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Instructional Practices Guide
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Instructional Practices Guide, The MAA Instructional Practices Guide is an open access publication distributed in accordance with the Creative. Commons Attribution Non Commercial CC BY NC 4 0 license which permits others to distribute remix adapt. build upon this work non commercially and license their derivative works on different terms provided the original. work is properly cited and the use is non commercial See http creativecommons org licenses by nc 4 0. 2018 The Mathematical Association of America Inc,Electronic ISBN 978 1 61444 325 4. Print ISBN 978 0 88385 198 2,Printed in the United States of America. Instructional Practices Guide,Project Leadership Team. Martha L Abell Georgia Southern University,Linda Braddy Tarrant County College.
Doug Ensley Mathematical Association of America,Lewis Ludwig Denison University. Hortensia Soto University of Northern Colorado,Published and Distributed by. The Mathematical Association of America, The MAA Notes Series started in 1982 addresses a broad range of topics and themes of interest to all who are involved. with undergraduate mathematics The volumes in this series are readable informative and useful and help the math. ematical community keep up with developments of importance to mathematics. Council on Publications and Communications,Jennifer Quinn Chair. Notes Editorial Board,Michael C Axtell Editor,Crista L Arangala.
Suzanne Hamon,Hugh Howards,David R Mazur,Elizabeth W McMahon. Dan Sloughter,John M Zobitz, 14 Mathematical Writing by Donald E Knuth Tracy Larrabee and Paul M Roberts. 16 Using Writing to Teach Mathematics Andrew Sterrett Editor. 17 Priming the Calculus Pump Innovations and Resources Committee on Calculus Reformand the First Two Years. a subcomittee of the Committee on the Undergraduate Program in Mathematics Thomas W Tucker Editor. 18 Models for Undergraduate Research in Mathematics Lester Senechal Editor. 19 Visualization in Teaching and Learning Mathematics Committee on Computers inMathematics Education Steve. Cunningham and Walter S Zimmermann Editors, 20 The Laboratory Approach to Teaching Calculus L Carl Leinbach et al Editors. 21 Perspectives on Contemporary Statistics David C Hoaglin and David S Moore Editors. 22 Heeding the Call for Change Suggestions for Curricular Action Lynn A Steen Editor. 24 Symbolic Computation in Undergraduate Mathematics Education Zaven A Karian Editor. 25 The Concept of Function Aspects of Epistemology and Pedagogy Guershon Harel and Ed Dubinsky Editors. 26 Statistics for the Twenty First Century Florence and Sheldon Gordon Editors. 27 Resources for Calculus Collection Volume 1 Learning by Discovery A Lab Manual for Calculus Anita E Solow. 28 Resources for Calculus Collection Volume 2 Calculus Problems for a New Century Robert Fraga Editor. 29 Resources for Calculus Collection Volume 3 Applications of Calculus Philip Straffin Editor. 30 Resources for Calculus Collection Volume 4 Problems for Student Investigation Michael B Jackson and John R. Ramsay Editors, 31 Resources for Calculus Collection Volume 5 Readings for Calculus Underwood Dudley Editor. 32 Essays in Humanistic Mathematics Alvin White Editor. 33 Research Issues in Undergraduate Mathematics Learning Preliminary Analyses and Results James J Kaput and Ed. Dubinsky Editors,34 In Eves Circles Joby Milo Anthony Editor.
