The two questions of Calculus

At last on day 113, I dive in into Calculus. What I write bellow is my interpretation of what I have read me researched. If you are using this post as reference, I suggest you double check my statements. I am by no means a reference source on calculus, just another student trying to learn it.

 

Calculus, from what I have learned far, seems to be the study of change. It deals mostly with two major subjects differentiation and integration. If fact I finds some sources that specifically reference differential calculus versus integral calculus.

 

If I use Professor Edward Burger's approach to explain calculus from one his Thinkwell videos, then calculus is the mathematical discipline created to answer two mayor questions: What is the instantaneous velocity of an object? And What is the area or volume of an exotically-shaped object? The first question is in the realm of diffential calculus and the second belongs to integral calculus.

 

All courses I am reviewing right now, start with differential calculus.

 

In differential calculus I read that the original question that started the discipline was finding the slope of tangent line of a curve. I found this surprising because I remember drawing or working with tangent lines and alopes back in intermediate school geometry. However, when I read deeper I was blown away with the reason the slope of a tangent was such a problem.

 

A tangent is a line that intersects an object, like a circle or a curve, at one (and only one) point. That's the rub. Way back in my geometry class, I learned that you needed at least two points to make a line. The difference between this two point will give us the rise and run of the line, which is its slope. So how can Impossibly find the slope of a tangent line to a curve if a tangent is only one point? The answer, learn calculus. The short answer, and the heart of all calculus it seems, is to find another point in the curve that is sooooooooooooooo close to the first point that the distance between them is infinitesimal and therefore negligible.

In The chart opposite, as point point B in the curve get closer and closer to point A that is also in the curve, the line that passes through them looks more and more like the tangent of the curve.

 

Now imagine the curve actually represents a car's velocity traveling on a straight line. Then point A is we're the car is at time x. In algebra we can find the average velocity between A and B1 by subtracting the miles traveled by the time taken to travel those miles. However, how can I find the exact velocity (Instantaneous velocity) of the car at point A? If I make the time interval between A and B small enough as to make close to an instant, then I can Algebraically compute an approximation of instantaneous velocity.

 

In integral calculus, the questions searched are a little different.

 

What is the area of an exotic shape? And from what I have studied, exotic shapes just mean shapes other than the ones we have formulas for. We have formulas for squares, circles, triangles, cubes, spheres, pyramids and if my memory doesn't fail me, cones. I bet we also have formulas for many other shapes, but how about the area of the shape in the chart below.

 

How do we measure this shape's area?

 

Well, we could inscribe it in a grid of squares of a given size and count the squares that the shape fills.That would give us an approximation of the area we are looking for.

If we start to make the squares smaller and smaller, more of the shape is inscribed within it.

We can repeat the process of making the squares smaller and smaller, to get better approximations. If we make those squares infinitely small, the value of the area of the shape will be so close to its real area that the difference would be negligible.

 

Therefore, the previous act of getting the tangent of the curve and now the act of getting the area of an exotic shape, relied on the same procedure to get answered. We used infinitesimally small numbers. In fact, I have run across various references to calculus as infinitesimal calculus. If we want to make the jump from algebra, which can only give us approximations of the answers to these questions, to calculus , where we can get the exact answers we are looking for, we must go through Limits.

 

And so must I, next time.

 

What do to think?

 

 

 

Beginning Calculus in fits and starts

After 10 weeks of taking a class, it has been hard for me to figure out a way to start Calculus. The past few days I've been collecting resources to help me study. I have videos, ITunes courses, websites and more. What I now need is structure.

 

I guess I got too used to being given material instead of going to look for it. Maybe I am just holding myself back in fear of what's to come. Two posts ago I, finished with a line that sums up my feelings. I illustrate it bellow.

According to some accounts, ancient map makers used to put inscriptions at the edges of their maps that read: "beyond here there be (insert your mythological beast here)". I guess it was their way of saying, "don't know what's beyond so it must be sea monsters, dragons and some other weird stuff." It could have also meant they feared what was there.

 

Do I fear calculus?

 

A little. But I fear not knowing what's beyond even more.

 

It's about time Imjust dive in. Training's over. A part of me knows I am ready for this.

 

So here it goes...

 

Statement of accomplishment

I got my statement of accomplishment from the UCIRVINE pre-calculus class I took at Coursera. It looks clean and simple. It has my name and the signature of both professors in the class. The only grievance I have is the note at the bottom.

