Primary Maths and Problem Solving with Logo

Primary Mathematics and Problem Solving with Logo

Anthony Jones and Liddy Nevile
La Trobe University, Barson Research

For many years Logo users have made claims about the positive benefits of Logo in the learning of mathematics. In some cases these claims have been proven by well designed and executed research, but often the claims have been purely anecdotal. Papert, in some of the AI Laboratory memos (eg Papert 1973) and again in Mindstorms (Papert 1980), makes it quite obvious that much of Logo is mathematical in content and concept. One of the earliest papers on Logo (Feurzeig et al, 1969) discussed Logo as "a conceptual framework for teaching mathematics."

An approach to problem solving that involves students programming computers is based on the idea that the thought processes required to write a program assist in the development of problem solving skills. In this approach the emphasis is on students learning through their personal involvement in planning, writing, testing and debugging programs.

In this paper we will provide some background material for the

1. What mathematics can be learned through Logo?

2. What is problem solving and can it be taught?

3. How can Logo be used for mathematical problem solving?

While questions 1 and 2 will be considered incidentally in the conference workshp, the emphasis will be on participants experiencing a range of Logo activities designed to promote problem solving in primary mathematics.

What mathematics can be learned through Logo?

Given the comments about Logo and mathematics in the opening paragraph of this paper, it is not surprising that some teachers and researchers have determined specific mathematical topics that seem well suited to learning with or through Logo. In Edinburgh, Howe and his associates have had more than a decade of experience in using Logo programming to teach mathematics. They have discussed and evaluated the teaching of topics such as equations, ratios, co-ordinate systems, geometry and some algebraic concepts such as the distributive law.

Finlayson, in a paper given at the 1985 Logo and Mathematics conference, suggests that the types of mathematical concepts best suited to development through Logo are " Number,Directed number; Angle; Shape; Variable." (p. 84). She provides two microworlds as examples of how particular topics can be presented to students as a series of investigative Logo activities. One microworld:

concerns the geometry of any right-angled triangles, and could be used as a basis for Pythagoras' theorem, similar and congruent triangles and a few other triangle theorems, and trigonometry. (p. 85).

The other microworld is about the geometry of circles, and involves the concepts of diameter, circumference and arc. It is worth noting that Finlayson talks about using these microworlds with 9-11 year old children. What is not discussed is just how formally these mathematical concepts are are developed. There are many aspects of mathematics with which primary school children can have informal experiences without ever needing formal proofs or explanations. Finlayson's microworlds incorporate some of these aspects of mathematics as well as some of the knowledge and skills that usually occur in primary courses.

There are also some parts of primary mathematics that do not lend themselves to development through Logo programming. Although the basic arithmetic operations are a part of most versions of Logo, there seems to be no realistic way of using them to assist children learn the four operations with whole numbers or develop basic concepts of fractions.

What is Problem Solving and can it be taught?

While problem solving has always been a part of school mathematics, teachers have given it much more of their effort and attention in the past few years. In particular, the NCTM's (1980) recommendations for changes to school mathematics to suit the 1980s, provided an impetus for increased classroom application and research in problem solving. Of the eight recommendations put forward by the NCTM, the first was that "problem solving must be the focus of school mathematics in the 1980s" (p. 2) and the third that "mathematics programs take full advantage of the power of calculators and computers at all grade levels" (p. 8).

There have been many attempts to clearly define "problem solving." For the remainder of this paper we will consider a problem to be some task or challenge that has at least one solution that is attainable, even though the solution, and the method of solution, might not be initially apparent. Once we know what a problem is, we can take problem solving to mean the method or methods to used to obtain a solution to some problem.

Much research has been published since 1980 about teaching  children how to solve problems. For example the NCTM 1980 articles that relate directly to teaching problem solving as  a part of primary or secondary school mathematics.

Reference is often made to Polya (1973) and his four stages of problem solving. This series of steps, together with a collection of appropriate problem solving strategies, can be taught to primary school children. Teachers need to provide their students with many varied experiences in which the students can try different strategies. Suydam (1980) notes that research studies agree that children are better able to solve problems when they are taught a wide range of strategies, as this provides the child with a repertoire from which they can select for different problems. She also notes the finding that "when heuristics are specifically taught, they are then used more and students achieve correct solutions more frequently." (p. 43)

How can Logo be used for mathematical problem solving?

As we have established what problem solving is,that it is a legitimate part of mathematics, and that it can be taught in the mathematics classroom, we now turn our attention to the marriage of Logo and problem solving. Papert (1980) contends that Logo is at least partly based on the Piagetian concept of how children develop intelligence. This has led to many claims about the use of Logo programming or microworlds as an aid to developing various thinking skills.

There are many different approaches to teaching students to use the turtle graphics aspect of Logo. McDougall (1985) describes three major approaches currently used in schools. The first of these is the 'synthetic' approach, which usually involves students learning commands, practising and mastering these commands before being introduced to further commands. This bottom-up style of approach may be an efficient method of teaching Logo programming, but it does not seem the best approach for problem solving.

