Learning mathematical thinking and reasoning is a main goal in mathematical education. Instructional tasks have an important role in fostering this learning. We introduce a learning sequence to approach the topic of integrals in secondary education to support students mathematical reasoning while participating in collaborative dialogue about the integral-as-accumulation-function. This is based on the notion of accumulation in general and the notion of accumulative distance function in particular. Through a case-study methodology we investigate how this approach elicits 11th grade students’ mathematical thinking and reasoning. The results show that the integral-as-accumulation-function has potential, since the notions of accumulation and accumulative function can provide a strong intuition for mathematical reasoning and engage students in mathematical dialogue. Implications of these results for task design and further research are discussed.
A primary teacher needs mathematical problem solving ability. That is why Dutch student teachers have to show this ability in a nationwide mathematics test that contains many non-routine problems. Most student teachers prepare for this test by working on their own solving test-like problems. To what extent does these individual problem solving activities really contribute to their mathematical problem solving ability? Developing mathematical problem solving ability requires reflective mathematical behaviour. Student teachers need to mathematize and generalize problems and problem approaches, and evaluate heuristics and problem solving processes. This demands self-confidence, motivation, cognition and metacognition. To what extent do student teachers show reflective behaviour during mathematical self-study and how can we explain their study behaviour? In this study 97 student teachers from seven different teacher education institutes worked on ten non-routine problems. They were motivated because the test-like problems gave them an impression of the test and enabled them to investigate whether they were already prepared well enough. This study also shows that student teachers preparing for the test were not focused on developing their mathematical problem solving ability. They did not know that this was the goal to strive for and how to aim for it. They lacked self-confidence and knowledge to mathematize problems and problem approaches, and to evaluate the problem solving process. These results indicate that student teachers do hardly develop their mathematical problem solving ability in self-study situations. This leaves a question for future research: What do student teachers need to improve their mathematical self-study behaviour? EAPRIL Proceedings, November 29 – December 1, 2017, Hämeenlinna, Finland
The implementation of the new mathematical knowledge base in Dutch teacher education institutes for primary education raises a need for curriculum development. Teacher educators have to raise student teachers’ subject matter knowledge to a higher level. In working on this aim teacher educators experience that student teachers often feel uncertain about their mathematical skills and are not very interested in formal and abstract mathematics. Student teachers prefer to focus on mathematical pedagogical content knowledge. This paper presents two design studies that try to tackle this problem. The first one targets the development of student teachers’ specialized content knowledge (SCK) and the second one focuses on their horizon content knowledge (HCK). Both studies target developing student teachers’ mathematical subject matter knowledge in the perspective of teaching mathematics in primary school. In the studies we established student teachers’ learning environments that kept them involved and motivated, even when they found the mathematics hard to do. Primarily, this attitude supported their mathematical growth, while it also developed their pedagogical skills and insight. INTRODUCTION
