From policy to resources: programming, computational thinking and mathematics in the Danish curriculum

Abstract

This article investigates the relations between mathematics and programming and computational thinking (PCT). In scholarly knowledge, PCT is juxtaposed as an aid to mathematical problem solving, but also integrated as a collection of practices common to both domains. This knowledge is being turned into curriculum and teaching resources developed for Danish compulsory schools (students aged 6–15), in the context of a pilot project to embed a new technology comprehension subject into mathematics. In the curriculum, PCT is being juxtaposed to mathematics. The teaching resources are predominantly integrated, but lacking connections to mathematical problem solving and modelling. These misalignments are both missed opportunities and a leeway for a cautious integration in teaching practice.

Full text

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Elicer, R. & Tamborg, A. L. (2023). From policy to resources: programming, computational
thinking and mathematics in the Danish curriculum. Nordic Studies in Mathematics Education,
28 (3-4), 221–246.
From policy to resources:
programming, computational
thinking and mathematics in
the Danish curriculum
     
This article investigates the relations between mathematics and programming and
computational thinking (PCT). In scholarly knowledge, PCT is juxtaposed as an aid
to mathematical problem solving, but also integrated as a collection of practices
common to both domains. This knowledge is being turned into curriculum and
teaching resources developed for Danish compulsory schools (students aged 6–15),
in the context of a pilot project to embed a new technology comprehension subject
into mathematics. In the curriculum, PCT is being juxtaposed to mathematics. The
teaching resources are predominantly integ rated, but lacking connections to mathe -
matical problem solving and modelling. These misalignments are both missed
opportunities and a leeway for a cautious integration in teaching practice.
In recent years, we have seen a renewed international interest in how
programming and computational thinking (PCT) can be converted into
teachable competencies across primary, secondary and tertiary educa-
tion levels (Bocconi, Chioccariello, Kampylis et al., ). In many coun-
tries, this has led to curriculum revisions, in which PCT either has been
implemented as a new independent subject (e.g. the case of England;
Department of Education, ) or as integrated into other subjects (e.g.
the case of Sweden, see Heintz et al., ). When PCT is integrated into
existing subjects, mathematics is often argued to be a relevant site to
which these new PCT components should be added, because computer
science and mathematics have shared foundations in logic and proof, and
have formal means to model situations from other disciplines (Modeste,
; Gadani-dis et al., ). Moreover, when computational thinking
was introduced by Papert (), it was mainly described as a new and
efficient way for students to learn mathematics.
Raimundo Elicer, University of Copenhagen
Andreas Lindenskov Tamborg, University of Copenhagen

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There are, however, substantial differences in how PCT is integrated into
mathematics across different Nordic countries (Bocconi, Chioccariello
& Earp, ; Helenius & Misfeldt, ). This is evident in terms of
whether there is, in fact, a relation between PCT and mathematics in cur-
riculum revisions, what PCT and mathematical content are connected,
and to what extent such links are explicit (Misfeldt et al., ). More-
over, research shows that teaching materials do not necessarily comply
one-to-one with what is stated in the curriculum (Bråting & Kilhamn,
). The envisioned integrations between mathematics and PCT as
they are described in both curriculum documents and in teaching mate-
rials are important to understand since they are likely to play a significant
role in mathematics teachers’ practice. This is particularly the case since
PCT is new and unfamiliar to many mathematics teachers (e.g. Misfeldt
et al., ; Nordby et al., ). Previous studies have either focused on
describing the relationship between PCT and mathematics at the minis-
terial guidelines level (e.g. Helenius & Misfeldt, ; Misfeldt et al., )
or in curriculum material developed to support teachers in adopting new
PCT elements in the mathematics curriculum (e.g. Bråting & Kilhamn,
). In this paper, we investigate both the prescribed relation between
PCT and mathematics at the curricular level in the context of Denmark.
Our outset is a recent pilot project in which a new subject called tech-
nology comprehension (TC) was implemented at  schools across the
country. This pilot project involved the development of a new mathema-
tics curriculum to include TC, as well as a number of teaching resources
to support mathematics teachers in teaching the new TC components
as part of mathematics. It is this curriculum and the teaching materials
that constitute the empirical basis of this study. However, we limit our
analysis to investigating the envisioned relations between the TC com-
petency areas of modelling, programming, and data, algorithms and struc-
turing, thereby leaving out digital design, and design processes; and user
studies, and redesign. Our reason for this choice is that these two latter
PCT competency areas are unusual components in an international
context, which have no immediate relation to mathematics.
We analyse these data from the perspective of the anthropological
theory of the didactic (ATD). More specifically, we draw on the notion
of external didactical transposition to describe the new TC curricu-
lum and the mathematical curriculum resources developed to integrate
mathematics and TC. We begin the paper by briefly outlining existing
research on the relation between PCT and mathematics in curriculum
documents and resources, and we position our contribution in relation to
this body of knowledge. Next, we describe our theoretical framework and
the empirical foundation in greater detail before we present the results

Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
from policy to resources
223
of our analysis. We conclude the paper by pinpointing the interrelation
between mathematics and PCT in the curriculum and the tasks developed
to accompany the curriculum and by discussing potential implications.
Related Work
As stated above, this paper studies the relation between PCT and mathe-
matics in the Danish mathematics curriculum and teaching resources
developed to fit the curriculum. In recent years, we have begun to see an
increased interest in these matters in the international literature. The
literature in this area is however still sparse, since little time have passed
since such teaching resources have been published. In this section, we
begin by reviewing studies that address the scholarly aspects informing
PCT as part of mathematics education curricula and resources. Next, we
review literature that specifically studies the relation between mathe-
matics and PCT from an ATD perspective, since this is the theoretical
approach on which we draw in this paper. We conclude the section by
situating the contribution of this paper in relation to existing work.
A recent study that have investigated curriculum material that inte-
grate PCT and mathematics is that of Bråting and Kilhamn (). Their
study concerned Swedish mathematics textbooks and set out to investi-
gate programming and mathematics concepts and programming actions
in  tasks. Although these authors exclusively studied textbook mate-
rial, they noted a disconnect between how programming in mathematics
is described in the curriculum and how it is integrated into the tasks. To
”follow a procedure” is by far the most frequent while creating algorithms
is at the core of the curriculum for algebra in grades – and – and of
problem solving for grades –. Tasks more often connect to geometry
or arithmetic (Bråting & Kilhamn, ), even though programming
is described as part of algebra (Swedish National Agency of Education,
). Furthermore, Bråting and Kilhamn () suggest that curricu-
lum materials do not necessarily comply fully with the curriculum. Our
research questions aim at closing these gaps and shedding light on how
policy decisions manifest in concrete materials. Since few mathematics
teachers are trained to teach PCT, these resources are likely to play a
significant role in mathematics teaching.
Another branch of research in this area has set out to study the relation
between mathematics and PCT in mathematics curricula or resources
from a theoretical framework similar to that of this paper, namely that
of ATD. Because of the relative novelty of the inclusion of PCT in school
mathematics, these are, however, recent and few. Modeste () reports
a study drawing on ATD’s praxeologies and the external didactical

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transposition. France had included informatics or computer science in dif-
ferent ways since the s (Gueudet et al., ), which allowed Modeste
to conduct his analysis with a historical sensibility. One key contribution
of this study is to draw upon specific epistemological consideration of
the relationship between mathematics and computer science (Modeste,
), namely: () foundations; logic and proof; () continuity and inter-
faces; () computer-assisted mathematics and experimental dimensions;
() modelling, simulation, and relation with other disciplines.
There are a few comparative studies on the didactical transpositions
of making programming or algorithmics part of the official curriculum
and accompanying resources, following an ATD approach. Modeste and
Rafalska () illustrate the influence of institutions, traditions and
history in the inclusion of algorithmics in France and Ukraine. In the
former, algorithms are part of mathematics, whereas, in the latter, it is
a subject in itself. Helenius and Misfeldt () show that the Swedish
revisions of the mathematics curriculum mainly concerned a transposi-
tion of programming into mathematics. In contrast, the Danish trans-
position concerned a much broader area of technology comprehension,
drawing on knowledge from not only computer science but also sociology
and ethics. Their study also shows differences in the extent to which cur-
ricula in the two countries specify an explicit relation between program-
ming/TC and mathematics. These studies also include analyses of the
internal didactical transposition. The analyses at this level mainly func-
tion as exemplary tasks that illustrate the two different external didac-
tical transpositions. The benefit of this is that they are able to display
how different curricular priorities and approaches manifest in concrete
implications for curriculum materials.
Building from Helenius and Misfeldt (), Tamborg et al. ()
took a closer look at the external didactical transposition of PCT into
the Danish mathematics curricular guidelines. Their study takes a more
comprehensive and structural approach, revealing that transposed TC
knowledge has no immediate relation to definitions in the literature.
More importantly, TC competencies are barely juxtaposed to mathe-
matics, leaving most of the responsibility for integrating these domains
coherently to teachers and material developers. We contribute to this
work by systematically investigating both curriculum, tasks and the
relation between mathematics and PCT in these data sources.
Theoretical Framework
ATD has been developed as an epistemological perspective of the teach-
ing and learning of mathematics (Chevallard & Sensevy, ). It focuses

Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
from policy to resources
225
on the nature of mathematical knowledge regarded as a human activity,
and how it is disseminated and taught (Bosch & Gascón, ).
Although ATD commonly addresses the nature of mathematical
knowledge within the discipline, many studies have taken a bi-dis-
ciplinary scope, such as the combination of mathematics and history
(Hansen & Winsløw, ) and mathematical modelling in scientific
contexts (Jessen & Kjeldsen, ) and engineering problems (Schmidt
& Winsløw, ). Addressing a diversity of domains calls for specific
attention to the level of coordination between the elements from dif
-
ferent sources. The aforementioned coordination of multidisciplinary
elements can happen on different institutional levels, be they academia,
policymakers, teachers and students (Schmidt & Winsløw, ).
On those grounds, the theoretical element we draw from ATD is
the didactic transposition (Bosch & Gascón, ), which refers to the
processes by which scholarly knowledge is transformed into something
taught and learned. Two main such processes can be identified: the exter-
nal didactic transposition, from scholarly knowledge to that which is to
be taught, and on the internal didactic transposition, from knowledge
to be taught to teaching practice. This framing ”underlines the institu-
tional relativity of knowledge” (Bosch and Gascón, , p. ), acknow-
ledging that these processes are determined by institutions. Scholarly
knowledge is determined mostly by academic and professional institu-
tions who produce and use the knowledge. The knowledge to be taught is
determined by the myriad of institutions that configure the educational
system and society or noosphere. The knowledge that is actually taught
and learned is determined by particular classrooms and communities of
study.
At the time of writing this article, TC has not been not implemented
in the compulsory Danish educational system, and therefore, little can
be said about classroom practice. Our study focuses on knowledge to be
taught as depicted in the new curriculum and accompanying teaching
resources. We thus focus on the external didactic transposition to address
the following research questions.
RQ: What is the relation between mathematics and PCT in the
external didactical transposition when focusing on the newly
developed curriculum to integrate TC and mathematics?
RQ: What is the relation between mathematics and PCT in the
external transposition when focusing on the teaching resources
developed to accompany the newly developed curriculum?

