Developing Dynamic Sketches for Teaching Mathematics in Basic Schools

Transcription

1 Developing Dynamic Sketches for Teaching Mathematics in Basic Schools Valentina Dagiene, Prof., PhD Vilnius University, Faculty of Mathematics and Informatics Naugarduko str. 4, Vilnius, LT-035 Lithuania Egle Jasutiene, junior research Institute of Mathematics and Informatics Akademijos str. 4, Vilnius, LT Lithuania Abstract Reflecting on the actions and activities that are enabled by a new technology can catalyze a reconceptualization of the content and methods of teaching mathematics. Software might provide tools that enhance students actions and imagination. The five years long research has been developed on two phases. The first phase was to analyze problematic dimensions of teaching mathematics in schools using computer-based technologies and searching the most suitable software for the National curriculum of mathematics. The next step was to investigate (also to localize) the Geometer s Sketchpad and to built the various sets of dynamic sketches for teaching and learning mathematics in basic schools. More than 800 dynamic sketches have been developed within 9 th and 10 th grades (years 16 and 17) mathematics curriculum. Two CDs and descriptions have been prepared and published. The paper explores the main questions of developing dynamic sketches for mathematics curriculum of basic school in Lithuania. 1. Introduction Together with the rapid increase in the number of computers in schools a similar increase in the number of software products of the new technology has come. Mathematics is one of the main subjects in schools which require a lot of students efforts. Using technology while teaching mathematics is not only necessary but rather inescapable [1, 6, 11]. Regarding the Strategies of Information Technologies Implementation of Education in Lithuania [13] a wide attention is paid to educational software s implementation to curricula of various subjects: schools are supplied with such software and prepared methodical materials on it, besides different kind of workshops to teachers to introduce them the software are being held. That is one of the most important means to direct the computerization of schools towards the positive direction of upbringing improvement. Mathematical literacy in school is continuously gaining the stronger emphasis that is one of the aims suggested by the politicians of the European Union. All

2 pupils have to perceive the basic elements of mathematics. That s why they need a fair motivation. In this case the implementation of information technologies is one of such inviting solutions. Computers have much more to offer than drill and practice; in fact, they can be used in conjunction with all parts of the constructive learning process, when embedded in classroom culture where there is communication and cooperation. There are several ways in which computers can be used; for example in practicing of skill in a way that incorporated understanding or in simulations that enhance concept building []. Dynamic sketches created by computer provide the possibility of a deeper acquaintance with mathematical definitions, theorems, and properties. Often geometry is presented in static form in which the true and deeper meaning of a theorem does not get the true exposure it should. Thus the sketches developed by using computer-based technology help to look deeper to theorems of mathematics of secondary education [10]. D. Tall in his paper stresses that students using paper and pencil drawings of graph saw them as geometric shapes rather than a process of inputting x and outputting y [14]. The field of dynamic experimentation has been opened to new objects, the conics, where one can manipulate hyperbolas, ellipses, parabolas or their equations.. The main properties of software that supports teaching of mathematics When implementing IT in Lithuanian schools the computer applications that could be helpful for the implementation of the purposes, aims and didactical attitudes that are introduced in National Curricula and Educational Standards, and at the same time simple to use and handy for introducing of the wider scope of mathematical topics were searched. With reference to these criteria in 001 the educational software Geometer s Sketchpad 3.11 [ was bought to all Lithuanian schools; it was localized, and in 004 the localization was cardinally updated presenting the Lithuanian version of Geometer s Sketchpad 4 [5]. This educational software helps to implement the purposes, aims and didactical attitudes that are introduced in National Curricula and Educational Standards. According to [5] Geometer's Sketchpad 4 is "a software system for creating, exploring and analyzing a wide range of mathematics. You can construct interactive mathematical models ranging from basic investigations on shape and number to advanced, animated illustrations of complex systems." It allows an organized set of primitive actions to be turned into complex one using macroconstructions. The drawing produced at the surface of the screen can be manipulated by grapping and dropping around any point having sufficient degrees of freedom. [9]. Therefore teaching pupils to draw sketches helps to develop their creativity, algorithm thinking, carefulness, accuracy, and mathematical skills. The sketches created by pupils or teachers may be used for demonstration or research purposes. Dynamism of sketches created by the software may replace multiplex actions of drawing geometrical shapes on paper or on the desk.

