Describe, Define, Detail
DESCRIBE, DEFINE, DETAIL
- a didactic alternative to DOODLE, DESIGN, DEBUG.
Liddy Nevile when? post 1989??
At one level this paper contains some procedures and ideas for bringing students into contact with the fraction relationship. It aims to foster the development of investigations which will ultimately lead to understanding and some useful techniques for handling fractions (algorithms)1
At another level, the paper aims to make explicit one process for learning with Logo as suggested by Professor Uri Leron in a workshop recently held in Melbourne (1989). This process brings the Logo computer into play as a medium in which some already DESCRIBED concepts are DEFINED at a level of abstraction2 which suits the learner, possibly even containing some undefined details. In the third phase of the process, the DETAILs are either supplied or defined. Each of these phases is likely to include the whole process being described here: the process is recursive.
It is always very tempting for teachers to offer answers to their students. According to the learning model on which this work is based it is the students' task to construct the definitions and operations and the teacher has the task of creating the environment in which this will happen. This paper aims to help prepare a teacher for the ideas and incidents which will arise so that the teachers are able to take full advantage of them as 'teaching moments' (Mason et al, 1989).
In the work on teaching moments being done at the Open University, Mason and others have pointed to the need for teachers to learn to recognise when a moment has arisen which can form the basis for some fruitful discussion. This is a three-step process in which teachers first come to recognise that a special moment has arisen, secondly recognise the possibilities which the moment makes available and thirdly how to take advantage of those moments.
A typical moment is the one when the student says "I'm stuck". Instead of responding with a solution to the problem, the teacher can choose to use the moment as one in which to focus on ways of coping when one is stuck. Such moments are likely to occur in a free discussion about fractions: "I don't know why ...." or "that's because ...." can provide moments for noting ideas for later investigation and moments for considering how the particular hypothesis might be tested.
SETTING THE SCENE
The explicit topic of the unit is fractions and these are introduced by a discussion of sharing and cutting - open and descriptive with the aim of raising interest in the topic.
The open-endedness is to foster interest beyond the immediate concrete examples and to prepare for explorations to find ways of determining what might happen in cases when it is not easy to perform empirical tests. The openness is also designed to encourage students to think freely about fractions rather than merely to say what they 'know'. The descriptive nature of the discussion should help students describe fractions in everyday language which also will help them access their intuitive understanding of fractions. In harmony with these ideas, it is recommended that a particular goal not be set for the 'lesson': two good reasons being that by setting and stating a goal the teacher may set up the game of 'can the students guess what the teacher wants' and because anyway it may be that the lesson, if well conducted, does not flow according to a predetermined plan or schedule.
Within a short time a discussion about fractions is likely to include ideas about both fractions and operations on fractions such as: as the number of parts into which something is divided increases the size of the parts gets smaller. A useful activity to encourage is the paper and scissors testing of the students' ideas. Why not have squared paper, mark out squares to represent the 'whole' and cut out some squares to represent the 'parts':
MO-
a 'whole' is
being divided in 22 'parts'
There might be some discussion about the validity of this method as well as what might be discovered from its use. Again, the discussion will be more fruitful for the students if the teacher does not `deal with' or provide answers for these questions.
One idea is to start a class list of things which need to be investigated. Then later it will make sense to encourage students to contribute some 'what we found' and 'why we believe it' sheets of paper.
DESCRIPTIONS
From the discussion, the students and teacher together can develop some DESCRIPTIONS of fractions and fraction operations. One way to help this process is to test the descriptions with reference to a number of examples.
Some time in the discussion about dividing things up it is likely that a moment will arise when the names numerator and denominator can be considered. Why not use the moment without being prescriptive about numerators and denominators and integers? Is there any reason why the teacher should preclude the possibility that a numerator or denominator is itself a fraction? Perhaps this becomes another of those questions for investigation.
And how do we write fractions? It would be helpful if at this stage students could not only gather up the traditional forms of representation but also invent a few new ones.
A class discussion is likely to flow better in circumstances where precision is not demanded. Postponing the need for precise definitions and for explanation of details maintains the level of abstraction in the discussion. Leron's model (1987) draws on the computer science model of abstraction which can be illustrated in this case as follows:
gmneral discussion - in English
commands and operations in programming language (I ngo)
definition of new Logo 'primitives' for use of student programmer
the underlying Logo language
2
I All credit for the ideas on which this paper is based should go to Dr Rina lazkis and Professor Uri Leron of the Israeli Logo Centre in Haifa.
2 defined as it is in computer science
--------------
In the description phase the students articulate their ideas at a very abstract level - that is, there is no concern for detail so that how things might be implemented is not an issue.
DEFINITIONS
In the next phase of the lesson, the students will reduce their descriptions to definitions in Logo. This process is recognised as a distinctly mathematical activity in which intuitive ideas become formal mathematics.
