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Tiles as a visual aid in teaching mathematics

A unique portuguese legacy to the didatics of mathematics

September 29, 2014

Visuals supports such as diagrams or illustrations were and still are an important resource in learning mathematics because they facilitate the understanding of mathematical concepts, propositions, etc.

From a historical standpoint, Portugal has a unique legacy regarding visual resources used to teach mathematics. A unique collection of tiles was found, dating from the late seventeenth century or early eighteenth century, used once as a resource in teaching mathematics. As Professor Henrique Leitão (1) says , these mathematical tiles were the “Powerpoint” of the seventeenth century, and by being so, assumed a very diferent function than tiles usually had (and have): to decorate.

It is believed that the mathematical tiles were ordered by the Jesuits in following up the “Ordinances to encourage and promote the study of mathematics in the Lusitanian Province” written by Tirso González (1692), General of the Society of Jesus, in which he reinforced the need for improve the level of mathematics teaching in Lusitanian lands.

In this document, Tirso González recommends, particularly, the use of Tacquet’s version of The Elements, which It’s exactly the version that is used as reference for the production of diagrams of the tiles. According to António Leal Duarte, Professor at the University of Coimbra, the diagrams of the tiles are faithful reproductions of illustrations existing in Tacquet’s version (first published in 1654).

Tacquet’s edition of “The Elements” (source: here)

Among the various tiles of the collection, that once had more than 500 tiles, you can find:

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Tile with diagram illustrating Proposition 1, Book III: Given a circle to find its center. (source: here)



Captura de ecrã 2014-09-24, às 15.28.46
Tiles with diagram illustrating Proposition 20 (case 1), Book III: The angle at the center is double the angle BAC in BFC circumference with the same base BC (source: here)


Captura de ecrã 2014-09-24, às 15.30.40
Tile illustrating Lemma 2, Book V: If two quantities A and B have a common measure C, A will be added as many times as many C fits into B equal to B added as many times as many C fits into A. (source: here)


The researchers believe the mathematical tiles would be on the walls of the rooms of the Jesuit College at Coimbra, and that they were removed when the Society of Jesus was banned by order of the Marquis of Pombal.

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Coimbra Jesuit College (1732) (source: here)

The Colégio dos Nobres, created by the same Marquis, adopted R. Simon’s edition of the Elements and so did other institutions during the Pombaline Reform.  Thus, as the illustrations of R. Simon’s and Tacque’s versions were distinct, the tiles were no longer useful as a teaching resource.

Although the first public allusion to the existence of the tiles was is made by a national newspaper several decades ago (1980s), only in the recent years the collection of tiles became accessible to the general public. It is now exhibited at the National Tile Museum (Lisbon) and the National Museum Machado de Castro (Coimbra).




(1)  Auxiliary Investigator of theAutonomous Section of History and Philosophy of Science (SAHFC), FCUL. One of his research interests is the History of the Exact Sciences in the XV-XVII centuries




[2] Henrique Leitão & Samuel Gessner,’Euclid in tiles: the mathematical azulejos of the Jesuit college in Coimbra’, Mathematische Semesterberichte, 61.1, pp 1-5, 2014

[3] António Leal Duarte & Carlota Simões (ed.), Azulejos que Ensinam, Catálogo de Exposição, Universidade de Coimbra, 2007 (available here)


Written by Susana Pereira

Susana Simões Pereira, maths teacher and PhD in science teaching and communication. I enjoy games and photography and I'm passionate for science and art, specially when together in the same context. You can follow my updates on Twitter, LinkedIn and Facebook.