conceptual model based problem solving

Conceptual Model-Based Problem Solving

Teach students with learning difficulties to solve math problems.

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• Preliminary Material Pages: i–ix
• Conceptual Model-based Problem Solving Teach Students with Learning Difficulties to Solve Math Problems Pages: 1–9
• COMPS Program Pages: 11–17
• Representing Part-Part-Whole (PPW) Problems Pages: 19–34
• Solving Part-Part Whole (PPW) Problems Pages: 35–47
• Representing Additive Compare (AC) Problems Pages: 49–68
• Solving Additive Compare (AC) Problems Pages: 69–81
• Solving Mixed PPW and AC Problems Pages: 83–95
• Representing Equal Groups (EG) Problems Pages: 97–106
• Solving Equal Groups (EG) Problems Pages: 107–114
• Representing Multiplicative Compare (MC) Problems Pages: 115–125
• Solving Multiplicative Compare (MC) Problems Pages: 127–139
• Solving Mixed Equal Groups and Multiplicative Pages: 141–158
• Solving Complex Word Problems Pages: 159–172
• Solving Mixed Multi-Step Problems Pages: 173–182
• Connection between Mathematical Ideas Extend Multiplicative Reasoning to Geometry Learning Pages: 183–189
• Student Worksheets Pages: 191–226
• Reference Guide Pages: 227–265
• References Pages: 267–268

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Conceptual Model-Based Problem Solving pp 1–9 Cite as

Conceptual Model-Based Problem Solving

Teach Students with Learning Difficulties to Solve Math Problems

• Yan Ping Xin 2  

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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).

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Purdue University, West Lafayette, USA

Yan Ping Xin

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© 2012 Sense Publishers

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Xin, Y.P. (2012). Conceptual Model-Based Problem Solving. In: Xin, Y.P. (eds) Conceptual Model-Based Problem Solving. SensePublishers, Rotterdam. https://doi.org/10.1007/978-94-6209-104-7_1

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The purpose of the Conceptual Model-based Problem Solving (COMPS): A Response to Intervention Program for Students with LDM is to develop a web based computerized mathematical problem-solving tutoring program that addresses the needs of second- and third-grade students with learning disabilities or difficulties in mathematics (LDM) in addition and subtraction word problem solving.

Accordingly, the project’s cross-disciplinary team has developed a functional prototype of a web-based intervention tutor, COMPS-A ( additive ). The COMPS-A program utilizes a mathematical model-based problem-solving approach that emphasizes understanding and representation of mathematical relations in algebraic equations and, thus, will support growth in generalized problem-solving skills.

The project team has conducted a series of studies to test the effectiveness and possible impact of the COMPS-A program (e.g., Xin et al., 2020a; 2020b). Students who used the COMPS-A program showed significant improvement in mathematics problem-solving performance. For instance, in a randomized control trial study, COMPS-A was compared to a typical, “business-as-usual” condition. Findings of this study indicate that students in the COMPS-A condition improved their mathematical problem-solving performance on a researcher-developed criterion measure with an effect size of 1.08 favoring the COMPS-A condition (Xin et al., 2020). In addition, about 60% students, after working with the COMPS-A computer program, showed enhanced problem-solving performance on a commercially published standardized test. The findings from these studies indicate that elementary students with LDM are able to think mathematically, engaging in model-based problem solving, rather than relying on “keyword”-based or story-based problem-solving strategies that may not involve mathematical reasoning and thinking. This project is expected to have far reaching impact in the field of special education and beyond as the COMPS program advances students’ reasoning and problem solving above and beyond the concrete level of operation to the abstract level of thinking and generalized problem solving.

As part of the Conceptual Model-based Mathematics Intervention Tutoring (COMMIT) program ( https://laurencelei36.wixsite.com/comps-rti-purdue ), the COMPS-A, coupled with the product of our previous NSF-funded project, PGBM-COMPS (a tutoring program for multiplication and division word problem solving), will address the needs of today’s inclusive classrooms by providing tools that adapt to each individual student's needs. The COMPS-A computer program can be used with an entire class during the day and with individual students (at school, at home). We thus anticipate this project to impact mathematics education practice for students who are struggling in mathematics, by reducing the gap between students with LDM and their normal-achieving peers.

