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Middle-Grade Preservice Teachers’ Mathematical Problem Solving and Problem Posing

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Mathematical Problem Posing

Part of the book series: Research in Mathematics Education ((RME))

Abstract

Empirical data were gathered from 51 middle-grade preservice teachers who were randomly assigned into one of two groups. The first group solved a task and then posed new problems based on the given figures, and the second group completed these activities in reverse order. Rubrics were developed to assess the written responses, and then thoughts and concerns related to problem-posing experiences were collected to understand their practices. Results revealed that the preservice teachers were proficient in solving simpler arithmetic tasks but had difficulty generalizing and interpreting numerals in an algebraic form. They were able to pose some basic and reasonable problems and to consider important aspects of mathematical problem solving when generating new tasks. Thus, teacher educators should provide substantial educational experiences by incorporating both problem-solving and problem-posing activities into engaging instruction for preservice teachers.

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Author information

Authors and Affiliations

  1. The National University of Malaysia, Bangi, Malaysia

    Roslinda Rosli

  2. Texas A&M University, College Station, TX, USA

    Mary Margaret Capraro, Dianne Goldsby, Elsa Gonzalez y Gonzalez & Robert M. Capraro

  3. Sam Houston State University, Huntsville, TX, USA

    Anthony J. Onwuegbuzie

Authors
  1. Roslinda Rosli
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  2. Mary Margaret Capraro
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  3. Dianne Goldsby
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  4. Elsa Gonzalez y Gonzalez
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  5. Anthony J. Onwuegbuzie
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  6. Robert M. Capraro
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Corresponding author

Correspondence to Roslinda Rosli .

Editor information

Editors and Affiliations

  1. Department of Educational Sciences, University of Ploiesti, Bucharest, Romania

    Florence Mihaela Singer

  2. Department of Mathematics, Illinois State University, Normal, Illinois, USA

    Nerida F. Ellerton

  3. Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA

    Jinfa Cai

Appendices

Appendix A

Problem-Solving Task

Please work individually on the task given below.

Look at the figures below.

figure c
  1. i.

    How many blocks are needed to build a staircase of five steps? Explain how you found your answer.

  2. ii.

    How many blocks are needed to build a staircase of 20 steps? Explain how you found your answer.

  3. iii.

    Based on part i. and ii. write a rule to generalize the solution for any number of steps or describe in words how to find the numbers of blocks used in each step.

Problem-Posing Task

Please work individually on the task given below.

  1. i.

    Look at the figures below.

    figure d

    Pose three (3) mathematically appropriate problems for middle-school students in the space provided that are based on the above figures. Use your creativity and originality when you pose your new problems. Do not solve them.

  1. ii.

    What did you think about when you had to pose your own mathematical problems?

  2. iii.

    What were your concerns when you posed your own mathematical problems?

Appendix B

Problem-Solving Task Rubric

Elements

Performance Indicators

Unsatisfactory (one point)

Minimal (two points)

Satisfactory (three points)

Extended (four points)

Understanding of the problem

• Limited understanding of the mathematical concepts used to solve the problems

• Some understanding of the mathematical concepts used to solve the problems

• Substantial understanding of the mathematical concepts used to solve the problems

• Deep understanding of the mathematical concepts used to solve the problems

Strategies/procedures

• Applies incorrect strategies/procedures

• Applies inefficient strategies/procedures

• Applies appropriate strategies/procedures

• Applies efficient and effective strategies/procedures

Clarity and completeness of presentation

• Little/unclear explanation of attempts/solutions

• Presentation/description/diagram is not completely clear/complete

• Clear explanations

• Clear and complete explanations and reasoning

Appendix C

Problem-Posing Task Rubric

Elements

Performance indicators

Unsatisfactory (one point)

Minimal (two points)

Satisfactory (three points)

Extended (four points)

Problem structure/context

Replication/trivial changes to problem structure/context or trivial problem structure/context (for G2)

Moderate trivial changes to problem structure/context or minimal level of thinking required (for G2)

Some modifications/extensions to problem structure/context or good problem structure/context (for G2)

Major structural/contextual changes or higher level problem (for G2)

Understanding of problem

Does not fit the required mathematical concepts

Uses similar/basic mathematical concepts

Uses other basic mathematical concepts

Uses more advanced mathematical concepts

Mathematical expression

Unclear statement of problem (language)

Fairly clear statement of problem (language)

Moderately clear statement of problem (language)

Clear and precise statement of problem (language)

Ineffective/inaccurate use of mathematical expressions

Somewhat effective/accurate use of mathematical expressions

Moderately effective/accurate use of mathematical expressions

Effective and accurate use of mathematical expressions

Appropriateness of problem-posing design

No consideration of whether situation is feasible (workable), realistic, and engaging

Somewhat feasible (workable), realistic, and engaging

Moderately feasible (workable), realistic, and engaging

Feasible (workable), realistic, and engaging

  1. Note: G2 = Group B

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Rosli, R., Capraro, M.M., Goldsby, D., y Gonzalez, E.G., Onwuegbuzie, A.J., Capraro, R.M. (2015). Middle-Grade Preservice Teachers’ Mathematical Problem Solving and Problem Posing. In: Singer, F., F. Ellerton, N., Cai, J. (eds) Mathematical Problem Posing. Research in Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6258-3_16

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