Tuesday, January 14, 2014

Reflections on a First Semester Teaching

Currently Reading: Recountings: Conversations with MIT Mathematicians

It has really been a long time since I have updated this blog. There's been some reasons for this, most of which are sad, but I'm feeling willing to take up this torch again. I have successfully completed my Masters in Applied Mathematics. So, on an academic level, I guess I'm a "mathematician" of sorts. I still feel overwhelmingly clueless about the greater scheme of the subject. I found it hard to approach this blog, since I really wanted to share my joy of the subject, but I was not feeling very joyful towards it at the end. I found my degree very hard to complete, especially when I had to take classes that I found very uninteresting. Towards the end I really felt like I just wanted to quit at times and I was also realizing that, in dismay, I was not skilled enough to go into a PhD program. Watching your peers move onto brighter things is difficult, especially when you realize that you probably shouldn't be joining them. For quite some time I was pretty bitter about this and spent a summer not reading and not doing any kind of math.

Despite all this I did realize that I had a decent knack for teaching. I was immediately hired by my University to take on the role of adjunct. I was given three courses to teach in the Fall semester, which I think went okay for my first time teaching a subject I did not like very much. I tried to make the best of it though and I find working with students can be it's own reward no matter the subject matter.

So here are some things I dealt with...

Student's education level:
The class I was scheduled to teach is something my University calls "Management Pre-Calculus." What does that even mean? Well, basically, it's what you'd expect out of a Pre-Calc course, but without the Trigonometry. ...And at the end they do stuff with matrices out of nowhere. Either way it's the lowest level math course we offer.

I find the structure of this course rather frustrating and a lot of it has nothing to do with the Math department's approach. The things they make students do is mind boggling. I literally had students in that class that had taken Advance Placement Calculus in high school, but because of their chosen major the advisers would not let them test out of the subject. This blew my mind, because we give all incoming Engineering and Science majors assessment tests for this. Turns out my saving grace for an uninteresting subject was one motivated student that had taken AP Calculus in high school. She turned out to be a very good student and hopefully a promising future mathematician. I recommended she change majors immediately, because anyone who can grasp the process of Riemann Sums as a further motivation to the process of integration really has no business wasting their time in such a class/major. Plus she really enjoyed doing the higher level math, so I always say you should try to do what you love... and accounting just wasn't that exciting!

Now let me remind you, Management Pre-Calculus is basically considered the lowest level math class the university offers. So amidst students that already had some notion of integration I had students that flat out didn't understand adding fractions, never mind factoring polynomial functions. I had some students that were repeating this remedial course for the third time! Some students came from very strong school systems, but I would say it was about 50/50 on those who could do much. 50% clearly had the ability to grasp what I taught them, which was probably a higher level than when they walked in the door. The other 50% had to basically start to learn mathematical structure for the first time. And I do mean structure, a lot of my students could easily solve problems like 2x - 1 = 3 and come up with a correct answer. However, writing out every step to show me they understood what it meant to solve the problem mathematically was a massive battle.

Fractions, the typical bane for students, was like pulling teeth. The real reason this was such a massive issue for me is because they didn't know the nine rules that created the structure of the algebraic field. The whole notion that "a x 1 = a" and 1 can equal a lot of things was truly hard for them to see. The real reason it was hard for them to recognize is likely due to the fact that they rarely were made to show this in problem solving. Not having that idea at your fingertips makes solving rational functions a real chore. The whole method of completing the square was very hard and I literally had to go over this twice, because it is a really useful problem solving technique. Then it was never even on the department final... I was so angry. It's such an elementary and beautiful technique, just looking at the geometry of it is wonderful.

In the end I hope students got something out of the course. I tried to break things down into enough detail that anyone could solve them. But that whole problem solving sophistication where all you need to know is the technique and you can solve another problem that looks like it... I think that skill set was very hard for some students to grasp. Many students became baffled once the problem changed slightly. I honestly don't know how to overcome this, I feel like this was something I rarely struggled with, so I have no idea why my brain didn't care about the change and theirs does.

Textbooks and Online Learning:
A good text book is hard to find. Almost all of them are crap. The textbook I had to use was virtually unreadable. Examples weren't completely filled out and in one instance on Power Modelling it literally gave a table and said "clearly x^2 is the correct power model". Let's keep this in context, the students I have are considered being on the lower level, how are they possibly going to see that?! I was so enraged I just wanted to throw the thing out the window. But this book had a nice fancy cover, colored graphs and lots of pictures. It also painstakingly tried to develop "real world problems" for students to solve. The first equation a student sees in this book is literally this awful thing with all kinds of decimal points... way to make someone feel this will be accessible. The best books I've read in math have none of this. There is no "touch" of a marketing department. The covers are often blank and unassuming, but these are well written. They aren't written by a team of mathematicians. They are written by one or two authors at most, so you never have a break in a writers continuity. I really resent the textbooks that are being used today. There are a few publishers I really love and trust like Springer and Wiley, but a lot of others are just trying to outsell their competition by making a flashier product.

