Mathematical Investigations Using Logo: Part One

Mathematical Investigations Using Logo: Part OneAuthor(s): Ken BrownSource: Mathematics in School, Vol. 15, No. 3 (May, 1986), pp. 39-42Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214086 .

Accessed: 22/04/2014 07:44

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 82.16.238.55 on Tue, 22 Apr 2014 07:44:45 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=mathas

http://www.jstor.org/stable/30214086?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Mathematical

Investigations

using

LOGO

by Ken Brown Boney Hay Middle School,

Staffordshire.

IC/_./

,co

Part One

Problems and Investigations Using the Micro During a terms secondment to the Faculty of Education at Birmingham University I have been developing and testing some ideas for investigation and problem solving in mathematics using a computer with 11-13 year old middle school children. The medium used for the investigation was the computer programming language LOGO. All the children involved had acquired what Paper et al. define as "...a coherent and accessible minimum core knowledge of programming",'. The work was pursued using an Acorn/BBC micro, LCSI LOGO and a single disc drive. The children worked in matched pairs, each having roughly equivalent levels of ability. Each pair spent one day, (approx 4 hours) on a variety of mathematical tasks.

An important concern was to see what the children could achieve using the computer as a basic and very flexible tool, in the same way that one might regard the pocket calculator as a tool, for investigating mathematics. I believe that this is a fundamentally more pedagogically useful approach to the use of the computer in the mathematics classroom than a concentration on the use of CAL packages which are somewhat transient in nature, (OK until something better

comes along.), sometimes imposing a particular route or routes to solutions and often being little more than an extension to an expository style of teaching, (which has an important role to play in mathematics education but seems inappropriate to the promotion of investigation by children.).

In order for the children to use a computer for investigations they needed to be able to operate within some form of high level programming language that allowed them to utilize the machine's facilities. The reasons LOGO was chosen was that in my view it is at present the language which allows children the easiest access to these facilities. At the beginning it is easy to learn and provides a flexible environment for programming, allowing in more complete implementations full access to all the features of the machine.

One of the main advantages of using a computer in this way is that through programming the children are able to make the solutions their own. I believe this will promote a better understanding of the mathematics involved.

Although an important part of any modern implement- ation of LOGO is turtle graphics, the language was origin- ally conceived without it. Even lacking this feature it remains, and indeed was designed as, a powerful investigative tool for mathematics2. I sought to provide a mixture of topics for investigation which included some which did not need turtle graphics for their solution.

Throughout the study the children were closely ob- served. The types of open-ended investigation attempted could become very time consuming for the children, and yet I felt the time spent in this way was extremely valuable and very much to their benefit in terms of the greater mathematical understanding engendered. They struggled and

Mathematics in School, May 1986 39



wrestled with some demanding problems both in terms of the mathematics involved and in coming to terms with the programming tasks associated with it. They needed time to think, produce programmes, detect errors and make alter- ations. Some investigations ran out of steam after a fairly short period (up to 1/2 hour), certainly within a normal lesson period. Others sustained interest for longer and I believe that where investigative work is concerned teachers need to become very flexible in organising the use of the computers they have at their disposal. Obviously the more computer rich the environment the easier this becomes. One might aim to cover perhaps three mathematical investigations of this type per computer per year with a typical class group (25-30 children?).

The computer should not in my view come to be seen as a substitute teacher. The children need and require just as much attention as they might expect in a normal classroom situation. The benefit of the computer lies in the opportun- ity it provides for promoting better understanding of mathematical principles through allowing the student to operate within a mathematical environment where ideas can be tested. The ideas thrown up, the questions promoted, are likely to be more challenging if anything than those that were produced by our teaching in the past. They will put greater demands on teaching skills, not less! Consequently during this study I attempted to give the children attention in the following ways.

(i) The initial problem was described, the appropriate requisite prior knowledge was checked. Examples were given where appropriate.

