September 2, 2021.  Still pandemic!  Still isolating but now trusted friends gave us the gift of covid variant delta.  Fun times and we are in the middle of it.

Pandemic!

June 27, 2020

It's been over three months since my university went to online teaching for the rest of the Spring semester, classes ended and I started on finishing a book contract and a research grant.  On track to retire at the end of July!  But nowhere to go.  I think it will be at least 1 1/2 years before my wife and I can go out again!

I am working on a series of new books on analysis.  I received the contract from
CRC around January 2017.  I am always under a lot of pressure to perform at a high level
and so in the Summer of 2016 I decided to finally try to do something about my many notes on
analysis.  I had been moving them into typed form anyway and I wanted to be able to say
I had finished them before I retired.  I had just turned 65 and so it was time.  I also try to
find outside agencies to fund my research when I think there is a good match.  I don't pretend
to do work in some area just to get a funding nod so I don't always find a reasonable place to send
my ideas.  Well, one popped up in the Summer of 2016 and by the Summer of 2017 I had
a new three year grant for my work on complex systems (brain, immune system and path planning)
as well as a big book contract for five books.  So I have been busy with all of this
and I put off retirement until my grant runs out in July 2020 although I still have ideas
I am working on so I am thinking I might try to get another grant and stay longer.  We'll see.

Anyway, the editor of the analysis book series says I can use my own paintings for covers
so in the Summer of 2017 over a period of about 6 weeks I painted the five I wanted to use.
They all tell a story of how I helped train new generations of cephalopods to learn mathematics.
This post I will go over the first two books in the analysis sequence.

The first one is

The cephalopods wanted to learn advanced mathematics and decided to contact Jim.

and the book title is "Basic Analysis I: Functions of a Real Variable".  I study basic analysis in this text.

This is the analysis a mathematics major or

a student from another discipline who needs this

background should learn in their early years.

Learning how to think in this way changes one

for life.  From this point on, the reader will always know how to look carefully at

assumptions and parse their consequences.

That is a set of tools one can use for their own profit

in the future.  In this text, the reader learns a lot about how

functions which map numbers to numbers behave

in the context of the usual ideas from calculus such

as limits, continuity, differentiation and integration.  However, these ideas

are firmly rooted in the special properties of the numbers themselves

and we take great pains in this set of notes to place these ideas

into the wider world of mappings on sets of objects.

This is done anecdotally as we cannot study those things

in careful detail yet but we want you exposed to more general things.

The text also covers ideas from sequences of functions, series of functions

and Fourier Series.

This is a critical course in the use of abstraction and it is just one

in a sequence of courses which prepare students to become

practicing scientists.  It is important to balance the theory and abstraction

with clear explanation and argument so that students who are from

many areas can follow this text and use it profitably for self study

even if they can not take this material as a course.  Many professionals

need to add this sort of training to their toolkit later and

they will do that if the text is accessible.  

Presentation lectures are available, along with sample exams and so forth, at

Advanced Calculus One and Advanced Calculus Two.  This book is for a two semester

course.

The First Semester:

This semester is designed to cover through the consequences of

differentiation for functions of one variable and to cover

the basics leading up to extremal theory for functions of two variables.

Along the way, a lot of attention is paid to developing the theory

underlying these ideas properly.  The usual introductory chapters

on the real line, basic logic and set theory are not covered as

students have seen that before and frankly are bored to see it again.

The style here is to explain very carefully with many examples

and to always leave pointers to higher level concepts that

would be covered in the other primers.

The study of convex functions and lower semicontinuity

are also introduced so that students can see

there are alternate types of smoothness that are useful.

The number e is developed and all of its properties from the

sequence definition $e = \lim_n( 1 + 1/n)^n$.  Since ideas from

sequences of functions and series are not yet known, all of

the properties of the exponential and logarithm function

must be explained using limit approaches which helps tie

together all of the topics that have been discussed.

The pointer to the future here is that the exponential

and logarithm function are developed also in the second

semester by defining the logarithm as a Riemann integral

and all the properties are proven using other sorts of tools.

It is good for the students to see alternate pathways.

Note that this text eschews the development of these ideas

in a metric space setting although we do talk about metrics and norms

as appropriate.  It is very important to develop

the derivative and its consequences on the real line

and while there is a simplicity and economy of expression

if convergence and so on is handled using a general metric,

the proper study of differentiation in $\Re^n$ is not as amenable

to the metric space choice of exposition,  That sort

of discussion is done in a later primer.

Chapters 1 - 14 are the first semester of undergraduate analysis.

The Second Semester:

The second half of this primer is about developing the theory of

Riemann Integration and sequences and series of functions carefully.

Also, the students horizons are expanded a bit by showing them

some basic topology in two D and three D and revisiting sequential

and topological compactness in these higher dimensions.  This ties

in well with the last chapter of semester one.  These ideas are used

at the end to prove the pointwise convergence of the Fourier Series of a function

which is a great application of these ideas.    In the text, Chapters 15 - 28 are for the second semester.

The current table of contents is given below as snapshots from the book pdf.

The next book uses this cover:

The cephalopods were eager to learn.  Now both squid and octopi are coming to the lectures.

and the title is "Basic Analysis II: A Modern Calculus in Many Variables".  This book introduces

more ideas from multidimensional calculus.

