Science and Mathematics Education

Articulating Teachers’ Creation of Technological Pedagogical Mathematical Knowledge (TPMK) for Supporting Mathematical Inquiry with Authentic Problems Abstract Research studies have found that mathematics teachers face challenges when designing technology-integrated lessons to support mathematical inquiry with authentic problems. This suggests that teachers need to develop technological pedagogical knowledge (TPACK) or their professional knowledge for technology integration. Drawing upon the concept of TPACK, this study defines teachers’ professional knowledge for mathematics technology integration as technological pedagogical mathematical knowledge (TPMK) and examines the different kinds of TPMK that are needed to support the design of technology-integrated lessons for mathematical inquiry with authentic problems. Using content analysis, the design talk of three mathematics teachers designing a series of technology-integrated lessons to teach the concept of average through the inquiry of authentic problems was examined to articulate their associated TPMK. Thematic analysis was then used to derive five pathways used by the teachers to create the TPMK for mathematical inquiry with authentic problems throughout the design meetings that the teachers held across 2 months. These pathways articulate the process of teachers’ pedagogical decision-making as they create TPMK. The implications of these findings for future development of teacher knowledge for technology-supported mathematics inquiry are discussed.

15/16-Year-Old Students’ Reasons for Choosing and Not Choosing Physics at a Level Abstract Participation in post-compulsory physics is a matter of longstanding concern from both economic and equity perspectives. In considering this issue, this study draws upon Bourdieu’s theory of social practice, particularly notions of the ‘cultural arbitrary’, to explore what insights into post-compulsory physics choice might be provided by students who could have chosen physics, but did not, opting for other sciences instead. Utilising survey data from over 13,000 year 11 (ages 15/16) students in England, as well as qualitative interviews with 70 students of the same age, findings reinforce the key role of individual aspirations in subject choice. However, they also highlight the influence of the cultural arbitrary of physics (e.g. as difficult, masculine), which leads many students to conclude that physics is not ‘for me’ and hence choose other paths. This finding emphasises the entrenched nature of the challenges facing efforts to increase equity in post-compulsory physics participation.

Examining Technology-Mediated Communication Using a Commognitive Lens: the Case of Touchscreen-Dragging in Dynamic Geometry Environments Abstract In this paper, I introduce Sfard’s discursive framework to examine secondary-school calculus students’ communication during exploratory activities mediated by the use of touchscreen dynamic geometry environments (DGEs). Six pairs of secondary-school students participated in an open-ended task to explore calculus relationships using touchscreen-DGEs. Qualitative data capturing the students’ linguistic communication (speech) and hand movements (gestures and dragging) were analysed when the students interacted with the touchscreen-DGEs used during the task. Findings suggest that new forms of communication were mobilised through the act of dragging on a touchscreen-DGE. In particular, routines for comparing, reasoning, conjecturing and verifying emerged within the use of the haptic DGE interface. In this paper, potentials of the touchscreen-DGEs in facilitating new forms of gestural thinking, as well as theoretical and methodological considerations that recognise the changing ways in which the hand and media interact, are discussed.

Expert-Based Cognitive Model and Student-Based Cognitive Model in the Learning of “Time”: Match or Mismatch? Abstract Cognitive model [CM] is the description of individual’s problem-solving method on selected tasks at a precise level to make meaningful inference on his/her strengths and weaknesses. CM serves as a foundation in developing a cognitive diagnostic assessment [CDA] that links the interpretation of test score to the attributes. This study employed the CDA involving “Time” which was developed under the framework of assessment triangle and Attribute Hierarchy Method [AHM]. In this study, the expert-based cognitive model was generated by 7 experienced primary mathematics teachers while the student-based cognitive model was generated based on 259 Year 6 pupils’ responses in the CDA. The consistency between these cognitive models was calculated by using Hierarchical Consistency Index [HCI] and triangulated by pupils’ interview transcript. Hence, this paper aims to discuss the consistency of structure between the expert-based cognitive models and the student-based cognitive models. The extent of match between the expert-based and the student-based cognitive models is important as it determines the validity of the expert-based cognitive model from students’ perspective. In general, all the 3 expert-based cognitive models were excellently fit with the student-based cognitive models. However, the result shows that the cognitive models of those pupils who scored 50% and below exhibited poor fit to the expert-based cognitive models. Results of the study implied that the proposed CMs are able to predict pupils’ responses and there is a need to construct CM which is suitable for low ability pupils in order to help them in studying the topic of “Time.”

Curriculum Matters: What We Teach and What Students Gain Abstract This study examined and compared the intended mathematics curricula according to topic coverage, focus, coherence, and learning progression in grades 1 – 12 within China and the province of Ontario, Canada. The findings show that the overall topics in the two curricula are similar: Chinese curriculum covers 78 topics out of 79 while the Ontario curriculum covers 76. The two curricula also share a similar general sequence such as topics for transfer stages from numbers to functions by starting with constant mathematics, then variable mathematics, and finally functions. However, the detailed topic design of the 2 curricula differs markedly. The Chinese curriculum includes few topics in each year, a short duration or span of each topic, and a fast-paced topic progression. The Ontario curriculum, in contrast, includes more topics each year, longer duration of many topics, and a small pace of topic progression in grades 1 – 8 and a fast pace of topic progression in grades 9 – 12. Because of different curriculum designs in grades 1 – 12, the intended curriculum may influence students’ cognitive structures of mathematics, learning behavior and thinking, learning efficiency and achievement, and teachers’ professional development. This calls for more refined and advanced research on the defined list of topics, topic organization, and terms to study curriculum in order to increase student learning opportunities.

