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.
Order of Operations
In algebra, there is a certain order in which operations must be done. If you’re reading this site, there’s a fair chance that you’re so familiar with this concept that it’s basically subconscious. If not, you’ve probably heard of the mnemonic BEDMAS, meaning brackets, exponents, division/multiplication, and addition/subtraction. These rules are formulated so that there is logical consistency in algebra.
I believe there is also an “order of operations” for how we learn science and mathematics. It’s not quite as strict as a set of rules, but there are definitely optimal paths to take.
In my mind, this is usually the best path to take: question, analogy, example, rigorous explanation.
A good way to start learning about a new concept is to ask a question. This forces a student to reflect on what would be the best way to tackle such a question. At the very least, it primes them to be receptive to the strategy. Additionally, relating the new concept to an older concept (when possible) is great because it gives the student a link to what they have already learned.
A good example that I recently went through is in my Electricity and Magnetism (E&M) class. I first learned about electric fields before moving on to potential difference. These two concepts are obviously related, so it made sense for my teacher to relate them. Plus, our knowledge of the former made it easier to ask starting questions about what to do to find the latter (such as the back-and-forth of integrating and taking the gradient of the electric field).
Asking a question as the first step makes the transition from known knowledge to unknown knowledge more seamless.
Depending on the context, an analog can be well suited to familiarizing students with a new concept. An analogy is powerful because it once again links something that a student knows with something a student doesn’t know. Therefore, employing the analogy early on gives the students a way to understand the new concept.
Reaching back again to my E&M class, this was well done with the concept of equipotential maps. These are lines that show the potential of various points in a given region of space (usually with a few charges in the space). The lines on the map represent points on the map where the potential is equal. It’s basically a contour plot if you’re mathematically inclined.
The analogy for this concept was a topographic map, which have the exact same kinds of lines as equipotential maps. The lines represent a certain elevation, and so one can deduce where the high points on the map are.
This is a powerful analogy because it immediately places the student in the context of topography, which people seem to intuitively understand. Therefore, this sort of abstract notion of a potential field can be linked to a very concrete idea of topography.
Much like a question, a good analogy gives students a chance to get to the same “level” of thinking as the teacher.
I’ve found that a lot of concepts tend to be overwhelming at first. Even with a teacher posing great questions and giving a masterful analogy, it can sometimes be difficult to understand what the heck is going on with the mathematics of the situation.
In this scenario, I’ve found it helpful to jump straight into an example. Doing so gives students direct manipulation into the equations or procedures that one needs, instead of going through a bunch of theory first. I like theory as much as anyone else, but it can be debilitating if one spend forty minutes going into deep theory and then suddenly jumps to an example of something looked at long ago. Therefore, jumping into an example early on rids the student of this long stretch of theory.
I’ve found this step to be especially useful in mathematics, when the concepts are nearly always abstract and so make it difficult to follow every little thing that is going on in a lecture. Additionally, many things that are written out in mathematics tend to be more difficult to understand than when actually applying it to an example, so it makes sense to start with an example. I like doing this in order to get myself use to the mathematics, so that when I start taking notes on the theory behind it, I can actually follow along with what I am writing.
Giving students an example near the beginning of the subject allows them to concretely understand what is going on before the theory is presented to them, dodging the whole bit where students become note-taking zombies.
The final piece of the puzzle is the in-depth explanation. I save this for last for a simple reason: students should be able to follow along during these explanations, and the previous steps give them the best chance of following these explanations without issue.
This is the place where formal definitions will be given and equations will be derived. The idea is that once a student has a sense of what is going on or is trying to be achieved, then one can make an effort to formalize the concept using mathematics. By taking the informal approach first, students stand a better chance of remembering what is happening.
The best test I know of to see if students really understood a concept: do they have to flip back through their notes in order to read everything that was done? If so, then there’s a good chance a lot of the explanations went over their heads.
Start with questions, analogies, and examples, and students will understand what is happening more quickly.
Appreciating the Why
If I wanted you to calculate a bunch of derivatives for me, I could simply show you the algorithmic approach to the process and leave you to do it. With some practice, you’d get better and better at taking derivatives and could even become better than a mathematician.
However, the drawback is that you wouldn’t necessarily understand why this algorithm worked. You would get the correct answer all the time, but you wouldn’t know why it was workings. In essence, you would simply be defaulting to the process and trusting that this works. In a way, you’d be taking what I say is the method to perform derivatives on faith. Since you trust what I say, you would believe that what you’re doing is indeed taking derivatives of functions.
Unfortunately, this decision to “take the equations on faith” is a reason why so many people in our society look down at scientists and mathematicians who say, “look at the equations” when posed a question. When giving this response, we know that people will mostly not take the time to understand what the equations mean. Instead, they will either take it on faith, or accuse us of taking equations too seriously. This can be seen from those who say scientists are slaves to their equations and that nothing else can change their minds.
Whenever this kind of sentiment is expressed, I try to counter it. The reason is simple: the person making that claim doesn’t fully understand what equations are (at least, in science).
Often (and particularly with phenomena that are discussed in one’s school year), equations don’t just come out of the blue. They aren’t simply handed down from the sky above and become law for the rest of time. Instead, equations are consequences that stem from the relationships of different variables. They aren’t put together because they look nice, or even that they necessarily produce the correct result. Each component of an equation means something very specific. That is, if you want to prove the equation, you need to explain why everything is as it is. There aren’t any free passes. As such, an equation doesn’t force nature to be that way. It is the result of the pattern that variables tend to follow.
This is why I always want to understand how the equations I learn work. I’m currently in a statistics class where all the formulas are given. And while computation is the point of the class, I’d like to understand why they are as they are. This isn’t within the scope of the class, and it has bugged me all the time.
Realistically, I don’t need to fully understand a proof if I just want to get a result. It’s not necessary, but it is nice, because understanding the mechanics of an equation means it is easier to work with and modify. That’s why I try and take the time to really understand a proof when presented with something new.
I don’t trust my equations blindly. I use them because they work and because they mean something. If I had it my way, I’d make every equation super simple. Unfortunately, the reality is that many equations are not like that, and instead are giant messes. However, as my teacher likes to say, “I’d rather have it be ‘right’ than ‘nice’ every day of the week.”