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| 11:00am - 1:00pm | Future Trends I/II: Future Trends in Interactive Geometry (working group) Session Chair: Masataka Kaneko | |||
| VSP 1.04 | ||||
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Using Dynamic Geometry as a Robotics Interface 1PH Ludwigsburg, Germany; 2University of Halle-Wittenberg, Germany
Lego Mindstorms is an easy to use construction kit for robotics that perfectly fits to educational purposes. There are several programming environments available that have different advantages and disadvantages. Graphical programming environments like Lego NXT-G are easy to use but often limited in the possibilitys of implemented projects. On the other hand, librarys for common programming languages like BricxCC or LeJOS provide a wide range of functions for flexible progamms, but deeper programming skills are required before one gets started.
The interactive geometry software Cinderella comes with an easy to learn but powerfull functional scripting interface CindyScript to control geometric constructions and perform mathematical calculations. But using an internal timer integrated in Cinderella's scripting environment even sophisticated interactive animations ans simulations can be implemented with only a little programming effort. We use a Cinderella plugin that wraps LeJOS commands to CindyScript functions to controll the Mindstorms NXT motors and sensors. Connecting the framework with geometric construction elements, one gets a new kind of interactive robot remote control. Vivid visualization of sensor results are possible. In this talk we present some applications developed during a course for teacher students in computer science. The examples arise from the boarderline of mathematics, physics and computer sience and cover aspects of mathematical areas like functional thinking or analytical geometry. Sketchometry - Dynamic Mathematics on Mobile Devices University of Bayreuth, Germany
The internet goes mobile and mobile devices like tablet computers and smartphones are entering the classrooms. How do these developments influence learning? What opportunities are opened for mathematics education?
These questions are the introduction of the talk about sketchometry, a new kind of dynamic mathematics software, especially designed for mobile devices. Lines, circles or triangles are simply sketched with the finger on the screen. The software transforms them into geometric objects. The fingers become compass or ruler. In contrast to PCs, software on mobile devices is ready-to-use right after power-up. Creating constructions with sketchometry in mathematics lessons becomes as easy as the usage of pocket calculators. The mobile devices can be used point by point by the students. It is not necessary to go to a computer lab for the whole lesson. For a gainful integration of these advantages in mathematics education, it is necessary to rethink didactical and methodical concepts. Traditional media, as text books or printed worksheets, can be used together with sketchometry on the mobile device. The usage of computers does no longer happen isolated but integrated.
Cross-browser graphic user interface for interactive geometry University of East Sarajevo, Bosnia and Herzegovina
This presentation will introduce a new cross-browser dynamic geometry system with graphic user interface, based on JSXGraph. JSXGraph is library for interactive geometry completely implemented in JavaScript and it does not rely on any other library. Developed cross-browser interactive geometry system depends only on JavaScript and it does not require any additional plugins. It has a very small footprint and works on all devices, including multi-touch devices running iOS, Android, firefoxOS and Windows 8. This system is an excellent solution for the development platform-independent interactive web elements for geometry constructions.
Geometric Algebra – A foundation for the combination of Dynamic Geometry Systems with Computer Algebra Systems? TU Darmstadt, Germany
Geometric Algebra is a very general mathematical system including many other systems such as linear algebra, complex numbers, Plücker coordinates, projective geometry or quaternions.
The specific compass ruler algebra, for instance, is very well suited to compute similar to working with compass and ruler. Geometric objects such as circles and lines as well as geometric operations with them can be handled very easily inside of the algebra. A circle, for instance, can be described based on the outer product of three points of the circle. The compass ruler algebra is a 4D algebra describing the 2D plane with two additional basis vectors representing the origin and infinity. You are able to directly compute with infinity, for instance, when expressing the center point of a circle as the inversion of infinity in the circle. Gaalop is an easy to handle tool in order to compute and visualize with compass ruler algebra. While in the background a computer algebra system is responsible for the symbolic computations, its visualizing component offers basic DGS functionality. Based on this combination of geometry and algebra Gaalop is also very well suited for proving of geometric relations.
