Abstract
We investigated students’ understanding of proportional, inverse proportional, and affine functions and the way this understanding is affected by various external representations. In a first study, we focus on students’ ability to model textual descriptions of situations with different kinds of representations of proportional, inverse proportional, and affine functions. Results highlight that students tend to confuse these models and that the representational mode has an impact on this confusion: Correct reasoning about a situation with 1 mathematical model can be facilitated in a particular representation, while the same representation is misleading for situations with another model. In a second study, we investigate students’ ability to link representations of proportional, inverse proportional, and affine functions to other representations of the same functions. Results indicate that students make most errors for decreasing functions. The number and nature of the errors also strongly depend on the kind of representational connection to be made. Both studies provide evidence for the strong impact of representations in students’ thinking about these different types of functions.
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In the test, the word “representation” was replaced either by “graph,” “formula,” or “table,” each in one third of the cases (see also the example items in Fig. 1)
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Appendix
Appendix
List of the 12 Descriptions of Realistic Situations as Used in Study 1
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Situations with underlying proportional model
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Jennifer buys minced meat at the butcher’s shop. Which representationFootnote 1 properly denotes the relation between the amount of minced meat that Jennifer buys and the price she has to pay?
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John fills up the tank of his truck with fuel. Which representation properly denotes the relation between the amount of fuel filled and the price he has to pay?
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A cruise ship crosses the ocean at a constant speed. Which representation properly denotes the relation between elapsed time and the distance travelled by the cruise ship at that time?
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Situations with underlying inverse proportional model
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During the war, butter was rationed. Each week, butter was delivered and fairly distributed among the people. Which representation properly denotes the relation between the number of people who wants butter and the amount of butter everybody receives?
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For a benefit an action committee wants to peel a full container of potatoes. This job will take them several hours. Which representation properly denotes the relation between the number of committee members who collaborates and the time needed to finish this job?
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A group of friends participates in a gambling game. When they win some money it will be shared equally among the friends. Which representation properly denotes the relation between the number of friends and the amount of money each person will receive?
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Situations with underlying affine model with positive slope
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A new phone company charges for service by applying a fixed monthly subscription cost and an amount per number of minutes talked. Which representation properly denotes the relation between the number minutes talked and the monthly invoice amount?
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At a refinery fuel oil is pumped into a tank truck at a constant flow rate. This is done on a balance to prevent overloading of the truck. Which representation properly denotes the relation between the time fuel oil is pumped and the weight of the tank truck?
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A taxi company charges for a night ride a fixed fee upon entree and an amount for each kilometer driven. Which representation properly denotes the relation between the number of kilometers driven and the total price for the night ride?
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Situations with underlying affine model with negative slope
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A chemical concern has a big cistern with hydrochloric. This morning they started to pump with a constant pace all hydrochloric out of this cistern. Which representation properly denotes the relation between elapsed time and the amount of hydrochloric that is still in the cistern?
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In case of an electrical breakdown a hospital needs to switch to emergency generators. These generators are connected to large fuel tanks. Which representation properly denotes the relation between the time the generators are running and the amount of fuel in the tanks?
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Thom has a mobile phone subscription, but uses prepaid reloadable cards. Per minute talked the uploaded sum decreases by a fixed amount. Which representation properly denotes the relation between the number of minutes talked and the remaining sum on the card?
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De Bock, D., Van Dooren, W. & Verschaffel, L. STUDENTS’ UNDERSTANDING OF PROPORTIONAL, INVERSE PROPORTIONAL, AND AFFINE FUNCTIONS: TWO STUDIES ON THE ROLE OF EXTERNAL REPRESENTATIONS. Int J of Sci and Math Educ 13 (Suppl 1), 47–69 (2015). https://doi.org/10.1007/s10763-013-9475-z
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DOI: https://doi.org/10.1007/s10763-013-9475-z