**Designing Logo Interactive Activities for the ** **Mathematics Programs of the Mexican School ** **System **

**Ana Isabel Sacristán, ***asacrist@cinvestav.mx*

Dept. of Mathematics Education, Center for Research & Advanced Studies (Cinvestav), Mexico

**Nicolás Tlachy, ntlachy@yahoo.com **

EFIT-EMAT, Instituto Latinoamericano de la Comunicación Educativa (ILCE), Mexico

**Rocío Escobedo, rocio_tierra@hotmail.com **

EFIT-EMAT, Instituto Latinoamericano de la Comunicación Educativa (ILCE), Mexico

**Abstract **

In the past decade, the Mexican Ministry of Education has been making intense efforts for incorporating digital technologies. One of such efforts is the Enciclomedia programme which provides teachers with a system of computer interactive resources and activities designed to be used mainly on electronic interactive whiteboards. We were asked to design Logo interactive activities for such a system, to be used in the lower secondary mathematics programs of Mexico.

This presented many technical and didactic challenges, the foremost of which was the difficulty of preserving the spirit of Logo, and its benefits as a programming and constructive environment, in a situation where the interactive activities have to be used as self-contained “instructional”

presentation tools by the teacher on an interactive whiteboard, with limited typing possibilities.

The didactic design was thus crucial. We give an overview of the interactive activities we designed (such as the one in Figure 1), and exemplify the didactic design through the detailed description of one of the activities.

*Figure 1. Scenes from the Logo interactive activity “Randomness and Probability” which includes a turtle *
*race: each turtle goes forward depending on what number is generated by a dice. But the race is unfair, *

*and users have to predict the rules that make each turtle go forward, and modify the rules to make the *
*race fair. *

**Keywords **

Mathematics; interactive activities design; Logo; interactive whiteboard; Enciclomedia

**Introduction: Technology in the Mexican School System **

Since 1997, the Mexican Ministry of Education has been making intense continuing efforts for
incorporating digital technologies into the classrooms of the basic education system (primary
and lower secondary levels)^{1}*. Some of the largest projects in this effort are the Teaching *
*Mathematics with Technology (EMAT) and Teaching Physics with Technology (EFIT) *
programmes for lower secondary schools (children 12-15 year-olds), which began in 1997; and
*the Enciclomedia programme for primary schools, which began in 2003. The EMAT programme *
provides activities and a constructivist pedagogical model for incorporating the use of
technological tools in classrooms in order to enrich the teaching and learning of mathematics
(Ursini & Rojano, 2000). The EMAT model promotes student-centred exploratory and
collaborative activities in computer laboratories (or with graphing calculators). The main tools
currently used in EMAT are Spreadsheets, Dynamic Geometry, Logo and CAS activities with the
TI-92 calculator. On the other hand, Enciclomedia –which has been massively implemented in
all public primary schools^{2} in Mexico in the past two years— aims to help teachers by providing
resources, computer interactive activities and strategies –designed to be used mainly on
electronic interactive whiteboards— through links in an enhanced digital version of the
mandatory textbooks (Lozano et al., 2006).

In 2005-06, an extension of the Enciclomedia model to the lower secondary level was
*considered; and particularly, incorporating the model to the “Tele-Secondary” (Telesecundaria) *
School programme. The latter programme –which began four decades ago, in the late sixties, as
a very innovative project (Castro et al. 1999)— is an educational model of the Ministry of
Education that aims to reach the wider community (e.g. in rural areas) that may not have access
to regular lower secondary schools: in a Telesecundaria school, learning has traditionally been
structured through three types of educational materials: learning guides, content guides, and
television programs; with one teacher-promoter for all subjects. Despite its successes, the
Telesecundaria programme –because of, for instance, fixed transmission schedules of the
television programs— didn’t allow for many opportunities for students to express, interchange
and develop ideas. The Telesecundaria model has recently been renewed: the vision has been
to design learning activities that promote discussion, collaboration and critical analysis through
the use of a variety of resources and didactic materials, with the teacher acting as a link between
the students and the knowledge. In this renewed model, information and computing technologies
(ICT) are seen as the potential agent for change, not only to enrich the teaching and information
resources and forms of representation, but also to create situations that promote discussion and
communication practices in the classroom and in which the student can have a more active role
in his learning. At a first level, Telesecundaria schools are being equipped with the Enciclomedia
hardware: that is, a computer for use in the classroom with projection equipment for multimedia
material (e.g. on an interactive whiteboard) thus giving the possibility to carry out interactive
activities. At a second level, the vision is that these schools will have media labs with computers
and/or other tools like graphing calculators and sensors.

