Math Expressions
Developing Student Thinking and Problem Solving Through Communication
If ordering from outside of Canada, please call customer service. 1 (800) 361-6128
Copyright
2012
Delivery Method
Province
National
Imprint
Pearson Canada
"Many people believe that mathematics is a subject that consists only of formulas and right-or-wrong answers. They are of the opinion that it does not require discussion, debate, or exploration and that it offers little in the way of meaningful connection to their daily lives. These misconceptions are often due to how people experienced mathematics in social contexts or in school. By infusing a variety of learning experiences into our mathematics program, however, we can help students develop an understanding that mathematics has relevance in their lives."
Cathy Marks Krpan, Math Expressions
Research is clear: communication is an essential mathematical process. This book provides all the tools to make communication come alive and to ensure the classroom is a vibrant, collaborative learning environment. Centred around three main sections—Mathematical Discourse, Reading in Mathematics, and Writing in Mathematics—Dr. Cathy Marks Krpan provides practical suggestions on how to create such an environment.
Each section includes:
- What the Research Says
- Collaborative Skills and Structures
- Teaching Strategies
- Assessment Tips
- Supports for English Language Learners
- Canadian Student Samples with modelled Teacher Feedback
- Line Masters and a Companion Website
Authors
Cathy Marks Krpan, Ph.D. teaches courses in the Initial Teacher Education Program and graduate program at the Ontario Institute for the Studies in Education (OISE) of the University of Toronto. Cathy is passionate about seeing students learn—all students! And at the core of this passion lies a deep belief in the power of communication as an essential tool to enhance understanding, improve problem solving, and create a community of learners. Based on her research with diverse student populations, and her classroom experiences as an educator and student, Cathy believes that at the heart of all equitable math classrooms is the empowerment of student voices. It is her hope that all students will have the opportunity to share their insights and be part of a supportive mathematical community in which they feel comfortable taking risks and conducting mathematical investigations.
Cathy’s many speaking engagements and consultations across North America have provided her with numerous opportunities to collaborate and learn from students, teachers, and administrators alike. Her research over the past ten years has focused on student communication, graphic organizers, and the development of metacognition and collaboration skills. In addition, she is currently investigating teaching strategies to assist English Language Learners and those who struggle in mathematics. Since her research is field-based, she continues to collaborate closely with students and teachers in the field.
On February 21, 2021, Cathy received the 2021 Margaret Sinclair Memorial Award Recognizing Innovation and Excellence in Mathematics Education from The Fields Institute. The Institute writes “Supporters of Dr. Marks Krpan’s nomination noted her affirmation that any student can engage in deep and complex mathematics. Some of the innovative research she has undertaken include her work with elementary school teachers around mathematical argumentation and proof, which are rooted in theory while still providing practical support, thus allowing teachers to bring rich mathematical experiences into their classrooms”
Cathy is also the recipient of several OISE Student Union Initial Teacher Education Teaching Awards. She has also been nominated for the OISE Award for Excellence in Initial Teacher Education, 2011, 2012. Dr. Marks Krpan has authored numerous publications, including The Write Math, My Math Journal, BoldPrint books, and a set of math readers, co-authored with Larry Swartz, which received the 2006 American Teachers’ Choice Award.
In her current book, Teaching Math With Meaning, Cathy believes that through competency-based learning, students and teachers alike can deepen their mathematical understanding and share and impart that knowledge in and out of the math classroom. Teaching Math With Meaning takes a practical approach to embedding this deep learning in K to Grade 8 mathematics classrooms.
Should you wish to continue the dialogue beyond the pages of this book, or inquire about keynote, consultation, and/or workshop opportunities, you can contact Cathy at cathy.marks.krpan@utoronto.ca.
Companion Content
Introduction: A Culture of Mathematical Expression
Becoming a Facilitator of Mathematical Expression
What Is the Role of the Teacher?
