Why it is important to teach geometryfrink.machighway.com/~dynamicm/importance-of-geometry.pdfWhy it is important to teach geometry Gerry Leversha Mathematical Association Annual Conference

Why it is important to Why it is important to teach geometryteach geometry

Gerry Leversha

Mathematical Association Annual Conference

Cambridge April 2009

a

b

43°

46°c

D

B

A

C

GCSE ProblemGCSE Problem

Find the angles a, b and c .

a

b

43°

46°c

D

B

A

C


Find the angles a, b and c - giving reasons.

a

b

43°

46°c

D

B

A

C


Find the angles a, b and c - giving reasons.

ABC is isosceles - tangents from a point to a circle.

a = 67º (angle sum of triangle).

b = 67º (alternate segment theorem).

c = 70º (angle sum of triangle).

Assessing the ProblemAssessing the Problem

►Because three angles have been listed, the Because three angles have been listed, the problem has been broken down into easy problem has been broken down into easy steps.steps.

►This might discourage alternative ways of This might discourage alternative ways of approaching the calculation.approaching the calculation.

►The geometrical results (angle sum of The geometrical results (angle sum of triangle, alternate segment,...) are given triangle, alternate segment,...) are given equal statusequal status..

43°

46°c

D

B

A

C


Find the angle c

giving reasons.

This version has more mathematical content, since it forces the student to devise a plan of attack on a multi-stage problem.

WHO CARES?


►The new version is not so suitable for The new version is not so suitable for examination, but this does not mean that examination, but this does not mean that practising this sort of multi-stage problem practising this sort of multi-stage problem should not be part of the teaching process.should not be part of the teaching process.

►However, it does seem to be simply an However, it does seem to be simply an unmotivated calculation. Is geometry just a unmotivated calculation. Is geometry just a process of pointless calculation?process of pointless calculation?

46°c

D

B

A

C


The triangle ADC is isosceles.

Find the angle c (giving reasons).

Tangents to a circle.

Alternate segment.

Isosceles triangle.

Alternate segment again.

c = 46º.

SO WHAT?

46°c

D

B

A

C



Find the angle c .

There is something interesting happening here!

The angle at B was also 46º.

Was this a fluke?

Does it always happen?

bc

D

B

A

C



Prove that b = c .

This problem involves much more sophistication.

It is asking for a proof of a general property of the configuration.


►What is present in this version of the What is present in this version of the problem is an element of problem is an element of universalityuniversality. The . The result is true for any angle result is true for any angle b b ..

►What about the 'proof'? Clearly angle-What about the 'proof'? Clearly angle-chasing still works. chasing still works.

►But does this really But does this really explainexplain what is going on what is going on here?here?

►Is another approach more Is another approach more illuminating illuminating ??

bc

D

B

A

C



Prove that b = c .

The triangles BAC and ADC are both isosceles.

They have equal base angles (from the alternate segment theorem).

Hence they are similar.

But c is the angle at A.

And this is equal to b.



Prove that b = c .

Now we change the diagram so that BC is no longer a tangent.

Alternate segment

BAC = ADC

Isosceles triangle

BAC = ADC = ACD

AB is parallel to DC

c b

D

C

A

B



Prove that b = c .

Now we change the diagram so that BC is no longer a tangent.

Notice that this proof does not use the assumption that BC is a tangent at C.

This is a better proof of the previous result since it does not use an unnecessary assumption.

c b

D

C

A

B


►These problems are not an appropriate These problems are not an appropriate exercise for public examinations (such as exercise for public examinations (such as GCSE)GCSE)but there are two essential characteristics of but there are two essential characteristics of good mathematics teaching:-good mathematics teaching:- a a multi-stepmulti-step argument. argument. a recognition of a recognition of generalitygenerality..

►and there is even an aesthetic element - and there is even an aesthetic element - which proof is best?which proof is best?


►This analysis of a simple problem suggests This analysis of a simple problem suggests some reasons why geometry is worth some reasons why geometry is worth teaching as part of the school curriculum.teaching as part of the school curriculum.

►It is time to It is time to set out the stallset out the stall!!

