ResearchtheAppli_省略_daryMathTeaching_Yan.doc

2012 International Conference on Computer, Control, Education and ManagementLecture Notes in Information Technology, Vol.27Research the Application of the “Geometers Sketchpad” Software to Elementary and Secondary Math Teaching Yang Huansong, Shen ChuxianHangzhou Normal University, China a , Keywords: Geometers Sketchpad, elementary,mathematics teaching,research. Abstract.The “Geometers Sketchpad” software (GSS) is an excellent teaching software for math inelementary and secondary schools. Taking full advantage of its characteristics of convenient,operative and dynamic, one can achieve a multiplier effects with less efforts, giving rise to theinnovation of elementary and secondary math teaching (ESMT).1.IntroductionA mouth, a piece of chalk and a blackboard cannot meet the needs of the current ESMT. Theappearance of the GSS has greatly propelled the reform and brought the new spring for ESMT.What we should research into is how to apply the software correctly and efficiently to ESMT.2.Apply GSS to ESMT2.1 Stimulate the interests of math learning effectivelyGSS can not only create context and make students participate in it initiatively, which has thefeature of the game that can stimulate studentslearning interests. Therefore, it can turn the abstractand boring math concepts into intuitive and visual, so that students will convert the fearful anddisgust feeling of math learning into love and willing to learn. A dancing triangle, a scrolling wheel,a beautiful snowflake curve, and a retractable tree of Pythagorean all deeply attract the attention ofstudents. According to the survey, the students have showed great interests in GSSs features of“accurate, dynamic, intuitive, experimental, efficient and process”. In addition, GSS is conductiveto cultivate students math aesthetic ability, by fully displaying the beauty of dynamic and variableof geometric figures.2.2 Raise the efficiency of ESMTAs the conventional teaching method with a mouth, a piece of chalk and a blackboard, the teacherswould take a great amount of time and efforts to deal with the tasks about drawing figures, diagramsand processing data, which are overloaded and hard to be accurate. However, teachers are able tosave teaching time, increase teaching capacity and improve teaching efficiency, if taking advantageof the GSSs functions of drawing, calculating, transforming and recording. Only gently drag thecomputer mouse, the varying of figures, the transforming of function figures and the making ofcomplex figures can be easily achieved.2.3 Form the relevant math concept faster for studentsThe students with little math experience can hardly understand the math concept due to the highlyabstractness. While with the dynamic geometer circumstance and the cognitive circumstance ofoperational activities, GSS can make the forming and learning of math concepts easier. For instance,978-1-61275-030-9/10/$25.00 2012 IERICCEM 2012262 when it comes to the concept of right triangle, we always draw some typical right triangles on theblackboard. Once drawing another different angle, students have to distinguish them again. GSSallows students to operate figures and display the various modes of them. During the above process,students can observe the right triangles in difference shapes and the comparison among righttriangles, acute triangles and obtuse triangles. The dynamic operating creates space and condition forstudents comparison and abstraction. Thus, students can abstract the figures, then grasp the abstractconcepts and achieve math experiences.Examples:Formation of the concept of “angle”The concept of “angle”1(static state): the figure consist of two rays which have a public endpointcalled angle.The concept of “angle”1(dynamic state): the geometric figure formed as a ray rotating round itsendpoint called angle.To the students with little math experience, the dynamic concept is very abstract. We can teachthem with the operations from the static to dynamic. First, we draw two rays with a commonendpoint and measure its degree. Then, drag one of the rays to rotate round the endpoint. During therotating, various angles appear. Here are 00 angle, acute triangle, right triangles, obtuse triangle,straight angle, reflex angle and round angle (figure 1).mAOB = 90.0mAOB = 0.0mAOB = 43.6mAOB = 142.0BBBBOAOAOAAOmAOB = 180.0OABBOABOAFig.1. types of anglesIn the above process, students have seen the “rotation”, the “transformation” and “combination ofnumbers and figures”. It fills up their blanks of math experience and then forms the concept of“angle”. Furthermore, students can experience the forming process on their own operation.Formation of the concept of “fixed point”Students may get confused about such an abstract concept as a function figure that contains a “fixedpoint”. “Fixed” is a concept relative to “moving”. Lacking of the ability of spatial imagination,Students cannot be able to regard a static function relationship as a dynamic function figure, so theywould feel nothing about the “fixed point”. GSS can visualize the abstract concepts to help studentsfind them.For example: ask for the fixed point that the figure of function y=kx+1 contains. We can creationthe figure of function y=kx+1, then drag the parameter k to let students observe the dynamicchanging. Students will find that a family of straight lines is rotating around point (0,1) with thechanging value of k. Students shout“weve find the line”, which is“tied” (go thought) to thepoint (figure 2). Then, we study how to work out the question of fixed point with parameter in thefunction figure. Students find many methods (such as changing into“0,0”type, particular valuemethod, figure method and so on) to solve the problem. From now on, the students have the mathexperience of“fixed point”.263 Fig.2. function y=kx+12.4 Break through the teaching difficultyGSS can transform static into dynamic, abstract into intuitive, and the concepts that not easy tograsp into easy ones. The place that students thoughts may not able to reach also can be reachedwith the help of GSS.Studying the functional relationship in a dynamic figure and discussing the issues in segmentsare very hard for students to learn and for teachers to teach. While, if use GSS, teachers will find itmuch easier to teach.Example: as the