Conceptual Model-Based Problem Solving

Teach Students with Learning Difficulties to Solve Math Problems Yan Ping Xin Purdue University, West Lafayette, USA Are you having trouble in findin...
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Teach Students with Learning Difficulties to Solve Math Problems Yan Ping Xin Purdue University, West Lafayette, USA

Are you having trouble in finding Tier II intervention materials for elementary students who are struggling in math? Are you hungry for effective instructional strategies that will address students’ conceptual gap in additive and multiplicative math problem solving? Are you searching for a powerful and generalizable problem solving approach that will help those who are left behind in meeting the Common Core State Standards for Mathematics (CCSSM)? If so, this book is the answer for you. • The conceptual model-based problem solving (COMPS) program emphasizes mathematical modeling and algebraic representation of mathematical relations in equations, which are in line with the new Common Core.

Conceptual Model-Based Problem Solving

Conceptual Model-Based Problem Solving

• “Through building most fundamental concepts pertinent to additive and multiplicative reasoning and making the connection between concrete and abstract modeling, students were prepared to go above and beyond concrete level of operation and be able to use mathematical models to solve more complex real-world problems. As the connection is made between the concrete model (or students’ existing knowledge scheme) and the symbolic mathematical algorithm, the abstract mathematical models are no longer “alien” to the students.” As Ms. Karen Combs, Director of Elementary Education of Lafayette School Corporation in Indiana, testified: “It really worked with our kids!” • “One hallmark of mathematical understanding is the ability to justify,… why a particular mathematical statement is true or where a mathematical rule comes from” ( Through making connections between mathematical ideas, the COMPS program makes explicit the reasoning behind math, which has the potential to promote a powerful transfer of knowledge by applying the learned conception to solve other problems in new contexts. • Dr. Yan Ping Xin’s book contains essential tools for teachers to help students with learning disabilities or difficulties close the gap in mathematics word problem solving. I have witnessed many struggling students use these strategies to solve word problems and gain confidence as learners of mathematics. This book is a valuable resource for general and special education teachers of mathematics. - Casey Hord, PhD, University of Cincinnati

ISBN 978-94-6209-102-3


Yan Ping Xin


Spine 14.834 mm

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

Printed on acid-free paper

All Rights Reserved © 2012 Sense Publishers No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.




About the Author


1. Conceptual Model-based Problem Solving: Teach Students with Learning Difficulties to Solve Math Problems Introduction Algebra Thinking in Problem Solving Mathematical Modeling Theoretical Framework: Conceptual Model-based Problem Solving SBI that Emphasizes Semantic Analyses and Representation of the Problem COMPS that Emphasizes Algebraic Expression of Mathematical Relations Summary Program Components The scope and sequence of the program Target Audience and Users of the Program

1 1 2 3 4 4 6 7 7 8

2. COMPS Program Introduction Singapore Bar Models (SBM) to Facilitate the Transition to Mathematical Models Word Problem [WP] Story Grammar (Xin et al., 2008) A Cognitive Heuristic DOTS Checklist (Xin et al., 2008) Additive and Multiplicative Word Problem Structure and its Variations Instructional Phases

13 16

Part 1: Additive Problem Solving Unit 1: Representing Part-Part-Whole (PPW) Problems Lesson 1: Introduction Lesson 2: Part-Part-Whole Problem Representation Unit 2: Solving Part-Part Whole (PPW) Problems Lesson 3: Solving PPW Problems Unit 3: Representing Additive Compare (AC) Problems Lesson 4: Representing AC-More Problems Lesson 5: Representing AC-Less Problems Unit 4: Solving Additive Compare (AC) Problems Lesson 6: Solving Mixed AC Problems

19 21 21 26 35 35 49 49 60 69 69


11 11 12 13


Unit 5: Solving Mixed PPW and AC Problems Lesson 7: Solving mixed PPW and AC Problems

83 83

Part 2: Multiplicative Problem Representation and Solving Unit 6: Representing Equal Groups (EG) Problems Unit 7: Solving Equal Groups (EG) Problems Unit 8: Representing Multiplicative Compare (MC) Problems Unit 9: Solving Multiplicative Compare (MC) Problems Unit 10: Solving Mixed Equal Groups and Multiplicative Compare Problems


Part 3: Solving Complex Problems Unit 11: Solving Complex Word Problems Unit 12: Solving Mixed Multi-Step Problems

159 161 173


Connection between Mathematical Ideas: Extend Multiplicative Reasoning to Geometry Learning

97 99 107 115 127


Appendix A: Student Worksheets


Appendix B: Reference Guide






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).



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 Research). 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.