Thus, in the past two years, intense efforts have been made to develop activities for both the possible lower secondary Enciclomedia programme, and the new model of Telesecundaria. In the case of mathematics, since the use of the EMAT tools and materials is expanding in the country (and a great proportion of lower secondary schools already use them), and it is hoped that eventually the Telesecundaria schools will also have the possibility to use them, it was

1 According to the official statistics of the Mexican Ministry of Education (http://www.sep.gob.mx/work/appsite/nacional/index.htm, retrieved 30 March, 2007) in the academic year 2005-2006, there were over 14.5 million students registered in primary schools;

and almost 6 million students in lower secondary schools, out of which 1.2 million (20%) were in Telesecundaria schools.

2 Mexico has over 90,000 public primary schools.

decided to try to develop some activities that would create a bridge between the EMAT activities, and the interactive activities for the Telesecundaria and lower secondary Enciclomedia programmes. In particular, since Logo is one of the EMAT tools, we were asked to design interactive activities using Logo for the mathematics programme of Telesecundaria (and possibly of the regular lower secondary schools) to be used with the Enciclomedia system.

**The Challenge: Teacher-centred Logo Interactive Activities **

### The didactic challenges: preserving the spirit of Logo

The interactive activities using Logo had to be designed to be used mainly by the teacher in an interactive whiteboard environment. We were thus faced with a great challenge. The Logo philosophy promotes student-centred exploration and construction; a constructionist approach to learning where “programming itself is a key element of this culture” (Papert, 1999; p. xv).

The Logo EMAT activities, while trying to comply with a set school curriculum (see Sacristán, 2003), adhered to the Logo philosophy principles: For the incorporation of Logo into that project, we placed emphasis on a constructionist approach (Harel & Papert, 1991) where mathematical learning could be derived from student-centred programming activities.

With the interactive whiteboard we had to design activities to be used by the teacher that still provided some of the advantages of Logo. If we were going to use Logo, we needed to preserve the spirit and benefits of this tool, and not use it simply to present something that could be done just as well (or better) with another piece of software.

In fact, some early activity proposals –made before we were asked to join the design team—

used Logo only as an underlying platform on which to create animations for some instructive presentation. The criticism was that there was no difference between those Logo-based proposals and equivalent ones using other animation software like Flash; so, why use Logo?

Again, if Logo was to be used, its didactic advantages had to be exploited. But how?

The didactic design was thus crucial, and so was the choice of themes.

The first thing was to try follow, in general, the Logo philosophy. Thus, for the didactic design we
tried to have activities that would engage the whole classroom in collaborative activities of
*exploration and, as far as possible, construction. And related to the latter, we were also *
concerned with not “betraying” the potential of Logo as a programming language. We will try to
illustrate, in a later section, how we tried to include this in the didactic design, through the
example of one of the activities.

Also, we had to take into account that the teachers would not be familiar with Logo, and that in the case of Telesecundaria they may also not be mathematically proficient (since in that system the “teachers” are guides, and not necessarily well-trained mathematics teachers). This meant that the interactive activities had to be self-contained, and self-explanatory (more on this later).

But we wanted to use this “limitation” to our advantage, since, in Papert’s words, “a crucial aspect of the Logo spirit is fostering situations which the teacher has never seen before and so has to join the students as an authentic co-learner”; that is create a “relationship of apprenticeship in learning”, where “the student should encounter the teacher-as-learner and share the act of learning” (Papert, 1999; p.ix).