35 Youre the Professor What Next Ideas and Resources for Preparing College Teachers The Committee on Prepara. tion for College Teaching Bettye Anne Case Editor, 36 Preparing for a New Calculus Conference Proceedings Anita E Solow Editor. 37 A Practical Guide to Cooperative Learning in Collegiate Mathematics Nancy L Hagelgans Barbara E Reynolds. SDS Keith Schwingendorf Draga Vidakovic Ed Dubinsky Mazen Shahin G Joseph Wimbish Jr. 38 Models That Work Case Studies in Effective Undergraduate Mathematics Programs Alan C Tucker Editor. 39 Calculus The Dynamics of Change CUPM Subcommittee on Calculus Reform and the First Two Years A Wayne. Roberts Editor, 40 Vita Mathematica Historical Research and Integration with Teaching Ronald Calinger Editor. 41 Geometry Turned On Dynamic Software in Learning Teaching and Research James R King and Doris Schattschnei. der Editors, 42 Resources for Teaching Linear Algebra David Carlson Charles R Johnson David C Lay A Duane Porter Ann E. Watkins William Watkins Editors, 43 Student Assessment in Calculus A Report of the NSF Working Group on Assessment in Calculus Alan Schoenfeld. 44 Readings in Cooperative Learning for Undergraduate Mathematics Ed Dubinsky David Mathews and Barbara E. Reynolds Editors, 45 Confronting the Core Curriculum Considering Change in the Undergraduate Mathematics Major John A Dossey.
46 Women in Mathematics Scaling the Heights Deborah Nolan Editor. 47 Exemplary Programs in Introductory College Mathematics Innovative Programs Using Technology Susan Lenker. 48 Writing in the Teaching and Learning of Mathematics John Meier and Thomas Rishel. 49 Assessment Practices in Undergraduate Mathematics Bonnie Gold Sandra Z Keith and William A Marion Editors. 50 Revolutions in Differential Equations Exploring ODEs with Modern Technology Michael J Kallaher Editor. 51 Using History to Teach Mathematics An International Perspective Victor J Katz Editor. 52 Teaching Statistics Resources for Undergraduate Instructors Thomas L Moore Editor. 53 Geometry at Work Papers in Applied Geometry Catherine A Gorini Editor. 54 Teaching First A Guide for New Mathematicians Thomas W Rishel. 55 Cooperative Learning in Undergraduate Mathematics Issues That Matter and Strategies That Work Elizabeth C. Rogers Barbara E Reynolds Neil A Davidson and Anthony D Thomas Editors. 56 Changing Calculus A Report on Evaluation Efforts and National Impact from 1988 to 1998 Susan L Ganter. 57 Learning to Teach and Teaching to Learn Mathematics Resources for Professional Development Matthew Delong. and Dale Winter, 58 Fractals Graphics and Mathematics Education Benoit Mandelbrot and Michael Frame Editors. 59 Linear Algebra Gems Assets for Undergraduate Mathematics David Carlson Charles R Johnson David C Lay. and A Duane Porter Editors, 60 Innovations in Teaching Abstract Algebra Allen C Hibbard and Ellen J Maycock Editors. 61 Changing Core Mathematics Chris Arney and Donald Small Editors. 62 Achieving Quantitative Literacy An Urgent Challenge for Higher Education Lynn Arthur Steen. 64 Leading the Mathematical Sciences Department A Resource for Chairs Tina H Straley Marcia P Sward and Jon. W Scott Editors, 65 Innovations in Teaching Statistics Joan B Garfield Editor. 66 Mathematics in Service to the Community Concepts and models for service learning in the mathematical sciences. Charles R Hadlock Editor, 67 Innovative Approaches to Undergraduate Mathematics Courses Beyond Calculus Richard J Maher Editor. 68 From Calculus to Computers Using the last 200 years of mathematics history in the classroom Amy Shell Gellasch. and Dick Jardine Editors, 69 A Fresh Start for Collegiate Mathematics Rethinking the Courses below Calculus Nancy Baxter Hastings Editor.
70 Current Practices in Quantitative Literacy Rick Gillman Editor. 71 War Stories from Applied Math Undergraduate Consultancy Projects Robert Fraga Editor. 72 Hands On History A Resource for Teaching Mathematics Amy Shell Gellasch Editor. 73 Making the Connection Research and Teaching in Undergraduate Mathematics Education Marilyn P Carlson and. Chris Rasmussen Editors, 74 Resources for Teaching Discrete Mathematics Classroom Projects History Modules and Articles Brian Hopkins. 75 The Moore Method A Pathway to Learner Centered Instruction Charles A Coppin W Ted Mahavier E Lee May. and G Edgar Parker, 76 The Beauty of Fractals Six Different Views Denny Gulick and Jon Scott Editors. 77 Mathematical Time Capsules Historical Modules for the Mathematics Classroom Dick Jardine and Amy Shell Gel. lasch Editors, 78 Recent Developments on Introducing a Historical Dimension in Mathematics Education Victor J Katz and Costas. Tzanakis Editors, 79 Teaching Mathematics with Classroom Voting With and Without Clickers Kelly Cline and Holly Zullo Editors. 80 Resources for PreparingMiddle School Mathematics Teachers Cheryl Beaver Laurie Burton Maria Fung and Klay. Kruczek Editors, 81 Undergraduate Mathematics for the Life Sciences Models Processes and Directions Glenn Ledder Jenna P Car.