CONCEPTUAL MODEL-BASED PROBLEM SOLVING Teach Students with Learning Difficulties to Solve Math Problems


Although American students are struggling with many aspects of mathematics, the National Mathematics Advisory Panel has identified “algebra as a central concern” (National Mathematics Advisory Panel, 2008, p. xiii). Interestingly, American students tend to enjoy school mathematics during the early elementary grades. However, they begin to experience difficulty in and come to dislike mathematics after fourth grade when learning becomes more abstract or symbolic and involves more algebraic thinking (Cai, Lew, Morris, Moyer, Ng, & Schmittau, 2004). In particular, students with learning disabilities or difficulties in mathematics (LDM) are falling further behind their normal achieving peers as they move from elementary to secondary schools. A majority are essentially failing the secondary math curriculum. According to the Panel, mathematics achievement in the U.S. decreases significantly in the late middle grades when students are expected to learn algebra, which raises the essential question: How can students, including those with LP, “be best prepared for entry into algebra?”(Panel, p. xiii). No doubt, the Panel’s report underscores the importance of algebra-readiness instruction. The purpose of this curriculum book is to present a Conceptual Model-Based Problem Solving (COMPS) approach to the teaching of elementary mathematics problem solving. It emphasizes the teaching of big ideas in mathematics problem solving and making connections between mathematical ideas including the connection between arithmetic and algebra learning. In this chapter, I will first briefly characterize algebraic thinking in problem solving. Next, I will present a framework for mathematical modeling. Then, I will introduce the COMPS approach that emphasizes mathematical modeling involving algebraic thinking and readiness. Finally, I will provide a brief review of relevant research in word problem solving with students with LDM, and illustrate the distinctive features of COMPS and its advantages with the support of scientificbased research. Algebra Thinking in Problem Solving Problem solving is a relevant and significant perspective and context through which to introduce students to algebra (Bednarz & Janvier, 1996). With respect to the 1


elementary school curriculum, algebra is essentially “a systematic way of expressing generality and abstraction” (National Research Council [NRC], 2001, p. 256). In algebra, the focus is on expression or representation of relations (Carpenter, Levi, Franke, & Zeringue, 2005). Through translating information from real world situated word problems into symbolic expressions and equations that may involve one or more unknown quantity, such representation is considered one type of activity that involves algebraic thinking (NRC, 2001). Within the context of arithmetic problem solving, algebraic thinking “involves the use of symbols to generalize certain kinds of arithmetic operations” (Curcio & Schwartz, 1997, p. 296) and to represent relations (Charbonneau, 1996). Algebra is “a cluster of modeling” that serves as “a domain for making, expressing and arguing generalizations” (Kaput & Blanton, 2001, p. 4). Mathematical Modeling Recently, Blum and Leiss (2005) provided a framework for modeling (see Figure C1-1). In this modeling cycle, one must (1) read and understand the task, (2) structure the task and develop a real situational model, (3) connect it to and/or represent it with a relevant mathematical model; (4) solve and obtain the mathematical results, (5) interpret the math results in real problem context; and (6) validate the results (either end the task or re-modify the math model if it does not fit the situation). In light of research in mathematics education, many students have difficulties in making the transition from a real situational model to a mathematical model; and it is a weak area in students’ mathematical understanding (Blomhøj, 2004).

3 real model 2 real situation

mathematical model situation model



6 real results rest of the world

mathematical results 5


Figure C1-1. Blum and Leiss (2005) Framework for Modeling.

In short, modeling involves translation or representation of a real problem situation into a mathematical expression or model. Mathematical models are an essential 2


