TOOL USE AND CONCEPTUAL DEVELOPMENT: AN. EXAMPLE OF A FORM-FUNCTION-SHIFT. Michiel Doorman. +. , Peter Boon. +. , Paul Drijvers. +.
TOOL USE AND CONCEPTUAL DEVELOPMENT: AN EXAMPLE OF A FORM-FUNCTION-SHIFT Michiel Doorman+, Peter Boon+, Paul Drijvers+, Sjef van Gisbergen+, Koeno Gravemeijer* & Helen Reed+ +
Freudenthal Institute, Utrecht University, The Netherlands
* Eindhoven School of Education, Technical University Eindhoven, The Netherlands Tool use is indispensable both in daily life as well as in doing and learning mathematics. Research suggests a close relationship between tool use and conceptual development. This relationship is investigated for grade 8 students’ acquisition of the mathematical concept of function. The study includes two teaching experiments and qualitative as well as quantitative data analyses. The results show that a form-function-shift can be recognized in the process of tool acquisition and learning about functions. We conclude that this process can be fostered by an instructional sequence in a multimedia environment that supports the co-emergence of external representations and mathematical concepts.
1. THEORETICAL FRAMEWORK It is generally acknowledged that tools influence the process of students’ mathematical sense making. Cobb (1999) describes the dynamic interplay between students’ ways of symbolizing and mathematical reasoning in relation to tool use. In his study, computer minitools supported students’ reasoning in the domain of statistics. He noted that this support is not a property of a tool that exists independently of users, but rather is an achievement of users. In this view, learning is characterized as a process in which students construct mental representations that mirror the mathematical features of external representations. This is related to the distinction between artefacts – material objects – and tools as instruments constructed by users in activity. The instrument comprises the cognitive schemes containing related techniques and conceptual aspects. The constructive and dynamic process of tool appropriation in mathematics education can be conceived as a process of instrumental genesis (Artigue, 2002) and progressive mathematization (Gravemeijer, 1999). Research has shown that the concept of emergent modeling can function as a powerful design heuristic for creating such
processes in mathematics education (Gravemeijer, 1999). Open questions cast in realistic problem situations offer students opportunities to develop tentative situation-specific external representations. These representations give rise to (informal) models and new mathematical goals emerge. This is guided by the teacher with questions posed in situations that shift focus from real life problem solving to generalizing and mathematical reasoning (e.g. Doorman & Gravemeijer, 2009). We investigated the relationship between tool use and conceptual development in the domain of functions. With respect to the concept of function, Sfard pointed to the dual nature of mathematical conceptions (Sfard, 1991). She distinguishes operational conceptions (related to processes) from structural conceptions (related to objects). In the process of concept formation, the operational precedes the structural, but can not completely be separated as a result of this dual nature. Sfard uses the history of the function concept to argue that it evolved from calculations with inputoutput relationships to studies of sets of ordered number pairs. The dual nature of functions appears difficult for students and most students do not reach that level of understanding. We expect to be able to foster the operational and structural aspects of functions with the availability of computer tools. The main research question of our project 1 is: How can computer tools be integrated in an instructional sequence on the function concept, so that their use fosters learning? The principle of a form-function-shift of notations in use (Saxe, 2002) is particularly suitable for analysing the interplay between tool use and conceptual development. This shift describes the interplay between cultural forms (external representations) and cognitive functions (purposes for structuring and accomplishing practice-linked goals). In the domain of functions, students can be supposed to use representations in a computer tool for investigating input-output relationships. As these investigations initially build on the imagery of repeated calculations, a form-function-shift might take place when the representations start to signify structure and dynamics of the relationships’ co-variation. This paper deals with the role of a form-function-shift in analysing the interplay between tool use and conceptual development. 1
The project is supported by the Netherlands Organisation for Scientific Research (NWO) with grant number 411-04-123. The project’s website is http://www.fi.uu.nl/tooluse.
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2. METHODS The research method followed is characterized as design research and consists of creating and analysing educational settings. In its cyclical process three phases are distinguished: instructional (re)design, teaching experiments and data analyses (Cobb et al, 2001). We created an educational setting in which we could analyse and come to understand the interaction between the instructional activities, tool use and students’ learning. Instructional design The instructional goal was to facilitate students’ ability to understand and investigate dynamic co-variation of input-output relationships. We aimed at a process of concept formation, within which formal models are built upon situation specific reasoning. Consequently, the instructional design had to provide opportunities for the learner to develop self-generated calculations that can progressively be mathematised into representations of functions. The instructional sequence starts with three open-ended activities to reveal the students’ current thinking, to evoke the need for organising series of calculations and to provide for opportunities for the teacher to introduce the computer tool AlgebraArrows (AA). This tool primarily supports the construction of input-output chains of operations. The chains can be applied to single numerical values as well as to variables. Tables, formulae and (dot) graphs can be connected to the chains (Boon, 2008; Drijvers et al, 2007).
