# The Pythagoras Theorem Kokkaliari 2013 IOE

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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
Faculty of Culture & Pedagogy
MA Coursework cover sheet
Student’s name
MAGDALINI KOKKALIARI
Course of study
MA Mathematics Education
Module title
Learning Geometry For Teaching:
Widening (Mathematics Education
Practitioners') Geometrical Horizons
Module code
Date work submitted
MMAMAT_06
22/07/2013
Title of Assignment
The Pythagoras Theorem: Is the Methuselah theorem still alive?
Word length: 5125
Failed to submit by September 2012
Statement:
I confirm that I have read and understood the Institute’s Code on Citing Sources and
Avoidance of Plagiarism. I confirm that this assignment is all my own work and
conforms to this Code. This assignment has not been submitted on another occasion.
Signed: …………………………………………..
Date: London, 22 July 2013
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
The Pythagoras' Theorem: Is the Methuselah theorem still alive?
“Geometry has two great treasures. One is the Theorem of Pythagoras; the other,
the division of a line into extreme and mean ratio. The first we may compare to a
measure of gold; the second we may name a precious jewel.”
J.Kepler
INTRODUCTION
I have always liked the Pythagoras' theorem; I especially like to teach it. I don’t know
why; perhaps my attraction to it is due to its long life, or to its wide popularity, or to the
plethora of its applications. As a teacher, I have experienced seeing the contrast between the
emotions expressed by the eyes and mouths of 13-year-old students. When they are first
involved with learning it, they express their happiness when playing with shapes and ropes, but
during their last lessons on the Theorem they express pain and sorrow when they meet the
rigid notion of the irrational numbers, requisite for calculating the length of one side of a rightangled triangle given the lengths of the other two. This sudden change in their feelings reminds
me the shock that the discovery of the irrational numbers is said to have caused Pythagoras and
his students. The shock of disappointment is believed to have been so unbearable, that
Hippasus, the Pythagorean who by some scholars is credited with the discovery of irrational
drowned at sea.
And this is the crucial point for the teacher: how his/her student’s sacrifice, to be
drowned at the Mathematics Sea as a contemporary Hippasus, can be avoidable? In which
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
ways can he/she lead to students' development of cognitive strategies? An answer can be that
the conclusions from research on his/her own experiences of doing mathematical work may be
a) A SUBSTANTIAL PIECE OF MATHEMATICAL WORK IN GEOMETRY
Young students usually have the feeling that mathematics teachers know everything
about mathematics; or, even if they do not, they have their own secret ways and learning it is
such an easy process for them. They are unable to understand that learning and doing
mathematics demand similar efforts and process for anyone, even for a mathematician who is
involved with Pythagoras' theorem, a theorem which is expected to be too famliar:
Generalising the Pythagorean theorem in three dimensions:
De Gua’s theorem:
The square of the first face or base - opposite the right-angle corner - area of an orthogonal
tetrahedron is equal to the sum of the squares of the other three face areas.
Proof:
(hint1)
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
therefore, their areas can be found as the product of the two perpendicular sides. Then (ADB)=
1
2
1
1
x y, (BDC)= 2 y z and (CAD)= 2 x z, therefore, (ADB)2 + (BDC)2 + (CAD)2=
1
( x2 y2+ y2
4
z2+ x2 z2).
Now, we can find out the length of the three sides of the triangle (ABC) using the Pythagorean
theorem. That is: AB= x2  y2 ,BC= z2  y2 and CA= z2  x2 .
By Heron's formula the area (ABC)=
s=
s(s  x2  y2 )(s  y2  z2 )(s  z2  x2 ) , where
x2  y2  y2  z2  z2  x2
is the semi-perimetre of the triangle ABC.