Manifesto for real geometryManifesto for real geometry

►Geometry is worth its place in the school Geometry is worth its place in the school mathematics curriculum because :mathematics curriculum because : it has an it has an architecturearchitecture which gives it authority. which gives it authority. it relies on it relies on precisely definedprecisely defined concepts. concepts. it allows the construction of it allows the construction of multi-stage multi-stage

argumentsarguments to solve simple problems. to solve simple problems. it emphasises the need for clear and coherent it emphasises the need for clear and coherent

proofproof.. it includes results and arguments with real it includes results and arguments with real

aesthetic appealaesthetic appeal..


►But how can we implement this manifesto?But how can we implement this manifesto?►It will involve a significant investment in It will involve a significant investment in

teaching time and effortteaching time and effort on the part of both teachers and students.on the part of both teachers and students. which would not necessarily produce quick which would not necessarily produce quick

improvements in GCSE performance.improvements in GCSE performance. whose benefits are tangible only in the whose benefits are tangible only in the longer-longer-

termterm. .


►However, if, as a result, we decide to teach However, if, as a result, we decide to teach something which something which is challenging and stimulatingis challenging and stimulating develops genuine transferable skillsdevelops genuine transferable skills allows your pupils a chance to construct allows your pupils a chance to construct

mathematical pathways for themselvesmathematical pathways for themselves

then it is probably worth investigating.then it is probably worth investigating.


►A further issue, of crucial importance, is A further issue, of crucial importance, is keeping the students on your sidekeeping the students on your side. .

►How do you make such a course accessible How do you make such a course accessible without sacrificing its integrity?without sacrificing its integrity?

►How do you plan it, and where do you begin?How do you plan it, and where do you begin?►It is worth spending some time looking at the It is worth spending some time looking at the

sort of geometry encountered at primary level sort of geometry encountered at primary level (and perhaps up to year 8).(and perhaps up to year 8).

Geometry as Science Geometry as Science ►In their first encounters with geometry, In their first encounters with geometry,

pupils learn topupils learn to recognise and name basic shapes.recognise and name basic shapes. measure lengths and angles with rulers and measure lengths and angles with rulers and

protractors.protractors. appreciate properties of shapes - parallel sides, appreciate properties of shapes - parallel sides,

equal sides, equal angles, right angles, angle equal sides, equal angles, right angles, angle bisectors, bisection of lines, symmetry, etc.bisectors, bisection of lines, symmetry, etc.

use ruler and compasses for drawing.use ruler and compasses for drawing. build up a vocabulary - diagonal, vertex, etc.build up a vocabulary - diagonal, vertex, etc.

Geometry as Science Geometry as Science

►In this sense geometry is a sort of In this sense geometry is a sort of natural natural historyhistory . .

►A squareA square has four equal sides.has four equal sides. has four right angles.has four right angles. opposite sides are parallel.opposite sides are parallel. diagonals bisect one another.diagonals bisect one another. diagonals are perpendicular.diagonals are perpendicular. diagonals bisect angles at the vertices.diagonals bisect angles at the vertices.


►A square is now something which has all A square is now something which has all these properties at oncethese properties at once four equal sides.four equal sides. four right angles.four right angles. opposite sides parallel.opposite sides parallel. diagonals bisect one another.diagonals bisect one another. diagonals perpendicular.diagonals perpendicular. diagonals bisect angles at the vertices.diagonals bisect angles at the vertices.


►This is achieved through observation, This is achieved through observation, measurement, experimentation, listing.measurement, experimentation, listing.

►These are aspects of These are aspects of scientific methodscientific method but they are not yet but they are not yet mathematicalmathematical in nature. in nature.

►It might be possible that someone could It might be possible that someone could draw a shape which has some, draw a shape which has some, but not allbut not all, , of these properties. Is it then still a square?of these properties. Is it then still a square?


►Another feature of natural history is Another feature of natural history is taxonomy or taxonomy or classificationclassification.. All squares are rectangles. All squares are rectangles. All squares are rhombuses.All squares are rhombuses. All squares are quadrilaterals.All squares are quadrilaterals. Only some parallelograms are rectangles.Only some parallelograms are rectangles. If something is both a kite and a rectangle, it If something is both a kite and a rectangle, it

must be a square.must be a square.


►Another feature of natural history is Another feature of natural history is taxonomy or taxonomy or classificationclassification…………and there is the associated recognition and there is the associated recognition that, if we know something to be true about that, if we know something to be true about quadrilateralsquadrilaterals, it must also be true about , it must also be true about parallelogramsparallelograms…………but not every property of a but not every property of a squaresquare is true is true of a of a rhombusrhombus. .