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| 2:15pm - 4:15pm | AKMUI I: Vorträge – Lehrertag | |||
| VSP 1.04 | ||||
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Historische astronomische Daten und moderne CAS-Rechner MLU Halle, Germany
Die Modellierung realer Zusammenhänge ist oft mit einer typischen Aufgabe verbunden: Aus Messdaten soll ein analytischer Ausdruck abgeleitet werden, der den Daten „gut angepasst“ ist. Der Vortrag greift dies für ein historisches Astronomie-Problem auf: Die Frage der Funktionsanpassung für originale Messwerte wird für den Kometen von 1618 betrachtet. Die reale Datensituation wird genutzt, um für Schüler Funktionsapproximation mit einem CAS-Rechner erleb- und nachvollziehbar werden zu lassen.Die Modellierung realer Zusammenhänge ist oft mit einer typischen Aufgabe verbunden: Aus Messdaten soll ein analytischer Ausdruck abgeleitet werden, der den Daten „gut angepasst“ ist. Der Vortrag greift dies für ein historisches Astronomie-Problem auf: Die Frage der Funktionsanpassung für originale Messwerte wird für den Kometen von 1618 betrachtet. Die reale Datensituation wird genutzt, um für Schüler Funktionsapproximation mit einem CAS-Rechner erleb- und nachvollziehbar werden zu lassen.
Unterrichtsmaterial vor dem Hintergrund verschiedener Werkzeuge – einige Beispiele der Bildverarbeitung
„Wem, wie, wann, wo und warum nutzen Werkzeuge?“ Unter anderem auf diese Fragen der Tagungseinladung möchte ich – am Beispiel der für den Mathematikunterricht reduzierten grundlegenden Konzepte der Bildverarbeitung – einige Antworten anbieten und diese zur Diskussion stellen.
Ich habe mich damit auseinandergesetzt, welche Möglichkeiten die Bildverarbeitung bietet, alternative Zugänge zu vielen Teilbereichen der Schulmathematik zu schaffen. Die dabei benutzten Werkzeuge sind Maple 18, MaplePlayer und Excel. Mit meinem Vortrag will ich Ihnen – möglichst unterrichtsnah – einige Beispiele meiner Arbeit vorstellen und Material zur Verfügung stellen, das – mit bzw. ohne den Einsatz der oben genannten Werkzeuge – zum Unterrichtseinsatz genutzt werden kann. Digitale Werkzeugkomponenten
Über welche Kompetenzen sollen Schülerinnen und Schüler zum Abitur bzw. nach Abschluss der Sekundarstufe I beim Umgang mit digitalen Werkzeugen verfügen? Inwiefern geht es dabei um etwas anderes bzw. um mehr als um die Bedienung von Software und Hardware? Die Bildungsstandards der KMK lassen da einiges offen. Eine gemeinsame Arbeitsgruppe von MNU und T3 beschäftigt sich seit 2013 mit der Fragestellung, was unter ›Digitalen Werkzeugkompetenzen‹ zu verstehen ist. Erste Ergebnisse werden im Vortrag vorgestellt und anhand von Aufgabenbeispielen zur Sekundarstufe I und II konkretisiert. Damit verbunden wird der Frage nachgegangen, wie Lernende ihren Einsatz von digitalen Werkzeugen im Arbeitsprozess und schriftlichen Überprüfungen dokumentieren sollten.
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| 11:00am - 1:00pm | Automated Deduction: Automated Deduction in Dynamic Geometry Tools (working group) Session Chair: Predrag Janičić | |||
| VSP 1.04 | ||||
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Automatic theorem proving in Dynamic Geometry contexts: what is it good for? Two diverse points of view 1Universidad de Cantabria Spain; 2Architectural technical high school, Serbia
The talk will be dual, in the sense of addressing the same issue from two different perspectives. One, that of a secondary mathematics teacher, with experience on the introduction of ATP tools (GCLC and GeoGebra) at the classroom. Then, the point of view of a university ATP researcher and developer, with sustained involvement in math education. In both cases the main point will be to provide some answers to the key question:
What is ATP good for? The most efficient ATP systems in geometry are usually algebraic. Although these provers do not generate classical readable proofs, they give an yes/no answer and can generate non-degeneracy conditions (NDGs) that need to hold for the statement to be satisfied. Although ATPs can be applied to check the main statement of the theorem, we advocate that it is much more beneficial for pupils to apply automation on intermediate steps, and so to verify if their conclusions are correct and if they are on the right track to prove the main theorem. In such setting, pupils still need to produce global proof steps, while the ATP system takes care of details that are usually straightforward, but tedious to justify. Pupil cooperates with the machine and relies on its help, but his understanding of the geometry problem is still crucial for success. Trough this activity student develops his thinking and analytical skills, but, unlike with classical pen-and-paper proofs, in a very rigorous setting, since every conclusion must be precisely formulated and pass the automated test. Also, a careful analysis of NDGs may reveal many subtle issues about the theorem that is being proved. In this talk we will present some of our experience and ideas about using these techniques (implemented both in GCLC and GeoGebra) in high-school geometry classes.