Figure 1: The computer tool AlgebraArrows
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The initial computer activities are meant to show that AA is a useful tool for repeated calculations. The arrow chain resembles these calculations and input-output results obtained in prior activities. To structure inquiry of dependency relationships, the next step in the computer sequence is the use of tables and graphs and scroll and zoom tools (see Figure 1). After the computer lessons the students carried out a matching activity. In this activity, students sort cards showing various representations into sets which represent one single function (c.f. Swan, 2008). This activity was designed to offer opportunities to reflect upon the various representations which they had encountered in the computer environment. Teaching experiments The first teaching experiment was conducted with three grade 8 (13-14 year olds) classes at three different schools and the second with 23 classes from 8 schools. In both experiments teaching sessions and group work in two classes were videotaped, and screen-audio-videos of three pairs of students working with the computer tool were collected. Students’ answers to the computer activities were saved on a central server. In addition, the researchers collected students’ written work and the results of a written assessment at the end of the instruction experiment. Analyses of the first experiment showed that the principle of a formfunction-shift (Saxe, 2002) was helpful in describing what happened in the learning process of two students. We revised the instructional sequence to be able to investigate this shift quantitatively by strengthening the goals of the sequence and redesigning two activities for a second analysis. In addition, we questioned to what extent the final performance of the students was dependent on the use of the available computer tool. Therefore, we added a computer test as well as a written test at the end of the instructional sequence. For the analyses of the second teaching experiment we used data of 155 students from 3 schools. These schools concluded the experiment with both the computer test and the written test. Data analyses The analyses started with organization, annotation and description of the data in a multimedia data analysis tool. Initially, the tasks in the instructional sequence served as the unit of analysis for clipping the videos. Codes were used to organise and document the data, and to produce conjectures about patterns in the teaching and learning process following the principles of grounded theory (Strauss & Corbin, 1998).
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In analysing the first teaching experiment, we reconstructed the learning process of two students. This storyline motivated the use of a form-functionshift to describe the instrumental genesis of the computer tool. We identified characteristics of two computer tasks that represented different solution strategies. These characteristics were used to construct strategy codes with respect to the use of representations in the computer tool. In the second teaching experiment, the screenshots of 155 students’ answers to the specific two tasks were coded with the strategy codes. Good interrater-reliability was achieved (with a Cohen’s Kappa of 0.79) on 55 out of 306 items (18%). For an analysis of the relation between results on the written test and the computer test we used a paired t-test.
3. RESULTS The results of one of the open-ended activities (comparing two cell-phone offers) are illustrative of attempts by the students to organize the situations mathematically, i.e. organize repeated calculations, construct variables and use various representations (e.g. see Figure 2).
Figure 2 Posters of students’ calculations
The teachers used various ways to introduce and guide the computer activities (Drijvers et al, in press). With respect to the interplay between tool use and students’ conceptual development we analysed the computer work of a pair of students (Lina and Rosy). This resulted in a storyline of their learning process with respect to the computer tool. One of the first activities is similar to the cell phone activity and concerns determining when it is wise to change from one phone company offer to the other. Lina and Rosy built a calculation chain from the input box by adding operations and finally connecting an output box. They started to reason by initially focusing on specific cases. They entered numbers like 100 and 50 successively for comparing the differences between the two offers.
