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 1 . 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 domainspecific 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.
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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
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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.
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Figure 2 An input – output assignment
The applet allows not only a change of input value, but also the study of the covariation 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
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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-improveprocess 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
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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 dependencyrelationships. 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 formfunction-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 formfunction-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
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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
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