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Method
Addressing the research questions requires scrutinising the interplay
between PCT and mathematics as part of scholarly knowledge and com-
paring it with how Danish curricular guidelines and available teaching
materials go about it. As a point of departure, we mirror the approach
taken by Jessen and Kjeldsen (). To study the external didactic trans-
position of mathematical modelling in scientific contexts, they applied
the same categories to analysed selected examples from th-century
scientific developments, ministerial guidelines excerpts, teaching mate-
rials and exam tasks. In our study, we aimed for a systematic approach to
describe the knowledge at these levels. Below, we describe the data sources
on which we draw and how we approached the analyses of these sources
through the lens of two categories: the extent of integration or juxta-
position of PCT and mathematical knowledge, and a characterisation
of integrated subject-matter areas from each domain, when applicable.
Data Sources
In their study, Smith et al. () describe how TC is the result of a partici-
patory endeavour involving a broad diversity of societal actors, with their
corresponding perspectives and agendas. Helenius and Misfeldt ()
point out; in particular, t hat in D enmark as it is in Sweden the integration
of PCT into mathematics requires the mobilisation of knowledge from
several domains, including the humanities, the technological industry
and computer science, among others. Therefore, elements of PCT cannot
be mapped directly from computer science as a scholarly discipline, let
alone when intertwined with mathematics as a domain of knowledge.
Our basis for portraying scholarly knowledge relies on research that
has dealt with this problem, by investigating the interplay between
PCT and mathematics in industry and academia. Specifically, we draw
on one recent and prominent study by Kallia et al. (), who aimed at
characterising computational thinking in mathematics education. This
choice is without loss of generality, since they first conducted a systematic
literature review selecting  papers for full screening. This p ool includes
other prominent studies that, aside from their own literature reviews,
have relied on a variety of documents released by educational policy-
makers and appropriate professional organisations (e.g. Pérez, ), and
observations and interviews with mathematicia ns, computer and natural
scientists (e.g. Weintrop et al., ).
Aside from the literature review, Kallia et al. () refined and
validated their characterisation through a Delphi study (Vernon, ),

Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
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227
in which experts in mathematics and computer science answered three
open-ended questions:
() What characterises computational thinking in mathematics
education?
() What are the common aspects of computational thinking and
mathematical thinking?
() Which aspects of computational thinking can be addressed in
mathematics instruction? (Kallia et al., , pp. –)
We narrow down to the insights gained from the first two questions,
which aim at illustrating how PCT is currently part of mathematics edu-
cation () and its commonalities with mathematical thinking (). Ques-
tion  is more aspirational and is out of the scope of scholarly knowledge.
Within the knowledge determined by the noosphere, we draw upon a
revised mathematics curriculum and curriculum materials developed as
part of a pilot project in Denmark that implemented the new TC cur-
riculum in Danish schools. The pilot project was launched in  by the
Danish Ministry of Education, in which  schools across the country
were to implement this new subject. The pilot project sought to gain
initial experiences with two different models of implementing TC to
systematically research the effects of these approaches and, ultimately,
inform a future, national-scale curriculum revision (BUVM, b). The
two strategies were ) technology comprehension as a subjec t and ) tech-
nology comprehension as an integrated part of existing subjects, such as
Danish, mathematics, social sciences, science, physics/chemistry, craft
and design, and the arts (BUVM, ). Both implementation strategies
began with a curriculum, which was developed as an initial part of the
project. Moreover, it was decided that both implementation strategies
should address the same curriculum components and that the curriculum
for a subject in its own right should be developed first (Tamborg, ).
Afterwards, the individual components of this curriculum were then
distributed among the subjects into which it should be integrated. This
curriculum was developed by an expert writing group, which developed
a purpose declaration for technology comprehension and described four
competency areas that it should address. These four competency areas
were as follows:
Digital Empowerment: the critical and constructive exploration and
analysis of how technology is imbued with values and intentions,
and how it shapes our lives.