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During the third computer lesson we observed quite another strategy by these students with AA while solving a similar kind of activity about two offers by contractors. The use of the representations in the tool as well as the focus of their reasoning seemed to have changed from case-by-case calculations to modeling with input-output chains and using the table tool to investigate the dynamics of both relationships. Lina starts the discussion and operates the mouse and the keyboard. L: Company Pieters charges start costs and an hour rate … that is R: plus 92, times 30 … [L drags an input and an output box into the drawing area and creates a label for the in-box] L: First company Pieters … I write only Pieters here, eh? R: euro … plus costs per hour Pieters … and the total costs will come in here. [The mouse moves from in-box to out-box and she creates a label and types the name of the out-box. The variables are identified.] L: Yes, here are the costs. [The chain of operations is constructed and connected to the input- and output-box] (…) L: And … the other is TweeHoog … [The inbox and outbox are positioned in the drawing area. Labels are created and names of the corresponding variables are typed in the labels.] R: Strange costs per hour 32,75. [The arrow chain of operations is constructed.] R: Now you have to connect it to a table. [They start to scroll in the table of the input values and quickly see that the break-even point is at 18 hours.] (…) L&R: From 18 hours. [They type their answer ‘from 18 hours’ in the answer box of question b.]
This vignette illustrates how the operation and function of the arrow chain representation changed in comparison to the previous computer activity. Lina and Rosy organized dependency relationships by immediately positioning and labeling boxes for input and output variables and then filling the gap with operations. For investigating the dynamics, they successfully 6
operated the table tool in the software. This way of reasoning is precisely what we aimed for: an immediate identification of the variables, a clear construction of a dependency relationship with the computer tool and finally a purposeful and productive use of the table tool to investigate the dynamics of both relationships. The second teaching experiment was on an increased scale, and one of the research goals was to obtain quantitative support for the conjectured formfunction-shift during the computer activities. The solutions of 155 students of three schools to two specific computer activities were coded with the strategy codes. These results offered significant evidence to support our conjecture. We found that initially 130 (out of 155) students used the tool only for calculating successive input-output values, while in the end 89 (out of 152) students used the tool for structuring and investigating the dynamics of the relationships. In a paired t-test we found no significant difference (p=0.200) between the final scores on the written test and the computer test. The correlation of 0.38 between these scores was moderate (p=0.001). These results indicate that the students were able to transfer their functional thinking in the computer environment to the paper-and-pencil environment.
6. CONCLUSIONS We investigated how computer tools can be integrated in an instructional sequence on the function concept. We conclude that a form-function-shift can be used to describe the process of tool use and conceptual development. The shift took place when students started using representations of functions as tools for investigating relationships. As the investigations with arrow chains initially built on the imagery of repeated calculations, final operations on representations resembled an object conceptualization of functions. The performance of students appeared to be independent of the computer tool. The emerging and shifting mathematical goals that fostered the formfunction-shift mirrored the instructional sequence. During the introductory activities, students invented tentative representations that prepared for the successive computer activities. The representations in the tool linked up with these prior activities and offered opportunities to shift goals from solving situation specific problems to operational and structural aspects of functions. These shifts accomplished mathematical development as a step in the process of progressive mathematization and instrumental genesis.
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REFERENCES Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning 7, 245-274. Boon, P. (2008). AlgebraArrows. Retrieved http://www.fi.uu.nl/wisweb/en/welcome.html.
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Cobb, P. (1999). Individual and Collective Mathematical Development: The Case of Statistical Data Analysis. Mathematical Thinking and Learning 1, 5-43. Cobb, P., Stephan, M., McClain, K. & Gravemeijer, K. (2001). Participating in Classroom Mathematical Practices. Journal of the Learning Sciences 10, 113163. Doorman, L.M. & Gravemeijer, K.P.E. (2009). Emergent modeling: discrete graphs to support the understanding of change and velocity. ZDM-The International Journal on Mathematics Education 14, 199-211. Drijvers, P., Doorman, M., Boon, P., Van Gisbergen, S., Gravemeijer, K., & Reed, H. (in press). Teachers using technology: orchestrations and profiles. Paper submitted to the PME33 conference, 19-24 july 2009, Thessaloniki, Greece. Drijvers, P., Doorman, M., Boon, P., Van Gisbergen, S. & Gravemeijer, K. (2007). Tool use in a technology-rich learning arrangement for the concept of function. In Pitta-Pantazi, D., & Philippou, G., Proceedings of CERME 5, 1389-1398. Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning 1, 155-177. Saxe, G. B. (2002). Children’s Developing Mathematics in Collective Practices: A Framework for Analysis, Journal of the Learning Sciences 11, 275–300). Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics 22, 1-36. Strauss, A., & Corbin, J. (1988). Basics of qualitative research. Techniques and procedures for developing grounded theory. Thousand Oaks, CA: SAGE Publications. Swan, M. (2008). A Designer Speaks. Educational Designer, 1. Retrieved from: http://www.educationaldesigner.org/ed/volume1/issue1/article3.
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