1. The document describes how to use mental imagery to teach mathematical concepts like fractions, ratios, and algebraic expressions. It presents a series of figures and asks the reader to imagine moving points on lines and consider their relationships. 2. It then uses these figures to introduce topics like distributivity, literal calculations, products of sums and differences, and ratios. Students are guided to express relationships first using specific values then more generally using letters. 3. The goal is to build familiarity with mathematical operations and concepts through dynamic visualizations, then transfer this understanding to algebraic expressions and problem solving. Students learn to perceive relationships that hold true regardless of specific values.

1. http://www.uneeducationpourdemain.org
Page 1 sur 7
Mental Imagery in Mathematics
Maurice Laurent
Introduction
Words have no meaning in themselves: orally they are only arbitrary juxtapositions of sounds.
However they trigger off in us images which, generated by our perceptions, carry meaning.
Where do these images come from? First of all, from the faculty we have of perceiving the
world around us and states or various movements inside us, that is to say of the faculty we
have of being sensitive to the energy impacts coming from the outside in different forms and
to movements of energy inside us. Next, from the power we have to conserve structured and
objectified traces. Finally from the power we have to recall these traces, to evoke them.
Since creating images and evoking them are powers of the self at our disposal all the time,
and since perception is the source of meaning, it makes sense to take them into account when
we want to help others learn, and more particularly in our teaching of mathematics, if we want
this activity to make sense to those who must do it. In fact, it only means allowing others to
transfer the powers they already have to a type of activity where they very clearly have their
place.
Based on a precise example, the object of this article is to show how it is possible to solicit in
our students the power of creating images so, educating it all the while, they put it to the
service of true mathematical activity.
1. How to create a mathematical being to order (Figure 1)
Imagine a rectangle with one horizontal side. Call A the vertex which is at the top left. If you
run your eyes clockwise starting from A around the perimeter of your rectangle, you will find,
in order, the vertexes B, then C, then D. Look at and name, starting from A but going anti-
clockwise, the series of consecutive vertexes.
Now imagine that a red point called V, placed on A has the freedom - and that freedom only -
to go to B in a straight line and then to return to A the same way. We can say then that it
follows - or describes - a line segment [AB] and that it belongs to the line segment [AB].(1)
.
Imagine now that another red point, V', which belongs to [DC], has the same freedom to go
along it from one extreme to the other, in both directions and at any speed. Make it move,
speed it up, slow it down and stop it between D and C ...
Now put V and V' respectively on A and D. In such a situation we say that they are
respectively merged with A and D. Imagine that they start to move simultaneously, that they
do so at the same speed and that their speed does not vary, remains constant.
Where is V' on [DC] when V, leaving from A, has covered half of [AB]? one quarter of

2. http://www.uneeducationpourdemain.org
Page 2 sur 7
[AB]? 3/5 of [AB]? Where is V on [AB] when V', leaving from D, has covered 1/3 of [DC]?
still has 2/7 of [DC] to cover to reach C?
You are aware, from now on, of the fact that the movement of V on [AB] is linked to that of
V' on [DC] and vice versa, in other words, that the movement and the position of one
determines the movement and the position of the other.
Now trace the line segment whose endpoints are V and V'. Consequently, new questions can
be asked of you. What figure does [VV'] scan when V follows [AB]?(2)
What are the elements
of your figure which have the same direction as [VV']? What are the elements of your figure
that have perpendicular directions? And whatever position V is in between A and B, is it
always true?(3)
How do you know?
2. Dealing with distributivity (Figure 2)
Now we are going to look at the same situation from another angle.