The cross-disciplinary nature of this project allowed research team members from the disciplines of special education, mathematics education, and computer science to develop new perspectives and apply them to the construction of computer technology for improving mathematics education and education in general. The research team in this project explored a novel visualization method to analyze and present student problem-solving data. With the visualization method, a student’s problem-solving process data (e.g., eye movement when interacting with the computer tutor, time for task completion, accuracy of responses) is presented graphically. This representation provides teachers with a  holistic view of a student’s problem-solving process as well as a view of  strategies used during the process. Consequently, the teacher will be able to use evidence from student problem solving to adapt interventions to student needs thereby enhance program effectiveness.

Xin, Y. P., Kim, S. J., Lei, Q., Wei, S., Liu, B., Wang, W., Kastberg, S., Chen, Y., Yang, X., Ma, X., Richardson, S. E. (2020a). The Impact of a Conceptual Model-based Intervention Program on math problem-solving performance of at-risk English learners. Reading and Writing Quarterly: Overcoming Learning Difficulties, 36(2), 104-123, published online April1st, 2020. https://www.tandfonline.com/doi/full/10.1080/10573569.2019.1702909

Xin, Y. P., Kim; S., Liu, B., Lei, Q., Wei, S., Wang, W., Kastberg, S., Chen, Y., Richardson, S. E. (2020b). The effect of a web-based computer tutor on improving math problem solving of students with learning difficulties. Paper presented at 2020 AERA Annual Meeting, San Francisco, CA.

Last Modified: 12/26/2020 Modified by: Yan Ping Xin

Please report errors in award information by writing to: [email protected] .

Xin’s math problem-solving program featured by Made for Math

Do you ever struggle with math? Perhaps this mathematics problem-solving intervention program created by a Purdue University College of Education professor could help you.

Math tutoring company Made for Math recently featured Yan Ping Xin ’s  Conceptual Model-based Problem Solving  (COMPS) on its website as an evidence-based mathematics problem-solving intervention program. It is designed to help at-risk students solve math problems.

Xin, a professor in the Department of Educational Studies’ Special Education program, pioneered Conceptual Model-based Problem Solving in 2008 (Xin, Wiles, and Lin, Journal of Special Education ). Because many scholars, educators, and practitioners in the field wanted more program details, she compiled this work in a book, Conceptual Model-Based Problem Solving (2012).

The program is also included in a chapter of the National Council of Teachers of Mathematics ’ publication Useful and Useable Research Related to Core Mathematical Practices to “inform teaching practice in K-12 mathematics classrooms and beyond” (Silver & Kenney, 2015).

Xin researches mathematics problem-solving involving at-risk students. Her work has been referenced in sources such as the National Mathematics Panel Final Report , the What Works Clearinghouse , and the Institute of Education Sciences Practice Guide .

“My research in math problem-solving has been funded by the National Science Foundation (NSF) for over a decade,” Xin said. “Through collaborative work, my research team has created two math problem-solving computer tutoring programs. One addresses  multiplicative reasoning/problem-solving  and the other focuses on  additive reasoning/problem-solving . I have also been funded by  NSF’s Innovation Corps (I-Corps)  program to promote the marketing of the computer tutors my research team and I have created.”

Xin collaborated on the research with Dr. Signe Kastberg, professor in the College’s Department of Curriculum and Instruction; Dr. Ron Tzur, formerly a professor in Curriculum and Instruction; Dr. Victor (Yingjie) Chen, associate professor in the Purdue Polytechnic Institute; and Dr. Si Luo, a former professor in Purdue’s Department of Computer Science.

Adrianne Meldrum, the CEO & Founder of Made for Math, interviewed Xin about her research and the COMPS intervention program in Dyscalculia & Word Problem Solving  (Episode 16).

