I remember when I first took a class in linear algebra, and we were talking about vector spaces. In addition to the definition of a vector space, we were also given multiple axioms that define what the structure of a vector space looks like. This included a bunch of boring things, like the fact that if you have a vector v, you should have a corresponding vector -v such that v+(-v)=0, where 0 is the zero vector. There are eight of these axioms, and together they describe exactly what can be called a vector space.
If you look at the axiom above, you might think that’s obvious. Of course I can go backwards from my vector v to get to zero. Similarly, you would probably balk at the notion of most of the other axioms, such as commutativity (a+b=b+a) or that there is a zero vector such that v+0=v. These properties seem more than obvious to you. They are ingrained. Why do these define a vector space specifically? Doesn’t this always hold for any kind of space?
Well, not quite. If we just focus on commutativity, this is something that isn’t always true. To take an example that isn’t part of mathematics, but is something else we do in our everyday lives. Imagine you were eating a bowl of cereal in the morning. After taking out your bowl and putting cereal in it, you then pour your milk, followed by eating the cereal. But what it you did those last two steps in the opposite order? Now, you eat your cereal, and then you pour your milk in the bowl. Evidently, your cereal-eating experience won’t be the same in both cases. This illustrates something we don’t often appreciate in mathematics. Order is important. It’s just that we tend to work in spaces that have this nice and special property, so we start to think that all spaces do.
If you want a more mathematical example of commutativity not holding, the classic example is that of matrices. This is another part of linear algebra, and while taking the course you quite quickly figure out that the order does matter when multiplying matrices. Suffice to say, bad things happen when you start assuming things will work out without knowing if your assumption is correct.
Why is this important? We often go through a lot of our mathematical lives without considering our assumptions. Instead, we focus on solving particular problems, and then comparing with a known answer. Furthermore, we tend to get used to thinking about the particular spaces we work in, and assuming that these nice things will hold for all other spaces. This lack of diversity creates bad habits in our mental models, creating prejudice for one specific way of things working.
Remember, axioms are the foundation for basically everything you do in mathematics. If you keep on digging, you should always hit a floor of axioms. This is simply because the goal of mathematics is to discover truth from those axioms. These axioms can’t be proven. They are taken for granted. But the surprising thing is that the rules and elements we take for granted on a day-to-day basis within mathematics aren’t axioms. They are actually statements that can be proved using other axioms.
For example, take an innocent-looking statement that almost no one will blink an eye at while reading:
If a ≠0 and ab=ac, then b=c.
This is known as the law of cancellation. We get quite good at doing this in secondary school, where we learn how to simplify fractions and other expressions. This naturally comes in handy, because it lets us avoid working with larger numbers than we have to do. If 2x=8, we can “cancel” a factor of two from both sides of the equation and obtain x=4.
So far, so good. This is harmless, right? How could a simpe rule like this possibly go wrong?
The problem is that you’re not thinking about this in general. The subtlety here is that it matters where the elements a and b come from. If they are part of the real numbers, then fine, that’s not a problem. But it takes a particular type of “setting” to have this cancellation rule. In particular, you need to be in an integral domain, which is a space that is basically defined by this property.
And no, there are very real examples that aren’t integral domains. For example, let’s do some arithmetic in modulo eight. This just means that we reduce any number until it’s the remainder upon division by eight. In this setting, we then have that 6 ≡ 14. But look what happens here. We know that 2⋅ 3 = 6, and 2⋅ 7 = 14, but even though 2⋅ 3 ≡ 2⋅ 7, we certainly don’t conclude that 3 ≡ 7. When you divide both of these numbers by eight, the remainders are three and seven respectively, so they are not the same. As such, you can immediately see that the cancellation doesn’t work all the time.
What this hints at is that we can often be surprised by our prejudices. The cancellation rule is very common and is acknowledged throughout secondary school, so you might almost feel cheated that this is suddenly not taken for granted. This is a normal reaction, and it’s one you will slowly learn to ignore.
This is why learning and remembering those “boring” axioms is so important. They are the building blocks in which we base our structures on, so it’s vital that you remember them. Additionally, you should frequently pose yourself questions such as, “Why do we require this axiom? Can I get all of the results I want without it?” Remember that we are frequently interested in generalizing notions in mathematics, so we don’t want to carry along “extra baggage”, if you will. We want enough axioms to give us some constraint and structure (or else we won’t be able to generate results), but we don’t want to assume anything that we aren’t absolutely forced to. It’s about minimality, and only adding structure when you need it. So all of those axioms you learned in various classes, were important, because there are other settings where these aren’t taken for granted.
It’s a mathematical jungle out there, so best to keep in mind your foundational axioms while learning a new topic.
There’s a saying among students regarding preparing for an exam. In short, it goes like this: Study a lot before the test, and then you can forget most of what you know.