In addition to the didactic design, we needed to find mathematical themes that would provide

“powerful ideas” (Papert, 1980) and benefit from a Logo-based presentation; but that we could also incorporate easily into teacher-centred interactive activities. One obvious choice for us, was the inclusion of activities on fractals, because fractals are so easy to construct and describe using Logo, due it simultaneously providing an accessible language and recursive capabilities. In fact, we were specifically requested to design interactive activities on fractals and infinity, so that these topics, that are normally not part of the school curriculum, could also be explored in the lower secondary schools. For other activities, we looked through the EMAT activities (Sacristán

& Esparza, 2006) and chose some of the richest activities mathematically speaking. We give a detailed description of the themes and activities we have chosen and designed, further below.

### The technical challenges: the version of Logo, and the system limitations

The choice of themes and the didactic design that preserved some of the spirit of Logo, were not the only challenges. We have also been faced with other technical challenges.

One of these, is the problem of the version of Logo to be used for the interactive activities. The EMAT programme uses MSWLogo; one of the main reasons for that choice was that MSWLogo is an open source version, available in a Spanish, that we could upgrade and make it freely available (see Sacristán, 2003, for further details). For the new interactive activities we were again restricted to the use of some freeware version of Logo and preferably available in Spanish.

That meant that we couldn’t use commercial versions of Logo with integrated easy-to-use design and multimedia capabilities like Imagine, or Microworlds Logo. We looked into several freeware versions of Logo, but, in the end, it was decided that if EMAT used MSWLogo, and if one of the aims was to link the new activities with some of the EMAT ones, we should use the same version as EMAT, despite the many technical restrictions it imposed.

However, after almost two years of working in the design, we have realized that, in some ways,
the choice of using MSWLogo has also affected the didactic design. One of the arguments for
using MSWLogo in EMAT, besides the main reasons explained earlier, was that the simple
interface could make students focus more on the programming aspect through the writing and
debugging of procedures and that we did not need, or even want advanced features, nor to
develop sophisticated microworlds or environments (Sacristán, 2003). But in our current
*situation, the latter is exactly what we are doing: we are creating sophisticated interactive *
environments and we have been restricted by the possibilities of the software. Nevertheless, we
have made the best of the software we are working with, and know that the use of the same
version for the interactive activities and EMAT will be easier for the users in the cases when both
models are used in a school.

Another challenge is that the interactive activities have to be “inserted” within the Enciclomedia system. That means that, not only are the activities opened through links in the Enciclomedia system, but the entire activity is executed within a frame of the system (with the Windows platform actually hidden from the users). This is why most of the non-Logo interactive activities that have been designed for this project, are Java applets. In our case, we had the problem that the MSWLogo software had to be loaded unto the machine that will be using the activities; we sorted this out, by adding the installation file of MSWLogo to the installation program of the Enciclomedia system.

We also faced many other technical requirements, such as using a maximum screen resolution of 800 X 600, the use of specific colours, and the need to comply, as much as was possible, with the established look-and-feel for all the interactive (non-Logo or Logo) activities being designed.

Other challenges had to do with the use of the interface, since the activities would be presented on an interactive whiteboard. This meant that we needed to restrict the amount of typing to a minimum. This was something that we considered mainly in the didactic design, but also tackled from the technical perspective:

To give commands to Logo’s turtle, we created a button command window (see Figure 2) with
the basic graphic primitives (forward, right, repeat, penup, etc.). (Note: the commands are given
without abbreviations, so that they can be self-explanatory.) When the ‘repeat’ button is pressed,
a window appears asking what commands the user wants to repeat; these commands can be
inserted into the window using the other command buttons. Also, whenever any of the buttons
are used, we made sure the corresponding commands appear in Logo’s text screen. We were
*also requested that we add an ‘undo’ [deshacer] button; we didn’t like the idea much (after all, *
we want the users to reflect on what the turtle had done to deduce what needs to be typed to
correct a path), so this button only undoes the last command by giving the turtle the inverse
instruction to the prior one.