How many rectangles do you see when V is joined neither with A nor with B? Name all three
of them. What does [AB] measure if [AV] measures 7 cm, if [VB] measures 4 cm, and if 7
and 4 are the only numbers you are permitted to use to express it? If [AD] measures 8 cm,
then what is the area of ABCD? of AVV'D? of VBCV'? Is there any relation between these
three areas?`
If you first look at [DA], then at [AV], then at [V'V], then at [VB], you immediately see that:
A(ABCD) = A(AVV'D) + A(VBCV') 8 x (7 + 4) = 8 x 7 + 8 x 4
And if you look at the two small rectangles before the big one, you immediately see that:
8 x 7 + 8 x 4 = 8 x (7 + 4)
In what order must one look at the elements of the figure in order to read:
(7 + 4) x 8 = 7 x 8 + 4 x 8 8 x (4 + 7) = 8 x 4 + 8 x 7 (4 + 7) x 8 = 4 x 8 + 7 x 8 7 x 8 + 4 x 8 =
(7 + 4) x 8 ...
Such exercises - you only need to change the numerical values to create others - can be given
until they become so easy for the pupils that they are no longer of interest. In doing them, they
practise the property of distributivity of multiplication over addition, to the left and to the
right, the symmetry of an equation, and create a new mental structure. But what's more here,
leading the pupils to ease, is leading them to indifference of numerical values and putting the
accent on operations, in other words giving them access to algebra.
What will permit this is the awareness of the fact that, whatever the dimensions of [AD],
[AV] and [VB] are, the relation between the three areas in permanent.
Call the length of [AD] x, of [AV] a and of [VB] b. What segments of the figure are the same
length as x? as a? as b? Express the relation between the three areas using x, a and b in eight
different ways:
x (a + b) = xa + xb (a + b) x = ax + bx ... xa + xb = x (a + b) ...
After having read, said and written down one of the eight equations, for example: x (a + b) =
xa + xb, is it possible to find the seven others without evoking or "reading" the figure? And
by using which properties of addition and multiplication?(4)
3. Double distributivity (Figure 3)
In the same way that you linked the movement of V, V' and [VV'], now link H, a point
belonging to [AD], H', a point belonging to [BC] and [HH'].

3. http://www.uneeducationpourdemain.org
Page 3 sur 7
Now imagine that V and H are on A and that V takes the same amount of time to follow [AB]
as H does to follow [AD], whatever the length of the two segments is.
Where is H on [AD] if V has covered half of [AB], one third of [AB], three quarters of
[AB]?... Where is V on [AB] if H has covered two fifths, three sevenths, of [AD]?... Where is
V'... Where is H'... How many rectangles do you see if V is neither on A nor on B? Knowing
that [VV'] and [HH'] intersect at I, name them, the nine of them, because that is how many
there are.
Suppose that AH = 3 cm, HD = 4 cm, DV' = 5 cm, V'C = 6 cm.(5)
Look at, read and express
the fact that the area of the big rectangle is equal to the sum of the area of two rectangles, of
two other rectangles, of four rectangles... Name the length of [DV'] a, of [V'C] b, of [CH'] c
and of [H'B] d. Read and write down the identical elements that you see, if V is neither on A
nor on B.
As in (2), we easily succeed in going from numerical expressions to literal expressions, from
the expression of particular cases to that of all cases...
(a + b)(c + d) = a(c + d) + b(c + d) (a + b)(c + d) = (a + b)c + (a + b)d (a + b)(c + d) = ac + ad
+ bc + bd ... (c + d)(b + a) = ... ... a(c + d) + b(c + d) = ... ...
The different forms of equality are more numerous here. Write them all down and/or
determine how many there are if your heart so desires!
The most important awareness here is that all these equations are at one's disposal, for they
are all perceptible, tangible, differing one from another only by the order which I choose to
look at them.
4. An introduction to literal calculation
On the basis of the dynamic mental images formerly structured - Figure 1 and Figure 2 -
numerous exercises, varied and graded in difficulty, can be proposed to pupils, which they
will do easily and which will introduce them to literal calculations on the basis of meaning
and where the adequate terminology is introduced. We will give here only some of the
possible orientations. First evoke Figure 2. Let it be known that, once and for all, one unit of
measure has been chosen: cm, dm... or any other segment unit.
