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.

Can You Change Your Mind?

Despite people in the sciences are supposed to be rational, changing one’s mind on a topic is just as difficult (if not more difficult) as in non-scientific settings. Often, I’ll watch some sort of academic debate where the debaters will talk for over an hour on a topic, yet they still won’t listen to each other in a way that accomplishes the goal.

In my mind, a debate or talk is supposed to introduce new information to a person, which then will hopefully inspire a new perspective on a topic. If one enters a talk with someone else and doesn’t gain any new perspective or has none of their views stretched or challenged, there’s a fair chance the conversation wasn’t productive.

This drives me nuts. I’ll hear an argument from one person, and then the other person won’t actually interact with that argument, and just say that it’s wrong or that the person should think of something else. It’s infuriating because there is little interaction of ideas. Instead, it’s a conversation in parallels, with each person seeing the other but not directly interacting with them.

In this scenario, no one will ever change their minds. Additionally, one might even strengthen their old beliefs, which doesn’t do someone any good.

The sign of a great thinker (particularly in the sciences) is to not be married to any one idea. As soon as that happens, you’re biased and your judgement can become clouded. Furthermore, it becomes a challenge to interact with others, since they don’t subscribe to all your ideas.

Of course, I’m just as bad as anyone else. I can sit here and write about changing one’s mind as if I do it all the time, but the truth is that we are all biased in some way. We can’t escape this, so the best we can do is put our biases to use. In my case, this means use the fact that I believe science is the best route to discover the mysteries of the universe. Therefore, my interactions and beliefs will be based within that mindset. However, I try to keep in mind that when there is scientific uncertainty, everyone will have an idea for the underlying cause of a phenomena, and I should keep an open mind to all of them. Then, instead of shooting down an idea because it doesn’t fit in nicely with my idea, I should consider the weight of the argument on its own. From there, I can make a decision.

Too often, I find, we say we are rational, yet we don’t change our minds for anything that comes up. We may say that our reasons are scientific, but a lot of the time it comes to not wanting to accept that your personal idea is wrong, and that’s a problem.

As such, my personal goal is to seek to change my mind as often as I can. That is the sign of learning.

What’s in a Good Question?

As a science student, I’ve taken many tests over the years. The staple of a science class is the tests that are spread out over a semester, so they are to be expected. Consequently, I’ve answered many questions on tests, and so I have a fair idea about which questions are actually good questions to ask on tests, and which ones seem like there is no point.

Why have tests?

First of all, I should establish why I think we even have tests. I’m pretty sure it has to do with the fact that most science classes are lecture based, meaning there is a lot of information coming in to the student. To be sure that this information is retained by the student, tests are given (and are the easiest way to do so).

I have no problem with the concept of taking tests. However, I do take issue with the way some of the tests are structured. Certain types of questions fit better in certain situations than other types.

Multiple choice

The almost ever-present feature of any test is the multiple choice section. Many questions are asked, and the only thing that is graded are the answers.

First of all, it seems to me that this is an incredibly dumb way to administer tests in subjects like mathematics or physics (at least, most of the time). The reason is that teachers from very early on tell students to show their work. This is always supposed to be heart of a solution, the process of how one got to an answer. Then, teachers turn around and give their students multiple choice questions where they cannot show their work as they’ve been told numerous times to do. It’s a bit of a confusing situation. Therefore, the multiple choice format does not really fit the objective of the test. When a multiple choice question is simply a short answer question with four options for answers listed, the question fails in my mind. It’s leaving out some students who may do everything right, yet make a small calculator error. And if we still want to take away points for that, then we have to consider the way we correct all the other questions too.

On the flip side, there are instances where multiple choice questions work a bit better. The example I can think of is my biology class, where the theory wasn’t usually mathematical, but more about reasoning with information that is given. In this case, I found the multiple choice format worked well, since the question doesn’t have much “work” that can be shown. Sure, you could argue that the logical reasoning could be shown, but I think that is stretching the idea of showing one’s work a bit much. For these types of questions, it’s okay to use multiple choice.

Here’s a quick example of such a question:

Given a situation X, what is the likely type of microevolution that occurred with this species?

