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Concept image revisited

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Abstract

Concept image and concept definition is an important construct in mathematics education. Its use, however, has been limited to cognitive studies. This article revisits concept image in the context of research on undergraduate students’ understanding of the derivative which regards the context of learning as paramount. The literature, mainly on concept image and concept definition, is considered before outlining the research study, the calculus courses and results which inform considerations of concept image. Section 6 addresses three themes: students’ developing concept images of the derivative; the relationship between teaching and students’ developing concept images; students’ developing concept images and their departmental affiliation. The conclusion states that studies of undergraduates’ concept images should not ignore their departmental affiliations.

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Acknowledgements

This work was supported by a grant from the Turkish Ministry of Education. An early form of this article appeared in Simpson (2006). Some of the data used in this article also appears in Bingolbali et al. (2007). Thanks, without responsibility, to Shlomo Vinner for comments on the early article.

Author information

Authors and Affiliations

  1. Gaziantep Universitesi, Egitim Fakultesi, Ilkogretim Bolumu, Gaziantep, Turkey

    Erhan Bingolbali

  2. University of Leeds, Leeds, UK

    John Monaghan

Authors
  1. Erhan Bingolbali
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  2. John Monaghan
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Corresponding author

Correspondence to Erhan Bingolbali.

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Appendix

Appendix

1.1 Question 1 (Q1)

Line L is a tangent to the graph of y = f(x) at point (5,3) as depicted in the graph below.

  1. a)

    f′(5) = ?

  2. b)

    What is the value of the function f(x) at x = 5.08? (be as accurate as possible)

1.2 Question 2 (Q2)

  1. a)

    Find the equation of the tangent to the curve \(y = x^3 - x^2 + 1\) at (1,1)

  2. b)

    Find the equation of L by making use of the graphs given below.

1.3 Question 3 (Q3)

Find the rate of change with respect to the given variable of the following functions at the values indicated.

  1. a)

    \(f\left( x \right) = x^2 - 7x\), when x = 3

  2. b)

    \(g\left( x \right) = \left( {x^2 - 1} \right).\left( {x + 1} \right)\), when x = 1/3

1.4 Question 4 (Q4)

For a certain period the population, Y, of a town after x years is given by the formula \(Y = 1000\left( {50 + 2x - {{x^2 } \mathord{\left/ {\vphantom {{x^2 } 6}} \right. \kern-\nulldelimiterspace} 6}} \right)\). Find:

  1. a)

    The initial population,

  2. b)

    Its initial rate of increase,

  3. c)

    The time at which the rate of increase is 1,000 people per year,

  4. d)

    The time at which the population stops growing and its value at this time.

1.5 Question 5 (Q5)

What is the meaning of a derivative? Define or explain as you wish.

1.6 Question 6 (Q6)

Two university students from different departments are discussing the meaning of the derivative. They are trying to make sense of the concept in accordance with their departmental studies.

Ali says: The derivative tells us how quickly and at what rate something is changing since it is related to a moving object. For example, it can be drawn on to explain the relationship between the acceleration and velocity of a moving object.

Banu, however, says: I think the derivative is a mathematical concept and it can be described as the slope of the tangent line of a graph of y against x.

  1. a)

    Which one is closer to the way of your own derivative definition? Please explain!

  2. b)

    If you had to support just one student, which one would you support and why?

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Bingolbali, E., Monaghan, J. Concept image revisited. Educ Stud Math 68, 19–35 (2008). https://doi.org/10.1007/s10649-007-9112-2

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