Geometry grows up Geometry grows up ►The move from The move from science science to to mathematicsmathematics is an is an

essential essential rite of passagerite of passage..►ObservationObservation is replaced by is replaced by proofproof..►Real world shapes Real world shapes are replaced by are replaced by mental mental

objectsobjects..►Classification Classification is refined using is refined using precise precise

definitiondefinition..►Random facts Random facts are organised into a are organised into a logical logical

hierarchyhierarchy..

The importance of definitions The importance of definitions

►A measure of the maturity of geometry is A measure of the maturity of geometry is the the need for careful definitionneed for careful definition..

►Handout 1Handout 1 lists some definitions. lists some definitions.►These are These are minimalminimal definitions - they give the definitions - they give the

smallest possible number of facts that much smallest possible number of facts that much be checked to be sure that something is a be checked to be sure that something is a rectangle, say.rectangle, say.

►To illustrate the reason for this, take a look To illustrate the reason for this, take a look at an 'unhelpful' definition.at an 'unhelpful' definition.

The importance of definitionsThe importance of definitions

►SMP definition of a SMP definition of a parallelogram parallelogram :-:-

Start with a (scalene) triangle.



Start with a (scalene) triangle. Rotate it through a half-turn about the midpoint of a side.



Start with a (scalene) triangle. Rotate it through a half-turn about the midpoint of a side. Combining both images gives a parallelogram.



Note first that this is better than no definition at all! We are not simply recognising a parallelogram by intuition.



The advantage of this definition is that it gives all the symmetry properties of a parallelogram immediately.



The disadvantage is that it is hard to check that part of a diagram is a parallelogram. (It isn't enough to say that opposite sides are parallel.)



If you wanted to be sure, you would need to check that it could be made by rotating a scalene triangle.


►The usual definition of a The usual definition of a parallelogram parallelogram is:-is:- a quadrilateral with two pairs of parallel sidesa quadrilateral with two pairs of parallel sides

►It is now easy to check that something is a It is now easy to check that something is a parallelogram.parallelogram.

►Several theorems show that the other Several theorems show that the other properties of parallelograms follow from this.properties of parallelograms follow from this.

►A rectangle, being a type of parallelogram, A rectangle, being a type of parallelogram, also has all these properties.also has all these properties.


►Other 'theorems' present alternative Other 'theorems' present alternative conditions for a parallelogram -conditions for a parallelogram - a pair of opposite sides equal and parallela pair of opposite sides equal and parallel two pairs of opposite angles equaltwo pairs of opposite angles equal the diagonals bisect one anotherthe diagonals bisect one another

►or for other quadrilateralsor for other quadrilaterals if the diagonals bisect one another at right if the diagonals bisect one another at right

angles, it is a rhombusangles, it is a rhombus if the diagonals bisect one another and are if the diagonals bisect one another and are

equal, it is a rectangle.equal, it is a rectangle.

The architectureThe architecture

►The precise definitions are powerful The precise definitions are powerful because they are part of a because they are part of a rigorous logical rigorous logical structurestructure..

►Without an appreciation of this, our Without an appreciation of this, our insistence on precision seems to be merely insistence on precision seems to be merely pedanticpedantic..

►How can this logical structure be How can this logical structure be presented?presented?


Worksheet 2: basic angle factsWorksheet 2: basic angle facts►The course which I use in teaching begins The course which I use in teaching begins

with a simple, well-known result: with a simple, well-known result: angles on a line are supplementaryangles on a line are supplementary

►Note that, almost immediately, a related Note that, almost immediately, a related result - the result - the converseconverse - is introduced. - is introduced.

►Deduce the converse Deduce the converse from the original from the original resultresult


►Show that the converse of angles on a Show that the converse of angles on a line follows from the original resultline follows from the original result..

ba

A C

B

180a b+ = o

Start by drawing A, C and B so that they do not form a straight line.

dab

A C

B


►Now show that the converse of angles at a Now show that the converse of angles at a point follows from the original result.point follows from the original result.

180a b+ = o

Now complete the straight line and use the first result

180a d+ = o

Hence CB is along the dotted line.

b d∴ =

ConversesConverses►Already we have moved into a different Already we have moved into a different

realm from that of GCSE, where the realm from that of GCSE, where the converse results are converse results are not mentioned.not mentioned.