A Holistic Picture of Physics Student Conceptions of Energy Quantization, the Photon Concept, and Light Quanta Interference Abstract A detailed investigation of student conceptions of quantum phenomena is needed, both to characterize student understanding of quantum concepts and to inform how we might teach quantum mechanics (QM). In this vein, in-depth semi-structured interviews were conducted with 35 students who majored in physics and received university-level QM instructions. Interview protocols were used and based on three quantum contexts: the quantization of energy in explaining the blackbody radiation, the photon concept in explaining the photoelectric effect, and light quanta in explaining the gradual formation of an interference pattern in the cases of low-intensity light beam. By applying a developmental phenomenographic analysis of the interview responses, three description categories (i.e., general patterns behind the conceptual understandings used in explaining each quantum contexts) were identified. These categories of descriptions revealed that most students’ thinking regarding foundational concepts in QM ranged from naïve and deficient descriptions based on classical ontologies to simple hybrid and/or mixed models of classical and quantum conceptions. Regarding learning QM, the study found that the perspective of naïve and classical ontologies in explaining quantum phenomena influenced students’ responses; they made incorrect generalizations and/or inappropriate links to the concepts learned in classical physics; and patterns of incorrect notions of QM are analogous to those that were documented. Besides, the study confirmed that students’ conceptual difficulties with QM are real, stable over time, and cross-cultural. It seems that the challenge to make QM interesting, effective, and relevant for physics students is a universal concern that knows no boundaries.

Middle Schoolers’ Biases and Strategies in a Fraction Comparison Task Abstract The present study uses a short, computerized task to investigate individual differences among middle school students in terms of the intuitions and strategies that they use to compare fractions. To tap into their intuitions about fractions, students were presented with pairs of fractions on the screen for a limited time of 10 s. Fraction pairs to be compared were controlled as to whether fractions shared or not a common component (numerator or denominator), as well as whether the greater fraction was the one with greater components or not. Data from about 500 students were processed with a clustering analysis based on these four item types, revealing distinct patterns of answers that we interpret in terms of strategies for answering the task. Half of the sample followed a simple larger-component-larger-fraction strategy as suggested by a congruency-based implementation of the natural number bias, although several clusters showed opposite patterns. These results show that a short, simple task can provide valuable information about students’ individual differences in comparing fractions, useful for both research and practice in mathematics education. We discuss implications of these results for research and practice related to mathematics cognition and education.

Supporting Elementary Teachers’ Planning and Assessing of Mathematical Reasoning Abstract Attention to mathematical reasoning in curriculum standards is part of an international trend, but identifying and understanding reasoning continues to challenge teachers.We report on one component of an Australia-wide initiative supporting teachers to implement innovative pedagogies. This paper contains insights from design research that focused on trialling classroom materials to support elementary teachers in their planning and assessment of mathematical reasoning. Findings confirmed planning is a critical step to developing learning experiences that elicit student reasoning, including consideration to task modifications and teacher questioning. Teachers’ capacity to assess their students’ reasoning was explored using the purposefully designed Assessing Mathematical Reasoning Rubric. The results reveal the complexity involved in constructing accurate judgements of students’ reasoning capabilities, particularly appreciating the non-linear nature of mathematical reasoning and the need to draw on multiple sources of evidence. Implications for supporting teachers in their planning for, and assessing of, mathematical reasoning are raised.

The Impact of Digital Divides on Student Mathematics Achievement in Confucian Heritage Cultures: a Critical Examination Using PISA 2012 Data Abstract This study critically examines if digital divides, comprising access to and use of information technology (IT) in two spheres (schools and at home), affect student achievement in Confucian heritage cultures (CHCs). The sample comprised 38,158 students from 1030 schools in seven CHCs who participated in Program for International Student Assessment (PISA) 2012. Markov chain Monte Carlo multiple imputation, hierarchical linear modeling (HLM), and latent class analysis (LCA) were employed in the analysis. Results showed that home (but not school) IT use benefited student mathematics achievement, and students with the overall least IT resources were most academically successful. These results indicate the importance of understanding the nuanced effects of digital divides in different contexts.

Effectiveness of Design-Based Science on Students’ Learning in Electrical Energy and Metacognitive Self-Regulation Abstract This study investigates the effect of design-based science (DBS) instruction on students’ learning of electrical energy and students’ metacognitive strategy use in science. Two classes of seventh-grade students participated in the study. DBS instruction was used in the treatment group, while the comparison group was taught the unit with curriculum-oriented science instruction (COSI). Mixed between/within subjects, ANOVA results suggest that at the beginning of the study, there was no difference between the treatment and comparison groups in terms of electrical energy learning or metacognitive self-regulation. At the end of the study, students in the DBS group outperformed students in the COSI group in electrical energy learning. Students in both groups had increased understanding of electrical energy; however, the improvement levels were much higher in the DBS group than in the COSI group. At the end of the study, the DBS group also reported higher levels of monitoring strategy use than the COSI group, while there was no difference between the groups in terms of planning and regulation strategies. Furthermore, the DBS group showed improvement in all metacognitive learning strategy components, whereas no change was detected in the COSI group. Thus, it seems that DBS contributed to students’ learning in electrical energy and enhanced their metacognitive self-regulation in science more than COSI. Additionally, students in the DBS group generally thought positively about the influences of DBS instruction on their science learning. They identified similarities between DBS processes and activities engineers use in real life.