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

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.

Computer modeling is a significant indicator of professional competence for future civil safety specialists, whose professional activities include an ability to apply computer modeling to solve various professional problems by choosing appropriate computer systems.

The above said determines the content of educational process of future civil protection specialists in computer modeling teaching, which goes beyond the applying of ready-to-use computer models and involves thorough improvement of computer modeling skills required for the implementation of all phases of construction of the model and its research.

The system of computer modeling skills formation, which covers all periods of educational process (basic mathematics level, interdisciplinary level, professional level).

Development and use of computer simulation has a positive impact on the training of future professionals in general, bringing the motivational component in the learning process, facilitating the mapping of interdisciplinary connections, the integrated use of knowledge of the various sciences, enhancing the importance of self-learning and cognitive and research students.

Fractional calculus, i.e., calculus of derivatives and integrals of fractional order are getting more and more important in applications, in particular in oscillation theory, biology, etc… However these notions are not part of any standard university curricula, mainly due to the deep mathematical theories needed.

In our talk, we will present a series of dynamic demonstrations developed in Mathematica and Geogebra. We give an interactive introduction to different definitions, properties of “diffintegrals” by simple examples to both math and applied students.

The interactive demonstrations will be available on our website www.model.u-szeged.hu.

The article presents authors' teaching experience at a lower secondary school where teaching materials based on Archimedes' Book of Lemmas were presented to the students.

Book of Lemmas consists of 15 propositions concerning a circle/semicircle some of which are possible to use in geometry teaching at lower secondary schools. The rest of the Lemmas can be presented in higher secondary school classes. Dynamic geometry software plays an important part in these lessons.

The article gives an idea of the importance of DGS in geometry teaching and also links Archimedes' propositions to geometry topics at lower secondary school, describes actual course of the classes and other findings from the lessons.

We consider two kinds of algebraic exercises in Basic course of Mathematical Logic:

1) Truth-table exercises (filling the truth-table, checking of tautologicity, satisfiability, equivalence and inference, building a formula with given truth-column),

2) Formula manipulation exercises (expression using given connectives, normal forms).

Starting from early nineties, our students have solved these exercises in computerized environments that check each step in the solution, give error messages and require correction before the next step. The programs diagnose and count separately errors in order of operations, truth-value/equivalence, syntax, and answer dialog. The truth-table environment also enables to establish the penalty for each type of error and counts the points automatically. The final grading, however, is done by our instructors who are able to take into account additional aspects:

1) What part of the task is solved (if the solution is incomplete),

2) Errors,

3) Solution economy/conformity with the algorithm.

For this task the instructors use two additional programs to

a) Find the shortest formula for a given truth-column,

b) Identify and count inexpedient steps in formula manipulation tasks (24 types of inexpediency).

Note also that the formula manipulation environment contains an automated Solver that provides step hints and can be used for finding the ‘ideal’ number of steps.

In the paper we identify initial variables for human-like determination of grade for both kinds of exercises and show that they can be obtained by adding only fairly simple components to our existing programs. Further we describe how the teacher can specify the assessment algorithm by entering weights for parts of the task, basic penalties for error types, and spreadsheet-like formulas for possibly nonlinear calculation of penalties from the numbers of errors. Alternatively, the teacher could use a selection of pre-specified grading principles.

Students face many linguistic challenges in mathematics classrooms that use CAS: Not only do they need to use the mathematical language adequately, in addition to their everyday language, but they also need to master the technical language of their digital tool. These challenges become especially material when students have to document their processes and their results. There have already been important results (e.g. Ball 2003) that emphasize the extent to which CAS changes written records, and the need to learn to use the CAS syntax adequately for those written records (Ball & Stacey 2005). In this context, there has been a focus on normative questions on students’ documentation – e.g. emphasis was put on normative questions regarding what might be an adequate documentation for tests (Weigand 2013) or which means may help to structure students’ documentation (Ball 2003).

Since the distinction between CAS syntax and non-CAS syntax seems to be empirically necessary but not sufficient when looking at students documentation, there is a need for a qualitative analysis of different forms of language used in a mathematics classroom that uses digital tools.

This contribution will present results of an empirical study that works out different categories that students use in order to document their work. Therefore, different forms of documentation using technical, school (subject-based) and everyday language will be descriptively analyzed.

The qualitative study was conducted with 60 students in the 10th grade attending an upper secondary highschool in Germany. In different phases within a school year, after recieving a new CAS, the students worked on paper pencil tests which served as a foundation of the empirical material. Also, clinical interviews were conducted in order to find out more about the different uses of certain registers within a problem solving process. All exercises were within the context of functional reasoning.

Tarski has shown that formulas of first order predicate logic over certain fields can be decided algorithmically and algorithmic progress, especially the method of algebraic cylindrical decomposition . Tarski himself noted that this leads to a decision procedure for elementary geometry as well. Furthermore it gives a systematic way to solve systems of polynomial inequalities over R. Many notions from calculus that are expressed in terms of quantifiers can be formalized and decided for purely algebraic functions. This shows that the method of quantifier elimination is suited for several classes of problems that are relevant in math education at various levels. Thus the question arises, whether this method can be used as a teaching tool. One may hope that having access to quantifier elimination in a computer algebra system may give students the opportunity to explore the mentioned fields of application. Especially one may hope that this may provide a playground to exercise the formalisation step in mathematics. E.g. one may have an intuitive idea of what it means for a function to be convex on an interval but it is a crucial further step to be able to formalize this in the language of predicate calculus. We give examples of all kinds of didactically relevant applications and especially example on the formalizations of notions. Based on this example set we systematize the potential and the inherent problems of quantifier elimination a s a teaching method.