In addition to classical approaches, I also discuss

things like one forms and 

a reasonable coverage of one of the multidimensional versions

of the one variable Fundamental Theorem of Calculus ideas.

I also continues your training

in the abstract way of looking at the world.  I feel that is a most important

skill to have when your life's work will involve quantitative

modeling to gain insight into the real world.

The first text essentially discusses the abstract concepts underlying the study of calculus on

the real line.  A few higher dimensional concepts are touched on such as the development

of rudimentary topology in two D and three D, compactness and the tests

for extrema for functions of two variables, but that is not a proper study

of calculus concepts in two or more variables.  A full discussion of the  n dimenisional

based calculus is quite complex and even this second primer can not cover all the

important things.  The chosen focus here is on differentiation in n dimensions

and important concepts about mappings from n dimensions to m dimensions such

as the inverse and implicit function theorem and change of variable

formulae for multidimensional integration.  These topics alone require

much discussion and setup.  These topics intersect nicely with

many other important applied and theoretical areas which are no longer

covered in mathematical science curricula.  The knowledge here allows

a quantitatively inclined person to more properly develop multivariable

nonlinear ODE models for themselves and physicists to learn the proper background to

study differential geometry and manifolds among many other applications.

However, this course is just not taught at all anymore.  It is material

that students at all levels must figure out on their own.  Most of the

textbooks here are extremely terse and hard to follow as they assume

a lot of abstract sophistication from their readers.  This text is designed

to be a self - study guide to this material.  It is also designed

to be taught from, but at my institution, it would be very difficult to

find the requisite 10 students to register so that the course could be taught.

Students who are coming in as Master's Students generally do not have

the ability to allocate a semester course like this to learn this material.

Instead, even if they have a solid introduction from the the first course on analysis,

they typically jump to what is done in the third book, which is an introduction to 

very abstract concepts such as metric spaces, normed linear spaces

and inner products spaces along with many other needed deeper ideas.

Such a transition is always problematic and the student is always

trying to catch up on the wholes in their background.

Also, many multivariable concepts and the associated theory

are used in probability, operations research and optimization

to name a few.  In those courses, n dimensional based ideas

must be introduced and used despite the students not having

a solid foundational course in such things. Hence, good students

are reading about this themselves.  This primer is intended to give

them a reasonable book to read and study from.

There is also provide an introduction to winding numbers for sets in two D

which uses line integrals and Green's Theorem in the plane.

New models of signals applied to a complex system use these

sorts of ideas and it is a nice way to plant such a pointer to future things.

Presentation lectures will be available for this course although not written yet.

Hence, this text would be a primary text for a senior year course that follows the first course

in  basic analysis.  However, this course is not taught very often now,

so it is designed for self-study and recommended as the appropriate primer

for anyone whose work requires that they begin to assimilate more

abstract mathematical concepts as part of their professional growth after graduation

or as a supplement to deficiencies in undergraduate preparation that

leaves them unprepared for the jump to the first graduate level analysis course,

which is book three.

A rough table of contents is this:

First the ideas of compactness are extended to two D and three D

along with basic ideas of convergence and continuity.  Hence, the notion

of the consequences due to continuity on a compact set emerge early.

There is an extensive study of mappings from

n dimensions  to m dimensions culminating in proofs of the implicit and inverse

function theorem.  There is therefore a complete discussion of differentiation in n dimensions

to allow the proper development of these ideas.  

There is reasonably deeper look at extremals in n dimensions which castes

extremal conditions in terms of positive definiteness and matrix minors

which is not usually discussed but which is very important.

Riemann integration is then developed in two D and higher dimensions

and characterization of Riemann integrable functions in terms

of their sets of discontinuities is included.  This allows the important ideas

of sets of measure zero to be broached which is a nice pointer to

the text in measure theory in book four.  The text finishes with

a good discussion of winding numbers, line integrals and 

Fubini Theorem type results and change of variables

formulae.

The extensions of the Fundamental Theorem of Calculus

to two D and three D, surface integrals and the like are not discussed with the exception of Green's Theorem.   In many ways,  material  of that nature should be covered using ideas from differential geometry and manifold theory

and more general ideas about integration.  

The current table of contents is this (although still in flux):

That's all for today.

I have done a number of paintings over the years which I use in my lecture notes.
Some of my lecture notes have turned into books and I try to have these paintings
either as covers for the books or as insert pages.  The earlier paintings are done in acrylics
and the later ones are done in water based oils.

The Astronautical Years:

This is my Jupiter painting which was done in 1981 for my wife Pauli.

I am using this as the cover for the first volume of my notes on programming for complex models.
The title is "Complex Models On Graph Based Topological Spaces IIA: Basic Algorithms
and Their Implementation in Fortran, C and C++" and I'll be putting it on Lulu soon.
Here, I talk about learning to program in Fortran, C ( to properly introduce pointers) and then C++ to begin a study of class organization.  

This is my Saturn painting which I think was done in 9183 or 1984.

I am using this as the cover for the second volume of my notes on programming for complex models.
The title is "Complex Models On Graph Based Topological Spaces IIB: The Implementation of Neural Codes in MatLab, C and C++".  This book is an
attempt to show students with an interdisciplinary focus how to build implementations of a number of
neural codes.  I begin with standard versions of ODE solving code in both C and C++ and discuss solving Hodgkin - Huxley models.   Then I show how to implement Fast Fourier Transform ideas to make polynomial multiplication more efficient. These codes are just to warm the reader up to the neural code challenge. I implement code for the chain network and lagged network also.
All of this will prepares the reader for the even more sophisticated task of implementing
general graph models of networks.