 

It reads:

"PLEASE NOTE: THE ONLINE OFFERING OF THIS CLASS DOES NOT REFLECT THE ENTIRE CURRICULUM OFFERED TO STUDENTS ENROLLED AT THE UNIVERSITY OF CALIFORNIA, IRVINE. THIS STATEMENT DOES NOT AFFIRM THAT THIS STUDENT WAS ENROLLED AS A STUDENT AT THE
UNIVERSITY OF CALIFORNIA, IRVINE IN ANY WAY. IT DOES NOT CONFER A UNIVERSITY OF CALIFORNIA, IRVINE GRADE; IT DOES NOT CONFER UNIVERSITY OF CALIFORNIA, IRVINE CREDIT; IT DOES NOT CONFER A UNIVERSITY OF CALIFORNIA, IRVINE DEGREE; AND IT HAS NOT VERIFIED THE IDENTITY OF THE STUDENT."

This is my first Coursera course so I do not know if all other universities use the same language. If they do, shame on them. I understand all the caveats they must state, specially the part about this course being in no way eligible for college credit and the fact that in the free version of the course they could not verify your identity. I guess what bothers me is that they focused on all the negatives, none of the positives.

 

I will make a suggestion in how to phrase the bottom of those certificates.

Please Note: This online offering while covering essential topics of the subject studied is different from the curriculum offered to students enrolled at UC Irvine. The statement is given to the student named above under the assumption that said student abided by the honor code explained in the syllabus and handed in their own work. Unfortunately we cannot verify this students's identity. This course is not eligible for college credit, grade or degree at UC Irvine. This statement does not affirm that this student was enrolled as a student of UC Irvine.

But maybe I'm just overly sensitive. Good news is I passed the course...and at least I can verify my identity. At least I hope I can. Let's check:

 

Fernando Santiago = the guy who spend 10 weeks doing an UC Irvine precalculus class at Coursera.

Me=Me

 

Identity verified.

 

This post should give me closure.

 

Your thoughts?

Autopsy of a test result: How panic, exhaustion and arithmetic don't mix

Since my school days I've had this rule on test results: If you can understand why your answers are incorrect, you are OK. The reasoning behind it is that if you are able to figure out what went wrong and why, that means the learning had taken place but there was a mistake along the way to execution. On the other hand if you have no clue as to why your answer is incorrect, that should send you back to review the concepts being tested.

 

As I related in my previous post I scored a 24.5 out of 34 in my Pre-Calculus final. The one that was 2.5 hours long and I had to take twice back-to-back because I thought I had set the time running on the final attempt by mistake. I took this week to understand what wheat wrong in those 10.5 items I got wrong. The answers surprised me. But they really shouldn't have.

 

In one of my Encouragement pictures posted here there is a quote that reads: 10 out of 9 times it's arithmetic that gets you. No one seems to get the joke... But this test is testament to that. 9 out of the 11 items missed were due to arithmetic, but not the way that you think. Here is the list of reasons my answers were incorrect.

 

1. Wrote the wrong sign when copying problem.
2. Used full angle when I needed the co-terminal angle. Which I knew was only in Q4 or Q1.
3. Wrote the wrong sign when copying problem.
4. Wrote a 28 that looked like a 78, I couldn't read my own writing!
5. This time I used a sign incorrectly (summed -3+10 and wrote -7).
6. Used a parenthesis instead of a bracket in an interval, even when I knew the number was included in the interval.
7. Error copying line in a problem, I wrote a 5 instead of a 1 in the line below.
8. I got the answer write but the program would only accept the variables py as p*y. A clearer head would have realized this.
9. Boldly stated that the square of 9 was 49...which is the square of 7.
10. Forgot to find the squarer root of the hypotenuse in a Pythagoras theorem solve. I am actually good at solving these problems, it was a huge oversight.
11. I was too exhausted to simplify the solution of a half-angle cos identity.

With the possible exception of 2 and 6, the rest of my mistakes were due to my exhaustion and my panic to solve the test. Most of them were due to sloppy, hurried writing. The others were due to foggy thinking. I could find none that showed a flawed understanding of the material. In fact when I got back to doing these again for review, the moment I found my mistake was invariably followed by a "well, duh!", or "that was a stupid mistake". None of them were followed by a "why is this incorrect".

 

Before you think I am saying that I deserved a higher score, rest assured I feel I got the score I deserved. The mistakes I made were mistakes nonetheless, and they prove I need to pay more attention to what I am doing. And prove that I shouldn't take 5 hour long tests at 3am on a Saturday!

 

What do you think?

 

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.