Mathematical Understanding for Secondary Teaching

A Framework and Classroom-Based Situations

Edited by:
M. Kathleen Heid, The Pennsylvania State University
Patricia S. Wilson, University of Georgia
Glendon W. Blume, The Pennsylvania State University

Published 2015

A perennial discussion about teacher development is the optimal content background for teachers. In recent years, that discussion has taken center stage in the work of mathematics education researchers, mathematicians, mathematics professional developers, and mathematics education policymakers. Much of the existing and prior work in this area has been directed toward mathematical knowledge for teaching at the elementary level. The work described in this volume takes a sometimes-neglected approach, focusing on the dynamic nature of mathematical understanding rather than on a stable description of mathematical knowledge, and on mathematics for secondary teaching rather than mathematics for teaching at the elementary level.

The work reported in Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations is a practice-based response to the question of what mathematical understandings secondary teachers could productively use in their teaching. For each of more than 50 events, our team of almost 50 mathematics educators who were experienced mathematics teachers developed descriptions of the mathematics that teachers could use—each of those descriptions (consisting of the event and the mathematics related to the event) is what we call a Situation. We developed our Framework for Mathematical Understanding for Secondary Teaching (MUST) based on an analysis of our entire set of Situations. We call the work practice-based because the MUST framework is based on actual events that we witnessed in our observations of secondary mathematics practice.

Groups of mathematics teachers can use this volume to enhance their own understandings of secondary mathematics. School leaders and professional developers in secondary mathematics will find our MUST Framework and Situations useful as they work with teachers in enhancing and deepening their understanding of secondary mathematics. Mathematics teacher educators and mathematicians who teach mathematics to prospective and in-service secondary teachers will be able to couch their mathematical discussions in the Situations—examples that arise from secondary mathematics classrooms. They will be able to use this volume as they design courses and programs that enhance mathematics from the perspectives identified in the MUST framework. Policymakers and researchers can use our MUST framework as they consider the mathematics background needed by teachers.

