Tooluse supporting the learning and teaching of the Function concept

Tooluse supporting the learning and teaching of the Function concept

Paper for ISDDE2008

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

1. Introduction

Tool use is indispensable both in daily life as well as in doing and learning mathematics. Research suggests a close relationship between tool use, cognitive

development and social practice. The ToolUse-project at the Freudenthal Institute focuses on the use of computer tools for grade 8 (13-14 years) students’ acquisition of the

mathematical concept of function, in an effort to identify the relationship between the use of technological tools and the learning of mathematics

. The main research question aims at understanding how computer tools can be integrated in an instructional sequence on the function concept, so that their use fosters learning. This is studied in terms of the notion of instrumental genesis (Kieran & Drijvers, 2006, Trouche, 2004) and domain- specific theories on the teaching and learning of mathematics within the philosophy of realistic mathematics education (Freudenthal, 1991). The research setup includes a cyclic process of instructional design, teaching experiments and data-analysis (Gravemeijer, 2004).

2. Theoretical framework

We investigated an instructional approach for the function concept that aims at a learning process in which the mathematics is build upon everyday intuitions. This is also in line with the objective of realistic mathematics education (RME), where instructional design is aimed at creating optimal opportunities for the emergence of formal

mathematical knowledge from situation specific reasoning. In order to achieve this, contextual problems are cast into situations that are experientially real for the students.

During the process of teaching and learning students can preserve the connection between the mathematical concepts and what is described by these concepts. Students’ final

understanding of the formal mathematics should stay connected with their understanding of these experientially real, everyday-life phenomena (Freudenthal, 1991).

Well-chosen context problems offer students opportunities to develop informal, highly context-specific models and solving strategies (Doorman et al., 2007). These informal solving procedures then may become subject of formalization and generalization to constitute a process of further abstraction, in RME dubbed as: progressive

mathematization. The instructional designer tries to construct a set of problems that can

1

The project is supported by the Netherlands Organisation for Scientific Research (NWO) with grant

number 4121-04-123. The project’s website is http://www.fi.uu.nl/tooluse.

lead to a series of processes that together result in the reinvention of the mathematics aimed at.

Research on the design of primary-school RME sequences has shown that the concept of emergent models can function as a powerful design heuristic (Gravemeijer, 1999). Context problems are selected, offering the students the opportunity to develop situation-specific representations and strategies. Students’ reasoning gives rise to (informal) models during classroom discussions and subsequent activities. In this sense, the models emerge from the activity of the students. Even if the models are not actually invented by the students, great care is taken to approximate students’ inventions as closely as possible by choosing models that link up with their learning history and with their contributions. Another important criterion is in the potential of the models to support mathematization towards the intended concepts. The idea is to look for models that can be generalized and formalized to develop into entities of their own, which then can become models for mathematical reasoning (e.g. Doorman, 2007; Gravemeijer, 2004;

Rasmussen & Blumenfeld, 2007).

An additional theoretical foundation of the research we discuss here is the use of computer minitools in mathematics education. In general, tools influence the process of students’ mathematical sense making. Cobb (1999) illustrated this by describing the interplay between students’ ways of symbolizing and mathematical reasoning. In relation to this he referred to the notion of affordances (Cobb, ibid). The computer minitools afforded the students’ reasoning on statistical problems. Affordances are not properties of tools that exist independently of users, but are achievements of users in activity. As a consequence, instructional designers should take into account how students might act and reason with the tools as they participate in a sequence of mathematical practices.

We adopted these ideas for integrating a computer tool in the instructional materials for the teaching and learning of the function concept. The tool was meant to create opportunities for students to develop understanding of the input-output relationship and the related dynamics in graphs and tables.

The instructional design: AlgebraArrows

In the design phase of the project, teaching materials and a computer tool were developed. The computer, AlgebraArrows (AA), primarily supports the construction of input-output chains of operations (Boon, 2008). These chains can be applied to single numerical values as well as to variables. In the latter case, tables, formulas and (dot) graphs can be shown. The chains can be extended, linked, compared and compressed.

The applet is embedded in an electronic learning environment, allowing students to save final situations of each activity and to continue working in any location with internet access. The researchers, as well as the teacher, are able to monitor the answers of the students with all-class result overviews.