Automated algebraic calculations of geometric figures in dynamic geometry systems University of Education Weingarten, Germany
Using Methods based on Automated Deduction in Geometry (ADG), it is possible to perform algebraic calculations on interactively constructed figures. Thus, a parameter of such a figure can be calculated algebraically as a function of other parameters of this figure. This opens up a new computer-assisted connection of synthetic elementary geometry to algebra. On the other hand, the question: "WHAT HOW depends on WHOM?" is cleared up when a figure is dynamically varied. - In this lecture there are given some selected examples in a constraint-based DGS developed for educational purposes. Their didactic relevance is explained and some resulting mathematics education problems are discussed.
Teaching loci and envelopes in GeoGebra 1University of Vigo, Spain; 2Johannes Kepler University, Austria
GeoGebra is open source mathematics education software being used in thousands of schools worldwide. Since version 4.2 (December 2012) it supports symbolic computation of locus equations as a result of joint effort of mathematicians and programmers helping the GeoGebra developer team. The joint work, based on former researches, started in 2010 and continued until present days, now enables fast locus and envelope computations even in a web browser in full HTML5 mode. In conclusion, classroom demonstrations and deeper investigations of dynamic analytical geometry is ready to use on tablets or smartphones as well.
In our talk we consider some typical grammar school topics when investigating loci is a natural way of defining mathematical objects. Such topics include definition of a parabola and other conics in different situations like synthetic definitions or points and curves associated with a triangle. In most grammar schools, however, no other than quadratic curves are discussed, but generalization of some exercises and also every day problems will introduce higher order algebraic curves. Thus our talk will mention the cubic curve ``strophoid'' as locus of heights of a triangle when one of the vertices moves on a circle. Also quartic ``cardioid'' and sextic ``nephroid'' can be of every day interest when investigating mathematics in a coffee cup. We will also focus on GeoGebra specific tips and tricks when constructing a geometric figure to be available for getting the locus equation. Among others, simplification and synthetization (via the intercept theorem) will be mentioned.
Extending the range of computable objects in Dynamic Geometry by using Quantifier Elimination University of Vigo, Spain
Traditionally, loci in Dynamic Geometry Systems (DGS) have been displayed either tracing the locus point, either through a special command that, roughly, enhances such tracing procedure. Some DGS incorporate more sophisticated approaches being able to return algebraic knowledge about loci. Nevertheless, these enhancements deal with a restricted type of loci, those where the position of the locus points is completely determined by another point, the mover, which must lie on a linear path. If the mover point is not unique, or it is not bound to a line, standard DGS can only offer a tracing strategy, just returning a graphical answer. Consider, for instance, two circles and a point on each circle. The locus of their midpoint is a 2-dimensional region, that cannot be easily described with current DGS.
In this talk I illustrate these limitations when computing loci. Furthermore, a discussion on the application of Cylindrical Algebraic Decomposition software will be given. | |||
| 2:15pm - 4:15pm | Workshop 2 & 3 | |||
| VSP 1.04 | ||||
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Mapping mathematics learning resources University of Education Weingarten, Germany
In this workshop, we shall work collectively to write down a map of the sources of learning resources, be them open or not, adaptable or not, nicely made or not.
The aim is to assemble a broad panorama of the places where mathematics learning resources can be found. We shall do so in a web-based map which can be followed live during the workshop. The wish is to leverage the diversity of attendees at the conference so as to obtain some clarity in the local practices in teachers of their surroundings. The result of this workshop would support such harvesting projects as Open-Discovery-Space, i2geo.net, probably Elixier, and other initiatives. Moreover, it would help users of such repositories to get a broader overview and thus decide better what to visit at their next resourcing excursions. Controlling Lego Mindstorms robots by Cinderella's scripting interface PH Ludwigsburg, Germany
By installing the Mindstorms plugin, the interactive geometry software Cinderella can be used as a remote control for Lego Mindstorms robots.