3 3. Modeling and using dynamic sketches 3.1. Drawing of sketches The experience of other countries indicates that sketches are often used to solve particular task, for example, to demonstrate Pythagorean theorem, the Golden Rectangle Revisited, and so on [3]. There is, however, a group of sketches created to study particular mathematical or even physical topics. In order to construct a meaningful sketch you need: 1) to choose a topic which visualization can be supported by The Geometer s Sketchpad possibilities, ) to model the sketch, and 3) to construct the sketch. The Geometer s Sketchpad is convenient to introduce approx. 50% of math topics introduced in secondary schools, i.e. plane geometry, plane analysis, basics of geometrical functions and their charts, basics of mathematical analysis, trigonometry, part of stereometry (part since the software does not support 3D system), differential equations directional fields, number line and basic arithmetic operations, as well as vectorial algebra and complex numbers. After the proper topic of mathematics is chosen it is needed to analyze for which purpose the sketch will be created, i.e. which definition, property, or theorem must be demonstrated by the sketch. Special regard has to be paid in order to avoid the situations when the created sketch could serve as an obstacle of learning process or could bring confusion to pupils minds. The modeling of sketch should begin after these considerations are taken into account. The modeling of sketch is quite difficult stage and requires different skills and knowledge: 1) knowledge on mathematics theory, ) skills of methodology of mathematics teaching, 3) deep sophistication on software, and 4) ability of information structuring. Possibilities of the Geometer s Sketchpad allow creating sketches that universally approach geometrical objects and their relations. For example, functions may be analyzed regarding two different aspects: changing functions coefficients as parameters, selecting concrete and definite values or using the scroll bar for changing coefficients and observing function s chart as well as changing function s chart (dragging one of the chart s points) at the same time observing how the coefficients of function are changing. Therefore when modeling the sketch on paper the common picture of mathematics is needed to be seen and the context of the future sketch has to be anticipated. After getting acquaintance with the software, the creation of dynamic image of the modeled sketch becomes not so complicated. However, programmer, who is intended to create the sketch, besides the understanding of the software, has to obtain deep knowledge of algebra, geometry and methodology of teaching mathematics as well as skills on algorithm approach. Principle of the Geometer s Sketchpad is rather simple: we have an empty sheet of paper, ruler, pencil, calculator, and several drawing commands, thus we have to create. Very often quite complex dynamic images have to be created by using the

4 merest means. In such case quite a few steps have to be performed. For example, to create a decision model of inequality the algorithm of approx. 00 has to be implemented. The Geometer s Sketchpad does not limit the possible number of algorithm steps. It rather depends on the computer facilities as well as person s invention. When creating a sketch the ordinary means that are not included in the software often may be necessary, e.g. the scroll bar for changing coefficients or angles marking arc. Such means may be created by user and then implemented in various sketches. However, when looking at the final sketch all drawing steps remain invisible. In most cases just desired result, i.e. the complete image, is displayed. Thus, to create a sketch there is a need of time, knowledge of the theory and teaching of mathematics as well as familiarity with software possibilities. 3.. Set of dynamic sketches for mathematical lessons In 003 the research on software implementation in Lithuanian comprehensive schools has been performed and it has revealed that just 7% of schools are actually using the Geometer s Sketchpad during lessons [1]. It is not an easy task to the teachers themselves to develop sketches. The main reasons of this are the following: lack of time to properly prepare (teachers have many lessons), fear of technology and insufficient computer skills. Therefore the decision was made to help teachers to create sketches that are needed according to the National mathematics curriculum and to provide instructions on implementing those sketches in their lessons. Senior grades (16-17 year) were selected as target group of the research since: 1) the curricula of these grades embrace the most part of mathematics topics that can be visualized by dynamic sketches, ) in these grades the major part of new definitions, properties, and proofs (although most of them do not need to be demonstrated, their sense is still obscure to students; students remain not persuaded in their correctness) are introduced, 3) in these grades the summarized course, which has influence to further studies, is provided. Regarding these criteria the curricula of mathematics in 9-10 grades was analyzed and the topics that can be directly visualized by the Geometer s Sketchpad were selected. For 9 th grade the following topics were chosen and sets of sketches were developed (the number of sketches are presented in brackets): linear function (146), quadratic function (90), systems of linear equations (1), similarity of triangles (78), solution of quadratic equations (19), and circle and circular disk (116). Similar actions were applied to 10 th grade curricula after analysis the following topics and sketches were selected: graph of a function (95), set of equations and inequalities (3), quadratic inequalities (39), trigonometry functions of acute angles (83), and exploration of triangles (74). Although the Geometer s Sketchpad possibilities depend on invention of user, there was no visualization provided for combinatorics, probabilities, solid geometry, and percentage. Visualization of these topics using the Geometer s