▪ If a has the value of 2, do you see that: x (2 + b) = xx 2 + x x b = 2x + bx ? x x2 + x x b = x
x (2 + b) = x (2 + b) ?
▪ If you imagine that a and x are the same length, do you see that: x(x + b) = x x x + x x b =
x2 + bx ? x2 + bx = x x x + x x b = x(x + b) ?
▪ If b is the unit of measure of the length, do you see that: x(a + 1) = x x a + x x 1 = ax + x
? ax + x = x x a + x x 1 = x(a + 1) ?
▪ And if x = 2a?...
▪ And if x = a = b?...
▪ 7, ...n) And if ...
Now evoke Figure 3. Draw it, if need be.
▪ If a = 2c and if d = 2b, do you see that: (2c + b)(c + 2b) = 2c2 + 4bc + bc + 2b2 ? = 2c2 +
5bc + 2b2?
▪ If b = 1 and if d = 1, do you see that: (a + 1)(c + 1) = ac + a + c + 1? 3, 4, ...n)
▪ And if...
The generally used conventions of reading and writing and how to write them down can be
introduced little by little during these exercises, when they are needed for concision, ease and
elegance of expression: the place of numerical coefficients in the expression of monomials (x
x 2 = 2 x x = 2x), letters ordered alphabetically (x x b = b x x = bx), notation of the products

4. http://www.uneeducationpourdemain.org
Page 4 sur 7
of identical factors in the form of powers ((a x a = a2)... Also to be introduced are the words
which relate the algebraic transformation which a product or a sum can undergo, in order to
give them another appearance: the multiplication of a monomial by a binomial, the addition of
similar monomials, the factorisation of a polynomial... These exercises will be done with
sureness if they continue to be done on the basis of perception.
5. The products of sums and differences
Evoke Figure 2 again. x remaining the length of [AD], call the length of [DC] a and that of
[V'C] b. Then express the length of [DV'], then the area of the rectangle AVV'D. Whatever
the position of V is, is it always true? Do you see that:
x (a - b) = xa - xb = ax - bx? (a - b) x = ax - bx?
Now evoke Figure 3. Conserve DV' = a and V'C = b. Let CB = c and H'B = d. Do you see
that:
(a + b)(c - d) = a(c - d) + b(c - d) ? (a + b)(c - d) = (a + b)c - (a + b)d ? (a + b)(c - d) = ac - ad
+ bc - bd ? (a + b)(c - d) = ac + bc - ad - bd ?
Recall Figure 3 once again and modify it as necessary to express: (a - b)(c + d), in four
different ways.
Finally you would like to express (a-b)(c-d).
If DC = a, V'C = b, CB = c, H'B = d, is it really the expression of the rectangle DHIV'?
Do you see that
if we take the rectangles ABH'H and BCV'V away from the rectangle ABCD, we take away
twice the rectangle VBH'I,
if we want to conserve only the rectangle DHIV, that is to say, (a - b)(c - d)?
So, you see that:
(a - b)(c - d) = ac - ad - bc + bd
6. Lengths, areas and fractions (Figure 4)
Evoke Figure 3 again. Transform it such that [AB] and [AD] are the same length. What figure
has the rectangle ABCD become?
Whatever the position of V between A and B, what segments are the same length as [AV]
now? and as [VB]? Consequently, what do the rectangles AVIH and CV'IH' have in
particular? If V, starting from A, has followed less than half of [AB], what can you say about
the area of AVIH, compared with that of CV'IH'? What condition must be imposed upon the
position of V between A and B - or on H between A and D - if we want the area of AVIH to
be greater than that of CV'IH'? And what happens when V is on A? on B? in the middle of
[AB]?(6)
When V is one third the way along [AB] starting from A, what distance is left for it to cover
to reach B, in relation to what it has already covered? Do you see that it is the double, that is
to say two thirds of [AB]? What fraction of [AB] does V have left to cover if it has already
covered 1/4 of [AB]? 2/7 of [AB]? 5/9 of [AB]? a/n of [AB]?