Environmental Inquiry: A Pedagogical Framework, Natural Curiosity: A Resource for Teachers, The Laboratory School at the Dr. Eric Jackman Institute of Child Study. Ontario Institute for Studies in Education, 2011
http://www.naturalcuriosity.ca/environmental.php?pgcat=branch1&sfield=eight
Graduating from Teacher to Facilitator in the Technology-Rich Classroom
Carol M. Webb, 2001
http://www.4teachers.org/testimony/webb/
Supporting Learners with Math Anxiety
Equity for All Students in the New Millennium: Disabling Math Anxiety
Joseph M. Furner and Mary Lou Duffy, Intervention in School and Clinic, November 2002
http://isc.sagepub.com/content/38/2/67.extract
A Teacher’s Quick Guide to Understanding Mathematics Anxiety
DesLey Viator Plaisance, LATM Journal, Louisiana Association of Teachers of Mathematics, 2011
http://www.lamath.org/journal/vol6no1/anxiety_guide.pdf
Study Suggests There May Be a Neurological Basis for Math Anxiety
From the Stanford University press release, 2012
http://therapytoronto.ca/news/?p=1972
Math Anxiety
Counseling Center, Texas State University
http://www.counseling.txstate.edu/resources/shoverview/bro/math.html
Supporting English Language Learners
Equity of Literacy-Based Math Performance Assessments for English Language Learners
Clara Lee Brown, Bilingual Research Journal: The Journal of the National Association for Bilingual Education, 2005
http://www.tandfonline.com/doi/abs/10.1080/15235882.2005.10162839#preview
Multilingualism in the English-Language Classroom: Pedagogical Considerations
Jim Cummins, TESOL Quarterly, June 2009
http://onlinelibrary.wiley.com/doi/10.1002/j.1545-7249.2009.tb00171.x/abstract
TIPS for English Language Learners in Mathematics, Grades 7, 8, 9 Applied, 10 Applied
Ontario Ministry of Education, 2005
http://www.edugains.ca/resourcesELL/Resources/TIPSforELLMath4All.pdf
Understanding and Supporting Children’s Mathematical Vocabulary Development
Rheta Rubenstein and Denisse Thompson, Teaching Children Mathematics, October 2002
http://www.nctm.org/publications/article.aspx?id=21326
What Can a Mathematics Teacher Do for the English Language Learner?
Texas Comprehensive Center, 2006
http://txcc.sedl.org/resources/ell_materials/mell/
Assessing Math Learning
Growth for Every Student
Limestone District School Board, Ontario, 2008
http://www.limestone.on.ca/programs/elementary/Math%20Resources/Growth%20for%20Every%20Studentfinal.pdf
Growing Success: Assessment, Evaluation, and Reporting in Ontario Schools, First Edition, Covering Grades 1 to 12
Ontario Ministry of Education, 2010
http://www.edu.gov.on.ca/eng/policyfunding/growSuccess.pdf
Chapter 1: A Culture of Collaboration and Inquiry
Cooperative Learning
Exploring Mathematics Talk in Elementary Classrooms
Cathy Mark Krpan, Partnerships for Professional Learning: Literacy & Numeracy Initiatives, Ontario Institute for Studies in Education, 2009
http://www.oise.utoronto.ca/oise/UserFiles/File/LNS%20PUBLICATION_FINAL.pdf
Supporting Struggling Learners: Collaborative Problem Solving in Mathematics
Cathy Marks Krpan, Inquiry into Practice: Reaching Every Student Through Inclusive Curriculum, Ontario Institute for Studies in Education, 2009
http://www.oise.utoronto.ca/cms/oisefiles/ITE%20PUB%202011_COMPLETE/HTML/index.html#/103/zoomed
Cooperative Learning Instruction Strategies
Teachnology: The Online Teacher Resource
http://www.teach-nology.com/currenttrends/cooperative_learning/
Indicators of Success for a Collaborative Mathematics Classroom
.PDF 22kb 1pg
Observation Checklist for a Collaborative Mathematics Classroom