►Why are converses important?Why are converses important?►The role of The role of conversesconverses in geometry is in geometry is

similar to that of similar to that of inverse operationsinverse operations in in arithmetic and algebra.arithmetic and algebra.

►This is identified as a key principle in good This is identified as a key principle in good mathematics teaching.mathematics teaching.

ConversesConverses►If P, then QIf P, then Q►If Q, then PIf Q, then P

►Here we are talking about an inverse Here we are talking about an inverse operation operation in logicin logic..

►It is worthwhile emphasizing this It is worthwhile emphasizing this distinction using simple examples from distinction using simple examples from arithmetic and algebra, as well as from arithmetic and algebra, as well as from 'informal' geometry.'informal' geometry.

►Have a look now at Handout 3Have a look now at Handout 3

ConversesConverses

►I would claim that understanding the I would claim that understanding the difference between a result and its difference between a result and its converse is possibly more important as a converse is possibly more important as a ‘life skill’ than being able to solve a ‘life skill’ than being able to solve a quadratic equation.quadratic equation.

►For instance, do we make a decision based For instance, do we make a decision based on evidence or do we assess the evidence on evidence or do we assess the evidence based on a decision we have already based on a decision we have already made?made?


►The course I follow then lists two further The course I follow then lists two further results, concerning angles at a point and results, concerning angles at a point and vertically opposite angles.vertically opposite angles.

►It is immediately emphasised that these It is immediately emphasised that these two results are two results are consequencesconsequences of the first of the first one, angles on a line.one, angles on a line.

Show that this is the case.Show that this is the case.


►The first of these demonstrations requires The first of these demonstrations requires a a constructionconstruction and, strictly speaking, a and, strictly speaking, a generalisationgeneralisation of the first result. of the first result.

►The second is much more straightforward.The second is much more straightforward.

c2

c1

ed

ba


►The aim, therefore, is to present geometry The aim, therefore, is to present geometry as a logical discipline, with certain results as a logical discipline, with certain results stated as the stated as the foundationsfoundations and with the and with the theorems established as theorems established as consequencesconsequences..

►There is, therefore, a There is, therefore, a hierarchyhierarchy..►For instance, the alternate segment For instance, the alternate segment

theorem is theorem is more advancedmore advanced than the angle than the angle sum of a triangle.sum of a triangle.


►We have looked at a number of features of We have looked at a number of features of the architecture:the architecture: theorems form a hierarchy, with firm theorems form a hierarchy, with firm

foundations.foundations. in the process of building, established results in the process of building, established results

are used to prove new results.are used to prove new results. theorems and converses are distinguished.theorems and converses are distinguished. definitions are explicit, precise and minimal, so definitions are explicit, precise and minimal, so

that they can be used effectively.that they can be used effectively. there is an emphasis on generality.there is an emphasis on generality.


►Actually, the method I recommend is a Actually, the method I recommend is a compromisecompromise. There is no statement of . There is no statement of Euclidean axioms and I am not strictly Euclidean axioms and I am not strictly formal about the development.formal about the development.

►But what I am recommending is But what I am recommending is a world a world apartapart from the geometrical experience from the geometrical experience offered by the national curriculum.offered by the national curriculum.


►After this, the course covers familiar topic After this, the course covers familiar topic areas such as parallel lines, angle sums of areas such as parallel lines, angle sums of polygons, congruent and similar triangles, polygons, congruent and similar triangles, areas, Pythagoras and so on.areas, Pythagoras and so on.

►However, they are tackled with the same However, they are tackled with the same philosophy as set out in the manifesto.philosophy as set out in the manifesto.


►Another essential feature is an emphasis Another essential feature is an emphasis on problem-solving.on problem-solving.

►As soon as a new result is introduced, the As soon as a new result is introduced, the students must apply it to solve a carefully students must apply it to solve a carefully graded set of exercises.graded set of exercises.

►It is very important to ensure that some, It is very important to ensure that some, at least, of these problems involve at least, of these problems involve surprising results and elegant arguments.surprising results and elegant arguments.

►Similar triangles figure prominently in Similar triangles figure prominently in GCSE, along with other important work on GCSE, along with other important work on scale factors of areas and volumes.scale factors of areas and volumes.