This is my asteroid painting also done in probably 1984 or so. It shows a supernova burst which highlights an asteroid belt.

This book is the third of a collection of books which try to
train students and others how to program complex models
using first principles.  This one is called "Complex Models On Graph Based Topological Spaces IIC:
Factoring, Graphs and Basic Visualization". Previously, the first volume discussed
Fortran, C and C++ programming and the second volume
implemented a variety of more complicated things culminating
in codes for neural models.  This volume adds visualization
into the mix as learning about client - server event loops and
how they complicated programming really stretches one's
abilities.  It is the first time the reader really begins to think asynchronously! I also start the process of teaching you about dynamic binding
more carefully.  The graphical models lend themselves to this nicely.

This next image is not from a painting of mine. It is a photograph of art performance my son Quinn gave at the coffee house he was working at after he graduated from college. I cut out the outline of the character Schmoo you see from plywood, Quinn painted it and we transported it to the coffee house.  Customers  then worked on art and placed their finished pieces on the wall behind Schmoo. It was a lot of fun!

You can see all of his work at  The Art of Quinn Peterson.

This book is the fourth of a collection of books which try to
train students and others how to program complex models
using first principles.  It is called "Complex Models On Graph Based Topological Spaces IID:\\
Dynamic Programming, Visualization and Complexity Modeling".
Previously, the first volume discussed Fortran, C and C++ programming and the second volume
implemented a variety of more complicated things culminating in codes for neural models.  The third volume started discussing tree and graph code and the beginnings of visualization. This volume adds additional visualization tools into the mix as the reader continues to learn about client - server event loops and how this complicated programming really stretches your abilities.  It is the second time the reader really begins to think asynchronously as I started on that journey in volume three. I also continue start teaching about dynamic binding and the design of code for graphs of computational nodes and edges.  We also explicitly
discuss dynamic programming code with and without visualization components.

Just a hello after a long five years.

Thoughts On Designing Interdisciplinary Courses III

Hi all,

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. 

Thoughts On Designing Interdisciplinary Courses II

Hi all,

This continues the previous post.


2. Interdisciplinary Science

A good example of how many problems there are in trying to
build better freshman and sophomore courses that address
integration of material is to look at
my attempts to increase the integration of biology, mathematics and
computation in the training of new biological scientists.
This has made me focus on the much deeper questions of
how to train students to think in an interdisciplinary manner
not only in the biological sciences but also as part of a new
general educational curriculum.
I think it is clear that to do this, we will have to foster a new
mindset within both the students and the faculty related to interdisciplinary
interaction.  In addition, we will have to introduce new metrics for faculty
evaluation and reward systems to ensure that the hard work
and creative effort to build such a curriculum is acknowledged.
This means also that new discussions of the tenure process are inevitable.
The following discussion will add some meat to the bones of this tale.
Let's focus on the interdisciplinary triad of ideas from
Biology, Mathematics and  Computational tools
which for expositional convenience, I will denote by the symbol
BMC.  The implementation of BMC will require the building
of bridges between mathematical, computer and biological sciences departments.
For convenience, let's call this integrative point of view Building Biological Bridges
or B3.


The typical biological sciences major thus must have a deeper
appreciation of the use of BMC as bridges are built
between the many disparate areas of biology and the other sciences.
The sharp disciplinary walls that have been built in academia hurt our
students chances at developing an active and questing mind that is able
to both be suspicious of the status - quo and also have the tools
to challenge it effectively.  Indeed, I have longed believed that
all research requires a rebellious mind.  If a student revers
the expert opinion of others too much, they will always be afraid
to forge a new path for themselves.  So respect and disrespect are both part of
the toolkit of our budding scientists.
Blending many disciplines into one program, even when there are good reasons to
do so, is very hard.


2.1 Astrobiology As Metaphor For Biomath

Consider the
attempt to create a new Astrobiology program at the University of Washington.
Even a cursory examination of the new textbook
Planets and Life: The Emerging Science of Astrobiology
(W. Sullivan and J. Baross, editors. Planets and Life:
The Emerging Science of Astrobiology. Cambridge
University Press, 2007.)
illustrates the wealth of knowledge such a field
must integrate.  This integration is held back by students who are
not trained in both BMC and B3.
Let's paraphrase some of the important points made by the graduate
students enrolled in this new program about interdisciplinary training
and research. Consider what the graduate students in this new
program have said ( page 548 in Sullivan's book )

`` ...some of the ignorance exposed by astrobiological
questions reveals not the boundaries of scientific knowledge,
but instead the boundaries of individual disciplines.
Furthermore, collaboration by itself does not address this ignorance,
but instead compounds it by encouraging scientists to rely on each
other's authority.  Thus, anachronistic disciplinary borders
are reinforced rather than overrun.  In contrast
astrobiology can motivate challenges to disciplinary
isolation and the appeals to authority that such isolation fosters.''

Indeed, studying problems that require points of view
from many places is of great importance to our society and
from (page 548 in Sullivan's book ), we hear that

``many different disciplines should now be applied to
a class of questions perceived as broadly  unified and
that such an amalgamation justifies a new discipline
(or even meta discipline) such as astrobiology.''