.PDF 23mb 1pg
Indicators of Success for Descriptive Feedback
.PDF 32kb 1pg
All Line Masters (.PDF 63kb 4pgs)
LM 1.1 Cooperative Math Chart
.PDF 32kb 1pg
LM 1.2 Cooperative Group Roles for Mathematics [Primary]
.PDF 38kb 1pg
LM 1.3 Cooperative Group Roles for Mathematics [Junior/Intermediate]
.PDF 22kb 1pg
LM 1.4 Observation Checklist for Cooperative Work
.PDF 24kb 1pg
Chapter 2: Mathematical Discourse
Mathematical Discourse
Equity for English Language Learners in the Mathematics Classroom
Gina Borgioli, Teaching Children Mathematics, October 2008
http://www.nctm.org/publications/article.aspx?id=22261
Describing Levels and Components of a Math-Talk Learning Community
Kimberly Hufferd-Ackles, Karen C. Fuson, Miriam Gamoran, Journal for Research in Mathematics Education, March 2004
http://www.nctm.org/publications/article.aspx?id=17479
Unpacking the Nature of Discourse in Mathematics Classrooms
Eric Knuth and Dominic Peressini, Mathematics Teaching in the Middle School, January 2001
http://www.nctm.org/publications/article.aspx?id=19872
Exploring Mathematics Talk in Elementary Classrooms
Cathy Mark Krpan, Partnerships for Professional Learning: Literacy & Numeracy Initiatives, Ontario Institute for Studies in Education, 2009
http://www.oise.utoronto.ca/oise/UserFiles/File/LNS%20PUBLICATION_FINAL.pdf
Supporting Struggling Learners: Collaborative Problem Solving in Mathematics
Cathy Marks Krpan, Inquiry into Practice: Reaching Every Student Through Inclusive Curriculum, Ontario Institute for Studies in Education, 2009
http://www.oise.utoronto.ca/cms/oisefiles/ITE%20PUB%202011_COMPLETE/HTML/index.html#/103/zoomed
Principles and Standards for School Mathematics
National Council of Teachers of Mathematics, 2000
http://www.nctm.org/standards/content.aspx?id=16909
Fostering Mathematical Thinking and Problem Solving: The Teacher’s Role
Nicole Rigelman, Teaching Children Mathematics, February 2007
http://www.nctm.org/publications/article.aspx?id=22041
Using “Think-Time” and “Wait-Time” Skillfully in the Classroom
Robert J. Stahl, ERIC Resource Center, 1994
http://www.eric.ed.gov/PDFS/ED370885.pdf
Let’s Talk: Promoting Mathematical Discourse in the Classroom
Catherine A. Stein, Mathematics Teacher, November 2007
http://www.nctm.org/publications/article.aspx?id=19362
A Classroom Perspective on the Principle of Moderate Challenge in Mathematics
Julianne C. Turner and Debra K. Meyer, The Journal of Educational Research, August 2010
http://www.tandfonline.com/doi/abs/10.3200/JOER.97.6.311-318
A Teacher’s Meditation of a Thinking-Aloud Discussion in a 6th Grade Mathematics Classroom
Betina Zolkower and Sam Shreyar, Educational Studies in Mathematics, 2007
http://www.springerlink.com/content/8526484n24032244/
A Resource of Multilingual Math Words
English as an Additional Language
Emaths
http://www.emaths.co.uk/index.php/4-teachers/other-resources/eal
Bansho
Three-part Problem-Solving, Clip 3 Bansho
Webcast, Curriculum Services Canada, 2008
http://resources.curriculum.org/secretariat/may2_bansho.shtml
All Tools (.PDF 456kb 32pgs)
Indicators of Success for Mathematical Discourse
.PDF 22kb 1pg
Observation Checklist for Mathematical Discourse
.PDF 24kb 1pg
My Math Talk [Primary]
.PDF 31kb 1pg
Indicators of Success for Effective Mathematical Discourse [Junior/Intermediate]
.PDF 32kb 1pg
Talk Prompts
.PDF 129kb 1pg
Pattern Block Designs
.PDF 256kb 9pgs
Cards for Clothesline Activity
.PDF 62kb 10pgs
Word Problems for Math Congress
.PDF 28kb 8pgs
All Line Masters (.PDF 194kb 8pgs)
