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A Lesson on Fractions
Suzanne Lachaise
An account of what I saw in a lesson given by Maurice Laurent to two children during a
Teacher Training course in Maths from 2nd to 5th May, 1991 in Besançon. Christian
Duquesne helped me to remember the end of the session - corrected and completed by
Maurice Laurent.
The Children
A girl, Havoli, in Grade Three and a boy, Arona, in Grade Five. Their mother, a participant in
the Teacher Training course, introduced Arona as a child having a lot of difficulty
understanding fractions (he could just manage to understand what was going on when one cut
an apple in two or when one cut up a cake).
Havoli is bright, quick, sure of herself. She has a twinkle in her eye, loads of charm that she
knows how to use! Arona is more taciturn, not as sure of himself and, as far as one can judge,
convinced of the "superiority" of his sister, although this does not seem to affect him much...
This lesson took place on the second day of the training course. The participants had already
worked a lot on mental imagery. The importance of action and of perception for children had
already been talked about. Terms such as: temporal hierarchies, real and virtual action,
perception, evocation, another name for, changing points of view, elaborating appropriate
mental structures, algebra before arithmetic, mathematisation, and some others, had already
been put into circulation. Maurice Laurent would keep coming back to them throughout the
course, creating mathematical situations adequate for the students that we were.
The Lesson
The two children were seated facing each other. Maurice was at the end of the table with, in
front of him, a pile of white sheets of paper (21 cm x 29,7 cm). Because Havoli and Arona
were young, they were invited to engage in real actions: taking, folding, unfolding, showing,
touching etc.
Maurice gave each of them a sheet of paper and asked them to fold it in two. We could see
that the expression "fold in two " meant for them in two equal parts which could be
superimposed, and also that they showed, with respect to this action, standard behaviour of
folding the paper width against width.
Folding in two was easy; they were able to understand the instructions directly, they could do
it without looking as if they have to think about it -but they moved into it with care, Arona
being more careful than Havoli -this was part of what they brought with them. One can
construct on such solid bases. So they were asked to fold another sheet in 4. The same ease,


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founded on the experience that: folding in 4 is folding in 2 and again in 2.
When the sheets were folded, they were asked to unfold them and even to "iron " them so that
they would lie really flat!
Since they knew how to do this, they were able to observe the results and talk about them, but
when they counted the parts made evident by the unfolding process, they touched them at the
same time, with a finger or the whole flat of their hands: verbal expression was not
dissociated from action to begin with.
Then, folding in 3 put them in contact with the unknown; Havoli thought first of all and then
began and was able to do it; Arona tried to begin with what he knew how to do: fold in 2 and
then again in 2. The result surprised him; his second try was as unproductive as his first
although he modified his technique: having seen the results of Havoli's work, he folded a
small part of the paper and with successive folds he again finished with 4 parts. It was clear
that what he was asked to do did not create a structured mental image in him.
Maurice Laurent remarked about this: One cannot say that he cannot imagine a folded sheet
with three isometric parts (which can be superimposed). But what is sure is that his mental
image is not structured by adequate dynamics. In fact, to know how to fold in three, one must
be aware of the fact that if one folds 1/3 onto the rest of the sheet, this third covering a second
third leaves visible the third 1/3.
Strategy requires that one see the rotation of 1/3 coming on 1/3 and that, because of this, one
becomes aware that the two rectangles have the same dimensions. And the fact that one can
do it without hesitating means that one can do it mentally. Then, and only then, can we say
that folding in three is a feedback job which shows that the mental job has been done, as far as
this point is concerned.
And while Arona tried, concentrated on his job, Maurice left him alone and showed interest in
Havoli's work, Havoli who benefited from this intervention -which tended to make her more
present and more careful- and who... did not weigh on her brother: her looking down on him
and making demeaning comments were short-circuited without resorting to repression: this is
education at two levels. Havoli was learning to be more with herself and to leave her brother
his place, his right to make mistakes; whilst Arona was learning to take his legitimate place,
the right to make mistakes and the right to take the time he needs. Just after this, when
Maurice gave time to Arona, Havoli was extremely attentive to the educative action activity
in which Arona was the actor and participated "almost " without interfering.
Up to this point, the teacher has given instructions leading to correct actions. This time, one of
the children has a problem because the instructions do not send him back to a mastered
experience. This child is, therefore, in a learning situation and the consequence is that the
teacher has a real teaching problem: how to create a working situation which will allow the
child to generate the elaboration of an adequate mental structure the by-product of which will
be a strategy for correct action -or alternatively, how to transform a pre-existing mental
structure into a mathematical structure?
And the observers have the opportunity of watching how the teacher subordinates his teaching
to the learning of this student.
Maurice took a new sheet of paper and began with what Arona was trying to do: Folding the
sheet in the direction of the width, leaving enough paper sticking out for the following folds,
Maurice held the sheet, the width towards him, and began the folding movement by sliding
the edge closest to him in the direction of the opposite width. The sheet "rolled " in his hands;
the movement was slow enough to be able to generate in the child the idea of what was going
to happen (which is to say, he could foresee, anticipate), and the same time observe what was


