The effect of context on students’ reasoning about forces

When attempting to solve closely related problems in science, students will often respond to irrelevant contextual features in the questions rather than generalizing their conceptions over the range of relevant situations. In this study, a group of 40 students (one group of 15‐16‐year‐olds and another of preservice science teachers) was surveyed and interviewed to determine the effect of context on the reasoning which they used to solve problems concerning the forces acting on objects in linear motion. It was found that the younger group of students were influenced by contextual features such as the speed, weight and position of the moving object, the direction of the motion and their own personal experience of the context. There were clearly two types of contextual effects ‐‐ primary and secondary, which are described. The older group of students was generally less affected by context and thus more consistent in their reasoning.     

Teachers attending to students’ mathematical reasoning: lessons from an after-school research program

There is a documented need for more opportunities for teachers to learn about students’ mathematical reasoning. This article reports on the experiences of a group of elementary and middle school mathematics teachers who participated as interns in an after-school, classroom-based research project on the development of mathematical ideas involving middle-grade students from an urban, low-income, minority community in the United States. For 1 year, the teachers observed the students working on well-defined mathematical investigations that provided a context for the students’ formation of particular mathematical ideas and different forms of reasoning in several mathematical content strands. The article describes insights into students’ mathematical reasoning that the teachers were able to gain from their observations of the students’ mathematical activity. The purpose is to show that teachers’ observations of students’ mathematical activity in research sessions on students’ development of mathematical ideas can provide opportunities for teachers to learn about students’ mathematical reasoning.     

The role of interactive dialogue in students’ learning of mathematical reasoning: A quantitative multi-method analysis of feedback episodes

Interactive dialogue within feedback episodes is essential for developing primary school students' mathematical reasoning competence. Our goal was to better understand the nature of the associations between observed dialogue, teachers' formative feedback, and students' mathematical reasoning. We applied a two-step approach, first constructing a video-analysis instrument for assessing the quality of interactive dialogues and then combining the interaction data with student and teacher questionnaire data from 804 students in 44 fifth and sixth grade primary school classes. The quality of the observed dialogues predicted class differences in students’ self-efficacy for explaining but not in their reasoning competence, which was predicted by perceived formative feedback.     

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.     

Forms of mathematical interaction in different social settings: examples from students’, teachers’ and teacher–students’ communication about mathematics

The study presented in this article investigates forms of mathematical interaction in different social settings. One major interest is to better understand mathematics teachers’ joint professional discourse while observing and analysing young students mathematical interaction followed by teacher’s intervention. The teachers’ joint professional discourse is about a combined learning and talking between two students before an intervention by their teacher (setting 1) and then it is about the students learning together with the teacher during their mathematical work (setting 2). The joint professional teachers’ discourse constitutes setting 3. This combination of social settings 1 and 2 is taken as an opportunity for mathematics teachers’ professionalisation process when interpreting the students’ mathematical interactions in a more and more professional and sensible way. The epistemological analysis of mathematical sign-systems in communication and interaction in these three settings gives evidence of different types of mathematical talk, which are explained depending on the according social setting. Whereas the interaction between students or between teachers is affected by phases of a process-oriented and investigated talk, the interaction between students and teachers is mainly closed and structured by the ideas of the teacher and by the expectations of the students.

Developing computer-aided diagramming tools to mine,model and support students’ reasoning processes

Educational technology research and development - Despite the last 40 years of research showing that computer-aided diagramming tools improve student learning, very little research reveals...     

Measuring STEM students’ mathematical identities

Studies on identity in general and mathematical identity in particular have gained much interest over the last decades. However, although measurements have been proven to be potent tools in many scientific fields, a lack of consensus on ontological, epistemological, and methodological issues has complicated measurements of mathematical identities. Specifically, most studies conceptualise mathematical identity as something multidimensional and situated, which obviously complicates measurement, since these aspects violate basic requirements of measurement. However, most concepts that are measured in scientific work are both multidimensional and situated, even in physics. In effect, these concepts are being conceptualised as sufficiently uni-dimensional and invariant for measures to be meaningful. We assert that if the same judgements were to be made regarding mathematical identity, that is, whether identity can be measured with one instrument alone, whether one needs multiple instruments, or whether measurement is meaningless, it would be necessary to know how much of the multidimensionality can be captured by one measure and how situated mathematical identity is. Accordingly, this paper proposes a theoretical perspective on mathematical identity that is consistent with basic requirements of measurement. Moreover, characteristics of students’ mathematical identities are presented and the problem of “situatedness” is discussed.     