2
Therefore,
x2  y2 
x2  y2  y2  z2  z2  x2
 x2  y2
2
y2  z2  z2  x2
2
x2  y2 
y2  z2  z2  x2
2
x2  y2 

x y 
y2  z 2
y2  z2  z2  x2
2
2
2
2
2
2
 x2  y2 
y z 
2
z x
2
2
2
y z 
2
y z 
2
2
x y 
2
2
z x
2
2
x y 
2
2
y z 
2
2
2
z x
2
2
(1)
2
z x
2
2

z x
2
2
=
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
As a student, I was unhappy to face confusing algebraic manipulations of this kind,
several times as a teacher, too. Although I am used to such situations, I must confess that I am
not patient any more, perhaps I never was. At the starting point I thought 'OK. Let's do it' but
eventually I gave up, thinking that it is of no use. It was only when I was about to fulfil the
assignment that I decided to confront the situation successfully in this way:
Let
2
2
x2  y2 = a, y  z =b and z  x =c, then from (1)
2
(ABC)=
4
1
4
=
2
ca b
c a b
a bc
2
2
2
2
( (a+b) 2 -c2 )(c2  (a-b) 2 ) =
(ABC)=
1
a bc

=
(a+b+c)(a+b-c)(c  a-b)(c  a+b)
1
=
2
1
4
1
4
4
( (a 2 +b 2  2ab-c 2 )(c 2  a 2 -b 2  2ab) and substituting again
(x +y  z  y  2 x  y
2
2
(2y  2 x  y
2
1
2
2
2
2
2
2
2
z  y )( 2 y  2 x  y
2
2
2 2
2 2
4
2 2
4
x z  y z  4y  x y  4y =
Hence, (ABC)2=
z  y -z -x )(z  x -x -y  z  y  2 x  y
2
2
1
2
2
2
2
2
z y )=
2
x z y z x y
2
2
2
2
2
2
1 2 2 2 2 2 2
( x y + y z + x z ) q.e.d. ▄
4
2
2
2
1
4
2
2
2
2
2
z y )
2
2
( 4( x  y )( z  y )  4 y ) =
2
2
2
2
4
2
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
A special case: De Gua's Theorem in a 'Unit' tetrahedron
Hence, (ABO)2 = (AOB)2 + (BOC)2 + (CAO)2
Proof:
(hint2)
Let D be renamed as O and suppose that AO=1, OB=1 and OC=1.
The triangles AOB, BOC, CAO are right angled isosceles
triangles, therefore, their areas are all equal to
Hence,  AOB 
2
 BOC 
2
  CAO  
2
1
.
2
3
(1).
4
By the Pythagorean theorem their hypotenuses are easily calculated AB=BC=CA= 2 ,
resulting in the triangle ABC being equilateral with altitude h=
formula (ABC)= a2
3
and area given by the
2
3
3 2 3
, hence, (ABC)2=(( 2 )2
)  (2)
4
4
4
(ABC)2=(AOB)2 + (BOC)2 + (CAO)2 q.e.d. ▄
After the previous example I decided to generalise the previous proof:
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
Another proof for De Gua's Theorem
Proof:
(hint3)
Let AO=x, OB=y and OC=z.
The right angled triangles AOB, BOC, CAO have areas
1
1
1
(AOB)= 2 x y, (BOC)= 2 y z and (CAO)= 2 x z,
therefore,
(AOB)2 + (BOC)2 + (CAO)2=
1 2 2 2 2 2 2
( x y + y z + x z ).
4
Now, let h be the altitude of the triangle (first face of the tetrahedron),
corresponding to AC and a and b the projections of the sides AB and BC
respectively.
