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.”
Building On Top of Each Other
I once had a mathematics teacher who would say something that bugged me: what we’re doing is easy. I am barely being hyperbolic when I say that this teacher would say this for every single concept we learned. Therefore, I couldn’t help but think that surely not everything could be this easy.
The saying would take on an ironic meaning to me, because even for the most difficult concepts we would learn in the class, my teacher still saw it as easy. I would frequently turn to my friend and exclaim, “How can it all be easy?”
He had no answer for me, and I thought that there wouldn’t be one. However, after finishing the course and reflecting on my experience, I can start to see what my teacher was talking about. It’s not that the concepts are super easy to understand. Instead, it’s that the leap from what we previously knew to what we learned is not huge, so it shouldn’t be too difficult to understand the new material.
Here is a concrete example. When I was in secondary school, I learned about algebra and functions. At the end of secondary school, I learned about more complicated functions and curves (quadratic, hyperbolic, and ellipses), plus I was introduced to the notion of vectors.
In the beginning of CÉGEP, I learned how to manipulate vectors by addition, subtraction, dot products, and cross products. This was mainly seen through the lens of physics, but also through my mathematics courses.
Fast forward to my final mathematics course (Calculus III), and I began to learn how to put these ideas together. Suddenly, vectors weren’t static anymore. They were affected by parameters such as time or angles, meaning they became curves.
Now, imagine trying to show this to a student who doesn’t already have a good grasp on vectors or basic functions. It would be basically impossible. The leap from knowing nothing about vectors to working with vector-valued functions is too drastic. However, if one progresses from simple functions and vectors to these more advanced topics, it’s more of a transition than a leap. The topic that proceeds makes sense given the previous material that was seen.
For myself in that class, it meant that once I learned the polar coordinate system and graphing in three dimensions, everything we did was both in Cartesian and polar coordinates. Taking derivatives or integrals was done with both systems because it was the obvious next step to take.
I can now say that I understand what my teacher meant by the material being easy. It’s not that it would be easy to anyone, but that it should be easy for us, given our progression (and since I had the same teacher for all my mathematics classes, I know they had a good idea for the progression).
The implication of this statement though is that the fundamentals are so important. If one wants to make the next logical transition to a new concept, the previous concept must be understood. By rewinding the clock all the way back, one arrives at the absolute beginning of learning the first important concept. If this concept isn’t understood well, then it can have consequences down the line in terms of why a newer concept doesn’t make sense.
Let’s face it: school isn’t made for us to fail. It’s designed in a way such that a student can succeed. Therefore, the progression should be appropriate, and the responsibility is on the student to understand the fundamentals before moving on. Yes, this can be difficult when one is in a class that seems to be moving on despite you not understanding, but that’s when either the teacher or the student needs to step up and take a moment to review. If not, a student is just pushing their problems down the road.
If you feel like you’re making a huge jump in your learning that you don’t understand, it’s likely that the fundamentals you learned aren’t completely absorbed. Strengthen them, and those leaps will become baby steps forward.
The Skill of Spotting
If I had to pick one skill that I’d say helps me out in science and mathematics the most, it would be the skill of “spotting”. Simply put, it’s how good one is at figuring out the essence of a question. Personally, it’s a skill that I cherish, because it allows me to do so well during my exams. Rarely do I read a question and go blank. Instead, I’ll immediately get to work, going through the steps needed to solve the problem.
Of course, there are really two ways to solve a problem: the “make it up as you go along” strategy, and the “spotting” strategy. The former can work, but it’s likely a longer method and can result in a lot of dead ends or useless work. However, the “spotting” strategy allows one to know exactly what the steps are to solve the problem.
For example, if I’m given a problem on mathematical induction, I know the steps I need to do. First, I write the base case out to show that it is indeed true, and then I do the calculation with $k=1$ and $k=2$, just to show the pattern seems to hold. Next, I write down the equation with k, and then I try to solve it by induction using k+1 in conjunction with my base case and assumed truth of the statement with k.
This is my “format” for solving induction. What you can notice is that there isn’t any numbers or specific examples in the above case. It’s just a template for solving problems involving mathematical induction. By “spotting” the problem, I can just apply this template and be confident that I will get the correct answer.
This is why the “spotting” strategy is so good. It takes the thinking out of the present moment by referencing a template for a specific kind of problem.
So how do you cultivate this skill? The best answer is that you have to do a lot of problems. By practicing over and over, you will begin to sort out problems into various categories. You’ll start recognizing exactly what you have to do when you get a problem with acceleration, velocity, and position, or when trying to find a tangent plane to a surface. The values and functions might be different, but the approach is the same. Slowly, you will amass an archive of examples that you can reference in your mind when faced with a new problem.
When I enter a test now, I always try to spot. If I’ve prepared well for the exam, this shouldn’t be a problem. After all, the content of the exam is stuff I’ve seen, so there’s no reason I shouldn’t recognize it. That’s not to say that a certain question won’t be difficult, but the odds are with me that I’ll be capable of spotting a problem.
This calms me down immeasurably during a test. Since I’m hyper-focused on getting a great grade, it’s a relief to me when I know exactly what I have to do to solve a question, and all that’s left is for me to go through the motions. Basically, I practice a lot so that I can offload the work on the test to my vast archive of previous examples in order to solve the question.
My advice to you is this: if you want to have an easier time during tests, learn to spot similar questions. It will save you the mental energy of always figuring out from scratch what you should do. It’s like using formulas. After you’ve been exposed to the proof, there’s no need to prove the formula you use on every question you answer. You just use it. Likewise, don’t waste time reformulating the same strategy over and over again to solve a question. Use the one that worked, and just apply it.
It will save you a lot of time and make tests go that much more smoothly.