The tale of a test twice taken

Well, the time had actually arrived. I took my pre-calculus test last Saturday. Before I let you know how I did, I must tell you the story of how I came to take a 2.5 hour test twice almost back to back.

 

In my previous post I was going on about assessment and self-esteem, of how I would feel if I did worse in the exam that I thought I deserved. My conclusion was that it would be hard to do badly but the most important thing is that I know I have learned. I had also made a point about tests being a tool to mostly measure teacher performance rather than student performance. In other words, the tests scores will give more information to the professor about what worked in class than I would to a single student about their performance. Fast-forward a week and you would find me sitting with my laptop, iPad, textbook and notebook ready to start my test...at 3:30am.

 

Why in earth would I be up at three in the morning to take my final? Simple, I had misread my watch and thought it was 4:30am. I was going to start the test at five, but when I had to check on my baby daughter, after she got out of bed, I decided to start the test early...that as you will see was a lucky mistake.

 

The final test was worth 80% of the class grade, so all those exercises I had been doing throughout the 10 week course (about 140 items) would count for 20% only. So, if O failed the test I would fail the class. The only solace I had was that the test could be taken twice at any time from Friday to Monday. My plan was to take it early on Saturday, then study up on the segments I had done badly. If I liked my score the first time around, I would take it once. My goal was no only to pass with the required 65% but to get a certificate of completion with distinction. I needed an 85% in the course to qualify for that.

 

After I have all my tools set, I start the test. The countdown read 2 hours 30minutes and started to descend a little faster that I wanted. I had 35 exercises to finish. My first wake up call came at question number 1. I had no idea what it was asking me to do. I flip through my notes and look through my reference pages to jog my groggy brain into gear. It occurred to me that an hour of sleep would have been welcome. But I soldiered on, I skipped the first couple of questions until I reached one that made sense. I picked up my pencil and calculated away. It would be a long two hours and a half. I could tell.

 

Halfway through my time I notice I am not hallway through my test yet. I start to get the feeling O will not finish all the questions, which would be bad because I wanted to use my first attempt as a reference and needed all answers graded. At least I had saved an HTML copy of the test in,y hard drive, so even if I do not finish all the items, I would know what they asked.I wanted to make a static PDF copy of the test, but found out my laptop did not have that capability. That would come and bite me later on.

 

As the clock winds down to the last 5 minutes I still have 4 or 5 exercises to go. I feel tired from all the calculations, and checking my notes, and finding reference pages to use. But at least I would get the majority of the except cowed marked one way or another. When the time stops I wold my breadth for the result.

 

I scored 15.67, just under 44%.

 

Feeling a little down, but not that much considering I had practiced very little for test, I proceeded to open the HTML copy I had saved in order to print the questions for review. And that's when disaster struck. As I looked at the HTML copy of the page I had saved, I saw something that made my heart stop. There was a clock at bottom counting down from 2 hours 30 minutes. It seemed I accidentally triggered my second attempt of the test.

 

On impulse I close the page. Then I freak out thinking I just lost my chance to take the test again. So I do the only thing a sleep deprived, exhausted human being would think of doing, I opened the page again. The clock started counting down from 2 hours 30 minutes once more. However, I could not know if this new clock was real or if the true countdown would be the one starting when I first opened the HTML file. Fully awake now, I figured I had to options: either hope it was a glitch, study all day for the test and retake it the next morning risking not having a second attempt to do; or suck it up, take the test again and try to finish it before that first countdown wound down.

 

Desperate, and not wanting my grade of this course which I had pit so much into, I decided to take the test again right then and there. Knowing full well that it might all be for nothing, since nothing could guarantee that upon hitting submit after completing all 35 exercises I would not get a message saying:"We are sorry but it appears you have already attempted this test twice." Regardless of that possibility I barreled on. It was already 6:00am.

 

The new test was slightly different from the first. It had the same questions but the variables and constants changed in most of them. Still, the second time around I was sufficiently awake to start remembering all I had learned in the class. Still, I knew my nervousness and agitation could make me make mistakes. And there was no fixing mistakes this time. It was now or never.

 

About an hour and a half into the test, at around 7:30am, my baby daughter awoke. She would be hungry and very curious about what daddy was doing. It was an eventuality I knew I would face. My plan was to leave the test were I was and get her breakfast. While she ate I could do some more exercises. Then my wife woke up. She instinctively noticed my predicament and told me to go on with my test. She would make breakfast. My wife is an angel. She woke up early on the day she could get to sleep late to help me pass a test that was, in the grand scheme of things, insignificant. All because she knew it was important for me. Thanks to her I was able to finish the test with 17 minutes to go. With trepidation I hit the submit button. I hoped against hope that the system would accept this attempt. It did.