part of all areas of mathematics including arithmetic and should be introduced to all age groups including elementary students (Mevarech & Kramarski, 2008). It should be noted that engaging students in the modeling process does not necessarily mean engaging students in the discovery or invention of mathematical models or complex notational systems; however, according to Lesh, Doerr, Carmona, and Hjalmarson (2003), it does mean that when such models or systems are given to the students, “the central activities that students need to engage in is the unpacking of the meaning of the system” (p. 216), representation of the real problem situation in a mathematical expression or model, and the flexible use of the model to solve real world problems. Theoretical Framework: Conceptual Model-based Problem Solving Contemporary approaches to story problem solving have emphasized the conceptual understanding of a story problem before attempting any solution that involves selecting and applying an arithmetic operation for solution (Jonassen, 2003). Because problems with the same problem schema share a common underlying structure and hence require similar solutions (Chen, 1999; Gick & Holyoak, 1983), students need to learn to understand the structure of the mathematical relationships in word problems and should develop this understanding through creating and working with a meaningful representation of the problem (Brenner et al., 1997) as well as mathematical modeling (Hamson, 2003). The representation that models the underlying mathematical relations in the problem, that is, the conceptual model, facilitates solution planning and accurate problem solving. The conceptual model should drive the development of a solution plan that involves selecting and applying appropriate arithmetic operations. According to Lesh, Landau, & Hamilton (1983), a conceptual model is defined as an adaptive structure consisting of the following primary components: (a) a within concept network of relations; (b) a between-concept system that links and combines within-concept networks; (c) a system of representations (e.g., written symbols, pictures, and concrete materials); and (d) systems of modeling processes. The first two components address students’ understanding of the idea or underlying structure of the concept. The third component concerns different representation systems, and the fourth component deals with modifying the situation to fit the existing model or changing existing model to make it applicable to a given situation. Based on Lesh et al. (1983), in applied problem solving, important translation and /or modeling processes include (a) simplifying the original problem situation by ignoring irrelevant information in the problem, and (b) “establishing a mapping between the problem situation and the conceptual models used to solve the problem” (p. 9). Building on metaanalysis (e.g., Xin & Jitendra, 2009) and cross-cultural curriculum evaluation (e.g., Xin, 2007), as well as empirical studies of intervention strategies (Xin, 2008; Xin et al., 2011; Xin, Wiles, & Lin, 2008; Xin & Zhang, 3


2009), I have developed the Conceptual Model-based Problem Solving (COMPS) program that is consistent with the theoretical framework of mathematical modeling and conceptual models (e.g., Blomhøj, 2004; Lesh et al., 1983). One distinguishable difference between the COMPS approach and prior research in word problem solving by students with LD (e.g., schema-based instruction [SBI]) is that the former focuses on representing the word problem in a defined mathematical model (the stage of “mathematical model” as it is presented in Blum and Leiss’s mathematical modeling cycle, see Figure C1-1), which is expressed in an algebraic equation that directly drives the solution plan. In the next section, I will provide a brief review of intervention research with students with LDM using SBI and more recently Conceptual Model-based Problem Solving (COMPS) in facilitating elementary students’ ability to solve mathematics word problems. SBI that Emphasizes Semantic Analyses and Representation of the Problem During the past decade or so, schema-based instruction (SBI) has shown potential benefits for teaching mathematics problem solving to students with and without disabilities. Jitendra and Hoff (1996) examined SBI that emphasized semantic analysis of various additive word problems and the mapping of these problems into schematic diagrams (adapted from Marshall, 1995) that are specific to different problem types (i.e., change, group, and compare. See Table C2-1 in Chapter 2 for examples of these problem types). The semantic analysis of word problems and categorization of problem types are originated from the framework of Cognitively Guided Instruction (CGI) (Carpenter, Fennema, Franke, Levi, & Empson, 1999). The study was conducted with three third and fourth grade students with learning disabilities using a single subject design. Later, Jitendra and colleagues extended this single subject design study to a group comparison study (Jitendra et al. 1998) and implemented SBI in regular classroom settings that involved students with and without disabilities (Jitendra et al., 2007). The SBI strategy used in the studies was similar in that they all emphasized semantic analysis of the problems by which students make distinctions among Change, Group, and Compare problem types and then map the problem into respective schematic diagrams. Afterwards, students are expected to create a math sentence for the solution with the help of solution rules such as “Total is not known, so add” or “Total is known, so subtract” (Jitendra, 2002, p. 36). COMPS that Emphasizes Algebraic Expression of Mathematical Relations Emerging from SBI, COMPS has transformed semantic representation of additive problems in various diagrams (as in SBI) to a single mathematical model to facilitate solution planning and accurate problem solving. With the COMPS approach, the focus is not on semantic analysis of the word problems, rather, it 4


emphasizes an algebraic representation of mathematical relations in equation models (e.g., “Part + Part = Whole” for additive word problems; “Unit Rate × Number of Units = Product” for equal group structured multiplicative word problems). Borrowing the concept of story grammar from reading comprehension literature, I have created the term Word Problem [WP] Story Grammar to denote the symbolic representation of mathematical relations in problem solving. Although story grammar has been substantially researched in reading comprehension (e.g., Boulineau et al., 2004), WP story grammar has never been explored in math word-problem understanding and solving. Rather than focusing on the textual analysis of story content as emphasized by the story grammar in reading comprehension, the WP story grammar emphasizes the analysis of mathematics problem structures. Subsequently, I developed WP story grammar questions for prompting learners to identify elements of problem structures to be represented in model-based diagrams, thereby linking problem representation to solution. To investigate the effects of COMPS, I, along with my colleagues, have conducted a series of research studies. For instance, Xin, Wiles, and Lin (2008) examined the effects of teaching word problem (WP) story grammar (see Figure C1-2 for an example) to five 4th- and 5th-graders with LDM, with a purpose to help their representation of problems in mathematical model equations (e.g., “Part + Part = Whole” for additive problems, and “Factor × Factor = Product” or “unit rate × # of units = product” for multiplicative problems). The results indicated Equal Group (EG) An E G problem describes number of equal sets or units Unit Rate

# of Units X

Product =

EG WP Story Grammar Questions Which sentence or question tells about the Unit Rate (# of items in each unit)? Find the unit rate and write it in the Unit Rate box. Which sentence or question tells about the # of Units or sets (i.e., quantity)? Write that quantity in the circle next to the unit rate. Which sentence or question tells about the Total (# of items) or ending product ? Write that number in the triangle on the other side of the equation.