Then AB= x2  y2 , BC= z2  y2 and a + b=AC= z2  x2 (1)
by Pythagorean theorem for the right-angled triangles ABE and BCE:
a2+h2=( x2  y2 )2  a2+h2= x2  y2  h2= x2  y2 - a2(3)
b2+h2=( z2  y2 )2  b2+h2= z2  y2 (2)
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
From (1) b=a- z2  x2 (4)which we substitute at (2)
and (a- z2  x2 )2+h2= z2  y2 
a2-2a z2  x2 +z2+x2+h2= z2  y2  (from (3))
a2-2a z2  x2 +z2+x2+ x2  y2 - a2= z2  y2 
-2a z2  x2 +2x2=0  a=
z2  x2
which we substitute in (4) b=
b=
z2
z x
2
2
x2
z x
2
2
- z2  x2 
and from h2= x2  y2 - a2(3) where we substitute a,
h2= x2  y2 - (
 h2= x2  y2 -
But (ABC)=
x2
x2
z x
2
2
)2
x2 y2  y2 z2  x2 z2
x4

h=
x2  z2
z2  x2
1
1
AC*h= (a + b)*h=
2
2
x2 y2  y2 z2  x2 z2
x2
z2
1
(
+
)*
=
x2  z2
2
z2  x2
z2  x2
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
1
2
z2  x2 *
(ABC)2=
x2 y2  y2 z2  x2 z2
1
 (ABC)=
2
2
x z
2
x2 y2  y2 z2  x2 z2 
1 2 2 2 2 2 2
( x y + y z + x z ) q.e.d. ▄
4
By the time I was doing this proof it occurred to me that the relevant theorems of a right- angled
triangle connected with the Pythagorean theorem can be expanded to 3 dimensions. For example, we
know that the length of one of the perpendicular sides is the geometric mean of the length of the
hypotenuse and the length of its projection to the hypotenuse. (quoted below as hint 5). I stated the 3dimensional analogous (case 1 in hint6). I omitted the proof but I noticed that from the latter, an 'elegant'
and easy proof for Pythagorean theorem arises (hint5) that can be extended in 3 dimensions having
another proof for De Gua's Theorem:
Proof:
AAOB2=AABC*AABH (1)
Respectively: ACOB2=AABC*ACBH
ACOA2=AABC*ACAH
(2)
(3)
Adding (1) (2) (3) by parts
AAOB2 +ACOB2 +AAOC2=AABC*AABH +AABC*ACBH +AABC*AACH
= AABC*(AABH + ACBH + AACH)= AABC2 q.e.d. ▄
Now we know that the volume of a tetrahedron is given as the product of its base times the corresponding
altitude, hence we can find it in two ways V= AABC*h
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
or V= AAOB *c=a*b*c
therefore we proved (3) from (hint 6) q.e.d. ▄
Now from AAOB =AABC*AABH  AAOB
2

2
V2 2
h =AABC*AABH h  2 h = AABC*AABH h2
c
2
2
1 AABC * AABH
=
V2
c2
Similarly
1 AABC * AACH
=
V2
b2
A *A
A *A
1 AABC * ABCH
1 1 1 A *A
=
and by adding by parts: 2 + 2 + 2 = ABC 2 ABH + ABC 2 ACH + ABC 2 BCH
2
2
V
V
V
V
c
a
a
b
by De Gua's Theorem
1 1 1 AABC 2
1 1 1 1
+ 2 + 2 = 2  2 + 2 + 2 = 2 q.e.d. for the (4) of (hint 6) ▄
2
c
c h
V
a
b
a
b
At this phase of my work I was happy of having stated a theorem on my own which later I found
(hint6)
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
My wandering however did not stop at this point, as the 'unit' tetrahedron brought to mind the
Cartesian coordinate system and vectors. And I went on to prove De Gua's theorem using vector
theory. This demanded a different kind of work from me as for many years I have not worked
with the cross product. So I had to study almost from the beginning to achieve to make sense of
the most brief proof I could find as a solved problem and I copy it here because part of my work
was revising vectors so that I could understand this proof.:
(hint4)
Let O (the vertex opposite the first face) be the origin of the coordinate system and OA= a ,
OB = b , OC = c the vectors of the vertices opposite to faces A,B,C respectively. We want to
prove that D2=A2+B2+C2
Then the area A of face with edges has b and c is A=
B=
1
b  c and respectively
2
1
1
c  a , C= a  b .
2
2
The face of area D is the triangle spanned by b  a , c  a ,
so that D =
1
1
b  c  b  a  a  c) = b  c  a b  ca .
2
2
The vector b  c is a scalar times a , a b is a scalar times c and ca is a
scalar times b mutually perpendicular because of the perpendicularity of a , b , c .
