Mathematical Investigations Using Logo: Part 2

Mathematical Investigations Using Logo: Part 2Author(s): Ken BrownSource: Mathematics in School, Vol. 15, No. 4 (Sep., 1986), pp. 35-39Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214111 .

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Mathematical

Investigations

using

LOGO Part Two

The second of a series begun in the May 1986 edition

by Ken Brown Boney Hay Middle School, Staffordshire

Investigation 2 The production of lists of multiples is not a new idea for investigation using a micro. It is a problem clearly suited to a computer. Whilst the production of such a list might seem relatively easy, this proved not to be the case, even for quite able children.

Indeed this exercise served to highlight the lack of understanding many children appeared to have about the mechanics of producing series. The computer demands a high quality of "explanation" for a given solution in order for it to output a correct series. In this case the solution is an easy one for most children to check since they probably know the first few multiples of a given number as "facts". Many teachers may accept that children know about multiples because they are able to recite them and because they can work out larger multiples by what might be best described as a form of "Formula Cranking" (see H. A. Cohen 19741 for a discussion of this approach to problem solving.). As Hart 19812 points out, showing a child "how to do it" in any mathematical situation and then setting several exercises repeating the teacher method seems to be of limited value.

Hart goes on to note that if children could recognise that they had made an error themselves they might do some-

thing about correcting it. As I have mentioned above all the children involved in this investigation were able to easily evaluate the correctness or otherwise of their results as output. However I feel it is important to note that some of the children were loath to tinker with their procedures and seemed afraid of producing "wrong" answers. Some children would repeatedly ask when making some change to a procedure, "Is that right?", (needless to say I merely repeated the original problem in these situations). These types of reactions seem to me to be symptomatic of the deep seated insecurity many children have when dealing with numbers; this appears to stem from a very early age. More research is needed in this area!

Two girls of average ability when asked to produce a procedure to print out a list of any set of multiples first of all produced:

TO TI "A PR :A TI :Ax 7 END

Which when called with 7 produced a list of powers of 7.

This was changed to:

TO TI "A PR :A TI :A x 1 END

Mathematics in School, September 1986 35



Which when called with 7 produced a list of 7s.

After first of all going back to their first solution the girls decided to try:

TO TI "A PR :A TI :A + 7 END

This procedure was first called with 1 and produced 1 8 15 etc. After some discussion they decided to try the procedure with an input of 0 and succeeded in producing multiples of 7.

The girls seemed satisfied with this so I restated the original question, i.e. to produce a list of multiples of any given number. They adapted their procedure as follows:

TO TABLE "A "B PR :A TABLE :A+:B :B END

This is an acceptable solution. Particularly noteworthy here is the fact that both Samantha and Amanda opted for, and had no trouble with, a tail recursive solution to the problem; that is to say that they chose to create a repetition by producing a procedure that calls itself on the last (tail) line. Imbedded recursion is where the procedure calls itself at some other point in its execution. These two children had had some experience of using tail recursion previously. An interesting discussion of childrens perceptions of recursion in LOGO procedures is given by Kurland and Pea 1984.3

Children of comparable ability who used MAKE and REPEAT statements produced equally valid solutions. Marie and Sharon for example created a global variable as part of a solution that was not placed inside a procedure to begin with;

MAKE "R 1 REPEAT 12 [PR :R MAKE "R :R -x7]

Their first attempt at multiples of 7 produced the sequence; 1 7 49 343 etc. This was followed with:

MAKE "R 1 REPEAT 12 [PR :R MAKE "R :R + 7]

Giving the series; 1 8 15 22 29 36 etc. Interestingly this is a similar mistake to one made by Samantha and Amanda above. Marie and Sharon decided on the basis of their figures that they were "one out" and so made the following adjustment:


Giving: 1 7 13 19 25 31 37 etc. Finally after a long period of silence Marie tried;


Which produced the result that the children wanted.

Because of their comparative lack of programming experience these two girls had some difficulty in converting their solution into a procedure. Nevertheless I felt that their work up to that point had been just as valuable for them as for any other pair of children. Furthermore it serves to demonstrate that sophisticated programming skills are not necessary for children to pursue investigations using LOGO.

One pair of girls spent a very long time huddled around a piece of paper on this problem. After about ten minutes they came up with a potentially correct solution first time!

TO TABLE "R "N MAKE "R :R +:N PR :R TABLE :R :N END

However they chose 1 as their input to :R. It took them several minutes longer to realise that the input should be 0.