I think of tests as barriers that I have to get past. For example, I took seven classes this past semester, and so I had many tests and final exams. The back half of my semester involved a lot of studying in order to get over these barriers of tests. This often involved studying a lot before the test, and much less after.
For a while now, I’ve been quite disillusioned about the nature of tests and grades. I know I come from a place of a lot of privilege, since I’ve benefitted quite a bit from having good grades (through scholarships), but I can’t help but notice the dismal state of affairs with regards to this system. I think of myself as someone who loves to learn, and I use a good chunk of my day to study and do homework. Yet, I can’t say that I don’t share the mindset that I wrote above concerning tests. I will study a lot before a test, and then let a lot of that specific knowledge drift away after the test. Of course, I know that I retain some of the skills and knowledge that I’ve learned, but I’m not going to delusion myself into thinking that I don’t see tests as annoying barriers to get by. Tests (and the proxy of getting good grades) is the current “carrot” in the system.
However, I’ve found that this is completely misaligned with some other, more important, incentives. For one, I would ideally want to be capable of learning a subject and explore any particular rabbit hole that catches my intention. Instead, we have to go through a certain course outline and check off a bunch of things before the end of the semester. There’s also the fact that tests often aren’t a good reflection of learning the content. Instead, I would argue that they show that one can write solutions to problems in a limited amount of time in a high-stress environment. This is the easy choice if we want a system capable of comparing students and giving them “rankings”, but I would argue that it isn’t useful to learning the subject.
The other thing that bugs me more and more is how we often say that science should be a collaborative process that is better off when people work together, yet we will give examinations where there are very tight restrictions on what is allowed. For example, it’s almost a no-brainer that students can’t talk amongst themselves during an exam to discuss the problems. Furthermore, and suggestion of being able to use electronic devices would be met with strong opposition. But why is that?
Of course, some will argue that this would lead to easy cheating. Student’s will be able to “get through” the filter of a test without actually knowing the subject. This is true, but would that really be a problem? If a student is motivated enough to cheat, then they will pass the course. But down the road, this lack of knowledge will catch up with them. Once we get to post-secondary education, I’d say that those who are there should be motivated enough to learn such that they wouldn’t cheat1.
Additionally, I don’t think this would magically raise everyone’s grades to perfect marks. My argument here is based on the way assignments are done in classes. What I’ve observed in my own classes is that there is a spectrum of students. Some don’t really care about how much effort they put into assignments, while others will spend hours on them. (I recently spent nearly forty minutes going over one part of a problem in which I couldn’t find my error. I’m certain that not everyone is as patient as myself.) This is reflected in the marks students receive on these assignments.
But here’s the funny thing: it’s not like students are banned from discussing the problems with others, using online resources, or even asking the professor for help. If a student really wanted to, they could easily go online and find the answers to just about any textbook question, with a detailed solution. Suffice to say, students aren’t exactly in a test situation while doing assignments. Yet, I always see a distribution in terms of grades when we receive our assignments back. So what gives?
One aspect admittedly is that assignments aren’t weighted as much as the final exam (and other tests). This means that some students have a lower incentive to actually complete them. However, I want to submit that this isn’t entirely what is happening. In addition to students not caring as much about assignments because of the weight, students don’t write solutions to assignments in the same manner. This might seem trivial at first glance, but it’s really a huge aspect. If you want to convince someone you are correct, the best way to do that is through a thought-out argument. Sure, you can probably get the same answer while doing some sloppy work, but it’s not going to be as polished and strong as others. This is the important bit in education. It’s not about if you can get the right answer. Education is about being able to think critically and present your arguments in a compelling way.
I’ve talked to several of my fellow students who do marking as a side job for professors, and they will often say that the quality of a student’s work is highly variable. Some have obviously copied their answer from other places, while some only scribble the minimal amount of work without any indication as to what means what. On the other hand, you will find students who frequently write out detailed and clear solutions. Mind you, this doesn’t necessarily imply that longer is better. Being succint can be just as great, as long as one can follow the argument. The point is that we can start grading with respect to this variable, which is something that is only partially done, and with a lot of deference to the correct answer.
When I write my solutions to problems, I like to think that they are detailed and clear. However, this almost entirely goes out the window when a test arrives. Time constraints are an enormous stress on a student, and it makes the work worse accordingly. This is tragic, because the most important problems in a class are those in which the students don’t get time to mull over the presentation of their work. Think about it. The weighting of a course grade is almost always heavily skewed towards tests and exams, where time constraints are often strict and cause students to rush. I write these words hours after I’ve finished my last test before my final exams (long into the past by the time you are reading this), and the time limits definitely affected my clarity. When you are under time constraints, it’s easy to rush into solving a problem without taking a moment to fully pause and ponder what you are doing. Every moment spent idling is one moment less to write the rest of the test. This is not a good recipe to getting students to learn and understand in the long term. It might be a good method to get used to being under duress, but I would argue that this isn’t what we should be searching for.
