In the last two questions of this week's blog, I have asked you to explain what echelon form is to a high school student and to find an alternative name for it.

It has been quite a few semesters since I have started asking these questions and I have received a couple of comments from students who did not like this style of questions. Although a couple out of several hundred students is a small percentage, I think it is worth pointing out why I think these questions are beneficial.

I have moved to Connecticut in 2019 from Toronto, ON. Aside from the fact that there is no universal healthcare, one thing that scared me most was the drivers in CT. (I am used to it now). If you have ever driven a vehicle in CT, you will know that a very big percentage of drivers do not signal here when they turn. However, we will not focus on them. We will focus on the drivers who start signalling just before they start turning. The point of signalling is to inform other people that you will make a turn. If you do not do this in a timely manner, for instance before you start slowing down, what purpose does it serve?

I have spent months complaining to my wife about this behavior. She also has a PhD in math. And she also teaches courses. At the end of the day, we have come to the following conclusion: these drivers know that they should signal. They do not know why they should signal. We realized that this is a very common pattern in our education systems. This is the same in Turkey, Canada, United States and more.

In this course, a very big number of questions reduce to setting up a matrix, reducing it to its echelon form and analyzing the result. From spanning to linear independence, from eigenspaces to determinants. However, if you just memorize that aspect of these concepts without understanding why you are doing this:

• I don't think you will have much fun or if I have to use fancy words you will not be satisfied intellectually;
• you will only learn a skill which is mostly useless in real life because we already have computer programs that do these computations for us, we need people to understand the meanings of the processes and the results.

With this in mind I prepared my lecture notes, this is why I am asking these questions. I have had students giving me the "correct answer" without understanding any of my learning goals just because they were able to memorize they have to put some numbers in a rectangular array and do some arithmetic operations. This was obviously my fault. So, I am changing (or rather diversifying) the type of questions I'm asking. When I ask you "What is linear independence" or "Explain it to a high school student the meaning of eigenvalues" what I aim to do is to see what exactly you understand from these concepts rather than whether you can solve a specific type of question. And don't worry, in these blogs unless you completely miss the point, you will not lose marks. And if you completely miss the point, we will try to fix the lost marks.

The same applies to "Give a different name" questions. If I was the one who invented this concept, what would I call it? To be able to answer this question, I need to know what the concept really is about and find a name that fits it. In the past, there were students who called echelon form a chocolate bar form or who tried to explain vector space axioms using analogy to pasta sauces. Mathematics is not pushing around some numbers until you get to a certain answer. It is about ideas. My main purpose will be to pass this feeling to you throughout the semester.

Also, it has been more than 10 years but I think I learned about this giving the correct name idea from Hermann Hesse's Siddhartha. If you haven't read it I highly recommend.