Xin, along with co-editors Tzur and Thouless, recently published another book,  Enabling Math Learning of Struggling Students , which has been adopted as the required textbook for a core course, “Math Curricula for Learners with Exceptionalities” in the College’s  Special Education online master’s  program. “Including the chapters authored by myself as well as Professor Kastberg, math instructional programs included in this book will impact the teaching practice of preservice and in-service teachers worldwide,” Xin said.

Source:  Yan Ping Xin, [email protected]

Yan Ping Xin

• Conceptual Model-based Problem Solving (COMPS): A Response to Intervention Program for Students with Learning Difficulties
• http://nmrsld.education.purdue.edu
• Purdue University
• Yingjie Chen
• https://polytechnic.purdue.edu/profile/chen489
• Assistant Professor
• Signe Kastberg

Conceptual Model-based Problem Solving (COMPS): A Response to Intervention Pr...

NSF Awards: 1503451

The “Conceptual Model-based Problem Solving (COMPS): A Response to Intervention Program for Students with Learning Difficulties (LDs)” is an NSF funded project. This project is to create a cross-platform COMPS intervention program to address the burning needs of second- and third-grade students with LDs in meeting the challenging curriculum standards for Mathematics—particularly in the area of additive word problem solving. The objectives of this COMPS project include:

• Create the curriculum content and screen design for four modules of the COMPS program in the area of additive word problem solving (COMPS-A);
• Design and develop the cross-platform COMPS-A computer application that can be ported as web-based, iPad, Android, or windows app, which can run on different kind of computers and be accessible to students with various social economics status in a range of environment;
• Conduct field-test studies to evaluate the potential of the program in enhancing students’ word problem-solving performance.

This COMPS-A program represents a shift from traditional problem-solving instruction, which focuses on the choice of operation for solution, to a mathematical model-based problem-solving approach that emphasizes an understanding and representation of mathematical relations in algebraic equations and therefore promotes generalized problem-solving skills. This project is expected to make a broader impact due to (a) its emphasis on concept formation and additive reasoning and problem solving, which would benefit students with LDs as well as average or below average performing students, and (b) this tool’s flexibility with group or one-on-one instruction in and out of the classroom settings.

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Welcome to our NSF supported COMPS-RtI project showcase video! Your favorable voting is appreciated! We welcome your constructive/insightful feedback and comments—Thank you!

As I am currently traveling in China and I may not have access to this website, my responding to your posts may be delayed.  Thank you for your patience!

Andrew Izsak

Sybilla and I were intrigued by parallels with our own work in multiplication. You use a single mathematical model equation to represent diverse problem types, and you distinguish the original model from the equation that computes the solution. We do something similar in the context of multiplication when working with future middle and secondary grades mathematics teachers.  

I am thrilled to hear that you and Sybilla’s work in multiplication is parallel to our work. In fact, Sybilla and I were presenting in the same working group at ICMI Study 23 in Macau. We are looking forward to learning from you and Sybilla’s work.

Wendy Smith

Your project looks very interesting. Does your program have a feature that reads the problems aloud for students for whom reading is difficult (either through learning disabilities, non-native English speakers, or other reading issues)? What kinds of support are there for teachers and students to learn to use your software? How successful do you find it to teach students the language of different problem types (such as part-part-whole)?

Thank you for your interest in our work. Yes, we do have the feature that reads the problems aloud for students. In fact, most part of the tutoring is realized through on screen display/animation along with the voice output.

Towards the end of the project, we will develop a teacher’s manual to go along with this COMPS-RtI tutoring program so that the teachers, who use this program, will understand the underpinning theoretical framework as well as the tasks /skill sets addressed in each of the four modules of this tutoring program. In addition, general tips /guidelines for using the program will also be included in the teacher’s manual.

Whenever possible, we take advantages of the computer technology to create visual animations to go along with the voice/language so that the learners understand the meaning of the language we use to describe the concept of “part and part makes up the whole” as well as key mathematical concepts (e.g., the concept of composite unit) and vocabulary (e.g., “less than,” “more than”).

I hope above answer is helpful.

