This continues the previous post.
2.4: A Philosophy of Modules
Still, it is instructive to go over the philosophy the few of us who have been doing this have
towards the projects that teach a lot of interdisciplinary
science (each of the above spills out into many areas) and require mathematical
and computational tools to gain insight and illumination, we have
worked hard to find a working philosophy towards their development.
A All parts of the modules must be integrated, so we
do not want to mathematics, science or computer
approaches for their own intrinsic value.
Experts in these separate fields must work hard
to avoid this. This is the time to be generalists
and always look for connective approaches.
B Models must be carefully chosen to illustrate the basic
idea that we know far too much detail about virtually
any biologically based system we can think of. Hence,
we must learn to throw away information in the search of
the appropriate abstraction. The resulting ideas can
then be phrased in terms of mathematics and simulated
or solved with computer based tools. However, the results
are not useful, and must be discarded and the model changed,
if the predictions and illuminating insights we gain from
the model are incorrect. We must always remember that
throwing away information allows for the possibility of
mistakes. This is a hard lesson to learn, but important.
C Models from population biology, genetics, protein interaction, cognitive dysfunction,
regulatory gene circuits and many others are good examples to
work with. All require massive amounts of abstraction
and data pruning to get anywhere, but the illumination payoffs are potentially
quite large. In all of these examples, we must
address fundamental principles for model construction, parametrization, and
validation. However, our enthusiasm for these projects must
be tempered by the understanding that these are students
who are beginners at modeling tasks.
Consider the process by which a given mathematical and biological
model is converted into a UGRE embedded module for deployment into
these quantitative courses. This basic flowchart
is also applicable to such development in Quantitative Biology although that
is specifically a junior level course.
A: Protein Synthesis:
To develop this module, we must ask several important questions:
what science, mathematical and computational background do the students have?
In the case of a protein production model, at the freshman level
we must discuss basic facts about how a gene is converted into an amino acid
chain via the standard mRNA -> tRNA -> protein sequence.
Since the students come into the courses with varying backgrounds, this
is not a trivial exercise. Next, we must decide how much mathematical
background can be assumed and how much must be explained. In the context
of Biology I, prior to or concomittant with Calculus for Biologists exposure, such mathematical training may be minimal and so the textual material that accompanies this set of lectures
must include preliminary mathematical concepts. In Calculus for Biologists, we can assume
the students have seen first order linear models and the tools for their solution
prior to the protein model discussion. Either way, there is a significant
amount of planning that must be done.
B: Kinetic Proofreading:
In this module, we must decide how much the student knows about a simple model
of the ribosome factory. Then, two different models are presented. The first
uses equilibrium analysis to find that the error rate is .01, a hundred times
too large. The tools needed here are algebra and a knowledge of what equilibrium means
but knowing how to solve first order linear differential equations is not necessary.
However, there are sophisticated biological concepts and we must carefully discuss
why we approximate the messiness of the real biology as we do with the model.
To introduce the concept of kinetic proofreading we must alter the ribosome
production model to add intermediate molecular states to the tRNA + Amino Acid
complexes previously used. This new complication allows us to determine the
error rate is actually .0001 which aligns nicely with experimental evidence.
We can do a similar analysis for any sort of module we plan on using. We must determine
A What is the appropriate biology level? Do we need to write
explanatory material to supplement our lecture?
B What is the appropriate mathematical level? Again, do we need
to develop textual materials to supplement our lecture?
C What is the appropriate computational tool level? Do we write
our own simulations in MatLab, Stella or something else?
Do we need to write explanations of the use of MatLab, Stella
and so forth for use within the lecture or the lab?
In our experience so far, it takes a minimum of 4 - 6 weeks of hard work
to develop this material even in preliminary draft form. It then needs to be deployed within a classroom
situation to make sure it works in that context. A typical UGRE module is then covered
within a 2 - 3 lectures or perhaps in a laboratory setting. In Calculus for Biologists, the protein
synthesis module is covered in 2 lectures.
2.5 Current Problems
We had been fairly successful with the freshman level of our
integration plan with enrollments rising from 20 per semester in 2006
to 100 per semester in 2012 - 2013. However, I have not been able
to build a cadre of other faculty to teach the course.
There are many reasons, but some are
A it requires a large new preparation and asks mathematically
trained faculty members to embrace biological concepts.
There are not that many well-trained interdisciplinary members in
my department and as I have mentioned before, there is more
reward for getting involved with engineering interactions.
B there is a lot of anxiety in some of the biological students
who take the freshman Calculus for Biologists course which translates
into a lot of office hours for those students. Not all faculty members
want to do this as they are primarily rewarded for papers and grant
applications. There is no question that being around for students
costs one in the research arena.
