Visualization

Challenge & Selection  The challenge for that I used Visualization, or Visual Thinking, was to achieve a deep understanding  of advanced analysis for approximately 200 students in a university course. I was teaching a huge  tutorial lesson for them in a German university. The actual lectures were done by a professor who  was very abstract and explained math very high level, because from his viewpoint, the intuitive part  about it was anyways clear. He did not notice that we were facing a bunch of undergrads who had  learned at school and in previous university courses that math mainly was about juggling with  formulas.

My aim was that as many students as possible gain a deeper understanding of mathematics plus to  enable them to travel through these abstract worlds laying in front of them with as less formula  luggage as possible. But I only had 90 minutes each week to achieve this.

That is why I decided for the tool of visualization. For me, even if math gets more abstract, it seems  like a big landscape that one can discover. And instead of explaining to the students how to move  from A to B in this landscape, I tried to give them a mental map that enables them to go from any  point they want to any other point they want.

For this, I decided to break down connections they seemed to find difficult into pictures, sketches,  that they can remember quickly. I used mathlab to generate some of these pictures and I drew many  pictures on the board – very classically with several chalk colors – and tried to connect these pictures  for them with the formulas and theorems they described.

Application  Whenever I had to explain another theorem or formula to the students, I asked them why somebody  invented it. I tried to break things down to easy connections they could see in the world, like corn  fields in the wind to describe vector fields. I encouraged them to forget about the severness they had  learned to apply for their studies and to go back to their childhood and try to imagine. I wanted them  to feel what we are doing.

If I was explaining to them what different differential operators were deduced from, I had prepared  pictures within mathematica that explained the connection easily. If it was about less complex topics,  I plotted the easiest example I could think of to the board and tried to make the students come up  with the right idea themselves by asking them questions. Whenever they got a step closer, I plotted  that part of the answer to the board. If plotting was too static, I tried to use my body language to  make them see the context.

A nice example of this was the proof of the hairy ball theorem. It says that – if the world was a ball –  there is always at least one point on the world where there is no wind blowing. For that, I visualized  different proofs and methods for proving, e.g. we cut balls open into two halves, then pressed them  flat to gain something like a round plate and then we walked around that plate. This one worked out  so well that it was even possible to explain it to school students who had just turned 12 years old.

This kind of visualization motivated most of the students to learn the formulas belonging to the  picture as well. And even more, they didn’t have to learn them by heart anymore, because they could  remember the visualizations of each formula element and the full picture and could puzzle it  together.

Insight & Approach  What I have learned from conducting this way of teaching is that it is very important to have a close  look especially at the weaker students whether they get the connection between the visualizations in  my class and the abstract formulas in the professor’s class. But I also learned that it helped the  average students to understand the content of the abstract class much better and that it encouraged  the strong students to learn even more, to take their phantasy into their studies more than before  and to evolve to – in total – much stronger math students.

I also learned that this way of teaching brought me much closer to the students then teaching them  in formulas. Enabling them to “see” what they are actually doing seemed to take the fear away to ask  me questions. The nicest moments were moments when students with questions got up in this huge  hall, walked to the board and towards me, asked me for the chalk and dared to plot their question to  the board. Even if they could not express themselves so quickly in the formula language, the fact that  we communicated with visualization enabled them to express their ideas and uncertainty much  faster. It also helped me to understand with which parts they actually had trouble understanding and  what I could explain better or more closely to enable them to get the full picture and a much better  insight.

Meanwhile I am working in economics and not at university anymore. Also here, I use the tool of  visualization frequently to solve more complex problems and to enable a group of people to gain a  common understanding of what it is we are actually talking about.

If I should ever teach high level math again, I would always choose the tool of visualization again to  explain the connections students have to discover. I would even go so far that if math was only  teached in this way, most students who failed in math at school would not have had any trouble and  I can only advice this tool to anyone who has to explain more abstract content to a group of people.  It prevents frustration of not understanding in the audience, helps people to remember and makes  talking about actual understanding deviations much easier.

