Few thoughts on math puzzles and computational problem-solving

Recently I read the book The Power of Computational Thinking: Games, Magic and Puzzles to Help You Become a Computational Thinker by Paul Curzon and Peter W. McOwan. I got me thinking about math, problem-solving, and computation. Not least what it takes for young children to grasp some of the concepts, that is involved in computational thinking.

Working computationally involves a basic understanding of symbol manipulation and modeling. This requires that the pupils become aware that e.g., coding and programming are subject to certain kinds of logic and terminologies. The pupils must learn special grammar (certain syntaxes) and think in specific ways before they can begin concrete computational work. For example, abilities within mathematical logic, decomposition, and pattern recognition are part of the prerequisites pupils must possess to work computationally.

These can be taught in many ways, but a well-known and well-tested method is to have pupils solve mathematical puzzles. One of many examples is Cut Hive Puzzles (, which in short, are about having a pattern (cubes) where some walls are marked with thicker lines. There are two rules:

  1. Each marked area must contain numbers from 1 to the maximum number of cubes in the marked area (4 in the example below).
  2. The same number may not appear within the same marking, and the same number may not appear in two cubes that are in contact with each other. 

By working with this type of task, the pupils learn basic principles that can later be transferred to computational methods, such as the fact that writing programs consist of certain logical rules and patterns.

Although computation historically has its origins in mathematics, CT is not limited to mathematical problem-solving. The pre-computational perspective considers that pupils do not necessarily have the necessary prerequisites to think computationally but that this, along with reading and arithmetic, must be trained and developed. In the CT literature, the abilities to code and program are often compared to literacy skills that are needed for complex mathematical problem-solving in the natural sciences and high-level literature analysis in the language subjects. Therefore, pupils need a basic understanding that precedes the computational.

CT embeds a perspective that is based on the pupils’ ability to think abstractly about phenomena in the world and translate these in ways that can be processed computationally and rule-based. With the Cut Hive example above in mind, this means going from being able to solve a puzzle to writing down rules for solving it.

A completely banal example is:

IF <a given input>
THEN <a specific action>.IF <a hexagon with area 2 contains the number 1 or 2>
THEN <the second hexagon contains the second number>

Writing down rules for programs includes abstractions, generalizations, and pattern recognition. Pupils must be able to exclude other parts of the puzzle to simplify and generalize while simultaneously comparing the generalized rules with the rest of the game’s rules (pattern recognition).

The example here shows how the abstract perspective is fundamental in CT.

What distinguishes this approach to problem-solving in school from others is that, in most cases, it is a question of thought processes that seek to reduce the complexity and interpretation possibilities of phenomena in such a way that a machine (that cannot make decisions based on intuitive interpretations and emotions) can solve tasks that would be too difficult, or take too long for humans to complete.

Book Chapter, Publication

New publication on Multimodality

Today I received my copy of the new handbook “Erhvervsdidaktisk opslagsbog” [Roughly translated: “Handbook for Vocational School” Didactics].
Together with my good colleague Dorthe Carlsen, I wrote a piece on “Multimodality” and multimodal learning design.
The book contains 58 chapters from 44 authors covering the fundamental concepts and issues concerning the vocational education.
I hope you will enjoy reading it.

Book, Poster Presentation

CT for All?

Together with a good colleague – Stig Børsen Hansen – I presented a poster on “Computational Thinking in Language Arts”. The poster session was held online at the ISLS (International Society of the Learning Sciences) Annual Conference 2022.

Click on image to get the PDF

Stig Børsen Hansen also wrote a really nice piece on Philosophers of Technology unfolding the understandings of technology by philosophers Langdon Winner, Albert Borgmann, Karl Marx, Herbert Marcuse, Martin Heidegger, John Dewey, and Bruno Latour.

Book Chapter


Computational Literacy in Scenario-Based Teaching Practices

Today I contributed with a chapter for a new Danish handbook on Scenario-based Teaching.
The purpose of the chapter is to contribute to the field of scenario-based teaching and computational literacy as an emerging field in primary school. I do this by exemplifying a concrete and tested scenario fragment where the students solve an educational escape puzzle through analysis and manipulations of technologies. The activity forms the starting point for a close investigation of students’ computational problem-solving strategies, as a focal point in computational literacy. The analysis is based on a socio-cultural view seeing tangible artifacts as mediating tools for actions unfolding in the given practice. From here the chapter deals with computational issues and the significance of artifacts as mediators for scenario-based activities that have computer literacy as part of the subject domain. A specific emphasis here is on the opportunities for socio-epistemic negotiations, the tangible artifacts creates. On a more practical level, the article seeks to deliver a well-described example that can inspire practitioners and support pedagogical decisions when designing for computational literacy.

