Teaching Mathematics and Its Applications (2009) 28, 69^76 doi:10. 1093/teamat/hrp003 Advance Access publication 13 March 2009 GeoGebra c freedom to explore and learn* LINDA FAHLBERG-STOJANOVSKAy Department of Mathematics and Computer Sciences, University of St. Clement of Ohrid, Bitola, FYR Macedonia Downloaded from http://teamat. oxfordjournals. org/ at University of Melbourne Library on October 23, 2011 VITOMIR STOJANOVSKI Department of Mechanical Engineering, University of St. Clement of Ohrid, Bitola, FYR Macedonia [Submitted November 2008; accepted January 2009]

We start by visiting the maths section of the web site answers. yahoo. com. Here, anybody can ask a question from anywhere in the world at every possible level. Answers are given by anyone who wants to contribute and then askers/readers rate the responses. A brief look here and it is starkly clear that our young people are struggling and their ability to think logicallycthat is understand a problem, organize data into knowns and unknowns, explore possibilities and assess solutions is definitely on the decline. In our opinion, this is more insidious than the actual decline in their overall mathematics skills.

Further, one is struck by the fact that technology seems to be contributing to this decline when in fact it should be the opposite. We then examine two question/answer cycles in detail and show how the freeware GeoGebra (www. geogebra. org GeoGebraWiki: www. geogebra. org/ wiki GeoGebraForum: www. geogebra. org/forum)cwhich gives the freedom to explore and learn to everyone, everywhere and at any timeccan be of tremendous value to pupils and students in their understanding of mathematics from the smallest ages on up. 1. Introduction

There is no question that there is a decline both in the skill and interest level in mathematics and science among our young people. Many argue that these skills are not needed in everyday life and others argue that without these skills, we cannot compete in a global economy. Regardless of our thinking on this, the following is certainly true. Given any problem in life, one needs to be able to think about it logically. This means, understand what the problem is, organize data into knowns and unknowns, explore possibilities and assess solutions. These are crucial life skills.

By its very nature—the process of learning and doing mathematics should significantly increase these capabilities. However, it seems that our students are not learning these skills and the integration of technology has not increased the development of these skills, as we had hoped. Why do we say this? What is the reason for this? What can we do to reverse this process and (hopefully) at the same time increase both their interest and actual skills in mathematics? *For additional information see http://math247. pbwiki. com/GeoGebra y Email: [email protected] k ? The Author 2009 Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. . All rights reserved. For permissions, please email: journals. [email protected] org 70 L. FAHLBERG-STOJANOVSKA AND V. STOJANOVSKI 2. Facing the Facts: answers. yahoo. com Where do our young people go for help with their mathematics? Check out the mathematics questions at answers. yahoo. com. The questions come in every minute and range from the absolute most simple to complex questions from calculus and beyond.

Answers are given by anyone who wants to contribute and again the range in quality and quantity is immense. Finally, the askers and readers rate the answers. This is a breathtaking view into what is happening with our young people and their mathematics education throughout the world. It is also an incredible opportunity for all kinds of mathematical exploration and we will show this through detailed consideration of two examples. Downloaded from http://teamat. oxfordjournals. org/ at University of Melbourne Library on October 23, 2011 3. Sample Problem 1

Find equation of a circle with center on the line y = 3x and tangent to y-axis on (0,2) (answers. yahoo. com, Find equation of a circle? http://tinyurl. com/58cueo). Within minutes, there were two answers—both obviously incorrect. While getting ready to respond, a third answer appeared. This third answer was long and detailed and eventually chosen as ‘best answer’. Like all answers at answers. yahoo. com, it is written in text math so it is a bit difficult to follow. The answer is: 3×2 + 3y2A2xA6y = 0. Our first response was to check this answer.