So, when we multiply out D2=
1
(b  c)2  (a b)2  (ca )2 =A2+B2+C2 ▄,
4
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
(www.math.ubc.ca/~feldman/m152/geometry
)
b) PERSONAL REFLECTIONS ON DOING THE MATHEMATICAL WORK, ITS
RELATION TO TEACHING AND LEARNING, AND QUESTIONS TO RESEARCH
OR INVESTIGATE COMING FROM THESE REFLECTIONS
What could a substantial piece of mathematical work in geometry be? The first
'frustrating' thing was the choice of the work. The choice of the mathematic topic raises the first
issues. What a human being chooses to learn, why and how? My first attempt to clarify the
implications behind these questions is to try to remember and outline the conditions under
which this choice surfaced. Doing mathematical work starts from the moment you accept being
triggered by it. My belief is that teacher's first concern should be the student's well being and
Mathematics teacher's second concern should be inspiring the student to 'do' mathematics.
Therefore, it is of great importance for teaching and learning to investigate and answer the
questions above.
During the second session of the module ‘Teaching and Learning Geometry’, when we
were discussing about the Pythagorean Theorem, we were asked to suggest proofs working
collaboratively. As a maths teacher of Greek post compulsory education, where Euclidean
Geometry is taught widely, I was familiar with the proof:
By the similarity of the triangles ABC,
ACX and BCX we take
(hint5)
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
a x

c a
 a2=c x
b cx

 b2=c (c-x)
c
b
Hence, a2+b2= c x + c (c-x)=c2 ▄,
which I suggested without second thoughts confusing my colleagues. I tried to make it clear for
one of them, but was not successful. Another asked whether I could remember a very nice
proof that he could not recall. These two snapshots led me as homework to work on recalling
other proofs that I had met before, more fascinating or more simple. Later I came up with the
idea to find some more relevant information from my favourite book 'Great Moments in
Mathematics' By Howard Eves (Dolciani Mathematical Expositions). Proofs is an obsession
mathematicians of Hellenic culture, so it was normal that I was challenged by the quotation 'We
leave the matter of proof to any enterprising reader' (Eves H. 1983 p. 38) that comes after De
Gua's Theorem.
I could not understand why you should be an enterprising reader for such an easy proof,
just calculating the areas of four triangles. Without thinking I drew a draft diagram of a
tetrahedron with front face A and the others A1, A2, A3 on a paper and started with the areas of
the three right angled edges: A1=
ab
bc
ca
, A1= , A1= , hence, (A1)2+(A2)2+(A3)2=
2
2
2
a 2b 2  b 2 c 2  c 2 a 2
a 2b 2  b 2 c 2  c 2 a 2
. The only thing now was to prove that A=
(from hint1). I
4
4
stopped working for a while in order to think how I could find the area of the base. The only
idea was to find the sides of the base -by the Pythagorean theorem as hypotenuses of the right
angled triangles- and afterwards, to apply Heron's formula for the base. It is not my favourite
engagement with all these square roots, but I had nothing better in mind and the result is
mentioned above. After a reasonable time I gave up.
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
Later, I thought that I reacted as students do: they try to answer mathematical tasks by
hand, as I like to say to them, not by brain. So I started again trying this time to think what this
theorem really means. I tried to find a visual representation such as the one we use to teach the
Pythagorean theorem in two dimensions.
For example
or the famous 'Behold' http://britton.disted.camosun.bc.ca/geometry/behold.html
in an inspired and inspiring animation. Unsuccessfully, of course! The reason for this is
because squaring the length gives area in two dimensions, but squaring areas leads to four
dimensions, where we can work only with intuition. When I tried to find any relation to the
Pythagoras' Theorem in 3 dimensions used for calculating the diagonal of a cube, I came to the
decision of constructing a paper tetrahedron to make things visible (the one I use as a diagram
in (hint2) and (hint3)).
The benefit of this was that one thing led to another. The construction of the tetrahedron
transformed the rigid mathematical work to a pleasant game. Acting again as a child I
disregarded the squares and I started to doubt the validity of the theorem. How the sum of the
areas of these three triangles can be equal to the area of the fourth triangle? An idea to verify,
then came to my mind. It is what I called 'unit' tetrahedron before and this verification (hint2)
finally led me to a proof of De Gua' s theorem (hint3). This, in turn, reminded me of the proof
that I suggested in the module session (hint5), which gave birth to the idea of other analogous
to three dimensions, the justification of which revealed, all of a sudden, a new way of proving
De Gua' s theorem and so on. Like a usual mathematics fairy tale! Alice in mathematics-land!
