Fifth Central- and Eastern European Conference
on Computer Algebra- and Dynamic Geometry Systems
in Mathematics Education

Technical innovations are a ubiquitous phenomenon in our time. While new possibilities emerge, at the same time old possibilities disappear. Most probably internet based communication of mathematics in the future will observe several significant shifts in various categories: from PCs to tablets, from mouse to (multi-)touch and from Java to JavaScript.

The talk discusses implications on the design of interactive math that come along with these changes and tries to exhibit positive and negative aspects of these changes. The talk will be illustrated by demonstrations of various software projects done by the author including the interactive mathematics software Cinderella, the iOS App iOrnament and several interactive installations in Museums and public exhibitions.

Structured derivations is a new method for presenting mathematical arguments. It is a further development of Dijsktra's calculational style reasoning, and can be used for all kinds of mathematics: proofs, calculations, geometric constructions, etc. It combines the three main proof paradigms, equational reasoning, forward reasoning (Hilbert style) and backward reasoning (Gentzen style), in one single proof format. The use of structured derivations in high school mathematics education has recently been piloted in the E-math project (an EU project 2011 - 2013). The project has created new mathematics textbooks based on the systematic use of structured derivations. These textbooks cover the whole national mathematics curriculum for first year in high school, in Finland, Sweden, and Estonia. The textbooks have been implemented on a new software platform for interactive e-books created in the project. This platform has special support for displaying and writing mathematics on a computer. The new math e-books have been piloted in 15 high schools in 2012 - 13, with some 1 000 students participating in the pilots. The talk will present the main findings from the E-math project, and discuss conclusions that can be drawn from the pilots.

Dynamic geometry is designed as a helpful tool for mathematics comprehension. This can be thought as a single direction: from dynamic geometry to mathematics. But, conversely, some non elementary mathematics seem to be required to understand (and to improve) dynamic geometry performance.

Noticing this mutual interaction is neither very popular, nor strictly new (e.g. consider the paradigmatic case of Cinderella). On the other hand, we would argue in the talk how to trace back the origins of such interaction in order to include some fancy names such as, say, Babbage or Watt, ending up with Nash.... Yet, we think it is perhaps convenient to insist now and again on the importance of traveling back and forth along this two way track, for the benefit of mathematics education.

This would be the main idea in my lecture, exemplified by some situations I have been recently dealing with, in order to improve dynamic geometry features for locus computation and automatic theorem proving.

“If production of knowledge is understood in this way, what constitutes a ‘‘problem’’ will depend on the nature of the humans-with-media collective. A problem that needs to be solved, or that puzzles someone, may not be a problem when a search software tool like Google is available. Similarly, a real problem for collectives of humans-with-orality may not constitute a problem for a collective of humans-with-paper-and-pencil.” (p. 804, Borba (2012)

In this talk I will unpack the above quote from a recently published paper on ZDM. I will discuss first the way Internet and mobile telephones in particular, and digital technology in general, are changing the nature of what it means to be a human being (Castells, 2009; Borba, 2012). I will present to the reader my view regarding four phases of the use of digital technology in mathematics education (Borba, 2012) in order to discuss how interaction occurs in presence of such technology. I will then discuss what can be labeled “emergent classrooms” within the fourth phase. I will focus on how social networks such as Facebook and other features of this phase are transforming interaction in the classroom, and perhaps even creating new images of what a classroom may be. Examples from pre-calculus/early calculus will be provided.

References

Borba, M.C. Humans-with-media and continuing education for mathematics teachers in online environments. ZDM Mathematics Education 6(44) p. 801-814. (2012).

Borba, M. C. Potential scenarios for Internet use in the mathematics classroom. ZDM, 41(4), 453–465, 2009.

Borba, M. C., Villarreal, M. E. (2005) Humans-with-media and the reorganization of mathematical thinking: information and communication technologies, modeling, visualization, and experimentation. New York, Springer.

Castells, M. (2009) Communicating Power. London: Oxford University Press

Levy, P. (1993) As Tecnologias da Inteligência: o futuro do pensamento na era da informática. Rio de Janeiro: Editora 34.

The next generation mathematics education software should take advantages of the state-of-the-art research in the fields of automated reasoning. The new tools should be able to automatically solve different sorts of mathematical problems, provide understandable solutions, guide the users through the solving process, check if their solutions are correct, provide an appropriate support for interactive theorem proving, etc. In this talk, we will discuss these and other challenges for the next generation mathematics education software, primarily for geometry. For geometry education software, some of the specific challenges are defining appropriate foundations for high-school geometry, automated proving of theorems with human-readable proofs, automated solving of construction problems, linking theorem proving with dynamic geometry tools, automated discovery of theorems, automated discovery of loci, etc.