So what would seven ninths look like? It is not very convenient to write a fraction such as 7/9 in Logo because the computer immediately performs the division and replaces the vulgar fraction by the decimal 7.7777777. Rather than tell the students this, the teacher may choose to use another teaching moment, "What happens if a Logo user types 7/9?" After allowing the students time to think about this problem, it may be necessary to suggest that as Logo respects lists and does not change their form unless commanded to do so, the fraction could be represented in Logo as a list, ie [7 9). The square brackets show that the fraction is an ordered list of two elements where the first is the 'numerator' and the second (or last) the 'denominator'.
Now, one way to define a fraction in Logo is to write a procedure which makes a list from a given numerator and denominator and gives back a Logo-type fraction. In Logo terms, this will be a procedure which operates on two numbers to convert them into a Logo-type fraction:
insert image
Thus the procedure will expect two inputs, numerator and denominator, and will output one thing, the fraction:
TO LOGO.FRACTION :NUMERATOR :DENOMNATOR
OP SE :NUMERATOR :DENOMENATOR
END 3
and the complementary procedures which draw the numerator or denominator out from a given fraction:
TO NUMERATOR :FRACTION TO DENOMNATOR :FRACTION
OP FIRST :FRACTION OP LAST :FRACTION
END 4 END 5
3 To use this type SHOW LOGO.FRACTION 7 9
4 To use these type SHOW NUMERATOR [7 9] eg.
5 For students who are inexperienced Logo programmers, it may be helpful to type the programs being suggested into Logo and the students can just use them as they would any other Logo primitives. To do this in Logowriter, type the words into a page and then save it with the name DEFNS. Then create another page called FRACT'NS and write a little procedure on that page of the type:
TO STARTUP
GETTOOLS "DEFNS
END
The students will need to leave the little STARTUP procedure on the page called FRACT'NS.
DETAILS
The programming technique which produces the procedures just shown can be avoided at this stage by making pre-written procedures available or the 'teaching moment' can be seized. Such details as the way Logo procedures handle variables, the difference between commands and operations, the use of SE to make a list from some elements are just a few aspects of this.
Another area for exploration at this level concerns the fractions themselves. It is possible to create fractions in this Logo notation which are syntactically correct but which have zero as the denominator. Similarly, some of the Logo fractions might have fractions as the numerator or denominator. A 'filter' can be constructed to test if a fraction is of the type which is to be accepted:
TO FRACTION"? :X
OP (AND (LIST? X) (2-COUNT :X) (NUMBER? NUMERATOR :X) (NUMBER? DENOMINATOR X) (NOT EQUAL? 0 DENOMINATOR X) )
END
It should be noted that this filter accepts as 'fractions' lists which have fractions as numerator or denominator. It requires that a fraction is expressed as a list, that it has two elements, that each element is a number, and that neither number is zero. It can be modified to exclude 'fractions' with fractional numerators and denominators. Teachers will be able to judge at what stage of the activities each of the fraction criteria need to be included.
OPERATIONS
So far we have only worked on the definition of fractions. What about operations involving fractions, like addition, subtraction, etc? This could well be the point for many students at which the real fun starts.
Again the describe, define, detail process is recommended. The discussion will be open again and focus on the intuitive ideas about operations on fractions. Definition of a first Logo procedure can follow the same lines:
TO ADD.FRACTIONS :LOGO.FR1 :LOGO.FR2
OP LOGO.FRACTION (NUMERATOR :LOGO.FR I) + (NUMERATOR :LOGO.FR2) DENOMINATOR :LOGO.FR 1 END
(What about subtraction of fractions with the same denominator? This is merely a clone of the ADD.FRACTIONS procedure.)
Then there will be some details which will need to be dealt with. How might two fractions with different denominators be made to have the same denominators? Let. us try with 1/2 and 3/4? What about things like lowest common denominator? etc. If the .,.,i.udents are working from intuition, this may not arise immediately.
So again, the first step is to DESCRIBE what happens if two fractions have different denominators and the students want them to have the same denominators. A statement like 'we can multiply each denominator by the other one' will suggest one possible solution to the problem.6 the definition of the statement is easy (if long):
6 It is often difficult for teachers to resist the temptation to 'guide' students towards the `correct' solution. In cases such as the one presented here, a valuable teaching moment can be missed if a student suggests a plausible answer which could be investigated by students who are not sure of its validity.
TO SAME.DENOM :LOGO.FR1 :LOGO.FR2
IF EQUAL? DENOMINATOR :LOGO.FR1 DENOMINATOR :LOGO.FR2 [OP SE :LOGO.FR1 :LOGO.FR2 STOP]
OP SE LIST LOGO.FRACTION (NUMERATOR :LOGO.FR I) * (DENOMINATOR :LOGO.FR2) (DENOMINATOR : LOGO.FR1)* (DENOMINATOR :LOGO. FR2) LOGO.FRACTION (NUMERATOR :LOGO. FR2) • (DENOMINATOR :LOGO.FR1) (DENOMINATOR :LOGO.FR2) * (DENOMINATOR :LOGO_FRI )
END
Now we will have a list of two fractions with the same denominator. To do ADD.FRACTIONS to this list will cause an error because the ADD.FRACTIONS procedure wants two inputs. Here again there is a need for more detail. It will be necessary to take the fractions back out of the list or modify the original ADD.FRACTIONS procedure.
