HOW MUCH MEANING CAN WE CONSTRUCT AROUND
GEOMETRIC CONSTRUCTIONS?
Snezana Lawrence and Peter Ransom
Bath Spa University, England
This paper describes a way in which the mathematical heritage can be used to
identify potentially ‘rich’ tasks undertaken by student teachers to deconstruct, and
subsequently better understand, the meaning of mathematical concepts they already
know and are expected to teach. It is based on a small-scale project undertaken in the
South West of England and which is proposed as a pilot for a larger project to map
the curriculum against such topics. The project has been generously supported by the
National Centre for Excellence in the Teaching of Mathematics (NCETM).1
Of course, mathematics involves deductivity. Working with the
slide rule and the protractor is no mathematics, measuring areas
and volumes is no mathematics. But accepting a deduction is no
mathematics either, unless you adhere to the interpretation of
mathematics as a ready made subject.
Hans Freudenthal, Geometry between the Devil and the Deep
Sea, 416.
1. THE CURRICULUM CHANGES AND THE CULTURAL HERITAGE
Within the past few years a great number of changes have been initiated in the
mathematics education in Great Britain.2 The first of the most dramatic two of those
changes was certainly the introduction of the new curriculum which was brought in
as the old government prepared to leave the scene and the new was preparing to step
in.3
One of the most important things for schools to get right is being identified now to be
mathematics education.4 But is it mathematics that we as a society are interested in,
or the results of the PISA study5, the content of the curriculum, the pedagogy, the
level of difficulty of our qualifications, or the power of teachers in classrooms?6 We
will argue in this paper that none of the above gives satisfactory answers to the better
teaching and learning of mathematics, and that the disengagement of teachers from
the actual mathematical content is the most important reason for disengagement of
our students.7 We will also argue that this is so because the teachers are not inducted
in any way in the mathematical culture which they are supposed to transfer and
transform through their own practice, either by ‘doing maths’ or by inducting their
own students into that practice. It is our fear that whatever curriculum we have the
same will be valid in this regard: when the student teachers engage in doing maths,
they follow the ‘heritage’ path and are rarely aware of the origin of concepts and
therefore their meaning. This paper will argue that the awareness of mathematical
cultural heritage, and a set of skills in identifying such heritage, is a necessary
component in any preparatory teacher education. This process we identify also as a
possible way into introducing the students to the history of mathematics which brings
out all the well known benefits to their subsequent practice. (Grattan-Guiness, 2004).
2. THE RESEARCH IMPETUS
Grattan-Guiness (2004, 174) describes the type of mathematics education ‘very much
guided by heritage.’ But this heritage he identifies thus as one which brings out the
‘…reactions of students – including myself, as I still vividly recall – are often distaste
and bewilderment; not particularly that mathematics is very hard to understand and even
to learn but mainly that it turns up in “perfect” dried-out forms, so that if there are any
mistakes, then necessarily the student made them.’
In this way described, the heritage in secondary school is the tradition of learning
Euclid by rote, or in more recent times, following algorithmic learning and teaching
of famous examples such as completing the square or the theorem of Pythagoras
without the historical accuracy evident in teaching, or the real understanding reported
by students. Although not entirely the same, this reminds us also of the Freudenthal’s
anti-didactical inversion: in fact we are dealing here with taking the ‘mathematical
activities of others… as a starting point for instruction’ (Gravemeijer & Terwel,
2000, 780).
We argue however, that the heritage, and its dried-out mathematics which does not
engage students, has a role to play in teacher training if only as a way of exploring a
culture of mathematics and a possible initiation route into the historical study of
mathematical concepts. In other words, we should use the ‘heritage’ approach of the
ready-made mathematics of the prescribed curriculum(s) to identify the concepts
which may be rich in their potential to search for ‘meaning’, so that dealing with
concepts and doing mathematical problems or exercises which use them, becomes a
meaningful activity. The use of curriculum in such a way provides a comfort zone for
teachers to structure their exploration in a sustainable way, through gradual learning
process rather than sweeping enthusiasm which ends up with not having enough time
to engage with the ‘new’ material through research and reading, and subsequently can
end as a project never to be completed.8
In order to use ‘heritage’ in this way we propose that answers to the questions:
• Why do we call this (concept/process/tool) by this name?
• What does it mean and/or how does it work?
must be possible to answer. For example, Euclidean tool or construction, and a
Pythagoras’ theorem would be starting points for an investigation of such type in
which heritage would be used to explore the history and gain the meaning about a
mathematical concept.