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
And then, after a short respite, I reached .the fascinating Cartesian coordinate system
and the short, smart, simple proof (hint4), which was not my own work, but which I had come
across (www.math.ubc.ca/~feldman/m152/geometry.pdf) during my research.
Although it occurred to me that I could go on with Linear Algebra, I thought it would be
more interesting to try find applications in real life problems. It led to the notion of tetrahedron
as a 3-simplex with applications in industrial statistics and other things I am totally unfamiliar
with and I confess uninterested in. What I found challenging was this new notion of simplex as
a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions, the number of
faces of a simple that forms Pascal's triangle without the left diagonal and lots that it is of no
means to be mentioned here and the artistic results of Petrie polygons, which I left to be my
mathematical work in the future (https://en.wikipedia.org/wiki/Simplex).
1
2
3
4
5
6
7
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The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
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For the time being the only thing I wanted to do was going back to Pythagorean
theorem and try to find some other aspects of its generalisations. 'Generalizations are the lifeblood of mathematics' (Mason J et al, 2010 p.8), mathematical thinking implies examining 'all'
the cases.
In Euclidean Geometry (4) in any right-angled (1) triangle (3), the area of the square
(2) whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are
the two perpendicular sides,
or as n equation: a2+b2=c2 or as a diagram
1. only for right-angled triangles?
If the triangle is not right-angled? Then we have the cosine rule and the consequence of
its inverse is a nice way to find the kind of a triangle with given the lengths of sides. On the
other hand Pappus of Alexandria extended Euclid' proof in two ways: the triangle is not
required to be right-angled and the shapes built on its sides are arbitrary parallelograms instead
of squares. Instead of the formal proof I give a diagram for thought and an animation, which I
think fosters reasoning. It is a good example of how DGS (here Geogebra) can be used to
foster the interaction of seeing on the computer (visualisation) -by dragging the vertices of the
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
triangle or the parallelogram the student comes to the conclusion that it is valid for any triangle
and parallelogram- and justifying by means of theoretical arguments (Laborde, 2001) -the areas
of the parallelogram are constant because in any case the length of the base and the altitude
remain the same-.
http://www.enallax.com/exams/geogebra/theorems/pappus.html
2. only for squares or parallelograms?
Of course not (I use the examples of the table below in the third section as an example
of insight for similarity).
It stands for other triangles
for semicircles
Maor E.(2007 p. 124)
even for this
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
http://danpearcymaths.wordpress.com/
3. only for triangles?
De Gua' and one more example with volumes
It is a generalisation of Pappus' theorem for three dimensions the
volume of the tetrahedron ANOPΓΒ is equal to the sum of the three
triangular prisms (Eves 1983)
4. Is it only a Euclidean one?
In spherical, hyperbolic and differential geometry it is given by the formulas:
Maor E.(2007 p. 170)
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Magdalini Kokkaliari
11094464
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
(http://en.wikipedia.org/wiki/Pythagorean_theorem)
and
Last but not least is a proof for Pythagorean theorem from
5. 'Origami' Geometry or paper folding:
the of the red square is equal to
the sum of the areas of green and pink
squares
This is a picture from a construction used as a warrant for a lesson on Pythagorean Theorem In
B Gymnasium In the Greek Secondary School of London (Kokkaliari, 2011)
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Magdalini Kokkaliari
The Pythagoras' Theorem:
Is the Methuselah theorem still alive?
11094464
c) A SYNTHESIS OF RESEARCH, CULTURE, HISTORY, PHILOSOPHY ETC. THAT
ARE
USED
TO
THE
QYESTIONS
ARISING
FROM
THE
MATHEMATICAL WORK.