Now simply replace the key word astrobiology
by biomathematics or  B3 and reread the sentence again.
Aren't we trying to do just this when we design these new interactions?
Since we believe we can bring additional
illumination to problems we wish to solve in
biological sciences by adding new ideas from mathematics and
computer science to the mix, we are asking explicitly for
such amalgamation and interdisciplinary training.
We thus want to create a cadre of biomathematically literate majors
who believe as the nascent astrobiology graduate students do
( page 549 in Sullivan's book ) that

``Dissatisfaction with disciplinary approaches to fundamentally
interdisciplinary questions also led many of us to major in
more than one field.  This is not to deny the importance of
reductionist approaches or the advances stimulated by them.
Rather, as undergraduates we wanted to integrate the results of
reductionist science.  Such synthesis is often poorly 
accommodated by disciplines that have evolved, especially in
academia, to become insular, autonomous departments. Despite the
importance of synthesis for many basic scientific questions, it is
rarely attempted in research...''


To paraphrase ( page 550 in Sullivan's book ), we believe that

``[B3 enriched by  BMC] can change this by challenging the ignorance
fostered by disciplinary structure while pursuing the
creative ignorance underlying genuine inquiry. Because of its
integrative questions, interdisciplinary nature, ...,
[B3 enriched by BMC] emerges as an ideal vehicle for scientific
education at the graduate, undergraduate and even high school levels.
[It] permits treatment of traditionally disciplinary subjects
as well as areas where those subjects converge (and, sometimes, fall
apart!)  At the same time, [it] is well suited to reveal
the creative ignorance at scientific frontiers that drives discovery.''

To address these needs and concerns,
a Quantitative Emphasis Area within the Department of
Biological Sciences at my university has been developed.
I have designed the mathematical components of this area
mostly alone because there is little interest within my
own Department of Mathematical Sciences for this particular
interaction.  At my university, a reorganization some years
ago placed Mathematical Sciences into the Engineering College
and this profoundly altered my department's perceptions
about college level interaction.  Biology is in another college
and, although it shouldn't make a difference, it does.
Interdisciplinary interaction is defined in adhoc
ways using engineering examples and since tenure and promotion
decisions are within the engineering college, there is a general
movement towards work with an engineering character.
With this said, my colleagues in biology and myself have labored to
design this area which will be a linchpin in our plan for the
interdisciplinary training of biomathematical majors.
My colleagues and I
believe,as do the astrobiology students, ( page 550 in Sullivan's book )
that

``The ignorance motivating scientific inquiry will never
wholly be separated from the ignorance hindering it.
The disciplinary organization of scientific education
encourages scientists to be experts on specialized
subjects and silent on everything else.  By creating
scientists dependent of each other's specializations,
this approach is self-reinforcing.  A discipline of
[ B3 enriched by BMC] 
should attempt something more ambitious: it
should instead encourage scientists to master for themselves
what formerly they deferred to their peers.''


Indeed, there is more that can be said ( page 552 in Sullivan's book )

``What [B3 enriched by BMC] can mean as a science and discipline is yet
to be decided, for it must face the two-fold challenge of cross-disciplinary
ignorance that disciplinary education itself enforces.  First, ignorance
cannot be skirted by deferral to experts, or by other implicit
invocations of the disciplinary mold that [B3 enriched by BMC] 
should instead
critique.  Second, ignorance must actually be recognized.  This is not
trivial: how do you know what you do not know?  Is it possible to understand
a general principle without also understanding the assumptions and
caveats underlying it?  Knowledge superficially ``understood''
is self-affirming.  For example, the meaning of the molecular
tree of life may appear unproblematic to an astronomer who has
learned that the branch lengths represent evolutionary distance, but will
the astronomer even know to consider the hidden assumptions about
rate constancy by which the tree is derived?  Similarly, images
from the surface of Mars showing evidence of running water are prevalent
in the media, yet how often will a biologist be exposed to alternative
explanations for these geologic forms, or to the significant
evidence to the contrary? [There is a need] for a way to
discriminate between science and ... uncritically accepted results of 
science.''

A first attempt at developing a first year
curriculum for the graduate program in astrobiology
led to an integrative course in which specialists
from various disciplines germane to the study of
astrobiology gave lectures in their own areas
of expertise and then left as another expert took over.
This was disheartening to the students in the program.
They said that (page 553 in Sullivan's book)

``As a group, we realized that we could not speak the
language of the many disciplines in astrobiology and that
we lacked the basic information to consider their claims
critically.  Instead, this attempt at an integrative approach
provided only a superficial introduction to the major contributions
of each discipline to astrobiology.  How can
critical science be built on a superficial foundation?  Major gaps
in our backgrounds still needed to be addressed.  In addition, we
realized it was necessary to direct ourselves toward a more specific
goal.  What types of scientific information did we most need?
What levels of mastery should we aspire to?  At the same time,
catalyzed by our regular interactions in the class, we students
realized that we learned the most (and enjoyed ourselves the most)
in each other's interdisciplinary company.  While each of us
had major gaps in our basic knowledge, as a group we could
begin to fill many of them.''