A better approach, although still basically bottom-up in style, is Nevile's (1983) 'turtle humming.' This involves students exploring and experimenting with fragments of Logo code, without initially having a specific goal or direction. Turtle humming appears to enable students to work independently earlier than other methods, and considerably restricts the amount of teacher direction and intervention.

The 'analytic' or top-down approach starts with students being given selected Logo procedures to experiment with, to alter and to extend. It obviously involves much more teacher intervention than turtle humming, but it does allow students access to teacher selected microworlds, and this can enable the teaching of problem solving.

Gallini (1985) reviews research on problem solving, and  proposes an instructional framework within which Logo can be used to generate a computer-based problem solving environment  that has the potential for enabling students to transfer the skills they acquire to other areas. She hypothesises that it is essential for any programming to be taught as an adjunct to the subject content of the lesson. For example, when using LOQO to explore some mathematical relationships between the sides and angles of polygons, the focus should be on the geometric properties of the polygons and not on the Logo commands that are used to move the turtle around in order to draw the polygons. It is also essential that a form of 'guided discovery' be used to make the student aware of the problem solving process. Using the previous polygon example, this would mean that the teacher would prepare a series of questions or prompts to steer the students in the desired direction. The top-down approach to Logo programming fits in with both these conditions.

As well as these two essential conditions, Gallini identifies a number of elements that are inherent in a problem solving task. Firstly the student must be aware that a problem exists, and that this will involve recalling what is already known in the content area of the problem. A problem exists for a student when the knowledge and rules they are familiar with are not sufficient to find a solution. The existing familiar information has to be combined with new rules that have to worked out. As a direct consequence of finding a solution to a problem, students should have acquired new or higher order rules that become part of their repertoire.

As an example of the way that Logo can be used within this instructional framework, consider the following lesson based on a procedure that draws polygons:

The problem is to deduce the relationship between the number and the size of the exterior angles of regular polygons, using a Logo procedure with the angle as a variable.

"Instructional Conditions.

The two conditions proposed for a computer environment are reflected in the Logo activity in that

1. Logo programming is used within a content domain (i.e., teaching the relatioship between the number and size of exterior angles of a regular polygon) and

2. questions are raised to make the learner aware of how Logo programming skills help him or her in solving the problem and can generalize to other problem situations." (Gallini p. 10)

Elements of the task.

1. Defined problem. The student is to discover a simple mathematical relationship that is true for any regular poiygon. This tasK is often presented using the square as an example, which can lead students to confuse the interior and exterior angles and the effect of altering their size. ...

2. Familiar rules. The student will already know the size of the interior angle of some regular polygons and the number of angles of the most common polygons.

3. Combining known and new information. To find the relationship students have to combine what they know about drawing one polygon, say a square, and modify this to draw another regular polygon, say an equilateral triangle. To channel their thoughts in the desired direction the teacher can ask questions such as:

1. When the turtle turns through an angle, is this angle inside or outside the polygon?

2. To draw a triangle does the turtle turn through a bigger or a smaller angle than for a square?

3. If you drew a polygon with six sides, would the angle the turtle turns through be bigger or smaller than for a square?

4. To draw a square you need 4 turns of 90. To draw an 8 sided polygon you would need * turns of **.

4. Outcomes. By exploring the angle required to draw various polygons students should develop the  relationship between the number of turns and the size of each turn. This exercise has used simple closed polygons. Extension activities could include exploring different stars to look for a similar relationship.

REFERENCES

Feurzeiq, 4.,Papert, S.,Bloom, M.,Grant, R. and Solomon, C. (1969) Programming languages as a conceptual framework for teaching mathematics. Report  No 1899, Cambridge, Mass.: Bolt Beranek & Newman Inc.

Finlayson, H. (1985) Logo as an environment for learning mathematics. In C.Hoyles and R.Ncss (Eds) Proceedings of the LOGO and Mathematics Education conference. London: University of London Institute of Education, 82-87.

Gallini, J.K. (1985) Instructional conditions for computer-based problem-solving environments. Educational Technology, 25(2), 7-11.

Howe, J.A.M.,O'Shea,T. and Plane,F. (1980) Teaching mathematics through Logo programming: an evaluation study. In R.Lewis and E.Tagg (Eds.), gmuter assisted learning: scopes progress and limits. Amsterdam: North-Holland.

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NCTM (1980) An agenda for action: Recommendations for school mathematics of the 1980s. Reston: National Council of Teachers of Mathematics.

Nevile, L. and Dowling, C. Let's talk Apple turtle: teachers' and parents' edition. Melbourne: Prentice-Hall.

Papert, S. (1973) Uses of technology to enhance education. AI Memo No 298. Cambridge, Mass.: MIT.

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Suydam, M.N. (1980) Untangling clues from research on problem solving. In S.l(rulik (Ed.) Problem solving in school mathematics: 1980 yearbook. Reston: National Council of Teachers of Mathematics, p.34-50.