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1. Assessment of Mathematical Modelling
and Applications
Henk van der Kooij
OW&OC. Tiberdreef 4.3561 GG Utrecht, The Netherlands
Abstract: In The Netherlands mathematics education is changing from learning about
structures and mechanic manipulation of formulas into learning how to formulate, how to
structure. The so called realistic approach to mathematics education is strongly process oriented
in stead of product oriented.Students (and teachers) are active in (re)inventing mathematics for
themselves.ln this kind of education important skills to be tested are: choosing suitable
strategies for solving problems, modelling real problems and critical analyzing given models
and given solutions. The HEWET and HAWEX-project have shown that there are appropriate
ways for assessment of mathematical modelling and applications, even in time restricted written
tests (like the final examination).
Keywords: realistic mathematics education, mathematical modelling, applications, HEWET
project, HAWEX project, final examination
The HEWET-project resulted in a new curriculum for the upper grades (age 16-18) of the pre-
university level of secondary education in The Netherlands: mathematics-A. The curriculum is
aiming at those students who are preparing for a study in social sciences at university. Since
1987 nationwide the final examinations are based on this curriculum.
The HAWEX-project led to new curricula (mathematics-A and -B) for the upper grades of
Havo (higher general education, a middle level of Dutch secondary education, see Appendix 1).
Mathematics-B is aiming at the students who are preparing for any technical study in higher
vocational education. Mathematics-A is aiming at those students who are preparing for any
social study, but also at those students who go to work after finishing school.
The first experimental examinations were held in 1989 and 1990.
Although the curricula are different because of the different groups of students they are
aiming at, all three of them are based on the same idea: the realistic approach of mathematics
education. In this approach modelling, applications and problem solving are very important
ingredients. Consequently the assessment of this kind of education must contain aspects of
modelling and applications.
Before turning to the assessment we will first look at some aspects of the education itself.
The general ideas of the realistic approach of mathematics education and the consequences for
the HEWET mathematics-A curriculum are described extensively, respectively by Treffers [3]

2. 46
and De Lange [2]. We will look at the way the ideas of the realistic approach work out in the
classroom.
Mathematization in the Classroom
In the realistic approach mathematization plays a very imponant role. Real world problems are
explored intuitively, resulting in mathematizing that real situation.Imponant activities in this pan
of the process of mathematization are: Organizing, structuring, schematizing and visualizing.
As soon as the problem has been transformed to a mathematical problem it can be attacked
and treated with (more or less advanced) mathematical tools. Keywords of this pan of
mathematization are: representing relations in formulas, proving, refining and adjusting models,
combining and integrating models, formulating a new mathematical concept, generalizing.
A beautiful example of a problem that asks for a number of these activities from students is
the 'Rat-problem', taken from one of the HEWET-booklets. The problem stans with a text from
'Rats', a book written by a well known Dutch novelist Maarten 't Han:
GROWTH OF RAT POPULAnONS. As regards the progeny of one pair of rats
during one year the numbers given vary considerably. In the next chapter I shall
discuss the scanty information supplied by research into the fenility of rats in
nature, but at this point it might be interesting to estimate the number of offspring
produced by one pair under ideal conditions. My estimate will be based on the
following data. The average number of young produced at a binh is six; three out of
those six are females. The period of gestation is twenty one days; lactation also lasts
twenty one days. However, a female may already conceive again during lactation,
she may even conceive again on the very day she dropped her young. To simplify
matters, let the number of days between one litter and the next be fony. If then a
female drops six young on the first of january, she will be able to produce another
six fony days later. The females from the first litter of six will be able to produce
offsprings themselves after a hundred and twenty days. Assuming there will always
be three females in every litter of six, the total number of rats will be 1808 by the
next first ofjanuary, the original pair included.
This number is of course entire fictitious. There will be deaths; mothers may reject
their young; sometimes females are not in heat for a long time. Nevenheless, this
number gives us some idea of the host of rats that may come into being in one
single year.
The simple question on this text:
Is the conclusion that there will be 1808 rats at the end ofthe year correct?
Although the question is simple, the problem turns out to be very difficult.Given as a
homework task no more than 5 or 6 students out of a group of 30 are successful in solving it.
They have to demonstrate and explain their solutions to the other students.