*Figure 2. The button graphic commands window. *

Although we tried to restrict typing as much as possible, the need to type commands at some point was inevitable (and, after all, we still wanted to have some programming activities). At first we only considered restricting typing from the perspective that we wanted the activities to flow easily in the teacher-in-front-of-classroom situation, and we didn’t think that some typing would be a problem. Then we realized that we also had to consider how to input the characters. We knew that interactive whiteboards have a virtual keyboard, and we thought that would be sufficient. But when we tested the activities with a real interactive whiteboard, the use of the whiteboard’s virtual keyboard was extremely difficult. Two possible solutions for this are: (i) To recommend to the teachers that they use the computer’s physical keyboard; this is the easiest but at the same time often impractical solution (e.g. the computer may be far away from the interactive whiteboard, etc.); although a wireless keyboard could in theory be used. (ii) The second solution we considered was to program an MSWLogo virtual keyboard. “Fortunately” for us, the problems with the interactive whiteboard’s virtual keyboard was a generalized problem not exclusive to the Logo activities, so the Enciclomedia system developers are now adding to the Enciclomedia toolbox a link to open and use Windows XP’s On-Screen Keyboard. We have not yet tried this out with an interactive whiteboard, but we hope it will work better.

Finally, we had the problem of how the Text/Command window of MSWLogo normally appears, as it includes several buttons which were confusing for non-trained teachers and users (such as Pause, Trace, etc.). We also had a problem of an overlapping and an overcrowding of windows.

For these reasons, we modified the Text/Command window of MSWLogo to make it smaller and
*so it would only include the “Run” [Ejecutar] button. The new Text/Command window can be *
seen in the upper left corner of Figure 5.

**The interactive activities **

*Figure 3. The opening screen of the “Fractals” interactive activity *

In the following sections we will try to illustrate the didactic design of the interactive activities.

### The general design of the activities

Each activity is opened through a link in the Enciclomedia system and begins with an opening
*screen (see Figure 3). After pressing the “Begin” [Inicio] button, the activity begins. *

At the bottom of the screen there is a bar with buttons for Instructions, Purpose (or Objectives), Didactic suggestions, Activities and Exit (see Figure 4). All of the interactive activities (Logo and non-Logo) have to have these required links, because the activities have to be self-contained;

these instructions and objectives relate to the entire activity.

*Figure 4. A scene from the interactive activity “Frequencies and probability”. Notice the button bar at the *
*bottom with the Instructions, Purpose, Didactic suggestions, Activities and Exit links. *

Each interactive activity is composed of sequence of “scenes” or sub-activities; by pressing the

“Activities” button, the user can see which are the sub-activities and jump to a different scene; if its not appropriate to be able to jump to another scene (e.g. if a previous sub-activity needs to be completed) then this button appears dimmed. If the sequence of sub-activities is followed sequentially, then a “Continue” button takes the user from one scene to next.

### The different interactive activities

Earlier we mentioned that one of the challenges for the interactive activities, was the choice of themes. So far, we have developed activities in the following themes, mostly based on activities from the Logo EMAT materials (Sacristán & Esparza, 2006):

The construction and properties of regular polygons: This activity is illustrated in a section below.

Randomness and probability: This activity (see Figure 1) has as aim to introduce the concept of randomness, and to analyse probabilities. It includes a turtle race where each turtle goes forward depending on what number is generated by a dice. But the race is unfair, and users have to predict the rules that make each turtle go forward, calculate the probabilities that each turtle has for going forward, and modify the rules so that the three turtles have equal probability to advance; i.e. to make the race fair.

Probability and frequency distribution: This activity (see Figure 4) has as aim to analyse the frequency of times that each number “falls” when a dice is thrown, and calculate each number’s probability. The frequencies and probabilities are analysed for a 6-sided dice, for a 12-sided dice, and the latter compared with those of two 6-sided dice. To help in the analysis, frequency graphs and percentage tables are used.

Ratio and proportion: In this activity, users are given a procedure to produce an “L” letter.