Now we can place this comment into the biomath world by simply
paraphrasing as follows:

``Students can not speak the
language of the many disciplines B3 enriched by BMC
requires and they
lack the basic information to consider their claims
critically.  An attempt at an integrative approach
that provided only a superficial introduction to the major contributions
of each discipline can not lead to the ability to do
critical science.  Still, major gaps
in their backgrounds need to be addressed.  
What types of scientific information are most needed?
What levels of mastery should they aspire to?  
It is clear the students learned the most (and enjoyed themselves the most)
in each other's interdisciplinary company.  While each
has major gaps in basic knowledge, as a group they can
begin to fill many of them.''

In many ways, our quantitative emphasis area is trying to address these
concerns.  However, it requires a lot of infrastructure and culture changes
to make an impact.  For example, simply writing lectures is a challenge.
From the comments above, it is clear we must introduce and tie disparate threads of material
together with care.  I tend to favor theoretical approaches
that attempt to put an overarching theory of everything together into a discipline because it
offers insight into the basic building blocks hidden inside complexity.
So I pay a lot of attention to books that
approach various aspects of biologically theoretically
in order to enrich a biomathematically interdisciplinary approach.
Reading these books gives me insight into the design of biologically inspired algorithms
and models which, in later presentation, I use frequently
when I design undergraduate lecture and research experiences.  A selection
of these books includes

Theoretical systems biology (U. Alon. An Introduction to Systems Biology: Design

Principles of Biological Circuits. Chapman & Hill:

CRC Mathematical and Computational Biology, 2006.);

Theoretical genomics (E. Davidson. Genomic Regulatory Systems: Development and Evolution. Academic Press, 2001 and  

E. Davidson. The Regulatory Genome: Gene Regulatory Networks in Development and Evolution. 

Academic Press Elsevier, 2006) 

and 

(M. Lynch. The Origins of Genome Architecture. Sinauer Associates, Inc., 2007);

Theoretical Neuroscience

(J. Kaas and T. Bullock, editors. Evolution of Nervous

Systems: A Comprehensive Reference Editor J. Kaas

(Volume 1: Theories, Development, Invertebrates).

Academic Press Elsevier, 2007),

(J. Kaas and T. Bullock, editors. Evolution of Nervous

Systems: A Comprehensive Reference Editor J. Kaas

(Volume 2: Non-Mammalian Vertebrates). Academic

Press Elsevier, 2007),

(J. Kaas and L. Krubitzer, editors. Evolution of Ner-

vous Systems: A Comprehensive Reference Editor J.

Kaas (Volume 3: Mammals). Academic Press Else-

vier, 2007) and

(J. Kaas and T. Preuss, editors. Evolution of Nervous

Systems: A Comprehensive Reference Editor J. Kaas

(Volume 4: Primates). Academic Press Elsevier, 2007); 

Models of the thalamus (S. Murray Sherman and R. Guillery. Exploring The

Thalamus and Its Role in Cortical Function. The MIT Press, 2006);

and 

Theories of organ development (A. Schmidt-Rhaesa. The Evolution of Organ Systems.

Oxford University Press, 2007.);

and

(J. Davies. Mechanisms of Morphogenesis. Academic Press Elsevier, 2005);

All have helped me to see the big picture and have helped motivate

our design of interdisciplinary training techniques.

However, it is not clear other mathematicians would be willing to invest

this much time into the development of the lecture material.

In fact, when I sent my book to a former mathematics mentor of mine, he was

quite dismissive of my efforts: 

``no other mathematician would be willing to do this, Jim''.

But I am pretty stuborn so I forged ahead anyway.  The problem,

of course, in order for a new approach to take off, more than one person

such as myself must become engaged fully.  And my former mentor's comment

did not bode well!   For now, let's end this section with

a nice quote from the astrobiology community (page 552 in Sullivan's book) 

``Too often, the scientific community and academia facilely

redress ignorance by appealing to the testimony of experts.

This does not resolve ignorance.  It fosters it.''

This closing comment sums it up nicely.  We want to give the students

and ourselves an atmosphere that fosters solutions and the challenge is

to find a way to do this.  The interdisciplinary training  needed to foster thus

requires all of us to be aware of these underlying philosophical 

changes that must be made.  

2.2 The Quantitative Emphasis Area

Since 2006, my colleagues and I have developed a plan to enable an integration

between biology, mathematics and computational tools for placement

within a typical existing department of biological sciences.

Although we have built this program at my university,

we are aware that our experiences need to have a national perspective

and we have planned accordingly.

We posit that in a traditional college entity

80% to 90% of the tenured faculty have little interest

in the use of  BMC in their own course development

and teaching. Hence, there are great challenges in implementing

this plan that revolve around both faculty retraining and

student training issues.  In this traditional college

department, we see incoming majors who have an uneven training in mathematics

and computer science and who may even be biased against

mathematics and computer science as part of the discipline of

biology.  There is also an existing mathematics requirement of two semesters of

engineering based calculus which does not serve the interests of biological

sciences well.  Note, recent events in the Spring of 2013 have taken away the requirement for

the second semester calculus course for biology students which has essentially completely

undermined all these efforts!

Nevertheless, in the remarks that follow, we will be very detailed in our descriptions

of what we did in order to develop these courses.  Hence, there is a fair

bit of mathematical detail, but if that is not one's cup of tea, just

try to get the overall flow.  Note how complicated it is to merge

as seamlessly as possible, mathematical and computational ideas into

a traditional biology department's curriculum.

