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Pre-service Mathematics Teachers’ Understanding of Rate of Change Throughout a Model Development Sequence

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Abstract

This study examined a cohort of middle school pre-service mathematics teachers’ understanding of the rate of change as they engaged in a model development sequence. By adopting a design-based research perspective, a model development sequence on the concept of rate of change has been designed and implemented as part of a mathematical modeling course for pre-service teachers. The data were collected from twenty senior year middle school pre-service mathematics teachers (PSTs) through questionnaires, modeling activities, reflection papers, and semi-structured interviews. The data analysis showed that PSTs have difficulties conceptualizing the rate of change and conceiving it as a multiplicative comparison of changes in two covarying quantities. As they frequently employed its percentage interpretation, PSTs experienced additional difficulty conceiving the conventional meaning of rate of change in a population growth context. PSTs generally used motion context as a reference while explaining the rate of change in different non-motion contexts. In general, PSTs developed their conception of the additive rate of change throughout the model development sequence. However, for some PSTs, difficulty in ratio-based reasoning on the rate of change in different non-motion contexts prevailed. We provided some arguments concerning the teaching and learning of rate of change.

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References

  • Ärlebäck, J. B., & Doerr, H. M. (2018). Students’ interpretations and reasoning about phenomena with negative rates of change throughout a model development sequence. ZDM–Mathematics Education, 50, 187–200.

    Article  Google Scholar 

  • Ärlebäck, J. B., Doerr, H. M., & O’Neil, A. M. (2013). A modeling perspective on interpreting rates of change in context. Mathematical Thinking and Learning, 15(4), 314–336.

    Article  Google Scholar 

  • Bezuidenhout, J. (1998). First-year university students’ understanding of rate of change. International Journal of Mathematical Education in Science and Technology, 29(3), 389–399.

    Article  Google Scholar 

  • Bingolbali, E. (2008). Türev kavramına ilişkin öğrenme zorlukları ve kavramsal anlama için öneriler [Learning difficulties regarding the concept of derivative and suggestions for its conceptual understanding]. In M. F. Özmantar, E. Bingolbali, & H. Akkoç (Eds.), Matematiksel kavram yanılgıları ve çözüm önerileri (pp. 223–255). Pegem A.

    Google Scholar 

  • Byerley, C., & Thompson, P. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48, 168–193.

    Article  Google Scholar 

  • Carlson, M. P., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378.

    Article  Google Scholar 

  • Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understanding learning processes. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education (pp. 68–95). Routledge.

    Google Scholar 

  • Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.

    Article  Google Scholar 

  • Cobb, P., Jackson, K., & Dunlap Sharpe, C. (2017). Conducting design studies to investigate and support mathematics students’ and teachers’ learning. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 208–233). National Council of Teachers of Mathematics.

    Google Scholar 

  • Coe, E. E. (2007). Modeling teachers’ ways of thinking about rate of change (Publication No. 3258070) [Doctoral dissertation, Arizona State University]. ProQuest Dissertations & Theses Global.

  • Collins, A., Joseph, D., & Bielaczye, K. (2004). Design research: Theoretical and methodological issues. The Journal of the Learning Sciences, 13(1), 15–42.

    Article  Google Scholar 

  • Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2/3), 134–165.

    Google Scholar 

  • Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26, 66–86.

    Article  Google Scholar 

  • Cooney, T. J., Beckmann, S., & Lloyd, G. M. (2010). Developing essential understanding of functions for teaching mathematics in grades 9–12. National Council of Teachers of Mathematics.

  • de Beer, H., Gravemeijer, K., & van Eijck, M. (2015). Discrete and continuous reasoning about change in primary school classrooms. ZDM-Mathematics Education, 47, 981–996.

    Article  Google Scholar 

  • Doerr, H. M., & O’Neil, A. H. (2011). A modelling approach to developing an understanding of average rate of change. In M. Pytlak, T. Rowland, & W. Swoboda (Eds.), Proceedings of the Seventh Conference of European Research in Mathematics Education (pp. 937–946). University of Rzeszów.

  • Doorman, L. M., & Gravemeijer, K. P. E. (2009). Emergent modeling: Discrete graphs to support the understanding of change and velocity. ZDM–Mathematics Education, 41, 199–211.

    Article  Google Scholar 

  • Feudel, F. (2017). Students’ interpretation of the derivative in an economic context. In T. Dooley & G. Gueudet (Eds.), Proceedings of the tenth congress of the European Society for Research in Mathematics Education (pp. 2049-2056). DCU Institute of Education & ERME.

  • Goerdt, S. L. (2007). The effect of emphasizing multiple representations on calculus students’ understanding of the derivative concept (Publication No. 3277946) [Doctoral dissertation, University of Minnesota]. ProQuest Dissertations & Theses Global.

  • Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39, 111–129.