3. 47
Four examples ofwell structured schema:
(A)
(B)
(C)
t
-1
0
1
2
3
4
5
6
7
8
9
0
2
3
4
5
6
7
8
9
N(t)
2
2 6
2 6 6
2
2
2
4
4
total number
8
2
8
2
4
8
146
278
536
974
1808
N(t) _ 2
t - -1 0 1 2 3 4 5 6 7 8 9
adults = 4 7 10 22 43 73 139
(D)
adults young] youngO t number offemales
1 0 0 -1
1 0 3 0 1 + 3 =
1 3 3 1 4 + 3
4 3 3 2 7 + 3 =
7 3 12 3 10 + 3x4 =
10 12 21 4 22 + 3x7 =
22 21 30 5 43 + 3x1O =
43 30 66 6 73 + 3x22 =
73 66 129 7 139 + 3x43 =
139 129 219 8 268 + 3x73 =
268 219 417 9 487 + 3x139 =
Figure 1
I
4
7
10
22
43
73
139
268
487
904

4. 48
When all solutions are put on the blackboard the solutions themselves become subject of study.
It is obvious that only those students who found some kind of structuring, are successful
in finding the complete solution. Another surprising fact is that no two solutions are similar (or
at least look the same). Schema like (C) and (D) are very transparent, because they neglect the
male rats.
Each of these two structures is leading to a further going schematizing.
Schema (C) can be translated into a graph and/or a matrix:
from
Yo Y1
a
Yo
to :: C: )
Figure 2
Refinement of the idealized model (see the text of Maarten 't Hart) is possible by changing the
weights in the graph and in the matrix.
Schema (D) leads to a recurrence relation:
N(t+1) =N(t) + 3 x N(t-2), with initial values: N(-1) = 1, N(O) = 4, N( I) = 7
This relation can easily be put into a computer, with the possibility to look at the growth of
the population for a much longer period. The growth turns out to be exponential.
The Rats problem is not an easy one. It appears at the end of the two years students are
working on mathematics A (pre-university level). But throughout the course there are many
opportunities to become familiar with mathematization activities on (more or less) complex
problems taken from the (more or less) real world.
Mathematization activities are not reserved to real world problems only.
The next problem, taken from a booklet on mathematics B, is a purely mathematical one:
Given the straight lines 1: y =2x - 10 and m: y =10 - O.5x
A is on line 1 and B on line m, so that AB is horizontal and AB = 6
Calculate the coordinates of A and B.
(Hint: let xB = x; express YB, xA and YA in x)
The students who tried to solve the problem using the hint, did not succeed (see figure 3).
They did not understand our formal, algebraic approach, typical for mathematicians!
Those who looked for a solution in their own way, did a very good job. Look at the four
following solutions. Each one is based on knowledge the students already have from
earlier lessons.

5. 5
(a) Algebraic Approach
x
A =0.5y + 5
xB =20 - 2y
49
Figure 3
xB = 2.5y - 15 = 6 ,soJA = 8.4 and x
A = 9.2 ;xB = 3.2
Figure 4
(b) Dynamical approach.
Translation of line lover vector (-6 ,0) gives line 1': y =2x + 2.
The intersection of I' and m is B:
2x + 2 =10 - O.Sx , so it follows that x =3.2 ,etc.
(c) Using the meaning of slope
Using the slopes
I :y=2x -10,soifAy=1thenAx =0.5
m :y = 10 - 0.5 x . If Ay = 1 then Ax =-2
So a vertical step AY = 1 causes a horizontal widening Ax = 2.5
From AB = 6 = Ax it follows that AY = = 2.4
Conclusion: yA =y B =8.4 and xB =8 - 2x2.4 = 3.2
and xA =8 + 0.5x2.4 = 9.2
Figure 5

6. (d) Geometrical solution
h 6 SO· h 36 2 4
"6=T5' . =T5= .
so y = 8.4; and xB follow by
substitution of y = 8.4 in the
formulas
50
6
Figure 6
The HEWET and the HAWEX-project have shown that by the realistic approach of mathematics
education students become good (and sometimes very professional) problem solvers, who are
fully aware of what they are doing and for what purpose. They are critical of the way they are
attacking problems, and also of the way other people do. Very often they surprise their teachers
with more elegant solving strategies than the ones the teachers have in mind.
In order to enable them to arrive at that level , it is very important that every student is
given the opportunity to create and to use his/her own solving strategies, at least until other
strategies (of other students or the teacher's one) are accepted as being more suitable.
A more extensive discussion about these ideas can be found in Gravemeijer [1].
The Assessment
It may be clear that students who have learned mathematics this way, may not be "punished" by
examinations in which only technical and algorithmic skills are tested.
In The Netherlands the final examinations are time restricted written tests. Traditionally the
stated problems are of a very technical kind. The students have to show that they are well
trained in using standard algorithms. This kind of testing only shows what a student does not
know and he is punished for that.
The new curricula are asking for tests in which students can demonstrate they are able
- to choose an appropriate strategy for solving a problem,
- to criticize a given model,
- to integrate different mathematical models.
Part of the HAWEX-project was the description of general goals, knowledge and skills that
can be tested in the final examinations (see Appendix 2).This was not done in the HEWET-
project. It appeared that many teachers did not take these higher goals for granted
Of course very open problems, like the Rat-problem, don't fit in written tests like the final
examinations. Three examples of problems are given to show some possibilities for testing the
stated goals within the final examinations.