In this workshop we will explain how to install and use the plugin. We will give a short introduction to the CindyScript programming language. Simple examples will show how to read sensor values from a Lego NXT and use them to manipulate geometric constructions. Vice versa, interactive geometric constructions are used as remote controls for motors. There will be a few Mindstorms controller on location for own explorations. | |||
| 4:45pm - 5:45pm | Collaborative: Collaborative use of DGS and KETpic (working group) Session Chair: Setsuo Takato | |||
| VSP 1.04 | ||||
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KETCindy - Fine Combination of KETpic and Cinderella 1Toho University, Japan; 2Kogakuin University, Japan; 3Nagano National College of Technology, Japan; 4Kisarazu National College of Technology, Japan; 5Graduate School of Information Science, Nagoya University, Japan
According to the result of our questionnaire survey, one major opinion of collegiate mathematics teachers in Japan is that there is no necessity to use high-quality graphics in education. However, from our experience, graphics use seems to play a crucial role in some classroom situations. Though we have been attempting some statistical efficiency assessments for teaching materials containing graphics, it is not so easy to separate the effect of using graphics from other factors like context of classroom or communications between students. The aim of this research is to verify the effect of using high-quality graphics in collegiate mathematics education through some cognitive scientific experiments. The experiments were designed in the following two ways:
(1) Detecting the change of students’ brain activity after their seeing effective figures through EEG (ElectroEncephaloGram) measurement (2) Observing the difference in time needed for students to answer a question between before and after their seeing effective figures In fact, we picked up the case of the comparison of growth degree between exponential function y=2^x and power function y=x^4. We prepared some graphs of these functions by gradually changing the scale in y direction, so that students can recognize that the growth of y=2^x is greater than that of y=x^4 when x becomes sufficiently large. We showed these graphics step by step to three students and detected their brain activities through EEG (ElectroEncepharoGram) measurement. As a result, the judgment of these students changed when they saw a triggering figure, and some change in the trend of EEG signal and solution time was observed at that time. These results indicate that using effective figures should have great influence on students’ reasoning processes.
A Collaborative Laboratory for Geometry: A Case Study at Portugal and Serbia 1CISUC/University of Coimbra, Portugal; 2Faculty of Mathematics/University of Belgrade, Belgrade, Serbia; 3CISUC/Department of Mathematics, University of Coimbra, Coimbra, Portugal; 4School of Science and Tecnology/University of Trás-os-Montes e Alto Douro, Vila Real, Portugal
The Web Geometry Laboratory} (WGL) platform is a collaborative blended-learning Web-environment for geometry, it integrates a dynamic geometry systems (DGS) and it provides a collaborative environment for students and teachers. Its use is possible in the context of a classroom or remotely. Apart from its development its evaluation is being done through a series of case studies, sustained through a qualitative approach (interpretive research), being conducted in Portugal and Serbia (Prototype in hilbert.mat.uc.pt/WebGeometryLab).
An initial case study in Portugal, with groups of secondary students (17 years old) was done, using various gathering information techniques: quizzes; tests; direct observation; record interactions on the platform; challenges. We analysed the use of the WGL collaborative environment by the students. Another case study, in Serbia, was conducted in the context of remote access to the platform (homework). The study included 50 secondary students (15 years old). All students attended the traditional classes in school. Half of the students used WGL platform for homework and the other half did their homework the traditional way. We investigated the impact of collaborative work to the motivation level and level of achievement. Using an action research approach, the platform is being developed. These studies revealed some aspects that could be enhanced, e.g. a chat feature. More and wider case studies are being prepared allowing the validation and further development of the WGL platform. These studies also indicates that there is a significant improvement in the motivation of students and a slight improvement in their achievement when using the WGL platform. The WGL platform will include in future stages of development the implementation of an adaptive environment allowing the construction of students' profiles and learning paths. A final stage will be the integration of a geometric automated theorem prover and its use in the learning process.
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| 11:00am - 1:00pm | Edu. Resources: Open Educational Resources in Mathematics (working group) Session Chair: Paul Libbrecht | |||||||
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Adaptations to a DGS learning resource University of Education Weingarten, Germany
Learning resources have been created to represent digital units of exchangeable materials that teachers and learners can "pull from" in order to support the learning processes. Leveraging the web, one can often find learning these resources; but what characteristics does it need in order to be easily exchangeable? Although several investigations have explored the ways to publish and, in particular, the ideal best practice to do so, few have considered realistic use cases, where the technical competency of the teacher is counterb
In this paper we consider a simple language-crossing situation which was stimulated by the i2geo.net platform. A learning resource has been re-purposed from a French context to a German context, from one school and its software reality to another (Cabri to DynaGeo, Windows French to Windows German, Google Earth to Google Maps), from one set of learners' competency to another. In neighbouring fields, practices such as that of the PhET repository of physics simulations, which collects simulations as "center-points" of the learning resources' exchange process, while "scenarios" are satelite resources, are described as candidate sharing mechansims. This need for adaptations is not uncommon. Until a large diversity in the available resources can be achieved and searched through, it will remain the mainstream practice of teachers that attempt to employ learning resources. However, quite an amount of learning resources' publishers do not make it fully possible. This research highlights a few learning resources' characteristics that support this, from the legal and technical point of views, from the ease of perception to the ease of adaptation.