How else can teachers themselves impart the ‘meaning’ of concepts they are dealing
with onto their pupils? This may seem like an obvious fact, but the research presented
2
here will show that imparting the all elusive ‘meaning’ is not a part of everyday
practice, and that teachers in education are prone to unquestionably take concepts in a
heritage-like, ready-made mathematics way. Should every teacher be aware of why
we call certain constructions ‘Euclidean’ (In fact how many teachers actually do so?)
and why a certain way of solving equations is done through the method of
‘completing the square’? How can a teacher use these concepts with the secondary
age pupils in a meaningful way without herself/himself really being sure what they
mean? Not knowing these simple facts by teachers approaches the near-total
disengagement with a mathematical culture that the teachers are trying to somehow in
turn engage their students with. It may not then come as a surprise to hear that the
perception that many young people in Britain have of mathematics is that it is “boring
and irrelevant” as a consequence (Smith, 2004, 2).
3. FROM EXPERIENCE TO EXPERIMENT
While exploring the possible ways in which mathematics ‘subject knowledge’ can be
revitalised through the Initial Teacher Education course for Secondary and Middle
Years Mathematics Students, we came across the barrier of students wanting to
engage with simple exercises and working things out for themselves. Doing
mathematics is different to teaching mathematics, but can mathematical concepts be
re-examined for the purposes of education without doing the mathematics one already
‘knows’? This was another of the questions that puzzled us as we tried to understand
both the wide-spread lack of knowledge about the origin of certain concepts that we
teach at secondary level, and the lack of questioning from prospective teachers as
they train to use these in their craft, and in their (what it seemed) ultimate goal of
covering the prescribed curriculum. In fact, should we be asking the questions: ‘Do
they actually care?’ and ‘Does it make a difference to their teaching and students’
learning?’ However those questions are beyond the research of this paper.
Watson (2008, 7) documents the way that we have in fact structured our curriculum
through examinations in the recent past:
Questions involving application of theorems can be avoided in UK national tests at 16+
and students still be awarded the highest grades. Theorems and proof of any kind, let
alone geometrical contexts, do not play a part in higher school examinations.
The lack of questioning of the premises upon which we should build some
mathematical understanding by new graduates in this context does not then seem
puzzling. The difference between doing mathematics (seen in all its diverse cultural
interactions, a mode of intellectual enquiry but also a mode of intellectual
communication), and doing the ‘school’ mathematics, is described by Watson (2008,
2) in the same paper:
Learners… have a different experience to those taught with a more abstract view, but
solving realistic and everyday problems need not lead them to understand the role of
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mathematics beyond providing ad hoc methods for real problem-solving, or as a service
subject which holds tools for moving forward in other domains.
How it is possible that such practice is embedded in the system of education becomes
apparent when at the start of such practice it is evident, as will be shown, that the
enquiry stops at the gate of the curriculum temple.
The project described here was a pilot project to deal only with the single issue of
Euclidean constructions. The question was ‘How much do teacher students
understand what a Euclidean construction is and how could they employ such
understanding to teach the topic from the curriculum?’ The aim was to:
• find the level of understanding of Euclidean constructions and their historical
origins;
• engage the student teachers in exploring the possibilities to do mathematics but
also to think about how they would teach it;
• find the ways in which this process could be modelled for other topics from the
curriculum.
The pilot will serve as a basis to plan a larger project to examine the issues of
• overcoming the disengagement of teacher students from the meaning of
mathematical concepts through
o doing mathematics first through a heritage-like way
o identifying the ‘heritage’ elements of mathematics thus ‘done’
o going onto the exploration of the history of the concept (with all
subsequent benefits, but primarily: aiming to understand the way of
mathematical thinking associated with the concept, and aiming to
systematize the interconnectedness with other mathematical discoveries
and concepts)
• devising a system to identify the topics from the curriculum which can be rich
to offer such explorations by the teachers in training and
• attempting to define ‘how’ and ‘how much’ history of mathematics should and
could be incorporated into the teacher education.
This paper therefore does not give a full and/or comprehensive list of answers to the
question of how and why to introduce the teachers in education and training to the
history of mathematics. But it does trace a project which shows the increased
engagement and motivation giving teachers the confidence to use the syllabus in a
creative way and to explore the concepts they are meant to teach, which would
otherwise be made into an empty list to be ticked off as they go into the lessons. It
also evidences the students’ increased capability to search for cultural and historical
roots and construct the meaning around mathematics they are teaching.
