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This is the published version of a paper presented at Mathematics Education in the Digital
Age (MEDA), Linz, 16-18 September, 2020.
Citation for the original published paper:
Borg, A., Fahlgren, M., Ruthven, K. (2020)
Programming as a mathematical instrument: the implementation of an analytic
framework
In: Ana Donevska-Todorova, Eleonora Faggiano, Jana Trgalova, Zsolt Lavicza, Robert
Weinhandl, Alison Clark-Wilson, Hans-Georg Weigand (ed.), Mathematics Education
in the Digital Age (MEDA) PROCEEDINGS (pp. 435-442).
N.B. When citing this work, cite the original published paper.
Permanent link to this version:
http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-80187
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435
Programming as a mathematical instrument: the implementation of
an analytic framework
Andreas Borg, Maria Fahlgren, and Kenneth Ruthven
Karlstad University, Department of Mathematics and Computer Science, Sweden
andreas.borg@kau.se
This paper relates to an ongoing project using design-based research as a methodol-
ogical approach in which students with no prior experiences of using programming as
a mathematical tool are observed trying to solve mathematical problems with the help
of programming. The Instrumental Approach is used as conceptual framework in which
the concept of instrumental genesis describes the process where the programming
environment as an artefact together with student-developed mental schemes forms an
instrument in order to solve mathematical problems. The development of schemes is of
special interest in this paper where Vergnaud’s components of a scheme provide a
framework for analysing transcripts of talk between student pairs and the
programming code that they generate.
Keywords: mathematics education, computer programming, instrumental genesis.
INTRODUCTION
During the past decade, there has been a renewed recognition of programming as an
important digital competence to be developed as part of the general education of all
students, and of its particular relationship to mathematical competence. This has been
recognized in changes to curricula in many countries: in France, Finland, and Sweden,
for example, programming is included in mathematics curricula. In Sweden, where the
study described in this paper took place, programming was in 2018 included in
mathematics from year 4 in lower secondary school. In upper secondary school,
programming is to be used as a tool for mathematical problem solving.
Papert (1980) argues that using programming in school and in mathematics could have
positive effects on children’s learning and could help students to develop new cognitive
skills. According to Hoyles and Noss (2015), the use of programming in mathematics
education is seen by most students as an engaging activity where they can
independently “build, learn from feedback and debug” (p. 7). Programming is also a
means for developing creativity and ability in problem solving (Romero, Lepage, &
Lille, 2017) and offers a natural opportunity for students to be exposed to mathematical
concepts closely related to programming, e.g. iterations (Noss, 1986). But although the
introduction of programming has the potential of offering new possibilities for
learning, Drijvers and Gravemeijer (2005) argue that the integration of new
technologies in mathematics education can be complicated and that it would be naïve
to believe that “we can separate techniques from conceptual understanding and that
leaving the first to the technological tool would enable us to concentrate on the latter”
(p. 164). Instead they argue that machine techniques and conceptual understanding
must be interwound and be developed simultaneously. Drijvers and Gravemeijer
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(2005) consider this interwinding as a fundamental part of The Instrumental Approach
which will be described in the following section.
THE INSTRUMENTAL APPROACH
The Instrumental Approach originates from the field of cognitive ergonomics
(Rabardel, 2002) and considers the process where a subject, involved in a goal-driven
activity, uses an artefact (a material or abstract object) to act towards a given objective.
During the process when the subject appropriates the artefact to her/his needs and
integrates the artefact with her/his activity the subject develops mental utilization
schemes associated both with use of the artefact and with the objective of which the
artefact should act towards (Rabardel, 2002). These schemes can be usage schemes –
directed towards the artefact itself - or instrumented action schemes - directed towards
the object of the activity. The artefact together with the associated mental schemes
constitutes an instrument for the subject, where the instrument is regarded as a
psychological construct. The process through which the instrument is formed is called
the instrumental genesis and is, in this ongoing research project, followed with special
interest when studying how students (the subjects) use a programming environment
(the artefact) when solving a mathematical problem (the objective).
The development of schemes
In order to study the instrumental geneses of students, the development of mental
schemes is therefore of special interest. Vergnaud (1998) argues that mental schemes
can be divided into four different components; goal and anticipations, rules of action,
operational invariants, and possibilities of inferences and this paper will focus on
students' use of different rules of action and operational invariants. The rules of action
are considered by Vergnaud (1998) as the generative part of the scheme, directed by
operational invariants (Buteau, Gueudet, Muller, Mgombelo, & Sacristán, 2019).
