Only Part of the Picture
I remember when I was in secondary school, my class learned about conjectures in mathematics class. This section was probably one of the most confusing part of my mathematics class because there was nothing definite about it. I found it strange how we went about doing these problems, and I was happy that we didn’t spend too much time on it (a sign that I probably should have spent more time on it).
Just to give a quick explanation of a conjecture, it’s essentially a statement that does not have a proof, but states a certain property or relationship between things. For example, most of us have seen some form of proof for the Pythagorean theorem, involving squares and geometry. However, suppose you never had seen the proof and was told that the square of the sums of two legs of a triangle is the same as the squared length of the hypotenuse, that would be a conjecture.
If you thought it was false, you’d try and find a counterexample. After trying a couple of different triangles, you’d see that they do indeed fit the conjecture. At one point, you may throw your hands up in frustration and say, “All these stupid triangles fit the conjecture!”
You haven’t actually proved anything, but you now have a stronger suspicion that the conjecture is correct. You couldn’t just keep using triangles with different lengths if you wanted to prove this relation. You would have to take a triangle with arbitrary lengths and do some kind of proof to get to the conjecture. Then you could be sure of the conjecture, and it would become a proved statement.
Instead of learning that though, what I learned was that you needed to either find a counterexample (in which case you were then done), or you had to do three examples of the conjecture and show that it does indeed give the correct answer each time. Basically, it’s a glorified version of arithmetic, with only a bit of thinking involved.
I never got why we were doing this, nor how we were supposed to form a strategy for these conjectures. Was I just supposed to pick random numbers and see if they worked or not? Was I supposed to try and think of a counterexample?
What I think would have made much more sense now was if I was shown that conjectures are like “proofs-in-waiting”. They are ideas looking for proofs. Additionally, I don’t get why we were supposed to just examine the conjectures. After all, they are pretty boring on their own. Instead, we should have had to use those conjectures to learn how to construct proofs. It would have been a great example of the power of proving something.
Unfortunately, the closest I’ve come to proving something as an exercise on my own (read: not through the teacher’s notes on the board) was when I learnt how to do mathematical induction. The idea is that if your premises are true, than this is what you get. It was a powerful way to prove things that happen in series (such as repeated matrix multiplication). It gave me a taste of what mathematics can do.
Therefore, I believe we need to do give more effort to show young students the trajectory of what they will do with the tools they are learning now. Maybe it isn’t always necessary to complicate their lives, but I think it’s important to show them that these ideas will be used in the future. It gives them the sense that mathematics isn’t just a series of hurdles to jump through.
The secret to my skill in running is simple: I practice every day. Virtually every day I go for a run, which means I give my body more practice to become familiar with the act of running. Assuming I’m not solidifying bad running habits, it’s simply a matter of time that I improve. Since I dedicate so much time to running, my body is never “surprised” by the act of running.
However, this attitude is not often taken for school. Instead, we assume that once a student has passed a class, they have learnt that material from that moment onward. This isn’t necessarily a bad thing for those who have good marks, but when someone barely passes a course, it is probably unfair to say that they have learnt and absorbed all the content.
Personally, I find my mathematics classes tend to stay with me the longest, because they are used in all of my other classes and they “pile up” on top of each other as I delve deeper into mathematics. Still, there are some ideas that slip through the cracks and don’t get used often. Once that happens, it’s easy to forget how to deal with that kind of situation.
This is why I think it’s great to periodically look back and ask yourself if you’ve lost anything important that you had once learnt. If the answer is yes, take some time to go back and understand the idea. By doing this, you’re setting yourself up to forget less of the content you learned years ago. Additionally, it’s a good idea because older ideas you’ve learnt tend to crop up later on in different courses, so it never hurts to learn some material another time. Just a quick refresher can make a concept that much clearer in your mind once again, and it’s generally easier to relearn it than to go through it the first time.
Above My Level
As of this moment, I’m in my second year of undergraduate studies in physics. Therefore, I’ve seen a bunch of classical topics (electricity and magnetism, mechanics, wave motion, elementary astrophysics) while only seeing a snippet of “modern” physics (special relativity, mostly in the form of time dilation). I haven’t taken any classes on quantum mechanics, so I’m definitely a novice there.