7. 51
Example 1. Mathematics-A, Pre-University Level.
In a thesis about juvenile criminality a researcher assumes that 30% of the students at secondary
school occasionally have committed shop-lifting. A headmaster wants to know whether the
percentage of 30 is also true for the 1200 students of his school. Assume that indeed 30% of the
students of this school have shop-lifted.
A random sample of 15 students is taken.
»1. Calculate the probability that at least 5 students in the sample have ever committed
shop-lifting.
Assume that 6 out of a class of 20 students have ever committed shop-lifting.
»2. Calculate the probability that in a random sample of 10 students out of this class,
less than 3 students have ever committed shop-lifting.
A teacher decides to make a thorough investigation by questioning all students of the
school. He knows that in such a study not everybody will tell the truth, so a method must be
used in which it is not always necessary to answer truthfully. He makes use of the following
method:
- every student is asked: "have you ever shop-lifted?";
- before answering the student has to throw a die; the result of the throw remains
unknown to the teacher;
- the question now must be answered in the following way:
ifyou have thrown:
1,2,3 or 4
5
6
then your answer must be:
the truth "yes" or "no"
always "yes"
always "no"
The student is the only one who knows whether the given answer is given by chance or
according to the truth. This method of questioning, known as 'randomized response technique',
makes it possible to draw conclusions by studying all the given answers.
Of the 1200 given answers, 416 were "yes". The teacher estimates that the number of
students that ever committed shop-lifting is 324.
»3. Explain how he arrives at this estimation.
The number of 324 students is obviously smaller than the 360 students you could expect
according to the thesis. Of course the students of this school are not a random sample of all
students in secondary education. Therefore the assumption of the researcher may not be rejected
because of this sample.
>>4. Verify whether or not the assumption ofthe researcher should be rejected in case of
a random sample where exactly 324 students did ever commit shop-lifting. Use a significance
levelof5%.
The randomized response technique is discussed in the mathematics lesson. One student
proposes a much simpler method with the following instruction:

8. 52
ifyou have thrown: then your answermust be:
1,2 or 3 the truth
4, 5 or 6 the opposite to the truth
In this case the privacy ofevery person is also guaranteed, he says.
»5. Is this variant ofthe randomized response technique a useful one?
The students are accustomed to questions like 1,2 and 4. Questions 3 and 5 are highly
original. They have never seen something like that before. These questions really ask for skills
that surpass the level of algorithms and techniques. Just try to solve it for yourself, before
looking at the solutions of students! Students (and teachers) who never are given room to
develop their own strategies for solving problems, will fail on answering this kind of
questioning.Three solutions of students:
(a) P(5) = 116, so 200 persons (=1200/6) always answer "yes".
P(I,2,3,4) = 4/6 so there are 800 answers according to the truth.
416 - 200 = 216 and therefore 216 out of 800 answer "yes", according to the truth.
Conclusion: P ("yes") = 216/800 and so the expected number is 1200 x 216/800=324.
(b) Let X be the number of students who have ever.committed shop-lifting, then:
(5/6)X + (116)(1200 - X) = 416 and that gives X = 324.
(c) A visualisation of the problem in a tree:
1200
800 200 200
yeV'.no Iyes lno
m 200 200
Figure 7
The total number of "yes" is 416, so ? = 216. That means P("yes") = 216/800. The expected
number of 'criminals' will be about 1200 x 216/800 = 324.
Example 2. Mathematics-A, Higher General Education.
One of the exhaust gasses emitted by a car, is carbon-monoxide (CO). The amount of CO (the
so called CO emission) depends on the temperature of the engine and on the driving speed. That
appears from an article in the magazine Verkeerskunde.
The article was illustrated by the graph on figure 8:
The CO emission for a wann engine is given by the formula:
(1) e = 4.4 + 196.0/v e in g/km
v in km/h
»1 The emission decreases when the speed increases. How can you see this in the
formula?