2.1 Student Discourse Observation Sheet
.PDF 24kb 1pg
2.2 Interview Sheet
.PDF 20kb 1pg
2.3 Concept Attainment: Shapes with Line Symmetry [Yes Cards]
.PDF 63kb 1pg
2.4 Concept Attainment: Shapes with Line Symmetry [No Cards]
.PDF 67kb 1pg
2.5 Concept Attainment: Shapes with Area Less Than or Equal to 6 Square Units [Yes Cards]
.PDF 41kb 1pg
2.6 Concept Attainment: Shapes with Area Less Than or Equal to 6 Square Units [No Cards]
.PDF 49kb 1pg
2.7 Concept Attainment: Prime Numbers [Yes and No Cards]
.PDF 34b 1pg
2.8 Wanted!
.PDF 40kb 1pg
Chapter 3: Reading in Mathematics
Reading in Mathematics
The Answer Is 20 Cookies. What Is the Question?
Angela T. Barlow and Janie M. Cates, Teaching Children Mathematics, December 2006/January 2007
http://www.nctm.org/publications/article.aspx?id=22030
A Problem-Solving Alternative to Using Key Words
Lisa Clement and Jamal Bernhard, Mathematics Teaching in the Middle School, March 2005
http://www.nctm.org/publications/article.aspx?id=20390
The Thought-Filled Curriculum
Arthur L. Costa, Educational Leadership, February 2008
http://www.ascd.org/publications/educational_leadership/feb08/vol65/num05/The_Thought-Filled_Curriculum.aspx
Mathematics and Cognition
Arthur Hyde, Educational Leadership, November 2007
http://www.ascd.org/publications/educational-leadership/nov07/vol65/num03/Mathematics-and-Cognition.aspx
Knowledge Change as a Function of Mathematics Experience: All Contexts are Not Created Equal
Nicole M. McNeil and Martha W. Alibali, Journal of Cognition and Development, 2005
http://www.tandfonline.com/doi/abs/10.1207/s15327647jcd0602_6
Reciprocal Teaching as a Comprehension Strategy for Understanding Mathematical Word Problems
Delinda van Garderen, Reading & Writing Quarterly, 2004
http://www.tandfonline.com/doi/abs/10.1080/10573560490272702
Online Mathematics Dictionary
A Maths Dictionary for Kids
Jenny Eather
http://www.amathsdictionaryforkids.com/
All Tools (.PDF 874kb 26pgs)
Indicators of Success for Reading in Mathematics
.PDF 22kb 1pg
Observation Checklist for Reading in Mathematics
.PDF 23kb 1pg
My Math Reading: Word Problems [Primary]
.PDF 31kb 1pg
Indicators of Success for Understanding Word Problems [Junior/Intermediate]
.PDF 31kb 1pg
Mathematical Picture Book List
.PDF 65kb 5pgs
Word Problems
.PDF 27kb 8pgs
Math in the Environment
.PDF 511kb 7pgs
Mathematical Recipes
.PDF 61kb 2pgs
All Line Masters (.PDF 209kb 7pgs)
3.1A Biography Glyphs
.PDF 69kb 1pg
3.1B Biography Glyphs
.PDF 52kb 1pg
3.2 Train Recipe Card
.PDF 45kb 1pg
3.3 My Train
.PDF 48kb 1pg
3.4 Mathematical Recipes [Whole Numbers]
.PDF 39kb 1pg
3.5 Mathematical Recipes [Fractions and Percents]
.PDF 47kb 1pg
3.6 Mathematical Recipes [Algebra]
.PDF 42b 1pg
Chapter 4: Writing in Mathematics
Writing in Mathematics
Investigating Mathematical Thinking Using Open Writing Prompts
Leslie Aspinwall and Julie Aspinwall, Mathematics Teaching in the Middle School, March 2003
http://www.nctm.org/publications/article.aspx?id=20106
Blueprint for Writing in Middle School Mathematics
Juliet A. Baxter, John Woodward, Deborah Olson, and Janet Robyns, Mathematics Teaching in the Middle School, September 2002
http://www.nctm.org/publications/article.aspx?id=20047
Beyond Essentialism: The Complexity of Language in Teaching Mathematics to Latina/o Students, American Educational Research Journal, December 2002
Rochelle Gutiérrez,
http://aer.sagepub.com/content/39/4/1047.abstract
Academic Emotions in Students’ Self-Regulated Learning and Achievement: A Program of Qualitative and Quantitative Research
Reinhard Pekrun, Thomas Goetz, Wolfram Titz, and Raymond P. Perry, Educational Psychologist, 2002
http://www.tandfonline.com/doi/abs/10.1207/S15326985EP3702_4#preview
Graphic Organizers
Q & A: What’s the Big Deal with Graphic Organizers?