They are then asked to draw similar “L” letters of different sizes, and then to modify the procedure so that it creates an “L” of another size. Then the users go through the same sub-activities for producing the drawing of a house. In the final scenes, the aim is to modify the “LetterL” and “House” procedures using variables so that they become general procedures.

Sequences and recursion: This activity aims to introduce recursion. It has sub-activities presenting rotating figures and others of sequences of numbers.

Geometric sequences: The aim of this activity is to study simple geometric sequences through their graphic representations and their Logo instructions that define them (such as bar graphs and spirals), and reflect on “what happens at infinity”. The activity ends with the example of how a fractal tree can be produced.

Fractals: This activity (see Figure 3) presents different examples of fractals. It also includes sub-activities where the perimeter and area of Koch’s snowflake is analysed.

(The last three activities, although also included in the EMAT materials –Sacristán & Esparza, 2006— are based on the work by Sacristán, 1997).

**The “Regular Polygons” activity **

Below, we present a detailed description of the activity on regular polygons. This was the first activity we developed and we based the design of all the other activities on what we learned from designing this one. Its didactic design should illustrate how we tried to find solutions to the technical and didactic challenges presented earlier.

The regular polygons activity consists of six “scenes” or sub-activities. We will present each of them in turn.

### Scene 1: Examples of regular polygons

*Figure 5. The first scene or sub-activity of the “Regular Polygons” activity *

Since this has to be a self-contained, instructional activity, after the opening screen (not shown),
the first scene (see Figure 5) gives examples of regular polygons, with buttons for the most basic
ones (triangle, square, pentagon), and a space for specifying the number of sides of any other
polygon that the users wish to draw. Whenever the turtle draws a polygon through the buttons of
*this scene, the commands that the turtle is executing (e.g. forward 100) are written in the *
text/command window, so that the students can see how the polygon is being created.

### Scene 2: Construction of regular polygons

In the second scene (see Figure 6), the users have to draw a randomly generated polygon –in direct mode using the command buttons or directly typing the commands— over a dotted line;

besides the dotted line polygon, only the size of the side is given, but not the angles which the users have to figure out.

*Figure 6. The second scene or sub-activity of the “Regular Polygons” activity *

When the polygon is finished, three consecutive pop-up windows appear asking for the number of sides that the polygon had, the internal angle, and the rotation angle; if the inserted values are correct, they are recorded in a table, as shown at the right of Figure 6.

After drawing and analysing at least three polygons in this way, the user can press the

“Continue” button. Before going on to the next scene, a “Discussion” window (see Figure 7) is displayed, that asks to reflect on the relationship between the internal angle of the polygon and the rotation angle used. We consider these discussion windows to be an important pedagogical tool that induce classroom debate and reflection on the mathematical relationships.

*Figure 7. Sample “Discussion” window, asking a question for classroom analysis and debate. *

### Scene 3: Construction of regular polygons 2

The aim of the third scene (see Figure 8) is to use a full line of Logo instructions to draw a
polygon. Users are asked to fill-in the values for the number of sides of the polygon –as the input
*to the ‘repeat’ instruction— and the rotation angle –the input to the ‘right’ [giraderecha] *

instruction, in the given window with the instruction line: repeat <> [forward 50 right <>]^{3}. Above
the space for each input, we have added short labels describing each: e.g. “Number of sides”,

“Rotation angle”, so that the meaning of the command inputs is self-explanatory. After the instructions are filled-in, by pressing a “Draw” button the instructions are run and the figure drawn.

If the resulting figure is a correct closed polygon, with the turtle returning to the initial position, then a pop-up window –as is shown in the upper left-corner of Figure 8— asks for the value of the rotation angle, and a follow-up table to the one from the previous scene is filled in (i.e. the values of the number of sides and the rotation angle for each of the polygons investigated so far, are contained in the table so that it can be analysed).

3* In Spanish Logo, the window shows: repite <> [avanza 50 giraderecha <>]. *

*Figure 8. The third scene or sub-activity of the “Regular Polygons” activity *

If the figure generated is not a correct closed polygon, then a window asks to observe that fact and to reflect on why that is the case.

