Uso de applets GeoGebra no auxílio ao pensamento funcional
[Tradução (ainda não iniciada) do GeogebraBook - "The use of GeoGebra applets to support functional thinking"] In teaching a variety of difficulties arise when working with functions. By the use of technology, functional thinking must not be limited to thinking in and with functions but it can be extended to a thinking in changes of objects. What is the influence of technology-based material to the individual conceptions of students of lower secondary school 1? Are dynamic worksheets able to support the development of skills? In this talk, specific dynamic GeoGebra-Applets, which are intended to support the development of appropriate skills, as well as the research design and first results of a corresponding qualitative study are presented.
Table of Contents
- Introduction
- Introduction
- Applets
- Hill
- Valley
- Billiard
- Billiard dynagraph
- Triangle
- Triangle dynagraph
- Geoboard picture
- Research
- Research questions
- Research design
- Student responses
- Task "Area"
- Task "Skier"
- References
- References
Introduction
Students have various difficulties in working with functions. There the question arises, what is the influence of dynamic GeoGebra-Applets on the individual conceptions of students of lower secondary school. Based on this question, my PhD-thesis “the use of GeoGebra-Applets to support functional thinking in lower secondary school” has developed.
Functional thinking
But what is meant with „functional thinking“? Vollrath (1989) describes it as a way of thinking that is typical for dealing with functions, and he mentions 3 different aspects concerning this. [br] [br]Malle (2000) formulates the following aspects of functional thinking in a slightly altered version:[br][list][*][b]Relation aspect[/b]: Each argument x is associated with exact one value f(x).[br][/*][*][b]Co-variation aspect[/b]: If the argument x is changed, the value f(x) will change in a specific way and vice versa. [/*][/list]
Problems
In the literature are various difficulties of students in the context of functional dependencies described. Especially the following are prominent:[br][br][b]Poorly developed co-variational aspect[/b][br]The co-variational aspect is very important for working with functions in practice. But empirical studies show that especially this aspect is underdeveloped with students (De Bock, Verschaffel & Janssens, 1998; Malle, 2000; Hoffkamp, 2011).[br][br][b]Graph-as-picture error[br][/b]A poorly developed co-variational aspect can be seen among other things on the graph-as-picture error. This occurs in various forms and means that students see functional graphs as a photographic image of a real situation (Clement,1989; Schlöglhofer, 2000; Hoffkamp, 2011).[br][br][b]Illusion of linearity[br][/b]Another problem of students is the so called “Illusion-of-linearity”. This means that linear or directly proportional models are preferably used for the description of relations (DeBock, Van Dooren, Janssens & Verschaffel, 2002; Hoffkamp, 2011).[br][br][b]Slope-height confusion[br][/b]Difficulties arise also in the interpretation of slope and growth, for example if the point of maximum growth is confused with the largest function value. The confusion of height and slope leads also to difficulties in the interpretation of path-time graphs (Janvier, 1978; Clement, 1989; Hoffkamp, 2011).
Hill
(a) Click on [b]START[/b] and watch the animation several times. Describe in words how the speed of the ball changes on its path! Draw the corresponding graph in a coordinate system![br][br](b) Click on the check box "speed". Play the animation. What is displayed in the coordinate system? [br][br](c) Does the representation in the coordinate system approximately match with your solution from exercise (a)? [br][br](d) Solve using the representation in the coordinate system: At which position do you find the maximum speed? At which position do you find the minimum speed?[br][br](e) How does the speed change while the ball is moving? Can you read the solution off the graph?
Research questions
These problems and considerations lead to the following research questions.[br] [br][b]Question 1:[/b] What conceptions with particular attention to pre- and misconceptions emerge concerning functional thinking of students of lower secondary schools (grade 7/8)?[br] [br][b]Question 2:[/b] How should dynamic materials be designed to support students of lower secondary schools to develop appropriate mathematical understanding and thus to reduce errors due to pre-[br]and misconceptions concerning functional thinking?[br] [br][b]Question 3:[/b] What is the influence of dynamic materials on conceptions and internal representations of students of secondary schools particularly with regard to the understanding of[br]the co-variational aspect of functional thinking?