Figure C1-2. Conceptual Model of Equal Groups (EG) Word Problems (adapted from Xin et al., 2008). 5


that conceptual model–based representations prompted by WP story grammar improved students’ performance on arithmetic word problem solving and promoted prealgebraic concept and skill acquisition. To extend COMPS to more complex real world problem solving, Xin & Zhang (2009) explored the effectiveness of COMPS in solving problems that require sense-making of a decimal solution (e.g., “Marilyn is putting her CD collection of 152 CDs into cabinets. Each cabinet can hold 36 CDs. How many cabinets does she need?”), as well as problems that require background information, pictograph problems, and multi-step problems. A multiple probe, single subject design was used to examine the intervention effects across three 4th- and 5th-graders with LDM. The results indicated that the intervention improved student performance on researcherdesigned criterion tests and a norm-referenced standardized test. Recently, Xin et al. (2011) employed a pretest-posttest, randomized group comparison design to compare the effect of COMPS to general heuristic instruction (GHI) taken from the participating schools’ enacted curriculum and teaching practice. The results indicate that only the COMPS group significantly improved (with an effect size of 3.12 over the comparison group, Xin et al., 2011, p. 390) elementary students’ performance on the criterion test that involved multiplicative word problems as well as the pre-algebra model expression test (taken from the school-adopted math curriculum). In summary, preliminary findings indicate that the COMPS program, with a focus on representing the problem in its mathematical model (Blum and Leiss, 2005), seems to enhance elementary students’ problem solving skills. SUMMARY

Most of the existing research in SBI, including Fuchs and colleagues’ recent work (e.g., Fuchs et al., 2008; Powell & Fuchs, 2010), in elementary math word problem solving in particular, has a focus on semantic analysis and classification of word problem types on the basis of CGI’s framework (e.g., Carpenter et al., 1999), and representing the problem in a diagram or equation that is associated with each of the problem types. Students then rely on solution rules, taught through explicit instruction, to create a math sentence or set up an equation for solving problems. In contrast, with the COMPS approach students are not required to make fine-grained distinctions between sub-problem types on the basis of semantic analysis of story feature (e.g., whether there is a change in time, for instance, “past to present,” to differentiate the “Change” problem type from the “Group” as well as the “Compare” problem types). Further, COMPS makes the connection between mathematical ideas through representing variously situated problems (either additive or multiplicative) in one cohesive mathematical model equation. By representing problems in mathematical model equations (e.g., part + part = whole, or unit rate x number of units = product), students do not have to memorize numerous rules to make decisions on the choice of operation for finding the solution; rather, the mathematical models, 6


which depict mathematical relations involved in the problem, provide students with a defined algebraic equation for solution. PROGRAM COMPONENTS

The Scope and Sequence of the Program This program addresses elementary word problem solving including four basic operations. It may serve as a supplemental program with an aim to help students with LDM learn big ideas in elementary math problem solving that involve four basic operations: addition, subtraction, multiplication and division. This program will cover additive problem solving (e.g., part-part-whole and additive compare problems), and multiplicative problem solving (e.g., equal groups and multiplicative compare problems). The tasks involved in this book are those typically found in elementary math textbooks, which roughly represent about 67% of the elementary math content. The COMPS program involves three parts: (1) five units on additive word problem solving that involves addition and subtraction; (2) five units on multiplicative word problem solving that involves multiplication and division; and, (3) two units on solving complex word problems such as those involving pictographs, irrelevant information, and mixed additive and multiplicative multi-steps. In both Part I and Part II, the first Unit (i.e., Unit 1 or Unit 5) engages students in learning mathematical models through representing the problem in the model equation (see Figure C1-2 in page 5 for an example: unit rate × # of units = product). During the model equation representation stage, word problem stories with no unknowns will be used. The purpose of presenting story situations with no unknowns is to provide students with a complete representation of the problem structure so that mathematical relations in the problem are clear to the students. In addition, self-regulation questions pertinent to Word Problem Story Grammar (Xin, Wiles, & Lin, 2008) (see Figure C1-2, lower panel) will be used as a heuristic to help students analyze various situated word problems/stories and represent the information from the problem in either the Part-Part-Whole (PPW) model equation or the Equal Groups (EG) model equation. The second Unit in Parts 1 and 2 (i.e., Unit 2 and Unit 7) engage students in solving either PPW problems (Part 1) or EG problems (Part 2) with an unknown, after they learn the problem structure representation in Unit 1. The third and fourth Units in Parts 1 and 2 (i.e., Unit 3 & 4; and Unit 8 & 9) introduces a variation of either the PPW or the EG problem structure. In particular, students will learn to represent (Unit 3) and solve (Unit 4) additive compare problems using a variation of the PPW model equation, or represent (Unit 8) and solve (Unit 9) multiplicative compare problems using a variation of the EG model equation. The fifth unit in Parts 1 and 2 (i.e., Unit 5 and Unit 10) will engage students in solving mixed additive (Unit 5) or mixed multiplicative problems (Unit 10). 7