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Digital Design and Design Processes: the ability to frame problems
within a complex problem area and, through iterative processes,
generate new ideas that can be transformed into form and content
in interactive prototypes.
Computational Thinking: the ability to translate a complex problem
into a possible digital solution, and the abstraction of phenomena
and relationships in the world and the computer’s ability to process
this information.
Technological Agency: the ability to understand and use digital
technology as a material for developing digital artefacts.
(Smith et al., , p. )
For RQ, we analyse the official curriculum published for TC embedded
in mathematics. This curriculum includes the four competence areas
described above, further defined by – subject matter areas presented
as pairs of skillsets and knowledge. These competency areas were then
added to the otherwise maintained mathematics curriculum. The pre-
vious mathematics curriculum was organised as a description of com-
petence goals organised into four competence areas: () mathematical
competencies (see Niss & Højgaard, ); and subject-matter areas, ()
numbers and algebra, () geometry and measure, and () probability and
statistics. In the experimental version of the curriculum, TC was added
as a fifth competence area (see table ).
Competence area Skillset and knowledge goals
Mathematical competencies Description of six competencies: problem
handling; modelling; reasoning and thinking;
representation and symbol treatment;
communication; aids and tools
Algebra and numbers Description of five areas: numbers; calcula-
tion strategies; equations; formulas and
algebraic expressions; functions
Geometry and measurements Description of three areas: geometric proper-
ties and relationships; geometric sketching;
placement and movements; measurement
Statistics and probability Description of two areas: statistics,
probability
Technology comprehension Description of six areas of comprehension:
digital design and design processes; model-
ling; programming; data, algorithms, and
structures; user studies and redesign;
computer systems
Table . Overview of how TC was added to the description of the mathematics
curriculum for Danish K–9

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In the case of the mathematics curriculum, six TC components were
integrated: digit al design and design processes; modelling; programming;
data, algorithms and structures; user studies and redesign; and computer
systems. This is illustrated in table , below.
As described, our analysis exclusively focuses on the PCT aspects of the
TC curriculum, leaving out digital design and design processes, user
studies and re-design, and computer systems.
To address RQ, we draw upon  teaching resources designed for
the pilot study in grades –. These resources have been developed to
support the integration of TC into mathematics teaching and are pub-
licly available online

. The materials include explicit declarations of what
mathematical competencies, mathematical subject matter areas and
TC components Skillset Knowledge
Digital design
and design pro-
cesses
The student can design
digital artefacts through an
iterative design process that
will benefit the individual,
the community and society.
The student has knowledge
about complex problem
solving and iterative design
processes.
Modelling The student can construct
and act on digital models of
the real world and assess the
range of the model.
The student has knowledge
about how models of the
real world can be used to
describe and treat this.
Programming The student can modify
and construct programs for
solving a given task.
The student has knowledge
about methods for stepwise
development of programs.
Data, algorithms,
and structures The student can recog-
nise and utilise patterns in
the structuring of data and
algorithms with a departure
point in specific problems.
The student has knowledge
about patterns in structur-
ing data and algorithms.
User studies and
redesign The student can plan and
carry out investigations of
users’ perspectives and appli-
cations of digital artefacts.
The student has knowledge
about users’ perspective
and application of digital
artefacts.
Computer
systems The student can assess dif-
ferent computer systems’
possibilities and limitations.
The student has knowledge
about how number systems,
encryption mechanisms,
and network protocols
affect the basic construc-
tion and mode of operation
of computers and networks.
Table . The six TC components related to the mathematics curriculum for Danish
K–9 (our translation from Danish)

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TC competencies and sub-areas they combine. A full overview of how
these competencies are combined throughout the sequences is available
in table .
Area Mathematics Technology Comprehension
Grade
Course title
Numbers a nd algebra
Geometry and measure
Statistics
Digital design and design
processes
Modelling
Programming
Data, algorithms, structuring
User studies and redesign
Computer systems
What can a robot do? • •
Polygons’ geometrical
properties • •
Design the class’s new
clock • •
Concept of chance • • •
Descriptive statistics with
help of aliens • • •
Design and re-design of
containers • • • •
Robots and trajectories • • •
Play yourself healthy • • • • •
Next steps w ith micro:bit • • • • • • •
Red ears in the common
room • • •
Can you play yourself
skilled in mathematics? • • • •
Safety in the local com-
munity • • • • •
WEB  • • •
Packaging design a nd
development • • • •
Statistics with bias • •
Update dice • • •
Can I count on the
machine? • • •
How does a computer
”think”? • • •
Table . Overview of the available teaching materials and the content from
mathematics and TC they address

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The overview in table  shows how these courses aim at teaching
mathematics and TC, and eventually more than one sub-area of each
are involved, marked with dots. As the courses entitled ”Design the
class’s new clock”, ”WEB ” and ”Packaging design and development” do
not address PCT content – marked with grey shading – , these are not
included in our analysis. Consequently, our analysis of tasks concerns
 didactical sequences. The sequences are structured into three phases:
introduction, construction and challenges, and outro. Within each phase,
activities take the form of feedback and subject (faglige in Danish) loops
(BUVM, a). However, the materials signpost tasks with different
names, such as activities and concrete challenges. We therefore desig-
nate the units of analysis as each task included in the  aforementioned
didactical sequences, for a total of  tasks.
Approach to analysis
As described above, we draw upon different data sources to address our
research questions that are different in nature. Accordingly, we processed
and analysed them differently. However, addressing the external didac-
tic transposition means investigating the coherence between knowledge
defined by different institutions using a common framework (Jessen &
Kjeldsen, ). Our framework consists of two overarching dimensions:
. Juxtaposition versus integration. This dimension follows the episte-
mological consideration of continuity and interfaces (Modeste, ),
that is, the fact that ”the frontier between Mathematics and Com-
puter Science is impossible to draw” (Modeste, , p. ). We aim
at laying out the extent to which mathematical and PCT
knowledge are displayed separated or combined.
. Co-occurrence of PCT and mathematics. Within the extent to which
mathematics and PCT are integrated, we take a deeper look into
how they do so. This dimension relates to two other epistemologi-
cal considerations (Modeste, ): computer assisted mathematics
and experimental dimensions, and modelling, simulation and relation
with other disciplines. These aspects account for more specific inter-
actions between computational and mathematical elements. That
is, we address which competencies and subject-matter areas from
each domain are most commonly integrated.
In what follows, we elaborate on our approach for doing so for each source.

