EARLY EXPERIENCES IN DYNAMIC VISUAL REASONING IN A COMPUTER ENVIRONMENT

1. EARLY EXPERIENCES IN DYNAMIC VISUAL REASONING IN A COMPUTER ENVIRONMENT • This poster presents a computer environment called Tarta, a tool built with two goals: a) analysing the effects of different learning environments at the elementary school level; b) understanding the visual–spatial thinking of learners. That computer environment was carefully designed so that emphasizes the exploration and understanding of geometric transformations, isometries (translation, reflection and rotation) through the rigid motions (“slide”, “flip” and “turn”). General description TARTA is a microworld built over the Megalogo language[1] where two basic shapes can be controlled with 20 easily accessible simple commands. The word TARTA is a nickname for TARTARUGA (the portuguese word for TURTLE, used in a cartoon series the NINJA TURTLES, very popular in Portugal). The two objects are the Tarta (a right triangle) and the Azulejo (a tile, a square with two triangles inside). These two objects can be easily moved (slided in horizontal and vertical directions, turned around a point, flipped around a line). The user may change the position, size and colour of the Tarta and the Azulejo . In TARTA microworld we can easily define points and lines, but the angles must be multiples of 15º and there are some restrictions on the movement of the objects. The TARTA provides the learner with a set of auxiliary tools, for instance: “undo the last built figure” (desfaz, d); “erase the previous figure” (ApagaAnterior, apa); “erase auxiliary points and lines” (refaz). The basic commands have the following meaning: T - Build a simple object Tarta (with smaller leg equal to 30) Tarta :n - Build an object Tarta, with smaller leg equal to n, 10  n  300. A - Build a simple object Azulejo, with side 30. Azulejo :m - Build an object Azulejo with side m, 10  m  300. limpatudo - Erases all the screen and all the objects from memory. passo (pa)[2] - Shows the possible steps of the objects to be moved: integers from 1 to 300; the great leg measure of Tarta (CM); the diagonal measure of the Azulejo (CD). linha ( l) - Draw a line. ponto (p) - Choose point on the screen (then use the arrow keys to place the point). vira ou reflecte - Flip an object around a line but after choosing the flip line. roda :m - Turns right an object m units (360 units is a full turn) after choosing the turn centre. rodar :n - Turns left an object n units after choosing the turn centre. deslizad (dd) - slide right an object, passo (pa) units. deslizae (de) - slide left an object, passo (pa) units. deslizac (dc) - slide up an object, passo (pa) units. deslizab (db) - slide down an object, passo (pa) units. How the environment works? Firstly there is an introduction screen. The user should press enter or any key to continue. The second screen allows to enter in the environment in different modes: as a teacher (Professor), as a new pupil (novo aluno), as a pupil returning to continue the previous work. In this environment a learner acts on each representation by writing a symbolic command and sees immediately the result of each action. The learners can use this feedback to correct their own mistakes so the responsibility of correction is put on the learner. There are specific error message boxes which help the learners to identify incorrect commands and data. The TARTA also provides help and control to the learners (pupils or teachers) through a window called “Information” (Informação, I). The teacher has access to more features. [1]Megalogo is another Comenius Logo reference, Comenius University-Bratislava. [2] There is a box on the screen with this same name which point out the step prompt to the object is going to be moved. Types of learning sought: - Build, in an informal way, understanding about the characteristics of geometric transformations (translation, reflection and rotation) through the motions slide, flip and turn. These motions are used in a friendly environment and somehow connect mathematics with art. - Develop concepts such as: angle, measurement, centre of rotation, axis of reflection and congruent figures. - Familiarize with geometric symmetry and patterns (tessellation), recognise and create patterns with rotational and reflectional symmetry. - Develop visual-spacial reasoning: a) perform visual-spacial mental operations (to build relationships between visual images); b) develop different modes of visual-spacial thinking; c) exercise imagery using concrete, kinestesic or dynamic images, manipulating images and motions, mentally planning and using mental enactment; d) use abilities such as: visual estimation, structuration, coordination, verbalization, translation. Uses of the computer environmentwith pupils of 4th grade The TARTA was used in a computer room of a School of Education[1] by eighteen pupils (nine or ten years old) of a group of 4th grade of a Coimbra elementary school, 1996, during five sessions (45 minutes each one). This group had not any previous experience with computer in the classroom and the microworld was used only after the pupils worked with the motions (flip, turn and slide), with other manipulative materials. The pupils were divided in groups of two. In the beginning of each session, the tasks were carefully explained to the group by the researcher. The researcher and the pupils´ teacher directed the pupils´ attention to central ideas.The tasks given to the pupils were only related with one of the objects, the Tarta. In the task denominated “borders” pupils use the motion “slide” for the first time in the computer. The commands relatives to motions “turn” and “flip” had been already used by the pupils in another tasks. This task involves 3 geometric designs connected with roman mosaics and the degree of difficulty is increasing. In the task called “flower of the Indian” the pupils were already familiarized with the all features of the microworld.  