Part 3 will engage students in analyzing and solving more complex problems (e.g., problems involving irrelevant information, pictographs, comprehension of decimal answers, and multi-steps). Following an introduction in Chapter I and the COMPS program in Chapter II, lastly, Chapter III will extend the big idea of multiplicative reasoning to the learning of elementary geometry with the intention of teaching students the connection between mathematical ideas that both strengthen their knowledge base and promote generalizable problem solving skills. Overall, the program involves the features below: 1. The COMPS program teaches model-based rather than solution-rule-based problem solving. It emphasizes the representation of mathematical relations in algebraic model equations. COMPS aims to promote conceptual understanding of big ideas in additive and multiplicative problem solving, as well as the connection between mathematical ideas. As such, it is in line with the Common Core Standards. 2. The program addresses the National Mathematics Advisory Panel’s concerns about algebra readiness and the National Council of Teachers of Mathematics (NCTM)’s call for algebra as a K-12 enterprise (Mathematics Advisory Panel, 2008; NCTM, 2000). 3. The program addresses the connection between concrete/semi-concrete representation and abstract mathematical model expression. Singapore Bar Model will be used to facilitate the transition from the concrete/semi-concrete model to the symbolic/abstract COMPS model equation in solving additive and multiplicative word problems. 4. Borrowing the concept of story grammar from reading comprehension literature (e.g., Dimino, Gersten, Carnine, & Blake, 1990), the program focuses on word problem story grammar (Xin, Wiles, & Lin, 2008) in facilitating students’ understanding of the mathematical structure of the problem and transformation from real situation model representation to mathematical model expression. 5. The scope-and-sequence of the program is in line with the NCTM mathematics curriculum standards. 6. The Program incorporates the best practice in mathematics problem solving intervention with students with LD. 7. To show the connection between mathematical ideas, the program extends the big idea of multiplicative reasoning to the learning of basic concepts in elementary geometry. Target Audience and Users of the Program Addition and subtraction problem solving covered in Part 1 are consistent with the math content typically presented in second or third grade math curricula. Therefore, it can be used for third or fourth grade students with LDM (or even older students with LDM) who have not mastered additive problem solving. Multiplication and 8


division in Part 2 are consistent with the math content typically presented in third and fourth grade math curricula and can be used by 4th or 5th grade student with LD (or even older students with LDM) who have not mastered multiplicative problem solving. This program can serve as a supplement to regular school math instruction. As this program teaches big ideas in additive and multiplicative problem solving, students will be equipped with a tool to learn math problem solving systematically and hopefully catch up with their normal-achieving peers within a short period of time as supported by previous research (Xin & Zhang, 2009; Xin et al., 2011). The COMPS program can be used as Tier II or Tier III intervention models within the context of Response to Intervention (RtI) model. It can be easily integrated into regular inclusion classrooms as part of Tier I instruction. The COMPS program can be used by regular classroom math teachers (special education or regular education teachers), Instructional Supporting Team interventionists, school psychologist, tutors who work with students with LDM in after-school programs, and anyone who works with students with LD in math problem solving. The COMPS program is also useful for professional development and for the preservice training of prospective elementary teachers, special education in particular, to enhance their content knowledge in elementary mathematics problem solving.