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3.1 The results of the experiment
The study concentrated on analysing a group of students at a university in South West
England. They were all mathematics specialists: 24 who enrolled on a postgraduate
course leading to the Postgraduate Certificate in Education (PGCE) preparing them to
teach at secondary level (11-18 year olds) and 12 students preparing to teach in
middle-schools (8-14 year olds), but all working on the mathematics related
pedagogy for 11-14 year olds.
The students were given a task to complete on geometrical constructions. They
needed to:
• research the
constructions
requirements
of
the
curriculum
regarding
geometrical
• find as many constructions as possible, appropriate for the curriculum levels
and execute them themselves
• devise a learning activity for their prospective pupils.
The first problem that teacher students encountered was the material from which to
source their own, and then their students’, learning. While many school textbooks
deal with geometrical constructions, they do so in a haphazard way without sufficient
explanation as regards to the context in which these arise, or were conceived, and
often do not provide any underlying conceptual understanding. The history of the
concepts is often entirely disregarded, even though one of the most widely available
textbooks insists on mentioning ‘Euclidean constructions’.9 This has been identified
by authors in other countries as well, such as in recent study of Nicol and Crespo
(2006).
At the end of the task, the students were asked to say what constructions they learnt
and were able to execute without resource to repeated instruction. The percentages of
students being able to complete various constructions are given in table below.
However, of more interest was the
subsequent students plotting of the
constructions against the curriculum for
11-14 olds; the students were asked to say
how many topics from the curriculum they
could teach (partially or in full) through
geometrical constructions. Percentages of
student teachers who believed they could
teach topics listed are given below.
In this survey, some interesting results came through:
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• 70% of students originally said
they could construct triangles, but
only 10% declared they could see
how geometric construction of the
triangles would be relevant to the
curriculum topic
• Furthermore, only 20% of students
were able to identify or recognise
that you could teach geometric
constructions and loci (a topic in
the curriculum) by actually doing geometric constructions.
When students were asked in a whole-group discussion to make sense of these two
answers, they further clarified:
• They could not see how contructions of triangles aided the conceptualisation of
various properties of triangles and/or geometrical reasoning related to this topic
• Whilst they recognised that the geometric constructions and loci were part of
the curriculum, they believed this to be an ad-hoc and not important part of
actually ‘doing’ mathematics on one hand, and on the other, that engaging with
constructions was not going to cover the curriculum topic in its entirety.
In our opinion the application of geometrical constructions is important to school
students. Many texts and exercises consist of the pure geometrical constructions first,
before looking at applications of these constructions. Even then the applications are
thinly disguised (e.g. boat sailing between two rocks). Our work with 13/14 year old
students suggests that by putting constructions into a larger piece of practical work
(usually a 2½ hour single session, though this can be spread over three single lessons)
more is appreciated of the use of such constructions and there is then time to explain
more of the geometrical proof of why the construction works. One example used in
the classroom (and shown here) is on how to find true north/south using a vertical
pointer. Other examples can be found at Ransom (2004, 22-26) and the student
teachers involved in this research project worked through these examples. It was
interesting watching them work through the materials since their first reactions to
drawing a perpendicular from a point was to use a protractor, just as school students
tend to do. This does, however, allow the point to be made that all measurement is
approximate (to varying degrees of accuracy), yet the construction method is
theoretically accurate. This concept of accuracy does not appear to be important to
many.
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Deconstruction and discussion
Without doubt the most important part of the experiment was the further
deconstruction of the students’ learning process, which they used to improve the
design of their learning task. Van Maanen (1992, 223) has claimed that the use of
constructions begins only when these are to be designed:
Clearly the crucial point is to find out which construction steps are needed to solve the
problem (the analysis stage). When these steps have been discovered the proof of the
correctness of the construction (the synthesis stage) is usually easy.
Through this analysis, and subsequent synthesis of the Euclidean constructions, the
students first made the discovery that geometry and measurement are not necessarily
the same; this led some students to delve further into the history of measures as well
as measuring devices on the one hand, and constructions and mathematical
instruments on the other.