►But similar triangles are hardly ever used But similar triangles are hardly ever used in constructing geometrical arguments.in constructing geometrical arguments.

►A particularly useful result is the A particularly useful result is the midpoint midpoint theoremtheorem . .

Developing proofDeveloping proof

►Do some of the problems on Handout 4.Do some of the problems on Handout 4.


�

B'

�

C'

�

A

�

B

�

C

►Exercise 3 is quite Exercise 3 is quite interesting.interesting.

►Care must be taken Care must be taken over drawing the over drawing the diagram in the most diagram in the most general case: it is not general case: it is not necessarily an necessarily an isosceles trapezium.isosceles trapezium.


S

RQ

P

D

B

C

A

►Now use the midpoint Now use the midpoint theorem twice.theorem twice.

►PS parallel to AD and PS parallel to AD and half its length.half its length.

►QR parallel to AD and QR parallel to AD and half its length.half its length.

►That makes PQRS into That makes PQRS into a parallelogram.a parallelogram.


S

RQ

P

D

B

C

A

►PS parallel to AD and PS parallel to AD and half its length.half its length.

►PQ parallel to BC and PQ parallel to BC and half its lengthhalf its length

►So PQ is equal to PS So PQ is equal to PS and perpendicular to it.and perpendicular to it.

►That guarantees a That guarantees a square.square.


S

RQ

P

D

B

C

A

►Finally, note that the Finally, note that the diagram can be re-diagram can be re-interpreted as being a interpreted as being a three-dimensional three-dimensional figure.figure.

►Is the result still true?Is the result still true?►Does the proof still Does the proof still

work?work?


S

RQ

P

D

B

C

A

►Now we encounter a long list of circle Now we encounter a long list of circle theorems. These are familiar to all GCSE theorems. These are familiar to all GCSE students, but are they ever used for anything students, but are they ever used for anything apart from rather silly calculations?apart from rather silly calculations?

►Part of the reason for this is a Part of the reason for this is a significant significant absenceabsence - a group of results which is entirely - a group of results which is entirely lacking from the GCSE diet!lacking from the GCSE diet!

►As I have already mentioned, they are the As I have already mentioned, they are the converse theoremsconverse theorems..

Developing the structureDeveloping the structure

Converse circle theoremsConverse circle theorems

►There are two reasons why these are There are two reasons why these are important in 'real geometry'.important in 'real geometry'.

►The first concerns the The first concerns the method of proofmethod of proof..►Try to prove the theorem: if the opposite Try to prove the theorem: if the opposite

angles of a quadrilateral are supplementary, angles of a quadrilateral are supplementary, then it is cyclic.then it is cyclic.

►(You may, of course, assume the original (You may, of course, assume the original result!)result!)

►We start with We start with ABCD so that the ABCD so that the angles B and D angles B and D add to 180add to 180º.º.

►We shall make the We shall make the assumption that assumption that this is this is notnot a cyclic a cyclic quadrilateral.quadrilateral.

Converse circle theorems Converse circle theorems

A

B C

D

A

B C

D*D

►Now we draw the Now we draw the circumcircle of circumcircle of ABC, which does ABC, which does not go through D.not go through D.

►The angles at B The angles at B and D* add to and D* add to 180180º, so D and D* º, so D and D* are equal.are equal.


A

B C

D*D

►But this is absurd, But this is absurd, since it would since it would make the angle make the angle DCD* zero.DCD* zero.

►So our initial So our initial assumption was assumption was wrong and ABCD wrong and ABCD is cyclic.is cyclic.



►Proof by contradiction is such a stunning achievement of mathematics and logic that our pupils deserve to be given access to it.

►This is another 'transferable skill' which is This is another 'transferable skill' which is more 'relevant to life' than most of school more 'relevant to life' than most of school mathematics.mathematics.


'How often have I said to you that when you 'How often have I said to you that when you have eliminated the impossible, whatever have eliminated the impossible, whatever remains, however improbable, must be the remains, however improbable, must be the truth?'truth?'

AC DoyleAC Doyle The Sign of FourThe Sign of Four


►The second reason why these results are The second reason why these results are important is that they enable you to solve important is that they enable you to solve problems.problems.

►Using a converse theorem, identify a cyclic Using a converse theorem, identify a cyclic quadrilateral.quadrilateral.