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Literature analysis
Kallia et al. () reported aspects that characterise PCT in mathe-
matics education (question ) and that are common to mathematical
thinking (question ). They displayed the percentage of participants’
responses containing these aspects, and reduced the list based on a sta-
tistical analysis. For example, although some respondents mentioned the
use of digital technologies to solve problems, this aspect did not gene-
rate consensus, since many academics respondents view PCT as a way
of thinking detached from computer devices and put forward the value
of unplugged or analogue tasks (Caeli & Yadav, ; Li et al., ). We
first narrow down our analysis to aspects that reach consensus. From
there, our approach consisted of identifying the extent to which these
aspects are depicted in a well-defined divide between PCT and math-
ematics (juxtaposition) or not (integrated). For example, the abstrac-
tion aspect reaches a consensus as being part of mathematical and com-
putational thinking, which points to integration. At the same time, in
the bigger picture, abstraction is depicted as a computational thinking
process that aids mathematical problem solving, signalling juxtaposi-
tion. For the co-occurrence dimension, we inquired on which specific
elements from PCT and mathematics are combined when integrated. For
example, logical thinking appears as a common substantial element wh ich
may indicate the intersection between computer science and proposi-
tional logic. However, as Selby and Woollard () argue, logical think-
ing is seldom defined and broader than formal logic. In most cases, coding
co-occurrence is speculative rather than explicit in the study.
Curriculum analysis
Our analysis at the level of curricular guidelines concerns mainly how
the PCT components structurally relate to mathematics, building on our
previous work (Tamborg et al., ). We applied the two dimensions
of our analysis in two ways. First, we study how the newly added PCT
elements are structurally positioned in relation to the existing mathe-
matical content in the curriculum, that is, how PCT in the curriculum
goals relate to the remaining mathematical content in the curriculum.
This latter approach allowed us to infer whether PCT and mathematics
were juxtaposed or integrated. Second, we investigated how the added
PCT curriculum components are described in the curriculum, and how
these descriptions relate to mathematics. Here, we focus on the skillsets
and knowledge as they are described in the curriculum goals for pro-
gramming, modelling, data, algorithms and structuring and computer
systems. In particular, we study how the PCT goal descriptions relate

Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
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233
to mathematics to inform the external didactical transposition when
focusing on the curriculum. For example, the TC curriculum includes
the competency area data, algorithms and structures. Its associated skill-
set description specifies that students should be able to ”recognise and
utilise patterns in structuring of data and algorithms with a departure
point in specific problems”. We first identified that the goals for students’
skills were separate from the existing goals for students’ mathematical
skills, as oposed to revising mathematical skills to integrate the recog-
nition and use of such patterns. In the second dimension, we identified
that patterns, data and algorithms resembled mathematical content rep-
resented in other aspects of the existing mathematics curriculum such
as statistics.
Task analysis
To analyse the tasks, we expand the schemes developed by Bråting and
Kilhman () and Elicer and Tamborg (). These analyses involved
identifying and distinguishing between mathematical and PCT concepts
and actions. Respectively, these categories account for the know-what and
the know-how involved in the learning activities. The coding scheme of
the  tasks is as follows.
Mathematical and PCT concepts are coded as depicted explicitly in the
tasks. Implicit or potentially exploit able ideas are not coded. For example,
the course ”Red ears in the common room” declares the algebraic concept
”variable” in the briefing and the evaluation. However, no signposted
task that students ought to face makes mention of it, and thus it is not
coded. We identify mathematical concepts as those belonging to tradi-
tional school mathematics, whereas PCT concepts are those that convey
a computational idea. In case of ambiguity, the context of the task pro-
vides guidance. For example, the concept of ”model ” app ears in the course
”Design and redesign of containers”, not as a mathematical model but as
a physical D-printed prototype and as a computer (TinkerCad) model.
In that context, it is then coded only as a PCT concept. A concept can be
coded in both domains, as is often the case with ”data”.
PCT actions are mapped into the follow ing categories: follow (stepwise
instructions), figure out (a pattern), debug (or fix), program (or create),
explain, envisage (or predict), and bridge (to mathematical concepts).
Bråting and Kilhamn () proposed this analytical tool by adapt-
ing it from Brennan and Resnick () and Benton et al. (, )
for task analysis of Swedish textbooks. In our coding process, we allow
PCT actions to be coded in a task even if no PCT concepts are, since, as
a construct or framework, PCT tends to be outwards oriented (Pérez,