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happening simultaneously above and below (the part of the sheet which slid and the part
which remained immobile on the table).
The pressure of his hands on the paper and the movement generated are such that:
▪ the fold is not marked - the fold rolls,
▪ the movement is continuous,
▪ the movement is slow so that you can see moment by moment the relationship between the
part folded over and the part which is not yet covered, and that the part folded over is
above the covered part.
But Arona remains responsible for making the decision because Maurice asks him to say
when he should stop moving the paper before actually folding it.
Remarks
▪ The child's fruitless trial and error strategy was stopped because, in this particular case, it
stopped the child from "allowing the problem to educate him ".
▪ The teacher has not worked in place of the student; he has freed him from what was
hindering him by being his assistant, proposing a situation which would allow him to
solve the problem. The decision remained with Arona since only he could show when
the necessary mental strategies had been developed.
To make such a decision, one must have one or more criteria for judging. Arona has the
opportunity of observing the movement of the paper on itself and of seeing the part which
moves forward get bigger as the lower part visibly becomes smaller or, according to the
direction of the movement, of seeing the part moving backward get smaller while the part
moving forward gets bigger. (We were to see the same thing the following morning with the
rods when we were dealing with constant sums, but the situation here is a little more
complex...) There were a few slow movements backwards and forwards, stopping here and
there so Arona could make comments, estimations and even measurements with two fingers
stretched out. The decision is taken when the mobile part -and the part that it covers- is the
same dimension as the part that is not covered. (But this is not said in so many words, "this
part is the same as that one " is sufficient for the moment.) (And we had an extra lesson here,
an instrumental lesson on the discipline necessary for using this way of measuring (one
mustn't change the distance between the fingers). Arona "measures " a possible fold with his
thumb and index finger, then M. L. asks him to bring them down on the same segment. Arona
sees clearly that he has changed the spread of his fingers. A few attempts are necessary in
order to be sure that he is in control of the spread of his fingers and there he is with a decent
tool for measuring. This tool will allow him to compare two lengths, each measured against a
third; this is preparation for visual evaluation that he will do some time in the future (real,
conscious action creates mental equipment for virtual action in the future -in a moment or in a
year's time...).
During this process of evaluation, the children had to consider the folds made by Havoli and
those then being made by Arona. Once the first fold was made, the free edge of the paper
folded onto the sheet served to mark the next fold. Maurice's questions "Where will such and
such a point be when we have finished folding? " allowed the observers to see that the
children imagined the movement correctly, that they had the notion of invariable points in this
transformation (the points on the axis of the folds) and they knew where the image of such
and such a point would be after the folding. A possible direction for more work opened up
which would not be explored because of the discipline inherent in the exercise taking place,
and because one must choose, since one cannot do two things at once.
Each child now had one sheet folded in 2, one in 4 and one in 3. They could when asked, once
again, say for each sheet how many parts it had, the name of each part (the children know the
words half, quarter and third); they could show one of the thirds, another of the thirds and still


