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In: Hodgen, J., Geraniou, E., Bolondi,G. & Ferretti, F. (Eds.) (2022). Proceedings of the Twelfth Congress of the Euro-
pean Society for Research in Mathematics Education CERME12 (pp. 1891–1894). Free University of Bozen and ERME.
Introduction to the papers and posters of TWG11: Algorithmics
Christof Weber1, Janka Medova2, Maryna Rafalska3, Ulrich Kortenkamp4, and Simon Modeste5
1University of Teacher Education Lucerne, Switzerland; christof.weber@phlu.ch
2Constantine the Philosopher University in Nitra, Slovakia; jmedova@ukf.sk
3University of Côte d’Azur, France; maryna.rafalska@univ-cotedazur.fr
4University of Potsdam, Germany; ulrich.kortenkamp@uni-potsdam.de
5University of Montpellier, France; simon.modeste@umontpellier.fr
Keywords: Algorithms, algorithmic thinking, teaching and learning of algorithms.
In CERME12, our working group “Algorithmics” started its work as a newly established TWG. Since
algorithms have always been at the heart of mathematics and their importance has been steadily in-
creasing since the beginnings of theoretical computer science, the design and analysis of algorithms
– called algorithmics (Traub 1964, Knuth 1985) – lies at the intersection of mathematics and com-
puter science. For this reason, on the one hand, various algorithms and algorithmic activities have
their traditional place in mathematics curricula at all levels. At the school level, mathematics and
computer science have interacted since the 1980s, when many schools set up labs with computers
equipped with programming software. On the other hand, many questions arise in the context of
teaching and learning algorithms: a first, more applied group of questions aims at algorithms in math-
ematics education and curricula, a second, more theoretical group of questions seeks to clarify the
concepts of algorithm and algorithmic thinking.
Conference presentations
Due to the Corona pandemic, the conference was held as a virtual event. Nevertheless, a total of 11
papers and 7 posters were presented remotely by their authors at the conference, with a total of 24
group participants from 11 countries. The contributions were considered in four themes, as follows.
Theme 1: Beliefs and domains in which algorithmic thinking occurs
A first group of papers focuses on the place and importance of algorithms in mathematics in general
and arithmetic in particular. They assess the beliefs of experts about the role of algorithms in mathe-
matics and mathematics education or their role in mathematics courses.
· Lockwood, DeJarnette, Thomas and Mørken offer three perspectives on algorithms, particularly
in computational settings: an algorithmic approach in a mathematical example, the view of a math-
ematician, and the view of an undergraduate students taking a course in mathematics.
· Geraniou and Hodgen interviewed two mathematics educators who had experience using technol-
ogy to solve mathematical problems, and they, too, shared very different views on algorithms in
mathematics education, one even not seeing the use of algorithms as a mathematical activity.
· Kortenkamp analyzes an arithmetic course for pre-service primary teachers. He identifies several
algorithmic activities in the topics covered in the course, such as designing algorithms, specifying
algorithms, performing algorithms, proving their correctness, and comparing algorithms.
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· Leifeld and Rezat’s poster provides a thorough analysis of the possibilities of certain arithmetic
algorithms for addition and subtraction to deepen students’ understanding of inverse operations.
Theme 2: Teaching and learning of algorithmic thinking at primary level
Another group of papers focuses on teaching and learning algorithmic thinking in primary school.
They use different tasks with different goals: Some use algorithmic thinking as a means to an end (in
the sense of learning a new mathematical concept), some use algorithmic thinking as a goal (in the
sense of understanding a given algorithm, or developing an algorithm to solve a problem):
· Crisci, dello Iacono and Ferrara Dentice explore how primary school children can be stimulated
to learn new mathematical concepts by Scratch. In their report, they present a specially designed
task that required visual programming to complete a given figure so that it becomes axially sym-
metric. The children developed two different strategies to solve this task. For example, they found
out that points that are axially symmetrical to each other must be equidistant from the axis.
· Funghi and Ramploud are interested in how to teach the standard long-division algorithm so that
children understand why it “works.” To this end, they had fourth graders compare the optimized,
digit-by-digit long-division procedure with the procedure in which the divisor is repeatedly sub-
tracted from the dividend. Their analysis of class discussions suggests that this approach could
actually result in less rote learning, but in more conceptual learning.
· Zindel’s study wants children to acquire algorithmic thinking without using computers. In her
papers, children are instructed to decrypt and encrypt certain words. They had to articulate the
necessary steps themselves and record them in writing. Although these texts show great differ-
ences, the author succeeds in reconstructing some constituents of algorithmic thinking.
· In Gaio’s study, too, children are asked to develop algorithms in the sense of systematic proce-
dures, without the help of computers. Here, the tasks given to children of different school levels
(3rd to 8th grade) are in the context of sorting problems. As the author reports, he can see traces
of classical algorithms in the procedures that the children have worked out cooperatively.
Theme 3: Teaching and learning of algorithmic thinking at university level
Concerning the development of algorithmic thinking at university, the discussions showed two big
issues: the development of algorithmic thinking and algorithmics in mathematics, for students, inde-
pendently of their projects, and more specifically, the development of algorithmic thinking in math-
ematics for future teachers, and in particular future primary teachers.
Four papers dealt with algorithmic thinking in advanced mathematics, three at university, and one
concerning an education program for gifted students. Above them, three were interested in links with
discrete mathematics, combinatorics, graph theory, which illustrates the specificity of those mathe-
matical fields, at the interface with computer science:
· De Chenne and Lockwood explore the use of programming and computer science in solving basic
counting/combinatorics tasks in college, and how the knowledge of student in computer science
can influence their solving strategies and support their learning.