A major benefit of allowing children to pursue this type of activity is that it draws the teachers attention to such deficiencies in understanding that might be difficult to detect in more conventional work.

A pair of twin sisters from the 11-12 year age group with quite a lot of programming experience produced the following:

TO MULT "NUM PR :NUM :NUM END

followed by:

TO MULT "NUM "NU PR :NUM *:NU MAKE "NU :NU + 1 END

Which when called with inputs 5 1, output 5. Louise said, "I bet it'll work if we put repeat." And so they finished up with:

TO MULT "NUM "NU REPEAT 12 [PR :NUM -:NU MAKE "NU :NU +1] END

The use of repeat is particularly important in that its use allows the children to easily define which number in the series they want the computer to output up to. With recursive solutions such as those described earlier, specific numbers in the series could only be obtained by introducing a variable to count the number of recursive calls made and then making the program stop using an IF statement. This is quite a complex programming exercise for many children. An attempt at this is shown below (i.e. VAR and MULT).

Whilst some very able children had difficulty producing lists of multiples their greater confidence in using numbers enabled them to debug their work more effectively. My subjective observation was that the boys in particular seemed to show a greater willingness to have a go at changing values or program structures to find solutions. Girls on the other hand seemed to prefer to do most of their thinking off the computer using pencil and paper. One pair of boys came up with:

36 Mathematics in School, September 1986



TO MU LT "A MAKE "N 1 PR :Ax :N MAKE "N :N + 1 IF N = 10 [TOPLEVEL] MULT :A END

Calling MULT with an input of 5 output a series of 5s.

On looking closely at their procedure, (following the flow of the procedure using real numbers), they realised that they were resetting :N to 1 on every pass. The solution arrived at was to create a superprocedure to deal with the initial setting of :N to 1.N it will be noted becomes a global variable. (NB: This is not generally a good idea in LOGO programming. It is far better to keep variables local where- ver possible. For example the computer will store the value of N from the last time it was used in MULT below. Consequently if MULT is called on its own without resetting :N to 0 the series of multiples will start at whatever N was set to when the procedure was halted previously. This may be the source of some confusion.)

TO VAR "A MAKE "N 1 MULT :A END

TO MULT "A PR :A :N MAKE "N :N + 1 IF :N = 10 [TOPLEVEL] MULT :A END

Which produces correct results. This solution was quite quickly arrived at. This appeared to be due to the very systematic approach being adopted by the boys concerned.

In common with other groups these two children went on to attempt to produce a Fibonacci series. As they were of higher than average mathematical ability I asked them to solve the somewhat more complex question, "What is the result of dividing the 1000th by 1001st number in a Fibonacci sequence?" (the computer cannot cope with numbers that big!).

They were able to produce a correct Fibonnaci sequence first time with the following carefully thoughtout procedures:

TO FIB MAKE "D :B+:C PR :D MAKE "B :C+:D PR :D MAKE "C :D+:B PR :C FIB END

TO ONACH MAKE "D 0 PR :D MAKE "B 1 PR :B MAKE "C 1 PR :C FIB END

Once again global variables have been used. I was surprised when Mattie stopped the computer before it had had a chance to come up with the "NUMBER TOO BIG" message and said, "I'll bet its always the same!" Tried;

5/3 8/5 13/8 etc.

In command mode (if he had only tried the first few he would probably not have discovered how true his statement was. See Brett and David below).

One group of girls also came up with a similar solution. They had had the forsight (or good luck), to include the division in their procedure for producing the Fibonacci series. They could clearly see that the ratio became the same to 4 decimal places.

Whilst many pairs of above average children were able to produce the Fibonacci sequence the majority found it quite a difficult exercise. One pair of able children were able to produce the correct sequence using;

TO FIB MAKE "T 1 MAKE "A 0 REPEAT 1001 [MAKE "T :T+:A PR :A+ :T MAKE "A :T+ :A PR :A+:T] END

However they were entirely lost when it came to finding the result of dividing the 1000th by the 1001st number in the sequence. Their initial strategy consisted of trying to reduce :T to make the number "fit into the computer". They tried MAKE "T 0.001 , 1E-13 , 1E-26 (all resulting in NUMBER TOO BIG messages), and even 1E - 99 (which output a series of Os). Both David and Brett were very quiet for several minutes at this point; neither of them able to think of an appropriate strategy for continuing. At one point David did suggest looking to see if they were "the same distance apart". He typed 2/3, 3/5 and 5/8; Brett said they were not the same and they decided to go on to an entirely different line of thought. Due to lack of time the investigation was terminated at that point. This is a situation that emphasises the need for a teacher to carefully review the work of the children and iron out any misconcep- tions that may have been engendered by the investigation itself.