Before introducing the sample teaching script as a guide to facilitate the implementation of the COMPS program, I would like to introduce few salient components in the COMPS program, which include: (1) Singapore bar models (Singapore Ministry of Education, 1981) to facilitate the transition from the semi-concrete model to the abstract mathematical model; (2) word problem [WP] story grammar (Xin et al., 2008) self-prompting questions to facilitate problem representation using COMPS model diagrams; and (3) a cognitive heuristic DOTS checklist (Xin et al., 2008) to facilitate the entire problem solving process. Then, I will present a general description of the instructional phases when implementing COMPS, followed by a summary of various additive and multiplicative word problem situations. Singapore Bar Models (SBM) to Facilitate the Transition to Mathematical Models SBM refers to a visual representation of relations among quantities (including known and unknown quantities) in the problem using a rectangular bar. In particular, each quantity in the problem will be represented by a segment of bar, the size of which corresponds to the numerical value of that quantity in comparison to the other quantity involved. Students will then solve the problem through directly analyzing the relations depicted by the bar models. Similar line models, rather than bar models, appear in the Chinese math textbooks (Shanghai Elementary and Secondary School Curriculum Reform Committee, 1995) in teaching word problem solving. Regardless of whether bar models or line models are used, they are good tools for representing the concept of composite units, or units made of ones. The bar model can be used as a tool for nurturing and reinforcing the concept of composite unit and to facilitate students’ transition from counting by ones to operating by composite units. The bar model also bridges the conceptual gap between concrete modeling (operating on the ones) and abstract representation of mathematical models as presented in the COMPS diagram equations. As such, I will use the SBM in the beginning stage of the modeling and practice sessions to help students understand the relations among the quantities, and then map the information from the problem to the COMPS diagram expressed in an algebraic equation. In summary, the SBM makes the connection and transition between the concrete model (operating at the unit of Ones; see Slide1-1-2 in Unit 1 Lesson 1) and the symbolic equation model (see Slide 1-1-3 in Unit 1 Lesson 1). 11


Later, the bar model will be faded out and students will no longer rely on Bar models. That is, they will directly represent the problem in the COMPS diagram, with prompts from WP story grammar questions. Word Problem [WP] Story Grammar (Xin et al., 2008) In the early 1900s, anthropologists found that people follow a pattern when retelling stories they have read or heard regardless of age or culture. This pattern is referred to as “story grammar” (Dimino, Gersten, Carnine, & Blake, 1990; Mandler & Johnson, 1977; Stein & Glenn, 1979). In “story grammar,” “grammar” means “elements.” Therefore, “story grammar” addresses the elements of a story. Story grammar involves a set of expectations or knowledge about the internal structure of stories (which can be conceptualized as story schema; Rand, 1984) that makes both comprehension and recall more efficient. Story grammar instruction directs attention to key elements of stories and provides students with a specific structure for the organization of text information. Story grammar aims to improve students’ reading comprehension by giving them a framework they can use when reading stories (e.g., by asking a series of story grammar questions regarding who, what, where, when, and why). Consistent use of the same questions about the stories (e.g., Who? What? Where? When? Why?) equips students with the framework that they can apply on their own (Gurney, Gersten, Dimino, & Carnine, 2001). Story grammar can serve as tools assisting students with organizing and representing the internal structure of stories and therefore improve comprehension (Sorrell, 1990). Just as there is a common structure in narrative stories which is called “story grammar,” a word problem story structure that is common across a group of word problem situations can be defined as WP story grammar for the particular problem type. Corresponding to the COMPS model diagram equations, I designed a set of WP story grammar self-prompting questions to facilitate a meaningful and accurate representation of information from a real world problem into the COMPS diagram equation. For instance, in the part-part-whole problem types, basic WP story grammar questions such as “Which sentence tells about the whole or combined quantity?” and “Which sentence tells about one of the small parts that makes up the whole?” can aid in comprehension and representation of the underlying structure of a word problem in the conceptual model (i.e., part + part = whole), therefore facilitating solution planning. Emphasis on the meaningful representation of mathematical relations in problem solving is consistent with contemporary approaches to story problem solving that promote the conceptual understanding of story problems before considering the choice of operations. In addition, emphasis on representing mathematical relations in equations facilitates algebraic reasoning and thinking (Moses, 1997; NRC, 2001). This is consistent with NRC’s (2001) call that “the basic ideas of algebra as generalized arithmetic” (p. 419) and “algebraic ways of thinking” should be introduced “well before” students get to “the formal study of algebra” (p. 13). 12


