This summer I taught lower division differential equations. The purpose of this post is to share my experiences and recommendations for teaching mathematics with differential equations as an example.
In summary I emulated the things that I’ve seen John Baez do in classes I’ve taken from him. I consider the most important lesson I learned to be the way in which he does homework. Briefly, homework should be like a story that leads the reader into learning something new. I’ve attached my homework assignments at the bottom of the page for anybody who wants to see how it works without having to read my entire post where I’ll go into more detail.
I had been a teaching assistant for this class so I knew a little bit about what troubled students and about what I should cover. Additionally I had some discussions with another graduate student, Andrew Walker, who taught this course the previous summer. From our conversations and my personal experience I believe the following things to be useful notes.
1) Instructors artificially increase the difficulty of this class by making problems that require difficult integration techniques.
In my opinion the point of a basic differential equations class is to learn how to solve some differential equations, not how to solve difficult integrals. Difficulty can be increased in other more productive ways that deal with concepts strictly having to do with differential equations. Students get so bogged down that they leave the class without ever grasping what they were supposed to. It seems like many students finish without understanding what it means to solve a differential equation, what a solution is, or even what a differential equation is.
The solution to this issue is simple- when integrals are necessary for solving a technique make sure problems being done have easy integrals. It’s a bit painstaking to create problems which satisfy this condition, but I believe the payoff is absolutely worth it. (Note: Rarely tough integrals are unavoidable!).
2) Students only write down what you write down.
Even the best students can’t write as fast as someone can talk. Math instructors are too often guilty of haphazardly writing down concepts while verbally explaining important points or sloppily writing down a solution without a single word on the board. The verbal explanation can be enlightening and perfect, but if the students can’t copy it down then they probably won’t remember it!
While teaching differential equations I wrote down every single concept that I wanted students to understand in complete sentences and in complete detail. I wrote down every single example I did as complete thoughts and sentences which brings me to the next point.
3) Writing complete sentences to explain math is a must.
It’s best to illustrate this with an example. I may eventually upload my notes, but for now the following will suffice. Suppose we had a question that said, “Solve the differential equation ”
Many professors will teach this in the following way. They will say (but not write) that this differential equation is separable and explain things (verbally) about separable equations while simply writing on the board-
The student will then write the same 4 lines in their notes. Instead the instructor should write the following on the board-
is a separable equation, we can “multiply” to get
then we can integrate both sides to get
This problem is simple so it’s possible a student can have a perfect understanding of what happened without the words, but difficulty increases it becomes more necessary to write complete thoughts and sentences so you may as well make it a habit during the easier topics.
Writing out problems like this serves a few purposes-
-By doing this enough times they come to understand mathematics is not just lists of equations. It’s about presenting something true in a very logical way that flows from a clear beginning to a clear end. They work the same way sentences do. Sure I could read a sentence with a bunch of things missing and get the gist of it, but whoever wrote it doesn’t really understand how sentences are used to express their ideas. Getting the final answer correct is much less important than having an understanding of the concepts!
-When problems are more complex students have notes that make enough sense to be useful. Words as simple as “since, then, thus, if” etc. increase understanding by a huge amount because the notes become readable. When looking back at notes they can easily follow along from start to finish instead of just seeing a jumbled mess of equations. The reality is that most will never open a textbook so this is their textbook.
-When they do problems on homework, quizzes, and tests, their solutions look much better because they start to solve problems in the same logical way. Their thoughts are clearly expressed. The lack of complete thoughts being taught to them from elementary school onward is evident in the work they turn in. It looks exactly like what they are told to do. That is, write down messes of equations that make no sense on their own and then a box around some final answer.
Besides writing nice details one should also reiterate additional details (like context) verbally while writing down the nice solutions, but whatever the additional details are should have been written down somewhere earlier when the concepts were being taught. The verbal repetition is just there to nail the ideas (which had already been written down!) into their head.
4) Assign written homework which acts like a story.
Traditionally, instructors will teach a section and assign a huge amount of problems which require the student to do the same thing over and over that they learned in the section. The idea is to get some process stuck in their heads.
The problem here is that different students need different amounts of problems until things click. Thus assigning a certain fixed amount is usually very annoying for students. The ones who already get it will be annoyed that they have to keep doing something that know how to do (over and over and over). Often they’ll not really know how to do it and just keep doing problems incorrectly which creates a bad habit! The ones who don’t get it will be frustrated that they can’t even start the assignment and still have so many more problems left to do.
It’s pointless because once you can do a few problems in a section that usually means you can do all of them.
I’ll try to explain my solution to this issue, but it’s better to see an example in the links at the bottom.
Assignments should remind students about what they’ve been doing in class with a brief explanation. Questions should lead students through the step by step inner workings of solving a problem without telling them too much. After they’ve gone through questions the assignment should explain to them what they just did and how it relates to other things they know.
This type of homework is more engaging and interesting without being tedious and repetitive. Unfortunately though repetition is also necessary at some point. For this reason I also added some practice problems which are not required to be turned. My hope is without the burden of a due date the problems will seem easier. I also asked the teaching assistant to go over these during the discussion section in class to give them more help with the boring repetition.
I believe using the discussion section like this was very helpful, but I realize not all people have this at their schools. Speaking of discussion sections, I had them do 1 question quizzes at the end of discussion sections in order to force attendance. They were supposed to be similar to practice problems that the teaching assistant went over, but also way easier. The students knew it was coming so hopefully would pay attention to the practice problems and then be able to do a simple problem.
Finally, here are all of the assignments that I wrote. They work better with my specific notes because they tie into things that I did in class, but my notes probably won’t ever be typed up.
It’s possible that there are typos which went unfixed and I’m sure things could be explained better by someone more experienced. Additionally I would have liked to cover more topics, but it was my first time running a class so I didn’t have a good sense of how much time I had to work with.
If you are interested I can send you the latex files.