penter and Timothy D Comar Editors, 82 Applications of Mathematics in Economics Warren Page Editor. 83 Doing the Scholarship of Teaching and Learning in Mathematics Jacqueline M Dewar and Curtis D Bennett Ed. 84 Insights and Recommendations from the MAA National Study of College Calculus David Bressoud Vilma Mesa. and Chris Rasmussen Editors, 85 Beyond Lecture Resources and Pedagogical Techniques for Enhancing the Teaching of Proof Writing Across the. Curriculum Rachel Schwell Aliza Steurer and Jennifer F Vasquez Editors. 86 Using the Philosophy of Mathematics in Teaching Undergraduate Mathematics Bonnie Gold Carl E Behrens and. Roger A Simons Editors, 87 The Courses of History Ideas for Developing a History of Mathematics Course Amy Shell Gellasch and Dick. Jardine Editors, 88 Shifting Contexts Stable Core Advancing Quantitative Literacy in Higher Education Luke Tunstall Gizem Karaali. and Victor Piercey Editors, 89 Instructional Practices Guide Martha L Abell Linda Braddy Doug Ensley Lewis Ludwig and Hortensia Soto Proj.
ect Leadership Team,Manifesto A declaration of values. Success in mathematics opens opportunities for students A wealth of research literature exists on how. mathematics instructors can facilitate rich meaningful learning experiences and on what instructors can do. to improve teaching and learning at the undergraduate level Effective teaching and deep learning require. student engagement with content both inside and outside the classroom This Instructional Practices Guide. aims to share effective evidence based practices instructors can use to facilitate meaningful learning for stu. dents of mathematics Professional associations in the mathematical sciences along with state and national. funding agencies are supporting efforts to radically transform the undergraduate education experience it is. truly an exciting time to be a mathematics instructor. With that big picture in mind this guide is written from the perspective that teaching and learning are. forces for social change Beyond the confines of individual instructors classrooms beyond their decisions. about what mathematics to teach and how to teach it there are societal forces that call upon all mathematics. instructors to advocate for increased student access to the discipline of mathematics Inequity exists in many. facets of our society including within the teaching and learning of mathematics Because access to success in. mathematics is not distributed fairly the opportunities that accompany success in mathematics are also not. distributed fairly We in the mathematical sciences community should not affirm this inequitable situation. as an acceptable status quo We owe it to our discipline to ourselves and to society to disseminate mathe. matical knowledge in ways that increase individuals access to the opportunities that come with mathemat. ical understanding, Some of us have become reflective instructors over the course of our careers and our classrooms have. changed and improved as a result But if we truly want to effect change then we are compelled to extend the. reach of our efforts beyond our own students in our own classrooms It is our responsibility to examine the. system within which we educate students and find ways to improve that system It is our responsibility to. help our colleagues improve and to collectively succeed at teaching mathematics to all students so that our. discipline realizes its full potential as a subject of beauty of truth and of empowerment for all. Such a sea change will require transforming how mathematics is taught and facing our own individual. and collective roles in a system that does not serve all students well Societal norms tend toward a belief that. only a certain kind of individual can do mathematics and other kinds of people need not even try We in the. profession of teaching mathematics must look inward to determine if we are doing our part to dispel this. All instructors can facilitate student success in mathematics and we cannot underestimate the power. of the environment in our classrooms departments and institutions to positively impact student learning. Changing teaching practice is hard But those of us who do mathematics recognize the hard work required. to learn and understand it and we choose to do that hard work We can likewise choose to do the hard work. required to teach our beloved subject, Mathematics instructors stand at a crossroads We must gather the courage to take the difficult path of. change We must gather the courage to venture down the path of uncertainty and try new evidence based. viii MAA Instructional Practices Guide, strategies that actively engage students in the learning experience We must gather the courage to advocate. beyond our own classroom for student centered instructional strategies that promote equitable access to. mathematics for all students We stand at a crossroads and we must choose the path of transformation in. order to fulfill our professional responsibility to our students This Instructional Practices Guide can serve as. a catalyst for community wide transformation toward improved learning experiences and equitable access. to mathematics for all students Society deserves nothing less. Introduction to this guide, The most recent MAA documents Committee on the Undergraduate Programs in Mathematics CUPM Cur.