9. 53
I.
II
0---11......" •
• _WiII.......
,,,
! .
- - " .....11,1_1
Figure 8
Assume fonnula (1) may be used for a speed of60 km/h.
»2. Calculate the emission (in glkm)for this speed.
The CO emission for a cold engine is given by the formula:
(2) e =6.9 + 298.5/..
At a certain speed the emission ofa car with a cold engine was 14 gIkm.
»3. Calculate the speed ofthat car.
There also are fonnulas in which the CO emission is given depending on the drive length
and the duration of the drive. For a warm engine this fonnula is:
(3) E = 4.4L + O.054T
E = amount of CO in g, emitted during the drive
L = drive length in km
T = duration of the drive in sec.
»4. Calculate the total CO-emission (in g) for a drive of5 km in 8 minutes with a warm
engine.
»5. Calculate the total CO emissionfor this drive also withformula (1).
»6. Find a formula for the total CO emission E depending on drive length Land
duration T for a cold engine.
Mostly the students on the mathematics-A program of HAVO are very poor 'algebraists'.
Therefore the last question was only meant for the best students, to demonstrate they could do
more than they had leamed in the classroom. We thought the solution had to be something like
this:
formula (2): e.. 6.9 +
298.5
v
. L
" =3600 xT" (from hours to seconds!)
substitution in formula .(2) gives: .. 6.9 +
298.5 T
I =6.9 + 0.083 x r:
'f 3600
E =e xL - 6.9 xL + .0.083 xT
Figure 9

10. 54
Only one student answered the question this way, but there were many solutions to the problem
based on analogy:
(a) wann engine e =4.4 + 196.0/v gives E =4.4L + 0.054T, so
cold engine e =6.9 + 298.5/v gives E =6.9L + 0.082T
"0.082 because 196/0.054 = 3629.6296 and 298.5/3629.6296 = 0.0822398.
I've taken four decimals just like is done in the 'wann' fonnula."
(b) a short answer that shows insight in the structures of the fonnulas:
The fonnula is E = 6.9 L +0.083T
0.083, because you have to divide 298.5 by 3600 (60 x 60 from sec to hours)
(c) a very fine solution, using some information from earlier questions:
(4) E = 6.9L + x
With a cold engine you get e = 6.9 + 298.5/37.5 = 14.86
(he takes L = 5 km, T = 8 min and v = 37.5 km/h from »4)
so: E = 5 x 14.86 = 74.3. Substitution in (4): 74.3 = 6.9 x 5 + x so x = 39.8
39.8/480 = 0.0829 (because 8 x 60 = 480 sec)
Conclusion: E = 6.9L + 0.0829T.
Example 3. Mathematics-B, Higher General Education.
The church tower on the photo will be examined.
The plan of this tower is a square of 6 by 6 meters. The roof consists of four identical
rhombus. The lowest corners of the rhombus are 18 m above the ground. The top of the tower
is 26 m above the ground. The remaining corners of the rhombus are 22 m above the ground,
each on the line of symmetry of the four side walls.
Figure 10
On the worksheet ( see Appendix 3) you see a start of a drawing of the tower in a parallel
projection.
»1. Complete the drawing o/the tower.

11. 55
The quality of the bell ringing depends on the volume of the room in which the bell is
hanging. The floor of this room is 12 m. above the ground. The ceiling of that room can be
built on a height of 20, 22 or 24 m.
»2. Draw theform ofeach ofthese three possible ceilings (scale 1:100).
The ceiling is built on a height of 22 m.
»3. Calculate the volume ofthe room where the bell is hanging.
Space geometry is one of the two main subjects of the mathematics-B curriculum. The
students have to 'use their spatial imagination in an effective way' ( see Appendix 2). They need
that imagination for each of the three questions. Although the questions look very 'closed', the
students haven't learned a standard way to answer them. Consequently there is a large variety
of solutions.
Three examples of answers to question »3:
(a)
skelch, n
Figure 11
Four pyramids are cut off. The volume of one pyramid is
G x h/3 = 4 x ...J18 x ...J4.5 = 12
The volume is V =4 x 12 + 216 =264
(He oversees the volume of the 'square': 18 x 4 = 72).
(b)
4
Figure 12
A nice idea: The four pyramids seem to fill up some space, but the student didn't notice that
the inner space isn't filled up!

