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.

Table of Contents:
Preface. Chapter 1. Background for the Mathematical Understanding Framework, Jeremy Kilpatrick. Chapter 2. Mathematical Understanding for Secondary Teaching: A Framework, Jeremy Kilpatrick, Glendon Blume, M. Kathleen Heid, James Wilson, Patricia S. Wilson, Rose Mary Zbiek. Chapter 3. Development of Practice-Based Situations, Patricia S. Wilson and Rose Mary Zbiek. Chapter 4. From Situations to Framework, Patricia S. Wilson, M. Kathleen Heid. Chapter 5. Creating New Situations as Inquiry, Rose Mary Zbiek, Glendon W. Blume. Chapter 6. Summary of Uses of the MUST Framework and Situations, James Wilson. Chapter 7. Division Involving Zero, Bradford Findell, Evan McClintock, Glendon Blume, Ryan Fox, Rose Mary Zbiek, Brian Gleason. Chapter 8. Product of Two Negative Numbers, Ryan Fox, Sarah Donaldson, M. Kathleen Heid, Glendon Blume, James Wilson. Chapter 9. Cross Multiply, Rose Mary Zbiek, M. Kathleen Heid, Brian Gleason, Shawn Broderick. Chapter 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. Chapter 11. Modular Arithmetic, Pawel Nazarewicz, Glendon Blume, Heather Johnson, Svetlana Konnova, Jeanne Shimizu. Chapter 12. Absolute Value Equations and Inequalities, Shari Reed, AnnaMarie Conner, Sarah Donaldson, Kanita DuCloux, Kelly Edenfield, Erik Jacobson. Chapter 13. Absolute Value and Square Root, Heather Johnson, Shiv Karunakaran, Evan McClintock, Pawel Nazarewicz, Erik Jacobson, Kelly Edenfield. Chapter 14. Properties of i and Other Complex Numbers, Erik Tillema, Evan McClintock, M. Kathleen Heid, Heather Johnson. Chapter 15. Square Root of i, Heather Johnson, Shiv Karunakaran, Evan McClintock, Ryan Fox. Chapter 16. Exponential Rules, Erik Tillema, Sarah Donaldson, Kelly Edenfield, James Wilson, Eileen Murray, Glendon Blume. Chapter 17. Powers, Tracy Boone, Jana Lunt, Christa Fratto, James Banyas, Eileen Murray, Bob Allen, Sarah Donaldson, M. Kathleen Heid, Shiv Karunakaran, Brian Gleason. Chapter 18. Zero Exponents, Tracy Boone, Christa Fratto, Jana Lunt, Heather Johnson, M. Kathleen Heid, Maureen Grady, Shiv Karunakaran. Chapter 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. Chapter 20. Adding Radicals, Amy Hackenberg, Eileen Murray, Heather Johnson, Glendon Blume, M. Kathleen Heid. Chapter 21. Square Roots, Tracy Boone, Jana Lunt, Christa Fratto, James Banyas, Sarah Donaldson, James Wilson, Patricia S. Wilson, Heather Johnson, Brian Gleason. Chapter 22. Inverse Trig Functions, Rose Mary Zbiek, M. Kathleen Heid, Ryan Fox, Kelly Edenfield, Jeremy Kilpatrick, Evan McClintock, Heather Johnson, Brian Gleason. Chapter 23. Zero-Product Property, Jeanne Shimizu, Heather Johnson, Ryan Fox, Laura Singletary, Sarah Donaldson. Chapter 24. Simultaneous Equations, Dennis Hembree, Erik Tillema, Evan McClintock, Rose Mary Zbiek, Heather Johnson, Patricia S. Wilson, James Wilson, Ryan Fox. Chapter 25. Graphing Inequalities Containing Absolute Values, Shari Reed, AnnaMarie Conner, M. Kathleen Heid, Heather Johnson, Maureen Grady, Svetlana Konnova. Chapter 