riculum Guide to Majors in the Mathematical Science Zorn 2015 and A Common Vision for Undergraduate. Mathematical Sciences Programs in 2025 Saxe and Braddy 2016 serve as an impetus for this Instructional. Practices Guide The CUPM Curriculum Guide provides course recommendations along with sample syllabi. for mathematical science courses but does not provide specific teaching strategies faculty have found to be. effective with their students and Common Vision calls for the use of evidence based instructional strategies. by reiterating the call from the INGenIOuS project report Zorn et al 2014. We acknowledge that changing established practices can be difficult and painful Changing cultures of. departments institutions and organizations can be even harder But there is reason for optimism In. mathematical sciences research we are always willing even eager to replace mediocre or somewhat. successful strategies with better ones In that open minded spirit we invite the mathematical sciences. community to view this call to action as a promising opportunity to live up to our professional respon. sibilities by improving workforce preparation p 25. With this in mind this Instructional Practices Guide is designed as a how to guide focused on mathe. matics instruction at the undergraduate level It is based on the concept that effective teaching is supported. by three foundational types of practices classroom practices assessment practices and course design. practices all informed by empirical research as well as the literature on technology and equity In this intro. duction we describe the intended audience provide a brief overview of each practice and offer suggestions. on how to navigate this guide, The Instructional Practices Guide is founded on the belief that every student should have the opportunity. to engage in deep mathematics learning guided and mentored by their instructor It is intended for all in. structors of mathematics from the new graduate teaching assistant to the most experienced senior instruc. tor from the contingent faculty member at a two year institution to the new faculty member at a doctor. al granting institution from the instructor who wants to transform her own teaching to the mathematician. or mathematics educator facilitating professional development for graduate students or collegiate faculty. It is also intended for administrators who are in positions to work with their faculty to initiate systemic. change in their departments and across their institutions Administrators will recognize that many of our. suggestions are applicable to other disciplines in fact some of the suggestions are borrowed from research. in science education, The Classroom Practices chapter provides examples of teaching practices both inside and outside the. classroom that foster student engagement as well as a section on selecting appropriate mathematical tasks. that contribute to building a sense of community within the classroom The Assessment Practices chapter. builds on policy assessment documents from various associations including the National Council of Teach. ers of Mathematics the American Statistical Association and of course the MAA This chapter centers. on the interplay between formative and summative assessment to examine the teaching and learning of. mathematics with a strong focus on learning outcomes The Design Practices chapter provides the reader a. x MAA Instructional Practices Guide, brief introduction to instructional designs that help achieve desired learning outcomes based on theories of. design along with potential challenges and opportunities associated with instructional design. We acknowledge the suggestions in this guide are not exhaustive but we aim to include something of. interest for any reader to adapt for their own classroom Each of the practices informs the others and de. pending on readers experiences they might choose to read the guide in an order other than the one pre. sented We purposefully begin with the Classroom Practice chapter in an effort to engage readers who are. just beginning to transform their teaching As readers gain more experience with student centered teaching. practices they can navigate back and forth among the chapters as needed For example a reader more expe. rienced with student centered teaching might begin by reading the Design Practices chapter to prepare for. designing a new course then read the Classroom Practices chapter to prepare specific lessons and activities. then read the Assessment Practices chapter to garner formative assessment strategies redesign a lesson or. classroom activities based on the results of the formative assessment and then learn about novel summative. assessments The model shown below indicates the fluid way in which readers might utilize the guide. Assessment Classroom,Practices Practices, We also acknowledge transforming one s classroom practices takes time and we firmly believe specific. examples are helpful in facilitating such a transformation Throughout the guide we offer vignettes that are. both easy to follow and informed by the substantial body of research regarding effective teaching and deep. student learning, The crucial finding from the research upon which this guide is founded is that effective teaching and deep.