When the “Continue” button is finally pressed, another “Discussion” window is displayed, that asks to reflect on the relationship between the number of sides of the polygon and the rotation angle used.

### Scene 4: Predict what regular polygon will be produced

*Figure 9. The fourth scene or sub-activity of the “Regular Polygons” activity *

In the following scene (see Figure 9), the purpose is to give users explicit examples of the
relationship between the number of sides and the rotation angle needed to draw a regular
polygon. Thus, filled-in instructions are given to draw a polygon, where the input to the turning
*angle (the ‘right’ [giraderecha] command) is a function of the number of sides (i.e. as a *
relationship to the full turn of 360). That is, the instructions are given in the form of the following
example:

repeat 6 [ forward 50 right 360 / 6]

Users are then asked to predict what polygon the turtle will draw. They can verify their prediction
*by pressing the “Draw” [Dibujar] button. They can look at as many examples as they wish. *

### Scene 5: Draw any polygon

The following scene is very similar to third scene, with a window where users can fill in the inputs to the instruction line: repeat <> [forward <> right <>]. The aim of this scene is to allow users to freely play and draw examples of as many regular polygons as they wish, after hopefully having observed, in the previous scenes, how the number of sides of a regular polygon is related to turning angle. In this way, they can apply that knowledge.

### Scene 6: Generalising the construction of regular polygons

The last scene (Figure 10) has as aim to lead users to create a general procedure for a regular polygon. It also introduces how to create a Logo procedure, the editor and, very importantly from an algebraic perspective, the use of variables.

*Figure 10. The final scene or sub-activity of the “Regular Polygons” activity. In the screen capture on the *
*left (a) we see the editor open for completing the general procedure of a regular polygon. In the screen *

*capture to the right (b), we see how the Polygon [POLIGONO] procedure can be run. *

When the scene first opens, a window with the following text appears: “Generalizing: Complete the instructions to build a variable regular polygon” with a button that says “Press here to complete them”. This button opens the editor (see Figure 10a) where an incomplete general procedure for a regular polygon appears, with comments to help the user:

TO POLYGON :NUMSIDES :SIZE

;Complete the instructions by substituting the dotted lines with the input for the rotation angle in terms of the number of sides REPEAT :NUMSIDES [FORWARD :SIZE RIGHT …… ]

;Press Save and Exit when you are finished END

As further assistance, a help button (marked as “?”) explains that variables are written preceded without spaces by a colon (:).

When the editor is closed, a new window appears (see Figure 10b) with the instruction line POLYGON followed by blank spaces for each of its variable inputs, and a button “Run Polygon”.

There is also a help “?” button; when this button is pressed, labels with the names of the variables of the inputs of Polygon appear above the blank spaces. In this way, the users can try out their polygon procedure. If needed, there is a button to go back to edit the procedure.

This is the last scene, and the hope is that users will then play by modifying in different ways the procedure, perhaps also creating stars.

**Final remarks **

With this paper we tried to illustrate how we tried to preserve Logo’s philosophy and spirit in a design situation that presents many technical and didactic challenges. In particular, with the

“Regular Polygons” activity, we tried to exemplify how we tried to find solutions to those challenges:

from the challenge of an activity that needs a design as a self-contained “instructional” tool by the teacher (and that has technical limitations such as the typing restrictions);

to the difficulties of using Logo with no previous introduction to the language or environment;

to our attempts to provide users (the teacher and his/her students) with opportunities to explore, play and try out instructions; to learn together (in a relationship of “apprenticeship in learning”); and to reflect on mathematical relationships (access to “powerful ideas”) and discuss them collaboratively;

to how we tried to lead users to define a mathematical relationship and show them how this could be done;

to present users with an opportunity to generalize the mathematical relationship through the use of variables, and finally give them the possibility to complete a “half-baked” general Logo procedure and thus learn to program in Logo and complete a constructive process.

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**Acknowledgements **

The materials presented here belong to Mexican Ministry of Education (SEP), in partnership with
*the Instituto Latinoamericano de la Comunicación Educativa (ILCE). *