So, select the math text, Ctrl + C to copy, open the free math program GeoGebra, click in the input field, Ctrl + V to paste and hit Enter—a total of 10 s. From Fig. 1, it is obvious that the circle is not tangent to the y-axis. So this ‘best answer’ cannot be correct. 3. 1 Two questions Two questions immediately came to mind. The first question is the obvious—why did not either the answerer or asker do this? That is, spend 10 s and test the answer and see that it could not be correct? The second question is—how could one guide the asker to think about this problem that would (a) be immediately useful and (b) help in other problems?

FIG. 1. Input and check answer. (This figure appears in colour in the online version of Teaching Mathematics and its Applications. ) GEOGEBRA 71 3. 2 Answer A good starting point is GeoGebra and it is important that GeoGebra is freeware, very versatile and it is easy to use. Below, we will give more details about GeoGebra—but first let us just see it in action. 3. 3 Exploring Downloaded from http://teamat. oxfordjournals. org/ at University of Melbourne Library on October 23, 2011 How to teach exploring? What was exploring in the ‘olden days’ before technology? Let us look at the sample problem again.

What are the key phrases? —circle, center on line, line y = 3x, circle tangent to y-axis and on the point (0, 2).  We read: point (0, 2) and line y = 3x. We would have drawn the point and the line on paper.  We read: ‘circle with center on line and on (0,2)’ We would have taken our compass and drawn a couple of circles with the needle stuck in the line and going through the point (0, 2), looking for one that was tangent to the y-axis. This was the way we explored and learned. Let us do this in GeoGebra.  In GeoGebra, draw the point (0, 2) and the line y = 3x (Fig. 2). In GeoGebra, draw a point on the line (Fig. 2). Important: GeoGebra ‘knows’ that this point is on the line—it is a dependent object. Select the Move tool. Click and drag the point. The point will move but only along this line—just like moving our compass needle. This is the key to the exploring bit so we need to make sure that we (and they) understand that this point is not fixed, but a movable ‘exploring point’.  In GeoGebra, draw a circle with this point as center and through (0, 2) (Fig. 3). It is not tangent to the y-axis. But, we can slide this circle along line a.

FIG. 2. Point A, line a and Point B on a. (This figure appears in colour in the online version of Teaching Mathematics and its Applications. ) 72 L. FAHLBERG-STOJANOVSKA AND V. STOJANOVSKI Downloaded from http://teamat. oxfordjournals. org/ at University of Melbourne Library on October 23, 2011 FIG. 3. Circle with center B through A. (This figure appears in colour in the online version of Teaching Mathematics and its Applications. ) FIG. 4. Use move tool to click and drag B. (This figure appears in colour in the online version of Teaching Mathematics and its Applications. ) Select the Move tool. Click and drag the center point B. Now the circle also moves, but its center stays on the line and it always passes through the point (0, 2) (Fig. 4).  Move it along the line until the circle is tangent to the y-axis (Fig. 5). 3. 4 Thinking and algebra Now the mathematics thinking and algebra begins. Some good questions might be: What characterizes this circle that is tangent? Where is the center point? Can we find the coordinates? What is the radius of the circle? Finally, one requires that they do the math and find the equation of the circle itself.

Of course, we expect the student to check that his solution matches the one he gets in GeoGebra. GEOGEBRA 73 Downloaded from http://teamat. oxfordjournals. org/ at University of Melbourne Library on October 23, 2011 FIG. 5. Stop when circle is tangent to y-axis. (This figure appears in colour in the online version of Teaching Mathematics and its Applications. ) 3. 5 The whole process The idea is to get them started with things they know and leave them free to explore and think and then come back and get the mathematics out of them.

The first time this type of question comes up, we might help them build the GeoGebra file, but leave them alone to find a circle tangent to the y-axis. Since GeoGebra is freeware and available online and offline, we can send them home with the file or post it as an interactivity online ahead of time. That is—they can interact with the file freely and legally. We can give them a similar problem and they can design and post their own GeoGebra file. There are no restrictions on where, when and how they can use and share the files. 4. Sample Problem 2

At noon, ship A is 90 km west of ship B. Ship A is sailing south at 40 km/h and ship B is sailing north at 20 km/h. How fast is the distance between the ships changing at 2:00 PM? This is a related rates problem from calculus (Answers. yahoo. com, Calculus related rates problems? http://tinyurl. com/5jrxhp http://mathcasts. org/gg/student/calculus/RelatedRates/rates. html). Again within minutes, there were two answers—both algebraic/symbolic solutions with no visual aids. They will both turn out to be incorrect. We do not know the answer to this problem, but we are intrigued.

