Thursday, May 22, 2014

Teaching Calculus for the first time

Currently Reading: The Artist and the Mathematician (The story of Bourbaki)

As I mentioned in my update about four months ago... I was scheduled to teach Calculus in the following semester. I had toyed with the idea of updating this blog more regularly... but I had, honestly, forgotten how much work goes into working on a Calculus course... especially one that requires re-arrangement of material. That being said I taught the second sequence of the Management courses and this was "Management Calculus" and, once again, the curriculum was truly mind-blowing to say the least. I should point out here that in an odd turn of events, I took this course at the university I teach at over ten years ago now. The curriculum is entirely different and much more in-line with an actual calculus course. This, in turn, makes it more interesting for me, as the instructor to teach. I should also disclose that when I took this course I got around a C for my final grade. Whether or not this has to do with my poor mathematics ability at the time or the fact that my instructor spent the majority of his time flirting with the girls is up for debate. I think that teacher has really made an impression on me that I need to be a bit more of a no-nonsense kind of instructor when I present material.

From what I can gather, I am actually considered a fairly relaxed professor, but I think my students have the understanding that I take the subject matter quite seriously. I try to keep this in the range of a happy medium as much as possible. Most universities do a student feedback sort of thing before the final, and as an adjunct my job depends heavily on these. I do try to strike a happy medium between requiring rigorous work and having an enjoyable lecture. I had many of my students from last semester take this course. While I try to be friendly with everyone, you do get to know a select group of students better than others. One of the comments I saw show up frequently, and I mean it showed up maybe four times out of the 90 students I had in Pre-Calc, was that some felt I wasn't being objective in grading or something to that nature (I actually don't remember the exact wording). I tried not to give the impression I played any kind of favorites, for example, one of the students I got to know very well got a C+ at the end of the semester. They were perfectly happy with the grade and it was one of the best they had gotten in a math course, but other students don't see this. I try to grade as objectively as humanly possible.

In any event, this semester started off with a pretty wild bang. I was originally assigned to teach two sections of Calculus, and my reputation as a good instructor erupted like wildfire. My classes were swiftly overflowing and giving out permission numbers quickly made my two courses rise to the attendance level of 90 students between the two! This is partially my fault for giving them out, but before I had seen my room I thought it would be fine. The room I was assigned was extremely small and had a huge pole in it obscuring many students view of the board. To make matters worse there were no lights in this room! After the first day I requested a room change and my room was quickly changed to a larger room down the hall... but not too much better and it also had a huge pole in the middle of the room. At least I could fit all the students uncomfortably, but it would be a tough semester. A week into school I asked about this problem and the solution we came up with was to make an emergency third course in the morning and have some of the afternoon students go into that one. I said, okay, I can teach for four hours straight twice a week. This wound up backfiring entirely and before I knew it I had about 20 new students in the morning course! I got some students to move out of the later sections, but instead of two sections of 45 students I had two sections of 38 students and one section of 28 when all was said and done. I think starting off this way was exhausting and really disrupted my time, because now I had a new class and already a week behind into the semester! I had to slow down my other classes so the new one could catch up. I then lost another week to snow days and by the end I barely made it through all the material. I had to lecture on Integration in such a blitzkrieg fashion that I don't think they really got much out of it. There were days I honestly felt like I was assaulting people with mathematics and by the look on their faces I think they felt that way too. Everyone was wonderful though, they understood the position I was in and I did it all quite apologetically, so I felt, in the end, no one was really wronged from it all. This all lead me to an array of new problems to deal with...

Other People's Students:
First and foremost I am not going to bad mouth any of my colleagues. I'm not in their classroom and I have no idea what challenges they face. I have no idea how this student was with them and often times students just don't match up with the way a particular professor does something, and that is okay. My own students coming into the next course couldn't do some of the things I thought I covered to death, so a lot of this isn't a criticism of the professor they had before me and a lot of this is probably a terrible hold-over from High School's mistreatment of the subject matter. Also, my colleagues actively use the online homework method I railed against so hard in my prior blog, so I'm sure some differences are showing up there.

That being said, I had about 50% new students in my classes. Some were better trained than others, but I was surprised at the level of remedial work I had to cover in this. I don't think I will ever, fundamentally, understand why someone can't do the difference quotient. To me, it is literally just plugging things into a formula, but it is rarely seen that way. Even if I give them the formula a solid group still have no idea what to do. When dealing with the Power Rule for the first time I introduced the idea that you need to convert all your problems so they appear to look like x^n. So if you have the square root of x, you need to change it to x^(1/2). Once it was converted I found most people could handle the level of differentiation I required of them... but I honestly had to go back and blow out a whole class covering how to convert things like "1/x = x ^ (-1)". This is seriously Algebra 2 material. I just deal with it and get everyone up to speed as best I can so I can cover more difficult material. About half the students found this painfully elementary, but they understood that I had to help the other half of the class and sat quietly through a boring lecture. Most of my students from the prior semester understood these things fairly well... but even in Pre-Calc I think I took this knowledge for granted too much. I'm definitely going to spend some time doing, what I would consider, Algebra the right way... as Axler would put it!