Methods
To answer the research questions, the following methods are chosen.[br][br][b]Question 1[/b][list][*]Diagnostic test 1 [br][/*][*]Diagnostic interviews (Zazkis & Hazzan, 1999; Hunting, 1997)[br][/*][/list][br][b]Question 2[/b][br][list][*]Literature on design criteria (Clark & Mayer, 2008)[br][/*][*]Observation: audio-, video- and screenrecordings[/*][/list][br][b]Question 3[/b][br][list][*]Observation: audio-, video- and screenrecordings[/*][*]Diagnostic test 2[/*][*]Diagnostic interviews[/*][/list]
Task "Area"
In the piloting the students' responses to the task "Area" from diagnostic test 1 can be divided into 4 categories.[br][br][list=1][*][b]Graph-as-picture error[/b][br]Some students made a graph-as-picture error and reasoned their answer just with the shape of the graph. A typical answer was:[br][i]Because the drawing at the top looks exactly like the on I have circled.[/i][br][br][/*][*][b]Graph-as-picture error[/b][br]Some students already argued with the area but they still took the shape into account. Typical answer: [i]Because the area increases and then stays the same.[/i][br][br][/*][*][b]Correct answer[/b][br]... but wrong reasoning showed that the students did not interpret the diagram correctly.[br][br][/*][*][b]Correct answer[/b][br]Some students chose the correct diagram with an explanation as follows: [i]Because the grey piece is getting bigger[/i] or similar answers.[br][/*][/list]
References
[list][*]Clark, R.C. & Mayer, R.E. (2008). e-Learning and the Science of Instruction. Pfeiffer.[br][/*][*]Clement, J. (1989). The concept of variation and misconceptions in cartesian graphing. Focus on Learning Problems in Mathematics, Vol. 11, 1-2.[/*][*]De Bock, D., Verschaffel, L. & Janssens, D. (1998). The predominance of the linear model in secondary school student's solutions of word problems involving length and area of similarplane figures. Educational Studies in Mathematics, Vol. 35(1).[/*][*]De Bock, D.; Van Dooren, W., Janssens, D. & Verschaffel, L. (2002). Improper use of linear reasoning: an in-depth study of the nature and the irresistability of secondary school students' errors. Educational Studies in Mathematics, Vol. 50(3). [/*][*]Glaser, B.G. & Strauss, A.L. (1998). Grounded Theory - Strategien qualitativer Forschung. Verlag Hans Huber.[br][/*][*]Hoffkamp, A. (2011). Entwicklung qualitativ-inhaltlicher Vorstellungen zu Konzepten der Analysis durch den Einsatz interaktiver Visualisierungen - Gestaltungsprinzipien und empirische Ergebnisse. PhD thesis.[/*][*]Hunting, R.P. (1997). Clinical Interview Methods in Mathematics Education Research and Practice. Journal of Mathematical Behavior, 16(2).[/*][*]Janvier, C. (1978). The interpretation of complex cartesian graphs representing situations - studies and teaching experiments. PhD thesis, University of Nottingham.[/*][*]Malle, G. (2000). Zwei Aspekte von Funktionen: Zuordnung und Kovariation. Mathematik lehren, 103.[/*][*]Nitsch, R. (2015). Conceptual Difficulties in the field of functional relationships. codi-test.de (2015-07-06).[br][/*][*]Schlöglhofer, F. (2000). Vom Foto-Graph zum Funktions-Graph. Mathematik lehren, 103.[/*][*]Vollrath, H.-J. (1989). Funktionales Denken. Journal für Mathematikdidaktik No. 10.[/*][*]Zazkis, R. & Hazzan, O. (1999). Interviewing in Mathematics Education Research. Choosing the Questions. Journal of Mathematical Behavior, 17(4).[/*][/list]