Exploring alternate strategies to build trigonometric concepts in Logo
Silvia Branco Vidal Bustamante
Coordenadora do Centro de Informática Educativa
Fernando Braga de Almeida
Frederico Luis Cabral
Roni Cordeiro Zillig
Universidade Católica de Petrópolis
E.mail: cies@risc.ucp.br
Fax: 55 242 433745
RJ  Brasil
Keywords:
Logo; trigonometry; mechanical reasoning; comprehensive mathematics;bugs; reformulation; alternate strategies.
Abstract
This approach concerns to the use of computer learning environments to introduce mathematical understanding structures. Through this project developed by the Centro de Informática Educativa at the Universidade Católica de Petrópolis, Rio de Janeiro, Brazil, young students, between 14  16 years old, are involved with divergent ways to construct trigonometric concepts and to apply them in meaningful learning contexts. The main object of this study is to explore mathematical resources and overcome the traditional reasoning. Some observations reveal the lack of understanding in school mathematics and the potential of learners in thinking through hypothesis, verification and reformulation process which constitutes the basis of thinking. Moreover, the study points the importance to restore comprehensive mathematics to follow unusual ways to discover the formal knowledge meaning.
1. Introduction:
The work with mathematics and computer programming language at learning environments proposes the interactive use of computers in a different context from the one students and teachers are using technological resources. Behind the newness of cybernetic at school, the main design of mathematical learning environments is to reestablish the understanding of concrete and abstract elements which compose the formal thinking. To accomplish this project young students are invited to an exploratory use of computers through the Logo language resources which wake up the need of thinking about the contents they have just acquired by mechanical reasoning .
Two important points mark the proposal here introduced: the opportunity of no more mystifying the conventional use of computers and the need of restoring a comprehensive mathematics: a different mathematics without using mathematical ideas before 'understanding' them (Hoyles & Noss, 1994). These activities are both very difficult: by one side, because computers are commonly machines to make easier the results of some tasks; by the other, because it is conventional the teaching of mathematical contents without worry about the internal process of understanding their constitutive elements.
Exploring mathematics at an alternate context of learning introduces the supposition that teaching activities commonly aim to obtain results without understanding (Clements & Battista, 1989). Students with good results in exact sciences not necessarily knows about their own internal process to construct mathematical strategies and structures. At this kind of environment, students work with mathematics without thinking mathematically. They "know what" but they do not "know how" (Swann, 1991).
The project development supposes the pedagogical approach the teachers may acquire. It is the technical preparation of programming and mathematical resources. Some didactic sessions are also important to change the traditional paradigm of teaching learning process in a new environment where an open system of learning is priority. In this sense teachers are invited to build problem situations in mathematical contents. These problems do not constitute a software to be given in an explanation to the learner, but the research field and the teacher domain about learning situations to be explored as defy by the students. From depart of these learning proposals, the students are being invited to construct their own algorithmic or creative problem solution. The project is planned by teachers without imposing but proposing defies to the thinking students development. Through their own ability, the environment stimulates the zone of proximal development referred by Hoyles & Noss (1994).
The teacher intervention is step by step more interactive. The students have the opportunity of perceiving new problems in each solution already acquired. Working with many different students it is possible to identify divergent lines at the building process of each student. It is also possible the diagnostic of learning problems in mathematical contents, or deviation in the perceptive domain. The two groups observed in Lab activities are demonstrating abstract information about mathematics but, with exception of four students, they are inept to apply this abstract information in concrete skills to win the proposed problem. There is absence of mathematical thinking. The students generally activate abstract and mechanical connections without learning about the meaning of its concerning operations. In this sense, the hypothesis is that students memorize mathematics.
The evident absence of autonomy in the approach of the problem creates the need of the answer without thinking about the question. It reveals a symptomatic behavior of a cultural learning environment which encourages the prompt answer gave by the teacher in conventional school. It obstructs the chance of thinking, not explored but inherent to each student.
2. Exploring "bugs":
Introducing the problem and the project with support of Logo as appropriated to mathematical domain ( Blanchet, 1991), debugging activities wake up cooperative discussion about the procedure performance. Some initial troubles are made as follow:
to tr2 lt 90 fd 100 rt 90 fd 100 rt 135 sqrt 20000 end 
Fig. 1 
The initial phases exploring trigonometric relations with programming languages implies some confusion concerning the commands organization, producing failure in the graphic representation. In this case (fig.1), the student understood the need of sqrt without implementing the command fd to plot the hypotenuse. It is a typical example of an initial desequilibration between the mathematical problem conception and the programming syntax resources of problem solving. It involves hypothesis, bug, verification and reformulation processes including not just the calculus, but also the relation with concrete mathematics and geometric execution of the figure. It points the distance between formal and concrete concepts, establishing the need of retroaction ( Piaget, 1976) and dialectical thinking from the concrete to the objective building of abstract procedural orders. They are both, syntax and semantic, involved at the information process.