In the above example, the question doesn’t really require work. It simply asks one to recognize the type of microevolution taking place based on the given situation.

Therefore, the multiple choice format can work, but it should only be for certain questions. They don’t serve any purpose (except for allowing the teacher to correct less and give more questions) when the format is used for a question involving some sort of calculation. Additionally, there’s also the possibility that someone who uses the correct reasoning but makes a small mistake gets the incorrect answer, while the one who doesn’t have a clue how to solve the question guesses an answer and gets it right. I’m not sure how frequently guessing works, but I assume it should work a quarter of the time (if there are four possible answers). This is obviously something we do not want, so this is another reason to use multiple choice as little as possible.

Can you actually solve the problem at hand?

Another issue I have with questions is the fact that there are times when questions require one to have knowledge that doesn’t really relate to the specific problem at hand, but nonetheless needs to be known if the problem is going to be answered. I’ve written about this before (link), but this usually comes up in the form of formulas.

The prime example is in my mathematics classes, where formulas were never (or rarely) given. We needed to memorize everything, or else there was no hope for us on the test. This was obviously frustrating, because it created situations in which one would know what needs to be done, yet they couldn’t actually do the problem because they did not precisely remember the formula.

I remember this happening to me while I was taking a linear algebra class. The question was about the Cauchy-Schwarz inequality, and I just couldn’t remember if the two terms multiplied together or if they were summed. I knew exactly how to answer the question if I did know the formula, but since I didn’t all I could do was go with the one I thought was right (I was wrong).

In this situation, I would have never gotten into this situation if the formulas were provided to me. And why shouldn’t they be? The important part of the question was not to test if I knew what the inequality was. It was to see if I could use it, a subtle difference. Therefore, giving me the formula wouldn’t have solved the problem for me, and it would have allowed me to actually solve it correctly (like I knew how to do).

This is the problem with many tests I’ve taken. What is being tested isn’t the only thing required on the test, even though it should be. I just gave an example of not remembering a formula that I believe should have been provided, but there are other examples as well.

Here is another situation. In my physics class, I took a test in which there was a lot of work to be done. The class only lasts fifty-ish minutes, and everyone was working right up into the time limit. Basically, no one was really finished, and so there was frustration all around. (Since my physics teacher was very nice and accommodating, she adjusted the grades of all the students, giving us a bit of a boost since the test was so long.)

What wasn’t fun in this situation was that the questions weren’t all that difficult to solve. Instead, it was the sheer amount of work required combined with the time constraints, that made the test difficult. This is why the test was barely finished by anyone. We simply did not have the time.

In this situation, we couldn’t answer the questions to the best of our abilities because of the time limit. I’m not saying that every student should have an unlimited time to ponder over the questions, but I definitely think that students should have an amount of time that allows one to think through each question.

Put differently: a test does not have to “fill up” the fifty minutes of class with questions.

This was brilliantly done by my astrophysics teacher during our final exam. We had three hours to complete the exam, but in reality it only took myself about an hour and a half. Assuming I went a little bit quicker than the rest of my classmates, it could have taken them two hours to finish. Despite the longer time, they would still have an extra hour to go back over the exam.

In my mind, this is a fantastic idea. Just because there’s a lot of time in some exams, there doesn’t need to be question that fill that time gap. Instead, teachers need to think about what they want out of their students. Do they want well written solutions? Or do they want hastily scrawled answers because of the time constraints?

It’s a no-brainer, really, yet I still don’t see this happening in many classrooms. Instead, I see tests being jammed to the brim with questions, as if testing students with more content is going to be useful.

The pointless question

Finally, there is a category of question that I just despise: the definition or concept question. Basically, this sort of question entails one to respond by giving a definition or some other fact that could be easily looked up in a textbook.

For example, asking students to memorize the first twenty elements of the periodic table is just a waste of time. Sure, it’s important. But it’s also something that anyone working in a scientific field can look up at any moment in time. Furthermore, one will naturally become familiar with the periodic table as one works on questions concerning it, meaning there is no use to having students memorize the table. If they work with it a lot, they will remember it out of convenience. If they don’t remember, they can simply look it up.

