A Sense of Play
In mathematics, there’s a sense of play that must be achieved if one wants to really understand what is going on in mathematics. Contrary to what people might think, this sense of play and “just getting” mathematics is not some sort of genetic feature (or at the very least, not only genetic). It’s the result of immersing oneself in mathematics and freely playing with the concepts. After a long time, this play translates into what people call “intuition”.
Have you ever opened up a graphing calculator program and tried out different functions? Have you looked at their features and thought, “this is interesting”? Have you ever opened up your mathematics book and simply read it of your own accord to see if there were cool things you had missed?
That is what it means to have a sense of play.
Having a knack for mathematics is not so different from being good at a sport, or a musical instrument. Sure, having the right technical skills is important, but being really good at something requires one to experiment and refine one’s process. In this sense, trying new things and playing with numbers, functions, and other mathematical concepts is key to improving one’s “skill” in mathematics.
This sort of aptitude doesn’t necessarily translate to solving more complex problems or suddenly learning new concepts faster. What it does do is give one a better way to “see” a concept, making it more approachable and familiar instead of challenging because one does not know how to proceed.
Develop a sense of play in mathematics and test things. Take out a piece of paper and try some equations out just for the fun of it, and you may surprise yourself by the relationships and patterns you find.
One of the classic methods of teaching I’ve witnessed in a science classroom is the “finish the sentence” method. Essentially, it involves the teacher saying a sentence and trailing off at the end while raising their voice in order to make it sound like a question, which prompts the students to answer. In and of itself, this isn’t a bad strategy. It engages students and makes them participate in a class discussion instead of having a teacher lecture the entire class time.
However, there is a fundamental issue with the strategy: the teacher wants this interaction to go smoothly. Said another way, the teacher wants the students to be right the first time they answer. Consequently, I’ve witnessed teachers unconsciously bias this process by giving cues to the students about the answer they are expecting. This is not ideal, because it takes away the sense of reflection that the student is supposed to take. It encourages efficiency over deliberation, which can lead to stupid mistakes. Worse, these cues can end up giving the students the answer the teacher seeks, allowing the students to faithfully reiterate it back and satisfy the teacher’s expectation.
Personally, the most nefarious cue I’ve seen is when a teacher basically finishes the sentence that they were planning to allow their student to answer. It will usually take the form of answer, followed by the question, “Right?” As a result, the student will most likely say that they understand, even though they might have no clue. With the answer being supplied for them, they grasp onto it, even if they have no clue what’s happening. I know this because I’ve seen it happen to other students, and I’ve done it myself too. The end result is an interaction that looks like the student is engaging with the content, but really the teacher is just supplying the information.
On the other side of the equation, I’ve found myself guilty of committing this error in teaching as well. When I tutor students and ask them questions to make sure they understand what I’m talking about, I’ll often nod my head or finish a thought for them that seems easy to me but what very well may have been a struggle for them (something I unfortunately tend to notice only when it is too late). I’ll then hit myself mentally, because I know giving my students the answer will only make things easier in the short run.
Therefore, I try to avoid asking the “Right?” question, and instead try to give more substance to the question so that the student has to actually think about the question. Furthermore, I do my best to not make students finish my sentences, because it puts them on the spot and makes them embarrassed if they don’t happen to know the perfect answer I am looking for when I ask. I know this happens because I’ve experienced it myself when a teacher asks a question and I can tell they expect me to know it. Worse, I may actually know what they are asking, but I can’t answer because the wording of their question may be strange to me. That’s why I believe trying to “guide” students along the right path while they are learning something new is not something to be done by making them finish my sentences.
Reading a Solution
When you have a lot of homework, it’s awfully tempting to just look up the answers. After all, you’re only doing it because you want to get other homework done, right? And if you take the time to read through the solutions, it’s good enough. You aren’t just copying down answers. You’re following the work that is done.
While the above situation sounds good in theory, in practice it is a terrible way to try to learn. It’s easy enough to find solutions to mathematics and science problems online, but finding them obviously diminishes the value of the problems.