to triangulo :x :y lt 90 fd :x rt 90 fd :y rt 180  arctan ( :x / :y ) fd :x * :x + :y * :y * sqrt * 2 end 
Fig. 2 
The failure in this building (fig. 2) consists on mathematical syntax error, over and above a misconception about the programming syntax, specially linked to fd sqrt to plot the last side, although the use of more sophisticated resources as arctan. This deviation is discussed and reformulated among the students during the work to execute the correct figure. Therefore the use of commands instead of variables assumes a local value to the procedure execution. Besides hypothesis, bug and reformulating process, this example presents syntax mathematical failure and syntax programming difficulties of implementation, often verificable at the start of the process development in computer environments. It requests some mathematical and programming review during the work with Logo proposal. Moreover, it enhances the comprehension of the syntax to express formal contents and semantic resources to get the correspondence between the concept relations and the graphic meaning of the procedure.
to estrela make "a sqrt (10400  (4000*cos 22.5)) make "b sqrt (6800  (3200*cos 22.5)) make "alfa arccos ((9600 + ( :a*:a)) / (200*:a)) make "gama arccos ((:b*:b + 6000) / (160*:b)) make "delta arccos ((:a*:a  9600) / (40*:a)) make "teta arccos ((:b*:b  6000) / (40*:b)) fd 200 bk 100 rt 90 bk 100 fd 200 bk 100 rt 45 bk 80 fd 160 bk 80 rt 90 bk 80 fd 160 bk 80 rt 22.5 bk 20 fd 40 bk 20 rt 45 fd 20 bk 40 fd 20 rt 45 fd 20 bk 40 fd 20 rt 45 fd 20 bk 40 fd 40 lt (180  :delta) repeat 3 [ fd :a rt 180 rt 2*:alfa fd :a rt 180 rt :delta rt :teta ~ fd :b rt 180 rt 2*:gama fd :b rt 180 rt :teta rt :delta ] fd :a 

The third example (fig.3) shows a correct employ of variables without perceiving the essence of the command repeat: another very long way to develop the procedure does not damages the final result but involves some devaluation of a simple resource as iteration. As the figure was explored step by step, the hypothesis is that the last part of the star was ended by empirical strategy. This last example, introduces the particular thinking without passing for syntectic resources which allow the best performance at the procedure building. The thinking that the skill operates with, is yet at the same time concrete and abstract ( Swann, 1993), not presenting the entire comprehension of all language resources. In spite of this failure, the global variables are correctly declared before the procedure development, processing a good performance in relation to thinking anticipation. There is a more advanced conception of variable resources at programming skills also applying the creativity of their own project involving trigonometric concepts. It works with trigonometric functions applying them in an innovation figure through variables to be used in distinct situations.
Thinking through the bugs is an opportunity to verify the information and to reformulate cognitive strategies. This is the moment of a review of the process. Using some of these skills, the nature of the process is then explored by hypothesis, analysis, verification and reformulation besides other alternatives as discussion and choice of the best and not just the correct way to overcome the failure. Moreover, the creativity of deeper and divergent process is also possible in geometry and mathematics (Lichnerowicz, 1967), developing the sense of art and understanding involving formal structures.
3. Observing different strategies:
During the sessions, different cognitive styles are observed. Besides this general approach, working with initial defies, some different strategies of constructing problem solutions are planned and observed as:
to trapezio :h :b :R cs rt 90 pu fd ( :R / 2 ) lt 90 pd lt 90  (arctan (2 * :h) / ( :R  :b)) fd sqrt ((:h * :h)+(((:R  :b)/2) * (:R  :b) /2)) lt 90  arctan (((:R  :b) /2) /:h) fd :b lt 90  arctan (((:R  :b) /2) /:h) fd sqrt ((:h * :h)+(((:R  :b)/2)*(:R  :b) /2)) lt 180  arctan ( 2 * :h)/ ( :R  :b) fd :R end 
Fig. 4 
Constructing the trapezium the student makes the fig. 4 and just employ local variables. The used strategy access the processor to calculate each arctan value. Working with a machine without mathematic coprocessor it is possible to verify the slow processing. It supposes empirical and local domain of trigonometric resources, without exploring another more global dimension of variables use, which would enhance the procedure style and represent an economy of particular instructions. This correct procedure does not explore more general parameters which would simplify the work and the execution. The thinking remains local and specific to the encountered problem and should be debugged through some evolution in the variables use.