    Article  Google Scholar 

  • Herbert, S., & Pierce, R. (2008). An “Emergent Model” for rate of change. International Journal of Computers for Mathematical Learning, 13, 231–249.

    Article  Google Scholar 

  • Herbert, S., & Pierce, R. (2011). What is rate? Does context or representation matter? Mathematics Education Research Journal, 23, 455–477.

    Article  Google Scholar 

  • Herbert, S., & Pierce, R. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics, 81, 85–101.

    Article  Google Scholar 

  • Hoffkamp, A. (2011). The use of interactive visualizations to foster the understanding of concepts of calculus: Design principles and empirical results. ZDM–Mathematics Education, 43, 359–372.

    Article  Google Scholar 

  • Johnson, H. L. (2012). Reasoning about variation in the intensity of change in covarying quantities involved in rate of change. Journal of Mathematical Behavior, 31, 313–330.

    Article  Google Scholar 

  • Kertil, M., & Gülbağcı-Dede, H. (2022). Promoting prospective mathematics teachers’ understanding of derivative across different real-life contexts. International Journal for Mathematics Teaching and Learning, 23(1), 1–24.

    Article  Google Scholar 

  • Kertil, M., & Küpcü, A. (2021). Prospective mathematics teachers’ ways of reasoning on differentiability and corner points in a real-life context. International Journal of Mathematical Education in Science and Technology, 52(9), 1361–1384.

    Article  Google Scholar 

  • Kertil, M., Erbas, A. K., & Cetinkaya, B. (2019). Developing prospective teachers’ covariational reasoning through a model development sequence. Mathematical Thinking and Learning, 21(3), 207–233.

    Article  Google Scholar 

  • Kertil, M. (2014). Pre-service elementary mathematics teachers’ understanding of derivative through a model development unit [Doctoral dissertation, Middle East Technical University]. OpenMETU. https://open.metu.edu.tr/handle/11511/23389

  • Lesh, R., & Doerr, H. M. (2003). Foundations of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh, & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3–33). Lawrence Erlbaum.

  • Lesh, R., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003). Model development sequences. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 3–33). Lawrence Erlbaum.

    Chapter  Google Scholar 

  • Mkhatshwa, T., & Doerr, H. (2018). Undergraduate students’quantitative reasoning in economic contexts. Mathematical Thinking and Learning, 20(2), 142–161.

    Article  Google Scholar 

  • Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14, 235–250.

    Article  Google Scholar 

  • Rowland, D. R., & Jovanoski, Z. (2004). Student interpretation of the terms in first-order ordinary differential equations in modeling contexts. International Journal of Mathematical Education in Science and Technology, 35(4), 503–516.

    Article  Google Scholar 

  • Sahin, Z., Aydogan-Yenmez, A., & Erbas, A. K. (2015). Relational understanding of the derivative concept through mathematical modeling: A case study. Eurasia Journal of Mathematics, Science, and Technology Education, 11(1), 177–188.

    Article  Google Scholar 

  • Delos Santos, A. G., & Thomas, M. O. J. (2001). Representational fluency and symbolisation of derivative. In W. Yang, S. Chu, Z. Karian, & G. Fitz-Gerald (Eds.), Proceedings of the Sixth Asian Technology Conference in Mathematics (pp. 282–291). RMIT University; ATCM Inc.

  • Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.

    Article  Google Scholar 

  • Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Sage Publications.

  • Stroup, W. (2002). Understanding qualitative calculus: A structural synthesis of learning research. International Journal of Computers for Mathematical Learning, 7, 167–215.

    Article  Google Scholar 

  • Tall, D. (1992). Students’ difficulties in calculus. In C. Gaulin, B. Hodgson, D. Wheeler, & J. Egsgard (Eds.), Proceedings of working group 3 on students’ difficulties in calculus, ICME-7 (Vol 7, pp. 13–28). Les Presses de l'Université Laval.

  • Teuscher, D., & Reys, R. E. (2012). Rate of change: AP calculus students’ understandings and misconceptions after completing different curricular paths. School Science and Mathematics, 112, 359–376.

    Article  Google Scholar 

  • Thompson, P. W. (1994a). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229–274.

    Article  Google Scholar 

  • Thompson, P. W. (1994b). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181–236). University of New York Press.

  • Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics.

    Google Scholar 

  • White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus course. Journal for Research in Mathematics Education, 27(1), 79–95.

    Article  Google Scholar 

  • Wilhelm, J. A., & Confrey, J. (2003). Projecting rate of change in the context of motion onto the context of money. International Journal of Mathematical Education in Science and Technology, 34(6), 887–904.

    Article  Google Scholar 

  • Yoon, C., Dreyfus, T., & Thomas, M. O. J. (2010). How high is the tramping track? Mathematising and applying in a calculus model-eliciting activity. Mathematics Education Research Journal, 22(1), 141–157.

    Article  Google Scholar 

  • Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education IV (pp. 103–127). American Mathematical Society.