· Medová, Milicic and Ludwig study the competencies involved in algorithmic thinking for univer-
sity, and in particular abstraction, modelling, and visualization skills which are difficult to master
for students, and questions the development of computational thinking in mathematics.
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· Bóra and Gosztonyi analyze the place given to algorithms and what could be seen as algorithmic
problem solving, in Hungary’s advanced mathematics programs, questioning what can be consid-
ered traditionally as algorithmic in mathematics and its place according to mathematical culture
of the country.
· The paper of Calor, Palha and Kubbe concerns at the same time advanced mathematics and pre-
service secondary teachers’ education. It deals with analysis, and in particular developing instruc-
tional material concerning differential equations for algorithmic thinking and programming. First
results show that students did indeed develop algorithmic thinking in their work.
The two other contributions dealt with algorithmic thinking in and for primary teacher training:
· Weber’s paper examines primary teachers’ use of loops to solve a geometrical problem and their
conceptions of the loop construct. It elaborates some challenges in their conceptions and some
misconceptions that require deepening their understanding from a teacher training perspective.
· Dobgenski and da Fontoura’s poster presents and reflects on an experience of making pre-service
primary teachers deal with computational thinking using Scratch.
Theme 4: Concepts related to algorithmic thinking: computational thinking, algebraic think-
ing, problem solving, and mathematical literacy
The last group of contributions deals with different, no less relevant aspects of algorithmic thinking:
· Rafalska’s paper illustrates how tasks could be constructed in order to lead children in mathematics
lessons (without the use of computers) to algorithmic thinking in the sense of developing a solution
strategy and which individual learning processes can be triggered by these tasks.
· Pohlkamp and Lengnink’s paper takes a different look at algorithms: It discusses algorithms that
make decisions and are thus socially relevant. Addressing and studying them in the classroom
would mean taking more seriously the educational mandate to teach social skills as well.
Finally, two poster proposals deal with two concepts related to algorithmic thinking:
· Rekstad and Rasmussen investigate the question to what extent aspects of computational thinking
mentioned in the literature are also reflected in teachers’ beliefs when asked about the role of
computational thinking in mathematics education.
· The relationship between algorithmic and algebraic thinking is the subject of Müller-Späth, who
plans to investigate how algorithmic thinking (realized by an app) affects the development of the
ability to generalize and thus of algebraic thinking.
Conference discussions
As mentioned earlier, our working group has just begun its work, and a common understanding of
the concepts has yet to be developed: What does algorithmics mean in the context of teaching and
learning mathematics? What is algorithmic thinking? To this end, after the presentations in which
quite different views were expressed, we worked on the following three questions:
Question 1: Which mathematical algorithms could stimulate algorithmic thinking?
The discussion of this question revealed relatively unanimously five mathematical types of algo-
rithms: i. Algorithms based on the place value system (standard algorithms for addition etc., algorithm
for calculating logarithms), ii. graph-theoretic algorithms (shortest path problem, Königsberg
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problem), iii. approximation algorithms (Heron’s algorithm, Newton’s method, etc.), iv. sorting al-
gorithms (heap sort, bubble sort, etc.), and v. miscellaneous (Gauss’s Easter algorithm, etc.). We were
not in agreement of whether each procedure is also an algorithm. For example, everyday procedures
(tying shoes, making jam sandwiches, etc.) were not considered by all participants to be suitable for
addressing and promote algorithmic thinking in its “proper sense” because they show only one char-
acteristic feature of algorithms: the order of steps.
Question 2: Which mathematical topics could promote algorithmic thinking?
The discussion of this somewhat broader question also yielded five topics from which tasks could
come to stimulate algorithmic thinking: i. number theory (arithmetic, prime number tests, factoriza-
tion), ii. discrete mathematics (graph theory, combinatorics, counting problems, etc.), iii. geometry
(transformations, algebraic geometry), iv. computer science (cryptography, etc.), and v. games and
puzzles (Rubik’s cube, tower of Hanoi, etc.). One participant’s question about what properties these
fields would have in common was discussed intensively and controversially.
Question 3: What (human) activities with algorithms can we think of?
The activities discussed suggest a wide range of possible activities to deal with algorithms: i. creating
(developing algorithms, improving algorithms, debugging algorithms, optimizing algorithms, trans-
ferring algorithms to an analogous situation, etc.), ii. analyzing (effectiveness and proof, efficiency,
complexity, stability, similarity etc.), and iii. comparing (comparing different algorithms for the same
problem, comparing analogous algorithms for different problems, classifying algorithms, etc.). Alt-
hough executing an algorithm without any reflection would be a possible activity with algorithms,
most participants do not want this to be understood as algorithmic thinking.
Surely the reader can think of further examples or answers to these questions. In other words, the
three questions need to be discussed further and their answers are still quite open.
Outlook
As the overview of the contributions as well as the first answers to central questions show, there is a
great variety of approaches (theories, methods) and views (topics, perspectives) in our working group.
Given that we are entering a young (or at least long-neglected) area of research in mathematics edu-
cation and that we have just begun work in our TWG, it was to be expected that the results would be
disparate and sometimes controversial. However, in terms of a first step towards a robust and sustain-
able understanding of concepts, this diversity makes us confident that there are many more questions
around the challenging topic of algorithms and algorithmic thinking that are worth working on. With
this in mind, we look forward to CERME13 and hope for a fruitful continuation of the work we have
begun – and that it can then be carried out again as a physical conference.
References
Knuth, D. E. (1985). Algorithmic thinking and mathematical thinking. The American Mathematical
Monthly, 92(3), 170–181. https://doi.org/10.1080/00029890.1985.11971572.
Traub, J. F. (1964). Iterative methods for the solution of equations. Prentice-Hall.