to trapezio :a :b :c rt 90 fd :a make "hip sqrt ( (:c*:c) + ( (:a:b)/2 )*( (:a:b)/2 ) ) make "ang arctan ( :c / ( (:a:b)/2 ) ) lt 180  :ang fd :hip lt :ang fd :b lt :ang fd :hip lt 180  :ang end 
Fig. 5 
The same trapezium is here built (fig.5) creating and declarating global variables. Bringing into relation the fig.1 and the fig. 5, there is the variable "hip, which receives sqrt (( :c * :c) + (( :a  :b /2) * (( :a  :b /2)). Later on, he uses fd :hip to plot something like fd sqrt and its value. In this case, just two variables are used, involving a more elegant programming style and optimizing the procedure performance without accessing the coprocessor at each command use. The passage to global levels of thinking implies at the same time more efficient processing and result on procedure execution. Moreover, the variables are not yet declared at the start of the procedure as would be in an anticipation of what is used in the algorithmic structure. This procedure reveals global and abstract mathematical programming thinking applying it at distinct moments of the problem solution.
to trap :h :t :b rt 90 fd :b make "alfa arctan :h/((:b:t)/2) lt 180:alfa make "hipo sqrt((:h*:h)+((:b:t)/2)*((:b:t)/2)) fd :hipo make "beta 18090:alfa lt 90:beta fd :t lt 90:beta fd :hipo end 
Fig. 6 
Trapezium built with three global variables (fig.6) revealing an interesting thinking resource. The variable "beta at this procedure is an useless variable which would be expressed by lt :alfa. This figure presents the thinking design executing some no objective operations, but presents the deviation or the divergent strategy to construct the same result. This deviation may be by inobservation of unnecessary ways, as quite as using something which will be later used in opposite sense. This process presents no complete domain of procedural structure building, and although there is no bug in result, it is possible to identify bugs in its internal structure. Some other approaches or examinations in the same procedure would identify the useless way applied at this procedure, conducting to a logic reformulation process. It characterizes the steps behind pointed by Noss and Hoyles (1991).
to trapezio :x :y :z rt 90 fd :x make "alfa arctan(:z/((:x:y)/2)) lt (180  :alfa) make "hip sqrt ( (:z*:z) + ((:x:y)/2)*((:x:y)/2) ) fd :hip lt :alfa fd :y lt :alfa fd :hip end 
Fig. 7 
This trapezium building (fig7) presents the same design as the one in the fig. 5, just without declaring the global variables in order to anticipate general parameters to the procedure, including the calculus of lt (180 :alpha) between the two global variables. Over and above, with the procedure development, thinking will acquire a global style of programming overcoming particularities and presenting a general structure through an objective problem formulation and reformulation.
Working with local variables, the extented value to the procedure is bounded by the structure of itself. The importance of declaring global variables concerns to enhance these values showing another grade of abstract thought, applying them in distinct particular moments of the procedure. This strategy reveals general domain of variables function.
4. Alternate ways to explore mathematics:
All these distinct strategies result from different views and different ways of the knowledge access. It reveals that this appropriation is not built by the same linear process by different subjects. It is important to detach that although the result is the same, the procedure building is different for each student, with local or global variables. This point marks distinct ways to perceive the problem, to organize the elements and to provide a problem solution according to individual concrete or formal stage of understanding and operating resources. To define local and global frames involved in similar problems improves the activity of thinking mathematically.
Moreover, to join math formulas and programming syntax is a slow process. There is no previous living experience of computer language programming among these students. The purpose is not only to understanding trigonometric functions but to discriminate when, how and why to apply the abstract formula in a functional procedure of a programming activity.
Thus, to build knowledge structures implies investigation of distinct strategies. Teacher intervention consists on let students to go through empirical basis of mathematical resources, trying to do some formal representation of their own thinking. In a later phases, after the subjectivity of hypothesis, students discuss similarities and differences, usual or new strategies, debugg failures, verificate and reformulate the subjective approach searching for the objective one.
Cognitive and social discussion contributes to overcome the problem. In spite of this, it needs to discriminate if:
 the mathematical knowledge is correctly prescribed to
implementation;
 the programming algorithm is correctly designed;
 the creative structure is adequately implemented;
 the commands are appropriated to accomplish the intended
result;
 the procedural structure organizes the commands with syntactic
conditions to execute the semantic meaning of the problem;
 the graphic simulation reaches in all conditions the problem
specification.
Sometimes this goal accomplishment is only reach by local or determinate values. The application in the global aspect of the problem is difficult. In other words it is possible to work with specific values but is not realizable with general values. This point marks the need to apply formulas and programming resources with more abstract parameters which consider all the variables involved on the problem.
In this approach, to understand the mind working with different contents in similar problems is to know the essence of learning.
Besides distinct strategies, there is a reasoning to win the subjectivity of the hypothesis towards the objectivity of mathematics. Moreover the environment allows exploration and creativity (Lichnerowicz, 1967), the open investigation constitutes the chance of question mathematical contents in sense of overcome conventional information, discovering the thinking as a tool. Another important conquest is the learning through the "bugs" as an opportunity to verify the knowledge and reformulate cognitive resources to construct it.
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