We start at the freshman level and try to encourage

existing faculty and students into a new found appreciation of 

the BMC triad.  This then permits integration of these ideas from

the freshman to sophomore year and prepares students for quantitative

biology in the junior and senior years.   Our basic plan has

kept the existing First Semester Calculus Course

for engineers as a mathematical baseline (although, an argument

can be made for a retooling of this first course as well even though

we do not do that here).  Since different universities use

different course numbers, we will call this

course Math-One-Engineering for convenience.  However, we have

replaced the existing Second Semester Calculus Course, called here

Math-Two-Engineering, with a new

course specifically designed for Biology majors called

Calculus for Biologists.  

We have also   

added model based points of view to the first year

fundamental biology sequence which we call Biology-One

and Biology-Two. In these introductory biology courses,

the students write three papers per semester.  Four of these six papers 

have a required statistical analysis to determine the significance of 

the difference between two treatments.

At the moment, this is confined to  

the chi-square median test but other  

statistical tests using Excel spreadsheets could be done.

In addition, there are two labs on bioinformatics. One is a general overview  

that shows the students a lot of different tools, and the second one  

specifically asks them to use bioinformatics tools to look at the  

evolution of humans through mitochondrial DNA sequences.

Next, using Stella (ISEE Systems. STELLA, 2009),  students build models of an equilibrium  

chemical reaction and human weight regulation. In each of these  

examples, more complex Stella models are then used to explore more  

sophisticated questions.  

There are also several embedded modules which go beyond the traditional

lectures to expose the students to beginning research concepts. These

include 

A. the use of the Hardy-Weinberg  

      equation to predict equilibrium genotypic frequencies;

B.  a population genetics simulation used to study the effects of 

      selection and genetic drift on allelic frequencies;

C. an elementary treatment of exponential and logistic population growth models;      

D. simulation of complex populations with different age classes;

E.  a large unit on cladistics in both lecture and lab. 

So in general, despite the fact that in most topics the use of mathematics is minimized, 

the course is quantitative in philosophy.

The Calculus for Biologists course

uses discovery based learning and introduces biomathematical research based topics

that are appropriate to the level of these beginning students.        

It was run as a special

section of  Math-Two-Engineering called Math-Two-Engineering B and enrollments

climbed steadily.  Currently, we teach approximately 100

students per year in this course.  This experimental course

was formally approved (Fall 2009)

as the new course Calculus for Biologists. 

We developed this course using the following point of view. 

We feel to serve the needs of the BMC point of view,

mathematics is subordinate to the biology: the

course therefore builds mathematical knowledge needed to study interesting

nonlinear biological models quite carefully.  

We emphasize that more

interesting biology requires more difficult mathematics and concomitant

intellectual resources.

We choose a few nonlinear models and discuss them very 

thoroughly; e.g.,

logistics, Predator - Prey and disease models.

Since graphical interfaces lose value with more variables,  at the end of the

course, we explore

how to obtain insight from a six variable linear Cancer model.

We always stress how we must abstract

out of biological complexity the variables necessary to

build models and how we can be wrong.

We also focus on how to solve problems which have many steps

(so that in the modeling portions, typical assignments can have

3 - 4 pages per turned in per problem).  We believe strongly that

this is a first step towards training them to see the larger picture

for long term projects.  We therefore are preparing them for

extended research experiences.  

We are successful at developing very sophisticated mathematical

ideas in these students.  We achieve this by organizing our topics in

the following way. From the standard Math-Two-Engineering

we choose as topics simple substitution,       

the Fundamental Theorem of Calculus (FTC),

the Logarithm and Exponential Function developed using the FTC,

integration by parts and partial fraction decomposition methods

and the ideas of

approximating functions by tangent lines and the error that is made.

Since we use these mathematical tools right away in logistic models,

the students always see a correlation between the mathematics we choose

to cover and its usefulness in our model building.

We need concepts from linear algebra (here, a sophomore or junior level

course) as well.  So we cover 

vectors and matrices, linear systems of equations,

determinants and what they mean and

eigenvalues and eigenvectors for 2 by 2 matrices.

This prepares them to study more differential equations.

From a traditional sophomore level differential course,   

we can then cover      

simple Rate Equations and their applications,

the idea of an integrating factor,

building a Newton's Cooling model from data,

the logistics model

complex numbers and simple complex functions and

second order linear homogeneous equations.  Our discussion of the second order

linear differential equations cover all three cases: distinct roots, repeated roots

and complex roots.

In addition to these ideas, we have also been showing them the use of

computational tools a little at a time.  We have chosen MatLab, but other

interpreted tools are of course possible to use instead.  Hence, we 

discuss approximations with tangent lines, Euler's method and MatLab

implementations concurrently with these topics.  We also introduce

the ideas of Runga - Kutte methods as a more sophisticated numerical approximation

tool.  We finish our discussions of linear models with linear systems

of differential equations, although we restrict our attention to

the distinct real root case.  At this point, we also move toward qualitative

graphical means of understanding the solutions we are finding.  This is all done

by hand to let them grow insight.

We then segue into nonlinear differential equations.

Little of this material is covered in a traditional differential equation course.

We thoroughly discuss the 

Predator - Prey model without and with self interaction, a simple SIR disease model

and a cancer model at the end of the course.  We stress heavily that

models can have explanatory power but need not unless there

is a careful interaction between the science, the mathematics and the

computational tools. We note graphical analysis and 

numerical methods are not always useful and

determining the validity of the model is the most

important thing.  We help them to realize that interesting problems

are inherently difficult; realistic biology requires nonlinearity

which implies more sophisticated mathematical and numerical tools to be

brought to bear.

