## Pedagogy, experimenting with experiential in STEM

Learning mathematics by, doing, reading, searching and interacting.

There is a lot of interest in experiential   learning now-a-days, especially in STEM and engineering. There is a big debate for and against this pedagogy which is far from settled. The proponents of the method are up-beat about how this pedagogy solidifies knowledge, makes students independent thinkers, good learners and team-worker, who are aware of multifaceted real aspects of the world. The opponents talk about the methods being too focused in small areas of knowledge, students missing out on the variety of analytical methods and rigor and probably the most importantly, the methods not being scalable. The last point is especially important for India where there are usually hundreds of students in a classroom.

I have been experimenting with using a small component of projects in the courses that I teach at the School of Engineering and Applied Sciences at the Ahmedabad University. In this post I would like to talk about some of the student projects in my course “Calculus and differential equations” offered to the first semester students of engineering of three different streams, ICT, Mechanical and Chemical. My strategy is to teach them a about use of differential equations in understanding dynamics and in modeling and basics of numerical integration. I then let them come up with their own project idea, in which either they make observation of a real system and model it from the data, or study an applied problem that uses differential equations and try to find how general the application of the differential equations studied by them is by looking at variations of the system and related applications. Following is a list of some of the student projects over the last seven years that I have taught. As the project component was very small usually about 5 to 10% weight, the submissions were not particularly in-depth, however they were interesting in terms of opening up a new area for the students to think and learn and opened up discussions on new topics. There were four general types of project areas in which students worked. (1) Mathematical modeling (2) Numerical simulations (3) Studying an applied system (4) Geometrical models to understand a concept. The submission types varied from submission of physical models, computer codes, charts and documents.

Mathematical modeling: In these projects students usually collected data on their mobile devices and then tried to fit a function or a differential equation to the collected data and commented on how realistic their model is and why.

• Vibration of a scale.
• Rolling of objects down a slope.
• Projectile motions.

Numerical Simulations: In these projects students generally pick-up a system of differential equations and solve it numerically using usually the Euler method or the Taylor expansion.  Some examples are,

• The Lorenz attractor
• The Damped oscillator
• Cooling of objects, Newton’s law
• Bacterial growth
• Flue Epidemics

Study projects: Here students study an application or a research paper . Some examples are,

• Cancer growth models
• Chemical reactions
• Attenuation of light in liquid
• Rigid body dynamics
• Drug absorption in a body
• Emptying of water tanks
• Roller-coaster ride

Geometric Modeling: In these projects students make model of theoretical concepts studied by them, to gain visual insight in the theory. Some of the models prepared by students are given below.

• Models of optimization on a constrained surface to understand Lagrange’s multipliers
• Spherical and cylindrical polar coordinates
• Different types of surfaces, showing saddle points

My realizations in case of large classes when the project component is small are following,

• Between team interactions are limited, hence finding time and space for peer to peer learning gets highly constrained
• Because of coding and plotting skills needed, project submissions get postponed to the end of the course where due to impending exams the processes get rushed
• Finding sufficient time for providing feedback to the students in their ongoing project work is difficult mainly due to the large class size

On the whole implementing project based learning in a mathematics class has been quite a rewarding experience for me and I intend to make it much more effective during coming semesters. Your feedback will be very much appreciated.

## Encyclopedia of Ignorance

Since a long time I have been thinking of creating an encyclopedia of ignorance for the courses that I teach. When I correct papers, I see some standard mistakes students make.

Let me start randomly collecting them. May be some day I shall put them in order. I think I shall keep updating this page.

Physics:

1. The $\vec{E}$ and $\vec{B}$ field in an electromagnetic wave are point quantities. Text book diagram often makes people think about them as quantities extended in space.
2. Density is also a point quantity which is defined as the ratio of mass ina volume to the volume, in the limit volume going to zero. Though theoretical such definitions are very useful practically.
3. It is possible to have the function value zero at a point while its gradient is nonzero.Which also implies that the magnetic field at a point may be zero but the force on a dipole at that point may be nonzero.
4. Keeping a body magnetically levitated does not require work.

Mathematics:

1. $d^2y/dx^2$ is not the same as $(dy/dx)^2$, many student wrote this in their answers in the last exam. I tell them this would be like saying $a = v^2$ that usually makes sense to them.
2. Over and over again I have to remind the students to try to make sense of the equations they write. For example while applying Lagrange’s mean value theorem to $x \ln x$ on $[1, x]$ students wrote,  $\frac{x \ln x}{x-1} = 1 + \ln x$, replacing $\ln c$ on the RHS by $\ln x$. The formula when simplified gives, $\ln x = x-1$ which should raise a serious doubt, provided they stop doing mathematics mechanically and start thinking about meaning of the formulas.
3. Exponential processes in a variable x can also be of the form $a^x$, somehow many students think that only $e^x$ would be an exponential process but any given number $a$ can be written as $a= e^b$ hence  $a^x$ can be expressed as $e^{bx}$.