Whilst the students discussed the two technical meanings of the ‘construction’ – the
construction of the theorem and the construction in a form of a drawing (Martin,
1998, 3), the issues of the ‘Euclidean’ tools arose (as defined by Martin, 1998, 6, and
Holme, 2002, 48). It was this crucial piece of ‘meaning’, of the difference between
the measurement and the theorising through construction with Euclidean tools that
led students to a better understanding of the way that Greek mathematics dealt with
magnitudes as tools for understanding the relationships and building upon those
which have already been established. At this point in the discussion the student
teachers began to be truly engaged in an intellectually active way and see
mathematics as a possible way of sharing that intellectual dialogue with their
prospective students.
At the beginning of the experiment exactly 50% of the students had never heard of
Euclid and 100% didn’t know what Euclidean constructions were, although they were
happy to include mentioning of both Euclid and his constructions and tools in the
learning tasks they devised for their pupils. At the end however, 90% described that
their main motive for using Euclidean constructions would in future be in order
to engage the students in geometrical reasoning and proof.
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CONCLUSIONS
While small in scope, this experiment showed that the practical way of engaging
student teachers through:
• the process of doing (albeit some simple) mathematics;
• discussing the historical context and hence dissecting the meaning attributed to
some concepts (Euclidean tools);
• the learning about a topic they are attempting to teach, and a vision of how to
transfer that engagement in their classrooms.
It has been noticed, and Gulikers (2001, 224) gives supporting evidence, that there is
‘…growing interest among teachers in the history of mathematics. …results of two
questionnaire surveys… reveal that teachers are interested in the history of
mathematics, but at the same time, are not well resourced to actually use such
material in their own teaching’. While we agree with this, this small study and the
long term engagement with teacher groups from around England10 also gives cause to
believe that
• teachers do not have the full motivation and don’t actually see the history as a
necessary part of mathematics learning;
• they have not experienced those crucial insights that would make the history
of mathematics ‘necessary’ to the process of understanding and engaging with
a mathematical concept either for themselves or for their pupils.
At the end of the experiment, 75% of student teachers declared the desire to engage
more with the history of mathematics and 83% declared that they could see how to do
it.11
So, while it is worthwhile discussing the need to introduce the history of mathematics
into mathematics instruction, the student teachers need to have the experience of
how this is useful in their own practice. They are often bogged down with many
daily pressing issues, such as pupil motivation, behaviour management, getting the
hang of the mathematics curriculum, understanding the levels which are appropriate
for the classes they teach both in terms of age and ability, and finding enough
preparation time whilst learning how to plan effective lessons. Only after they have
understood a crucial piece of mathematics that they have ‘inherited’ and thus
practiced for many years (not very well if we are to judge by the results of the first
questionnaire) without questioning, do they begin to see the potential of the history of
mathematics.
The learning of mathematics includes various other activities that support learning
other than doing mathematics. Freudenthal identified those as ‘organising a subject
matter… from reality which has to be organised according to mathematical patterns if
problems from reality have to be solved. It can also be a mathematical matter, new or
old results, of your own or others, which have to be organised according to new
8
ideas, to be better understood, in a broader context, or by an axiomatic approach’
(Freudenthal, 1971). Others, like Lakatos for example, focused on the problemsolving as a way of reconstruction of a pure research mathematical discourse
(Lakatos, 1976).12 Both of these are valid ways of learning mathematics, but for those
who already ‘know’ and whose path to teaching is littered with ready-made
mathematics modules and heritage-style pictures of good mathematics like some
dusty and beautiful picture of a remote landscape in an old frame, the way of
rediscovery can simply be to engage with a question of when did we identify a
concept as a concept, why do we call it as we do and what does that mean. Only then
does the mathematics history become part of the mathematics education culture, and
the pleasure becomes the cultural one.13 And that is just for teachers, the kids will
follow.
REFERENCES
Barbin, E. (1994). The role of problems in the history and teaching of mathematics.
In R. Calinger (ed.), Vita Mathematica: Historical Research and Integration with
Teaching, MAA, Washington, 17-25.
Barbin, E. (1997). Sur les relations entre épistémologie, histoire et didactique. In
Repères IREM, 27, 63-80.
Davis, B. and Simmt, E. (2006). Mathematics-for-Teaching: An Ongoing
Investigation of the Mathematics That Teachers (Need to) Know. In Educational
Studies in Mathematics, 61 (3), 293-319.
Freudenthal, H. (1971). Geometry between the devil and the deep sea. In Educational
Studies in Mathematics, 3 (3-4), 413-435.
