Ann Kiefer

Mathematics is often perceived by pupils as an abstract discipline, disconnected from everyday reality. As a result, it is often seen as having little relevance to students’ lives outside the classroom. To remedy this perception, efforts are made to demonstrate the concrete applications of mathematics to real-world problems. However, these applications are often considered boring or out of context for teenagers’ lives (for example, adapting a recipe for four people to six, calculating interest rates, determining the height of buildings, etc.), or they are too complex for a secondary school student to really understand the underlying mathematical concepts (such as the mathematics of GPS or that used in artificial intelligence, for example).

In this module, we propose a mathematical application accessible to pupils: barcodes. The mathematics of barcodes is based mainly on the remainders of Euclidean divisions, a subject included in the Luxembourgish curriculum of 6e. This module is therefore an ideal lesson to practise Euclidean division, while at the same time discovering a practical application of a field that is often perceived as arid.

Although barcodes are not among the most fashionable technologies, they are omnipresent in our daily lives. We come across them every day. Furthermore, other items such as our bank accounts and cards, as well as our social security cards, are based on the same principle as barcodes. Even QR codes, although more complex, are based on a similar principle. For more information on this subject, we invite interested readers to consult Section 2.5: More on this topic.

Mathematics in information technology is invisible and therefore not real for teenagers, and people in general. The same applies to barcodes. They’re there, we scan them at the checkout counters, but who has ever wondered how and why they work? Coles, Barewell et al (2013) link this invisibility of mathematics and the ubiquity of technology with critical mathematics education:

The embedding of mathematics within information technology has a number of significant consequences. One is that mathematics is, in some sense, invisible. Interacting with software of IT systems does not generally give users access to the mathematical algorithms and models on which the system is based. This invisibility, of course, makes the system work more efficiently, but it also masks the role of mathematics and with it the human decisions that are made about which variables and parameters to include. Finally, information technology means that a mathematical ‘view’ of our world is deeply embedded in our society. Only things that can be measured and modelled can be included. A critical mathematical education, then, will include some attention to the formatting power of mathematics and the invisible role of mathematics in our lives.

The idea of critical mathematics education is not new and was developed by Skovsmose (1984, 1994, 2009). According to him, mathematics is not only a powerful means of interrogating the world around us, it is part of the fabric of our society (Coles et al., 2013).

Taking a real problem and using mathematics to find a solution is known as mathematical modelling. Conrad Wolfram (2020) has developed an entire curriculum around mathematical modelling, which is currently being implemented in schools in Estonia .¹

It is important to remember, however, that even in a lesson on mathematical applications, the focus remains on mathematical concepts. This is what Coles (2016) describes in his accomodation approach as follows:

As a teacher my focus remains firmly on the mathematics that students will learn. I use real-world examples whenever I can but the focus in each lesson perhaps quickly shifts away from the context and into a consideration of the underlying mathematics. The underlying mathematics have been determined by me in advance and is not open to question.

This is also the case in this module. Although the construction and understanding of the barcode mechanism is present throughout the module, the emphasis is on mathematics and, above all, mathematical thinking. If the ultimate aim was to explain how the EAN-13 barcode works, a 45-minute lesson would have sufficed: the mathematical formula behind the barcode is given and the students are left to work through a few examples. In this module, however, we’re trying to get students to understand the mathematical reasons behind the different ways in which a barcode works: why does it have a check digit, why is the check digit calculated from a sum rather than a product, why is Euclidean division by 10 used, etc.?

In conclusion, the barcode, although slightly old, is a perfect example of the mathematics hidden in our everyday lives, and leads to deeper reflections on the remainders of Euclidean divisions. This is also why the swiss mathematician Urs Stammbach (2006) chose this subject as the theme for his talk at the 17th Swiss Mathematics and Teaching Day (Schweizerischer Tag über Mathematik und Unterricht) at ETH Zurich.

1 See https://www.computerbasedmath.org/ for more information.

References :
1. Alf Coles, Richard Barwell, Tony Cotton, Jan Winter, Laurinda Brown. (2013). Teaching mathematics as if the planet matters. Routledge; 1st edition
2. Skovsmose, Ole. (1984). Mathematical Education and Democracy. Educational studies in mathematics 21: 109–128.
3. Skovsmose, Ole. (1994). Towards a Philosophy of Critical Mathematics Education. Dodrecht: Kluwer.
4. Skovsmose, Ole. (2009). In Doubt: About Language, Mathematics, Knowledge and Life-worlds. Rotterdam: Sense Publishers.
6. Stammbach, Urs. (2006). EAN, ISBN, CD, DVD: Von Prüfziffern zu fehlerkorrigierenden Codes. Schweizerischer Tag über Mathematik und Unterricht. ETH Zurich.
7. Wolfram, Conrad. (2020). THE MATH FIX. Kirkus Media LLC.