Historically (and at our present moment), this is something we aren’t doing well. We have misaligned our incentives such that they are easy to capture and process (grades on time-constrained tests), yet don’t give students enough time to fully think about what they are doing. One of my professors stated this misalignment in a way that stuck with me. He said, “You usually do more difficult work on the assignments because you have time and can work through the problems, but the tests are simpler.” I don’t know if he realized it at the time, but this is a crazy thing to say if our goal is to get students to understand subjects more deeply. It all comes down to honesty. Do we want students to have the required material internalized for a test, or do we want them to really think about what they are learning, past the final exam and into the future?
We are missing a huge opportunity here. I envision an alternative world where the objectives of a course aren’t measured through tests given at the end which are highly stressful on a student, but instead by evaluating students during the entire semester. It’s not that suddenly everything needs to be graded and you can’t get anything wrong. It’s about replicating the kinds of environments that reflect learning in the world, which aren’t closed-book exams under a time limit. Those are artificial constraints that serve to make students stressed, and it creates an incentive to study for that one test, versus thinking more deeply on the subject mattter.
The issue is alignment. Do we want to emphasize thoughtfulness and hard work during a course, or do we want students to prioritize one large “study period” at the end of the semester, and letting them forget a good chunk of the material after? If we ponder this question for a while, the answer becomes pretty clear. The current manner of doing things is great for efficiency, but it isn’t conducive to what we really purport to want with education. So let’s start trying to shift this trajectory, and get to where we want to be. It is possible, but we need to get past the inertia of the regular expectations.
By which I mean simply copying without actually understanding what is happening. ↩
Stating One's Goal Clearly When Solving a Problem
Everyone solves problems differently. Some like to work directly with the mathematics head on, while others prefer to have a more intuitive approach. This includes studying simpler cases of a problem, or looking at examples in order to really understand what’s happening. These are all valid approaches, but the point I want to highlight is that these are all strategies. There’s a certain method to tackling a problem. It’s not that you can’t solve a problem through trial and error, but if you want to solve more problems more quickly, your best bet is to figure out a strategy.
If there’s a “most difficult” part in solving a problem, it’s usually the first step. When you first encounter a problem, there are countless directions that you can go in. This can lead to paralysis, but it can also lead to trying things randomly and hopefully get something to work. In particular, I’ve long observed that secondary students come up with a strategy I like to call “combine the numbers”. The basics of this strategy are simple. When given a question (often applicable during a word problem), look at the numbers given in the text, and try and figure out how they fit together to provide an answer. Students know that the answer is usually some combination of simple arithmetic with those numbers, so there’s a good chance that they can solve the problem without completely understanding it.
I’m not just making this up: I’ve seen it happen many times with students. This is a problem, because they are learning a completely different skill than what we want. Instead of learning how to problem solve, students are learning how to find an answer given some numbers. Since that’s not what we want, we need to find a way to direct a student’s thinking to the actual problem.
The way I do this is through asking big-picture questions. When a student is working through a word problem for example, it’s extremely common for them to give it a single read, and then say that they don’t understand. I then ask them what the goal of the problem is. What do they need to find? Then, when the student has a good idea of that, I ask them how this can be done. What do they need in order to achieve the goal? Usually, this is where the student will start giving me a sequence of operations with numbers, but I’ll immediately direct them away from that. Instead, I ask them to tell me what they need to find in words. If the student says they need to find the cost by multiplying five and four together, I’ll insist that they say something along the lines of, “To find the total cost, we need to multiply the cost of one loaf of bread (four dollars) by the number of loaves we want (five).” I’m always trying to separate the particular problem from the more general method.
It’s not that I don’t think it’s important to solve that one problem, but in the grand scheme of things, it’s a lot less important to be able to quickly solve one particular problem versus solving a general class of problems. I want to help students develop tools that will let them solve a variety of problems, instead of only a certain one.
By getting students to be proficient in stating what they need to do in order to solve a problem, they will usually have an easier time starting the problem. I’ve personally found that clearly stating a problem and what I need to do to answer the question often gets me going and working towards a solution. Having a clear goal lets you break it down into subgoals. If I’m looking to prove a result, I ask myself, “What do I need to show in order to come to my conclusion?” This becomes the basis of the proof. It can also be useful to simply get accustomed to translating between the mathematics and describing the problem in words. This seems to get one to think with a slightly different mindset, which could help as well.
