I was in south India for an academic visit  and shared the university guest-house room with a lady, who was there for a conference.  She was possibly in late thirties. The first evening I was there, she asked me if I would accompany her to a relatives place, she had left some of her medicines there and she wanted to get them back.

Now, I am not particularly social, usually I am quite happy left alone. There is something good to read or a problem to tinker with. However since she had asked and I did not feel like saying no, so I agreed, a bit annoyed at myself and her. “Why on earth do people need company while traveling a short distance within city?” I ask you.  She also spoke very slowly and appeared somewhat dim-wit to me.

A little later in the evening, before bedtime, she removed her wig (I had no idea, she was wearing one). Then she told me about the brain tumor she had survived. How the doctors had said she may never be able to speak again after the surgery. That the medicines were the ones the doctors had prescribed after the surgery and she needed to take them regularly. She told me how her little daughter is proud of her, taking control of her life. It was now my turn to feel like a dim-wit.

We went to her relatives place, got the med.s the next day. The next day her colleagues were planning a day’s outing and at breakfast, someone asked her a question. She took a few minutes before replying and the questioner blurted out, “don’t keep staring at me, reply”. I wanted to yell at the guy but kept quiet. Aren’t we all dim-wits when we don’t know better?

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 $e^{bx}$.

Leave Me A Link and I’ll Share Your Page!!

Thanks to Danny for sharing knowledge.

As most of my followers know I am big into helping other bloggers gain more exposure. My goal has been to grow a community of like-minded people and I am part way to my goal.  I am bringing back the open call to leave a link and I’ll share it for you!!

To get a reblog you must do the following:

Each day I’ll randomly select 3 links to reblog.  I’m not sure it gets much simpler.  You can leave as many links as you want and I’ll cycle this post from day-to-day so more people can jump on board.

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Speaking Mathematics

The way we learn has changed drastically over the last decade or so. I consider myself fortunate for having access to so much high quality learning material with very little effort.  I am a physicist by training but have been teaching Mathematics to engineers for quite sometime now. There are many lines of thought on how Math should be taught to engineers. Three main views are

(1) Teach in the usual lecture, problem solving style with a lot of emphasis on concepts, rigor and formal methods.

(2) Classroom teaching in an intuitive manner with lots of games and real life examples involved.

(3) Teach Math hands-on while students work in the lab on real life projects.

A proponent of the first method is the well known computer scientist Djikstra He holds the view that learning formal methods without interpreting the symbols is a great way of preparing students to work with novel realities. In physics such an approach of working with mathematics without well decided interpretation (in terms of analogies) has been quite useful, especially in quantum mechanics.

The second view is a favorite of many educators, if one is to go by online platforms. John Conway is a proponent of this style of Mathematics teaching, where he tries to either develop or understand mathematics using a lot of analogy and tools. Conway

The third view, being followed by many educational institutes is that of project based learning, where the students work in teams to solve real life problems and pick up mathematical skills as and when needed PBL While some engineering institutes have used this pedagogy very successfully, when it comes to Math learning there are also voices that express doubts about effectiveness of this method.

I have used each of this style to some extent in my classes and here is what my experience tells me.

In an average classroom usually the second style of teaching mathematics intuitively works out very well. It gets students interested in learning and enthusiastic about the subject.

I believe the first style have many positives, for a higher level learner who has come to appreciate the abstract nature of Math and its power. In a regular classroom less than 10% of students usually have this appreciation. When mastered, this method can be utilized to application of mathematics in quite diverse fields.

The PBL method of learning is relatively new. It requires appropriate infrastructure and manpower. It has the advantage that the learning is driven by student motivation and it happens in the real world context. However this method also requires learning of new tools, machines and computational, searching for and using appropriate materials and working on open ended projects. These considerations mean that students get to focus only on limited aspects of mathematical details and learn fewer concepts compared to a student taught by the other two methods.

When it comes to my view, I would be flexible and change my methods based on the students involved, the motive of the course and the infrastructure available. When the students are advanced and have developed appreciation of abstract concepts and have the ability to apply their knowledge effectively, formal method is very good. In general it would work out very well for students wanting to major in Math. The second method would be fun and effective in an average classroom, whereas the PBL method is good for students who want to apply the knowledge gained in real life situations and are prepared to learn, take challenges and deal with uncertainties.