Main findings with “borders” - The pupils had no difficulties to build the first and the second border. But with the third border, it was different: almost all pupils when they had to make the Tarta with a 3nd different colour asked for help. To overcome this, the researcher suggested those pupils to simulate the Tarta´s motions on the screen, with a Tarta made in paper to identify the turn centre and the amount of turn of the Tarta. - The motivation and the level of participation of pupils were high. Main findings with “Flower of the Indian”: - Only a group of the pupils executed the task without any help. Four groups of pupils finished the task, asking for support. Three groups of pupils did not finish the task even with some help. - Some pupils had difficulty in the organization of the work (to draw a plan to make the task). - Sometimes pupils did not see the difference between what was made on the screen and the figure on the paper sheet, or they had some trouble to identify the Tarta as an element of the whole figure. - Some pupils had difficulties to anticipate the different motions to build the “flower of the Indian”. - The pupils used kinesthesic images to identify the motions. Obstacles - Some pupils show: *    Weak development of some mathematics concepts (horizontal and vertical directions and angle). *    Little familiarization with the motions. *    Troubles to cope with some features of TARTA, to define a point or a line, to change the steps of the objects. *    The shapes built on the screen sometimes were not recognized. - The motivation or the participation level of pupils decreased when: *    Collision arose between the peers of each group, due one of the pupils wanted to do everything and the activities fall under a unique pupil. *    The teacher or the researcher did not arrive at the right moment or the teacher was not able to help the pupils to overcome their obstacles. - Type of difficulties that pupils reported to have met. Three of the pupils reported they had no difficulties with the two tasks. Two of the pupils said they had some trouble with the task “borders”and fifteen of the pupils reported difficulties with “flower of the Indian” task. The indicators of the difficulties types are: 1) “borders” task: “To fill spaces” ; 2) “flower of the Indian” task: “to change the Tarta´position”; “ difficulties in passo command (2 pupils)”; “to do the lateral parts”; “to flip”; “in the top of flor do índio”; “I was nearly always writing refaz, desfaz, apa, etc; “To discover how to do”. [1]The pupils´ School was very near the School of Education. TARTA TARTA Mosaic Build the pattern below, with the help of the Azulejo. The idea of this mosaic comes from the House of the Mosaic Founatin in Pompeii where it is part of the floor of the atrium surrounding the impluvium. Borders Call the Tarta and build the borders you see. saw spine Tricoloured composition USESWith Teachers (graders k-4) in-service training The computer environment was used in the same School of Education in Coimbra, by two groups of experienced teachers, in courses about “Technology and Mathematics Education”. One of the groups included twelve teachers of grades k-4 in 1999, and the other group involved only seventeen kindergarten teachers in 2000. Some of the reasons why it was considered crucial to do those designed experiences: to see what really happens when the microworld is used, that is to say, to evaluate it, and mainly to infer how is the visual-spacial thinking of these teachers in that context. Those groups had not any previous experience with the computer in the classroom, they never used TARTA and it no preparation was made with respect to geometric transformations. The methodology used with 4th grade pupils is again repeated with these teachers, but each teacher had a computer set and the time spent were 279 minutes for one group and 200 minutes for kindergarten teachers. Also the teachers used both objects the Tarta and the Azulejo. Mosaic is one of their tasks. Main difficulties encountered by the teachers using TARTA: - “to imagine the procedures to obtain the desired image” - “to understand the command passo and CD” - “to define a line to flip and the command virar” - “to use the command apa” - “to define the turn centre and the angle of rotation” - “to use the turn motion” Observe that these difficulties are very similar to the pupils difficulties. Some indicators of teachers´ feelings relatively to the microworld and to the learning it stimulates: - “It has a lot of creative perspectives in terms of images” - “It is impelling and touching to the development of spacial sense” - “It supports the construction of mental images contributing to the development of visualization and spacial reasoning” - “It discovers and it develops the mathematical language” - “It is a good way to develop the aesthetic abilities” - “It gives possibilities to make very creative arrangements with only two basic elements” - “The TARTA can be used by children (five years old) using the object Azulejo and the motion “slide”. The activities with TARTA should always be integrated with activities directed to kindergarten such as playful activities and/or activities using pattern blocks” - “I learnt to use the commands and to think to use them” - “I learnt to use with greater precision the mathematics concepts” - “I learnt to imagine a picture before drawing on the computer”. TARTA Flower of the Indian Build the flower you see on this page, with the help of the Tarta. Conceição Costa (ccosta @esec.pt) Escola Superior de Educação Instituto Polictécnico de Coimbra