12. 56
(c)
Figure 13
Very clear and simple:
The volume of one pyramid is (0.5 x 3 x 3) x 4/3 = 6
So the volume is: V = 6 x 6 x 10 - 4 x 6 = 336
Alternative Tests
Final examinations are not the most appropriate tool to test the so called higher goals of the
curriculum.In the HEWET-project several ways of alternative testing are developed [2].
In the HAWEX-project a new phenomena is introduced: a practical examination.
Sixty mathematics-B students did a practical job on Space geometry,working in couples for
three hours. The concrete object was a lamp, built of 8 half cubes. Some of these half cubes are
rotative, so there are many different shapes possible. Each couple had 10 cardboard half cubes
to use, if necessary. Fourteen problems, starting simple and very complex in the end, had to be
solved. Photos and drawings of the different shapes of the lamp were the source for the
problems they had to solve (figure 14). Of course the couples were allowed to deliberate. They
also were allowed to consult the teacher.At the end of the three hours each couple had to hand in
one set of solutions to the 14 problems.
This way of testing offers some advantages over 'normal' tests.
The students have to cooperate. They must convince each other, for they had to hand in
only one solution.Of course this skill can never be tested in a written test.
The problems can be more complex than the ones in the written test, because the students
can deliberate, while the teacher can lend a helping hand, if necessary.

13. 57
Figure 14
The students did like this way of testing very much: "It's fine that you can talk to
somebody when you feel uncertain about how to attack a problem", "There is no trouble with
stress, the atmosphere is quite calm". The results were very good: only one of the thirty couples
failed.
Some Final Remarks
The realistic approach of mathematics education is strongly process oriented in stead of product
oriented. Therefore the traditional written test is not the best instrument to measure the
achievements of a student. Alternative tests are fitting better to the goals of the curricula of the
HEWET and HAWEX-project. The composition of such tests is not an easy job. Many teachers
in The Netherlands feel incapable of doing it, while others won't take (or have) the time for it.
Therefore this task should be done by professional teams.
From both projects it has become clear that the way students do attack problems differ from
the way most mathematicians do. A mathematician often starts solving a problem by translating
the problem to some algebraic form.Doing this, the problem becomes static. Most students
keep working within the context of the problem, only using (some) algebra when it is not
avoidable. This way of solving problems is much more dynamic. That's why students' solving
strategies often look more elegant, much more simple and straight to the point than the solution
of mathematics teachers. The student solves the stated problem, while the mathematician often
escapes from the uncertainty of the problem into the safety of the mathematical techniques.

14. 58
Appendix 1: A Global Picture of the Dutch School System, and the
Place of the Two Projects within it
university
(4·6years)
18 -1••-,••••..••••,-••-•••••-,••1
vwo
(high level)
HDO
higbervocatiooal
educalioo
(4years)
(middle level)
MDO
intennediate
vocational education
(4years)
mavo/tbo
(lower level)
secondary education
primary education
(age4·12)
Figure 15

15. 59
Appendix 2: Goals, Knowledge and Skills for the Upper Grades of
Havo
Goals:
Mathematics-B:
The curriculum is mainly focused on the use of mathematics in exact sciences.The students
have to have to their disposal algorithmic and geometric skills and have to be able to use them in
mathematical contexts and in geometric, natural scientific, technical and other situations.
Mathematics-A:
The curriculum has a general educational value and is focused on the use of mathematics in
society. The goal of mathematics-A is that students can understand and solve problems from
reality with use of mathematical tools.
Knowledge and Skills:
To achieve the mentioned goal, the students have to learn:
- to analyze problems and to show logical relations between data, statements and
results.(A,B)*)
- to choose a suitable mathematical method to solve a problem and to use algorithms
when solving.these problems.(A,B)
- to use calculators and computer programs when solving problems.(A,B)
- to analyze critically articles from news media with mathematical presentations,
reasoning or calculations.(A)
- to interpret mathematical solutions within the given context.(A)
- to recognize and extract mathematical essentials out of texts.(A)
- to use their spatial imagination in an effective way.(B)
- to combine and to integrate different solving methods.{B)
- to use new concepts or measures in new situations after a short description.{A,B)
- to present the choice of a method, the process of solving and the results conveniently
arranged in words or by use of suitable other representation fonns.{A,B)
*) (A): only for mathematics-A
(B): only for mathematics-B
(A,B): for both mathematics -A and-B
Main subjects
Mathematics B:
- Applied Calculus
- Space Geometry
Mathematics A:
- Tables,Graphs and Fonnulas
- Discrete Mathematics
- Statistics and Probability

16. 60
Appendix 3: Worksheet
Figure 16
References
1. Gravemeijer, K.,van den Heuvel, M., & Streefland, L. : Contexts, Free Productions, Tests and Geometry in
Realistic Mathematics Education. Utrecht: OW&OC 1990.
2. de Lange, J. :Mathematics, insight and meaning. Utrecht: OW&OC 1987
3. Treffers, A. & Goffree, F. : Rational Analysis of Realistic Mathematics Education. In: Proceedings of the
Ninth International Conference for the Psychology of Mathematics Education (L. Streefland, ed.), pp 79-122.
Utrecht: OW&OC 1985