In our implementation of this course,        

we therefore interweave the calculus, differential equations, MatLab

and numerical methods, linear algebra and modeling

material in non-traditional ways.  Roughly speaking

mathematics is introduced as needed for the modeling,

although not at all in a just in time paradigm.  We believe

firmly that these difficult tools need to be carefully brought out

over time, not quickly.

The first part of the course introduces

specific mathematical material as they are not

quite ready for models.  Hence, they find this more

challenging and great care must be taken to

keep them motivated. However, daily homework with lots of algebra and manipulation

helps as well as daily mention of where we will be going with

this material.  We follow the general principle that doing 1 or 2 problems encourages

mimicry, while doing 5 - 6 of each type builds mastery.

Graders help with the daily busywork.              

The ideas discussed above have been used to

develop a textbook which has evolved from handwritten notes 

in Spring 2006 to the fourth edition ( Calculus For Biology: A Beginning – Getting Ready For Models

and Analyzing Models. Gneural Gnome Press: www.lulu.com/spotlight/GneuralGnome, 2012 ) . 

2.3 The Current Quantitative Emphasis Approach

We have already begun the process of enhancing interdisciplinary training

by adding embedded undergraduate enhanced learning experiences 

in a variety of ways to the freshman and sophomore years. 

As we have discussed above, we achieve this by adding embedded undergraduate enhanced learning 

experiences to the courses

Calculus for Biologists , More Calculus for Biologists, Biology I and Biology II.

We have taken advantage of these new programmatic tools

to implement a new four year

Quantitative Emphasis Area within the Biological Sciences B.S. degree

program.   We think of it as a precursor to a new reasoned

approach to making these ideas a core part of the entire

biological sciences curriculum. In this section of the proposal, we specifically

address how we modify our current plan for the Quantitative Emphasis Area

to allow for undergraduate research experiences.

Since undergraduate research experiences will be a valuable tool for teaching the more 

quantitative biology of the coming decades and for increasing retention of students in biology,

mechanisms must be found for fostering their development within existing curricula.

This requires increased mentoring of undergraduates by faculty. 

Further, the traditional way to include such research experiences is to have

a capstone project in the senior year preceded by a quick research experience discussion.

Since the senior year is filled with many other requirements such as job search and

graduate school application preparation, this has also traditionally not been an optimal

path for many students.  To get students involved from the Freshman year onward will therefore

require altering some aspects of the program of study.  The Quantitative

Emphasis Area supports active student participation in the Freshman and

Sophomore year directly.  

As a starting point, consider

the details of the Quantitative Emphasis Area which has been approved,

as seen in Table I  and Table II below.  

These tables shows the

basic format; as always, there are caveats and substitutions, but there are

not essential for our arguments.   There are several important points to make here.

Note that essential ideas of Physics, say force and energy, which are so necessary to

many of the threads that run through quantitative biology ideas are not discussed formally

until the junior year.

Table I: A Typical Approved Quantitative Emphasis B.S. Freshman and Sophomore Year.

F and S denote the Fall and Spring semester, respectively.  The number in

parenthesis behind the course is the number of credit hours.

Year       Course                     Description

I (F)       BIOL 110   (5)      Biology I

            BIOSC 101  (1)      Frontiers in Bio I 

            CH 101     (4)       Chemistry I

            COMM 150   (3)    Communication

   MTHSC 106  (4)    Calc I (Eng)

   Total 17                 

I (S)       BIOL 111   (5)      Biology II

            BIOSC 102  (1)     rontiers in Bio II

   CH 102     (4)       Chemistry II

   ENGL 103   (3)      Composition

   MTHSC 111  (4)     Calc For Biology

    Total 17                           

II (F)      CH 223, 227  (4)    Organic Chemistry

            MTHSC 390       (3)    More Calc For Biology

            BIOSC 302, 303  (4)    Animal Diversity

            BIOCH 301, 302  (4)    Biochemistry

             Total 15                   

II (S)      BIOSC 443, 444  (5)    Ecology and Lab                 

   EX ST 301       (3)    Statistics I

   BIOSC 304, 308  (4)    Plant Diversity

   GEN 300, 301    (4)    Genetics

   Total      16    

Also, note the calculus in biology course gives students a full year of mathematical training during

their freshman year.  Then in the sophomore year, there is a more advanced mathematical modeling

course based on a partial differential equation point of view and the first semester of statistics.

However, formal mathematical training, per se, stops at that point.  Also, there is

no formal computer science course work which teaches some choice of programming language.

Hence, to develop the triad of mathematical, biological and computational science

absolutely requires large amounts of teacher based instructional materials and pedagogical

development to bridge the gap.

The junior year is quite crowded with necessary start up biology course work. 

Currently,

the real freedom in the schedule is in the senior year with 12 credit hours of

advanced biology course to be chosen.  We suspect this sort of schedule with

openings only at the end of the program is quite common. 

For concreteness, we show the typical Calculus for Biologist syllabus

in Table III and a version of the More Calculus for Biologists syllabus in Table IV.

Note built into these courses are modules that introduce the students

to the use of mathematics in biology at a level much more advanced than usual.