C since the class focuses on solving large problems with
many steps, this is completely counter to the scale-up philosophy
used in the other calculus courses for business and engineering.
So it is a big culture shock to approach the teaching this way.
I have been teach two sections of about 40 - 50 each in the Fall and Spring semester
which is 8 credit hours each semester.
As a research faculty member this is over my normal load of 6.
I have also developed the followup course on more calculus for biologists
and to do that I teach a version of it for free to 1 - 2 students each semester.
This enables me to get valuable feedback
from students. I have also do a variety of undergraduate research projects
with biology students. I have never had interest in doing this from
undergraduate mathematics students or graduate students.
As you can see, I am very busy, but none of it
is research oriented in the way my university requires. Hence, I am
not rewarded. Other faculty members see this and it definitely influences
whether or not they will get involved. Now if the biology department had not
made the Calculus for Biologists optional and effectively stopped these efforts,
enrollment might have continued to increase and we would have been forced
to add a third section each semester. Of course, I never did have a firm idea where
I would find an interested person to cover a third class. Finally, even though Biological
Sciences has about 1500 majors, even before the course was made optional
only about 100 of these either were required to take the course or wanted to
take it voluntarily. So covering Calculus for Biologists was possible as long as
I devoted my entire effort to it. If there had been a substantial interest in the
BIO 2010 ideas and all biology majors were required to take this class
as part of their major, we would need about 12 or more sections
per semester to cover the load. That would require significant resources
to be moved to this course and also significant interest from both the biology
department and the mathematics department.
Another huge problem has been that it is virtually impossible for biology
students to fit More Calculus for Biologists and Quantitative Biology
into their schedule. Biology majors just to
make it have a load of about 17 credit hours per semester. Most semesters they
have at least one lab which uses up a 3 - 5 hour block of time.
If you look hard at the possible curriculum ideas we have presented above,
you will see it is very difficult to fit the quantitative emphasis courses
into a schedule already bursting at the seams. So we have
decent success with the mandatory Calculus for Biogists course and virtually
no success with the other courses we wish to add.
A big problem is that even when the Biological Sciences department was
committed to this ( prior to the recent vote making all of this voluntary ), the Mathematical Sciences department
was neutral. It is clear all of the departments in an interdisciplinary
endeavor must be equally engaged and enthusiastic. Since this is not the
case, forward progress is slow and as in the case now, the entire effort can die
if both departments lack interest.
Another problem is the generation of the textual material for the courses.
I write all of this myself and I generate a
pdf document which is published using the Print On Demand (POD)
publishing site www.lulu.com.
This textbook currently places the UGRE modules as chapters in
the text. As new modules are written,
they are easily organized as chapters and I can
rapidly assemble a textbook for a new semester
using the new modules. In
essence, I can mix and match the developed text into books as I see fit.
The textbook for More Calculus for Biologists
(J. Peterson, More Calculus For Biologists: Partial Differential Equation Models,
www.lulu.com/spotlight/GneuralGnome, 2012)
evolved in a similar
way. Another possibility is to divide the Calculus for Biologists and More Calculus for Biologists textual material
into core textbooks (published Print On Demand
via the publishing site www.lulu.com) and UGRE modules published
separately in the same manner. For example, the Calculus for Biologists textbook
currently uses two chapters for the disease and cancer model,
respectively. Those chapters and additional ones on kinetic proofreading
and other possibilities could be managed as a separate book. The students
would then buy the core textbook and whatever version of the UGRE modules
I wish to deploy on a semester by semester basis. The POD
publishing route gives me extraordinary flexibility for my pedagogical needs.
However, it is not clear that anyone else wants to write such material
and so as long as I am interested in doing it, this project continues to
grow. Now, all of these comments are moot as even though I am interested,
there is no interest in the two departments.
Hence, this whole project will diminish.
3. A Discussion of General Education
It is very important to capture freshman and sophomore
interest in their course work and to design the classroom
experiences so that the students who enter the courses stay
for the full semester and learn the material well enough
to achieve a good grade. This is a problem common to all
universities and there are a variety of approaches to
improve what is known as the DFW rate. This is the
percentage of students in a course that either drop,
fail or withdraw. It is a measure, of sorts,
of pedagogical success. This is a very complex
issue and problem and there are a number of tools
which have been applied to improve the DFW rate.
In an effort to help students in
the engineering calculus sequence succeed, many departments at universities
in this country have
initiated the teaching of all
``student-centered, reduced-lecture'' teaching formats. This format is
modeled on North Carolina State's SCALE-UP program for its physics
courses. Briefly, using this format, courses are taught in rooms
containing five round tables, each of which seats nine students.
