Meta aspects of science in problem based project work
- experiences from Roskilde University
Tinne Hoff Kjeldsen
IMFUFA, Department of Science, Systems, and Models, Roskilde University
The meta-perspectives of mathematics and science form an integrated part of the two-year
introductory study programme in the natural sciences at Roskilde University. This programme is
characterised by its problem based, interdisciplinary project work. Meta-perspectives are taught
through the ‘reflection-project’, where students experience science as a cultural and social
phenomenon by asking and investigating meta-questions. The paper illustrates how this project
work can unite students’ meta-reflections about science with their learning of important subject-
matter. This is exemplified by means of a project in which a group of students investigated the
premises for a discussion between the physicist Nicolas Rashevsky and some biologists in 1934
about the usefulness of mathematical methods to explain cell division.
Discussions and problems using history in science and mathematics educations
Science and mathematics is generally known to be taught in a decontextualized manner with focus
on a narrowly defined content-driven curriculum of concepts and methods. Such teaching conveys
an image of science and mathematics that echoes the reputation of modern mathematics formulated
by Chandler Davis (1994, 132): “Most 20th –century mathematicians talk as if they had a subject-
matter outside of time and space. No wonder they seem snooty to others!”
Several problems with this portrait of science and mathematics have been discussed in the literature
in science and mathematics education: it de-humanises science and mathematics, it does not convey
the energy and life of scientific research and discussions, it lack motivational power. These
problems are often pointed out as the main reasons why students are inclined to think that science
and mathematics are boring and irrelevant, even though they acknowledge the importance of these
subjects in modern societies.1
History, or more generally, meta-aspects of science and mathematics has often been proposed in the
literature as having the potential to remedy some of these shortcomings by allowing students to see
another more human face of science and mathematics. Through its history the subject-matter of
science and mathematics get rooted in time and space to the work of particular scientists and
mathematicians – once real living persons – who themselves struggled to understand and develop
science and mathematics. Besides this, other reasons for including history have been given (1) as a
didactical means to motivate and to learn the content of science and mathematics, and (2) for its
own sake as an integrated part of science and mathematics education. Regarding (1), a problem
raised is that in order to be effective as a didactical tool history is often distorted and with respect to
(2), the problem is how to connect history of science with its content without reducing the history to
See (Klassen, 2006).
Meta-aspects of science in problem oriented project work
As described in the paper by Blomhøj in this volume one of the three themes for the project work in
the two-year introductory study programme in science at Roskilde University is devoted to
reflections on meta-aspects of science and mathematics. Together with the theme concerning the
application and functioning of science in technology and society, and the theme about the interplay
between theories, models, and experiments in the sciences this ‘reflection project’ serves the
purpose of exposing students to different viewpoints and mind-sets in and to science and
mathematics. The three themes constraining the students’ project work during the first three
semesters of their science studies constitute an ambition of perspective many-sidedness.
The purpose of the ‘reflection-project’, covering half of the students study time in the third semester
of the programme (15 ECTS), is to teach students about meta-aspects of science and mathematics
not as a unit disconnected from the content of science and mathematics but as an integrated part.
Students often choose to work with problems relating to the history and/or philosophy of science
and mathematics in their ‘reflection project’. Our experiences from over 25 years with this model
show that it is possible to unite the history of science and mathematics with its content in such a
way that the students learn subject-matters related to both areas.
The ‘reflection-project’ Rashevsky’s pride and prejudice
In the following I will illustrate how the students who carried out the ‘reflection project’
Rashevsky’s Pride and Prejudice learned specific subject-matters of physics, mathematics, and
mathematical modelling and used this to get insights into history and philosophy of science and
mathematics in a way that was neither distorting nor anecdotal.
The project was suggested by the author, who also supervised it, during a general presentation of
the ideas behind the ‘reflection-project’ as a problem-area that had the potential to develop into a
‘good’ problem for a meta-project. The background was a talk given by the physicist Nicholas
Rashevsky where he presented a mathematical model as an explanation of cell division for an
assembly of biologists in 1934. The talk was followed by a critical discussion where the biologists
questioned the validity of Rashevsky’s results, a discussion that indicated that what counts as a
legitimate argument may be different in different scientific disciplines.
The problem-area appealed to the students because it related to an experience they had had the
previous semester at the presentation of the theme for the second-semester project. Here the
students had noticed, with some frustrations, that the professors who had been assigned as
supervisors for the projects didn’t seem to agree on what should be understood as the method of
science. On a decontextualized level they all seemed to share a mutual understanding of the terms
experiments, models, hypotheses, and theories and the relationship between them, but when the
discussion addressed specific methodological issues in the various disciplines, disagreements
among professors from different scientific disciplines surfaced. The students were still puzzled
about this experience from the previous term.