The Text Book Problem:
I learned from the prior semester not to trust this thing. I didn't even bother trying to teach out of it, unless I came across some strange application I barely understood. But I recommended my students not even bother to deal with the book and most didn't. I just feel sick to my stomach when I see someone pull out a book covered in pictures and other doodads and makes me feel so wonderful when I sit down to read a Calculus book like Apostol's Calculus, which has one of the best deliveries of Set Theory in the introduction I think I've ever read. It's a quick and dirty explanation, but I think even my "low-level" students could understand it!

This semester all I did was look at my required syllabus for the topic and then teach it from memory when I learned it about four or five years ago now. The type of material they throw into this text is ridiculous though. They barely fill you in on the motivation of why the material is the way it is. There's no rigorous proofs for anything... I mean, why would I know why consumer surplus is related to integration? I studied Applied Maths related to physics, not the economy. I really couldn't care less about the economic applications.

The Sad Truth:
While I am required to cover this course and I truly love Calculus. I find it one of the more beautiful subjects I've studied and I am glad I got to share that beauty with someone, but at the end of the day they don't need this course. I went through the Business program at my University when I was my students age. It goes way out of its way to avoid using higher level mathematics, so unless things were overhauled since I was there ten years ago... I couldn't see why I covered the topics I did. Now, being a curious fellow, I did ask my students what kind of math they were using. The only thing I really ever remember using in my Financial courses was the exponential function to figure out Net Present Value or Future Value of investments. So, when I covered the chapter on the exponential function, I said it was very useful for such and such. Come to find out, the don't even model compounding interest that way in their courses. They seriously have the students put things into a spreadsheet to calculate daily interest rates and compound it that way. Sure, that's how the accounting program is actually doing it, but that really doesn't help students solve the problem in an elegant way. I also remembered going way out of our way to use only Algebra one, never mind Calculus. All of our functions were linear, the only time I ever saw a non-linear function was in Economics and those were never really discussed at a very rigorous level. Someday, perhaps, I will teach a course that is meaningful for people and I long for that day to come. I am still new though and I realized I will have to put in my time before that happens. Right now I'm trying to treat the course as a sort of Gen-Ed, while opening a new world of possibilities to my students as best I can. Perhaps their program won't use this, but someday they may be sitting at their desks and realize... that was a really interesting subject, maybe I should learn about that more deeply... just like I did when I couldn't handle working at the Financial firm I was at anymore.

This is a topic I wanted to cover in my last blog and forgot about, so I'm going to do a lengthy post here and include it at the end. Here's the ugly part of teaching that students don't really get to see. I, honestly, agonize over grades. I am fine grading home works and exams, but issue that final grade is so agonizing and heart wrenching. It's because I realize that GPA means so much for students now. You don't just do your best and if you get a C, but got a lot out of the course that's fine... no, nowadays this could lock someone out of getting that internship they truly want. Why am I responsible for this fate? For a course they technically never need? I decided, when going into this course I was going to do it justice and grade as I would any other mathematics course. I just wouldn't hold students to the highest degree of rigor is all and that seems to strike a happy medium for me, where I feel the students are getting a sense of how mathematics is structured and done in the real world. It's true that very few people fail my course, but I am not shy about giving out the dreaded D or C- either. I feel it is unfair to give everyone and A, even though everyone would like one. I believe the way I've come up with the grading scheme reflects how well a student can handle the subject matter. For example, a B to me, means you are decently competent with the material covered, a C means you are okay at it and so on. So who fails? Well that leads me to the following section.

The Homework Problem:
To this day the only people that have ever failed my course are people that didn't do their written homework. This means they started off doing nothing in the semester and I really don't care what your excuses are, but if you're not at least trying to keep up or communicating with me, then you are going to be in a sore spot by the time the first exam shows up. I have, once again, recommended people drop my course because they are in danger of failing it. When I look at why, it's almost entirely due to the fact that they didn't do the assignments. All subjects, to learn the meaningfully, require some intellectual exchange. You can't do nothing in a subject and claim to understand it. No one does that. I try to make my homework load reasonable, because I realize it's not a real requirement, but I try to give them just enough so that they engage and learn the material. I ask students to sometimes do about 5 or 6 problems of homework of varying challenging levels. I feel like this is entirely reasonable and if you can't be bothered to do 5 math problems a week, something is wrong, in my opinion. Maybe you shouldn't be taking that course right now? If something else is overwhelming your time and requiring your focus, go focus on that. I'm not going to be offended, but I do expect students to have the maturity to realize when they are in over their head. I realized also that you need to keep up with assignments in a math course. Calculus throws a lot of new notation at you and that requires a certain level of reading and understanding to wield correctly, so if you missed the first three assignments, you're not going to have a great time with the subject. You can't sit down and learn it all in one night, like some people think they can. I had one student claim "I'll to impress you", after me saying they should drop the course. Yes, that's true, you would have to impress me, but it's very unlikely you will. Their first exam came back nearly blank. I try to be very up-front about this and tell them the homework is the most important to do. For the students who did their homework it set them up to do great on the exams. People may not believe this, but in most of my classes my second exam average was in the 90's. This is almost unheard of... when I showed the exam to the department, since I was worried it was too easy, I got the response that "we have some engineering majors that would struggle with this." Which blew me away, but I didn't think it was that hard. I knew my students were doing great... but not that great.

Ah well... this is long enough. There are more reflections I have and maybe I will update again soon enough. I have some other topics brewing in my head and I'm feeling much more mathematically motivated. I'm also going to sit down and learn how to program in TeX so I can produce some great documents in the tutorial section!

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...