If we compare the learning that happens due to working through a problem and the learning the happens due to reading a solution, I’m sure our intuitions will be correct. Working through a problem is a much better use of our time, even though it isn’t particularly efficient.
We’ve all been there: gazing intensely at a problem, frustrated that the answer won’t present itself to us. Then, usually just a few centimetres away lies a device that can find the solution for us. Why struggle when you can get through the small hurdle and continue on the problem?
My answer: tests.
Sure, it is nice to be able to look up just a “hint” to help you complete the problem that has been frustrating you for the last half an hour. However, the issue is that the “hint” most likely gave away the crux of the problem. Rarely are problems created only to test your ability to calculate one thing. Instead, they’re usually made to tackle specific concepts and make sure a student can recognize and deal with them. By turning to hints and help every time those points of confusion are reached, you’re robbing yourself of the skill needed to come up with an insight during a test.
Let’s consider an even more extreme idea: only reading a complete solution to a problem instead of even attempting it. While I am certain you will understand the solution (as you should, since that’s the point of a well-written solution), your ability to have gotten to the answer without any help has not improved. Instead, you’re fooling yourself into understanding how to do a problem, when really you only understand someone else’s solution to a problem. Therefore, reading a complete solution will make you think that you understand everything, but it’s more likely that you don’t know how to do it on your own.
Still, these online resources are good for something. I use them to verify my answers when I am done working on a problem. I rarely do what I described in the preceding paragraphs, because I know that when I reach a point of difficulty on a test, I can’t just magically skip to the next line of my solution like I can do when reading a person’s full solution. That blank space needs to be overcome first, and making the effort to practice writing full solutions by yourself without help is the key to being able to solve problems on your own.
Don’t just read solutions. Get good at making them.
Expertise Does Not Imply Teaching Skills
One of the incorrect assumptions I’ve long made is about teaching. Basically, I would get excited by the prospect of teaching some sort of concept to another person, and so I would work very hard on the presentation. However, there would inevitably come a moment when I’d realize that I wasn’t actually an expert in what I was talking about, so I would decide to stop the project.
My thinking went something like this: if I couldn’t be an absolute expert on the subject, then I couldn’t teach it. And obviously, since I am just a student and it is very difficult to be an expert in anything, who was I trying to explain concepts to people?
This crippled me for a while, and it still does (to a certain extent). The difference is that I’m now more aware of the truth, which is that teaching means bridging the gap between one level of expertise (a higher one) to the other (slightly lower, who is looking to learn). Therefore, the only requirement to be a teacher to someone is to know more than they do.
This is truly a new perspective on how to view teaching. It’s not about a bunch of students who know nothing that are taught by these experts who know everything. Instead, it’s about teachers sharing what they know in the hopes that the students can get to the same “level” as them.
Additionally, teachers don’t necessarily start as experts. Imagine you wanted to drive from your home to a new place. The first time you take the route, you’re constantly unsure of yourself, continually checking the roads to make sure you’re still on track. Now imagine trying to show that person the route after that first day. It will be somewhat difficult, right? However, if you’re asked to show them after doing the route a hundred times, there’s a good chance you’ll be able to show them the route with zero difficulty. Repetition breeds expertise.
The problem is that students don’t see this repetition. Unless they get a teacher who is just beginning, the teacher already knows what they are doing. Therefore, it appears to the student as if the teacher has no difficulty at all with the content. If a student then wants to teach (not necessarily as a profession, but perhaps just by writing articles, or tutoring, etc.), the illusion they’ve seen is that teachers are experts in their subjects, therefore they need to be an expert as well.
I can attest to this experience. I looked at my teachers, as well as other science educators online, and couldn’t imagine myself being in the same league as them. Looking at their content, I seemed so unsure in my information about science and mathematics, while those I looked to seemed confident. This stopped me from thinking I could be as good of a teacher.
If there’s a piece of advice I could offer, it’s this: don’t give up on teaching/education because you don’t feel like you have enough expertise. If you’re really passionate about teaching, the knowledge of the material will come in time. It’s a matter of repetition. Keep at it, and you will get better. As long as you’re teaching someone that is one “level” below you, consider yourself a teacher.