Every action is built upon some information or concepts and Vergnaud (1998) thus
regards concepts-in-action as a vital part of the operational invariants. The second part
of the operational invariants consists of theorems-in-action, regarded as “proposition[s]
which [are] held to be true” (p. 168) by the subject when s/he acts. Vergnaud (1998)
argues that there is a relationship between concepts-in-action and theorems-in-action
since “concepts are ingredients of theorems” (p. 174).
Buteau et al. (2019) used Vergnaud’s (1998) four components of a scheme as an
analytic frame in order to analyse how university students engage in mathematical
inquiries using programming as a mathematical tool. They argue that their use of the
framework has “deepen[ed their] understanding of what is at stake in terms of students’
learning in this particular context” (p. 17) and has served as a means to illustrate
students’ instrumental geneses. In accordance with the work of Buteau et al. (2019),
Vergnaud’s (1998) components of a scheme will serve as an analytic framework for
this study and the research question that the study is addressing is: What are the
instrumental geneses of upper secondary school students’ use of programming
environments in trying to solve mathematical problems pre-designed to lend
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themselves to programming? The question of methodology which this specific paper
addresses is: How can Vergnaud’s (1998) components of a scheme be operationalized
for this study? Since the instrumental geneses in this study relate to students’ use of
programming environments as a mathematical problem-solving tool, the schemes
developed during the intervention relate to the problem-solving process as a whole and
not to specific parts of it. But due to limited space within this paper, only a specific
section of the problem-solving process will be described.
METHOD
The findings presented within the paper are part of a research project using design-
based research as the overarching research method. In 2019, 27 eleventh grade students
in a Swedish upper secondary school participated in the teaching intervention of the
first design cycle of a lesson in which students are intended to solve mathematical tasks
using a non-standard problem-solving strategy (e.g. an exhaustive trial) involving use
of the programming environment. The students were, at the time of the intervention,
taking the same introductory course in programming and thus had basic knowledge of
coding but no experience of using the programming environment as a mathematical
tool during their ongoing course in mathematics. Due to the students’ prior study of
coding, it was assumed that they had already developed basic usage schemes related to
the use of the artefact (programming environment). The focus of this study is thus on
the development of instrumented action schemes directed towards mathematical
problem solving.
Data collection and data analysis
During the intervention students worked in pairs and three of the pairs were followed
more closely through the use of screen-capturing software which also recorded the
conversation between students in each pair. This data was of special interest when
studying the development of schemes since it allowed the researcher to identify stable
behaviours important when analysing students’ instrumental geneses (Buteau et al.,
2019). The researcher (who acted as the teacher) also wore a microphone to record
conversations between students and the researcher. All voice recordings collected
during the teaching intervention of the first design cycle were transcribed using NVivo.
During the analysis, data from the recordings were grouped into themes relating to
different parts of the problem-solving process. Then these themes were coded in an
iterative process using Vergnaud’s (1998) components of a scheme. Both verbatim
abstracts and code generated by students have served as evidence when analysing the
development of students’ schemes.
RESULTS AND ANALYSIS
In this section, examples will be given of how verbatim abstracts from conversations
between students during the intervention have been used together with the generated
programming code to analyse the development of students mental schemes using
Vergnaud’s (1998) components of a scheme as the analytical framework. In this paper,
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the use of rules of action and operational invariants will be of special interest. The
mathematical problem will not be described in detail in this paper but concerns finding
the ages of three sisters. The problem can be mathematised algebraically in terms of
relationships involving the ages of the sisters but not in a manner which permits the
students to solve the problem using algebraic methods already known to them.
Therefore, it was anticipated that the students might use the programming environment
in order to conduct an exhaustive trial, a process which will be analysed in the
following sub-section.