►Then use the 'forward' theorem to show Then use the 'forward' theorem to show that something is true in the configuration.that something is true in the configuration.

►We will look at two examples.We will look at two examples.


►In a triangle ABC, In a triangle ABC, draw the altitudes draw the altitudes BE and CF, BE and CF, intersecting at a intersecting at a point H.point H.

►So we know that So we know that the angles at E the angles at E and F are 90and F are 90º.º.

HF

E

A

B C

HF

E

A

B C


►AFHE is cyclic. AFHE is cyclic. (why?)(why?)

X

HF

E

A

B C


►AFHE is cyclic. AFHE is cyclic. (why?)(why?)

►Hence Hence FAH = FAH = FEH.FEH.

►AH has been AH has been extended to meet extended to meet BC at a point X.BC at a point X.

X

HF

E

A

B C


►BFEC is cyclic. BFEC is cyclic. (why - and where (why - and where is the centre?)is the centre?)

X

HF

E

A

B C


►BFEC is cyclic. BFEC is cyclic. (why - and where (why - and where is the centre?)is the centre?)

►HenceHence FEB = FEB = FCBFCB

►SoSo BAX = BAX = FCBFCB

X

HF

E

A

B C


►SoSo BAX = BAX = FCBFCB

►HenceHence BXA = BXA = BFCBFC

= 90= 90ºº►and so AX is also and so AX is also

an altitude.an altitude.

D

HF

E

A

B C

Converse circle theoremsConverse circle theorems►We have shown that We have shown that

the three altitudes the three altitudes of a triangle are of a triangle are concurrent at H, the concurrent at H, the orthocentre.orthocentre.

►What is the What is the orthocentre of orthocentre of AHB?AHB? BHC?BHC? CHA?CHA?

Converse circle theoremsConverse circle theorems►Another elegant Another elegant

result is the result is the Butterfly theoremButterfly theorem..

►Begin with a circle Begin with a circle on diameter AB, and on diameter AB, and choose any point P choose any point P on AB.on AB.

BAP

C

BAP




►Draw any chord AC.Draw any chord AC.

D

C

BAP




►Draw any chord AC.Draw any chord AC.►Continue the zigzag Continue the zigzag

CPD and DB.CPD and DB.

F

E

D

C

BAP

Converse circle theoremsConverse circle theorems►Now draw a Now draw a

perpendicular to AB perpendicular to AB through P.through P.

►The segments EP The segments EP and FP are equal in and FP are equal in length.length.

►Prove it!Prove it!

F

E

D

C

BAP

Converse circle theoremsConverse circle theorems►Now draw a Now draw a

perpendicular to AB perpendicular to AB through P.through P.

►The segments EP The segments EP and FP are equal in and FP are equal in length.length.

►HINTHINT►See Handout 5 for See Handout 5 for

more problems.more problems.

Cinderella theoremsCinderella theorems

►Finally, I would like to mention a group of Finally, I would like to mention a group of results which are really obvious, but are very results which are really obvious, but are very neglected.neglected.

►Indeed, it is clear from marking scripts from Indeed, it is clear from marking scripts from the Intermediate Olympiad that these results the Intermediate Olympiad that these results are not known, even by very good students.are not known, even by very good students.

►They appear on They appear on Handout 5Handout 5..

The Cinderella theoremsThe Cinderella theorems►Exercise 5 is my Exercise 5 is my

favourite elementary favourite elementary result on circles.result on circles.

►As X moves around As X moves around the first circle, the the first circle, the segment YZ is of segment YZ is of constant length.constant length.

►I leave this for you to I leave this for you to do!do!

Z

Y

Q

P

X


►Geometry is worth its place in the school Geometry is worth its place in the school mathematics curriculum because :mathematics curriculum because : it has an it has an architecturearchitecture which gives it authority. which gives it authority. it relies on it relies on precisely definedprecisely defined concepts. concepts. it allows the construction of it allows the construction of multi-stage multi-stage

argumentsarguments to solve simple problems. to solve simple problems. it emphasises the need for clear and coherent it emphasises the need for clear and coherent

proofproof.. it includes results and arguments with real it includes results and arguments with real

aesthetic appealaesthetic appeal..

Documents

Why it is important to teach geometryfrink.machighway.com/~dynamicm/importance-of-geometry.pdfWhy it is important to teach geometry Gerry Leversha Mathematical Association Annual Conference