A Cognitive Heuristic DOTS Checklist (Xin et al., 2008) A cognitive heuristic DOTS checklist was developed to help students’ self- regulation of the problem solving process (please refer to the DOTS checklist in Unit 7). According to Polya (1957), such general heuristic procedures prepare students to develop good mental habits in the problem solving process. In the context of solving basic arithmetic word problems, it is important that students will first read and understand the problem as a whole. Based on their understanding of the problem, the learner needs to first detect whether the story or word problem is an additive structure (part-part-whole structure) or multiplicative problem structure (multiple equal groups) to which different mathematical models would apply. Although it is important to know the strategies, it is more important to know when to use what strategies and how to apply the strategy correctly. After students Detect the problem structure and apply an appropriate mathematical model, the rest of the problem solving process is about mapping information from the problem to the diagram. As only two basic models are necessary for most of the elementary arithmetic word problems that involve four operations (add, subtract, multiply, and divide), the WP story grammar described above will help students Organize or represent the information (from various structured additive or multiplicative problems) in either the additive or multiplicative model diagrams. It is important that students represent the problem in the diagram equation on the basis of a thorough understanding of the problem; this is where the WP story grammar plays a critical role in facilitating the conceptual understanding. After that, all a learner needs to do is Transform the diagram equation into a real algebraic equation (by “peeling off” the boxes and labels in the COMPS diagrams). The last step in the DOTS strategy is to Solve for the unknown quantity in the algebraic equation, provide a complete answer to the question, and check the accuracy (and meaningfulness) of the answer. Additive and Multiplicative Word Problem Structure and its Variations The additive problem structure includes a range of Part-Part-Whole and Additive Compare problem structures. A Part-Part-Whole (PPW) problem describes an additive relation between multiple parts and the whole (i.e., parts make up the whole). It includes problems such as combine (e.g., Christine has 5 apples. John has 4 apples. How many apples do they have together?), change-join (e.g., Christine had 5 apples. John gave her 4 more apples. How many apples does Christine have now?), and change-separate (e.g., Christine had 9 apples. Then she gave away 4 apples. How many apples does she have now?) (Van de Walle, 2004). Placement of the unknown can be on the part or on the whole (see eight variations of PPW problems in Table C2-1). An Additive Compare (AC) problem compares two quantities and it involves a compare sentence that describes one quantity as “more” (AC-more) or “less” (AC-less) than the other quantity (e.g., “Christine has 9 apples. She has 13


5 more apples than John. How many apples does John have?” or “Christine has 9 apples. John has 4 less apples than Christine. How many apples does John have?”). Placement of the unknown can be on the big, small, or difference quantity (see six variations of AC problems in Table C2-1). The most basic multiplicative problem structure includes various Equal Groups problem structures and various Multiplicative Compare (MC) problem structures. An Equal Groups (EG) problem describes a number of equal sets or units. The placement of the unknown can be on the unit rate (# of items in each unit or unit price), number of units or sets, or on the product (see three variations of EG problems in Table C2-2). A Multiplicative Compare (MC) problem compares two quantities and it involves a compare sentence that describes one quantity as a multiple or part of the other quantity. Placement of the unknown can be on the compared set, the referent set, or the multiplier (i.e., multiple or part) (see three variations of MC problems in Table C2-2). It should be noted that the MC problems in Table 2b only include those with multiple NOT part relations such as “2/3.” Table C2-1. Variations in Addition Word Problems (from Xin et al., 2008) Problem Type

Sample Problem Situations

Part-Part-Whole Combine Part (or smaller group) unknown

Whole (or larger group) unknown

1. Jamie and Daniella have found out that together they have 92 books. Jamie says that he has 57 books. How many books does Daniella have? OR Jamie and Daniella have found out that together they have 92 books. Daniella says that she has 35 books. How many books does Jamie have? 2. Victor has 51 rocks in his rock collection. His friend, Maria, has 63 rocks in her collection. How many rocks do the two have altogether? Change-Join

Part (or smaller group) unknown

Whole (or larger group) unknown

1. Luis had 73 candy bars. Then, another student, Lucas, gave him some more candy bars. Now he has 122 candy bars. How many candy bars did Lucas give Luis? 2. A girl named Selina had several comic books. Then, her brother Andy gave her 40 more comic books. Now Selina has 67 comic books. How many comic books did Selina have in the beginning? 3. A basketball player ran 17 laps around the court before practice. The coach told her to run 24 more at the end of practice. How many laps did the basketball player run in total that day?

(Continued ) 14


Table C2-1. Continued Problem Type

Sample Problem Situations Change-Separate 1. Davis had 62 toy army men. Then, one day he lost 29 of them. How many toy army men does Davis have now? 2. Ariel had 141 worms in a bucket for her big fishing trip. She used many of them on the first day of her trip. The second day she had only 68 worms left. How many worms did Ariel use on the first day? 3. Alexandra had many dolls. Then, she gave away 66 of her dolls to her little sister. Now, Alexandra has 63 dolls. How many dolls did Alexandra have in the beginning?