You see that when V is at 5/11 of [AB], 6/11 are left to cover: we say then that AV and VB
are in the ratio of 5 to 6 and that VB and AV are in the ratio of 6 to 5.

5. http://www.uneeducationpourdemain.org
Page 5 sur 7
What ratio are AV and VB in when V, starting from A, has covered 1/4 of [AB]? 1/5 of
[AB]?... 3/4 of [AB]?... 1/n of [AB]? a/n of [AB]? Do you see that if V has covered 13/29 of
[AB], AV and VB are in the ratio of 13/(29-13) or of (29-13)/13, according to whether V
started from A or from B? that if V has covered a/n of [AB], AV and VB are in the ratio of a
to (n-a) or of (n-a) to a?
And in each of the above cases, in what ratio are the areas of the squares AVIH et IH'CV'?
Note down, if necessary, your observations in a 4 column table:
AV VB AV/VB A(AVIH)/A (IH'CV')
1/4 de [AB] 3/4 de[AB] 1/3 1/9
You see that, whatever the fraction of [AB] covered by V starting from A is, you also see
what fraction is left for it to cover to reach B; that at the same time you know the ratio of AV
to VB, and immediately after, the ratio of the areas of the 2 squares that have respectively the
sides AV and VB... Because you have been lead, on the one hand to consider AV and VB
simultaneously, the ratio of one to the other and the ratio of the areas of the squares having
AV and VB as sides on the other hand, the awarenesses have been forced which allow you to
say:
a. If a point is situated at a/n of a segment, it divides it in the ratio of a to (n - a) or of (n - a) to
a.
b. If the 2 squares are such that the ratio of their sides is a to b, (b = n - a), the ratio of their
areas is equal to (a x a)/(b x b) = a2/b2; in other words the ratio of their areas is equal
to the ratio of the squares on their sides. Similar work based on imagery can be done
that will lead to comparing the areas of the squares and the rectangles of this Figure 4,
either one in relation to another or the others, or one in relation to itself when V
occupies a set of positions, or particular positions between A and B. Do it yourself if
you wish to, in order to be sure, for example, of the soundness of the assertions below:
c. If AV<VB,(7)
then A(AVIH) < A(HIV'D) < A(IH'CV').
d. When V follows [AB], from A to B, the area of the rectangle VBHI increases from 0 to
A(ABCD)/4 then decreases to 0.
e. If V is not in the middle of [AB], then A(AVIH) + A(IH'CV') > A(VBHI) + A(HIV'D).
f. If V is situated at a/b of [AB], the rectangle VBH'I has the same area as if V were situated
at (b - a)/b of [AB].
g. ad libitum...
7. Remarkable identities
Now that Figure 4 constitutes a mental image at our disposal, we have direct access to the
three basic special products, that are recognized to be true whatever the length of the side of
the square ABCD is, and whatever the position of V between A and B is.
Indeed, if AV = a and if VB = b, then AB = a + b. Consequently, look at your Figure 4 and
read:
A(ABCD) = (a + b) x (a + b) = (a + b)2 = a2 + ab + ab + b2 (a + b)2= a2 + 2ab + b2 (First
Remarkable Identity).
Verify also that:
a2 + b2 = (a + b)2 - 2ab (a + b)2 - (a2 + b2) = 2ab.
Now if you name the length of [AB] a and that of [AV] b, a x a = a2 expresses the area of the

6. http://www.uneeducationpourdemain.org
Page 6 sur 7
square ABCD and b2 that of the square AVIH.
Imagine that the square ABCD is white and we have superimposed the red square AVIH on it.
The area of the part that stays white, a2 - b2, is the difference of the areas of the 2 squares.
Look at the white part. Turn the rectangle VBH'I a quarter turn clockwise around its vertex H.