5 Sketchpad is too complicated and in some cases even impossible or inadequate in order to introduce certain topics in understandable way for students. All sketches were developed implementing solid methodology: 1) a short description, containing the information on what to do with the sketch and where attention should be paid, was provided together with sketch, ) sketches were dynamic, i.e. it s possible to drag objects, change parameters and therefore the possibility to go back to the initial state always remains, 3) there is a help provided to user and upon the demand the answers can be given as well. All sketches are provided in CDs with descriptions, that help to use the sketches, theoretical material of a textbook, and recommendations on how to solve certain tasks regarding mathematics textbook in efficient way. Two types of dynamic sketches have been developed: 1) visualizing theory and ) visualizing problems. The dynamic sketches that visualize problems have several properties: 1) one dynamic sketch embrace whole group of problems and ) in many cases they widen the problems conditions. The dynamism of Geometer s Sketchpad s sketches provides an opportunity to visualize a whole group of problems by using one sketch. Solution of quadratic equation may serve as an example. By solving a parametric quadratic equation the whole set of such equations may be solved. By creating sketches the complete image of mathematics had to be demonstrated and therefore sketches of problems provide more information than it is required regarding the condition. For example, a student is asked to calculate length and width of the rectangle when the area and the perimeter of the shape are provided; in this case the dynamic sketch provides the graphs of functions related to rectangle s perimeter and area as well. This is the way how the simple problem of solid geometry becomes related with function and the relation between different mathematical topics appears. Thus, the Geometer s Sketchpad helps to look at mathematics as an entirety rather than jumble of separated topics. However, when developing sketches their creators were avoiding to overweight them and were trying to organize material in such way that it wouldn t bother the main idea of a problem. In more than 800 dynamic sketches were developed: compact disks Mathematics 9 with Geometer s Sketchpad and Mathematics 10 with Geometer s Sketchpad [8, 9] Teaching by using dynamic sketches: an example There is an example of dynamic sketch that illustrates the 10 th grade topic system of equations when one equation is non-linear. According to the National mathematics curriculum [4] a student should: 1) approximately calculate the solution of system of linear equations containing two variables, ) solve ordinary equation systems where one equation is linear and another quadratic (either in graphic way or in alteration way).

6 For this topic three dynamic sketches have been developed: 1) graphical solution of the system of two linear equations, ) graphical solution of the system of two equations where one equation is linear one and another s graph is a circle, and 3) graphical solution of the system of two equations those graphs are circles. The first sketch analyses how the changes of the coefficients a, b, and c, result the graph of the equation ax+by=c and provides the graphical interpretation of system of two linear equations. By changing the coefficients of linear equations students examine the whole group of linear equations systems and may find the answer to the following sketch s problem question: how many solutions the system of two linear equations may have? This is the way to remind students the graphical method of solving the system of two linear equations, that was introduced them in the 9 th grade. The second sketch is intended to analyze: 1) how the change of the coefficients result the graph of equation ax+by=c, ) how the change of the values d, e, and f result the graph of equation (x-d) +(y-e) =f, and 3) the graphical interpretation of the solutions of the equation system a x+ b y= c, (x - d) + (y - e) = f (Fig. 1). Fig 1. The graphical solution of sets of equations: a) system of two linear equations, b) system of linear and circle equations, c) system of two circle equations Such dynamic sketch is not complicated to develop approx. 30 steps is enough, since the Geometer s Sketchpad possibilities provide an opportunity to develop graphs of functions directly [5]. The short description on what should be changed or moved and what to notice is provided together with the sketch. The sketch illustrates the whole group of systems of equations where one equation is linear one and another s graph is circle. When changing the values of equations coefficients-parameters, the obtained systems and the interpretation of their graphical solution may be observed. That is the most important feature of such sketch [8, 9]. To provide the entire picture of the topic the third dynamic sketch is developed; it illustrates: 1) how the change of values of the coefficients a, b, and c result the equation graph (x-a) +(y-b) =c, and ) the graphical interpretation of the solutions of equation system (x - a) (x - d) + (y - b) + (y - e) = c, = f. In fact, when examining the given