We decide to build a simulator with GeoGebra. It takes us 10 min. We post it online and link our answer to it. The simulator is interactive. It is adaptable and reusable, and we can legally post it on line, anybody can use it, download it, change it, translate it, etc. More importantly, is so easy to build and to understand that the student himself can do it. 4. 1 Making the simulator Draw point A = (0,0) and then draw B a point on the x-axis. Zoom out and move B to (90, 0). In this way, B will be a movable point east of A. Input t = 2, va = 40 and vb = 20.

We right-click on each of these and choose ‘Show object’. This makes them slider variables. Later we can adjust the intervals and increments. 74 L. FAHLBERG-STOJANOVSKA AND V. STOJANOVSKI Downloaded from http://teamat. oxfordjournals. org/ at University of Melbourne Library on October 23, 2011 FIG. 6. The simulator with values as in problem. (This figure appears in colour in the online version of Teaching Mathematics and its Applications. ) On this diagram, the position of boat A is a point At = A + (0,Ava*t) and the positon of boat B is a point Bt = B + (0,vb*t).

We draw ‘travel lines’, that is, a line segment a from A to At and line segment b from B to Bt. From here it is easy to see that the distance between the boats is the line segment c from At to Bt. Finally, we draw the line segment d joining A and B. We get the picture shown in Fig. 6. Now with this picture, it is easy to see that this distance c can be calculated using Pythagoras’ theorem. We use our dynamic variables ??????????????????????????????????? define the ‘distance function’ s(x). Input p d, va and vb to s = sqrt(d2 + (va + vb)2*x2) to get s? x? ? d2 ? ?va ? vb? 2 x2 . Here x is the variable for time.

We have already defined t as a number and t always has a ‘current’ value that is given in the Algebra window at left. So, t cannot be used as a variable. Indeed, the value of s(x) at x = t should give the ‘current’ distance, that is, the length of the segment c. This is a good way to check our function. Input dis = s(t) and check that dis = c. In the Algebra window, GeoGebra will show s(x) with the current values of d, va and vb. Changing these values (via the sliders or moving point B) will change this formula. 4. 2 Finding the answer We want the change in the distance s(x) with respect to time x.

That is, we want s0 (x), so we input sDer = Derivative[s]. GeoGebra will calculate s0 (x) and write the resulting function sDer(x) with the current values of d, va and vb. The answer to the problem is the value of the derivative when t = 2. So we input: ans = sDer(t) and then slide t to 2. This answer—that is, the value of ans—can be read in the Algebraic window at the left. For the given problem, ans = 48 km/h (Fig. 6). GEOGEBRA 75 Downloaded from http://teamat. oxfordjournals. org/ at University of Melbourne Library on October 23, 2011 FIG. 7. Simulator with changed values (answer not shown). . 3 Thinking and calculus This little simulator is fully functional; one can slide the values for time and speeds as well as move point B to change the east–west distance (Fig. 7). We can have them check the validity of their simulator by posing some good questions. Finally, we require that they do the mathematics. That is, they write down the function, find the derivative, calculate its value for the given t and check it against the simulator. 4. 4 Sample good questions Make the simulator simulate both boats starting at the same position. Set the speeds and animate the slider t from 0 to 5.