One other question that occurred to me: I wonder about your program's use of the "smaller" and "bigger" language attached to the part/whole. I understand what you are using it, but this is also a rule that "expires" as students begin to grapple with integers later. How does your project address the fact that you are teaching students something that will not be true later?

The concept of “Part and part makes up the whole” will not expire. We are not necessarily teaching the rule of “addition makes bigger.” The use of the words “smaller” and “bigger” is only to help students understand the mathematical relations described in additive comparison word problem stories. In particular, when two units (i.e., the compared , and the referent unit ) are being compared, it is important for students to understand who (which unit) has “more” or who has “less.” Given the “part + part = whole” model equation, the one who has “more” will always be represented or placed on one side of the equation by itself—which is comparable to the “Whole”; the one who has “less” will be considered as one of the parts, which will be placed on the other side of the equation along with the difference quantity. That is, “smaller quantity + difference quantity = bigger (larger) quantity.”

I hope this helps to address your concern.

Miriam Sherin

Providing students with new ways to explore relationships among quantities and to model these relationships is an important goal. I appreciate the work your project is doing to support this! I'm wondering if you can say more about how you imagine the materials being used by students. Do you expect this will be something that students use on their own? Will it supplement or replace curriculum materials? I'm also wondering what part of the materials in particular you think are valuable for students with learning difficulties. It seems to me that the modeling approach has the potential to also strengthen the math understandings of students who do not have difficulty with more procedural approaches.

First of all, thank you for your positive and insightful comments!

To answer your question—

Students will not be expected to work with the tutor completely alone. Teacher has a synergistic role in this process. That is, In case the learner is “stuck” in the program, the teacher will be called upon for help. Upon request, the computer program will provide the teacher with a brief report that indicates the task the learner is current working on as well as recent interaction/exchanges between the learner and the tutor—the information from the report will help the teacher identify the issue and make proper diagnostic decisions. The teacher will either redirect the learner to the proper module/lesson or provide remediation to address skill deficit or misconception so that the learner can continue in the tutoring program.

The Teacher’s Manual we will produce upon completion of the project will prepare the teacher for the use of this COMPS-RtI tutor. Re the Teacher’s Manual, please also refer to my answer to Wendy’s first question above. Thank you!

As this program is in line with elementary mathematics curriculum standards, this tutoring program not only can be used as a supplemental program for students with learning difficulties in mathematics, but also can be used as a tool to help math teachers’ teaching to meet the regular curriculum standards.

Thanks for this information. I'm wondering a bit about how easy it will be for teachers to review recent exchanges and then know how to proceed. If there are typical approaches you expect students to use perhaps that would be useful for teachers to have some information about?

Thank you so much for your insightful question/suggestion!

The Teacher’s Manual that the team will develop towards the end of the project will prepare the teacher with the knowledge in terms of "know how to proceed". Specifically--

If it is technical glitches, it is easy to be resolved—students will be monitored so that they follow the proper direction of the program. 

If it is conceptual deficits, the “node map” for each of the modules (which will be developed as part of the Teacher’s Manual) will help the teacher make decisions in terms of whether the student should be sent back to a specific module or lesson to make up the prerequisite skill/knowledge or whether the student should bypass a specific skill (e.g., use a calculator rather than mental math to solve for the unknown quantity).

I hope above explanation is helpful.

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

• ISBN-10 946209103X
• ISBN-13 978-9462091030
• Publisher Brill Academic Pub
• Publication date November 2, 2012
• Language English
• Dimensions 6.25 x 0.75 x 9.25 inches
• Print length 268 pages
• See all details

Product details

• Publisher ‏ : ‎ Brill Academic Pub (November 2, 2012)
• Language ‏ : ‎ English
• Hardcover ‏ : ‎ 268 pages
• ISBN-10 ‏ : ‎ 946209103X
• ISBN-13 ‏ : ‎ 978-9462091030
• Item Weight ‏ : ‎ 1.28 pounds
• Dimensions ‏ : ‎ 6.25 x 0.75 x 9.25 inches

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