Developments of schemes relating to the implementation of an exhaustive trial
During the intervention, the researcher asks a pair of students called Sophie and
Richard to describe their problem-solving strategy and how this had been implemented
using the programming environment. Sophie explains how the pair have used nested
loops in order to conduct an exhaustive trial:
Sophie: Yes, we have a nested for-loop so the first one... Eh... Or starting with a is
zero and every time it goes around then a increases by one. But before that
happens, b is set to a plus eleven and then comes the next for-loop which
then tests every age between a and b, which is then the mid sister. And if this
formula we came up with is true then the loops should stop. […]
The outer loop (Fig. 1) is thus used by the pair to systematically increase the value of
the variable a which concerns the age of the youngest sister. Within this loop the value
of b, the age of the oldest sister, can be calculated using a given relationship between
a and b. The inner loop is used to vary the variable concerning the age of the mid sister
and thus runs for integer values between a and b. Within the inner loop an IF statement
is used to test a mathematical condition within the task involving the ages of the sisters.
Figure 1: Screen shot visualizing the nested loops generated by Sophie and Richard
Based on the verbatim abstract above, several rules of action used by the pair could be
identified: (a) formulating the problem situation as amenable to solution through
exhaustive trial; (b) making use of programming to implement a solution strategy
based on exhaustive trial; (c) creating iterations through defining conditions for
loop(s); and (d) making use of the conditional operator IF to (i) evaluate given
conditions in order to (ii) perform different actions based on the validity of given
conditions. It could be argued that these rules of action are justified through several
concepts-in-action involving the ideas of (a) conducting an exhaustive trial; (b)
systematically combining variables; (c) establishing a loop relating to a variable; (d)
nesting loops (and statements within them) in order to achieve an appropriate sequence
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of variable-related actions; and e) using conditions within loops and conditional
operators in order to extract a solution within a given range. These concepts-in-action
form two theorems-in-action which guide the actions of the pair: (a) Systematically
combining variables serves as a means of achieving an exhaustive trial when more
than one variable is in play and (b) Establishing nested loops relating to the key
variables in play is a means of systematically combining these variables.
The ideas of students Emilia and Fredrik, on the other hand, are less developed. In
particular, they struggle to articulate – either orally or in code – how to systematically
combine variables. In the nested loops shown in Fig. 2, the outer WHILE loop is used
to check if the variable guldmynt_f (dependent on the sisters' ages, and recalculated
with each traversal of the loop) is less than or equal to 432 (a crucial value in the
problem). The inner FOR loop includes the same condition and initialises a control
variable i incremented on each traversal, which is then (mis)used within the loop,
apparently with the intent of systematically increasing the variable b, the age of one of
the sisters.
Figure 2: Screen shot visualizing the nested loops generated by Emilia and Fredrik
Fredrik then realizes that the other age variables a and f are never assigned new values
within the loops.
Fredrik: We should increase everyone? I think.
Emilia: No, but we just need... We just need to increase the age of Cinderella
(variable a) because the others increase automatically because we have
written a plus twelve there and a little something else as well.
Fredrik: Yes but... We still need to increase f.
Fredrik deletes the calculation of b in line 18 and inserts f = f + i instead. Later, Fredrik
returns discussion to the control variable i, which he relates to testing values associated
with an (unspecified) year and age:
Fredrik: But we need to find out what year it is... Because now they are zero years
old... That's why we have to test values all the time.
Emilia’s and Fredrik's way of coding indicates the use of concepts-in-action based on
the ideas of (a) conducting an exhaustive trial; (b) establishing a loop relating to a
variable; and (c) nesting loops (and statements within them) in order to achieve an
appropriate sequence of variable-related actions. But their failure to code loops which
will systematically increase the values of variables indicates that the pair, unlike Sophie
and Richard, have underdeveloped rules of action relating to how to use nested loops
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to combine variables in order to conduct the exhaustive trial. As a consequence, there
is also a lack of theorems-in-action guiding the pair’s actions.
A third pair of students, Christian and David, initially have a clear view about how to
use the programming environment as a mathematical tool.
Christian: Actually, you could set it up mathematically… Or... and then just use brute
force by testing lots of different combinations with the computer.
Although not made explicit by the verbatim extract above, the program structure (Fig.
3) later reveals the pair’s intention of conducting an exhaustive trial in order to solve
the given task. Unlike the other pairs, Christian and David also realize that they can
take advantage of the fact that the ages of the sisters must take integer values, and so
use this to establish an additional condition.
Christian: But yes, what we can do is just have a eh... WHILE TRUE and then we have...
And then we have... And then we check for that one so then you have like
eh... If this… and then you have like and-signs for... Eh.... And check if
something is INT (integers).