When I work with students, I only have a limited amount of time to have an impact on them. I don’t want to use that time to solve one problem with no transfer of knowledge to a new one. I want to get them comfortable with a whole range of problems, so they can work through whatever gets thrown at them. By having them clearly state the goal of a problem, my hope is that students will start to see the underlying similarity between lots of problems they work on. After all, there are only so many ways to ask an “applied” question about area in secondary school. I don’t them to understand the one particular problem we worked on together, and then be unable to do the one during a test because it wasn’t the exact same. Clearly stating what one needs to do solve a problem is a great way to combat that tendency.
Why is it that some students will pick up new subjects in mathematics and science relatively easily, while others will struggle for weeks on end getting just the simple concepts down? As a tutor, I see this all the time. In fact, I would almost wager to say that most students I work with aren’t actually having difficulty with the topic they say they don’t understand. So what’s going on?
If I ask them a question concerning this topic, they will often work slowly, but will get the answer correct in the end. The place where things get tricky is in the path to getting the answer. There always comes a step where the student, hesitates, unsure of how to proceed. It’s then my job to help them get past that hump and understand what is happening. However, those problems usually stem from past knowledge that has either been forgotten or poorly done in the past.
Here’s an example. Early on in secondary school, students begin extending the notion of a fraction. In particular, they learn that fractions are not unique, and that we can have (infinitely!) many fractions that all represent the same fraction. (Think of 1/2, 2/4, 3/6, and so on.) This is something the students will quickly understand. A few visual demonstrations also helps, and then they are sold on the idea. So far so good.
The next step is to introduce the concept of reducing fractions. If you have a large fraction, like 64/128, you want to rewrite it such that the numerator and denominator don’t have any more common factors. This lets us deal with “smaller” numbers in the numerator and denominator. Indeed, after a moment’s glance you will probably recognize that 64/128=1/2. The task is to get students to be able to go from these “larger” representations of a fraction to a smaller representation.
Once the mechanics of this process are hammered out, you would think that this would be a relatively simple thing for students to do. The actual “work” consists in factoring large numbers into their constituent parts. This is something that they have previously worked on, namely, while working on their multiplication tables in elementary school.
Yet this is where some hit a snag! It turns out that many student’s haven’t actually learned their multiplication tables, which makes reducing fractions go from being a straightforward task to one that is time-consuming and difficult. And yes, knowing one’s multiplication tables does essentially come down to memorization, but I would argue that is a very good use of memorization (unlike many other examples). Having a sense of how numbers “fit” together to produce other numbers helps in many other areas of mathematics, and it is very useful here.
For this specific example, the student will often have to work out each fraction individually, perhaps even making a factor tree for each number so that they are comfortable with how the number breaks up. Of course, there’s nothing wrong with this, but it makes it very difficult to finish a test within the allotted time when one has to work out the factors of a number one by one instead of knowing how they break up. This is only made worse with calculators, because it allows students to get by more easily, without knowing this basic concept. Then, when this need to be able to know the factors of a number comes up again, the student will have to pull out a calculator again, increasing their dependency on some outside source to help them.
This illustrates a broader point within mathematics in particular: most of the time, you can’t “escape” a concept by faking knowledge of it. Eventually, you will need to use that tool in some other problem, and you will be expected to know how it works. But, if the first time you learned it you did not take the time to understand and improve, that concept will come back to haunt you later on. This is simply a consequence of the fact that mathematics is cumulative, building on prieviously learned concepts. If a student didn’t understand something in the past, yet showed “enough” knowledge that they still continued on to new things, chances are they only got worse at the particular concept, since they never really worked at it again. This becomes a problem when it suddenly resurfaces and the student has to now learn both the new concept and the old one simultaneously.
This kind of knowledge (what I might call “peripheral” knowledge) is the kind that often causes the students I work with to struggle. It’s the simple reality that students start having difficulties because they’re trying to catch up on “old” material while also trying to get the hang of the new content. It’s no wonder that they struggle when this is happening!
There’s no easy way to fix this. The truth is that if the student wants to improve and really understand, they have to put in the work to catch up. If this means they have to work on their multiplication tables after school, then that’s what it takes. I try to make this very clear when I work with students and I see they are having these kinds of issues. I won’t be able to help them within a span of an hour or so every week, particularly when they bring work that addresses new topics, while I know that the underlying issue is the older topics. I do my best to suggest to them to do more practice on the underlying topic, because that’s where they will really see the largest returns.
Will it be easy? No. But it’s a lot better to be realistic about these kinds of issues than to simply “get a tutor” and hope that everything becomes better. In mathematics, the peripheral knowledge is what ends up separating those that struggle through every single problem and those that can work through them systematically. As such, I recommend to always be aware of your background within mathematics, and to acknowledge the areas in which you are weak and may need some work. Then, address those before automatically saying that you don’t understand the new material. Chances are, the new topics will make a lot more sense once you have worked through and are comfortable with the older topics.