Part (or smaller group) unknown

Whole (or larger group) unknown Additive Compare

Compare-more Larger quantity unknown

1. Denzel went out one day and bought 54 toy cars. Later, Denzel found out that his friend Gabrielle has 56 more cars than what he bought. How many cars does Gabrielle have? Smaller quantity 2. Tiffany collects bouncy balls. As of today she has 93 of them. Tiffany has 53 more bouncy balls than her friend, Elise. How unknown many bouncy balls does Elise have? Difference unknown 3. Logan has 117 rocks in his rock collection. Another student, Emanuel, has 74 rocks in his collection. How many more rocks does Logan have than Emanuel? Compare-less Larger quantity 1. Ellen ran 62 miles in one month. Ellen ran 29 fewer miles than her unknown friend Cooper. How many miles did Cooper run? Smaller quantity 2. Kelsie said she had 82 apples. If Lee had 32 fewer apples than unknown Kelsie, how many apples did Lee have? Difference unknown 3. Deanna has 66 tiny fish in her aquarium. Her dad Gerald has 104 tiny fish in his aquarium. How many fewer fish does Deanna have than Gerald? Table C2-2. Variations in multiplicative word problems (from Xin et al., 2008) Problem Type

Sample Problem Situations

Equal Groups Unit Rate unknown

A school arranged a visit to the museum in Lafayette. It spent a total of $667 buying 23 tickets. How much does each ticket cost?

Number of units There are a total of 575 students in Centennial Elementary School. If one (sets) unknown classroom can hold 25 students, how many classrooms does the school need?

(Continued ) 15


Table C2-2. Continued Problem Type

Sample Problem Situations

Product unknown Emily has a stamp collection book with a total of 27 pages, and each page can hold 13 stamps. If Emily filled up this collection book, how many stamps would she have? Multiplicative Compare Compared set unknown

Isaac has 11 marbles. Cameron has 22 times as many marbles as Isaac. How many marbles does Cameron have?

Referent set unknown

Gina has sent out 462 packages in the last week for the post office. Gina has sent out 21 times as many packages as her friend Dane. How many packages has Dane sent out?

Multiplier unknown

It rained 147 inches in New York one year. In Washington D.C., it only rained 21 inches during the same year. The amount of rain in New York is how many times the amount of rain in Washington D.C. that year?

Generally speaking, part-part-whole (or part + part = whole) is a generalizable conceptual model in addition and subtraction word problems where part, part, and whole are the three basic elements. In contrast, factor-factor-product (or factor × factor = product) is a generalizable conceptual model in multiplication and division arithmetic word problems where factor, factor, and product are the three basic elements. It should be noted that the three basic elements (in either the part-partwhole or factor-factor-product model) will have unique denotations when a specific problem subtype applies. For example, in a combine problem type (e.g., Emily has 4 pencils and Pat has 8 pencils. How many pencils do they have all together?), the number of pencils Emily has and the number of pencils Pat has are the two parts; these two parts make up the combined amount (i.e., “all together”) or the whole. In contrast, in an additive compare problem type (e.g., Emily has 9 stickers, Pat has 4 fewer stickers than Emily. How many stickers does Pat have?), the number of stickers Emily has is the bigger quantity (or the whole amount), whereas the number of stickers Pat has is the smaller quantity (or one of the parts) and the difference between Emily and Pat is the other smaller quantity (the other part); combining these two parts is the bigger quantity (or the whole). Instructional Phases Instructions to carry out COMPS will be delivered in two parts: problem structure representation and problem solving. During the instruction of problem structure representation, word stories with no unknowns will be used to help students understand the problem structure and the mathematical relations among the quantities. Specifically, students will learn to identify the problem structure and map the information from the problem to its corresponding COMPS diagram equation (see Figure C1-2 for an example: unit rate x # of units = product). During that stage, 16


as all quantities are given in the story (no unknowns) students will be able to check the “balance” of the equation to shape and reinforce the concept of “equality” and the meaning of an equal sign. Problem representation instruction will be followed by problem solving instruction. During problem solving instruction, word problems with an unknown quantity will be presented. When representing a problem with an unknown quantity in the COMPS diagram, students can choose to use a letter (can be any letter they prefer) to represent the unknown quantity. Students are encouraged to use the DOTS checklist (see Unit 7) to guide the problem solving process. Overall, the instruction requires explicit strategy explanation and modeling (see the Appendix for modeling worksheets for students to follow along during the instruction), dynamic teacher-student interaction, guided practice, performance monitoring with corrective feedback, and independent practice. During independent practice, students will be provided with an independent worksheet to solve either additive or multiplicative word problems (see the Appendix for independent worksheets) they have just learned. It is suggested that the COMPS model equations be provided on all modeling and guided practice worksheets, or even on independent practice worksheets in the beginning stage of the instructional program. However, they should be gradually faded out on the worksheet once students have internalized the models.