In the previous section 'Reflections on doing the mathematical work' I tried to write
down, in a personal tone, the whole experience of a mathematics teacher's -as a student- 'doing'
a piece of mathematical work and consequently learning mathematics. The main purpose was
to let questions emerge for further research on how teaching and learning occurs and how a
teacher can affect students' motivation, which is ‘sine qua non’ for learning.
My first concern is the research on the conditions under which the decision of doing the
mathematical work was made, what work was chosen to be done and how it was implemented.
A lot of research has been done on how an individual learns and how teachers' awareness of the
existing research in Mathematics Education can help their contribution on empowering
students' learning strategies. From the reflections it becomes obvious that the discussionoriented teaching support a motivated student, who learns in his/her effort to communicate
knowledge with, or attain knowledge from, the others working as a group or individually:
A. Individual work as a result of collaborative learning in a community of practice:
1. 'Classroom' as a community of practice with the interplay of 'teacher', 'students' and 'ego',
transforms learning from instructions and exercising to an active participation. Learning does
not come from the individual experience itself 'but from the experience of others, transmitted
through relationships and networks of social interactions' (Wenger 2000). The suggested proof
which does not make sense for the 'classmate', the other 'classmate' 's effort to remember a nicer
proof leads the individual to go home ready for further research on proofs, forgotten from past
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experiences or brand new ones, 'reinforced by a sense of membership in the group that affirms
and guides what any participant knows.' (Wenger 2000).
2. collaborative learning as a teaching method give students the chance to be engaged in
discussing and teaching each other, having the chance to share their ideas, realise their
advantages and disadvantages and try to improve their skills. Then, in Swan's words (Swan,
2005), 'Mathematics becomes an interconnected body of ideas and reasoning processes a
plethora of aspects are unfold raising questions'. Some of them are going to be answered
through discussion; the rest are going to remain unsolved challenging the learner to find the
3. 'homework or individual work', which is essential to help students find their own way to
attain learning. Working with others is challenging, but not sufficient for learner's deeper
learning. Deep learning presupposes learner's 'intention for action....which is driven,
subconsciously, by his/her feelings ' (Rodd M., 2003). When the learner works at home, in a
familiar place, without the pressure of the 'others', with resources available from prior work
experiences, he/she feels confident and capable of success. Research shows that 'studying
attentively in a quieter environment' (Plant, Ericsson, Hill, & Asberg, 2005) 'sparks students’
creative thinking' (Epstein J, 2001) and 'encourages (a) their academic learning and (b)
development of skills' (Warton P. 2001). Time available for homework helps insight, which is
essential for mathematical work. Insight does not emerge out of nothing; it arises after timeconsuming work by trial and error, research, mistakes, different approaches, breaks, 'respite',
feedback.
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Beyond these three components, the additional element of working with teachers from
different educational systems is valuable and helpful for the participant. Mathematics is
considered to be a universal and ubiquitous subject all over the world, but different societies
build different approaches. Knowledge is culturally connected with the society and individuals
adopt different styles of learning and have their own ways of achieving success or expertise.
Different approaches lead to a concurrent presentation of multiple representations, demanding a
quite new work to be familiar with. 'Learning to understand and be competent in the handling
of multiple representations can be a long-winded, context dependent, non-linear and even
tortuous process for students' (Arcavi A. 2003). Teachers from different countries working
collaboratively have the unique experience to see, through the eyes of 'others', problems, which
were a routine work for them, as quite new. This chance is beneficial from two points of view;
firstly because they become aware of the fact that reasoning is a matter of familiarisation,
independent of age and similar skills; secondly, because they can use multiple representations
of met-concepts as a help for themselves and their students in order to develop deeper, and
more flexible understandings.
My second concern is the meaning of doing mathematical work in geometry and why it
is central for teaching and learning, not only geometry, but, I dare say, every piece of
knowledge.