(ii) I interceeded when problems of syntax arose in programming, (some children had used other versions of LOGO).

(iii) Direct questions from the children were responded to, but direct help with solutions was avoided, (ideas and possible solutions were given after each session in order to clarify any misconceptions that had become apparent and to promote further work.).

(iv) The children were asked some direct questions about their work towards the end of each session.

In short, the children were given the sort of attention they might reasonably be expected to receive in a well organised maths lesson.

The Investigations The problems posed to the children were of four basic types

(i) Mathematical investigations involving the use of data directly collected from sensors attached to, and feeding data directly into, the computer.

(ii) The production and in some cases the graphical repre- sentation of number series.

(iii) Problems involving the production of plane shapes using turtle graphics.

(iv) Investigations involving the discovery of certain rel- ationships of slopes, lengths of lines, angles and right angle triangles.

The first of these was restricted to the 12-13 year olds. The rest of the investigations were tackled by all the children where considered appropriate.

In choosing the basic format of the investigations I tried to take full advantage of various intrinsic strengths of the computer and the LOGO programming environment.

Investigation 1 Computers are very good measuring tools when allied to appropriate peripheral devices. By using the analogue port on the BBC micro variations in light, heat, sound etc may be

converted into number which can be manipulated by children. A simple investigation was devised using two light dependent resistors attached to clothes, pegs, a length of metal guttering and a golf ball. See figure 1 and see (Nunns 1984)3, for an account of the circuitry involved.

Golf ball LDR 1

LDR 2

NB: LDR =Light Dependent Resistor

Fig. 1

The objective was to measure the speed of the ball as it rolled down a given slope. In order to do this using LOGO they had to be able to read the level of light from the two sensors. The facility to read the A/D port does not exist in L.C.S.I. LOGO in its basic format. However it is a fairly easy task to add new primitives to the language using simple machine code sub-routines written in assembler, (see below) (NB: An extension disc is now available from Logotron for LCSI LOGO which provides primitive equivalent to all those mentioned in this article). Con- sequently I was able to provide the children with SENSE1 and SENSE2 which output values from their respective light sensors in this case, although other types of sensor might equally well be used.

Two other new primitives were also provided; ZEROTIME to reset the computer's internal clock to 0 and RETIME to read the clock, (see appendix).

This investigation is one that is wide open to develop- ment over a long period and at many different levels. As with all the investigations I describe here I will not only discuss the reactions of the children to them but also discuss possible extensions, even if these extend to ideas perhaps more suitable for rather older children.

An interesting first step which some children took was to ascertain the meaning of the figure output by RETIME. One approach was to set the clock with ZEROTIME and then use a stopwatch to compare with the number returned by RETIME. Some children perceived a problem here in matching the button presses on the computer and stopwatch and devised a solution using the KEY primitive so that only one key needed to be pressed on the computer to read the time. Over a 30 second period on the stopwatch the computer output 3012; the two boys then decided (not surprisingly), that the computer was measuring 1/100 seconds.

The children had been shown how the numbers output by SENSE1 and SENSE2 varied depending on the bright- ness of the light detected by the light sensitive resistor. They now needed to decide on appropriate numbers to trigger the clock.

Most children tested the value output by the sensor by typing SENSE1 (or SENSE2), recording the number then placing the golf ball in front of the sensor and recording the new number. Since the numbers are varying all the time over a small range several tests were necessary to achieve values that would trigger operations reliably. One pair produced the following procedure to test various numbers;

TO TIME IF SENSE1 > 2000[ZEROTIME] IF SENSE2 > 2000 [PR RETIME STOP] TIME END

40 Mathematics in School, May 1986



This triggered both sensors on the first pass. Changes were made, first to 20000 then (eventually) to a final value of 10000. Changes were also made to the structure of the program to introduce features allowing them to trigger the loop as required;

TO TIME IF SENSE1 > 10000 [ZEROTIME] IF SENSE2 > 180000 [PR RETIME GOLF] TIME END

TO GOLF IF KEY? [TIME] GOLF END

This was perhaps the most sophisticated solution to the problem of reading the sensors produced by any pair of children. However all the children given the problem were able to come up with a method of solving it which worked. It is an example of a situation where a recursive solution is clearly appropriate.

