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Irrational gap: sensemaking trajectories of irrational exponents

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Abstract

We investigate how students make sense of irrational exponents. The data comprise 32 interviews with university students, which revolved around a task designed to examine students’ sensemaking processes involved in predicting and subsequently interpreting the shape of the graph of \( f(x)={x}^{\sqrt{2}} \). The task design and data analysis relied on the concept of sensemaking trajectories, blending the notions of sensemaking and (hypothetical/actual) learning trajectories. The findings present four typical sensemaking trajectories the participants went through while coping with the notion of an irrational exponent, alongside associated reasoning themes that seemed to have guided these trajectories. In addition to irrational exponents, the analysis revealed the participants’ reasoning and sensemaking of related mathematical ideas, such as rational exponents, approximations of irrational numbers by rational numbers, even/odd functions and numbers, and the meaning of exponentiation in general. The findings provide a step towards a better understanding of students’ conceptual development of irrational exponents, which could in turn be used for the refinement of tasks aimed at promoting students’ comprehension of the topic.

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Notes

  1. We note one possible explanation for the different results stems from the different settings of these two studies (interviews vs. written-response-tasks).

  2. In this paper, we discuss the case of real roots only and do not attend to complex numbers.

  3. We note that the graphs of y = x1.4 and y = x1.48 were drawn in a scale where −30 < x < 30 and −90 < y < 90 (see Fig. 1c), and accordingly, it was impossible to observe the intersection points at x = 1 and x =  − 1, and the changes occurring there (regarding which graph has larger values). However, this choice was made purposefully in order to keep the participants’ attention on the main issue of the task—irrational exponents.

  4. For example, the inclusion of the graph of y = x1.44 presents a new mathematical situation that is not illustrated by the other utilised graphs. This function (\( y={x}^{1.44}={x}^{\frac{144}{100}}={x}^{\frac{36}{25}}=\sqrt[25]{x^{36}} \)) is defined for x < 0 (as opposed to y = x1.5), though is an even function (vs. the odd functions y = x1.4 and y = x1.48).

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Authors and Affiliations

  1. Mathematics Education Unit, Department of Mathematics, Faculty of Science, The University of Auckland, 38 Princes St., Auckland Central, Auckland, 1010, New Zealand

    Ofer Marmur

  2. Faculty of Education, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

    Rina Zazkis

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  1. Ofer Marmur
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  2. Rina Zazkis
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Correspondence to Ofer Marmur.

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Marmur, O., Zazkis, R. Irrational gap: sensemaking trajectories of irrational exponents. Educ Stud Math 107, 25–48 (2021). https://doi.org/10.1007/s10649-021-10027-2

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