On this background the students asked the following questions to Rashevsky’s work (Andersen et
al., 2003, 2):
Why was Rashevsky unable to get through to the biologists of his time with his ideas?
Was it because the biologists could not accept Rashevsky’s scientific method?
If so, was this then caused by a fundamental difference in biologists and physicists
conception of biology and were/are controversies about the scientific method then a
manifestation of this difference?
To understand the biologists’ objections the students needed to understand the scientific content of
Rashevsky’s talk which was a major challenge.
Rashevsky believed that cell division could be explained by physical forces by way of deduction,
that cell division followed ‘logically and mathematically from a set of well defined principles’. He
argued that the problem of the complexity of a cell could be ignored because ‘we set out to
investigate the consequences of general principles’ so all properties not common to all cells could
be disregarded. He chose the process of metabolism as the most obvious property shared by all
cells, mathematized metabolism with the equation for diffusion and build a mathematical model to
explain cell division.
Without further explanation Rashevsky wrote the diffusion equation for steady state with respect to
D!2 c = q( x, y, z )
where D denotes the coefficient of diffusion, !2 the Laplace operator, c the concentration, and
q( x, y, z ) the rate of consumption of the substance in question at the location (x, y, z).2
Rashevsky then analysed what kinds of forces a metabolizing system produces. Besides osmotic
forces he considered forces of attraction between the molecules of the solvent and those of the
solute. In order to estimate these forces, Rashevsky considered the force exerted on a molecule (A)
of the solvent by all molecules (B) of the solute. To illustrate the work of the students I paraphrase
the explanation they gave in their report of how Rashevsky calculated this force:
Rashevsky has a figure that should help to calculate the total force exerted on A
(figure 1). We have made our own figures which in our opinion gives a better
understanding of what Rashevsky wants to show. The goal for Rashevsky is to find all
the forces of attraction exerted on a single molecule A by all the molecules B. … He
assumes that the A molecule is placed in the origin of the coordinate system:
First, Rashevsky finds all the forces exerted on A by all B-molecules lying between
two vertical planes through x and x+ ! x and bounded by the two cylindrical surfaces
r’ and r’+ ! r’ (figure 2).
(Rashevsky, 1934, 189).
Figure 1 (Rashevsky, 1934) Figure 2 (Andersen et al., 2003)
The B-molecules lying in this element of volume exert a force on A in the x-direction
that can be written as:
Nc( x) 2!Kr ' dr ' dx x N 2!Kc( x) xdxdr
M rn r M rn
Rashevsky does not explain how he reaches this but we have reached it in the
following way: The size of the element of volume lying between the planes x and
x+ ! x and the two cylindrical surfaces r’ and r’+ ! r’ is
dv = dr ' dx 2!r '
dr' dx is the area of the (infinitesimal) rectangle, 2!r ' the circumference of the circle
formed by rotation around the x-axis. The number of B-molecules in this volume is
dr ' dx 2!r '
N is Avogadro’s number, M is the molecular weight, and c(x) is the concentration.
Rashevsky has made the assumption that the concentration only varies along the x-
axis. The force exerted by these B-molecules on the A-molecule is
Nc ( x) K
Fres = 2!r ' dr ' dx
where we have multiplied with the force exerted by a single molecule K/r n, n being a
number between 5 and 8. Rashevsky has this number from a source that we couldn’t
find. We have now found the total force exerted on A by all the B-molecules lying in
dv. Rashevsky wants to find the force in the positive x-direction. We project Fres on
the x-axis and get Fres_x = Fres cos v, where v is the angle between the x-axis and r.
Since cos v = x/r we get
x Nc ( x) 2!Kr ' dr ' dx' x
Fres _ x = Fres =
r M rn r
If we prolong r with dr a triangle can be drawn that is similar to triangle ABx (figure
Figure 3 (Andersen et al., 2003)
From this we get
dr r '
= and thereby rdr =´r ' dr '
dr ' r
Hence, we have reached Rashevsky’s result (*):
Nc( x) 2!Kr ' dr ' dx x N 2!Kc( x) xdxdr
M rn r M rn
From here Rashevsky calculates the force exerted on A by all B-molecules lying
between the planes x and x+ ! x. Since this is the sum of the forces form the B-
molecules, that lie in the distance r = x to the distance r = infinity from A it can be
calculated by integrating Fres_x :
N dr 2$NKc ( x)dx
2$Kc ( x) xdx #r n
M (n ! 1) x n!2
By integrating this from x = d to x = infinity Rashevsky get the force exerted on A by
all B-molecules lying to the right of x = 0. (d is the smallest distance between two
molecules). In a similar way I calculated the force from the B-molecules to the left
from x = 0.3
(Andersen et al., 2003, 9-18).