An even better example is in one of my experimental physics quizzes. The question I had was: what does the term laser stand for? This was such a useless question. Who cares if it is light amplified by stimulated emission radiation? Knowing the name certainly wasn’t useful to anything I did concerning lasers (which has holography). The question was just a waste of time.

These kinds of questions aren’t always present on tests, but I’ve seen them enough that they deserve a mention. A question about a meaningless fact won’t give us much knowledge about science.

We need to be honest with ourselves about the kinds of questions we want to ask students. The truth is that many of the questions used today just don’t have a purpose for the student. The multiple choice format is a prime example of this, being used in a manner which is often not helpful to the student.

Additionally, exams tend to be constructed with the idea that one must fill the time up with questions, or else the test is too easy. However, if the goal of the test is to see if a student can clearly write a solution to a question, imposing a strong time limit is counterproductive to this goal. By all means, introduce a time limit. Just don’t make the stress of the exam be the time allotted to write the exam.

Finally, questions posed just to make students memorize useless facts aren’t helpful either. They just award grades for items that aren’t important, and therefore shouldn’t be kept.

By keeping these principles in mind, I think it’s fair to say that we can improve the questions science students have to answer. It might be tempting to include a lot of questions on a test, or to throw in questions about abbreviations and such, but they don’t actually help the student learn something about science. They are just little tidbits of information that must be recalled until the test, before they are forgotten.

Instead, let’s focus on giving students questions that challenge their problem solving skills, and make them think about what they are doing.

That would be a lot more helpful than a bunch of multiple choice questions about definitions.

Memory Aids

In secondary mathematics, students are allowed to prepare their own memory aid for an exam. It must be handwritten and can only cover one piece of paper. Usually, this memory aid is used to write formulas or challenging examples of different concepts so that one isn’t lost when writing an exam.

As a tutor, I have some experience advising others on their memory aids (as well as my own time being a secondary student, of course). I’ve seen some memory aids that were nearly blank and some that were filled to the brim. Both approaches can work, and it entirely depends on the person.

However, one thing I’ve seen time and time again is a tendency for one to write down the formula as provided by a formal definition. This is an important note, because a formal definition can be much more confusing to a student than the rigorous definition. And, as the students I’ve tutored tend to have difficulties in the subject of mathematics, these definitions can make little sense.

I remember an interaction I had with a specific student, who was having a trouble with a certain problem. I looked on his memory aid, and pointed out the exact concept to him. He looked at the writing as if written in a foreign language, and I could plainly tell he didn’t understand what was there.

Then it clicked for me.

“Do you understand the formula and definition you wrote down?” I asked, and he shook his head.

Now, I’m not trying to make fun of my student, but I’m trying to illustrate a broad point. Memory aids have turned into a piece of paper in which we write down formulas that we don’t even know what their functions are! My student simply wrote down the exact formulas from a generic copy of a textbook. Of course he wouldn’t be able to understand it when he needed to use the memory aid, he just copied a textbook.

With a memory aid, the goal should be to not forget any formulas. What I mean by this is that it’s nice to not lose points on a test because you’ve forgotten the exact composition of the quadratic formula. However, a memory aid should not be a used as a crash-course on a topic. It can be done, but the trouble is better spent trying to understand the topic before an exam. Once you’re in an exam, the memory aid should help remind you of the composition of formulas, not how to do a problem (though I have been in situations in which the example I wrote on my memory aid is indeed a test question). This comes down to the fact that mathematics and science is about solving problems, not memorizing formulas.

I’ve written a lot about memory aids, formulas, and memorization here already, but it’s a great topic to think about. As it stands, we send somewhat confusing signals to students: sometimes it is appropriate to have a memory aid, and sometimes it is not.

The truth, of course, is that the kind of knowledge we want to teach students shouldn’t be something we can place on a memory aid. For mathematics and most of the sciences, a memory aid can be completely appropriate, yet not give away the contents of an exam. To make sure this does not happen, there’s no problem to giving students pre-made memory aids to put them all on an even playing field.

What I want to avoid are situations in which students blindly copy formulas onto their memory aids without knowing what they actually mean. Whenever possible, write down formulas in ways that you understand, and it will save you from that terrible feeling in an exam where you do not understand what you wrote on your memory aid at all.