ICMT2 will take place on five days and therefore it is planned to have one plenary lecture each day. The opening lecture will be given by the Brazilian speaker.

A Multimodal Approach for Theorising and Analysing Mathematics Textbooks
Kay O’Halloran
Multimodal analysis has emerged in recent decades as an interdisciplinary area of research, arising out of linguistics, semiotics and drawing on other relevant fields (in this case, mathematics) to study the contributions and interactions of linguistic and nonlinguistic resources (e.g. spoken and written language, image, gesture, sound, page layout and website design) in the communication of meaning (Jewitt, 2014; Jewitt, Bezemer, & O’Halloran, 2016). From this perspective, mathematical knowledge is constructed using language, images and mathematical symbolism which integrate in specific ways in mathematical texts (O’Halloran, 2015a, 2015b). In this talk, I provide an overview of the semiotic nature of each resource, including their underlying organisation that enables them to fulfil different functions in mathematics. Following this, I undertake an analysis of a mathematics text in order to demonstrate how linguistic, visual and symbolic choices combine to solve mathematics problems, and the expansions of meaning that take place during this process. I conclude with a discussion of the difficulties of teaching and learning mathematics (O’Halloran, 2000, 2015a) with specific reference to mathematics textbooks.
References
Jewitt, C. (Ed.). (2014). The Routledge Handbook of Multimodal Analysis (2nd ed.). London: Routledge.
Jewitt, C., Bezemer, J., & O’Halloran, K. L. (2016). Introducing Multimodality. London & New York: Routledge.
O’Halloran, K. L. (2000). Classroom Discourse in Mathematics: A Multisemiotic Analysis. Linguistics and Education, 10(3), 359388.
O’Halloran, K. L. (2015a). The Language of Learning Mathematics: A Multimodal Perspective. The Journal of Mathematical Behaviour, 40 Part A, 6374.
O’Halloran, K. L. (2015b). Mathematics as Multimodal Semiosis. In E. Davis & P. J. Davis (Eds.), Mathematics, Substance, and Surmise (pp. 287303). Berlin: Springer.
Kay O’Halloran is Professor in the School of Education at Curtin University, Perth Western Australia. Her area of research is multimodal analysis which is concerned with the meaning arising from the interaction of language and other resources in multimodal texts, interactions and events. She has a specific interest in mathematics discourse, and has explored the historical evolution of the semiotics of mathematics, the semantic expansions of meaning arising from the integration of language, image and symbolism in mathematical texts, and multimodal approaches to mathematics, grammar and literacy, with a focus on the difficulties of teaching and learning mathematics stemming from its multisemiotic nature.

Mathematics textbooks for the million – Brazil’s mathematics textbooks assessment program.
João Bosco Pitombeira de Carvalho
We will discuss Brazil’s mathematics textbooks assessment program, which selects the textbooks freely distributed by the Ministry of Education. This program distributed 6 million mathematics textbooks among high school students in 2015. We will analyse the methodology used in the assessment program, its legal framework, how the programme’s results are known by mathematics teacher in schools all over the country and the way they choose the textbooks they will use in their classrooms. Finally, we discuss the programme’s influence on the textbooks — their quality, contents and formatting.
João Bosco Pitombeira de Carvalho received his PhD degree from the University of Chicago and was on the faculty of the Mathematics Department of the Pontifícia Universidade Católica in Rio de Janeiro, from which he retired as Emeritus, after 40 years of service. His fields of interest are the history of mathematics education, particularly in Brazil, the relations between history of mathematics and mathematics education. He is especially interested in the history of mathematics textbooks. For the last 15 years he has been actively involved in the national assessment textbook program of the Ministry of Education in Brazil.