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another, but also two of the thirds or two of the quarters on the corresponding sheet, or three
of the quarters. They could answer questions such as: "How many times can I ask you to
show me one of the quarters and then another of the quarters before you will have shown me
all of them? " They could also refer to a sheet which had not been folded and, by
experimenting and formulating, say that: "It's the same " to say 2 half sheets, or a whole sheet,
or 3 thirds or 4 quarters of a sheet. They were aware of "is another name for " (I don't change
the object, only my way of looking at it) and there has been a preparation for the awareness of
the notion of indifference, always present in Mathematics.
It is quite sure that we are working on the foundations of Mathematics, working on a specific
exercise, the children creating the mental structures, the future of which goes much further
than the present situation, the teacher putting them in contact with the mathematician in
themselves, using only what the students have brought with them: their experience, their
perceptive capacities, their bodies for action, their common sense, their capacity to abstract
(accentuating certain aspects while ignoring others), their judgement, their capacity for
expression, their faculty of creating dynamic images and acting on them, their aptitude to
control their muscle tone but also to suspend their judgement etc... (cf. the list of the attributes
of the self that the reader certainly remembers...)
Then, Maurice took a specimen of each sort of sheet, including one which had not been folded
under which he rapidly placed the sheets folded in 2, 3, 4..., in order, while the children
watched. Everything was there, but nothing was visible any more. Out of which came
questions on the invisible:
"What is there under this sheet? Just underneath? And what about underneath that? Which
one is the last one?..." This could all be checked, but it wasn't. We are in the domain of the
virtual: the mental images, heavily structured by personal actions and thinking -elaborated
step by step respecting the temporal hierarchies of the awarenesses which generated them- are
stable, printed for ever in their minds; evocation is easy, they can move about as easily in the
virtual ½, the 1/3 and the ¼ as they could before when they were reading the "real " ones.
And then a new question was put forth: "What would the next sheet be like?" The two
children replied immediately: "Folded in 5! " Maurice quickly folded a new sheet in 5: "I'm
folding this one."
(There are no distractions for an activity which will require all the presence of the two
children; knowing how to fold in 5 can be studied at some other time, it's a different problem;
whilst having a folded sheet serves as a bridge between the sheets folded by the children and
the following sheets folded only in their imagination: if I knew how to do it, the following
sheet would be folded in... 1 more than last time...)
(Havoli seemed to think that Maurice did the folding very quickly and suspected him of
having cheated and folded it in 6. She was happy only when she had counted the segments
along the edge of the sheet!)
"What's the name of each part?" "A fifth." "How many fifths are there in the sheet?" "Five!"
"What is the whole sheet called?" "Five fifths."
Questions on the following sheets which were completely virtual, and questions on the name
of the sheet folded in 9, in 15, in 43, in 100, in 'a', in... (other letters of the alphabet), which
forced the children to become aware of the way in which these words are constructed when
the automatic functioning misses a beat: folded in 'n' triggers 'enne néièmes' instead of 'enne
ennièmes', out of which comes a job of precision on the contents of words like 'cinquième'
and the expression 'cinq cinquièmes': there are 'cinq' and 'ième' in 'cinqième'. All this takes the
necessary time. Expressions such as 'ixe ixièmes' trigger: "Is that French? " This is a first
contact with algebra and indifference to numerical value. The important thing is to say the


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same word twice, as in: 'a aièmes'.
They explore the expression: "1 sheet is the same as 2 halves of a sheet which is the same as 3
thirds of a sheet... "
Summing up of the masteries taking place:
▪ The children know how to fold sheets in 2, 3, 4, 6, 8... parts.
▪ Effective special folding - in 5, in 7... - cannot yet be done.
▪ They can imagine folded sheets in any number of superimposable parts.
▪ They can name the parts and also the sheets in function of the way in which they are folded:
fractions equal to one.
▪ They can generate the corresponding correct mental images and act on them.
▪ They can express themselves according to their mental images: ¼ and ¼ and ¼ = ¾.
The teacher has made sure that the foundations exist and has furnished the materials for his
students to construct the edifice solidly for themselves on the basis of the human functionings
active at their age: action, perceptions, verbal expression describing these actions and these
perceptions and the affective states generated. All the time the work has been going on, he has
been making sure that bridges from one area to another exist and are solid, and that the
children can always return -through virtual or real action- to the place where the situation is
completely mastered.
Now it is time to go on to the next phase: symbolic notation.Maurice went to the board and
proposed writing what the children knew how to say:
1 sheet = (is the same as) 2 (two) /2 (halves of sheet) = (which is the same as) 3/3 (of a
sheet)...
Maurice was very demanding concerning the exact and complete expression of the
relationships symbolized by mathematical signs, and took the necessary time so that the
children acquired this discipline (cf. the indications in parentheses beside these signs).
Soon the board, elaborated by the children with rigorous discipline, showed:
one sheet = 2/2 = 3/3 = 4/4 = 5/5= / = 7/7 = 8/8 = ... 15/15 = ... = 43/43 = ... =111/111=...= a/a
Comments
After 5/5, Maurice wrote up the fraction bar alone, saying: "We won't write this, we don't
have time! " and he placed an = after which he waited for a new formulation which came out
without any problem: 7/7. After 8/8 = , he put a set of suspension points then = and the
children proposed 12/12, to which Maurice replied: "This one is folded in 15. 15/15 is
proposed, but I heard uncertainty in the child's voice concerning the value of "... ", the
suspension points.
At the next step, after another set of suspension points, Maurice announced a sheet folded in
43. The difference was so great that the children realised that this was not a mistake in
perception on the part of the teacher but a deliberate decision to indicate by the suspension
points a jump of some size.
Their voices were certain once again when they said 43/43 and, in their whole attitude, half
standing, leaning towards the board, I felt their attention stimulated, their excitement at the
idea of the mysteries which were sure to come, at the same time as the certainty of being able
to deal with the coming challenges. When 'a/a' came up, we were in contact with their
indifference to numerical value; it's 'something somethingths'. Here again is the algebra which
makes us perceive all the possibles in a single expression. It is this tremendous power of
algebra that the children are confronted with so quickly and with so much pleasure.


