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Creativity and Challenge: Task Complexity as a Function of Insight and Multiplicity of Solutions

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Mathematical Challenges For All

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

Abstract

Mathematical problem solving is the heart of mathematical activities at all levels. Problem-solving is both the means and the ends of the development of mathematical knowledge and skills as well as of the advancement of mathematics as a science. Researchers distinguish between problem-solving algorithms, problem-solving strategies and heuristics and problem-solving insight. Insightful and divergent thinking are at the base of mathematical creativity. This chapter analyzes the mathematical challenge embedded in problem-solving tasks from the point of view of evoked mathematical insight and the use of multiple solution strategies. While a variety of variables (such as conceptual density, level of concepts, length of solution or use of different presentations) determine the complexity of mathematical problems, the insight component and the requirement to solve problems in multiple ways increase the mathematical challenge of the task. Researchers distinguish between different types of mathematical insight as they relate to the distinction between mathematical expertise and mathematical creativity. In this chapter, we introduce a distinction between mathematical tasks that allow insight-based solutions and tasks that require mathematical insight. We provide empirical evidence for our argument that tasks that require mathematical insight are of a higher level of complexity.

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Acknowledgments

The research on tool development was supported by Eleusis Benefit Corporation, PBC.

Author information

Authors and Affiliations

  1. Department of Mathematics Education, Faculty of Education, RANGE Center, University of Haifa, Haifa, Israel

    Roza Leikin & Raisa Guberman

  2. AHVA Academic College of Education, Shikmim, Arugot, Israel

    Raisa Guberman

Authors
  1. Roza Leikin
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  2. Raisa Guberman
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Corresponding author

Correspondence to Roza Leikin .

Editor information

Editors and Affiliations

  1. Department of Mathematics Education, Faculty of Education, RANGE Center, University of Haifa, Haifa, Israel

    Roza Leikin

Appendices

Appendix 1

  1. (i)

    Puzzle problem (Problem 1 in this chapter).

    Davidson and Sternberg (2003) explain that “many people incorrectly assume that this is a ratio problem and, therefore, that they must compute the answer using the 4:5 information” (p. 157). We present the solution to Problem 1 in the “Research Experiment” Section.

  2. (ii)

    Unrealistic problem (Problem 2 in this chapter).

    Solution 1 (focusing on the right triangle – stereotypic and incorrect): In any right tringle two equal sides are legs of the triangle, then the hypotenuse is \( \sqrt{1.24^2+{1.24}^{2.}} \)

    Solution 2 (focusing on the perimeter): the third side is 3.72 − 2 × 1.24 = 1.24. The triangle is equilateral and thus cannot be a right triangle.

  3. (iii)

    The restriction consists of the examinees’ considering only convex quadrilaterals and supposing that the triangles ought to lie only “inside” the quadrilateral.

An image of a concave quadrilateral, a line passes through the bottom of the quadrilateral and cuts on four points.

Appendix 2 Model and Scoring scheme for the evaluation of creativity (based on Leikin, 2009b)

 

Fluency

Flexibility

Originality

Creativity

Scores per solution

1

Flx1 = 10 for the first solution

Flxi = 10 for solutions from a different group of strategies

Flxi = 1 for a similar strategy but a different representation

Flxi = 0.1 for the same strategy and the same representation

Ori = 10 if p < 15%or for insight/ unconventional solutions

Ori = 1 if 15 %  ≤ p ≤ 40%or for model-based/partly unconventional solutions

Ori = 0.1 if p ≥ 40 % or for algorithm-based/ learning-based conventional solutions

Cri = Flxi × Ori

Total score

Flu = n

\( Flx=\sum \limits_{i=1}^n Fl{x}_i \)

\( Or=\sum \limits_{i=1}^nO{r}_i \)

\( Cr=\sum \limits_{i=1}^n Fl{x}_i\times O{r}_i \)

  1. Note: Flu Fluency, Flex Flexibility, Or Originality, Cr Creativity
  2. n is the total number of correct solutions
  3. P = (mi/n) 9100% where mi is the number of students who used strategy j

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Leikin, R., Guberman, R. (2023). Creativity and Challenge: Task Complexity as a Function of Insight and Multiplicity of Solutions. In: Leikin, R. (eds) Mathematical Challenges For All . Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-18868-8_17

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  • DOI: https://doi.org/10.1007/978-3-031-18868-8_17

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