CONTENTS
Preface. 1. Background for the Mathematical Understanding Framework, Jeremy Kilpatrick. 2. Mathematical Understanding for Secondary Teaching: A framework, Jeremy Kilpatrick, Glendon Blume, M. Kathleen Heid, James Wilson, Patricia S. Wilson, Rose Mary Zbiek. 3. Development of practice-based Situations, Patricia S. Wilson and Rose Mary Zbiek. 4. From Situations to Framework, Patricia S. Wilson, M. Kathleen Heid. 5. Creating new Situations as inquiry, Rose Mary Zbiek, Glendon W. Blume. 6. Summary of uses of the MUST Framework and Situations, James Wilson. 7. Division involving zero, Bradford Findell, Evan McClintock, Glendon Blume, Ryan Fox, Rose Mary Zbiek, Brian Gleason. 8. Product of two negative numbers, Ryan Fox, Sarah Donaldson, M. Kathleen Heid, Glendon Blume, James Wilson. 9. Cross multiply, Rose Mary Zbiek, M. Kathleen Heid, Brian Gleason, Shawn Broderick. 10. Summing the natural numbers, Shari Reed, AnnaMarie Conner, Ryan Fox, Shiv Karunakaran, M. Kathleen Heid, Evan McClintock, Heather Johnson, Kelly Edenfield, Jeremy Kilpatrick, Eric Gold. 11. Modular arithmetic, Pawel Nazarewicz, Glendon Blume, Heather Johnson, Svetlana Konnova, Jeanne Shimizu. 12. Absolute value equations and inequalities, Shari Reed, AnnaMarie Conner, Sarah Donaldson, Kanita DuCloux, Kelly Edenfield, Erik Jacobson. 13. Absolute value and square root, Heather Johnson, Shiv Karunakaran, Evan McClintock, Pawel Nazarewicz, Erik Jacobson, Kelly Edenfield. 14. Properties of i and other complex numbers, Erik Tillema, Evan McClintock, M. Kathleen Heid, Heather Johnson. 15. Square root of i, Heather Johnson, Shiv Karunakaran, Evan McClintock, Ryan Fox. 16. Exponential rules, Erik Tillema, Sarah Donaldson, Kelly Edenfield, James Wilson, Eileen Murray, Glendon Blume. 17. Powers, Tracy Boone, Jana Lunt, Christa Fratto, James Banyas, Eileen Murray, Bob Allen, Sarah Donaldson, M. Kathleen Heid, Shiv Karunakaran, Brian Gleason. 18. Zero exponents, Tracy Boone, Christa Fratto, Jana Lunt, Heather Johnson, M. Kathleen Heid, Maureen Grady, Shiv Karunakaran. 19. Distributing exponents, Jeanne Shimizu, Tracy Boone, Jana Lunt, Christa Fratto, Erik Tillema, Jeremy Kilpatrick, Sarah Donaldson, Ryan Fox, Heather Johnson, Maureen Grady, Svetlana Konnova, M. Kathleen Heid. 20. Adding radicals, Amy Hackenberg, Eileen Murray, Heather Johnson, Glendon Blume, M. Kathleen Heid. 21. Square roots, Tracy Boone, Jana Lunt, Christa Fratto, James Banyas, Sarah Donaldson, James Wilson, Patricia S. Wilson, Heather Johnson, Brian Gleason. 22. Inverse trig functions, Rose Mary Zbiek, M. Kathleen Heid, Ryan Fox, Kelly Edenfield, Jeremy Kilpatrick, Evan McClintock, Heather Johnson, Brian Gleason. 23. Zero-product property, Jeanne Shimizu, Heather Johnson, Ryan Fox, Laura Singletary, Sarah Donaldson. 24. Simultaneous equations, Dennis Hembree, Erik Tillema, Evan McClintock, Rose Mary Zbiek, Heather Johnson, Patricia S. Wilson, James Wilson, Ryan Fox. 25. Graphing inequalities containing absolute values, Shari Reed, AnnaMarie Conner, M. Kathleen Heid, Heather Johnson, Maureen Grady, Svetlana Konnova. 26. Quadratic equations, Jeanne Shimizu, Sarah Donaldson, Kelly Edenfield, Erik Jacobson. 27. Graphing quadratic functions, Ginger Rhodes, Ryan Fox, Shiv Karunakaran, Rose Mary Zbiek, Brian Gleason, Shawn Broderick. 28. Connecting factoring with the quadratic formula, Erik Tillema, Heather Johnson, Sharon K. O’Kelley, Erik Jacobson, Glendon Blume, M. Kathleen Heid. 29. Perfect square trinomials, Bob Allen, Dennis Hembree, Sarah Donaldson, Brian Gleason, Shawn Broderick, M. Kathleen Heid, Glendon Blume. 30. Temperature conversions, Glendon Blume, Heather Johnson, Maureen Grady, Svetlana Konnova, M. Kathleen Heid. 31. Translation of functions, Bob Allen, Brian Gleason, Shawn Broderick. 32. Parametric drawings, Rose Mary Zbiek, Eileen Murray, Heather Johnson, Maureen Grady, Svetlana Konnova, M. Kathleen Heid. 33. Locus of a point on a moving segment, Rose Mary Zbiek, James Wilson, Heather Johnson, M. Kathleen Heid, Maureen Grady, Svetlana Konnova. 34. Constructing a tangent line, Pawel Nazarewicz, Sharon K. O’Kelley, Erik Jacobson, Glendon Blume, M. Kathleen Heid. 35. Faces of a polyhedral solid, Stephen Bismarck, Glendon Blume, Heather Johnson, Svetlana Konnova, Jeanne Shimizu. 36. Area of plane figures, Erik Tillema, Tenille Cannon, Kim Johnson, Rose Mary Zbiek. 37. Area of sector in a circle, Dennis Hembree, Sharon K. O’Kelley, Shawn Broderick, James Wilson. 38. Similarity, Evan McClintock, Susan Peters, Donna Kinol, Shari Reed, Heather Johnson, Erik Tillema, Rose Mary Zbiek, M. Kathleen Heid, Sarah Donaldson, Eileen Murray, Glendon Blume. 39. Pythagorean theorem, Patrick Sullivan, M. Kathleen Heid, Maureen Grady, Shiv Karunakaran. 40. Circumscribing polygons, Sari Reed, AnnaMarie Conner, Heather Johnson, M. Kathleen Heid, Bob Allen, Shiv Karunakaran, Sarah Donaldson, Brian Gleason. 41. Calculation of sine, Patricia S. Wilson, Heather Johnson, Jeanne Shimizu, Evan McClintock, Rose Mary Zbiek, M. Kathleen Heid, Maureen Grady, Svetlana Konnova. 42. Graphing sin(2x), Heather Johnson, van McClintock, Rose Mary Zbiek, Brian Gleason, Shawn Broderick, James Wilson. 43. Trigonometric identities, Bob Allen, Sharon O’Kelley, Erik Jacobson. 44. Mean and median, Susan Peters, Evan McClintock, Donna Kinol, Shiv Karunakaran, Rose Mary Zbiek, M. Kathleen Heid, Laura Singletary, Sarah Donaldson. 45. Representing standard deviation, Rose Mary Zbiek, M. Kathleen Heid, Shiv Karunakaran, Ryan Fox, Eric Gold, Sarah Donaldson, Laura Singletary. 46. Sample variance and population variance, Ken Montgomery, Sarah Donaldson. 47. Least squares regression, Susan Peters, Evan McClintock, Donna Kinol, Maureen Grady, Heather Johnson, Svetlana Konnova, M. Kathleen Heid. 48. The product rule for differentiation, Heather Johnson, Shari Reed, Evan McClintock, Erik Jacobson, Kelly Edenfield. 49. Proof by induction, Erik Tillema, Jeremy Kilpatrick, Heather Johnson, Maureen Grady, Svetlana Konnova, M. Kathleen Heid. 50. Situating and reflecting on our Situations work, M. Kathleen Heid.