Figure 3: Screen shot visualizing the single loop generated by Christian and David
This additional condition allows the pair to create a program (Fig. 3), which only uses
a single loop involving three variables corresponding to the age of each sister. The
variable controlling the loop is As (age of the youngest sister), initialised as 0, and, at
the start of each iteration, increased by 1. Thus, the loop systematically examines what
happens as As increases from 1. Within the loop, the Be (age of the oldest sister) is
then specified as Be = As + 11 and the third variable Fu (the age of the mid sister) is
calculated as Fu = 432 / Be (although it should be noted that the underlying definitions
of the ages used in these two calculations are not compatible with each other). The IF-
statement in line 12 checks three conditions which need to be met in order for the
combination of ages to be a solution to the problem. The first two conditions check if
Fu is the mid sister and the third condition checks if the value of Fu holds an integer
value. If Fu is an integer, the difference between Fu and the rounded value of Fu equals
zero. If all the three conditions are met the program should print the ages of the sisters.
The verbatim extract together with the code generated by Christian and David expose
how the pair has used several different rules of action during the development of the
program: (a) formulating the problem situation as amenable to solution through
exhaustive trial; (b) making use of programming to implement a solution strategy
based on exhaustive trial; (c) creating an iteration through defining conditions for a
loop; and (d) making use of the conditional operator IF to (i) evaluate given conditions
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in order to (ii) perform different actions based on the validity of given conditions.
These rules of action are justified through several concepts-in-actions involving ideas
of (a) conducting an exhaustive trial; (b) computing linked variables when more than
one is in play; and (c) using conditions within loops and conditional operators in order
to extract solutions within a given range. These concepts-in-action are related to a
theorem-in-action used by Christian and David stating that: Computing linked
variables when more than one is in play serves as a means of reducing the number of
variables to be systematically varied in an exhaustive trial.
The examples given in this section are small extracts from students’ problem-solving
process when trying to solve a mathematical problem using a programming
environment as a mathematical tool. Yet, they illustrate different approaches used by
students in order to conduct an exhaustive trial and also difficulties relating to
conceptual and computational understanding. The way Sophie and Richard try to
conduct an exhaustive trial differ from the method used by Christian and David. This
is illustrated by the components comprising their developed schemes, although some
generic components relating to the problem-solving strategy are common. Emilia’s and
Fredrik’s scheme could be regarded as deficient since it lacks several essential
components relating to the use of nested loops, an action often perceived as
conceptually difficult for novice programmers (Mladenović, Boljat, & Žanko, 2018).
DISCUSSION
Following Buteau et al. (2019), we have explored approaches to operationalizing
Vergnaud’s (1998) components of a scheme when studying students’ instrumental
geneses. Using conversations between students (and between students and the teacher)
together with their generated code has made it possible for the researchers to extract
different components of schemes explicitly stated by students (or shown in their
program structure). This in turn has presented a possibility for the researchers to search
for similarities and differences within different schemes as well as analyzing which
components are missing from deficient schemes. This is illustrated by the example
where Emilia and Fredrik, just like the other two pairs, had begun to develop an
instrumented action scheme directed towards mathematical problem solving, involving
the use of exhaustive trial to solve the mathematical problem. But the lack of well-
functioning rules of action relating to the use of nested loops, in order to systematically
combine variables, hindered this pair to implement their problem-solving strategy. This
deficiency within their instrumented action scheme illustrates that Emilia and Fredrik
may have had under-developed pre-existing usage schemes directed towards the
artefact itself relating to the use of (nested) loops.
We also argue that defining specific components of the scheme based on conversations
between students should not be seen as a straightforward process. In the analytic and
iterative process there has to be a balance between defining components, on the one
hand, generic enough to be able to search for commonalities across cases, but on the
other hand still specific enough not to lose the key characteristics of each case.
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Since most of the data involves conversations between students during an ongoing
problem-solving process involving a particular situation, it cannot strictly be argued
that the findings provide evidence of “the invariant organization of behavior for a
certain class of situations” (Vergnaud, 1998, p. 167). But, at the least, the data shows
how, in solving the task, students generate proto-schemes or schemes-in-progress
which together with the artefact start the formation of an instrument. Indeed, this is no
more than the notion of a dynamic, constructive process of instrumental genesis.
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