The Use of Computers for Calculus Teaching Pontifícia Universidade Católica, Brazil
This paper aims at analyzing the use of computers when teaching differentiability and continuity in real functions and real variable. The relation is approached in the case of a non-differentiable continuous function in the interval of straight lines. This example is found in an article written by David Tall and is used to evidence a way in which a computer helps the learning and teaching of concepts of Differential and Integral Calculus when didactic and meaningful materials are produced. Elements of Tall’s theory on the advantages of the use of computers in Education, as well as the historical importance of the development of an example of a continuous non-differentiable function are presented in this paper. Also, a case of a function defined as limit to a series of functions is explored. In addition, commands and tools which are available in the software named GeoGegra are presented. As a result, we present tools which will hopefully contribute to the practice as well as advancements in Mathematics Education at Higher Education level.
Using dynamic geometry in problem-based engineering courses University of West Bohemia, Czech Republic
It is a modern trend today when formulating the curriculum of a geometric course at the technical universities to start from a real problem originated in technical praxis and subsequently to define which geometric theories and which skills are necessary for its solving. Nowadays, interactive and dynamic geometry software plays a more and more important role in this scenario as it helps to think algorithmically, enables to discuss the solvability of the whole class of geometric problems from different point of views and mainly serves as a first step to variational geometry needed later in geometric modelling. This makes the teaching and learning process more efficient and also more interesting for students. In our contribution we will present this approach on several particular examples taken mainly from the courses for mechanical engineering, of course with emphasis on the application of dynamic geometry. All examples are characterized by a unique structure. First, we present a real engineering problem, then we subdivide it to a number of elementary sub-problems and finally we show how the curriculum and the corresponding teaching procedures are chosen to satisfy the goal to teach the students to solve the real-life problems. We will present our experience with the problem-based teaching of geometry using dynamic geometry software and discuss some aspects of this approach.
Isogonal and isotomial transformations of a triangle University of Miskolc, Hungary
The median and symmedian lines, the centroid and the symmedian point of a given triangle present interesting properties. Part of these properties can be formulated in a more general context for isogonals and isotomials, based on the trigonometric and algebraic forms of the Ceva’s theorem.
In a triangle the isogonal of a line passing through one of the vertices of the triangle is a line symmetric to the bisector of the given angle. Similarly a line passing through one of the vertices of a triangle and its isotomial intersect the opposite side of the triangle in two points which are symmetric to the midpoint of the given side. It can be proven that the three isogonals, respectively isotomials of three concurrent lines which pass through the three vertices of the triangle, are concurrent as well. This property serves as definition for the isogonal respectively isotomial transformation, the image of a given point in this transformation will be the intersection point of the three isogonals, respectively isotomials of the three lines which pass through the given point and the vertices of the triangle. The lecture is aimed to present some of the properties of the isogonal and isotomial transformations, and to visualize them using GeoGebra.
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| 2:15pm - 3:15pm | Workshop 8 | |||||||
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Dynamic Visual Resources for 16-19 Mathematics Institute of Education, University of London, UK
A resource using dynamic geometry software is being co-developed by teachers in Ontario and England for the new International Baccalaureate Mathematics at standard and higher level, but also useful for other mathematics courses at this level. The aim is to promote student exploration of mathematics in dynamic and visual ways. We will present some of the resources for Geometer's Sketchpad and Cabri - come and look at new ways to explore sequences and series, functions, vectors, calculus...
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| 11:00am - 1:00pm | Assessment: Assessment (general topic) Session Chair: Alla Stolyarevska | |||||||
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Problem-solving according to Archimedes University of South Bohemia, Czech Republic
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.
Designing human-like automated assessment to replace proportional penalties for error types University of Tartu, Estonia
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.
Student-Documentations in Mathematics Classroom Using CAS: Between Technical, Subject-Based and Everyday Language TU Dortmund, Germany
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.
Gains and Pitfalls of Quantifier Elimination as a teaching tool Goethe Uni Frankfurt, Germany
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.
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