 

My second score was 24.5 out of 35, exactly 70%.

 

I sighed with relief. A day ago, that score would have been a let down. But that day after 4 and a half hour of testing, and scribbling, and checking, and answering, I was exhausted and happy. My technical difficulties were overcome. I had passed the class. While my wife and daughter ate their pancakes, I raised both arms in triumph and gave a muted cheer. They cheered back.

 

I was done with my pre-calculus review. I would receive my statement of accomplishment a week later. Now, in the distance, through the wall I had just taken down , I could see a sign over the horizon that read: Beyond, there be Calculus.

 

Testing and self-esteem

This following week is my pre-calculus final exam. It will try to assess all that I have learned in the last 10 weeks. But can it?

 

Testing has been, in my opinion, one of those necessary "evils" of traditional education. They are necessary in order to keep a record, a benchmark, a log that can be revisited to understand the decisions made. I call it an evil because I know who the test is for: It's for the teacher, and for the school, not necessarily for the student. What I mean is that standardized assessment tools, like tests, work more as a measure of homogeneity in the learning of a group, that of individuals. Taking all the scores together a teacher can assess their own performance. However, gauging individual performances based on them can be tricky and the results can be deceitful. And before anyone thinks I am only writing this post out of apprehension of doing poorly in my test, I can tell you I am not alone in thinking this way.

 

However, I am worried about doing badly in my test. It would be really disappointing to have spent so much time and effort in class, and have nothing to show for it. Yet, do I really have nothing to show for it? Doesn't this blog chronicle all I have learned in 10 weeks better than any test could assess? Of course I do and of course it does. So why do I still worry? Because of my self esteem.

 

I recognize that I need that external evaluation to measure my performance. And that might not be a good thing. I shouldn't need a test to tell me how well I learned pre-calculus, but part of me does. And I know that a bad score in the test would take a lot of wind out of my sail. But why? Why should it? I guess because when people ask how you did on a task, the answers we give more often than not are the results, not the journey. We focus on the scores, the grades and the ranks to measure our selves. And what we usually measure our selves against is other people.

 

I've been doing some research on self-esteem, for other purposes, these last few weeks and I was surprised to find that my self-esteem is not as high as I thought it would be. In fact, my whole project on learning calculus could be seen as a why to correct issues stemming from low self-esteem or more likely low self-efficacy.

 

Self-efficacy is the confidence you have in the ability of doing something. Self-esteem is your perception of selfworth. And while I believed my self-esteem is always high, it takes me more than a while to get over setbacks, I expect perfection and I am my worst critic. Three factors that denote a less than high self-esteem. However, my self-efficacy had always seemed high because I have never had doubts on my ability to do anything or learn anything. With the exception of calculus. For the longest time I believed I could not do it. I am making that change this year.

 

So going back to the test, I start to understand where my apprehension lies. My self-efficacy tells me I can do this and that everything will be fine. My self-esteem is worried about the outcome and how to take it. My knowledge of assessment tells me the test is necessary as a tool for me to understand my weaknesses and strengths. My knowledge of test making tells me the test is a tool for quality assurance of course performance. When I take all these things together, I have to conclude that I am stressing over nothing...and that I need to monitor my self-esteem. No matter how I do in the test, I know I have made progress and I am proud of that. Besides, perfection is overrated.

 

Wish me luck, I'll keep you posted on the results.

 

 

Words of Encouragment

 

 

The Work

My favorite literary character is Sherlock Holmes. I have read the complete works of this series by Sir Arthur Conan Doyle at least twice and made a point of visiting 221b Baker Street on a vacation to London last year. I have also enjoyed the movies with Robert Downey Jr. and Jude Law. However, I must make special emphasis on my love for BBC's Sherlock series, which transports Holmes and Watson to present day London, but I digress.

 

What is commonly remembered of Sherlock Holmes is his power of "deduction" (actually it's induction), which is his ability of grabbing the minute details of a subject and arriving at a conclusion. For example: Holmes would take a look at a person's clothes and tell their occupation, errands of the day, mode of transportation, degree of anxiety and town of residence. He would do so by noticing things like tips of mid in the shoes, the pattern their clothes are crumpled, a ticket or stub protruding from a pocket, a button in the shirt not completely fastened, the know of a tie...etc. front hese actions people say That Sherlock Holmes is a brilliant man.