In Calculus for Biologists, we have some modules already in place

There is a  gene transcription rate model which shows

the students that a simple first order differential equation

model can give insight into why gene transcription has such

a low error rate.  We also have 

a standard SIR disease model with a discussion of

how to estimate the occurrence of an epidemic from hospital

admittance data.  Finally, we go over carefully a

colon cancer model which expands the students mathematical

modeling viewpoint by using 6 variables, albeit in a linearized way.

They are shown that interpreting such a model requires them to

think carefully about mathematics, the science of genes and computational

modeling tools.

Table II: A Typical Approved Quantitative Emphasis B.S. Junior and Senior Year.

F and S denote the Fall and Spring semester, respectively.  The number in

parenthesis behind the course is the number of credit hours.

Year    Course                     Description

III (F)   BIOSC 335           (3)   Evol. Bio.   

           ENGL 315            (3)   Science Writing

           EX ST 311           (3)   Statistics II

           PHYS 122, 124    (4)   Physics I and Lab.

           BIOSC 401, 402   (4)   Physiology  

           Total          17              

III (S)   BIOSC 428           (4)   Quant. Bio.

           BIOSC 461, 462   (5)   Cell Bio. and Lab

           PHYS 221, 223    (4)   Physics II and Lab.

           Social Science     (3)                           

           Total         16                                

IV (F)   BIOSC 493           (2)   Senior Seminar          

           GEN 440             (3)   Bioinformatics         

           Arts and Humanities (3)   Literature                       

           BIOSC 3xx-4xx       (7)   More  300 + Bio.

           Total          15                       

IV (S)  BIOSC 491           (1)   Under. Research      

           Arts and Humanities (3)   Not Lit. 

           BIO 3xx-4xx         (5)   More 300+ Bio.

           Social Science      (3)                      

           Total          12   

Table III: The Calculus for Biologists Syllabus (4 credit course)

UNITS   TOPIC                                         

1      Riemann Sums, Fund. Theorem      

1      Fundamental Theorem,  continued               

2      Defining ln and exp functions       

2      Basic Rate Differential Equations       

2      Gene Transcription Rates               

1      Partial Fraction Decomposition    

2      Integration By Parts        

2      The Logistic Type Model      

2      Euler's Method      

1      Matlab for First Order ODE

1      Complex Numbers   

4      Order 2 Linear ODE                          

6 Linear Systems          

2 Solution Linear System 

2       Matlab for systems

3 Phase plane analysis         

9 Predator - Prey      

3 Predator - Prey Self Interaction      

3 Epidemiological Model  

4 Cancer Model 

Table IV: The More Calculus for Biologists Course (3 credits)

UNITS  & TOPIC                                        

2       Multivariate Functions   

2       Partial Derivatives

3       Jacobians; Linearizations    

1       Complex Eigenvalues

1       Repeated Eigenvalues

2       Nonlinear ODEs again

4       Basic BioPhysics (Ion movement etc)

2       Partial Differential Equation Models      

4       Series and Series solutions   

4 Boundary Value Problems   

6 Matlab ( intertwined with other lectures)               

4 Hodgkin-Huxley Models   

4 Excitable neuron simulation                

In the junior year, Quantitative Biology, continues to expose

students to critical BMC concepts.  It applies quantitative methods to a wide range of biological problems. 

The main focus is on building modeling skills using population, physiological, genetic, and evolutionary problems. 

It also includes a review of statistical principles, and introduces basic bioinformatics techniques.

A typical syllabus is shown in Table V as a  3 credit lecture and 1 credit lab course.

Table V: Quantitative Biology Syllabus

UNITS   TOPIC                                   

1        Mathematical models    

2        Excel, Stella, and MATLAB    

5        exponential,logistic growth    

2        Population growth - Leslie matrices  

2        Optimal life history strategy    

1        Optimal diet (linear programming)      

2        Compartmental models (epidemics)   

4        Predator-prey Models    

2        Competitive interactions   

3        Biochemical mathematics  

3        Cell biology, development models   

6        Physiological, biomechanical mathematics 

2        Mathematics of genetics   

3        Evolutionary mathematics  

3        Statistical principles    

2        Bioinformatics tools

To fully energize and engage current biology and mathematical majors in

beginning research experiences and to potentially recruit new students

it is important to develop  a wide range of undergraduate research experience (UGRE) modules.

These can be deployed in a classroom lecture or a laboratory setting.

The development of such textual material is a highly nontrivial task

and requires a substantial investment of faculty time.

To make these new initiatives self-sustaining, it is important for these ideas 

to become firmly entrenched

within both the Department of Mathematical Sciences and the Department of

Biological Sciences.  This is both a retraining issue and an interest issue.

The courses Calculus for Biologists, More Calculus for Biologists, Biology I, Biology II  and Quantitative Biology are not taught in a standard manner and require much from the instructor.  To teach Calculus for Biologists effectively requires that the instructor both respects and loves both mathematics

and biology.  Since this is not true in general of any department's teaching staff,

this is a significant faculty and instructor training issue.

If Calculus for Biologists enrollments were to grow, a seasoned teacher

would have to develop a mentoring strategy so that new teachers of

this material could be trained.

Similar mentoring can be done in the teaching of More Calculus for Biologists,

Biology I, Biology II and Quantitative Biology.  

Indeed, training and mentoring would have to be a fundamental design component. 

Of course, all these concerns are moot now as with the recent decisions by my biology department,

the entire quantitative emphasis we have outlined here will receive no university support and hence,

die away.