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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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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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
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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.
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How has the population of Turkey changed over the years?
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What is the amount of change in the population growth between 1980 and 1985 and in other year intervals?
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What are the year intervals where the average rate of change in population growth with respect to time is maximum and minimum?
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What is the average rate of change in population growth with respect to time in the year 2000 approximately?
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What is the average rate of change in population growth with respect to time in the year 2004 approximately?
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How do the senior and young populations change in years?
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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?
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Age groups | |||||
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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 |
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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) |
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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 |
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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.
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What is the rate of change in the pressure with respect to altitude? Do not forget to state the unit of measurement.
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Between the 4th and the 5th measurements
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Between the 1st and the 6th measurements
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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.
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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.
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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?
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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.
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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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DOI: https://doi.org/10.1007/s10763-022-10324-z