An interesting feature of the work the children did was that some children had difficulty at first in realising the need to continually measure the light level in order that the computer could activate ZEROTIME or RETIME. Some thought they only needed to read the sensor once.

Although some children began by placing the sensors anywhere on the slope, they all eventually settled on a one metre spacing between the sensors as they decided this would make calculations easier.

A typical treatment of the recorded data was to take a number of readings, conveniently 10, and to find the mean of the values taken. Andrew and Jason decided to take 12 readings and discard the highest and the lowest.

The following example of readings and calculations by Mattie and James may serve to illustrate the type of work produced;

1/100 sees 284 313 322 294 324 342 260 313 253 331

They then found the mean of the last five numbers giving;

2.998 sec per metre

or;

2998 sec per Km

(They decided to express their result in Km per hr!)

2998/60 = 49.96 min per Km

60/49.96 = 1.2 Km per hr. (approx)

I then asked, "Is that the speed of the ball?" The reply was, "It's the average speed of the ball between the two sensors."

This is an interesting question, and an interesting point from which to develop the investigation. The original

Mathematics in School, May 1986

question which started the work was deliberately vague. Two children made the point that the result was not the speed of the ball at the bottom of the slope. When asked how one could measure the speed at that point Andrew suggested that a possible way might be to measure the speed over the lower half of the slope rather than over the whole distance. Jason said that they should measure the speed over the lowest part of the slope and make the gap between the sensors quite small.

The range of mathematical experience to be gained from this type of activity is quite broad; estimation, measure- ment, interpretation of data, tolerances, appropriateness of data, decisions as to the appropriateness of certain units, trial and error experimentation, concepts of speed, acceler- ation, and time, checking results, presenting results, etc.

Clearly there is the potential for some very interesting developments here but unfortunately there was insufficient time to pursue them during a single session of 1 hour. The potential for the derivation of mathematics from this particular investigation may warrant a longer period of time being spent on it. However, from a practical point of view a classroom teacher must always balance this against the availability of machines. Where computers are scarce this will always be a problem. I feel that this investigation would be particularly interesting for slightly older children ane possibly for those being introduced to calculus. The use of computer sensors rather than stop watches would help children to get more accurate measurements of time than might be possible using fingers, stopwatches and human reactions. Measurements could be made using slopes at different angles and different types of curves.

There were a number of interesting occurrences during the childrens work on this investigation. I was often bemused to see children working out quite difficult division problems using pencil and paper with a rather expensive "calculator" sitting in front of them. At the other extreme two able girls, Alison and Michelle, used the computer to add ten numbers together, getting a total of 16.73; and also to calculate 16.73/10 thus discovering that the ball travelled 1.17 m in 1.673 secs!

A method of adding new primitives to LOGO In order to add the extra primitives needed to access information from the two channels of the Analogue to Digital Converter used in the speed investigation, read and zero the clock through LCSI LOGO, please follow these steps closely, (remember this is written for the BBC micro).

Type in the following three assembler routines exactly as shown using BASIC. After typing each one type RUN, press RETURN and type NEW. These programs make use of memory usually allocated to the cassette filing system so a disc based system is required unless the routines can be changed to make use of other locations.

