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Using diagrams as tools for the solution of non-routine mathematical problems

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Abstract

The Mathematics education community has long recognized the importance of diagrams in the solution of mathematical problems. Particularly, it is stated that diagrams facilitate the solution of mathematical problems because they represent problems’ structure and information (Novick & Hurley, 2001; Diezmann, 2005). Novick and Hurley were the first to introduce three well-defined types of diagrams, that is, network, hierarchy, and matrix, which represent different problematic situations. In the present study, we investigated the effects of these types of diagrams in non-routine mathematical problem solving by contrasting students’ abilities to solve problems with and without the presence of diagrams. Structural equation modeling affirmed the existence of two first-order factors indicating the differential effects of the problems’ representation, i.e., text with diagrams and without diagrams, and a second-order factor representing general non-routine problem solving ability in mathematics. Implicative analysis showed the influence of the presence of diagrams in the problems’ hierarchical ordering. Furthermore, results provided support for other studies (e.g. Diezman & English, 2001) which documented some students’ difficulties to use diagrams efficiently for the solution of problems. We discuss the findings and provide suggestions for the efficient use of diagrams in the problem solving situation.

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Acknowledgment

The authors are grateful to the Editor and three anonymous reviewers for their comments on an earlier version of this paper.

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Authors and Affiliations

  1. Cyprus Pedagogical Institute, Nicosia, Cyprus

    Marilena Pantziara & Iliada Elia

  2. Department of Education, University of Cyprus, Nicosia, Cyprus

    Athanasios Gagatsis

Authors
  1. Marilena Pantziara
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  2. Athanasios Gagatsis
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  3. Iliada Elia
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Corresponding author

Correspondence to Marilena Pantziara.

Appendix

Appendix

1.1 Test B

  1. 1.

    A man planted four trees in a straight path.

    • The tree A was first in the row while the tree B was last in the row.

    • The distance between tree A and tree B was 25 m.

    • The distance between tree A and tree C was 15 m.

    • The distance between tree D and tree B was 15 m.

    Find the distance between tree D and tree A.

  2. 2.

    Mr. Andreas is standing in a rung of a ladder and cleans the building’s windows. Then, he stepped up three rungs to clean the rest of the windows. Next, he went down five rungs to clean other windows. Then he climbed up seven rungs to clean the rest of the windows and he was at the ninth rung of the ladder.

    In which rung did Mr. Andreas stand when he first started cleaning the windows?

  3. 3.

    A card company plans to make new boxes of greeting cards. In each box, there will be greeting cards that are:

    • Either green or yellow and have

    • Either Christmas greetings or Easter greetings and have

    • Either gold ribbon or silver ribbon.

    How many different cards will the company make?

  4. 4.

    Four teams competed in a knockout volleyball competition. Team A beat Team B and Team C lost to Team D. Who was the winner if Team D lost the final game?

  5. 5.

    Four friends, Anna, Vasos, Costas, and Dina like different kind of books. One likes comedies, one likes mysteries, one likes drama, and one likes adventures. Use the clues to help you find out which book each friend likes.

    • One of Anna’s friends likes mystery books.

    • Costas and Dina are not interested in adventure books.

    • Vasos likes drama books.

    • Dina does not like comedy books.

  6. 6.

    John sells ice cream. He sells three different types of cones, a white, a brown and a yellow one. He sells four different flavors, strawberry, chocolate, vanilla, and hazelnut. How many different combinations of cones and flavors can John do?

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Pantziara, M., Gagatsis, A. & Elia, I. Using diagrams as tools for the solution of non-routine mathematical problems. Educ Stud Math 72, 39–60 (2009). https://doi.org/10.1007/s10649-009-9181-5

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