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Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
234
). That is, PCT practices can be and are introduced for the sake of
disciplines other than computer science in itself (Weintrop et al., ).
To take advantage of the richness of these data, we add mathematical
competencies as declared in the courses to the coding scheme, drawing
directly from the curriculum: problem handling, modelling, reasoning
and thinking, representation and symbolism, communication, and tools
and aids. However, we only code a mathematical competency as being
activated when mathematical concepts are involved. Niss and Højgaard
() emphasise that mathematical competencies refer to different sorts
of mathematical situations, and thus this framework is inwards-oriented
(Pérez, ).
Once all the tasks were coded, we applied the two dimensions of our
the tasks in the following manner. We counted the number of tasks
that represent only mathematical knowledge (at least one mathematical
concept and no PCT concepts nor actions) and only PCT knowledge (at
least one PCT concept or action with no mathematical concepts). These
account for the juxtaposition of domains. In contrast, we count how
many tasks include at least one mathematical concept and at least one
PCT concept or action. These represent integration.
Among the integrated tasks counted in the first aggregation, we count
the tasks where the integration is through PCT concepts, actions, or both.
We analyse further into the data and identify prominent combinations of
actions and concepts from both domains. For this, among the integrated
tasks we count the share of which are combined with PCT concepts and
actions. We conduct this aggregation according to mathematical subject-
matter areas (reported in table ) and competencies (reported in table )
to account for the co-occurrence dimension.
Findings and discussion
Scholarly knowledge: PCT aids mathematical problem solving
We focus on the consensual aspects of PCT stemming from the first two
questions of Kallia et al.’s () literature-based Delphi study. Questions
 and  concerned, respectively, PCT in mathematics education and the
commonalities between PCT and mathematical thinking. Regarding the
first dimension of analysis, we find the interplay to be a mixed case of
integration and juxtaposition.
The case for an integration is grounded on the many consensual
common aspects to both types of thinking (question ) that characterise
PCT in mathematics education (question ): decomposition, pattern rec-
ognition, algorithmic thinking, modelling, abstraction, logical thinking,

Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
from policy to resources
235
and structured problem solving. The integration implies that many cha-
racteristics of PCT are an organic part of mathematical activity. However,
for question , decomposition, pattern recognition, algorithmic thinking
and modelling are followed by the expression ”to solve a problem” (Kallia
et al., , p. ). In that sense, PCT is a set of appropriate thinking
processes that aid mathematical problem solving. Moreover, an aspect
only raised in question  is that of ”being able to transfer the solution of
a mathematical problem to other people or machines” (Kallia et al., ,
p. ). Therefore, PCT in mathematics is somewhat juxtaposed, in the
sense of being a toolkit subordinated to mathematical problem solving
as a goal, with the additional solution strategies are formulated in such a
way that can be unambiguously communicated.
As for the second dimension, a few aspects derived from the ques-
tions hint subject-matter areas from mathematics. In particular, abstrac-
tion, generalisation, algorithmic thinking and pattern recognition can be
linked to algebraic thinking (Bråting & Kilhamn, ). However, this
is not exclusive to algebra. For example, seen as the selection of relevant
elements of a problem, is a step of any modelling practice (Ejsing-Duun
et al., ), mathematical, computational or otherwise. The recogni-
tion of visual patterns and generalisation are part of the development
of geometrical abstraction (Burger & Shaughnessy, ), particularly in
tasks aided by programming (Benton et al., ). Furthermore, abstrac-
tion and pattern recognition are essential data practices that relate to
statistics (Weintrop et al., ). Overall, the commonalities between
CT and mathematical sub-areas are not explicitly identified in Kallia et
al.’s () study.
Curriculum: a juxtaposition of PCT and mathematics
The experimental version of the mathematical curriculum positioned
the new TC content as a fifth competence area in the mathematics cur-
riculum (see table ). This implies that the mathematical goals from the
previous curriculum were not revised. Rather, new non-mathematical
PCT elements were added in a juxtaposed manner to the existing mathe-
matical components of the curriculum. This implies t hat PC T competen-
cies and learning goals included in the mathematic s curriculum described
above are structurally disconnected from the existing mathematical
competencies and learning goals. This implies that the new curriculum
is to include PCT in mathematics by rather superficially coordinating it
to the existing mathematical components of the curriculum.
In spite of the structural juxtaposition, it is worth examining the
descriptions of competency areas from TC that were selected to become

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Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
236
part of the pilot version of the mathematics curriculum. The compe-
tency areas of modelling and programming are described in gener ic term s
without explicit relation to mathematics, nor to the existing mathemati-
cal competency areas in the curriculum. The curriculum goals for model-
ling state that students should have knowledge of how models can be used
to describe the world and be able to construct and act upon models of the
world and assess their domain (BUVM, ). Mathematical content is
thus not explicitly mentioned. Similarly, the programming goals define
that students should have knowledge about methods and stepwise deve-
lopment of programs and be able to modify and construct programs to
solve a given task; no mentioning of mathematical knowledge nor skills.
Moreover, despite the fact that the curriculum specifies that students
should be able to both model and program concrete tasks in defined
contexts, mathematical content is, however, neither mentioned as a
context for the t ypes of tasks to solve nor as a component of modifying or
constructing programs.
The goals for data, algorithms and structuring and computer systems
are, however, more explicitly related to mathematics. The goals for data,
algorithms and structuring specify that students should acquire know-
ledge about patterNs in the structuring of data and algorithms, and that
they should be able to recognise and use patterns in the structuring of
data and algorithms in concrete problems. In the Danish mathemat-
ics curriculum, patterns are both part of algebra (number patterns) and
geometry (in relation to placements and symmetry). In this sense, the
data, algorithms, and structuring both include knowledge and skills
that relate directly to mathematics. The curriculum does not, however,
explicitly state these relations, which are up to teachers to infer.
Overall, there are great differences among the three PCT competency
areas described above. As seen in the case of modelling and program-
ming, it is not a given that the PCT competency areas in mathematics
include mathematical knowledge and skills. As seen in computer systems,
neither is it a given that it is made explicit how mathematical concepts
are to be activated by PCT skills. This implies that an important step in
integrating PCT and mathematics in the mathematics classroom is to
connect PCT competencies with mathematical competencies as part of
the internal didactic transposition. Thus, we consider, at the external
didactic transposition, the relation PCT and mathematics as juxtaposed.
Teaching resources move towards integration
In the realm of PCT, the coding is relatively balanced. Seventeen
percent of the tasks contain modelling concepts,  % include program-
ming concepts, and  % make explicit concepts of data, algorithms and

Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
from policy to resources
237
structuring. As for PCT action, the least present is to envisage ( %),
while the most frequently coded is to explain ( %). These results do
not suggest any strong preference or bias in the inclusion of PCT in the
mathematics teaching resources.
The following results from the analysis help us characterise the
teaching resources more globally:
. Out of the  coded tasks,  are only mathematical, and  are
only PCT. In contrast, a total of  integrate both domains. All in
all, teaching resources characterise predominantly an integration
( %) of mathematics and PCT, as opposed to the result of
juxtapositions thereof ( %).
. As for the co-occurrence analysis, we summarise the results in
tables  and . We remind the reader that the coding scheme is not
exclusionary; concepts from more than one subject-matter area can
be coded in the same task, as well as mathematical competencies
and PCT actions. We have added, at the bottom of each of these
tables, the total of coded tasks per column. These numbers can help
contextualise reported frequencies therein.
Considering the totals in table , integrated tasks can be characteri-
sed by an underrepresentation of numbers and algebra, compared to a
balance between geometry and statistics. The PCT subject-matter area of
modelling tends to be the least integrated in all three mathematical
Numbers and
algebra Geometry and
measuring Statistics and
probability
Modelling  
Programming  
DataAlgStruc   
Follow  
Figure out  
Debug   
Program   
Explain  
Envisage   
Bridge  
Total coded   
Table . Integrated tasks with PCT concepts and actions by mathematical subject-
matter areas

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238
areas. Though the general coding indicates explaining as the preferred
PCT action, this is only reflected in statistics, while programming is the
preferred action when integrating PCT with numbers and geometry.
The content of table  reflects the nature of the initial coding. The
mathematical problem-handling competency has no presence and the
mathematical modelling, and representation and symbolism competencies
are underrepresented due to the absence of explicit mathematical con-
cepts in the tasks. For this reason, many frequencies reported in table 
are two small from which to draw robust quantitative claims.
Nevertheless, a few points can be highlighted. The tasks do not par-
ticularly exploit the relation between computer modelling and mathe-
matical modelling. In relative terms, the few tasks integrating mathe-
matical modelling do so with no preference towards computer modelling
concepts.
PCT is integrated mostly with the mathematical communication com-
petency. Explaining is, unsurprisingly, one of the most integrated PCT
actions ( tasks), though programming is the highest ( tasks). To follow
a procedure comes next ( tasks).
The mathematical tools and aids competency comes in second place
as the most integrated with PCT. This is somewhat counterintuitive,
considering that TC in Denmark is a response to the irruption of digital
Problem
handling
Modelling
Reason-
ing and
thinking
Represen-
tation and
symbolism
Commu-
nication Tools and
aids
Modelling      
Program-
ming     
DataAlg-
Struc     
Follow      
Figure out      
Debug      
Program      
Explain     
Envisage      
Bridge      
Total
coded    
Table . Integrated tasks with PCT concepts and actions by mathematical
competencies

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239
artefacts in everyday life (Tamborg, ). Contrary to the general
coding, it is combined with more modelling concepts than data, struc-
tures and algorithms. Moreover, to program and to bridge are more often
combined with this mathematical competency than the general coding
that prefers explaining.
Concluding remarks
In this article, we aimed at understanding and characterising the inclu-
sion of PCT into the mathematics curriculum and corresponding
teaching resources in the external didactic transposition.
Our characterisation of this interplay in scholarly knowledge is based
on Kallia et al.’s () literature-based Delphi study. PCT is juxtaposed
to mathematics in the sense of being a collection of thinking processes
at the service of mathematical problem solving, with the particularity
of being able to communicate and transfer solution strategies to other
people or machines. To another extent, these domains are integrated,
in that the practices that aid problem solving are common to PCT and
mathematics. This integration, however, does not specify mathematical
subject-matter areas. Furthermore, it means that the historical and epis-
temological overlaps between mathematics and computer science as dis-
ciplines (Gadanidis, ; Modeste, ) do not coincide with the more
generic but subordinated connection between mathematics and PCT
as teaching-learning subjects (Kallia et al., ; Weintrop et al., ).
On this scholarly basis, we can now address our two research questions.
What is the relation between m athematics and PCT in the external didac -
tical transposition when focusing on the newly developed curriculum to inte-
grate TC and mathematics? PCT is part of the Danish national curricu-
lum, as part of the new subject TC in the form of three subject-matter
areas: modelling; data, algorithms and structur ing; and programming. These
areas have been added to the curriculum as part of a fifth competence
area, after mathematical competencies, algebra and numbers, geometry
and measuring, and statistics and probability. The descriptions of PCT
components suggest some relation to mathematics, in particular, data,
algorithms and str ucturing, given that computer science and mathematics
share several aspects (see e.g. Modeste, ; Bråting & Kilhamn, ),
and the associated skillset takes a point of departure in specific problems.
The inclusion results in a curriculum consisting of juxtaposed com-
ponents from PCT and mathematics, indicating that new mathematical
skills and knowledge are formed only to the extent that the TC compe-
tency areas include mathematics in themselves. They are, therefore, not
formed by integrating new PCT aspects into existing mathematical com-
petencies. This component of the external didactic transposition is here