The book is in danish, and can be found here:

Empirical Research

Poster Session – Fablearn Europe 2022

This year I had the pleasure of attending Fablearn Europe with a poster presentation of my work. The presentation highlighted the potential of Educational Puzzles in Computer Science education. Drawing on some of my empirical work, I highlighted how tangible computational things, hold the potential for learning the basics of problem-solving, decomposition, and algorithmic thinking.

Click the image to get the PDF

Empirical Research

Design activity #3: Final prototypes

In today’s design activity, we finished the first prototype and readied it for the first testing in school. We will present the prototype to the teachers who have volunteered to test the game in their classrooms.

Decisions and discussions on materiality
Lene and I have been discussing the materialities of different solutions, and whether the first prototypes should be made in e.g., cardboard or wood, the consistency in the graphical expression of the single pieces of the game, the correlation between the analog and digital part of the game and how to balance the game to both constrain the students into a specific fictional genre and afford openness to let the student’s imagination and creativity flourish.

Our considerations on materialities are also based on which kind of feedback we want to get from the teachers. For us, this concerns the balance between presenting a prototype, that looks done but not as a completely finished design. We don’t want the teacher to be concerned with the time and work put into the prototypes and thereby eventually limiting their evaluation. our hypothesis here is, that if the teachers think we have too much ownership or put to much work into it, then that will limit their evaluation and honesty towards concerns or critique of the design.

Lene checking the quality of the pieces and preparing them for the final finish
Design Experiments, Empirical Research

Design activity #2: First prototyping

After sketching the raw ideas of the game, my colleague (Lene Illum Skov) and I went through the first initiating design workshop. The aim of the workshop was to get closer to which elements we wanted to include in the game and how these would serve a specific purpose in regard to supporting the students’ creation of interactive stories.

The approach chosen in the project borrows its methodology from the field of participatory design research as described by Spinnuzi (2005):

As the name implies, the approach is just as much about design—producing artifacts, systems, work organizations, and practical or tacit knowledge—as it is about research. In this methodology, design is research. That is, although participatory design draws on various research methods (such as ethnographic observations, interviews, analysis of artifacts, and sometimes protocol analysis), these methods are always used to iteratively construct the emerging design, which itself simultaneously constitutes and elicits the research results as co-interpreted by the designer-researchers and the participants who will use the design.

Lene cutting and glueing pieces of the story to fit with specific tokens of the game

With that in mind, the design workshop is part of the research and the conversation unfolding during that process of great value for the choice made.
During the workshop, some important choices were negotiated, e.g.:

1) How did we want the tokens and pieces of the game to look?

2) Which text parts would be the best to include, keeping the storyline as open as possible?

3) Which materials would suffice and be suited to the job?

4) Simplicity in the expression of the artwork

Playfulness and game activities provide what Eva Brandt calls “dream material” (Halse, Brandt, Clark & Binder, 2010) that supports participants in playing out different versions of futures and outcomes. The process of designing the game could be seen as such, providing both Lene and me with opportunities to see meaningfulness and purpose while being engaged in developing ideas and artifacts.

The final outcome of the day. Not finished – but one big step closer
Design Experiments, Empirical Research

Designing for Computational Literacy in L1

Design activity #1: First ideas on designing a game for computational thinking

The mock-up and ideas

The basic ideas.
Today I started on the basic idea of the game for fostering algorithmic thinking and pattern recognition within the subject area of danish as a mother tongue (L1). Many thoughts have come up, but the basic idea is to combine tangible objects with digital artifacts in different ways to support the creativity and thinking skills of the students when writing stories. The tangible object (computational things) will function as structuring resources scaffolding the students when they outline their stories. Right now, I want to use the student’s mobile phones as well by using QR codes to give them excerpts from an existing story, to create basic plotlines. Using tangible objects, the students can fill out the missing parts linking the plots. To do this, the students involve (to some extent) pattern recognition and algorithmic thinking (my hypothesis).

From analogue to digital artifacts
After the students have finished the basics of the story, we want them to transform their ideas into interactive stories with the tool called Twine.
The great thing about Twine is that it both works as a tool for transforming the student’s creative ideas into a story others can engage in and also gives us researchers an opportunity to follow the students when they “program” a story in a tool, that uses basic programming syntax to function. E.g. creating a link requires the student to write a small command: [[ text ]]. Again the student also needs to involve both algorithmic thinking and creating logical patterns to create a coherent story.

Collaboration as a way to investigate how students think!
The game is designed towards as a collaborative enterprise. We think, that the only way to investigate the problem-solving and computational thinking strategies the students put to use is by letting the work collaborate and communicate their design ideas and actions.

Next steps:
The next steps will be a lot of further idea generation and sketching. Much is to be considered and learned. Both in regards to different properties of materials and how to design and manufacture the desired idea in the final steps. Right now, designing in Adobe Illustrator, Inkscape, and the like seems to be the next step, as well as trying to work with different forms and game formats.

A glimpse of the first drafts