This limits the size of each section to 45 students. During a
typical 50-minute class meeting, the instructor reviews material from
the previous meeting and then spends between 20 and 25 minutes
discussing the day's instructional objectives, providing examples and
modeling solutions. Finally, the students are divided into groups
of three and given ``learning activities'' that they are expected to
complete before the end of class. To assist these students, the
instructor and two graduate students circulate around the room
offering advice and encouragement. Students are then assigned
homework problems that are collected and graded at the end of the
week.
The objective in this effort is to increase the number
of students who earn an A, B or C in the course.
Our approach to the Calculus for Biologists course does not
follow ``student-centered, reduced-lecture'' format of SCALE-UP as the needs of the class are
different. One of our major goals is to provide an integration of mathematical and
computational tools into the traditional biological sciences curriculum.
Hence, we focus of putting many threads together simultaneously in order to present
whole models. This necessitates longer and more involved lectures
and homework exercises that combine many separate threads into an integrated whole.
The biology students in Calculus for Biologists thus see how the mathematical and computational
tools we ask them to learn will be used in models throughout the entire course.
This integrated approach has had a positive impact
on both the interest of the biology students and the retention of biological sciences students in their major.
3.1 What Is Our Product?
If we design and implement a new interdisciplinary course for first
semester freshman at a major university, we must back up and think hard
about what are assumptions are. There are many questions.
What is our product? I currently teach at a research university and so our
product is very mixed.
In terms of degree programs we have:
A The undergraduate degree which can be terminal so that the student
attempts to enter the work force and begin to assimilate into
society. However, such a student can also feedforward into a graduate program
to find additional training.
B The master's level graduate degree which is terminal with
the graduating student seeking employment. Again, there are students
who use this degree as an entry into the Ph. D. level programs.
C the Ph. D. program which can also be terminal with the student
looking for work in their area. However, even at this level, it can
be used as a stepping stone towards more advanced training.
To obtain a post in a Ph. D. granting institution in the sciences
and many other areas, a student now must first obtain a post-doctoral
appointment for approximately three years to show the proper seasoning
before they can apply for a permanent post. Hence, we have Ph. D. students
feeding forward to Postdoctoral programs.
D The Postdoctoral program in which the student works closely with
an established researcher to get more experience in what constitutes research
in their field.
The way we design course work is different for these targets. We can say similar
things about students who train for professional programs such as that of a medical
or dental school and so forth. After the undergraduate degree, there is medical
school, a residency and so forth before the student is fully vetted. In terms of time,
these long training programs are similar to those in guilds where a beginner apprenticed
to a master craftsman in the chosen area. After approximately 10 - 15 years of
work, the apprentice would be certified a master and would then be able to open a business
of their own. However, guild apprenticeship is also very different in that there is one
master who does the teaching and in a modern university, there are dozens involved in the
student's training; postdoctoral training with one mentor is, of course, closer to the spirit of
the guild apprenticeship.
To do a good job with our freshman interdisciplinary courses, we therefore need
to understand just what our product should be. A common thread throughout the education
process is the idea that we inculcate in our students the ability to think and reason
in new situations. This means we challenge them to help them grow intellectually.
But challenge inevitably means a higher risk of getting less than an A in a course.
In today's climate, freshman are already worried about getting into law and medical school.
We routinely see people drop courses now just because they are working at a B level.
As a freshman, they feel that one B will be the thing that keeps them from medical school.
So many of the freshman do not want intellectual challenge; instead, they want
a painless (i.e. not too much homework, not too many projects etc.) path to that A.
Nevertheless, we believe our product is a student who can understand the consequences of assumptions.
We all build models of the world around us whether we use mathematical, psychological,
political, biological and so forth tools to do this. All such models
have built in assumptions and we must train our students to think for themselves.
They must question the abstractions of the messiness of reality that led to the model
and be prepared to adjust the modeling process if the world they experience is different
from what the model leads them to expect. There are three primary sources of error when we
build models:
A The error we make when we abstract from reality; we make choices about
which things we are measuring are important. We make further choices about
how these things relate to one another. Perhaps we model this with mathematics,
diagrams, words etc; whatever we choose to do, their is error we make.
This is called Model error.
B The error we make when we use computational tools to solve our abstract models.
This error arises because we typically must replace the model we came up with
in the first step with an approximate model that we can be implemented on a computer.
This is called Truncation error.
C The last error is the one we make because we can not store numbers exactly
in any computer system. Hence, their is always a loss of accuracy because of this.
This is called Round Off error.