Some children attempted to produce the series of square numbers. Whilst more able children were able to do this without undue difficulty, children of average ability found the exercise quite a challenge. Marie and Sharon for example tried to adapt their multiples procedure producing:

TO SOUARENUM "SQU MAKE "R 0 REPEAT 6 [PR :R MAKE "R :R a:SQU] END

Which was not correct, followed by:

TO SQUARENUM "SQU MAKE "R 0 REPEAT 6 [PR :R MAKE "R :R :SOU + 1] END

Which output 15 21.




This pair of children were unable to produce a satisfactory procedure at this session. The investigation was terminated by me when it was clear that they were completely lost and making no headway.

The 11-12 year old twins on the other hand found their multiples procedure easy to adapt:

TO SONUM "NUM REPEAT 10 [PR :NUMx:NUM MAKE "NUM SNUM + 1 ] END

These children realised that the next square number in the sequence is the result of multiplying the next natural number by itself and that in order to generate that next natural number they only had to add on one to the last natural number. Trivial as this might appear to some, it seems to me that it is far from obvious to many children.

As Hunter and Cundy4 point out when looking at sequences of numbers it is important that it is stressed to children "... that a sequence is uniquely determined only when we have a formula for the general nth term, or some rule for the construction of this term, and that it is not enougth to provide the first few terms." By going through the process of producing LOGO procedures to output number sequences the children themselves have had to rigorously define such a rule for construction of the sequence.

The process of producing the simple number sequences discussed in the investigations above has had the clear benefit in my view of emphasising this, one of the most fundamental ideas in mathematics, which is often not stressed in school. If the pursuit of these investigations were only capable of making these processes clearer to children then they would surely be worth pursuing for this benefit alone.

Graphical Representations An interesting development of the work connected with investigating number patterns was the use of turtle graphics to produce a graph of the functions involved. The technique involved moving the turtle forward through the value of the dependent variable, returning the turtle to y = 0, turning right through 90 degrees, moving forward one unit, turning left, incrementing x and repeating the process i.e.:

FD :Y BK :Y RT90 FD 1 LT90Q MAKE "X :X + 1 etc. (using REPEAT or recursion).

Fig. 1

Children were shown how a value could be returned to FD and represented on the screen in this way. The had no difficulty in using the idea.

What is the purpose of this exercise? Graphs produced in this way have some interesting features. Firstly they allow

children to quickly and clearly see a pictorial representation of the number series they have been producing. Different graphs can easily be overlaid on the same graph and comparisons made. Different inputs can be tried and the graphical results seen accurately and quickly. Although none of the children in these investigations actually put axes on their graphs they could easily have been inserted with a simple procedure.

Two boys produced this solution to producing a graph of the square function, (see Fig. 1), they spent a considerable time exploring what happened when they tried variations of this based on other functions (e.g. 1/X, see Fig. 2), and even using negative integers for example:

TO GRASET "C H T PU BK 80 LT 90 FD 100 RT 90 PD WINDOW GRASTART :C END

TO GRASTART "C MAKE "A 0 MAKE "X 0 MAKE "D 0 GRASOR :C END

TO GRASOR "C MAKE "A :C x:C FD :A/100BK :A/100 RT 90 FD 1 LT 90 MAKE "D :D +:A MAKE "X :X + 1 PR :D IF :X= :C [PR :D WAIT 60] GRASOR :C+1 END

Notice how they used a scale factor to ensure the graph fitted on the screen. This program not only draws a graph of the function but it also outputs the cumulative lengths of the lines drawn to a given :X (set by the input :C), giving an approximate area for the region under the graph line to that point. (It is interesting to compare the results with those obtained from the formula:

A= X/3

With older children an approach based on these ideas might prove an interesting accompaniment to the introduction of finding areas by the integral calculus.) There is plenty of scope for discussion here about the "area" represented by the one pixel wide strips and the true area. Which is the greater? why? what is the approximate shape of the missing area for each strip? will it always be the same amount in each case? can we work out each missing area etc. Although as I have said this development of the investigation may be best left to older children none of the questions I have just suggested should in my view be beyond a capable 13 year old when set in this context.

I certainly felt that the majority of the children achieved some benefit from the representation of simple linear functions in this way. I shall return to this idea when discussing the investigation concerning the slopes of lines.