It takes the names of its vertexes with it. What can you say now about [CH'] and about [H'I],
the side of the rectangle that pivoted. Slide the rectangle IH'BV down until its vertex H' is on
C. Do you see that, consequently, its vertex I is on H'? Now you can see a new rectangle,
HVBD. Do not hesitate to repeat the operation until you are sure of your mental image...
Express the length and the width of this new rectangle using a and b, then the area. Thus, now
you see the area of the white part as (a - b)(a + b), whereas before you perceived it as a2 - b2:
(a + b)(a - b) = a2 - b2 (Second Remarkable Identity). (a + b)(a - b) + b2 = a2
For the perception of the last product, nothing more is demanded of you than to look at your
Figure 4 as you looked at your Figure 3 at the end of paragraph (5):
If AB = a and if VB = b, then AV = a - b and AH = a - b A(AVIH) = (a - b)(a - b) = (a - b)2
(a - b)2 = a2 - ab - ab + b2 (a - b)2 = a2 - 2ab + b2 (Third Remarkable Identity).
8. Some general considerations
a. The powers of the self, imaging and evoking, were called upon.
b. Words trigger mental images in those who read them or who hear them.
c. Words are arbitrary, but the images triggered by them are not: they are tangible and make
sense.
d. It is possible, through using words, to do things in such a way that others create
mathematical beings to order. (Note that the presence of pupils who, all along the way,
give you their feedback, makes the task much easier.)
e. The images created presented fixed elements and mobile elements: algebra exists in these
images because they are dynamic. Only movement allows one to become aware of the
permanent relations that structure a situation, in opposition to relations that are true
only in particular cases.
f. The images created are labile, ever transformable: it is possible to erase, to add, to
transform the elements..
g. The image at the start is so elementary - it demands practically to prerequisites - that
anyone can begin to study the situation.
h. Beginning from almost nothing, it is possible to get a lot done, in particular to force very
many awarenesses.
i. To modify one element alone, or to add one, or to make a change of point of view allows
one to begin a new "chapter": one single basic situation, correctly modified is
sufficient to consider multiple notions adequately.
j. As the mental image is "manipulated", its structuration, consequently the mastery one has
of it, increases and the length of time needed to see decreases: evoking images,
animating them, transforming them, becoming aware of the relations that structure
them, are educable activities.
Many pupils fail in mathematics because the words, the symbols and the notations remain, for
them, desperately empty of meaning because they evoke nothing when they read or hear
them. If we want our pupils to succeed and to appreciate mathematics, in other words if we
want them to know what they are doing and to be able to be in control, it is advisable, as we
have suggested, to give them the means to do so: the means to become autonomous,
independent and responsible.

7. http://www.uneeducationpourdemain.org
Page 7 sur 7
Notes
a. From now on, read "the line segment having the endpoints A and B" for [AB], and read
"the point V" for V, read "the point V prime" for V'.
b. Lines are said to have the same direction if they are parallel.
c. The age of pupils: With pupils who are beginning primary school, there is a step which
must precede such work on mental imagery. With pupils at the end of primary school,
the logic of perception can be accepted: at this age, action and perception - here virtual
action on the elements of a figure and perception of certain relations between them are
non-permanent or permanent - is the favoured way of knowing. With pupils in
secondary school, such questions will lead to exchanges which will demand of them,
in addition to the precise expression of their thinking, rigorous reasoning on the basis
of the properties of parallel and perpendicular lines and those of quadrilaterals.
d. The symmetry of an equation and the commutativity of addition and multiplication.
e. Read "the length of [AH] is 3 cm" for AH = 3 cm.
f. a and n represent any positive integer, but a remains inferior or equal to n
g. < is read: ... "is strictly smaller than" ... > is read: ... "is strictly greater than" ...
© Maurice Laurent Geneva, 1992 Translated by Donna L'Hôte
The Science of Education in Questions - No - 9 June 1993
"Mental Imagery in Mathematics" by Maurice Laurent is licensed under a
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.