Lead By Example
In my mathematics classes in CÉGEP, a lot of the content became more and more abstract and theoretical as I learnt more and more about calculus. As such, it became easy to lose perspective of what I was actually doing, since a lot of it was simply symbols. Fortunately, my teacher understood this and gave us plenty of examples to learn from.
I’ve been thinking about this topic in the context of first learning content. Here’s a question: should you start by giving students definitions, or examples?
Arguably, the former gives students a more complete view of the mathematics. After all, a definition is created precisely because people wanted to know when a certain concept applied or not. Therefore, a definition should be useful for students to figure out precisely what does and does not apply in the situation.
While that may be true, it’s probably not the way to introduce a new concept to students. I doubt I’m an outlier in saying that I always enjoyed seeing a graph or a picture of what we were doing in my mathematics class versus getting a rigid and complete definition. It’s not that the latter was bad, but that it didn’t give us any sort of intuition as to what we were doing.
For example, when I began to learn about derivatives and integrals in my calculus classes, my teacher didn’t immediately jump to showing the general definition of a derivative or integral. Instead, constant examples were provided, giving us a feeling for what these two mathematical concepts were. It wasn’t general by any stretch of the imagination. Instead, it was specific, and it gave us a foothold in the concept. Then, my teacher was able to step back and give us the general definitions.
In many disciplines, the opposite approach is what generally works. First, you focus on the general concepts, and then you drill down into the specifics for your goal. However, it’s easier to do the reverse in mathematics, because seeing an example (preferably, an easy one), allows a student to look at the example and think, I get the idea, how can we generalize it?
This is so crucial in the beginning of learning a mathematical concept, because definitions can be daunting. Often, they look like a bunch of symbols that just don’t make sense, whereas one can concretely get the idea of concepts like the directional derivative or the gradient vector by looking at examples or drawing graphs. Sure, they can also be understood by definitions, but it’s much more difficult to process. In general, we excel at absorbing information in a visual way, so it makes total sense that we would look at examples before generalizing.
The key idea here is that mathematics is about zooming out. We want to generalize information, formulas, and theorems as much as possible. It’s much more satisfying to have one equation that covers a whole gamut of possibilities instead of having one specific equation for each possibility. However, grasping these general equations aren’t immediately easy, since definitions are abstract. Therefore, the use of examples before broadening the picture allows students to understand what is happening for a specific concept, giving them a better idea of what the different components of a definition or theorem do.
If you just throw definitions and theorems at students, they will have a much more difficult time to grasp a concept than if you draw a simple graph or diagram.
Is always on the winner, on the best person or idea or organization. The person in the spotlight is the person who we think is the best.
However, what we often miss is the person just outside of the spotlight, the one who didn’t make it. The spotlight is narrow, covering only the best of the best, and nothing else. But others are still present, and have done things that are almost as good as the one with the spotlight, yet they are in the dark.
At first glance, the system seems fair. The best thing gets highlighted, and that’s that. But what we fail to see is how close a lot of these others are, and they don’t get the spotlight. By all accounts, these things are really good, yet they are all outshone by the one. A waste, really.
Of course, it’s fair that many will see this as the critique of someone who has come in second place and who is a sore loser. I can understand that. However, I think we need to be honest with ourselves when we lift one thing into the air above the rest, forgetting the others surrounding it.
One particularly salient application of this is in looking at people who are rewarded for being the “best”. This can be on academic grounds, or for any other reason. The result is the same: plenty of other hardworking people are left in the periphery, barely noticed or not even seen because of the spotlight on the best person.
What I’ve learned from this is that you cannot attach who you are to the kinds of results you get, particularly when they are compared to other people. That is a surefire recipe towards being discouraged for not being noticed.
Instead, I need to find meaning in the work I do, regardless of what kind of external circumstance given. Just like a race, I can control my effort, but I cannot control the environment and the weather. Similarly, I can do my absolute best to get the greatest results possible, but I can’t guarantee that this will make me the “best”. Therefore, comparing to others doesn’t matter (or, I’m trying to make it matter less and less to me).
The external variables aren’t controllable for us, so we need to let them go. By focusing on the work we do instead of if external variables align with our goal, we can make more progress. Furthermore, we can enjoy our own results more, instead of looking at the spotlight and being disappointed that it isn’t on us.
The spotlight is a very narrow instrument. It’s not made to highlight more than one person, idea, or organization. Therefore, it misses so much, which means attaching importance to only the spotlight is a misleading thing. Better instead to appreciate one’s own efforts, without looking at the spotlight to be that judge.
There always is a single “best”, but that doesn’t mean everyone else is unimportant. Just because one person wins a race, it doesn’t mean that the other runners did not have a good race themselves.
