effectiveness relative to a prior study, which is arguably due to improvement of the system based on the iterative system-engineering effort.

Learning by teaching SimStudent: an initial classroom baseline study comparing with cognitive tutor

This paper describes an application of a machine-learning agent, SimStudent, as a teachable peer learner that allows a student to learn by teaching. SimStudent has been integrated into APLUS (Artificial Peer Learning environment Using SimStudent), an on-line game-like learning environment. The first classroom study was conducted in local public high schools to test the effectiveness of APLUS for learning linear algebra equations. In the study, learning by teaching (i.e., APLUS) was compared with learning by tutored-problem solving (i.e., Cognitive Tutor). The results show that the prior knowledge has a strong influence on tutor learning - for students with insufficient training on the target problems, learning by teaching may have limited benefits compared to learning by tutored problem solving. It was also found that students often use inappropriate problems to tutor SimStudent that did not effectively facilitate the tutor learning.

A Type of Parental Involvement With an Isomorphic Effect on Urban Children’s Mathematics, Reading, Science, and Social Studies Achievement at Kindergarten Entry

Research showed that children’s school-entry academic skills are strong predictors of their later achievement, thereby highlighting the importance of children’s achievement at kindergarten entry. This article defines a particular type of parental involvement in children’s education and uses a representative sample of American urban kindergarteners to examine its effect on urban children’s mathematics, reading, science, and social studies achievement at kindergarten entry. The findings in this article are isomorphic in the different subject areas and show that children with more access to this particular type of parental involvement tend to have higher academic achievement than their peers.

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion, a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.

The Role of Instructional Engineering in Reducing the Uncertainties of Ambitious Teaching

Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious teaching is often confronted with uncertainties about how to advance students’ learning while also building on students’ contributions. In this article we propose a framework that aims to deepen understanding about the role of instructional engineering in helping reduce the uncertainties of ambitious teaching, particularly with regard to the design and implementation of task sequences that target academically important but difficult-to-achieve learning goals. To illustrate the framework, we consider how instructional engineering helped reduce the uncertainties in enacting ambitious teaching to advance university and secondary students’ understanding of what counts as “proof” in mathematics.

Principles of Task Design for Conjecturing and Proving

Principles of task design should have both the fundamental function of a clear relation to the learner’s rules, learning powers or hypothetical learning trajectories and the practical function of easy evaluation of many similar tasks. Drawing on some theories and practical tasks in the literature, we developed a total of 11 principles of task design for learning mathematical conjecturing (4), transiting between conjecturing and proving (2), and proving (5). To further validate the functioning of those principles, more empirical research is encouraged.

Teachers’ Professional Learning of Teaching Proof and Proving

This chapter reviews studies on teachers’ professional learning of teaching proof and proving. From them we conceptualise three essential components of successful teaching: teachers’ knowledge of proof, proof practices and beliefs about proof. With respect to each component, we examine research studies of primary and secondary teachers. We also discuss the challenges teachers may face in teaching proof and proving, as well as teachers’ professional learning activities. Throughout, we argue that the three components are interrelated in successful teaching of proof and proving. This argument raises a new challenge for further research.

Studying the Effect of a Competitive Game Show in a Learning by Teaching Environment

In this paper we investigate how competition among tutees in the context of learning by teaching affects tutors’ engagement as well as tutor learning. We conducted this investigation by incorporating a competitive Game Show feature into an online learning environment where students learn to solve algebraic equations by teaching a synthetic peer, called SimStudent. In the Game Show, pairs of SimStudents trained by students beforehand competed against each other by solving challenging problems to attain higher ratings. The results of a classroom study with 141 7th through 9th grade students showed the following: (1) Students improved their proficiency to solve equations after teaching SimStudent, but there was no observed improvement in their conceptual understanding. (2) Overall, the competitive Game Show promoted students’ extrinsic and intrinsic motivations—when the competitive Game Show was available, students’ engagement in tutoring (intrinsic motivation) was increased; students who arguably had a higher desire to win strategically selected opponents with lower proficiency for an easy win (extrinsic motivation). (3) The availability of the competitive Game Show did not affect tutor learning; there was no notable correlation between students’ motivation (intrinsic or extrinsic) and tutor learning. Based on these findings, we propose design improvements to increase tutor learning.