7 examples in the sketch the whole trick is to answer the question how many solutions such system can have. Developed dynamic sketches help to reveal the whole picture of mathematical topic and the sketch of extended course may incorporate itself in the common context of the given topic. Dynamic sketches are easy to control since all of them have a similar structure; by examining the first sketch it s already possible to guess what the second and the third one will be about. Usage of these dynamic sketches helps to examine the systems of equations and it brings valid convenience as well, i.e. teacher doesn t have each time to draw new examples of graphs of systems of equation on the desk. 4. Conclusions ICT enhance teaching mathematics and motivate students for investigation. The first step is to provide schools with the software proper to teach mathematics. The further step of particular importance is to prepare action plan that would embrace teacher training, information dissemination, development of methodological and educational aids, guidance for assessment, etc. In Lithuania the Geometer s Sketchpad country license was purchased in 001 and right after that teacher training and information dissemination were initiated, however that was not enough. Just small part of teachers began to use the software during their lessons. The main explanation of this was lack of time to prepare for the lessons. Regarding this the decision to reconsider the mathematics curriculum for basic school and relate it with scripts developed by using the Geometer s Sketchpad (dynamic sketches and proofs) was made. For 9 th and 10 th grades (16-17 years) more than 800 dynamic sketches and scripts according to the mathematics curriculum were prepared. Two CDs and descriptions of them have been developed. The model and examples described in the paper emphasize educational aids for basic school curriculum. The developmental model should be modified to other levels and grades. However, the basic procedures would be quite similar. 6. References 1. Balacheff N., Kaput J. J. Computer-Based Learning Environment in Mathematics. International Handbook of Mathematics Education, Kluwer Academic Publishers, USA, 1996, Becker Jerry P., Selter Ch. Elementary School Practices, International Handbook of Mathematics Education, Kluwer Academic Publishers, USA, 1996, Classroom Activities, Key Curriculum Press, index.php

8 4. General curriculum and education standards: pre-school, primary, and basic education, 003. Vilnius, Ministry of Education and Science of Republic of Lithuania [in Lithuanian]. 5. Geometer's Sketchpad 4 dynamic geometry software for exploring mathematics, Key Curriculum Press, Hoyles C., Jones K. Proof in Dynamic Geometry contexts, Perspectives on the Teaching of Geometry for the 1st Century, An ICMI Study, Kluwer Academic Pub., 11-18, (1998). 7. Howson G. National curricula in mathematics. Leicester: Mathematical Association, UK, Jasutiene E., Stepanauskiene L., Vanagas V. Matematics for 9th grade with the Geometer's Sketchpad. Vilnius: TEV, 003 [in Lithuanian]. 9. Jasutiene E., Stepanauskiene L., Vanagas V. Matematics for 10th grade with the Geometer's Sketchpad. Vilnius: TEV, 005 [in Lithuanian]. 10. Klotz G., Jackiw N. Geometer s Sketchpad (software). Key Curriculum Press, Berkeley, CA, Posmastier Alfred S., Advanced Euclidean geometry, Key College Publishing, Study on using and implementing educational computer aids. Ministry of Education and Science of Republic of Lithuania Summary of the Strategy for ICT implementation in the Lithuanian education, 00. Information technologies at school. Vilnius, (Dokumentai). 14. Tall D. Functions and Calculus, International Handbook of Mathematics Education, Kluwer Academic Publishers, USA, 1996, 89-35