All three of these errors are always present and so the question is how do we
know the solutions our models suggest relate to the real world?
We must take the modeling results and go back to original data
to make sure the model has relevance.
We agree with the original financial modeler's manifesto
(E. Derman and P. Wilmott. ”The Financial Model-
ers’ Manifesto”. Social Science Research Network, 2009) which takes
the shape of the following Hippocratic oath. We have changed
one small thing in the list of oaths. In the second item,
we have replaced the original word value by the more
generic term variables of interest which is a better fit
for our interests.
A I will remember that I didn't make the world, and it doesn't
satisfy my equations.
B Though I will use models boldly to estimate
variables of interest, I will not be overly impressed by mathematics.
C I will never sacrifice reality for elegance without
explaining why I have done so.
D Nor will I give the people who use my model false comfort
about its accuracy. Instead, I will make explicit its assumptions
and oversights.
E I understand that my work may have enormous
effects on society and the economy, many of them beyond my
comprehension
There is much food for thought in the above lines and all of us
who strive to develop models should remember them. We are all aware of how
poor assumptions in financial models led to our current ruinous state.
Indeed in our own work, many times managers and others who oversee what we are doing
have wanted the false comfort the oaths warn against. We should
always be mindful not to give in to these demands and we must train our students
to think deeply about assumptions at all times.
In addition, we must find a way to get all the students intellectually engaged in
our interdisciplinary course. In traditional engineering mathematics courses, students
are eventually going on to other courses that need all the things we are teaching.
The engineering, physics and other quantitative science students do not mind
learning this material even though they won't use it in their own problem domain
for several years. The biology students are not like this at all. They didn't
want to take the second semester of Calculus for engineers because they knew
they would never use this material again. Despite all of our effort in the design
and implementation of the Calculus for Biologists course, this is still true.
Once the students take Calculus for Biologists, they are no followup courses
other than the ones we have designed in
biology that use the material. Hence, the biology majors do not see this material
used on a regular basis. Instead, they see a dead end and a real impediment to their
medical school application. An interdisciplinary course must somehow
have relevance in addition to challenge in order to keep our students involved.
A good example is from the world of Processing
(C. Reas and B. Fry. Processing: A Programming
Handbook for Visual Designers And Artists. MIT Press, 2007).
The artists Reas and Fry developed a programming language for artists
starting about 10 years ago which is currently used by about 200,000 people
(at least) over the world. Reas and Fry were not classically trained mathematicians
or computer scientists, but they felt there was a lack of computer tools that
could be applied to the artistic process. So they implemented Processing.
Moreover, Processing allows anyone to design and build hardware
for use in art easily within the program (see the Arduino platform
(M. Banzi. Getting Started With Arduino. O’Reilly Media, 2008)
and ideas on programming interactivity
(J. Noble. Programming Interactivity. O’Reilly Media, 2009).
They teach semester courses using these tools in which the students in the class
learn the stuff they need to build art and then actually do art in the second
phase of the class. The thing is that the students are intellectually engaged.
If Reas and Fry can capture the interest of art students who usually don't care for
quantitative tools such as programming languages, mathematics and so forth, we feel
we can do this also.
Hence, the interdisciplinary course we field must apply what they learn to build
something. It can be hardware, a software model or a probing discussion of
ideas on the edge of what we know. We feel this is essential to grabbing and maintaining
student interest. After all, we want to more students take this one course.
So it has to mean something to them.
Finally, I must note that
for these interdisciplinary courses with a science, mathematical
and computational component, there will probably be about 3 - 5%
of the students with high levels of anxiety and other learning
issues. This is about 3 - 5 students who need a lot of office hours
each semester. Now I have 8 formal office hours for my two classes
of 40 per week. These 5 students, if they are actively seeking help,
can use up as many as 4 - 5 of those hours by themselves.
In addition, they may need to meet at times other than designated
hours which breaks up the day sometimes badly.
Also, when I use computational tools, they always confuse a portion of
the class and I have to allocate a lot of personal time to help them.
I need to go to the library or other computer areas and sit with them
and show them how to get started etc. This can easily be 20% of the classes
or about 16 students. Of course, this does settle down, but some weeks it can
get intense.
Finally, there is the emotional exhaustion that comes from helping students
who are upset, crying and so forth. I just try to be kind and attentive, but
it does rattle your own psyche a bit and it wears you down over the week.
So there is indeed a psychological cost to mentoring and helping the students
with anxieties of various kinds. Still, I think it is very valuable.
My university does have special facilities set up for this sort of thing,
but it is very impersonal and generally, the students will not go.
They would rather talk to their teacher. Another issue
is that it is often difficult to get the students who need help to come to your
office.