Edwin Ellis
http://www.graphicorganizers.com/About-MSS/q-a-a-whats-the-big-deal-with-graphic-organizers.html
What Are Graphic Organizers?
Instructional Strategies Online, Saskatoon Public Schools
http://olc.spsd.sk.ca/de/pd/instr/strats/graphicorganizers/index.html
Mind Mapping Resources
The University of British Columbia
http://wiki.ubc.ca/Mind_Mapping_Resources
Institute for Human and Machine Cognition Concept Map Tools
http://cmap.ihmc.us/
The Theory Underlying Concept Maps and How to Construct and Use Them
Joseph D. Novak and Alberto J. Cañas, Institute for Human and Machine Cognition, 2008
http://cmap.ihmc.us/publications/researchpapers/theorycmaps/theoryunderlyingconceptmaps.htm
What Is a Concept? …From a Concept Mapping Perspective
Alberto J. Cañas and Joseph D. Novak, Institute for Human and Machine Cognition, 2009
http://cmap.ihmc.us/docs/Concept.html
What Are Propositions? …From a Concept Mapping Perspective
Alberto J. Cañas, Institute for Human and Machine Cognition, 2009
http://cmap.ihmc.us/docs/Proposition.html
What Are Linking Words? …From a Concept Mapping Perspective
Alberto J. Cañas, Institute for Human and Machine Cognition, 2009
http://cmap.ihmc.us/Docs/linkingwords.html
All Tools (.PDF 695kb 15pgs)
Indicators of Success for Writing in Mathematics
.PDF 22kb 1pg
Observation Checklist for Writing in Mathematics
.PDF 24kb 1pg
My Math Writing [Primary]
.PDF 31kb 1pg
Indicators of Success for Writing [Junior/Intermediate]
.PDF 31kb 1pg
Five-Frames and Ten-Frames
.PDF 31kb 2pgs
My Math Pictures [Primary]
.PDF 31kb 1pg
Indicators of Success for Effective Pictures in Mathematical Writing [Junior/Intermediate]
.PDF 310kb 1pg
Mind Map Visual Overview
.PDF 330kb 1pg
Concept Map Visual Overview
.PDF 223kb 1pg
Mathematics is like…because…
.PDF 118kb 1pg
My Math Mind Map [Primary]
.PDF 31kb 1pg
Indicators of Success for an Effective Math Mind Map
.PDF 32kb 1pg
My Math Concept Map
.PDF 31kb 1pg
Indicators of Success for an Effective Math Concept Map
.PDF 39kb 1pg
All Line Masters (.PDF 185kb 14pgs)
4.1A VVWA [Primary]
.PDF 18kb 1pg
4.1B VVWA [Junior/Intermediate]
.PDF 18kb 1pg
4.2 Frayer Model
.PDF 18kb 1pg
4.3 Frayer Model
.PDF 18kb 1pg
4.4 Three-Part Concept Circle
.PDF 45kb 1pg
4.5 Four-Part Concept Circle
.PDF 42kb 1pg
4.6 Four-Part Concept Circle
.PDF 47kb 1pg
4.7 Five-Part Concept Circle
.PDF 50kb 1pg
4.8 Six-Part Concept Circle
.PDF 52kb 1pg
4.9 Eight-PartConcept Circle
.PDF 50kb 1pg
4.10 Sample Linking Words for Concept Maps
.PDF 19kb 1pg
4.11 Mind Map Elements
.PDF 18kb 1pg
4.12 Concept Map Elements
.PDF 18kb 1pg
4.13 Discovery Sheet
.PDF 18kb 1pg
Table of Contents
Introduction
A Culture of Mathematical Expression
Communication: The Critical Piece for Student Success
Challenges at Face Math Educators
Becoming a Facilitator of Mathematical Expression
Infusing Communication into Mathematics
Supporting Learners with Math Anxiety
Supporting Learners Who Struggle in Mathematics
