The Little Frustrations
I had the interesting realization recently while tutoring one of my students. As he worked on his solution for solving a system of equations, I suggested that his work would be much more clear if he outlined what he was doing and showed steps to his process instead of simply writing a bunch of equations and having the answer pop out at the end.
He bemoaned having to be so complete with the work, not seeing the point of having to add these small elements to his solution, but I held my viewpoint. I knew it was important for him to write the entire solution in a clear manner, but he didn’t seem to care about achieving this level of perfection. He thought the elements I suggested were more or less superfluous, since the core of his solution was correct.
So what was my realization?
It was that, ironically, I do the same thing as he does in my own mathematics classes.
The quest for one hundred
As I’ve mentioned a few times in my writing, I’ve had the same professor for all of my mathematics courses in CÉGEP. During this time, I had a bit of a personal goal: get a perfect score on one of my tests.
Unfortunately, I never met this goal, though I did get close. My best mark was 69.5/70, which is over 99%. Other than that, my tests usually hovered around the lower nineties.
What becomes clear though as you look at my exams is that, apart from silly errors, most of my work is correct, but is lacking a small clarification or a point of completion. As you can probably see (and like I said, I only just made the connection), my situation is remarkably close to that of my student who I tutor.
Every time I saw this kind of error on my exam, I’d shake my head, not really even knowing how I could have been expected to write that anyway. I took solace in the fact that almost no one in my classes ever included the little things, and so I was just like the rest of the class.
In regards to my own performance, I didn’t fully appreciate what these little clarifications for completeness would do to my solution. Instead, I told myself to be happy that I got the core part of the problem and just forgot a little thing of almost no consequence. This is obviously the wrong view to take, but I didn’t make the connection.
The reality is that mathematics is a sort of logic. Therefore, all parts of a solution to a problem are important if the final answer is to be believed. Whether or not the final answer works is entirely dependent on the small and big details of the solution. Omitting a detail means the foundation isn’t rock solid. Sure, it might happen to be the correct answer now, but it is only in spite of the lack of detail, not because of it.
For example, I lost points on one of my tests because I failed to identify the curve I was using as a cylinder. I was dealing with the classic $x^2+y^2=1$ representation of the cylinder, but I failed to mention that the previous equation was for all z values. I thought this was an implicit assumption from the equation, but it’s actually important to explicitly declare it. This is because the equation I wrote is actually a circle in two dimensions, yet is a sphere or solid in three dimensions.
Therefore, what I thought was a harmless extra point of completeness was actually very relevant to the solution. It was the difference between a circle and a cylinder. Consequently, I now fully understand why my professor took away points in the question. Of course she knew I was talking about a cylinder, but the equation I wrote was really that of a circle, so further clarification was needed,
What I’ve learned from this is that there is no small detail that you should leave out from a mathematics problem. Simply put, it’s much more advantageous to be explicit about the work you’re doing than to make a bunch of implicit assumptions and hope everyone gets it. You will rarely lose points for being complete, but you definitely will on the other side.
I’ve found that this occurs for me just as it does others. Even with over a decade of doing mathematics in class, I struggle to be perfectly complete. However, it does give me a bit of inspiration to include all the important points of a solution as they come up.
Remember: the little frustrations are annoying, yet they will instruct one to be more complete in their mathematical pursuits.
The Last-Minute Rush
Everyone is pooled outside of the classroom, anxiously waiting for the room to vacate so we can sit down. There’s a nervous energy in the air, permeating through even the most calm person. Many have their class notes out, mumbling about various facts and concepts. Others quiz each other, reciting definitions that I could say word for word, instead of giving their own “version” of the answer.
Every few minutes, my eyes drift to the clock. Two minutes left.
The door opens, and the time to check one’s notes evaporates. We all file in, and the exam begins.
I’m fairly certain you’ve had such an experience before. Depending on the type of person you are, this may have affected you more or less than it did I. Still, the last minutes before an exam are fairly similar for everyone. The age-old wisdom is still followed: study until the final minute.
I won’t lie: I do this all the time as well, and I don’t think it’s necessarily a bad thing. However, there’s a difference between looking something up just before an exam, and planning to remember new things right before you write the exam.
I’m reminded of a saying that, as runners, we use a certain expression: the hay is in the barn. This saying is supposed to calm one’s nerves before a race, where one might be doubting their training and preparation for the race at hand.
In the same way, this is how I try to position myself mentally before a test. I tell myself that I’ve done everything I could to prepare myself for the test, and that now I simply have to write it. Sure, I might be exaggerating in the sense that one could always do more, but I make sure to do the best I can before a test, and I tell myself that it’s enough.
What is my best? Well, in my science tests that means I will generally do all the practice problems suggested for me, plus I will go over all the content I’ve seen in class. In this sense, I feel as if I’m prepared, since I did everything suggested to do before the exam. And, it’s all optional, so I didn’t have to do it, but I chose to.
Getting ready for an exam is a stressful affair. I know some people who don’t or barely even study before an exam, and only start on the morning of, meaning they are trying to fill their heads with knowledge with only hours to go. This, I believe, is a losing strategy, because it means you aren’t familiar with the concepts at hand. In contrast, I usually enter exams confident that I know the material. I may have trouble on a question, but I know what I have to do.
Don’t fall prey to the rush of the last minute. By working on the material throughout your time, you’ll become much more familiar with it than if you worked on it for a day.
Jolting Your Ego
I’ve always considered myself to be good at physics. In every physics class I’ve taken, I’ve excelled at the subject and always enjoyed it. The blend of physical situations with the use of mathematics always enticed me. Consequently, my grades in nearly all of these physics classes have been great.