We expected the starting function image for most of the students to be the local

calculation procedure. This could be applied to one single input number at the time and

then result in an output number. The instructional design aimed at the transition from

these calculations to reasoning with a formula and its graphs and tables. The goal of the

learning arrangement with the computer tool was to keep the chaining of calculations

connected with formulas during the whole instructional sequence. The sequence started

with three open-ended activities (mobile phone offers, a quadrilateral and breaking distance) to reveal the students’ thinking and to provide for strategies that could be used by the teacher to introduce the computer tool AlgebraArrows. Both the purpose of and the representations in the tool were object of discussion (see Figure 1).

Figure 1 Poster of students’ calculations and a ´living operation chain´

The chain of operations in the AA applet, applied to a single numerical value, signifies the function as an input-output assignment signifying a list of calculations. This is an important step for the function concept, because it supports the importance of identifying the relevant variables and their dependencies. The step is certainly not trivial for the students. During a pilot experiment we noticed that, even after a few activities with AA, many students wanted to translate a formula like 2·x + 100 into a chain starting with input 2.

In Figure 2, two chains concerning different mobile phone offers invite a global

investigation of properties. The technique of putting numerical values in the input

window allows for comparison of function values. A variable input allows for a more

global comparison, in which graphs, tables and formulas can be involved.

Figure 2 An input – output assignment

The applet allows not only a change of input value, but also the study of the co- variation by means of tables and graphs. The chain in Figure 3 represents the cost of one particular mobile phone offer. The dynamics of the co-variation can be investigated by techniques of substituting different values, scrolling through the table, tracing the graph and studying the formula. The task for the students is to find the change of cost per minute change of the input variable – the number of minutes of phone calls.

Figure 3

The computer tool AA was supposed to support students’ investigations of many cases. The investigations by pairs of students were expected to foster a trial-and-improve- process and negotiation of meaning. During the investigations they could find patterns and develop strategies in dealing with dynamic input-output dependencies. Consequently, AA was expected to afford a means to structure inquiry about these relationships through the construction and modification of arrow chains, graphs and tables, and the generation and comparison of argumentations.

The conjectures concerning the role of AA in the instructional sequence were investigated with a design research method of iterative phases of instructional design, teaching experiments and data analyses.

Results

The teaching experiments took place in three classes at three different schools.

Video recordings were made of classroom teaching and group work (17 hours), and screen videos of three pairs of students working with the computer were captured (6 hours). Students’ final answers on the activities with AA were saved on a central server (41 pairs). We collected written work of the students as well as the results of a written assessment at the end of the experiment. Data analysis was carried out by qualitative analyses and coding of the data using the video analysis tool ATLAS.ti.

We started the analysis by watching and annotating the work of a pair of students (Lisa and Romy). We had screen videos (with audio) of the three lessons with the

computer tool. These screen videos were imported in ATLAS.ti. Initially, the clipping was guided by exercises as units of analysis. We described each exercise-clip with the comment-tool and added analytical notes. With these notes we wrote down a storyline of Lisa and Romy and their learning process with respect to the learning environment and our prior expectations. The clips were coded with the number of the activity as well as with their names. These codes could be used to browse through the data and to combine it with other data sources. We also imported and coded photocopies of written work and of screenshots of the computer activities of all the students which were saved on the central server.

We expected the two students to initially use the calculation chains in their reasoning about input-output-dependencies, and that this reasoning would change to using tables and graphs as basic structuring elements. These expectations were based upon the idea of emergent modeling, which is a characteristic of realistic mathematics education. One of the key aspects of emergent modeling is the development of a model and the ways of reasoning with the model through a sequence of manifestations

(Gravemeijer, 2004)

In the teaching experiment we provided the students with one tool for a sequence of ten lessons. The manifestation of the chain itself did not change, but the function of the representation changed. As a result, we didn’t see a clear development through a number of manifestations of a model, whereas the focus of reasoning with the model shifted from individual calculations to dependency relationships. Consequently, we noted in the work of Lisa and Romy that they reasoned with calculation chains during all the activities.

Moreover, their reasoning changed from referring to individual calculations to

characterizing dynamic processes. This storyline was discussed among the project

members to try to understand these findings and the relation to the instructional sequence.

During the discussions we successively clarified and assessed possible conjectures with the storyline and the references to the clips.

As a result of this analysis the idea arose that we could better use the principle of a form-function-shift (Saxe, 2002) for describing what happened in the learning process of Lisa and Romy. The arrow chain initially built on the notion of recording the structure of a series of calculations. The form-function-shift took place when the two students started using this record of calculations as a tool for investigating dependency- relationships. Simultaneously, as the investigations with arrow chains built on the

imagery of calculations with successive operations, constructing and modifying an arrow chain signified organizing a dependency-relationship. Calculation-routines turned into explorations, during which new narratives about relationships were constructed. The articulation of this shift seemed helpful in labeling different phases in the learning process with codes.