On the mathematics side, I’ve taken the usual classes of calculus, linear algebra, and differential equations. In a sense, I’m still learning the techniques before actually applying them to topics such as topology.
As you can see, I’m just starting to really dip my feet into my mathematics and physics training.
However, this hasn’t stopped me from pushing my boundaries and learning about other topics. I follow a variety of mathematics and physics blogs that I try and keep up with while reading. Additionally, I watch many video series which are part of the new swath of educational videos, such as PhysicsGirl, Infinite Series, 3Blue1Brown and PBS Spacetime (probably the best channel that I’ve found online). I also read books here and there that push my comfort zone in terms of physics and mathematics.
The point is that I don’t back down from being completely bamboozled. In fact, I want to find those places where I’m baffled, because they point to new opportunities to learn, which is always exciting. I also think that while I may not understand everything I read or watch, it can spark my curiosity to learn about it some more. Sometimes, it’s good to just have your head explode from not understanding.
At the same time, don’t just ignore what you can’t understand and move on. Instead, think about the problem and what’s fuzzy in your mind, and you’ll usually be able to form a plan to figure out what you need to learn. From there, you can give yourself a boost in your education without necessarily waiting for the appropriate classes at school. This is even more relevant when studying topics that aren’t part of your field, since self-study is the only opportunity you will usually get.
Take advantage of the wealth of information available to you (much of it free), and learn. It’s probably the best investment of time you can make.
When I was in secondary school, I really hated going to French class (I still wouldn’t particularly enjoy it). It wasn’t that the teacher was horrible or anything. Instead, it was simply because I didn’t like taking language classes other than English. I really excelled in English, and the huge chasm in my abilities in French versus English weren’t something I liked being reminded of. In English, not only was I well read and could write, I could also speak well. Conversely, speaking was my weakest link in French. It’s frustrating being able to know all the words that you want to say in your head (and even being able to think them), but not be able to actually express them. As such, I didn’t participate in class at all, preferring to only listen.
The result of this was that I came close to getting a horrible mark in the class for never participating. Since I didn’t want that, I changed my habits about halfway through the term and participated as needed. I even found that participating made me more engaged in the class, rather than just waiting for it to end.
The thing I see over and over again in a science or mathematics class now that I am out of secondary school and into higher education is the lack of participation in class. And I know it’s not because the students aren’t listening or are bored. Rather, no one participates because we are scared to be wrong. Almost every time a professor asks a question in my classes, they are met with silence, because we never want to answer a question and be wrong. I’m definitely a part of this group, and I struggle with this all the time.
Rationally, there’s no better place to be wrong than when giving answers to teachers. If the teacher is doing their job correctly, a mistake in an answer is an opportunity to expand someone’s mind and teach them a way to think. Done wrong, it makes a student feel unwelcome to share their answers again, most likely closing the door on future participation. I feel this all the time in class, and it’s why I don’t answer all that often. However, I’d almost wager that I am one of the most frequent contributors, showing just how infrequent we participate in class.
Being on the other end of the situation, I know that it’s not a fun feeling when no one wants to answer the question you pose in class. As such, I struggle not answering in class. After all, it makes no difference if I get something wrong in class. Who cares if others think I’m an idiot for asking the question? Who cares if it’s obvious to everyone else, but not me?
Personally, I’ve always struggled with this to some degree. I’m not actually shy, but I’m always so obsessed with being “good” at subjects that I don’t want to appear like I’m a novice. Therefore, I try to hide my lack of knowledge by doing the “rough work” at home, so I can appear to be knowledgeable during class. The truth, of course, is that the class is there for me to learn, not to know everything before we start.
This is why I encourage participation in mathematics and science class. We need to stop being worried about what others will think of our hypotheses and ideas. Yes, people will judge them, but who really cares? On an intellectual level, I know that asking questions and giving answers in class (even if I’m not sure about them) is a way to help me, not hinder me. As such, it’s one of my personal goals to work on, and I hope you can reflect on this as well. Do you participate in school, even when you aren’t sure about your answer? Do you ask questions, no matter how dumb? If the answer is “no”, I encourage you to work on this with me.
Participation in class is a key component to learning. Do what most people aren’t, and use it.