10 P%= &A0) 20 [.rtime 30 LDX #&80 40 LDY #80 50 LDA #81 60 JSR &FFF1 70 RTS 80 ] 90 *SAVE RTIMC AN0 A6F AN0

10 P%= &900 20 [.stime 30 LDA #80:STA &86:STA &87:STA &88:STA &89 40 LDA #2

41



50 a 60 a 70 JSR a 80 RTS 90 ]

100 *xSAVE STIMC 900 99A 900

10 P% = &380 20 [ 30 .radr LDA #&80 40 LDX #&l 50 JSR &FFF4 60 TXA:STA &70:TYA:STA &72 70 LDA #&80 80 LDX #&2 90 JSR &FFF4

100 TXA:STA &74:TYA:STA &76 110 RTS 120 ] 130 *SAVE ADVAMC 380 3DF 380

The above programs produce binary code which allows LOGO to read the internal clock, zero the clock and read the values output by the first two analogue to digital channels. If any problems are encountered using the machine code check that the first number in the bottom line of each program subtracted from the second number corre- sponds to the length of the program plus 1 in hexadecimal. Use TOP subtract PAGE to check this.

If all has gone well the machine code is now saved to disc. Type * LOGO and press RETURN. Now type * RUN RTIMC and press RETURN, * RUN STIMC and press RETURN and * RUN ADVAMC and press RETURN. The machine code should now be resident in your machine and accessible to LOGO.

Now type - FX16,2 to enable the Analogue to Digital channels.

All that remains is to define the new LOGO primitives for the children to use. They may be written as follows;

TO RETIME .CALL 2560 OP (65536*..EXAMINE 130) + (256

.EXAMINE 129) +.EXAMINE 128 END

TO ZEROTIME .CALL 2304 END

TO SENSE1 .CALL 896 OP (256 * .EXAMINE 114) +.EXAMINE 112 END

TO SENSE2 .CALL 896 OP (256 * .EXAMINE 114)+.EXAMINE 116 END

Now try typing in each procedure name to see what happens.

References 1. Papert, S., Weir, S., diSessa, A., Watt, D. (1979) Massachusetts

Institute of Technology (MIT) A.I. Memo 545 P. 1.5. 2. Feurzeig, W. (1971) Programming Languages as a conceptual framework

for teaching mathematics. 3. Nunns, T. (1984) Sensing and Control Projects for the BBC.

This work was undertaken during a one term secondment as a School Teacher Fellow at the University of Birmingham, Summer Term 1985. Part 2 will appear in the September edition.

An experience with (DARTS' by Gillian Oaks

S 20 1

' 7

C53

7'9 3

In trying to use the micro as an aid to the teaching of mathematics, I was experimenting with some of the "SMILE" programs (produced by ILEA), one of which produced an unexpected spin-off.

One day two fifth year, very low ability pupils were playing the game of "DARTS". This program is based on

the traditional game and can either be played as 'singles' or 'doubles' in which either side has the right to challenge their opponents score.

In the "doubles" challenge game, each team starts with 501 and has to state what they would like to score with their three arrows. The computer then randomly chooses to hit or miss the desired targets. In turn each person then needs to mentally calculate their score and subtract it from their teams running total to obtain the new score. The other team can then challenge if they consider that their opponents new running total is incorrect. Challenging a correct score warrants a penalty of 50 points.

The following is a reconstruction of a conversation that took place, quite naturally, between the two fifth-year boys and so the actual numbers used might be a little inaccurate. However they serve to demonstrate the method which Jonathan used to subtract two numbers.

At the start of the game both players had 501. Andy threw - Treble 20

Single 20 Treble 1

Heamentally added0his scorewto 83 and thensneeded to subtract 83 from 501. He was obviously struggling. Jonathan was becoming impatient - he wanted his turn! Andy struggled for several minutes during which time he muttered a few different numbers at which Jonathan had laughed. Then Jonathan decided to explain:

Jonathan "Look! - take 83 from 100 first - that's 17 isn't it? Then there's the other 401 to add on, so your score's 418!" Andy had screwed up his eyebrows and obviously hadn't understood Jonathan's logic. So he tried again.

42 Mathematics in School, May 1986



Documents

Mathematical Investigations Using Logo: Part One