We conjectured that in this form-function-shift the goal of the activities for the students shifted from calculating a specific value to reasoning about relationships, and the chain is used as a means for scaffolding and communicating ways of reasoning (emerging mathematical goals: Saxe, 2002). Next, based upon the analysis of the work of Lisa and Romy, we identified two illustrative and rather similar open-ended activities, one in the beginning of the sequence and one in the end. For these activities we characterized different solution strategies with the notion of the form-function-shift and constructed four codes for these characterizations (from “a record for calculations” to “as a tool for reasoning”). All the screenshots of the students were coded in order to try to get

quantitative support for the emerging conjecture. During the coding process we discussed a few cases for fine-tuning the code-definitions and the interpretations of the codes.

The result of this second analysis offered evidence for the conjecture of a form- function-shift in the students’ reasoning with chains of calculations in the computer tool.

We found that with the activity in the beginning 25 (out of 41) pairs of students used the chain as a record for calculating a specific value, while the activity at the end of the

sequence showed that 24 (out of 34) pairs used the chain as a tool for reasoning. However, during the coding we also observed that with the first activity students sometimes tried to use AA as a tool for reasoning, but were not able to really operate it sufficiently. This observation brought us back to a prior notion of the theoretical framework that tool-use development is closely related to a process of instrumental genesis.

Further research is needed to merge the notions of emergent modeling and form- function-shift with the notion of instrumental genesis to be able to better understand the interplay between tool use and mathematics learning.

Moreover, in the classroom learning processes, we noticed that students initially reasoned and wrote about mathematical and kinematical notions with a tentative language and inscriptions that were not as precise as the notions aimed at. This seems similar to Goldin’s description of three main stages in the development of representational systems:

(i) an inventive and semiotic stage, (ii) structural development and establishment of

relationships, and (iii) an autonomous stage (Goldin, 2003). During the inventive stage,

students tentatively used inscriptions and language to communicate their developing

ideas. In the autonomous stage the system can function flexibly in new contexts. We

observed that the teacher played an important role during this process. The teacher guided

whole class discussions and alternated between evoking students to present their solutions and focusing to the mathematics needed for what is to come (Doorman, 2007;

Sherin, 2002).

Redesign and further research should also reveal the criteria by which the teacher can make decisions during this balancing act of guiding students through a process for developing the function concept.

References

Boon, P. (2008). AlgebraArrows. Retrieved from (at june, 9

, 2008) http://www.fi.uu.nl/wisweb/en/welcome.html.

Cobb, P. (1999). Individual and Collective Mathematical Development: The Case of Statistical Data Analysis. Mathematical Thinking and Learning 1(1), 5-43.

Doorman, L.M. (2007). Wiskundeonderwijs met computeractiviteiten vraagt constructieruimte voor leerlingen. Pedagogische Studiën 84(5), 375-390.

Doorman, M., Drijvers, P., Dekker, T., Van den Heuvel-Panhuizen, M., De Lange, J., Wijers, M. (2007). Problem solving as a challenge for mathematics education in The Netherlands. ZDM-The International Journal on Mathematics Education 39, 405-418.

Freudenthal, H. (1991). Revisiting Mathematics Education – China Lectures. Dordrecht:

Kluwer Academic Publishers.

Goldin, G.A. (2003). Representation in school mathematics: a unifying research perspective. In: J. Kilpatrick, W.G. Martin & D. Schifter (Eds.) A research companion to Principles and Standards for School Mathematics. Reston: NCTM.

Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning 1(2), 155-177.

Gravemeijer, K. (2004). Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education. Mathematical Thinking and Learning 6, 105-128.

Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and- pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning 11, 205-263.

Rasmussen, C. & Blumenfeld, H. (2007). Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions.

Journal of Mathematical Behavior 26, 195-210.

Saxe, G. B. (2002). Children’s Developing Mathematics in Collective Practices: A Framework for Analysis, Journal of the Learning Sciences 11, 275–300).

Sherin, M. G. (2002). A balancing act: developing a discourse community in a

mathematics community. Journal of Mathematics Teachers Education 5, 205-233.

Trouche, L. (2004). Managing complexity of human/machine interactions in

computerized learning environments: Guiding students' command process through

instrumental orchestrations. International Journal of Computers for Mathematical

Learning 9, 281-307.