26. Quadratic Equations, Jeanne Shimizu, Sarah Donaldson, Kelly Edenfield, Erik Jacobson. Chapter 27. Graphing Quadratic Functions, Ginger Rhodes, Ryan Fox, Shiv Karunakaran, Rose Mary Zbiek, Brian Gleason, Shawn Broderick. Chapter 28. Connecting Factoring with the Quadratic Formula, Erik Tillema, Heather Johnson, Sharon K. O'Kelley, Erik Jacobson, Glendon Blume, M. Kathleen Heid. Chapter 29. Perfect Square Trinomials, Bob Allen, Dennis Hembree, Sarah Donaldson, Brian Gleason, Shawn Broderick, M. Kathleen Heid, Glendon Blume. Chapter 30. Temperature Conversions, Glendon Blume, Heather Johnson, Maureen Grady, Svetlana Konnova, M. Kathleen Heid. Chapter 31. Translation of Functions, Bob Allen, Brian Gleason, Shawn Broderick. Chapter 32. Parametric Drawings, Rose Mary Zbiek, Eileen Murray, Heather Johnson, Maureen Grady, Svetlana Konnova, M. Kathleen Heid. Chapter 33. Locus of a Point on a Moving Segment, Rose Mary Zbiek, James Wilson, Heather Johnson, M. Kathleen Heid, Maureen Grady, Svetlana Konnova. Chapter 34. Constructing a Tangent Line, Pawel Nazarewicz, Sharon K. O'Kelley, Erik Jacobson, Glendon Blume, M. Kathleen Heid. Chapter 35. Faces of a Polyhedral Solid, Stephen Bismarck, Glendon Blume, Heather Johnson, Svetlana Konnova, Jeanne Shimizu. Chapter 36. Area of Plane Figures, Erik Tillema, Tenille Cannon, Kim Johnson, Rose Mary Zbiek. Chapter 37. Area of Sector in a Circle, Dennis Hembree, Sharon K. O'Kelley, Shawn Broderick, James Wilson. Chapter 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. Chapter 39. Pythagorean Theorem, Patrick Sullivan, M. Kathleen Heid, Maureen Grady, Shiv Karunakaran. Chapter 40. Circumscribing Polygons, Sari Reed, AnnaMarie Conner, Heather Johnson, M. Kathleen Heid, Bob Allen, Shiv Karunakaran, Sarah Donaldson, Brian Gleason. Chapter 41. Calculation of Sine, Patricia S. Wilson, Heather Johnson, Jeanne Shimizu, Evan McClintock, Rose Mary Zbiek, M. Kathleen Heid, Maureen Grady, Svetlana Konnova. Chapter 42. Graphing sin(2x), Heather Johnson, Evan McClintock, Rose Mary Zbiek, Brian Gleason, Shawn Broderick, James Wilson. Chapter 43. Trigonometric Identities, Bob Allen, Sharon O'Kelley, Erik Jacobson. Chapter 44. Mean and Median, Susan Peters, Evan McClintock, Donna Kinol, Shiv Karunakaran, Rose Mary Zbiek, M. Kathleen Heid, Laura Singletary, Sarah Donaldson. Chapter 45. Representing Standard Deviation, Rose Mary Zbiek, M. Kathleen Heid, Shiv Karunakaran, Ryan Fox, Eric Gold, Sarah Donaldson, Laura Singletary. Chapter 47. Sample Variance and Population Variance, Ken Montgomery, Sarah Donaldson. Chapter 48. Least Squares Regression, Susan Peters, Evan McClintock, Donna Kinol, Maureen Grady, Heather Johnson, Svetlana Konnova, M. Kathleen Heid. Chapter 49. The Product Rule for Differentiation, Heather Johnson, Shari Reed, Evan McClintock, Erik Jacobson, Kelly Edenfield. Chapter 50. Proof by Induction, Erik Tillema, Jeremy Kilpatrick, Heather Johnson, Maureen Grady, Svetlana Konnova, M. Kathleen Heid. Chapter 51. Situating and Reflecting on Our Situations Work, M. Kathleen Heid.