 

What most people never realize is that Sherlock Holmes spent most of his time on study. He would famously analyze 140 different type of Tobacco ash. Therefore when he got on a scene and saw Tabacco Ash he could say what kind it was. In other words Sherlock Holmes was brilliant because he would do the work (and had an extraordinary brain that could make sense of all he learned and appy it as needed). Lacking the later I have taking to weeks to do the former with my trigonometry.

 

As promised, I took all this week to study up on trigonometric identities and the values of sin, cos and tan at various degrees. I had to do this because I felt I had reached a wall in my studies that was making it difficult for me to understand the newer concepts I was being given in Pre-calculus. This situation was creating a mental block that was giving me a chance to doubt my abilities to carry on my goal of learning calculus.

 

I am happy to say that, as I suspected and sincerely hoped, a serious attempt to memorize and analyze trigonometric properties and identities has helped me a lot.

 

I started with the basic stuff. It was taking me too long to grasp angles quoted in radians. Therefore, I spend a day with degree and radian conversions. I did not want to only memorize that π/6, π/4, π/3, π/2 and π were 30deg, 45deg, 60deg, 90deg and 180deg respectively. I wanted to understand it and see it. As it turns out, a quick drawing of an unit circle is very handy for this, as is knowing your multiples of 90, 180, 270 and 360.

 

Then I progressed with the value of trig functions at various degrees. The degrees I studied were of course the "nice degrees" as the people at mathmistakes.info call them. These are angles were sin, cos and tan (along with their reciprocal functions) have exact values that are easy to remember.

 

I cannot thank the people behind mathmistakes.info enough since they had in their site the very tool I needed to help me study. It was a chart with all the trig values in the easy angles for all quadrants. However, their chart has all the values hidden from view until you ask it to show them. It does not have a button for hide all or show all, so finding a pattern would be difficult, but it is a great way to test your knowledge. You can find the chart here.

 

Doing "THE WORK" helped me join together all the concepts I had studied so far in trigonometry. This might sound absurd since I have been doing trig quizzes for my pre-calc class the last 3 weeks, but it's true nevertheless.

 

All weeks prior, I've been relying on my memory and the examples the professors gave in class. It is very common that quiz questions can be answered using the exact process the Profesor used in class, and that was the case with my homework. Therefore, as long as I followed what the professor did step by step I would get a correct answer without any deep understanding of the process used. In othet words, I had gone through a lot of knowledge in pre-calculus class, but that knowledge didn't stay fresh in my mind for long. Case in point: reference angles.

 

A few weeks back I studied reference angles, which are acute positive angles that are coterminal to other angles and the x axis. This is a fancy way of saying that if you have an angle that is 150^o, it's reference angle is the smallest angle formed from the terminal side of that angle to the x axis. Therefore the reference angle of 150^o is 30^o. It is a lot easier to understand when you see it in the figure opposite.

The homework for that section was very simple and straight forward. I felt I was getting a free pass. If I had paid attention to the importance of reference angles in Trig values, it would have saved me a lot of headaches.

 

You see, It was not until I was studying the relationship between trig values and the unit circle that I finally understood the importance of reference angles. Once I had learned sin values from 0° to 90°, I went over to the other quadrant and thought to myself: Since trig functions are cyclical, sin of 120° must equal sin of 30° since 90°+30°=120°. When I went over to a reference chart to check this, I found out I was incorrect.

 

Sin of 30° was not equal to sin of 120°, sin of 30° is equal to sin of sin of 150°. That threw me for a loop until I put the values in an unit circle. Then it all made sense. Not only did I find a pattern for the values of sin, cos and tangent, but I also understood that the value of all of them is equal to the value of their reference angles before taking their sign value into consideration. Therefore since 30° is the reference angle of 150°, 210° and 330°, sin is 1/2 (or -1/2 depending of the quadrant) in all those angles.

 

It was an excellent week of study and discovery. Of course these discoveries are small compared to Sherlock Holme's deductions. If fact they are small in terms of what every high school student already knows of trigonometry. But for me that are huge and important because I found them on my own. Nothing can compare to the thrill of figuring things out for yourself.

 

Let me know how I'm doing or if I got something wrong.