B. Doing mathematical work in geometry
1. At the first place I' d like to comment on the word 'doing'. We usually say 'learning' or
'studying' or 'teaching' mathematics. What does 'doing' mathematics mean? 'Doing' can be any
activity that mathematicians do, such as solving a problem, dealing with a new theory,
answering an unsolved question, or even applying a piece of mathematics in a real world
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problem. In general, decision-making process is not an easy task, let alone the case of having
encountered such interesting new ideas as I had. My ultimate choice of the mathematical work
was not a fascinating new theory, but just the one I was mostly involved in. I decided to
choose De Gua's Theorem, as an example of a generalisation of the Pythagorean theorem, in an
effort to find out learner's actions and reactions when we have difficulties on doing
mathematics, in this case on proving a theorem and their implications on teaching and learning
geometry.
To take a simple difficulty of fulfilling a proof as a starting point, I wanted to analyse
the whole journey to 'met-before' or entirely new situations and draw conclusions on how
teachers can help their students cope with their difficulties on learning geometry.
2. Trying to give answers to the questions that arose above from the mathematical work on
geometry, we should first query what geometry is and how its teaching and learning can
contribute to develop mathematical skills. The main advantage of learning geometry is that ‘it
has direct applicability to everyday life and also that it can be understood with less intellectual
effort’ (Atiyah M. 1982). Even if it is not applicable or understandable, a diagram can help
give it tangibility, as it happened with the construction of tetrahedron. No matter whether
theorems for mathematicians can be a part of their real world, examples of physical world can
provide insight and shed light on obstacles.
In the same way the young student can be familiar with the Pythagorean theorem,
having fun by cutting and painting papers. I suppose that almost every student all over the
world has learned the relationship a2=b2+c2 by playing with the shapes below and verified it by
counting squares:
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or by cutting the parts of the squares of the two perpendicular sides and completing the square
of the hypotenuse, as below:
or, nowadays, by playing with the relevant puzzle available, as an applet, on the web. This first
contact between the student and the geometrical concept is very important, but does it ensure an
effective future relationship? Of course, not! However it could be a good model, if learned
properly, available for the child to build what he/she can learn, and how. As Papert S. (1980)
states: 'Thus the "laws of learning" must be about how intellectual structures grow out of one
another and about how, in the process, they acquire both logical and emotional form'. But how
it can be attainable? The role of the teacher is crucial. If we take into consideration that
'teachers formulate pedagogical points of view that are in part responses to personal educational
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experiences' (Smitherman S. E. 2006), they should take examples of their own experiences, to
conceive how they should facilitate their students to build their own effective learning models.
Afterwards the constructed tetrahedron became a tool for abstract reasoning, or in
Attiyah's words 'at a sophisticated level, geometry does involve abstraction. As the Greeks
recognised, the points and lines which we meet in the real world are only approximations to
some "ideal" objects, in an "ideal" world where points have no magnitude and lines are
perfectly straight. These philosophical reflections do not however worry the practitioner of
geometry, be he a school-child or a civil engineer and geometry at this level remains the
practical study of physical shapes.' (Attiyah 1982).
Does he imply that abstract reasoning has nothing to do with the student? Or if it has, at
what age should a student meet abstraction? For Greek educated students it happens at the age
of fifteen. Euclidean Geometry is one of the main subjects of the Greek Mathematics context.
Proofs and generalisations are part of their everyday life at secondary education. Are the
notions of generalisation and proof, I mentioned in the second section, inextricably linked to
abstraction? Could a visual proof be valid? Can the example of 'Origami' proof or the animation
argues that 'Mathematical proof is a concept that arises in the work of mathematicians, and as
such it has a sophisticated meaning' (David Tall 2002). From Melissa Rodd's point of view
separating proof from what she defines as warrants 'Proofs do not in and of themselves
constitute a warrant for belief in mathematical propositions because students are liable to
reproduce, rather than produce themselves, a proof as evidence of a proposition’s truth....... a
student’s mathematical knowing requires a mathematical way of thinking. In particular,
deductive argument will be a way in which a mathematically thinking student can attain
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personal conviction. Does this mean that the only way one can come to mathematical
knowledge is through a formal argument? Not if there exist other warrants, for example,
visualization.' (Melissa Rodd 2009)
And should it be harmful for the student, if, starting from Giokonda (generalisation 2),
then going on with the rest shapes, gains insight by visualisation (warrant) to come to the
conclusion of similarity? Deductive proofs for semicircles and triangles are of no interest or of
great difficulty? If a student is never engaged with deductive proofs during school years, will
he/she cope up with this kind of work later?
