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Examination of the constructed dynamic bridge between the concepts of differential and derivative with the integration of GeoGebra and the ACODESA method

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Abstract

The purpose of this study was to explore how university students construct the relationship between the concepts of differential and derivative with the integration of GeoGebra and the ACODESA method. The participants in this study were 33 pre-service mathematics teachers. An open-ended questionnaire, knowledge test, tasks, and participants’ dynamic constructions were used as data collection tools. The analysis of participants’ products was based on descriptive analysis and Toulmin’s model. As a result of the analysis, it was found that the participants constructed the relationship between the concepts of differential and derivative as well as the concepts of tangent and slope by using them within the geometric framework. Due to the integration of GeoGebra and the ACODESA method, the participants explored the relationships among true change, estimated change, and error by using the geometric interpretation of the concept of differential. It was found that with this method, they deduced that Δx and dx were two different symbols for the same variable.

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Acknowledgments

I am grateful to the editors and anonymous reviewers for their valuable comments on this paper.

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Authors and Affiliations

  1. Department of Mathematics and Science Education, Ziya Gökalp Education Faculty, Dicle University, 21280, Diyarbakır, Turkey

    Yılmaz Zengin

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  1. Yılmaz Zengin
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Correspondence to Yılmaz Zengin.

Appendices

Appendix 1. Task 1

At the end of the sixteenth century, Galileo explained, as follows, that when a stationary object is allowed to fall freely from a location close to the earth’s surface, the place the object will fall after t seconds is:

If y indicates the distance the object falls in meters, Galileo’s law states that: y = 4.9 t2

  • A piece of stone is falling off the top of a cliff. The average speed of a falling object over a period of time can easily be found. So, what do we have to do to calculate the speed at any given time? You can explain your claim by using GeoGebra.

  • Explain how there is a relationship between a line segment that joins two points of a curve and the average rate of change. Show why your explanations are correct with GeoGebra.

Appendix 2. Task 2

  • Explain how a secant line passing through points P and Q on the curve can be tangent to the curve at point P. Explain your claim with the help of GeoGebra.

  • Explain the relationship between the secant line and the tangent line in a dynamic mathematics environment. Indicate why your explanations are correct with the help of GeoGebra.

  • Construct a formal definition of the derivative in a dynamic mathematics environment. Geometrically construct the relation between the formal definition of the derivative and the tangent line.

Appendix 3. Task 3

  • In the differential approach, prepare a dynamic product by associating the error with the derivative concept.

  • Geometrically transfer the relationship among real change, estimated change, and error to GeoGebra software.

  • What is the relationship between dy and ∆y? Please explain your claim (you can use GeoGebra).

Appendix 4. Task 4

  • Construct the concept of the differential in a dynamic mathematics environment.

  • Explain the relation between approximate value and error by using derivative and differential concepts (when you explain, you can use this question):

    • Find the approximate value of \( \sqrt{97} \) by using the relationship between the concepts of differential and derivative.

  • Explain the relationship between the concepts of differential and derivative. Explain why your explanation is correct in a dynamic mathematics environment.

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Zengin, Y. Examination of the constructed dynamic bridge between the concepts of differential and derivative with the integration of GeoGebra and the ACODESA method. Educ Stud Math 99, 311–333 (2018). https://doi.org/10.1007/s10649-018-9832-5

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  • DOI: https://doi.org/10.1007/s10649-018-9832-5

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