The diagrams of Euclid’s Elements.
Ken Saito
Although Euclid’s Elements are no longer directly used in school mathematics, Euclid’s style deeply permeates in the education of mathematics. Our use of diagrams in elementary geometry, especially the use of labelled points to designate geometrical objects such as triangle ABC, comes from the Elements.
The study of diagrams in Greek mathematical texts is rather a new field, for diagrams called the attention of scholars only after 2000. The first result of these recent researches was that the diagrams we find in the translations of the Elements available today, are quite different from those in the best extant manuscripts, the sources of the text of the Elements. The former all depend on the critical Greek edition of the Elements by Danish scholar Johan Ludvig Heiberg, published in 1880’s. So Heiberg was by no means critical for the diagrams in his critical edition.
In the manuscript diagrams, we often see isosceles triangles in a proposition valid for any triangle, and rectangles where any parallelogram is intended. This tendency may be called overspecification (an angle which can be right is often drawn as a right angle, and two sides which can be equal are drawn as if they were equal).
Another characteristic typical to manuscript diagrams, which is somewhat contradictory to what is just said, is that they are indifferent to metrical accuracy: two equal lines or equal areas do not always appear equal in the diagrams. Sometimes a right angle may be drawn as acute or obtuse.
Heiberg’s diagrams are always accurate, and avoid overspecification. For example, in the socalled Pythagorean theorem, proposition I.47, the triangle is rectangular but not isosceles in Heiberg, while the manuscripts draw it as an isosceles triangle, mostly also rectangular, but sometimes obtuseangled.
We know that Heiberg copied the diagrams of geometrical books of the Elements from an edition of 1820’s, edited by Ernst Ferdinand August, a Prussian gymnasium teacher. August, without access to manuscripts, systematically eliminated overspecifications and metrical inaccuracy in the diagrams of previous editions, apparently from pedagogical intention.
So today we read the geometry of the Elements accompanied by the diagrams intended for gymnasium students of 19^{th} century.
As for the arithmetical books of the Elements, which are also accompanied by diagrams, the situation I have found is different from geometry, and is no less interesting. In short, Heiberg invented himself all the diagrams for arithmetic, but this was a disaster. They are much less helpful for the understanding of the text than those found in the manuscripts.
In my lecture, I will compare the diagrams of the Elementsin the manuscripts, in earlier editions, in Heiberg’s edition, and examine their expected roles in different periods.
Ken Saito is professor for history of mathematics at the Osaka Prefecture University since 1997. His main research interest is Greek mathematics. The method adopted in his research is to analyse the mathematical argument of each step of the proofs in the extant textbooks, and try to find out the ideas, which led the ancient mathematicians to write those proofs in their specific manner. The trend of research into Greek mathematics has considerably changed over the last decades. The mathematical texts, such as Euclid’s Elements, used to be read as if they were wholly genuine. Now scholars are more conscious of changes of text by later intervention; diagrams are a revealing indicator. Following this trend, he is undertaking syntactical analyses of the Greek text of Euclid’s Elements to identify the expressions typical of (or more frequent in) later commentators. A particularly innovative aspect of his research is to study the role of diagrams in the transmission of Greek textbooks; their high degree of variance reveals the relation between text and diagram as a key dimension of textbook analysis.

Electronic vs. Paper Textbook Presentations of Various Aspects of Mathematics
Zalman Usiskin
Textbooks now appear in paper form, in electronic form for computers, tablets, or phones, and in hybrids moving among the platforms. Whether an adaptation of a preexisting paper text or an entirely new book, some aspects of mathematics seem better suited for traditional paper book presentations while others seem better suited for certain kinds of electronic formats. Based in large part on our current work in adapting our own existing paper textbooks for secondary schools for several digital formats, this talk discusses the platforms with regard to the presentation of various aspects of mathematics, including: vocabulary and notation, deduction, modelling, algorithms, and representations.
Zalman Usiskin is a professor emeritus of education at the University of Chicago, where he was an active faculty member from 1969 through 2007. He continues at the university as the overall director of the University of Chicago School Mathematics Project, a position he has held since 1987. His research has focused on the teaching and learning of arithmetic, algebra, and geometry, with particular attention to applications of mathematics at all levels and the use of transformations and related concepts in geometry, algebra, and statistics. He has directed the writing (student and teacher materials) and development (from pilot through largescale classroom tests) of multiple editions of textbooks for all the secondary school grades known for their innovations in geometry, applications and modelling, and the use of the latest in technology.

An International Comparison on Selection of Contents and Difficulty Level of Examples in High School Mathematics Textbooks
Jianpan Wang
Based on our research project “A Comparative Study on High School Mathematics Textbooks of Certain Countries” (a China’s national key research project in educational sciences), this talk focuses on the following two aspects:
 On the selection of contents. The selection and presentation of contents has fundamental importance for mathematics textbooks. In this relation, the presentation will provide an analysis of the features of content selection in the textbooks of different countries with focus on four major areas of mathematics, i.e., algebra (including functions), geometry, probability and statistics, and calculus, illustrate (with tables) the similarities and differences of different countries in the selection of contents, and delineate (with examples) different ways of presenting a number of mathematics contents in the textbooks of different countries.
 On the difficulty level of mathematical examples. Mathematical examples are an important component of mathematics textbooks and their difficulty level largely reflects the expectation of the textbooks for the students. The presentation will provide a comparison of the difficulty level of mathematics examples provided in the selected textbooks from different countries, which is carried out in five dimensions including background level, mathematics cognitive level, reasoning level, operation level and knowledge coverage level. The talk will end with an analysis on the composite difficulty level of the examples presented in the textbooks.
References
Jiansheng Bao (2002). A comparative study on the composite difficulty of intended mathematics curriculum in Chinese and UK junior high schools. Global Education, vol. 31, no. 9. 48—52.
Jianpan Wang & Jiansheng Bao (2014). Crossnational comparison on examples in high school mathematics textbooks. Global Education, vol. 43, no. 8. 101—110. In Chinese.
Jianpan Wang & Jianyue Zhang (2014). An international comparison of core content in high school mathematics textbooks. Curriculum, Teaching Material and Method, vol. 34 no. 10. 112—119. In Chinese.
Jianpan Wang (ed.) (2015). An International Comparison on High School Mathematics Textbooks. Shanghai: East China Normal University Press. In Chinese.
Jianpan Wang, born January 1949, PhD of Math, a mathematics professor and a former president of ECNU (East China Normal University), as well as a former ICMI EC member. His primary research interest is the theory of algebraic groups and quantum groups. He is also deeply involved in math education, focusing on school textbook research and schoolteacher education. In the recent years, he leads a national key research project Comparative Study on High School Math Textbooks in Major Countries and a Shanghai local project On the Effective Design of School Math Textbooks. He is the convenor of ICME14 to be held in Shanghai in 2020.