Undoubtedly, nowadays, a new era for digital technologies forces a reform for school
geometry in all countries. 'Geometry offers a rich way of developing visualisation skills.
Visualisation allows students a way of exploring mathematical and other problems without the
need to produce accurate diagrams or use symbolic representations.' (Jones, K. 2002).
When I decided to work on proving De Gua's theorem and generalisations of
Pythagorean Theorem, I recalled David Tall's reference to a long conversation he had with the
Greek mathematician Stylianos Negrepontis, responsible for the education of Greek
mathematicians -one of them is me-. at the University of Athens for the last 40 years. The
Greek mathematician had began a long discourse about 'two' from his hero Plato's
philosophical point of view, when the British mathematician tried to put forward that 'the idea
of ‘twoness’ in the growth of a child could not be approached from such a lofty intellectual
position.' He narrates: 'He continued at length and, at a moment when he drew breath, I
interjected and said that, if he would let me make one comment in one sentence that I felt was
relevant, then I would remain silent so that he could complete the burden of his argument. After
some protestation from him, I got in my one sentence. It was this: ‘Plato was very young when
he was born.’
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Although I felt guilty for my students, I have not changed my mind about the chosen
work with the thought that culture is inseparable from one's preferences. But I hope that 'self'
research and awareness perhaps help me avoid sophisticated and confusing instructions to my
students.
After this 'Mathematics teacher's short apology' I' d like to say that I tried to describe a
mathematician's work that is focused on doing not reproducing, on studying not reading.
Human beings learn with actions, but things we just meet or read is a good legacy for future
work.
My last concern is learning mathematics through history. Although I have seen that
young people are not highly interested in mathematics history, I suggest that understanding
and learning mathematics as it was developed over the centuries, after it's been honed by
hundreds of people consists a nice way of studying. A student becomes aware of what people
were thinking, why they were asking certain questions, what their flow of the subject is across
time. Pythagorean theorem was undoubtedly the first major theorem in mathematics known to
Babylonians and Egyptians, who used 3-4-5 rop, since 2000 BC. Pythagoreans used it to
measure areas (they had not the notion of distance), Euclid quotes a very nice proof similar to
the one we quoted as animation, Descartes used it to define Euclidean distance and so
on....From measuring to abstract meaning from tool to notion. From Geometry to Algebra with
the Pythagorean triples and again back, from Euclidean to non-Euclidean geometries it seems
that appears unexpectedly to every modern or not facet of mathematics. A strong didactical tool
for teachers and students, since the first school years, justifies its popularity and value.
EPILOGUE
Although four thousand years old, the Pythagorean theorem appears vigorous and
ubiquitous. Its contribution to almost any kind of mathematics made Eves say that, if we want
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Is the Methuselah theorem still alive?
to construct a device indicating to possible outside observers that there is intelligence on our
planet, it would be a mammoth configuration illustrating the Pythagorean theorem because 'all
intelligent beings must be acquainted with this remarkable and nontrivial theorem of Euclidean
geometry, and it does seem difficult to think of a better visual device for the purpose under
consideration' (Eves H. 1983). To go further with it, I would add that 'all intelligent beings must
have experienced the joy that brings a sudden insight during school activities related to the still
alive Methuselah theorem.
ACKNOWLEDGEMENTS
I should like to thank most sincerely Melissa Rodd and Dietmar Küchemann for their inspired,
and inspiring, sessions of 'Learning Geometry For Teaching: Widening (Mathematics
Education Practitioners') Geometrical Horizons'. A special thank you to Piers Saunders for his
interesting presentation and his helpful instructions for DGS and last but not least, my dearest
colleagues who shared with me their knowledge and expectations.
MAGDALINI KOKKALIARI
KOK 11094464
MMAMAT_15
MA STUDENT
MATHEMATICS EDUCATION
INSTITUTE OF EDUCATION
UNIVERSITY OF LONDON
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Community
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