Whilst on the subject of graphs it is worth pointing to the potential for extending the simple "turtle graph" idea to

38 Mathematics in School, September 1986



Fig. 2

directly interpreting numerical inputs from sources such as the sensors discussed in investigation 1. Some interesting exercises in scaling would be necessary for the children to produce meaningful histograms.

Investigation 3 The production of plane figures using turtle graphics and turtle geometry is a topic that has been well aired in various books and journals over the past few years, (i.e. Mindstorms Seymour Papert 1980, Turtle Geometry H. Abelson and A. diSessa 1980 and Massachusetts Institute of Technology A.I. Memos 544-546). All of the children had had some experience of creating pictures using turtle graphics and also producing simple regular plane shapes. Some of them had participated in a long term project concerning the production of frieze patterns.

However some children had not previously attempted to draw a circle without the use of the ARCR and ARCL primitives available in the RML version of LOGO which enables the user to draw an arc of a given radius through a given arc. When asked to draw a circle most children were able to do so but they needed the lead in investigations of producing a triangle, square, hexagon, etc to get started. None of them seemed able to get started on this from "cold".

One pair of children, a pair consisting one above average 3rd year and one above average 4th year girl, not only were quick to develop the idea of a many sided shape, but were able to quickly to grasp the relationship between the angle turn given and the number of repeats in their procedures in order to produce state transparency, (i.e. getting the turtle back to the same position and heading that it started from, see Abelson and diSessa 19805.

After initially producing a procedure that would work, Debbie and Eldyne were able to go on to investigate which number affected the size of the circles drawn. They were able to discover without too much difficulty that they needed to change the length. From the procedure:

TO ELDE "L REPEAT 180 [RT 2 FD :L] END

They were then asked if they could produce a procedure to draw a polygon with any number of sides using variables: they came up with:

TO POLYGON "S "A REPEAT:S [RT :A FD 20] END

to which I said, "But you're doing all the work!". So they finally produced:

TO POLYGON "S REPEAT S [RT 360/:S FD 20] END

I felt that without the interventions on my part the children would not have been able to capitalise on their investigative momentum. This is the sort of situation where I feel teacher intervention is warranted, indeed should be expected, particularly where children without a wide experience of investigation and conjecture are involved.

I feel I should mention here the case of another investigation involving turtle graphics about which I have reserv- ations. It highlights one area where caution on the part of the maths teacher is called for. A few children who were capable of drawing circles using turtle geometry were asked if they could draw an ellipse. They were shown drawings of ellipses. Most of them failed completely within the time given. However one pair of children produced this procedure;

TO LIPS REPEAT 2 [REPEAT 120 [FD 2 RT 1] REPEAT 60

[FD 1 RT 1]] END

Which they both agreed formed an ellipse! Now this is ingenious, but it is not an ellipse. It may look to the naked eye like an ellipse but it does not conform to the mathematical properties of an ellipse (see figure 3). In this context does it matter? I would say that it does. The children's belief that they had produced an ellipse was a misconcep- tion that I felt it was the teacher's task to recognise and explain. The response of the children when told that it wasn't an ellipse was to protest "... but if it looks like an ellipse, it must be alright!"

This seems to me to be a clear case of a situation where children can be shown that a definition in mathematics has some precise meaning, and rigour is necessary in interpreting it. However it also highlights a fault on my part in that by only showing the children pictures of ellipses, (rather than giving them mathematical explanations which they would not in all probability understand, or examples of methods of drawing using two fixed points which seemed somewhat irrelevant in terms of turtle geometry), the children had considerable justification in protesting that their drawing looked like the drawings they had been shown. They had completed the task they had been asked to attempt!

On reflection this is not an investigation I would readily give to this age children again without a great deal more thought concerning the nature of the task I was actually asking them to perform and the mathematics I was expect- ing to arise from it. It also highlights a problem in evaluating whether geometric shapes drawn using turtle geometry are what they purport to be, and it begs the question as to how many teachers are capable of analysing a LOGO procedure to see if it is equivalent to the more conventional definitions of certain geometric shapes.

References 1. Cohen, H. A. (1974) "The Art of Snaring Dragons", MIT A.I. Memo

338. 2. Hart, K. (1981) Childrens Understanding of Mathematics 11-16 p. 214. 3. Kurland D. and Pea R. (1984) "Childrens Mental Models of Recursive

LOGO Programs", Bank Street College of Education Technical Report 10.

4. Hunter, J. and Cundy, M. (1978) The Mathematics Curriculum. 5. Abelson, H. and di Sessa, A. (1980) Turtle Geometry p. 83.




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Mathematical Investigations Using Logo: Part 2