Conceptual Model-Based Problem Solving: Teach Students with Learning Difficulties to Solve Math Problems

267 Pages·2012·6.58 MB·English

by  Yan Ping Xin

Conceptual Model-Based Problem Solving

Conceptual Model-Based Problem Solving
Teach Students with Learning Difficulties to
Solve Math Problems
Yan Ping Xin
Purdue University, West Lafayette, USA
A C.I.P. record for this book is available from the Library of Congress.
ISBN: 978-94-6209-102-3 (paperback)
ISBN: 978-94-6209-103-0 (hardback)
ISBN: 978-94-6209-104-7 (e-book)
Published by: Sense Publishers,
P.O. Box 21858,
3001 AW Rotterdam,
The Netherlands
https://www.sensepublishers.com/
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No part of this work may be reproduced, stored in a retrieval system, or transmitted
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TABLE OF CONTENTS
Acknowledgements vii
About the Author ix
1. Conceptual Model-based Problem Solving:
Teach Students with Learning Diffi culties to Solve Math Problems
Introduction 1
Algebra Thinking in Problem Solving 1
Mathematical Modeling 2
Theoretical Framework: Conceptual Model-based
Problem Solving 3
SBI that Emphasizes Semantic Analyses and
Representation of the Problem 4
COMPS that Emphasizes Algebraic Expression of
Mathematical Relations 4
Summary 6
Program Components 7
The scope and sequence of the program 7
Target Audience and Users of the Program 8
2. COMPS Program
Introduction 11
Singapore Bar Models (SBM) to Facilitate the
Transition to Mathematical Models 11
Word Problem [WP] Story Grammar (Xin et al., 2008) 12
A Cognitive Heuristic DOTS Checklist (Xin et al., 2008) 13
Additive and Multiplicative Word Problem Structure
and its Variations 13
Instructional Phases 16
Part 1: Additive Problem Solving 19
Unit 1: Representing Part-Part-Whole (PPW) Problems 21
Lesson 1: Introduction 21
Lesson 2: Part-Part-Whole Problem Representation 26
Unit 2: Solving Part-Part Whole (PPW) Problems 35
Lesson 3: Solving PPW Problems 35
Unit 3: Representing Additive Compare (AC) Problems 49
Lesson 4: Representing AC-More Problems 49
Lesson 5: Representing AC-Less Problems 60
Unit 4: Solving Additive Compare (AC) Problems 69
Lesson 6: Solving Mixed AC Problems 69
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TABLE OF CONTENTS
Unit 5: Solving Mixed PPW and AC Problems 83
Lesson 7: Solving mixed PPW and AC Problems 83
Part 2: Multiplicative Problem Representation and Solving 97
Unit 6: Representing Equal Groups (EG) Problems 99
Unit 7: Solving Equal Groups (EG) Problems 107
Unit 8: Representing Multiplicative Compare (MC) Problems 115
Unit 9: Solving Multiplicative Compare (MC) Problems 127
Unit 10: Solving Mixed Equal Groups and Multiplicative
Compare Problems 141
Part 3: Solving Complex Problems 159
Unit 11: Solving Complex Word Problems 161
Unit 12: Solving Mixed Multi-Step Problems 173
3 Connection between Mathematical Ideas: Extend Multiplicative
Reasoning to Geometry Learning 183
Appendix A: Student Worksheets 191
Appendix B: Reference Guide 227
References 267
vi
ACKNOWLEDGEMENTS
First, I would like to thank Nicole Spurlock, a pre-service elementary teacher
at Purdue University, for her great effort and contributions to the editing and
proofreading of this book. Her thoroughness and detailed comments helped me
tremendously in making my writing flow better and easier for readers to understand.
Second, I would like to thank Miss You Luo, a former elementary school
teacher, for her great contributions in field testing the Conceptual Model-Based
Problem Solving (COMPS) approach that integrates the bar model to facilitate
students’ transition from concrete or semi-concrete modeling to the use of abstract
mathematical models. In addition, Miss Luo contributed to the preparation of the
Student Worksheets and Reference Guide included in the Appendixes of the book.
Lastly, I would like to thank all of my Ph.D. students who have worked with me in
carrying out many research studies that examine the effects of the COMPS program
that I have been developing. Special thanks to Casey Hord, Ph.D., for his strong
interests in Geometry, which led me to expand my work to the area of geometry (see
Chapter III of the book).
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ABOUT THE AUTHOR
Yan Ping Xin, Ph.D., is an associate professor of special education at Purdue
University. She earned her Ph.D. in 2003 at Lehigh University. The focus of Xin’s
program of research is on improving mathematics performance of students with
learning disabilities/difficulties (LD). Her empirical work in (1) literature synthesis/
meta-analyses, (2) curriculum evaluations, and (3) intervention development has
led to theoretical contributions in conceptual model-based problem solving in
mathematics problem-solving instruction. Xin’s conceptual model-based problem
solving was recognized by the National Science Foundation [NSF] through a 5-year
research project (Xin, Tzur, and Si, 2008-2013) to support a multi-disciplinary
research project, directed by PI Xin, to develop an intelligent tutor to nurture
multiplicative reasoning of students with LD. In fact, Xin’s work in COMPS
(Xin, 2008) will be included in a new book, authorized by the National Council
of Teachers of Mathematics (NCTM), that summarizes selected research with the
potential to “inform teaching practice in K-12 mathematics classrooms and beyond”
in responding to the Common Core State Standards for Mathematics.
Xin has authored or co-authored about 50 publications including journal articles,
book chapters, refereed conference proceedings, and other publications. Xin
publishes in top-tiered journals in the fields of special education (e.g., The Journal of
Special Education [JSP], Exceptional Children [EC]), math education (e.g., Journal
for Research in Mathematics Education), and education (e.g., The Journal of
Educational Res earch). She has presented nationally and internationally on effective
intervention strategies in math problem solving with students with LD. Xin has
served on the editorial board of two flagship journals in the field of Special education
(EC and JSP) and served on NSF reviewing panel. Xin’s work in mathematics
problem solving has been referenced in prestigious sources including the National
Mathematics Panel Final Report (2008), the What Works Clearinghouse and the
Institute of Education Sciences (IES) Practitioner’s Guide, and many textbooks as
evidence-based or validated practices in teaching math problem solving to students
with diverse needs.
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