In the end Rashevsky reached an expression for the forces per unit volume produced in a cell by a
gradient of concentration. He then calculated the effects of those forces in a spherical, homogenous
cell. He was well aware that this was a ‘highly idealized’ system the results of which would ‘hardly
apply with any degree of precision to actual cells’ but would give ‘a general qualitative picture’.
Rashevsky’s talk was followed by critical comments and objections one of the problems being that
‘spherical cells isn’t the commonest form of cell’, and as one of the biologists asked:4
I realize that the foregoing objections do not invalidate the qualitative conclusion that
a dividing cell liberates energy, but would not these objections throw doubts upon the
validity of the mathematical equations?
In their report the students also compared Rashevsky’s foundational understanding of biology with
the understandings of the biologists and they concluded that
Rashevsky did not view the foundation of biology to be any different than that of
physics. He believed that constituent principles could be found in biology from which
all biological phenomena could be explained. He was of the opinion that those
principles were to be found in physics. … The biologists were opposed to his ideas
because they held the opinion that there exist phenomena in biology that cannot be
explained by physics.
The students concluded that the biologists disagreed with Rashevsky’s use of the mathematical
model to investigate cell-division because they questioned the applicability of the mathematical
method as a valid argument in biology. They concluded that the cause of the disagreement was
rooted in different views of the epistemological structure of biology. Their conclusions were based
on an analysis of Rashevsky’s model, historical studies into the ideas of biology in the 1930s and
the role of mathematics in biology up to that time, and finally on philosophical studies about
ontological, epistemological, and methodological reductionism and antireductionism, and the notion
As demonstrated, in the third semester project Rashevsky’s Pride and Prejudice the students’
reflections about meta-aspects of science were united with their learning of important subject-
matter. Through the reading of Rashevsky’s original paper the students learned about the Laplace
operator and they derived the general diffusion equation in physics in order to understand the
scientific content of Rashevsky’s paper. They got ‘hands on’ experiences with the mathematization
process of calculating for a small section and then adding up by integration which enhanced their
understanding of the concept of the integral and added to their modelling competencies on the
mathematization level. The students’ analysis of the discussion between Rashevsky and the
biologists supported their modelling competencies regarding interpretation and validity of the
results produced by a model.
Regarding the meta-aspects, the students engaged in reflections about what a valid argument means
in physics/mathematics and biology respectively. Through historical investigations of cell biology
in the 1930s and the role of mathematical modelling in biology at that time they were able to
(Rashevsky, 1934, 196).
understand the discussion that took place. With the help of (modern) philosophy of science they
uncovered (some) of the causes behind the conflict. During this project work the students
experienced a growing consciousness about history and philosophy of science that threw light on
and enlarged their understanding, tolerance and insights into the communication problems that can
be observed today between physicists, mathematicians, and biologists about ‘proper’ methods to
gain scientific knowledge.
One of the reasons why the third semester project at the introductory science programme at
Roskilde University provides opportunities for students to reflect on history and philosophy of
science in a neither distorting nor anecdotal fashion while learning scientific subject-matter is
because the point of departure for the projects are problems that are rooted in science and
mathematics and not a predetermined curriculum.
Andersen, L. D., Jørgensen, D. R., Larsen, L. F. and Pedersen, M. L.: Rashevsky’s Pride and
Prejudice, report, 3rd semester, Nat-Bas, Roskilde University, 2003. Danish.
Davis, C.: ‘Where Did Twentieth-Century Mathematics Go Wrong?’, in S Chikara, S Mitsuo, and J
W Dauben (ed), The Intersection of History and Mathematics. Basel, Boston, Berlin: Birkhäuser,
Klassen, S.: ‘A Theoretical Framework for Contextual Science Teaching’, Interchange, 37/1-2, 31-
Rashevsky, N.: ‘Physico-Mathematical Aspects of Cellular Multiplication and Development’, Cold
Spring Harbor Symposia on Quantitative Biology, II, Cold Spring Harbor, Long Island, New York,