Fujita, T and Jones, K. (2008). The Process of Re-designing the Geometry
Curriculum: the case of the Mathematical Association in England in the early 20th
Century. Paper presented to Topic Study Group 38 (TSG38) at the 11th
International Congress on Mathematical Education (ICME-11) Monterrey,
Mexico, 6-13 July 2008.
Grattan-Guinness, I. (2004). The mathematics of the past: distinguishing its history
from our heritage. In Historia Mathematica, 31 (2), 163-185.
Gravemeijer, K. and Terwel, J. (2000). Hans Freudenthal: a mathematician on
didactics and curriculum theory. In Curriculum Studies, 32 (6), 777-796.
Gulikers, I. and Blom, K. (2001). ”A Historical Angle”, A Survey of Recent
Literature on the Use and Value of History in Geometrical Education. In
Educational Studies in Mathematics, 47 (2), 223-258.
Heilbron, J. L. (2000). Geometry Civilized, History, Culture and Technique,
Clarendon Press, Oxford.
Holme, A. (2002). Geometry, Our Cultural Heritage. Springer, Berlin, Germany.
9
Lakatos, I. (1976). Proofs and Refutations. Cambridge University Press, Cambridge,
UK.
Lawrence, S. (2009). What works in the Classroom – Project on the History of
Mathematics and the Collaborative Teaching Practice. Paper presented at CERME
6, January 28th – February 1st, 2009, Lyon France.
Lee, S., Browne, R., Dudzic, S., Stripp, C. (2010). Understanding the UK
Mathematics Curriculum Pre-Higher Education. The Higher Education Academy,
accessed 29th September 2010,
http://www.heacademy.ac.uk/physsci/news/detail/2010/pre_he_maths_guide
Maanen, van J. (1992). Seventeenth Century Instruments for Drawing Conic
Sections. In The Mathematical Gazette, 76 (476), 222-230.
Martin, G. E. (1998). Geometric Constructions. Springer, NY, USA.
Nicol, C and Crespo, S. (2006). Learning to Teach with Mathematics Textbooks:
How Preservice Teachers Interpret and Use Curriculum Materials. In Educational
Studies in Mathematics, 62 (3), 331-355.
Ransom, P.(2004). The Maths Busters. In Mathematics in School, 33 (2), 22-26
Smith, A. (2004). Making Mathematics Count. Stationary Office, UK.
Watson, A. (2008) School mathematics as a special kind of mathematics. In For the
Learning of Mathematics. 28(3) 3-8.
1
For similar projects see Teacher Enquiry: Funded Projects
https://www.ncetm.org.uk/enquiry/funded-projects.
2
These are listed in the pre-university qualifications guide with a timeline of their introductions at
http://www.heacademy.ac.uk/physsci/news/detail/2010/pre_he_maths_guide.
3
The new curriculum was made effective from the beginning of academic 2008/9 for secondary
subjects.
4
From the speech of David Cameron, now the British Prime Minster, delivered on 2nd February
2009, accessed 20th September 2010 from
http://www.conservatives.com/News/Speeches/2009/02/David_Cameron_Conservatives_Maths_Ta
skforce_launch_with_Carol_Vorderman.aspx: “When it comes to what those disciplines are, and
how they are taught, I believe there few things more important than getting maths in our schools
right”.
5
Programme for International Student Assessment, by the Organisation for Economic Co-operation
and Development.
6
These are some of the most mentioned topics in the current British media; a separate study has
currently been undertaken by the author to identify these over the period of last three years.
10
7
Report by Adrian Smith, entitled ‘Making Mathematics Count’, was undertaken upon the
commission from the Advisory Committee on Mathematics Education, an independent body, based
at the Royal Society, London.
8
This has been further expounded upon in Gulikers & Blom, (2001).
9
For the matter of fairness the publisher and the book have not been mentioned.
10
See Lawrence, (2009).
11
As a small aside, around 17% of the students reported that they have acquired the employment
while in the course due to their meaningful use of historical context in the interview lessons.
12
Barbin (1996, 1997), who argues “that history of mathematics changes the epistemological
concepts of mathematics by emphasising the construction of knowledge out of the activity of
problem solving”, gives us a possible way of approaching the solution of how to introduce such
activities.
13
Heilbron (2000, 46) describes pleasures while researching geometry: “Finally the pleasure, or my
pleasure, has been cultural. Pursuing geometry opens the mind to relationships among learning, its
applications, and the societies that support them.”
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