Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving

This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.

Making Explicit the Analysis of Students’ Mathematical Discourses – Revisiting a Newly Developed Methodological Framework

Sfard and Kieran [Kieran, C., Educational Studies in Mathematics 46, 2001, 187–228; Sfard, A., Educational Studies in Mathematics 46, 2001, 13–57; Sfard, A. and Kieran, C., Mind, Culture, and Activity 8, 2001, 42–76] have developed a methodological framework, which aims at characterizing the students’ mathematical discourses while they are working in groups. In this study, I focus on an important aspect of this methodological framework, namely the interactive flowcharts. The aim of this study is to suggest two complementary analyses for the construction of the interactive flowcharts: an additional analysis by means of the analytical construct of contextualization as well as an analysis of types of mathematical discourses. Based on data from a study of how four groups of Swedish engineering students collaboratively construct concept maps in linear algebra. I show that the two complementary analyses make the construction of the interactive flowcharts more coherent and transparent, and hence, more reliable. Furthermore, the two complementary analyses dramatically changed the picture as to whether the studied discourses were to be seen as mathematically productive or not. In the end of the article, I discuss the possibilities of performing the suggested additional analyses within the original methodological framework.     

Parameter Recovery in the Graded Response Model Using MULTILOG

The graded response model can be used to describe test-taking behavior when item responses are classified into ordered categories. In this study, parameter recovery in the graded response model was investigated using the MULTILOG computer program under default conditions. Based on items having five response categories, 36 simulated data sets were generated that varied on true θ distribution, true item discrimination distribution, and calibration sample size. The findings suggest, first, the correlations between the true and estimated parameters were consistently greater than 0.85 with sample sizes of at least 500. Second, the root mean square error differences between true and estimated parameters were comparable with results from binary data parameter recovery studies. Of special note was the finding that the calibration sample size had little influence on the recovery of the true ability parameter but did influence item-parameter recovery. Therefore, it appeared that item-parameter estimation error, due to small calibration samples, did not result in poor person-parameter estimation. It was concluded that at least 500 examinees are needed to achieve an adequate calibration under the graded model.     

Improving the judgment of task difficulties: prospective teachers’ diagnostic competence in the area of functions and graphs

To teach adaptively, teachers should be able to take the students’ level of knowledge into account. Therefore, a key component of pedagogical content knowledge (PCK) is the ability to assume the students’ perspectives. However, due to the so-called expert blind spot, teachers tend to misestimate their students’ knowledge, such as when estimating the difficulty of a given task. This empirically well-documented estimation bias is predicted by Nickerson’s anchoring and adjustment model, which generally explains how people take on other people’s perspectives. In this article, we present an intervention study that aims to improve the accuracy of prospective teachers’ judgments of task difficulty in the area of functional thinking. Two types of treatments are derived from Nickerson’s model. In the first condition (PCK group), participants acquire knowledge about task characteristics and students’ misconceptions. The second condition (sensitizing group) serves to control the idea that potential improvements in the PCK group are not merely based on a pure sensitization of the expert’s estimation bias. Accordingly, these participants are only informed about the general tendency of overestimating task difficulties. The results showed that the PCK group improved both in terms of the accuracy of the estimated solution rates and the accuracy of rank order, whereas the sensitizing group only improved in regard to the former. Thus, the study shows that prospective teachers’ diagnostic judgments can be improved by teaching them relevant PCK in a short amount of time.     

Integrating History of Science in Science Education through Historical Microworlds to Promote Conceptual Change

This paper proposes a new way to integrate history of science in science education to promote conceptual change by introducing the notion of historical microworld, which is a computer-based interactive learning environment respecting historic conceptions. In this definition, “interactive” means that the user can act upon the virtual environment by changing some parameters to see what ensues. “Environment respecting historic conceptions” means that the “world” has been programmed to respect the conceptions of past scientists or philosophers. Three historical microworlds in the field of mechanics are presented in this article: an Aristotelian microworld respecting Aristotle’s conceptions about movement, a Buridanian microworld respecting the theory of impetus and, finally, a Newtonian microworld respecting Galileo’s conceptions and Newton’s laws of movement.     

The bright and dark side of writing motivation: Effects of explicit instruction and peer assistance

The authors investigated the impact of explicit instruction and peer-assisted writing on students' writing motivation and self-efficacy for writing. Eleven teachers and their 206 fifth- and sixth-grade students participated in a 2 (explicit instruction vs. writing opportunities without explicit instruction) × 2 (peer-assisted writing vs. writing individually) experimental intervention study with a pretest-posttest design. The four experimental conditions were compared with a business-as-usual (BAU) condition. The five-week interventions were implemented in authentic classes by regular class teachers, who received a prior professional development training. Multilevel analyses showed that students who wrote with a peer were more autonomously motivated at posttest than BAU students. Additionally, BAU students and students receiving explicit instruction were more controlled motivated than students who were offered ample writing opportunities while practicing individually. Theoretical and educational implications are discussed in view of realizing a bright pathway towards autonomous writing motivation.     

Nonlongitudinal Estimation of Growth Curve Parameters: The Cohort Growth Model

The cohort growth model (CGM) is a method for estimating the parameters of a latent growth model (LGM) based on cross-sectional data. The CGM models the interindividual differences in the growth rate, and it models how subjects’ growth rate is related to their initial status. We derive model identification for the CGM and illustrate, in a simulation study, that the CGM provides unbiased parameter estimates in most simulation conditions. Based on empirical data we compare the estimates of the CGM with the estimates of the LGM. The results were comparable for both models. Although the estimates of the (co)-variances were different, the estimates of both models led to similar conclusions on the developmental change. Finally, we discuss the advantages and limitations of the CGM, and we provide recommendations for its use in empirical research.     

Escape-from-fear performance as affected by handling method and an additional CS-shock treatment

Experiment 1 demonstrated that hurdle-jumping, escape-from-fear (EFF) performance was affected deleteriously when the handling of the rats was by the tail rather than the body and was facilitated by the administration, just prior to EFF training, of one additional CS-shock pairing along with the opportunity to jump the hurdle. Additional experiments elucidated the basis for the effect of these two variables. Experiment 2 indicated that tail handling degraded performance by decreasing reinforcement. In Experiment 3, the facilitatory effect of the extra CS-shock treatment was found to result from the occurrence of a response at the termination of the pairing and not from the CS-shock pairing per se.