Forms of Mathematical Knowledge: Learning and Teaching with Understanding

Description

What mathematics is entailed in knowing to act in a moment? Is tacit, rhetorical knowledge significant in mathematics education? What is the role of intuitive models in understanding, learning and teaching mathematics? Are there differences between elementary and advanced mathematical thinking? Why can't students prove? What are the characteristics of teachers' ways of knowing? This book focuses on various types of knowledge that are significant for learning and teaching mathematics. The first part defines, discusses and contrasts psychological, philosophical and didactical issues related to various types of knowledge involved in the learning of mathematics. The second part describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas in preservice and in-service education. The chapters provide a wide overview of current thinking about mathematics learning and teaching which is of interest for researchers in mathematics education and mathematics educators. Topics covered include the role of intuition in mathematics learning and teaching, the growth from elementary to advanced mathematical thinking, the significance of genres and rhetoric for the learning of mathematics and the characterization of teachers' ways of knowing. Front Matter....Pages i-iii Forms of Mathematical Knowledge: Learning and Teaching with Understanding....Pages 1-9 Intuitions and Schemata in Mathematical Reasoning....Pages 11-50 Intuitive Rules: A way to Explain and Predict Students’ Reasoning....Pages 51-66 Forms of Knowledge in Mathematics and Mathematics Education: Philosophical and Rhetorical Perspectives....Pages 67-83 Why Johnny Can’t Prove....Pages 85-109 Knowledge Construction and Diverging Thinking in Elementary & Advanced Mathematics....Pages 111-133 Beyond Mere Knowledge of Mathematics: The Importance of Knowing-to Act in the Moment....Pages 135-161 Conceptualizing Teachers’ Ways of Knowing....Pages 163-187 Forms of Knowing Mathematics: What Preservice Teachers Should Learn....Pages 189-208 The Transition from Comparison of Finite to the Comparison of Infinite Sets: Teaching Prospective Teachers....Pages 209-234 Integrating Academic and Practical Knowledge in a Teacher Leaders’ Development Program....Pages 235-252