    Google Scholar 

  • Zandieh, M., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. Journal of Mathematical Behavior, 25, 1–17.

    Article  Google Scholar 

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Acknowledgements

The research reported here is based on the first author’s Ph.D. Dissertation completed at the Middle East Technical University under the supervision of the second author. This research was partly supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) under grant number 110K250. Any opinions, findings, conclusions, and recommendations expressed herein are those of the authors.

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Authors and Affiliations

  1. Department of Mathematics and Science Education, Marmara University, Istanbul, Turkey

    Mahmut Kertil

  2. Department of Mathematics and Science Education, Middle East Technical University, Ankara, Turkey

    Ayhan Kursat Erbas & Bulent Cetinkaya

Authors
  1. Mahmut Kertil
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  2. Ayhan Kursat Erbas
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  3. Bulent Cetinkaya
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Correspondence to Ayhan Kursat Erbas.

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Ethical Approval and Consent to Participate

The study reported in this paper was conducted under the approval of the Human Subjects Ethics Committee of Middle East Technical University - Applied Ethics Research Center (UEAM) (B.30.2.ODT.0.AH.00.00/126/68-868). Informed consent was obtained from the participants included in the study.

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Appendix. The tasks used in the current study

Appendix. The tasks used in the current study

  1. 1.

    Population of Turkey

State Planning Organization (SPO) makes strategic plans for the next century. In this context, how the population of Turkey changes and the demographic structure would differ within the next 100 years are decisive for managing investment plans. Because investments are made according to the current conditions would become useless because of the changes in the population structures. SPO officials asked you to prepare a report answering the following questions. In the report, you need to investigate the changes in the population in the past and the formation of the population structure in the future.

  • How has the population of Turkey changed over the years?

    • What is the amount of change in the population growth between 1980 and 1985 and in other year intervals?

    • What are the year intervals where the average rate of change in population growth with respect to time is maximum and minimum?

    • What is the average rate of change in population growth with respect to time in the year 2000 approximately?

    • What is the average rate of change in population growth with respect to time in the year 2004 approximately?

    • How do the senior and young populations change in years?

    • Some experts claim that the population of Turkey will become stationary in the future. Do you think this is possible? If yes, when do you think this will happen?

  

Age groups

Years

Population*(× 1000)

0–19

20–39

40–59

60 or more

1980

45 586

22 556

13 745

6 756

2 529

1985

50 660

24 467

15 015

7 922

3 256

1990

55 971

25 981

17 500

8 456

4 034

2000

67 800

27 438

22 458

12 215

5 689

2007

70 786

24 938

23 604

15 176

7 068

2008

71 557

24 996

23 927

15 551

7 083

2009

72 641

25 133

24 225

15 832

7 451

2010

73 724

25 155

24 482

16 265

7 822

  1. *Mid-year population data (e.g., data for each year belongs to the sixth month of that year)
  1. 2.

    Follow-Up (Weather Balloon)

Weather balloons are launched daily to collect various meteorological data. As the balloon ascends, data is collected on pressure, humidity, and temperature through sensors at certain time intervals, and altitudes are transmitted to the mission control. The following table shows six weather balloon measurements collected on a mission.

Measurement #

Time (min)

Altitude (m)

Pressure (mbar)

1

0

Ground level

1000

2

360

260

925

3

650

440

850

4

1400

890

700

5

2750

1790

500

6

3600

2240

400

  • What is the rate of change in the altitude with respect to time between the 2nd and the 3rd measurements? Do not forget to state the unit of measurement.

  • What is the rate of change in the pressure with respect to altitude? Do not forget to state the unit of measurement.

    • Between the 4th and the 5th measurements

    • Between the 1st and the 6th measurements

  1. 3.

    Follow-up (Population Growth Rate)

    The population of a country is 50 million and 61 million in the years 2000 and 2010, respectively, as shown in the following graph.

figure a

  • Draw and interpret the population-time graph assuming that this country’s population increased at the same rate (in percent) every year between 2000 and 2010.

  • Draw and interpret the population-time graph assuming that this country’s population increased in equal amounts every year between 2000 and 2010. In this case, how does the annual population growth (in percent) change?

  1. 4.

    Follow-up of Tracking Track

    The population of a country is 50 million and 61 million in the years 2000 and 2010, respectively, as shown in the following graph.

A water tank is being filled with water at a constant flow rate. Figure 1 shows the height of the water as a function of volume. Figure 2 shows the derivative graph of the height-volume graph. What should be the units of the axes in Fig. 2? Write your answers into the given boxes in Fig. 2.

figure b

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Kertil, M., Erbas, A.K. & Cetinkaya, B. Pre-service Mathematics Teachers’ Understanding of Rate of Change Throughout a Model Development Sequence. Int J of Sci and Math Educ 21, 1769–1796 (2023). https://doi.org/10.1007/s10763-022-10324-z

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