Describe the rate of change in the distance between the boats. Change the speeds and test again. What can you conclude? First, they must think about moving B onto A. Then, by moving t, they see that the two ships are just moving directly away from each other. They should notice that the ‘answer’ is constant. They check the function and see d = 0 so s(x) = (va + vb)x (no square root). The derivative is constant sDer(x) = va + vb, that is, the answer is just the sum of the two speeds. This makes sense. They have checked that their simulator works properly in a ‘known’ situation. 4. 5 The whole process

Again, one of the great things about these simulators (Fahlberg-Stojanovska et al. , 2008), the students can build them themselves is the stimulator of Fahlberg-Stojanovska (2007). This teaches them to focus on what is really happening in the problem—crucial in visualizing and understanding the mathematics. Second, a dynamic simulator allows them to check that ‘known’ cases are being solved correctly. If it does not, their solution must be incorrect. This is a vital—but often severely neglected—step in any logical thinking process. And it is the beauty of freely available technology. 76 L.

FAHLBERG-STOJANOVSKA AND V. STOJANOVSKI 5. Summary In Section 1, we asked why are young people losing their ability to analyse, organize, explore possibilities, deduce and test conclusions—the crucial life skills for problem solving. And what can we—as maths teachers—do to reverse this trend and at the same time increase both their interest and actual skills in mathematics? We must change our ways of thinking about freeware, exploring and learning. If we live in a richer country, we have tended to get the ‘big math programs’ for our schools or require graphing calculators from our pupils/students.

Our students look only shallowly at ‘entering’ the problem—the technology does the rest. If we live in a poorer country, we have tended to disdain this ‘non-mathematical’ approach and require our students solve the problems in the old way algebraically/symbolically without access to true understanding that comes from visualizing the problem. Both of these methods are ‘solution-only’ approaches. A good freeware program solves this problem. It allows the student to explore the maths whether at school or at home. It allows conversation between students, students and teachers, online and offline conversations, devoid of any legal problems.

This freedom generates a ‘exploring and thinking approach’ to mathematics as opposed to a ‘solution-only’ approach. Properly encouraged, the student will explore visually/geometrically until the idea of how to solve the problem algebraically/symbolically comes to him. A bonus is that after solving, the exploratory approach also allows him to check his answer. This is the beauty of mathematics and its true value in the everyday lives of our students. GeoGebra is free and easy to use. It is very versatile, continually being expanded and upgraded.

There is an extremely well organized forum for asking and answering questions about using GeoGebra and an international wiki for publishing GeoGebra resources (GeoGebra: www. geogebra. org GeoGebraWiki: www. geogebra. org/wiki GeoGebraForum: http://www. geogebra. org/forum). In this article, we give two specific examples from actual questions asked and answered at answers. yahoo. com. Incorrect answers given to both of these questions could have been—but were not—easily verifiable with GeoGebra. Further, by using GeoGebra, true exploration and visualization was possible, leading to an understandable mathematics solution to both.

The creation of dynamic GeoGebra files by pupils and students to explore, discover and then understand mathematics is both viable and useful on a wide scale. GeoGebra has been translated into many languages and so can be used across the world in a single, copyright free environment. This makes GeoGebra good for individual, collaborative discussions and for classroom and asynchronous interactivities—the freedom to explore and learn. Downloaded from http://teamat. oxfordjournals. org/ at University of Melbourne Library on October 23, 2011 REFERENCES FAHLBERG-STOJANOVSKA, L. , STOJANOVSKI, V. BOCEVSKA, A. (2008) ICT in math education – small interactivities & specific goals. Proceedings of the 6th Intl CIIT 2008. Bitola, MK. FAHLBERG-STOJANOVSKA, L. (2003–2007) The Boat Landing Problem Presentation for Simlab 2007. Bitola, MK: DAAD (see http://www. mathcasts. org/mtwiki/GgbActivity/BoatLanding1 [accessed 9 March 2009]. Linda Fahlberg-Stojanovska is a professor of Mathematics and Computer Sciences at the University of St. Clement of Ohrid, Bitola, FYR Macedonia. Vitomir Stojanovski is a professor of Mechanical Engineering at the University of St. Clement of Ohrid, Bitola, FYR Macedonia.