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faithful to scholarly knowledge, to the extent of PCT and mathematics
being juxtaposed and the integration relying on commonalities between
them. This is illustrated by portraying modelling as a PCT competency
area. However, the main difference lies in that Kallia et al.’s () find-
ings position mathematical problems at the core of this relation, while
the pilot curriculum points to generic problems and makes no connect ion
to the mathematical problem-handling competency (Niss & Højgaard,
). Moreover, the curriculum does not mention the transferability and
reliability of solution strategies raised by scholars.
What is the relation between m athematics and PCT in the exter nal trans-
position when focusing on the teaching resources developed to accompany
the newly developed curriculum? The task analysis characterises teach-
ing resources predominantly as an integration of PCT into mathematics,
making this stage of the external didactic transposition rather conflicting.
Regarding subject-matter areas, this integration is biased against
numbers and algebra in the mathematical realm, and slightly against
modelling as a PCT area. In turn, algorithmic thinking, abstraction,
generalisation, pattern recognition – associated to algebraic thinking –
and modelling the are some of the common aspects between PCT and
mathematics highlighted in Kallia et al. ().
Concerning the PCT actions, explaining a code, procedure, pattern or
concept is the mo st integrated action, followed by programming, particu-
larly when combined with the communication, and tools and aids compe-
tencies. These two are the most integrated mathematical competencies.
These characteristics fit the particularity of PCT as an approach that
allows strategies to be unambiguously communicated and transferred to
other people or machines (Kallia et al., ).
One can spot some missed opportunities for this integration. For
example, the mathematical problem-handling competency is not declared
in any of the teaching materials. . The external didactic transposition
revealed in this study places the responsibility of developing meaning-
ful and coherent integrations on curriculum material developers and, in
part, in concrete teaching on the shoulders of teachers at the level of the
internal didactic transposition. According to the report on TC, Danish
teachers mostly rely on the supporting resources for such a new type of
subject (BUVM, b). Therefore, it is likely that teachers do integrate
PCT into mathematics (as opposed to juxtaposing it), but not subordi-
nated to problem-handling. We consider this misalignment problema-
tic since problem solving is considered the ultimate goal of mathema-
tics education where computational thinking is embedded in scholarly
knowledge (Pérez, ; Kallia et al., ). Our results show that the
interconnections between computer and mathematical modelling, as well

Nordic Studies in Mathematics Education, 28 (3-4), 221–246.
from policy to resources
241
as the handling of representations and symbolism from both domains, are
underexploited. However, these omissions may be interpreted as cau-
tious steps, since the apparent synonymity of concepts such as modelling
and function, and practices such as syntax and formalisms, can result in
more of a challenge than a tool in teaching practice (Bråting & Kilhamn,
). At a n international perspective, the English ScratchMaths project,
for example, aimed at bridging the separated computing and mathema-
tics syllabi, with no strong evidence of improving students’ mathematics
outcomes (Boylan et al., ). The Swedish Ifous project illustrated how
difficult it is for teachers to use PCT as a tool for mathematics learning
(Jahnke, ). Seen in this light, there is no evidence that a closer cor-
respondence between scholarly knowledge and knowledge to be taught
in fact will lead to improved mathematical learning on part of students.
This points to a need for further research on to what extent identified
potentials of PCT in mathematics education contexts can be exploited
and under which circumstances. That is, a necessary follow-up is
inquiring on the internal didactic transposition (Bosch & Gascón, ).
In spite that research on PCT in mathematics education has a long
history, we are currently not at a stage where the research foundation to
revise mathematics curricula to include are at hand. This calls for cau-
tious and, most likely, iterative approaches to implementing PCT in the
mathematical classroom, accompanied by ongoing research to study and
assess the relation between approaches and learning outcome.
Acknowledgement
This work is funded by the Novo Nordisk Foundation grant OC.
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 http://tekforsøget.dk/forlob

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Raimundo Elicer
Raimundo Elicer is a postdoctoral researcher at the Center for Digital
Education, Department of Science Education at University of Copen-
hagen. He is currently investigating the interplay between the teaching
and learning of mathematics and computational thinking. His earlier
research focused on critical issues of probability and statistics education.
reli@ind.ku.dk
Andreas Lindenskov Tamborg
Andreas Lindenskov Tamborg is assistant professor at the Center for
Digital Education, Department of Science Education at University of
Copenhagen. His research concerns implementation issues related to
computational thinking and digital education in the context of STEM
subject with as specific interest in mathematics education.
andreas_tamborg@ind.ku.dk