square AOBC (figure 3), point C has coordinates (4 2，0). Dynamic points Pand Q start from point O. P is moving along the direction of OACB one unit length per second. Q ismoving along the direction of OBCA two units length per second. P and Q stop moving till theymeet. Assume SOPQ=y and duration is t: (1) When and where will P and Q meet? (2) What is thefunctional relationship of y and t? (3) By drawing figure, evaluate the value of t when y has themaximum value.yAPxOCQBFig.3. movement of P, QThe functional relationship of y and t need to be discussed in segments during the movement,which is a high demand for students ability of spatial imagination.Many students may discuss the question incompletely or have no idea about why should segmentand how to do it, because of lack of math experience. Making the following figure (figure 4) canhelp solving the problems.AAAAAPP11PPP111QCCQOCOCOC222OO22-1Q-1-1Q-1-1BQBBBBFig.4. segmentsPoint Q moves along with point P. (This needs some skill to keep the speed ratio of points P and Qremaining 1:2 and enable the two points to turn a corner.) As soon as point P passes the midpoint ofOA, point Q moves to side BC. Students see exactly how the shape ofOPQ changes (from a right264 triangle to an acute triangle). After that, point Q moves to side AC, the moment point P moves toside AC though point A. Then, the shape ofOPQ changed again (from an acute triangle to anobtuse triangle). Through observation, students understand “why should segment and how to do it”and break though the teaching difficulty with the assists from GSS. At the same time, teachers donot have to draw and explain the figures on the blackboard. Dragging the computer mouse gently isthe only thing they need to do and all the cases are included with little efforts.2.5 Fully reveals the inner links among the knowledgeStatics figures make the relative knowledge apart and lose their inner links. Students only focus onthe fraction and neglect the whole part. Fortunately, GSS can overcome the obstacle. Example:Students will face such a proposition when learning the median of trapezoid as “half of thedifference of the lengths of the bases is equal to the length of the line between two midpoints oftrapezoids diagonals” (figure 5). Their knowledge is still and isolate, though they can prove it usingthe median theorem. In the teaching, GSS can make the following figures. We can operate it asfollowed: drag point A moving towards point D. When they coincide, the median of the trapezoidABCD becomes the triangle ABCs. Keeping moving point A across point D, EF becomes the linebetween two midpoints of the diagonals of trapezoid. During the above process, measuring thelength of segments EF, AD and BC and compare their links, we can unify the three lines of themedian of trapezoid, triangle and the one between two midpoints of the diagonals of trapezoid.Teachers can even raise the concept of “oriented segment” to contact them together.Fig.5. median of trapezoid2.6 Cultivate research ability and creative thinking of studentsGSS is a great researching tool, which can be used to discover, research, display and summarizemath disciplines. As the following figures 6: Firstly, randomly make three points A,B and C oncircle O, then the angle of circumference ABC appears, measuring its degree at the same time. Dragpoint B and ask students to observe the figure. They will find the value ofABC is fixed. Theywill wonder the item decides the degree ofABC. The location of Point B does not decide thedegree ofABC. What about anything else? What about the size of the circle or the arc interceptsABC? Asking the succession of questions, students are guided to do kinds of researches. Aftertrying, they are aware of that the arc interceptsABC changes along withABC. Thus, theythink the crucial factor to the degree of the angle of circumference is the arc intercepts it. Becauseof the radian is up to the central angle, students can associate the angle of circumference and thecentral angle, which intercept the same arc. They find that the degree ratio of the two angles is 1:2by measuring. Trying to drag point B to make side AB go though the center of circle, they find theequal radius constitute a isosceles triangle (the exterior angle of isosceles triangles vertex angle istwo times of the base angle). At last, students discover that the location of the center of circle hastwo different cases when it is in or out of the angle of circumference, which is the reflection of thefeature of isosceles triangles. The whole researching process above consists of observing, assuming,verifying and proving. It shows students initiative spirit and mathematical thinking activities.Not only can GSS provide students with exploratory learning circumstance, but also cultivatetheir creativity.265 Fig.6. angle of circumference2.7 Change the teaching mode and learning methodIt is a new teaching mode that allows students to participate in teaching directly in the computerroom. This new mode lets students learn math during operating GSS. The endless talking ofteachers will be replaced by doing experiments on students own. In this teaching mode ofexperiments, students get the results from their operating the software, observing the target,measuring, counting and analyzing the data and summarizing the varietal cases. It breaks the“teaching, practicing, and memorizing” conventional teaching mode and turns the process from“instructing-imitating and practicing-consolidating and memorizing-testing and assessing” into“question-assuming-experiment and verifying-proving and explaining-expanding”. In this way,students always keep the heavy interests of study, enhance the confidence of math learning andtruly enjoy in it. They no longer take math learning as burdens. Their abilities such as practicing,observing and summarizing are all well developed.3.Pay attention to the problems applying GSS to the elementary and secondary mathteaching3.1 GSS is just a assisting toolHow to take full use of GSS to optimize the math teaching? Besides enhance the operating ability,using correct teaching ideas to control it is more critical. GSS is just a tool for assisting the mathteaching. Taking good advantage of it can optimize the teaching quality. While the application isaimless against the teaching and cognitive rules, it may have the opp