Supporting English Language Learners
Three-Part Problem-Solving Lesson: A Structure for Communication and Investigations
Minds On/Getting Started
Working on It/Action
Consolidation/Re ecting and Connecting
Assessing Math Learning
Using Descriptive Feedback to Promote Strong Communication and Problem-Solving Skills
Chapter 1
A Culture of Collaboration and Inquiry
Indicators of Success
What the Research Says About Cooperative Learning
Elements of the Cooperative Learning Experience
Teaching Social Processes: Equity for All Learners
Positive Interdependence and a Common Goal
Individual Accountability
Face-to-Face Interactions
Facilitating Cooperative Learning
Cooperative Groupings
Assigning Group Roles
Promoting Equity
Assessing Cooperative Learning
Cooperative Learning Structures
Think-Pair-Share
Placemat
Three Stay, One Stray
Gallery Walk
Inside-Outside Circle
Chapter 2
Mathematical Discourse
Indicators of Success
What the Research Says About Mathematical Discourse
Assessing Mathematical Discourse
Facilitating Mathematical Discourse
Creating a Culture of Meaningful Talk
Encouraging Complex Discourse
E ffective Questioning
Supporting English Language Learners
Wait Time
Mathematical Discourse: Skills and Structures
Conversational Skills
Talk Prompts
Teaching Strategies
Pattern Block Talk
Clipboard Walk and Talk
It Is and It Isn’t
Concept Attainment
Mathematical Clothesline
Wanted Number
Math Congress
Bansho
Chapter 3
Reading in Mathematics
Indicators of Success
What the Research Says About Reading in Mathematics
Assessing Reading in Mathematics
Facilitating Reading in Mathematics
Infusing Reading into Your Math Program
Supporting English Language Learners
Supporting Readers Who Struggle
Building a Mathematical Word Wall
Using Mathematical Picture Books
Reading in Mathematics: Skills and Structures
Key Reading Strategies
Reading Word Problems
Reading Mathematical Symbols
Teaching Strategies
Picture Book Read-Aloud
Visualizing Mathematical Text
Mathematical Text Paraphrasing
Mathematical Text Sorting
What’s the Problem?
Show Me the Math!
Mathematical Glyphs
Mathematical Recipes
Chapter 4
Writing in Mathematics
Indicators of Success
What the Research Says About Writing in Mathematics
Assessing Mathematical Writing
Facilitating Mathematical Writing
Classroom Organization
Supporting English Language Learners
Encouraging the Use of Pictures
Modelling the Writing Process
Mathematical Writing: Skills and Structures
Cooperative Writing
Graphic Organizers
Teaching Strategies
Affective Writing Explorations
My Best Math Moment
De fining a Mathematical Term or Concept
Lesson Feedback
Exploring Mathematical Proofs
Lateral- Thinking Explorations
Lateral-Thinking with Pictures
Recording Mathematical Discoveries
Exploring Concept Circles
Exploring Mind Maps
Exploring Concept Maps
Mapping Your Learning
Math on the Back
Line Masters
References
Index
Virtual Samples
Math Expressions
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