learning require student engagement with mathematics both inside and outside the classroom Bringing. student ideas beliefs and practices into the direct view of peers and instructors enriches teaching and. learning and promotes community in remarkable ways The vast body of evidence strongly supports the. transformational power of these practices in prompting changes in instructors and students at all levels from. all demographic backgrounds, Indeed such transformation can promote diversity inclusion cultural responsiveness and social justice. within the mathematical sciences community Our task as a community is to create these meaningful and. inspiring mathematical experiences for all our instructors and students As such we conclude the document. with a brief discussion on cross cutting themes regarding technology and equity two important topics that. are intertwined in each of the other chapters We strongly encourage our readers to reflect on how they. integrate technology into each of the practices and how their practices promote equity in the mathematics. In summary this Instructional Practices Guide is a call to the mathematical sciences community to scale. up the use of evidence based instructional strategies and to collectively and individually hold ourselves. accountable as professional educators for improving the learning experiences of all undergraduate mathe. matics students,Acknowledgements, This large project could not have been completed without the hard work of many people within the mathe. matics community including MAA staff members and the project team is very grateful to all who helped. Of particular importance the project activities were supported in large part by the National Science Foun. dation Division of Undergraduate Education NSF 1544324 Any opinions findings and conclusions or. recommendations expressed in this material are those of the authors and do not necessarily reflect the views. of the National Science Foundation, The following people made direct contributions to the funded project and many more participated from. the initial discussion to reviewing the final draft. Project Leadership Team,Martha L Abell Georgia Southern University. Linda Braddy Tarrant County College,Doug Ensley Mathematical Association of America.
Lewis Ludwig Denison University,Hortensia Soto University of Northern Colorado. Project Steering Committee and Lead Writers,James Alvarez University of Texas Arlington. Benjamin Braun University of Kentucky,Elizabeth Burroughs Montana State University. Rick Cleary Babson College,Karen Keene North Carolina State University. Gavin LaRose University of Michigan,Julie Phelps Valencia College.
April Strom Scottsdale Community College,Project Advisory Board. Matt Ando University of Illinois,David Bressoud Macalester College. Marilyn Carlson Arizona State University,Annalisa Crannell Franklin and Marshall College. Tara Holm Cornell University,Dave Kung St Mary s College of Maryland. Rachel Levy Harvey Mudd College,Francis Su Harvey Mudd College.
xii MAA Instructional Practices Guide,Uri Treisman University of Texas Austin. Paul Zorn St Olaf College,Contributing Writers,Scott Adamson Chandler Gilbert Community College. Aditya Adiredja University of Arizona,Spencer Bagley University of Northern Colorado. Randy Boucher US Military Academy,Derek Bruff Vanderbilt University. Joe Champion Boise State University,Beth Cory Sam Houston State University.
Jessica Deshler West Virginia University,Jackie Dewar Loyola Marymount University. Jess Ellis Hagman Colorado State University,Angie Hodge Northern Arizona University. Brian Katz Augustana College,Elizabeth Kelly Berea College. Klay Kruzcek Southern Connecticut State University. Brigitte Lahme Sonoma State University,Luis Leyva Vanderbilt University. Rachel Levy Harvey Mudd College,Guadalupe Lozanoa University of Arizona.
Bill Martin North Dakota State University,John Meier Lafayette College. Victor Piercy Ferris State Uiversity,Mike Pinter Belmont University. Chris Rasmussen San Diego State University,Jack Rotman Lansing Community College. Behnaz Rouhani Perimeter College,Ay e ahin Wright State University. Milos Savic University of Oklahoma,Kimberly Seashore San Francisco State University.