University of Pécs launched Information Technology (IT) engineer MSc program in 2013. The curriculum involves Numerical Methods as a facultative subject. In my lecture our experience during the teaching of this subject is summarized.

The focus of the subject was solving model problems using Maple, a Computer Algebra System (CAS), sometimes substitute the exact mathematical proofing. During the solutions we have tried to take advantage of opportunities offered by the used Maple computer algebra system. While composing the topics of the course, the rapid development of computer algebra systems was taken into consideration (eg. different methods of solutions of linear equation systems). On the other hand, this way students with limited mathematical skill are also able to understand more complex tasks, such as solution of multivariate interpolations and regressions, or that of partial differential equations. In our lecture we present some real-life examples from the course material.

Most math students are familiar with classic Chebyshev polynomials T_n. They are usually introduced as a class of orthogonal polynomials wrt a certain weight function on a closed bounded interval [a,b]. However, it turns out that they can be also introduced as a class of extremal polynomials: namely, the nth (scaled) Chebyshev polynomial is the polynomial which deviates least from the zero constant polynomial on an interval among the monic polynomials with degree at most n. In this talk we investigate the explicit characterization of the generalized Chebyshev polynomials of low degree on some particular subsets of the complex plane: To give the coefficients, roots and norms of these polynomials can be computationally difficult. We consider some approaches to attack the problem by computer algebra and we sketch a pool of possible student projects that can be built around this topic.

The illustrative computational and graphical tools are developed in Mathematica by the author.

The functional line penetrates and closely interlaces with all areas of mathematics at different levels of studies, often determining their content and methods. Precalculus plays a special role in formation of the corresponding conceptual vocabulary. It is here that students become familiar with different properties, types and operations on functions, master the skills of "reading" graphs of functions, learn to recognize and take advantage of the functional dependencies "hidden" in a problem.

An adequate software, which concentrates students' attention not only and not so much on the demonstration of examples of the concepts being studied, as activates independent creative activity in detection and use of the corresponding properties of the studied material and the connections between them can significantly increase the strength and depth of understanding of the studied matter.

This report presents an approach to development and use of such software and its implementation in the author’s non-profit program VisuMatica. Various examples illustrate the technique of dynamic creation and evolutionary development of generalized models as result of live collaborative analysis of the needs and characteristics of the studied material, and proper activities of the students.

We discuss a new approach to didactics of mathematics triggered by technological innovation: Computer Theorem Proving (TP) attains increasing attention by application to large proofs, for instance to the Kepler conjecture. On the other hand, respective technology is still open source and used in several prototypes. Possible kinds of interaction in such prototypes is compared with possible interaction in chess software (where the latter usually is not TP-based).

Given TP-based educational software and the context of a problem in applied mathematics, then each input of the player/learner is checked reliably by the system (a move of a certain figure to a certain field on the chessboard / a formula or method promoting a calculation within the given context); and if the player gets stuck, the system can propose a next step (a move towards winning the game / a formula or method solving the problem at hand). The result of the interaction between learner and system on the screen is expected to be close to what is written on paper during an examination on applied mathematics.

The didactical analysis, which will be given for the above software-based approach, does not emphasize the fun of playing; rather, the strengths of reliability and of variability supported by TP technology will be emphasized: If an input is wrong, the system will state the incorrectness with the reliability of formal logic (while variants are handled with maximal generosity); explanations, of what is wrong in detail, can be given from TP-technology's feature of transparency. The variability of interaction follows from TP-technology’s power: if the learner is not satisfied with the progress of a calculation, she or he can go back a few steps and try another way. Or one can explore variants by going to different intermediate states and watch the system trying a solution.

The paper shows what a qualitative change in the effective use of proof is brought by contemporary mathematical software in the teaching of mathematics. Particular corresponding examples of school practice are presented. Such use of mathematical software, however, makes new demands on the teachers. They must for example choose suitable topics, adapt the lesson organization, change teaching methods and methods of evaluation. The paper brings the first results of the research that was done by the first author Irena Štrausová which focused on the role of the teacher when teaching mathematics using dynamic visual proofs at selected secondary schools.

Proving facts in school geometry involves several processes, among others: sketching, observing, stating hypothesis, checking them, and providing deductive argumentation. Nowadays technology provides considerable support to some of these processes, in particular dynamic geometry software is a valuable aid for sketching geometric configurations and empirical checking hypotheses. Currently, considerable effort is put into developing systems for automated proving.

In the presentation we shall explore the role of automated observation, i.e. using technology to detect properties of dynamic constructions. Observation is an essential part of analysis of a construction and enables the generation of hypotheses that possibly lead to synthetic or simple algebraic proofs. Automated observation is not only a powerful ‘geometric eye’ that spots hardly perceptible properties, it also gives rise to new ‘obstacles' in geometric thinking and calls for specific demands on dynamic geometry software. In this sense we shall present some solutions that are implemented in OK Geometry software. One of them, the implicit constructions, allows that geometric objects (in a dynamic construction) are specified by required properties and not (entirely) by construction steps. Automated observation of implicit construction may bring to light properties that lead to the solution of a problem related to the studied configuration.

Perhaps the most promising potential related to automated observation is the use of a database of (dynamic) geometric objects and operations. We shall present the implementation of a database related to the geometry of triangle, consisting of several thousands of characteristic points of a triangle (e.g. incentre, orthocentre), known as Kimberling centres, and a large number of lines, circles, conics, and geometric operations; many of these objects possess interesting geometric properties. In this way automated observation does not take into account only the objects of a studied construction but as well tries to relate them to the objects in the database.