Now another thing that is all the craze in higher education is using online programs. We use an online program system that basically just bombards students with problems. Instructors really see this as a wonderful thing, because the program tells students if they are correct or not and keeps track of their scores, so Professors don't even have to grade! My most major concern is a students ability to explain whether their logic is correct or not. I worry constantly if students are just going through the motions and not writing out every problem. If the programs are used correctly they can be quite valuable, but often I don't think students really know how to use them or use them in a lazy fashion, because no one has really told them otherwise. A lot of problem solving can be rather intuitive and you can look at a problem and solve it sometimes, but explaining why a solution is true can be far more difficult with rigorous mathematics. Also it didn't take my students to figure out that you can easily cycle through the problems and based on the number changes guess the answer correctly!

How did I fight against this? I honestly made up my own problems and assigned written homework. In the end my students seemed to really grasp what it meant to explain something and what was and was not well written. In a lot of cases I really felt like I was the first person ever bothering to teach them how to write anything. This is why math is so difficult for students. The simple fact that I collected papers and graded them myself every week went a long way, I realize professors aren't required to do this, but let's be honest, how else are we going to train people now? I heard numerous stories of high school instructors that just went around and checked off whether a student did their homework, never bothering to look at a students logic. And we wonder why mathematics is so misunderstood?

I hate these things. HATE THEM. They shouldn't be allowed. I was never allowed to use a calculator in undergrad or grad school. No cheat sheets half the time either on exams. The most you should ever need is a basic hand calculator for any level of arithmetic that might be needed. I am sympathetic on this end, because I don't really have a calculator brain, but I can certainly do Trigonometry and Calculus in my head. I can even do proofs! Very little of this is dependent on your ability to know what 7 x 43 is.

The purpose of learning mathematics, to me, is to train someone in some semblance of logical problem solving in a very particular way. Not all problems are mathematical, but I think it is valuable to at least have some notion of what it means to solve problems in that way. Just training yourself well enough to look at a problem and see where certain steps may or may not lead is invaluable. None of this is done on a calculator. We have calculators now that can solve our Algebra equations. I have a calculator that will do integration! Pushing buttons on a computer, I say, is not doing mathematics and as soon as you go down this road you are entirely defeating the purpose behind learning the subject. Mathematics is done with a paper and pencil. It's been that way for thousands of years and it really shouldn't be changing any time soon.

I took a very old school approach to the course and the response was incredible. I had students telling me that they though this was the first time they ever learned math. I don't think this has anything to do with my ability to lecture. I think a lot of it has to do with the fact that I took the time to grade their homework myself and write meaningful comments on how to solve the problems. The emphasis was on learning how to logically problem solve, not on getting the correct answer. Often times just solving the problem logically is enough to get you to the correct answer anyway, so we focused on that more than anything else. I also focused on writing a lot, which I think paid off in the end.

I'm scheduled to teach Calculus in the Spring semester... classes start next week. My classes are overwhelmingly filled up. I'm not a soft teacher either. The grade I gave out most last semester was a C. I think students genuinely value learning. Sure some care more about the grades... but the simple fact that they felt like they had bothered to learn something and had an instructor that cared about their ability to learn seems to go even further than a GPA. This is what university work is supposed to be about...


  1. I've always SUCKED at Algebra (well, at least intermediate/advanced Algebra), and I don't know why. Every other subject, literally every other subject, I can just sit in class and not take notes, and retain everything the teacher said and score well on tests. I'm going to attempt to teach myself calculus because I want to see if I can learn it (since it is the language of physics, a key interest of mine) but I'm scared of taking it in University in case I fail the class. Luckily, my best friend is an Aerospace Engineer by Degree, so he can help me if I get stuck. Keep up the blogging here!

  2. Maybe part of the issue is the way this subject is often introduced in school. Algebra is super important for dealing with Calculus and I'm not sure Calculus can be studied very well without knowing how to do Algebra. Maybe, instead of approaching Algebra from the high school system you should look for a good text book and try doing that. That's what I did. I sat down and read Holt Algebra and Holt Algebra 2 and I think these books were very well laid out for the most part. I'm not sure they gave me an incredibly deep notion of Algebra, but it was certainly a lot better than what I got out of high school and I was far more prepared for Calculus after reading those two books. I would probably do things entirely differently now if I could go back... but that's the point of my blog tutorials. Here's the way you can build good theoretical foundation for doing high level math/science.