Mary Shepherd Northwest Missouri State University,Robert Talbert Grand Valley State University. Diana Thomas US Military Academy,Christine von Renesse Westfield State University. Laura Watkins Glendale Community College, Claire Wladis Borough of Manhattan Community College. Phil Yates St Michael s College, Maria Del Rosario Zavala San Francisco State University. Manifesto A declaration of values v,Introduction to this guide vii.
Acknowledgements ix,Classroom Practices 1,CP 1 Fostering student engagement 1. CP 1 1 Building a classroom community 1,CP 1 2 Wait time 3. CP 1 3 Responding to student contributions in the classroom 5. CP 1 4 One minute paper or exit tickets 7,CP 1 5 Collaborative learning strategies 8. CP 1 6 Just in time teaching JiTT 15, CP 1 7 Developing persistence in problem solving 18. CP 1 8 Inquiry based teaching and learning strategies 21. CP 1 9 Peer instruction and technology 22,CP 2 Selecting appropriate mathematical tasks 26.
CP 2 1 Intrinsic appropriateness What makes a mathematical task appropriate 26. CP 2 2 Extrinsic appropriateness 27, CP 2 3 Theoretical frameworks for understanding appropriateness 28. CP 2 4 How to select an appropriate mathematical task 29. CP 2 5 Choosing meaningful group worthy tasks 30, CP 2 6 Communication Reading writing presenting visualizing 35. CP 2 7 Error analysis of student work 38,CP 2 8 Flipped classrooms 41. CP 2 9 Procedural fluency emerges from conceptual understanding 42. CP Conclusion 44,CP References 45,Assessment Practices 49. AP 1 Basics about assessment 49,AP 1 1 Assessment frameworks 49.
AP 1 2 Clearly specify learning outcomes 50,AP 1 3 Formative and summative assessment 53. AP 2 Formative assessment creates an assessment cycle 53. AP 2 1 Implementing formative assessment 54,xiv MAA Instructional Practices Guide. AP 2 2 Formative assessments to improve mathematical practices 57. AP 2 3 Formative assessment to influence students beliefs and motivations 58. AP 3 Summative assessment 59,AP 3 1 Assigning course grades 59. AP 3 2 Exemplary summative assessments 61, AP 3 3 Creating and selecting problems for summative assessment 63. AP 4 Assessments that promote student communication 66. AP 4 1 Writing assignments 66,AP 4 2 Oral presentations 68.
AP 4 3 Group projects 69, AP 5 Conceptual understanding What do my students really know 71. AP 5 1 What is conceptual understanding 71,AP 5 2 What are concept inventories 71. AP 5 3 Using items from concept inventories 73,AP 6 Assessment in large enrollment classes 75. AP 6 1 Online homework systems 75,AP 6 2 Classroom polling systems 77. AP 7 Assessment in non traditional classrooms 78,AP 7 1 Assessment in online courses 78.
AP 7 2 Assessing via technology 79, AP 7 3 Assessment in non traditional course settings 81. AP References 82,Design Practices 89,DP 1 Introduction to design practices 89. DP 1 1 Questions for design 90,DP 1 2 Considerations for design 92. DP 1 3 Designing for equity 92, DP 2 Student learning outcomes and instructional design 95. DP 2 1 Designing the learning environment 100, DP 2 2 Designing mathematical activities and interactive discussions 101.
DP 2 3 Designing homework 102,DP 2 4 Designing a flipped classroom 103. DP 2 5 Using formative and summative assessment in design 103. DP 2 6 Reflective instruction 103,DP 2 7 Students needing accommodations 104. DP 3 Challenges and opportunities 105, DP 3 1 Big picture challenges and opportunities 105. DP 3 2 Other challenges 105,DP 3 3 Embracing opportunities 108. DP 4 Theories of instructional design 109,DP 4 1 Backward design 110.
DP 4 2 Realistic mathematics education 110,DP 4 3 Universal design for learning 111.

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