METHOD I:
Just write out in full what is required:
eg, when calling ADD.FRACTIONS for fractions with different denominators, type
ADD.FRACTIONS FIRST (SAME.DENOM :LOGO.FR1 :LOGO.FR2) LAST (SAME.DENOM :LOGO.FR1 :LIOGO.FR2)
METHOD 2:
Making a more powerful ADD.FRACTIONS procedure. The original procedure looked like this:
TO ADD.FRACTIONS :LOGO.FR1 :LOGO.FR2
OP LOGO.FRACTION (NUMERATOR :LOGO.FR1 + NUMERATOR :LOGO.FR2) DENOMINATOR :LOGO.FR1
END
We now have a procedure for making the denominators the same so we can use it with a filter to
make an ADD.ALLFRACTIONS procedure:
TO ADD.ALL.FRACTIONS :LIST.LOGO.FRACTIONS
IF EMPTY? :LIST.LOGO.FRACTIONS [OP [0 I] STOP]
OP ADD.FRACTIONS
FIRST SAME.DENOM FIRST :LLST.LOGO.FRACIONS
ADD.ALLFRACTIONS BF :LLSTLOGO.FRACTIONS
LAST SAME.DENOM FIRST :LISTIOGO.FRACTIONS
ADD.ALLFRACTIONS BF :LIST_LOGO.FRACTIONS
END 7
The next step is likely to be concerned with simplification of fractions. This is a list processing exercise and should not be too difficult. It is wise to remember to DESCRIBE what is to happen, then to DEFINE a procedure with a few as yet undefined bits if necessary and then to go below the abstraction barrier to define those details which will make it a most effective procedure.
MULTIPLICATION AND DIVISION OF FRACTIONS
Multiplication should not provide difficulties so long as the DESCRIBE, DEFINE, DETAIL process is used. It may also be possible to use the procedures in the section on addition of fractions as models for those in multiplication and division.
7 To use this procedure it will be necessary to present the fractions to be added as a list eg ADD.ALLFRACTIONS [(3 4] [5 6] [3 81]
S
6
ome interesting ideas about division of fractions can arise if the fractions which are being used include ones in which either the numerator or the denominator is itself a fraction (see above). One way to make this acceptable to the Logo fraction machine is to provide another filter which makes the numerator, or denominator, operate like ordinary numbers. Why not merely reduce the LOGO.FRACTION to a decimal for the operation?
TO SAY.AS.NO :LOGO.FRACTION
OP (FIRST :LOGO.FRACTION) / (:LAST :LOGO.FRACTION) END
This would be sufficient if we did not have the problem that one of our LOGO.FRACTIONS might be a list of LOGO.FRACTIONS, eg [ [3 4]+[4 5] [8 9]*[3 4] ] or even zero.
So we had better make the procedure able to handle all possibilities:
TO SAY.AS.NO :LOGO.FR
IF EQUAL? 0 LAST :LOGO.FR [PRINT [This cannot be done.) STOPALL] IF AND NUMBER? FIRST ,ILOGO.FR NUMBER? LAST :L000.FR
[OP (FIRST :LOGO.FR) / (UST :LOGO.FR)]
OP (SAYASNO FIRST :LOGO.FR) / (SAY.ASNO LAST :LOGO.FR) END
Now, the procedures just developed are fairly difficult to write and may need to be given to the students.
CONCLUSION
The process has really just begun. The students will be able to go on thinking of refinements for their system for some time and as they gain proficiency in the system and programming in Logo they will be building a more complete understanding of fractions and their relationships.
In this paper the fraction exercise was used only as an example. It is not always easy to gain meaning from a theoretical statement and so this paper has been presented as an experience in this process to be shared with the reader. It aims to present some description and some definition of the process and to point to some details which are available in the literature (see REFERENCES below).
REFERENCES:
Abelson, H., L. Sussman, G., Structure and Interpretation of Computer Programs. Massachusetts: MIT Press, 1985.
Harvey, B., Computer Science Logo Style, Vol. 1, Interme Jiate Programming. Massachusetts: MIT Press, 1985.
Leroa, U., "Abstraction Barriers in Mathematics abd Computer Science" in Hillel, J., (ed.), Proceedings of the Third International Conference for Logo and Mathematics Education, p. 12. Montreal: Department of Mathematics, Concordia University, 1987.
Leron, U., "Implementing fraction algebra (rational number arithmetic) in Logo: an abstract data approach". Unpublished paper for Mathematics and Microworlds Workshop, Melbourne, September 25-7 1989.
Mason, 3., et al., "The study of teaching moments". Milton Keynes: Centre for Mathematics Education, Open University 1989.
Wills, S., "Doodle, Design, Debug" in Oakley, J., (ed.), Computers in Education: Issues and Applications, p. 219. Sydney: CEG NSW, 1985.
Appendix to Fraction Machine Paper
The fraction machine is itself built from a set of little machines (function boxes?). The machines can be labelled, cut out and assembled as required.
..■
Single input function machines:
Two-input machines: