The strange fate of abstract thinking Alexandre V Borovik University of Manchester The use of Information and Communication Technology (ICT) in mathematics teaching exploits the possibility of easy generation of concrete examples and encourages in students a bottom-up, inductive style of thinking. This increases the gap between mathematics as it is taught and the research level mathematics dominated by hierarchically structured top-down abstract thinking. Interactive representation of mathematics via user-friendly interfaces is made possible by advances in computer science. Paradoxically, computer science itself requires a level of abstract thinking that far exceeds the one normally acquired by mathematics graduates. The paper discusses a possible remedy: the use of high level functional programming languages such as Haskell as a tool for handling abstract mathematical structures. Introduction and disclaimers This short paper belongs to a series of publications (see, for example, Borovik 2011) informed by my work in the Education Committee of the London Mathematical Society [LMS] and the Research Committee of the Association for Learning Technology [ALT]. I support the LMS's Position Statement on ICT in Higher Education (London Mathematical Society, 2011). But, to avoid any ambiguity, I have to emphasise that the present paper is written by me in my private capacity and does not necessarily reflect views of the LMS, or of the ALT, or any other institution or organisation. The limited volume of the paper forces me to focus it on just one facet of the immensely complex problem of mathematical education in the modern world: development of abstract thinking. I restrict my recommendations to university level teaching of mathematics. Abstract thinking in modern society Although my paper is about university level teaching of mathematics, let us start by looking at school level mathematics and Information and Communication Technology (ICT). INTELLECT, a trade association for information technology industry in the UK (its members include Accenture and IBM, among others), recently proposed in their submission to the National Curriculum Review (Intellect 2011) that ICT as a statutory subject in its current form effectively discourages students from progressing to the more advanced computing courses [...] In place of ICT classes for all students, INTELLECT believes potentially gifted tech students should be encouraged into the core science, engineering and maths subjects. [...] INTELLECT has recommended to the DfE that computing should be a discrete subject available to pupils from Key Stage 3 onwards with options to follow a progression path where they learn increasingly more advanced skills. (Mitchel 2011) This statement is interesting because it sheds light on the intrinsic flaw at the core of the current concept of the use of ICT in teaching: it is the word “use”. The “use” produces “users”, not “producers” or “creators”. The IT industry discovered that teaching children to “use” ICT does not encourage them to become creators of ICT. The IT industry wants schools to teach actual computing, that is, development of algorithms and their subsequent implementation in computer code. (There are some interesting overlaps of INTELLECT statement and a recent policy statement of the ALT (2011).) The purpose of my paper is to highlight an important aspect of information and computer technology: learning modern computing (that is, computer programming) requires a very high level of abstract thinking. This position is actively promoted by Kramer (2007). An anonymous contributor to discussion of Kramer's paper in my blog Mathematics under the Microscope (Borovik 2007), a professional computer scientist, left the following comment: