The teaching and learning of
mathematical modelling
Mogens Niss
IMFUFA/NSM, Roskilde University
1
Introduction
• The use of mathematics in extra-mathematical
domains is always brought about by way of
mathematical models, whether explicit or
implicit.
• Mathematical models result from mathematical
modelling, either descriptive or prescriptive
modelling, both of which permeate the world.
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• We know from research that the ability to
conduct successful modelling is not an
automatic consequence of being well versed in
”pure” mathematics.
• Being a successful modeller does indeed require
mathematical competencies and knowledge, but
much more is needed. In other words: A solid
mathematical background is a necessary but
far from sufficient prerequisite for being a
capable modeller.
• However, the ability to model can be learnt!
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• If we want people to become able to model, we
have to teach it to them!
• Since mathematical modelling is highly
important within a multitude of extramathematical domains, we have to take
responsibility for developing the mathematical
modelling competency with our students, at all
levels.
***
Let us look at an example. It has been deliberately
chosen to be very simple, in technical terms.
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A modelling example
Paper sheet formats
• In many countries, organised series of rectangular
paper sheet formats are commonplace.
• Two such series, the A-formats and the Bformats, can be found in thousands of paper or
stationery shops around the world.
• Here are their official dimensions (in mm):
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A0: 1189841
A1: 841594
A2: 594420
A3: 420297
A4: 297210
A5: 210148
A6: 148105
A7: 10574
A8: 7452
A9: 5237
A10: 3726
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B0: 14141000
B1: 1000707
B2: 707500
B3: 500353
B4: 353250
B5: 250176
B6: 176125
B7: 12588
B8: 8862
B9: 6244
B10: 4431
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• Question: What are the patterns and
principles in and behind these sheet
formats? An intellectual, not a practical
question or problem.
• To answer this question, let us denote the side
lengths of An by ln (longer side) and sn (shorter
side), and the longer and shorter side lengths of Bn
by Ln and Sn, n  0.
• In other words we mathematise the siuation by the
ordered pairs (ln , sn) (An) and (Ln , Sn) (Bn),
respectively. What can we say about these?
• Observation 1: ln+1 = sn and Ln+1 = Sn
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• Observation 2 (hypothesis): It looks as if
sn+1  ln /2 and Sn+1  Ln /2
Combining Obs. 1 and 2: (ln+1 , sn+1) = (sn , ln /2) and
(Ln+1 , Sn+1) = (Sn , Ln /2),
which implies that A(n+1) and B(n+1) result from
folding An and Bn, respectively, along the mid-point
transversal of the longer sides
• However, this doesn’t explain the actual
dimensions of the paper sheets, only their
relationships. How to proceed?
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• Well, let us take a look at the side ratios of the
sheets:
• From Observation 2 we get
ln+1 / sn+1 = sn /(ln /2) = 2sn / ln = 2/(ln /sn )
and therefore also
ln / sn = 2/(ln-1 /sn-1),
which by insertion yields
ln+1 / sn+1 = 2/(2/ln-1 / sn-1 )) = ln-1 / sn-1.
Similarly for the B-series,
Ln+1 / Sn+1 = Ln-1 / Ln-1
• This leads us to stating
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as a hypothesis that the ratios between the sides
are the same for all sheets in each series:
ln+1 / sn+1 = ln / sn , Ln+1 / Sn+1 = Ln / Sn .
• This allows us to express (ln , sn) and (Ln , Sn) in
terms of (l0 , s0) and (L0 , S0), respectively.
For the A-series , we find that
ln / sn = sn-1 / (ln-1 /2) = 2/(ln-1 / sn-1 )
= 2/ (ln / sn),
i.e. (ln / sn)2 = 2,
whence ln = 21/2 sn for all n  0.
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Especially for n = 0: l0 = 21/2 s0.
Setting l0s0 = A, and inserting we obtain
21/2 s02 = A, i.e. s0 = A/21/4 .
This yields l0 = 21/2 s0 = 21/2 A/ 21/4 = 21/4 A .
In summary
l0 = 21/4 A, and s0 =A/21/4.
Next we have
l1 = s0 = A/ 21/4, s1 = l0/2 = 21/4 A/2 = A/23/4.
Proceeding by recursion and finishing by induction,
we obtain for any n  0:
ln = A/ 2(2n-1)/4 and sn = A/ 2(2n+1)/4
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For the B series, we obtain completely similar results,
setting L0 S0 = B:
Ln = B/ 2(2n-1)/4 and Sn = B/ 2(2n+1)/4
But we still don’t have specific dimensions for the
sheets, i.e. actual values of the side lengths. However,
we are pretty close:
• A = l0s0 = 1189841 mm2 = 999,949 mm2 
1,000,000 mm2 (= 1 m2) and
• B = L0 S0 = 14141000 mm2 = 1,414,000 mm2
(= 1.414 m2 )
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We are now able to conclude that the underlying
principles and patterns of the A- and B-formats are
as follows:
For the A-formats the side lengths fulfil the following
conditions:
• (a) The area of A0 is 1m2
• (b) All sheets are similar, i.e. they have same side
ratios
• (c) More specifically, the relationship between
A(n+1) and An is given by ln+1 / sn+1 = ln / sn = 2
and (ln+1 , sn+1)= (sn, ln /2), so that
ln = 103/ 2(2n-1)/4 mm, sn = 103/ 2(2n+1)/4 mm.
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For the B-formats the side lengths fulfil the following
conditions:
• (a) The shorter side of B0 is 1m
• (b) All sheets are similar, i.e. they have same side
ratio, namely 2
• (c) More specifically, the relationship between
B(n+1) and Bn is given by Ln+1 / Sn+1 = Ln / Sn = 2
and (Ln+1 , Sn+1)= (Sn, Ln /2), so that
Ln = 1.4141/2103/2(2n-1)/4 mm, and
Sn = 1.4141/2 103/ 2(2n+1)/4 mm.
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• It is now time to validate the model we have
established.
• We do so by confronting the model outcomes
with the official real-life data for both paper
series:
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Reality
• A0: 1189841
• A1: 841594
• A2: 594420
• A3: 420297
• A4: 297210
• A5: 210148
• A6: 148105
• A7: 10574
• A8: 7452
• A9: 5237
• A10: 3726
Model outcomes
• A0: 1189841
• A1: 841595
• A2: 595420
• A3: 420297
• A4: 297210
• A5: 210149
• A6: 149105
• A7: 10574
• A8: 7453
• A9: 5337
• A10: 3726
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Reality
• B0: 14141000
• B1: 1000707
• B2: 707500
• B3: 500353
• B4: 353250
• B5: 250176
• B6: 176125
• B7: 12588
• B8: 8862
• B9: 6244
• B10: 4431
Model outcomes
• B0: 14141000
• B1: 1000707
• B2: 707500
• B3: 500354
• B4: 354250
• B5: 250177
• B6: 177125
• B7: 12588
• B8: 8862
• B9: 6244
• B10: 4431
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• It is evident that the model outcomes are almost
identical with the real values.
• So, the model has been validated with
overwhelming success.
• What we have done here is to use mathematical
modelling to come to grips with an exisiting
reality – in this case certain given paper formats –
so as to uncover the principles and patterns behind
that reality. We have been engaged in descriptive
modelling.
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Analysing the process
First, we undertook pre-mathematisation – i.e.
prepared for mathematisation - by
• asking for the principles behind the formats
• collecting the official data for the rectangular Aand B- series sheets
• making Observation 1, that the longer side of
sheet no. n is the shorter side of the previous sheet
no. n-1
• making Observation 2, that the shorter side of
sheet no. n is (extremely close to) half the longer
side of the previous sheet, no. n-1.
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Next, we mathematised the situation by translating
it into a mathematical domain. This was done by
representing the two paper series by (infinite!)
sequences of pairs, (ln, sn) and (Ln, Sn) with positive
real components.
• Then we translated our observations into
mathematical relations
(1) (ln+1 , sn+1) = (sn , ln /2), (Ln+1 , Sn+1) = (Sn , Ln /2),
and further made the assumption that
(2) ln+1 / sn+1 = ln / sn , Ln+1 / Sn+1 = Ln / Sn .
• After setting (3) A = l0s0 and B = L0S0, we asked
the mathematical question: What sequences, if
any, satisfy conditions (1), (2) and (3)?
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Thirdly, after having completed the mathematisation,
we employ mathematical problem solving within
the mathematical domain of sequences of real-valued
pairs to determine the sequences that satisfy (1), (2)
and (3).
• We used algebra to realise that ln = 21/2 sn and
Ln = 21/2 Sn for all n  0,
and then algebra in combination with recursion
techniques to obtain – for all n  0:
• ln = A/ 2(2n-1)/4 and sn = A/ 2(2n+1)/4
• Ln = B/ 2(2n-1)/4 and Sn = B/ 2(2n+1)/4
Next, this outcome was justified by an induction
proof.
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Using the known data for A0 and B0 to compute the
areas A and B, and then the side lengths for both
paper series to obtain,
and
ln = 103/ 2(2n-1)/4 and sn = 103/ 2(2n+1)/4
Ln = 1.4141/2 103/ 2(2n-1)/4 ,
Sn = 1.4141/2 103/ (2n+1)/4 .
the mathematical treatment is completed by the
conclusion that the mathematised observations and
assumptions lead to these uniquely determined
answers to the mathematical questions posed.
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• In this case, de-mathematisation is almost trivial,
since it amounts to attaching units (mm) to the
numbers resulting from the mathematical
treatment, thus translating the mathematical
outcomes back to outcomes concerning the real
world domain of paper sheets.
• Validation of the present model consists in
confronting and comparing the model outcomes
with the real world data. In this case the outcomes
were in marked agreement with the real data. So,
the model is accepted as an excellent model of
paper sheet reality.
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• To be sure, the position taken in this descriptive
modelling activity was to consider the existing
reality of A- and B-paper formats as uncharted
land, in which no prior knowledge about the
systems behind the formats was invoked.
• Another position was possible: That of the paper
format designer. For a designer, the observations
and hypotheses of the descriptive modeller would
change to being requirements (axioms) that the
paper formats have to fulfil, if possible. This would
be an example of prescriptive modelling, in which
reality is created or organised. The outcome
would then be the uniquely defined specifications
of the two series of paper formats.
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The modelling competency
• The specific modelling process just conducted is an
instance of a general process, called the modelling
cycle, consisting of a number of characteristic phases.
• Modelling competency is the ability to carry out all
these phases in a multitude of different contexts and
situations.
• It should be emphasised that the modelling cycle is an
analytic construct, not a description of modellers’ real
life behaviour, step by step.
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Extramathematical
domain
Mathematical
domain
Idealisation
mathematisation
Specification
Idealised
situation cum
questions
translation
Mathematised
situation
cum questions
answers
Mathematical answers
de-mathematisation
interpretation
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The phases of the modelling cycle are:
• Pre-mathematisation, in which the situation in an
extra-mathematical domain D to be modelled is
specified by selecting relevant objects and relations
between them, making assumptions and stating
hypotheses, leading to an idealised situation for
which the model-generating questions can be
formulated.
• Mathematisation, in which the selected objects,
relations and hypotheses, and - above all - the
questions posed, are translated by some mapping f
into mathematical entities in some mathematical
domain M. The triple (D,f,M) is called the model.
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• The mathematical treatment – or problem
solving - takes place within M. It consists of
employing mathematical methods (calculations,
deductions, theories, theorems etc.) to answer the
mathematised questions and to justify the
mathematical conclusions obtained.
• De-mathematisation consists of translating the
mathematical answers obtained back into answers
to the extra-mathematical questions that gave rise
to the whole modelling endeavour in the first
place.
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• The final phase of the modelling cycle is validation
of the model. This consists in assessing the model
outcomes vis-à-vis the initial needs and wishes,
including the range, scope and solidity of the
answers obtained. Validation typically involves
confronting model results with known facts and
data concerning the extra-mathematical domain.
• In case the validation leads to rejection of the
model, a new or modified model will often have to
be constructed, thus giving rise to a new journey
in the modelling cycle – whence the term ”cycle”.
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• Mathematics education research provides
evidence that all five phases of the modelling
cycle are difficult to learners, depending on the
specific situation to be modelled. In some cases the
hardship is concentrated on one or two of the
phases whereas the others may be more tractable.
• However, the most demanding phase clearly
seems to be the mathematisation phase. And we
know pretty well why. Here is a brief account of
”why?”
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(1) Pre-mathematisation , i.e. structuring the extramathematical situation and preparing it for
mathematisation requires a first anticipation of
the potential involvement of mathematics, and the
nature and usefulness of this involvement,
including the mathematical domain(s) that might
be used to represent the situation and the
questions posed about it
Sometimes this is trivial, sometimes it is very
demanding!
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(2) When subjecting the prepared extramathematical situation to mathematisation, it is
necessary to anticipate specific mathematical
representations, suitable for capturing the
situation.
This requires knowledge of a relevant
mathematical apparatus as well as past experience
with the capability of this apparatus for modelling
similar situations.
Usually very demanding!
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(3) When anticipating a suitable mathematical
apparatus, it’s necessary to envisage the ways in
which this apparatus may provide answers to
mathematised questions.
This requires envisaging problem solving strategies
and procedures pertaining to the mathematical
problems arising from the mathematisation.
Sometimes easy, mostly very demanding!
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A common feature of (1), (2), and (3):
Successful mathematisation requires
implemented anticipation
of a rather involved nature:
At three key instances of mathematisation, the
modeller has to be able to anticipate (potentially
difficult) subsequent steps, and to implement this
anticipation in terms of decisions and actions that
determine the framework for the next step.
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The third step alone, in which mathematical
problem solving is planned and implemented, is
already very involved (see, e.g., earlier work by
Lesh, Lester, and Schoenfeld).
However, the first two steps are often much
more involved than the last one, as they aim at
creating links, not yet established, between an
extra-mathematical situation and mathematics, and
- when so doing – anticipating the last step.
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The itinerary of implemented anticipation
Idealised
extra-mathematical
situation cum question(s)
Mathematised
situation
cum
question(s)
Problem
solving
Mathematic
-al answers
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Approaches to teaching modelling
Two dual aspects of the teaching and learning of
modelling in mathematics education:
• Mathematics (also) for the sake of modelling
(fostering modelling competencies with students
is a goal in itself)
• Modelling for the sake of mathematics
(modelling can contribute to sense-making in and
learning of mathematics).
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• Both aspects are significant and have received
attention in mathematics education research.
However, here I focus on ”mathematics for the sake
of modelling”.
• In order to overcome the difficulties of modelling,
researchers have studied students’ work with the
entire modelling cycle as well as with each of the
five individual phases that represent subcompetencies of the overarching modelling
competency.
• This has given rise to three different approaches
to teaching modelling:
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• The holistic approach, in which students are
engaged in tasks that require invocation of the
entire modelling cycle, including all the extramathematical work needed along the road.
• The semi-holistic approach, which is the holistic
approach but applied to tasks in which some of the
phases are trivial or substantially reduced.
• The atomistic approach, in which students focus
on the individual modelling phase separately. This
approach is meant to develop each of the subcompetencies of the modelling competency in the
students.
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• Research findings suggest that neither of these
approaches can stand alone. The best strategy seems
to be a combination of the three approaches.
• However, this is time-consuming. If restrictions on
time are severe, it seems that particular emphasis
should be given to the hardest phase,
mathematisation.
• Whilst the bad news is that solid mathematical
knowledge is not enough to foster the modelling
competency, the good news is that this competency
can be developed by teaching – if you want to!
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Thank you
for your attention!

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