## Mondrian Blocks as a cognitive training tool

„The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together harmoniously. Beauty is the first test: there is no permanent place in this world for ugly mathematics (Godfrey Harold Hardy, A Mathematician’s Apology, London 1941). The Hungarian mathematician Paul Erdős had an imaginary book in which God wrote down the most beautiful mathematical proofs. When Erdős wanted to express his particular appreciation of proof, he shouted, „This one’s from the Book!”.

Mathematics is a basic tool of knowledge and science, while it is an art motivated by beauty, and an integral part of music, dance, fine arts, architecture – patterns, symmetry-asymmetry, golden ratio, polygons, polyhedrons, fractals. Education treats arts and mathematics separately, though mathematics and art both are based on spatial abilities and develop the brain in terms of the most important functions of human thinking: sensorimotor and executive functions, symbolic and abstract thinking.

It is no coincidence that spatial skills are the foundation for later successful cognitive abilities and learning (Lubinski, Benbow, 2006; Wai, Lubinski, Benbow, 2009; Freeman, Marginson, Tytler, 2019).

However, spatial-visual skills, and thus development and learning methods in this area, have still not gained their rightful place in teaching, although there are increasing attempts to do so. Typically, creative activities such as exploring the symmetry of tapestries, cutting geometric shapes, arranging mosaics, mirroring, rotating, and translating pictures link mathematics to the arts in teaching.

Artistic activities develop mathematical and creative thinking, most often with a focus on the creative side. However, teachers do not provide many opportunities for students to think creatively in the practice of teaching mathematics. This is largely due to the highly structured curriculum and math textbooks, which leave little room for such opportunities (Schoevers, Leseman, Kroesbergen, 2020).

The Mondrian Blocks Cognitive Training combines the benefits of mathematics and art by providing systematic developmental material through an art-based mathematical problem. Indeed, the famous, coloured rectangles of the Dutch painter Piet Mondrian form the basis of an interesting mathematical problem, a puzzle, which mathematicians are still trying to solve (e.g. Basen, 2016; Dalfó, Fiol, López, 2021). Mondrian’s mathematical problem consists in dividing a grid of size n x n into rectangles and squares such that the difference between the area of the largest and the area of the smallest rectangle is as small as possible.

The Mondrian Blocks Cognitive Training is based on an art-mathematical problem, and its developmental impact is manifold. Quantities are tangible, and palpable, through the use of rectangles in a mathematically meaningful context, while their place in the given space has to be found. Magnitude and direction relationships, estimation of dimensions, spatial orientation, spatial translation, and recognition of visual shapes are required, while the concept of number is imperceptibly shaped in the mind of a child. Mondrian Blocks tasks are solved through a series of meaningful trials, which in addition to the above develop risk-taking, error detection, failure tolerance, creativity, and critical thinking.

We hypothesize that the basic functions required for mathematics and reasoning can be developed and tested in at least three main cognitive domains by using Mondrian Blocks:

- Sensory-motor function – Spatial orientation; Spatial memory; Eye-hand coordination; Tactile processing; Sequential processing; Processing speed.
- Cold executive functions – Cognitive control; Cognitive flexibility; Working memory; Rotation (spatial working memory); Error detection; Performance monitoring.
- Hot executive functions – Emotional regulation; Reward processing; Delay discounting; Risky decision making.
- Mathematical Thinking – Reasoning; Number concept.

Development and testing are particularly important given the steep increase in the rate of children diagnosed with mathematical difficulties. Research by Agostini, Zoccolotti, and Casagrande (2022) has shown that children with mathematical difficulties are impaired in cognitive areas such as executive functions, attention, and processing speed. The Mondrian Blocks cognitive training may benefit these children more than the average.

The training and testing of relevant cognitive functions can and should be incorporated into models of education and development to identify

- the cognitive functions behind the solution of “Mondrian Blocks” tasks;
- the role of Mondrian Blocks in the development of relevant cognitive skills;
- individual differences which may exist behind effective task solving;
- the ways cognitive training could be the most effective.

References

Agostini F, Zoccolotti P, Casagrande M. (2022) Domain-General Cognitive Skills in Children with Mathematical Difficulties and Dyscalculia: A Systematic Review of the Literature. *Brain Sciences*. 12(2):239. https://doi.org/10.3390/brainsci12020239

Bassen H. (2016) Further insight into the Mondrian art problem.https://mathpickle.com/mondrian-art-puzzles-solutions/.

Dalfó, C., Fiol, M.A., López, N. (2021). New results for the Mondrian art problem. *Discret. Appl. Math., 293*, 64-73.

Freeman, B., Marginson, S., Tytler, R. (2019). An international view of STEM education. In: Sahin, A., Mohr-Schroeder, M. J. (eds.) STEM Education 2.0, Brill, 350–366 https://doi.org/10.1163/9789004405400_019

Lubinski, D., Benbow, C. P. (2006). Study of Mathematically Precocious Youth after 35 years: Uncovering antecedents for the development of math–science expertise. Perspectives on Psychological Science, 1, 316–345.

Schechter, B. (2000). My brain is open: The mathematical journeys of Paul Erdős. New York: Simon & Schuster. pp. 70–71. ISBN 0-684-85980-7.

Schoevers, E.M., Leseman, P.P.M., Kroesbergen, E.H. (2020). Enriching Mathematics Education with Visual Arts: Effects on Elementary School Students’ Ability in Geometry and Visual Arts. *Int J of Sci and Math Educ* **18, **1613–1634 https://doi.org/10.1007/s10763-019-10018-z

Wai, J., Lubinski, D., Benbow, C. P. (2009). Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance. Journal of Educational Psychology, 101, 817–835. https://doi.org/10.1037/a0016127