Therefore, I was quite confident that I would do well in a new book I found, called 200 More Puzzling Physics Problems. I figured they would be small puzzles that I could solve fairly easily after a bit of thought. With my grades, of course it was going to be a breeze!
Unfortunately, I received a rude awakening when I looked at the first problem and was stuck. I sort of understood what I needed to do, but not really. I ended up looking at the solution in order to solve the problem, and then moved on to the next one.
Same result. A question that seems simple enough on the surface, but has a complex answer. Once again, I looked to the solution, and saw the equations and definitions which were everywhere, and I realized that I had jumped into a level of difficulty I wasn’t prepared for. Even with the hint on each question, it was proving more of a challenge than I had anticipated.
Finally, I encountered a problem I was able to “partially” solve (which really means I couldn’t solve it, and had to look at how to set up the problem). A small bit of triumph exploded within me, and it has become my motivation for continuing. I want to solve more of the problems, even if my memory of some of my earlier physics classes is a bit vague. It has definitely been a hit to the ego when I could not solve a problem, but I realized that the better attitude is to look at these difficulties as learning opportunities. I don’t need to be perfect with all my answers, and the book is fleshing out some of the peripheral details that weren’t really explained in earlier classes. It’s good practice, and it has inspired me to improve my problem-solving skills.
When faced with a challenging situation for one of your passions. Don’t run away. Embrace it head on, and work to improve yourself. If you do that, you will be able to get through the situation.
Sometimes, it’s wise to leave your ego at the door.
Stages of Preparation for an Exam
The traditional science exam follows a predictable formula of how students will prepare in order to do well in an exam. Every class will have a mixture of every kind of student, but their are still general patterns.
Stage One: Announcement
About two or three weeks before the exam is written, the teacher will announce that a test is coming up. When this happens, some students will not even pay attention, figuring that there is plenty of time between now and the exam. Others will flip open their school planners and pencil the date in, organized much more than the average student.
Generally, the announcement of the exam generates a bit of buzz in the class, but not too much (since the threat is so far away).
Stage Two: One Week Out
As the exam looms nearer, review sessions are common. This is where people begin to really think about the exam (not just in abstract terms). During these times, the students will start to ask questions about concepts they aren’t familiar with. They will have plenty of practice for the exam, but if they are anything like me, they are still waiting to start most of the problems, because there is a bunch of time left before that exam.
An alternative: the beginning of panic. Depending on the actual content of the exam, panic can begin to blossom at this time. For some classes, the sheer amount of material becomes almost overwhelming. Personally, I’ve experienced this sensation for my mathematics class, where the amount of formulas and procedures we had to memorize and know how to use was a bit nuts.
Still, with one week left, the student will generally feel good about his or her chances on the test.
Stage Three: T-Minus Three Days
At this point in time, full panic has either set in, or the student knows exactly what they are doing. With only a few days left until the exam, students are in an awkward position in class. The exam is soon, but the teacher does not want to waste time reviewing old material (since that had already happened), so he or she moves on to new material. Therefore, the students have to try and absorb new material while simultaneously thinking about their upcoming exam. Evidently, what usually happens is that the new material is only somewhat understood and not contemplated for very long.
It’s at this stage in the game that I personally crack down on studying. Usually, this means going over the topics for the tests, and redoing any assignments that are part of the exam. I choose to do this three days and less before the exam because I feel like it gives my mind the “muscle memory” to know what to do when I encounter a similar question on an exam.
This strategy can somewhat backfire if there is a lot of material on the exam, because it means I’ll have to go through an extraordinary amount of work in only a few days. Consequently, the days leading up to the exam are quite stressful, and it is difficult to think about anything other than the exam.
Stage Four: One Day Until the Exam
As the penultimate day draws near, I have two possible reactions: confidence that I know the material well, or utter panic that I won’t be able to do as well as I would like on the exam. With so little time until the exam, those are the basic mindsets I have. If there are problems I try that day that aren’t making sense to me, it’s probably enough to send me down in a funk, certain I will do terrible on the test (usually, I don’t do horribly).
For others, it’s a day of making sure to read over the notes and testing one’s knowledge of the subject. I do this too, but I find it more useful in science exams to be practicing problems instead of theory (though both are indeed important).
Stage Five: Moments Before the Exam
As the hour draws near, many are beginning to get trapped in a mental panic. Questions are shot back and forth between students, as if they are part of a tennis rally. Answers are recited in textbook-like form, and procedures are gone over. For a few unlucky students, questions about concepts are still posed, which implies that there are still uncertain subjects left to the student.
When there are only ten minutes or so left before the exam, many students adopt a military-like approach. For one of my more difficult classes (not for me, but in general), my friend would say something to the effect: “Well boys, let’s get ready for war. See you on the other side.”
The stages of preparation for an exam follow this pattern pretty well, to the point that I know the routine to follow for each one. I always feel like it’s an endeavour that must one cross, like a huge hill that one has no choice to climb. As the climb approaches, I don’t necessarily want to climb it, but I resign myself to the fact that I must. Then, once I reach the top, I feel a huge sense of relief that it is all over.
An important thing to note is that these are the stages of a science exam when there’s only one coming up. At the end of the semester, the situation is a bit different. For one, the exams are all distributed at around the same time. Secondly, the exams are comprehensive, which means there is much more material than on an exam during the semester.
This means it is very difficult to wait until a few days before the exam to begin studying. It’s possible, but I definitely don’t recommend it (actually, I barely recommend any of my procedure). To deal with this, I try to begin studying sooner, and I try to address the most pressing exam first. It’s not the best strategy, but it’s the one that I use.
As you can see, the stages of an exam are filled with one thing: stress, which is why it feels so amazingly good to finish one of them.