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

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.