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What happened next
Then Maurice gave the following problem: "Someone has got one sheet of paper and another
one folded in five, but he's only looking at 3 parts of the second sheet. How can we write that?
He wrote: 1 sheet and (the 'and' was underlined twice and emphasized when Maurice said it),
Havoli completed what was necessary: 3/5 of a sheet.
Maurice: "Or?"
Both the children answered spontaneously: "1 + 3/5"
Maurice: Now someone has a sheet of paper folded in 11 but he's only looking at 7 of the
parts, and another full sheet.
He wrote on the board: 7/11 + 1 = ___ + ___ = ___
The children responded directly: "18/11", without saying the intermediary step (___ + ___).
Comments
It is obvious that they said it to themselves, and at the same speed, as Maurice wrote up the
fraction bars without saying anything. It can be said that the partial written representation
where the structure is present but not the numbers, allows the two pupils:
a. to make a correct mental formulation (their final answer is proof of this).
b. to work on the formula, the algebraic substratum, in a way that prepares them for and leads
them to indifference to numerical values, while they still need these values in order to
construct their knowledge.
c. to demonstrate to themselves that they are just as sure of the reality and the truth of their
mental propositions as they were earlier of their formulations and as they were even
earlier of their perceptions and actions, because they know they can go back through
the steps if they feel they need to, if there is any doubt in their mind at any level.
Next
Maurice : "I've got a sheet of paper and I'm hiding ¾ of it. " (written on the board: 1 - ¾). The
children immediately say "1/4 ".
Maurice : "I've got 2 full sheets of paper and another one folded in 5 but I'm only looking at 3
parts of it."
The children find 10/5 easily but are confused about what should come next. So Maurice goes
back and takes them through the steps again with the same rigor of expression that I will
explain here only for the first problem: "4 full sheets and 2/3 of a sheet is the same as 12/3 of
a sheet and 2/3 of a sheet, (the children are speaking while Maurice is speaking and writing),
which is the same as ?". The children say alone: "14/3". 4 + 2/3 = 12/3 + 2/3 = 14/3
The same thing for:
10 + 2/3 = ... = 32/3 100 + 2/3 = ... = 302/3 1000 + 1/3 = ... = 3001/3 1000 + 5/3 = ... =
5001/3
Now they can solve the problem from before: 2 full sheets and one folded in 5 of which I'm
looking at 3 parts:
2 + 3/5 = 10/5 + 3/5 = 13/5
Above all, they can move on easily to the usual way of writing things:
1 + 1/a = a/a + 1/a = (a + 1)/a and : 1 + 1/x = x/x + 1/x = (x + 1)/x


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Remark
(a + 1) and (x + 1) have been put in brackets to show that they are the numerators of the
corresponding fractions. Fractions with a horizontal bar cannot be written with this word
processor.
Conclusion
The children worked on the addition and subtraction of whole numbers and fractions. These
classic and important exercises encountered no resistance because the work was done as it
was, respecting the temporal hierarchies, this being the order of awarenesses to be had at each
level (action, perception, mental representation, oral expression, symbolization and access to
the concept). All this took just a bit more than one hour ... As far as the educative value of the
"lesson " is concerned, one only had to look at Arona: his face had changed; he no longer
looked like a submissive child; he was speaking out with assurance and often spoke ahead of
his sister, who, (surprised by this new brother?) started making mistakes because, in order to
try to maintain her prerogatives as the one who knows better and faster, she had to go so
quickly. Arona no longer let himself be pushed around; the children were on an equal
standing.
Thank you Maurice Laurent for the power of this lesson which was not a model lesson in the
traditional sense of the word, since it always depended on the reactions of the pupils
concerned. However it was a model of how to lead a lesson showing mastery of what was
being taught from the point of view both of the content and of the knowledge of the
awarenesses that need to be forced in each pupil ready to move into the field in question. This
was done with the openness of mind and the sensitivity necessary to stay in contact with the
inner states of the pupils as they moved into the new universe which was presented to them
for their discovery. We are - at last - in the universe of the Science of Education.
© Suzanne Lachaise Besançon, France, June 1991
The Science of Education in Questions - N° 5 - June 1991
"A Lesson on Fractions" by Suzanne Lachaise is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs 3.0 Unported License.

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