The spotlight will always be there, but you can choose the meaning you want to assign yourself, independent of the spotlight.
Have you ever asked yourself if you were all in? If you committed yourself fully to an endeavour, playing the long game and working to achieve big goals?
I’m sure that I am not alone in guiltily shaking my head at that one.
As much as I admire those who seem to go all in with what they love to do, I’ve always been too scared to do the same myself. If you’re anything like me, you worry a lot. Putting all your eggs in one basket is not what you do. Instead, you try and build fail-safes and backups, ensuring that failure isn’t too harsh.
This has manifested in my life through taking my passions only semi-seriously. I don’t commit fully. Instead, I dip my toes in on a bunch of things that interest me. This has the advantage of making my activities frequently appear novel, since I’m always switching from one to the other. On the downside though, I never am able to make the breakthrough that I want.
To get past the current plateau, this is an issue that needs to be resolved. For myself, that means picking a goal in running for 2017 and sticking to it. It means signing up for a race, and not “waiting until the race approaches to see if I am ready”. The point isn’t necessarily to be ready, but to compete on the given day and give what I can. Too often, I find myself getting psyched out because I’m thinking about what I want to do too much, to the point of detriment. What I need to do is decide, commit, and do. I need to think less and focus on the process. Long term goals are seldom accomplished by accident. They require a commitment to the process of improvement, and being only semi-serious means you are sabotaging your own goals.
Unfortunately, going all-in isn’t as simple as this. The difficulty is that fully committing makes you vulnerable. It’s saying, “I care enough about this goal that I am going to put myself out there and attempt it, even though I fully know I might not achieve it.” Even at this stage, my writing isn’t getting circulated at all. I write, but no one is reading. This is because I haven’t fully committed. I’m serious about writing, but not to the point that I’ve gone all-in. It’s a future want for me, but at the moment I’m working on other goals.
Committing isn’t easy, and it is extremely easy to talk yourself out of it by thinking too much. If I thought about how I run 130 kilometres every week, I don’t think I could commit. However, I can easily think about committing every day, so that’s what I do. The end result is 130 kilometres a week, but I’m not thinking about all the work that is left to do.
If you want to further your passions and become better at whatever you do, going all-in is the best way to improve. It’s not easy, and you will be making yourself vulnerable, but you’re also giving yourself a chance to improve. The key is to not overthink it. Jump right in, and focus on the process instead of the long commitment.
When I was in elementary, learning the basics of arithmetic was an important component of my mathematics education. I participated in various mathematics “challenges”, where I tended to do pretty good. I like the rush of having to beat someone to an answer in a competitive setting, so I became good at it.
Fast forward a few years (and even to today), and people seem to be astonished when I crank out answers to arithmetic faster than they can input the numbers into their calculator. Honestly, I’m not even that fast or that good, but knowing some simple patterns in counting allows me to appear as if I have super powers.
What I find is so interesting (and unfortunate), is the reaction that these people have. Once they see that I can compute quickly, they tend to say, “Wow, you’re pretty good at math!” This is a nice complement, but they miss the point of what it means to be good in mathematics. On the one hand, speed is important. After all, if you considered two people answering a multiplication question, I doubt you would say that the person who answers the question correctly, but slower, is the better person. You probably wouldn’t say that they have the same ability either, because the first person answered quicker.
This is true for brute mathematical calculations, and those become the work of computers. The faster, the better. However, the idea is that what you will focus on is answering questions that are much more deep and complex. You will need to develop your problem solving skills to figure out the answers to questions, something a computer won’t be able to do for you (at least, for now). You may not be faster than another person, but it doesn’t matter because you are tackling brand new questions that nearly nobody else thinks about. Yes, speed is important, but an awareness of the strategies needed to solve the problem are just as crucial.
In essence, being good at mathematics isn’t about crunching numbers quickly, it’s about knowing the process and being able to “spot” what tools are needed for the problem at hand.
Unfortunately, the present situation is that we call those who can quickly compute numbers early on as mathematical “geniuses” who are just so smart. What this does is encourages the ones who are doing well (which isn’t a bad thing), but discourages those who can’t calculate quickly yet (which is a bad thing). This distinction makes it possible to push students away from mathematics because they don’t feel they have the right “stuff” from a young age.
Instead, we need put forth the message that calculating quickly is great, but it’s much better to get the process right first. It’s better to get the right answer in a long time than the wrong answer in a short time.
Basically, speed matters in mathematics, but only for the sake of increasing productivity. While learning, it’s more important to teach the process and not alienate students because they aren’t deemed “smart” since they can’t speed through mental arithmetic. That’s not what mathematics is about.