The effects of computer programming on high school students’ reasoning skills and mathematical self-efficacy and problem solving

In this paper we investigate whether computer programming has an impact on high school student’s reasoning skills, problem solving and self-efficacy in Mathematics. The quasi-experimental design was adopted to implement the study. The sample of the research comprised 66 high school students separated into two groups, the experimental and the control group according to their educational orientation. The research findings indicate that there is a significant difference in the reasoning skills of students that participated in the “programming course” compared to students that did not. Moreover, the self-efficacy indicator of students that participated in the experimental group showed a significant difference from students in the control group. The results however, failed to support the hypothesis that computer programming significantly enhances student’s problem solving skills.     

A study of students’ reasoning about probabilistic causality: Implications for understanding complex systems and for instructional design

Understanding complex systems requires reasoning about causal relationships that behave or appear to behave probabilistically. Features such as distributed agency, large spatial scales, and time delays obscure co-variation relationships and complex interactions can result in non-deterministic relationships between causes and effects that are best understood statistically. Causal Bayesian Research (e.g. Gopnik and Schulz in Causal learning: psychology, philosophy, and computation, Oxford University Press, New York, 2007) suggests that summing across probabilistic instances is inherent to human causal induction, yet other research (e.g. Schulz and Sommerville in Child Development 77(2):427–442, 2006) suggests a human tendency to assume deterministic relationships. Classroom science learning often stresses the replicability of outcomes, putting this learning in tension with understanding probabilistic patterns in complex systems. This investigation examined students’ reasoning patterns on tasks with probabilistic causal features. Microgenetic studies were conducted in multiple sessions over a school year with students in kindergarten, second, fourth and sixth grades (n = 16) to assess their assumptions when dealing with tasks from four domains: social; games; machines; and biology. Later sessions attempted to scaffold students’ understanding using connection making and analogical reasoning. This paper reports on the overall patterns and trends in the data. Most students held a deterministic stance at the outset; however, at least one student at each grade level reasoned probabilistically from the start. All students except one eventually revealed at least one topic for which they held a primarily probabilistic stance. The results have implications for how students reason about complex systems and for how patterns of co-variation and evidence in science are discussed.     

Teachers’ reasoning about school violence: The role of gender and location

This inquiry uses a Cognitive Developmental Domain theory framework to examine how male and female teachers balance different moral and non-moral components when reasoning about hypothetical school fights. The potential impact of teachers’ attributions towards the gender of the intervening teacher and fighting students, and the location of the fight were examined. This investigation found that male middle school teachers expressed more conflict than female middle school teachers when reasoning about whether the gender of the intervening teacher or the fighting student impacted a teacher’s response. When asked to reason about a hypothetical fight, female middle school teachers were more conflicted than female elementary school teachers, particularly when the location of the fight was manipulated. Theoretical implications are discussed.     

Children’s conceptions of bullying and repeated conventional transgressions: moral,conventional, structuring and personal-choice reasoning

This study examined 307 elementary school children’s judgements and reasoning about bullying and other repeated transgressions when school rules regulating these transgressions have been removed in hypothetical school situations. As expected, children judged bullying (repeated moral transgressions) as wrong independently of rules and as more wrong than all the other repeated transgressions. They justified their judgement in terms of harm that the actions caused. Moreover, whereas children tended to judge repeated structuring transgressions as wrong independently of rules (but to a lesser degree than when they evaluated bullying) and justified their judgements in terms of the disruptive, obstructive or disturbing effects that the actions caused, they tended to accept repeated etiquette transgressions by arguing that the acts had no negative effects or simply that the rule had been removed. The findings confirm as well as extend previous social-cognitive domain research on children’s socio-moral reasoning.     

The effect of prior education on students’ competency in digital logic: the case of ultraorthodox Jewish students

In line with the growing interest in extending the diversity of CS students, we examined the performance of a unique group of students studying an introductory course in Digital logic: ultraorthodox Jewish men, whose previous education was based mostly on studying Talmud and who lacked a conventional high-school education. We used questions from the Digital Logic Concept Inventory . We compared the results to those of religious Jewish men with a conventional high-school education, and to the results reported in the literature. The ultraorthodox group performed better than the other groups in tasks that concerned number representation. No other statistically significant differences were found. Talk-aloud protocols revealed that the ultraorthodox students utilized a viable conceptual understanding in their performance. We can conclude that students’ unique, alternative prior education should not be merely viewed as an obstacle to their academic studies, but also as a potential source for strengths.     