 

The Wall

For a while, I have been able to see it from a distance. Every day I would check to see if I saw it moving closer, knowing that if it did, I would be in trouble. When I returned this week from vacation I was not surprised to see it right in front of me. I am talking about the wall. That seemingly insurmountable force that wants me to stop my quest to learn calculus by making it look daunting, foolish and, worst yet, insignificant.

 

I was expecting to have lingering doubts about how consequential my desire to learn calculus is. I knew a day would come when I would ask myself: "What am I doing? This is ridiculous. I am just going to waste my time with this." The posts on words of encouragement were a weekly reminder against this.

 

What I was not expecting was the attack on my intellectual ability to actually do this. When the wall appeared this week, it had an inscription that says: "You cannot understand any of this anymore. It will just get worse, I promise." It's hard to argue with the inscription when it rings so true. The last two weeks I have been struggling to truly grasp the concepts I have been reviewing. They are mainly pre-calculus properties and transformations. I have a hard time using radians and relating them to trig graphs and the unit circle.

 

Part of me knows that the reason i am struggling is that I have not done THE WORK. I have not consciously gone throughout the process of memorizing all the trigonometric properties and, more importantly, the values of sin, cos and tan for special angles in the unit circle. Still, a small but potent voice, tells me that I am just rationalizing the real problem: you are incapable of learning this. And for this week I have been to much of coward to find out which voice was right.

 

Until today.

 

Today I will begin THE WORK. I will set out to store in my long term memory all the trigonometric properties I can. I see cue cards and pop quizzes in my future. I am confident that the work I put in these days will be invaluable when I start with Calculus.

 

Wish me luck. I have a wall to turn down. Now, where is my sledgehammer?

 

Yes, today is π day 3.14

And to think I almost missed it when I have been battling with inverse trigonometric functions all week!

 

Check out this site for all things PI. http://www.piday.org/

I promise to post soon...vacations can put a hold on Math blogs!

 

On logarithms and funny notations

I remember having trouble with logarithms in high school because I had to do a transformation that hurt my head to understand. Before reviewing this section of pre-calculus, I consulted my memory bank's folio on logs. It read: "Logs will ask you to do stuff to convert them that will feel unnatural." I could not remember what it was exactly, but I could see the numbers bouncing around each other rearranging themselves. As I reviewed the chapter I remember what that is.

 

Logarithms are just the accepted notation to answer this question: to what power do I need to raise x number to give me answer y? In other words, I might want to know to what power should I raise 2 to get 64. In exponential form I would write 2^x=64.

 

However, in logarithmic form we would grab the x exponent throw it across the equal sign, then take the 2 shrink it, append it to the word "log" and finally take the 64 across the equal sign to join log₂. At the end we should have something like this: log₂64=x.

I hope you can see why it blew my mind as a teenager.

 

The Log transformation got me to think about math notation, or the way we represent mathematical concepts. I mean, if you think about it, someone at some point had to decide that the letter x would make a good place holder for a number we don't know; or that a cross would signify addition and a a horizontal line subtraction. I bet those first proponents had to explain what the symbols meant:

"Dear colleagues, whenever you see this notation of two parallel horizontal lines "=" it means that the terms on either side of "=" are equal"

I can almost hear the collective "Ahhh"s of fellow mathematicians reveling in not having to write the word "equal" anymore. I am of course being a little facetious. I am bet the introduction of the equal sign did not happen quite that way...but come on, I bet the person who came up with the radical sign (√) for square roots got a long of grief from the long division enthusiasts.

Getting back to logarithms, the logarithmic and exponential, as best as I can explain from what I understood, equations are crucial to describe rapid growth (or decline). Imagine you had a magic ball that would split into two magic balls every hour, then the new balls would also split every hour and so on. How many balls will you have in 24 hours? Pick a number in your mind and make it high. If you number is 16.7 million balls, you are right on the money.

The exponential or logarithmic graph has a segment rises or falls very quickly and another that gets progressively smaller but never touches the point it gets close two. That, if I remember correctly is an asymptote. A basic exponential graph has the x-axis as an asymptote, the logarithmic graph usually has the y-axis.

If you look at the graphs of log₂(x)and 2^x , the first is logarithmic and the second exponential, they will be mirror images of each other along the y=x line. This happens because in the logarithmic equation, the y values you get are the power you need to raise 2 in order to get the x values, while in the exponential one the y values you get are the result of raising 2 to the x power.

The cousera course videos keep telling me that exponential and logarithmic equations are important to calculus. For the time being, I guess I must take their word for it.

 

If anyone thinks I am missing something, let me know.

 

'Till next time.