Extended analyses: finding deep structure in standard high school mathematics

This paper describes an approach for pre-service or in-service mathematics education that teaches sophisticated mathematics using only the tools of high school mathematics. The idea is to start with a standard problem from high school mathematics and let the solution to this problem serve as a platform for asking good mathematical questions and searching for deeper mathematical structure beyond the obvious “answer” to the question. Surprisingly, it turns out that high school mathematics is remarkably open to this sort of analysis, and the results are more interesting mathematically than one might initially think.     

Connecting past and present through case-comparison learning in history: views of teachers and students

History education frequently aims at developing active citizenship by using the past to orientate to the present and the future. A pedagogy for pursuing this aim is making connections between the past and the present by means of comparing cases of an enduring human issue. To examine the feasibility and desirability of this case-comparison teaching approach, students (n = 444) and teachers (n = 15) who participated in an implementation study conducted in the Netherlands were questioned about their experiences and views. Results show that both students and teachers felt that case-comparison in the context of an enduring human issue is feasible and not more complex than the usual history teaching in which topics are studied separately without explicitly making comparisons between past and present, even if some students thought that taking account of episodes from different historical periods concurrently required an extra learning effort. Both students and teachers believed that connecting past and present in history teaching enhances engagement and meaning making. They suggested a curriculum combining the case-comparison approach with the type of history teaching they were accustomed to. Mixed methods were used for data collection. Implications for further research on case-comparison learning in history are being discussed.     

Embodied design: constructing means for constructing meaning

Design-based research studies are conducted as iterative implementation-analysis-modification cycles, in which emerging theoretical models and pedagogically plausible activities are reciprocally tuned toward each other as a means of investigating conjectures pertaining to mechanisms underlying content teaching and learning. Yet this approach, even when resulting in empirically effective educational products, remains under-conceptualized as long as researchers cannot be explicit about their craft and specifically how data analyses inform design decisions. Consequentially, design decisions may appear arbitrary, design methodology is insufficiently documented for broad dissemination, and design practice is inadequately conversant with learning-sciences perspectives. One reason for this apparent under-theorizing, I propose, is that designers do not have appropriate constructs to formulate and reflect on their own intuitive responses to students’ observed interactions with the media under development. Recent socio-cultural explication of epistemic artifacts as semiotic means for mathematical learners to objectify presymbolic notions (e.g., Radford, Mathematical Thinking and Learning 5(1): 37–70, 2003) may offer design-based researchers intellectual perspectives and analytic tools for theorizing design improvements as responses to participants’ compromised attempts to build and communicate meaning with available media. By explaining these media as potential semiotic means for students to objectify their emerging understandings of mathematical ideas, designers, reciprocally, create semiotic means to objectify their own intuitive design decisions, as they build and improve these media. Examining three case studies of undergraduate students reasoning about a simple probability situation (binomial), I demonstrate how the semiotic approach illuminates the process and content of student reasoning and, so doing, explicates and possibly enhances design-based research methodology.