Diversity courses and students’ moral reasoning: a model of predispositions and change

The purpose of this study was to examine how moral reasoning develops for 236 students enrolled in either a diversity course or a management course. These courses were compared based on the level of diversity inclusion and type of pedagogy employed in the classroom. We used causal modelling to compare the two types of courses, controlling for the effects of demographic (i.e., race, gender), curricular (i.e., previous course-related diversity learning) and pedagogical (i.e., active learning) covariates. Results showed that students enrolled in the diversity course demonstrated higher levels of moral reasoning than students enrolled in the management course. In addition, results show that previous diversity courses as well as current enrolment in a diversity course contributed to moral reasoning gains. Implications are discussed.     

An investigation of relationships between students’ mathematical problem-posing abilities and their mathematical content knowledge

The importance of students’ problem-posing abilities in mathematics has been emphasized in the K-12 curricula in the USA and China. There are claims that problem-posing activities are helpful in developing creative approaches to mathematics. At the same time, there are also claims that students’ mathematical content knowledge could be highly related to creativity in mathematics, too. This paper reports on a study that investigated USA and Chinese high school students’ mathematical content knowledge, their abilities in mathematical problem posing, and the relationships between students’ mathematical content knowledge and their problem-posing abilities in mathematics.     

Beyond Epistemological Deficits: Dynamic explanations of engineering students’ difficulties with mathematical sense-making

Researchers have argued against deficit-based explanations of students’ difficulties with mathematical sense-making, pointing instead to factors such as epistemology. Students’ beliefs about knowledge and learning can hinder the activation and integration of productive knowledge they have. Such explanations, however, risk falling into a ‘deficit trap’—substituting a concepts/skills deficit with an epistemological one. Our interview-based case study of a freshman engineering major, ‘Jim,’ explains his difficulty solving a physics problem (on hydrostatic pressure) in terms of his epistemology, but avoids a deficit trap by modeling the dynamics of his epistemological stabilities and shifts in terms of fine-grained cognitive elements that include the seeds of epistemological expertise. Specifically, during a problem-solving episode in the interview, Jim reaches and sticks with an incorrect answer that violates common sense. We show that Jim has all the mathematical skills and physics knowledge he would need to resolve the contradiction. We argue that his difficulty doing so stems in part from his epistemological views that (i) physics equations are much more trustworthy than everyday reasoning, and (ii) physics equations do not express meaning that tractably connects to common sense. For these reasons, he does not view reconciling between common sense and formalism as either necessary or plausible to accomplish. But Jim’s in-the-moment shift to a more sophisticated epistemological stance highlights the seeds of epistemological expertise that were present all along: he does see common sense as connected to formalism (though not always tractably so), and in some circumstances, this connection is both salient and valued.     

Extending students’ mathematical thinking during whole-group discussions

Studies show that extending students’ mathematical thinking during whole-group discussions is a challenging undertaking. To better understand what extending student thinking looks like and how teachers’ mathematical knowledge for teaching (MKT) supports teachers in their efforts to extend student thinking, the teaching of six experienced elementary school teachers was explored. During group discussions, all six teachers created opportunities for extending student thinking about important mathematical ideas and solution methods. Findings on the nature of these episodes include identification of individual instructional actions and the ways in which teachers’ MKT was connected to these actions.     

“It was smart when:” Supporting prospective teachers’ noticing of students’ mathematical strengths

Learning to name and notice students’ mathematical strengths is a challenging process requiring time and multiple iterations of practice for prospective teachers (PTs) to adopt. Mathematics teacher educators (MTEs) can approximate and decompose the complex practice of naming and noticing students’ mathematical strengths so PTs learn to teach mathematics while emphasizing what students know and can do. This study uses two tools MTEs can use to support PTs as they learn to name and notice students’ mathematical strengths: A LessonSketch experience, a digital platform with comic-based storyboards showing children engaged in a mathematics task, and a strengths-based sentence frame. Our study presents the findings from the 111 noticing statements from 18 PTs as they engaged in the LessonSketch digital experience and practiced making noticing statements about what children know about mathematics. The study found that after a sentence-frame intervention, the PTs are more likely to use strengths-based language and more likely to identify mathematical evidence in their noticing statements. Uncommitted language (statements that do not align with a strength- or deficit-based coding scheme), suggests a fruitful, yet complex space for supporting more PTs as they learn to name and notice students’ mathematical strengths. The paper concludes with implications for future research in teacher education.