I’m not a competitive person. Let me correct that. I try not to be a competitive person. I’ve recently been playing some of my favorite games from childhood like Monopoly, Chess, Spades, and Madden and I’ve been reminded of the competitive streak in me that hates losing. This streak has been relaxed in much of my adult life, and I tend to think that’s been to my benefit. Two pieces come to mind as I write this. One is a piece written by Slim Jackson of Single Black Male on the importance of asking questions. The second is what I heard on WAMU’s “Tell Me More” while driving home from work regarding the competitive nature generally assumed by (or dictated to) men. These two concepts go hand in hand in my opinion because I’ve found that at times where I see a person as “my competition” I’m less likely to seek advice or help from that person. Doing this limits my resources and the set of people who can help me. I know that one of the hardest things to do while playing a game like Madden is asking my competition “how’d you just make that play” or listen to this competition explain how he knew to make such a play. However, its just this ability to swallow my pride and ask these questions that I’ve gotten better in Madden. I will admit that I’m not the most humble person on the face of the earth and so there are some things that I haven’t asked how to do. Generally in those things, I’ve found myself repeatedly playing the game trying to figure things out for myself. There are applications of both sides of this to the real world. Some who love competition will say that it brings out the best in us, and they’d be sure to point out the feeling of satisfaction we get by working independently on a problem. As I look through my lists of inspirational quotes, I’m reminded of this with often repeated statements like “failure is the key to success”. As rewarding as a competition can be, there’s also an important saying that we don’t need to re-invent the wheel. And what often gets lost in the do-it-yourself nature of competition is the ability to utilize all possible resources available to us. In particular, the skills of how to acknowledge the things we do not understand and how to formulate questions aimed at gaining understanding. This brings us to is the other side of the competitive spirit. As rewarding as it is to be able to say, “wow, I can’t believe I was able to figure that out on my own”, it is also a stressful situation and there are many who are never able to say those words. Should these people be satisfied with failure? I do not ask this in some devil’s advocate type of way. I ask coming from the point of view of a mathematician, an educator, and as a former student. I had a pretty dark moment in graduate school where I realized that my mere “love” of mathematics would not get me through qualifying exams. It wouldn’t suddenly make text books and academic papers instantly understandable. Suddenly I was placed in an uncomfortable position. Instead of always being the one who was the first to get the concept and who was leading the study groups on it, I’d be the one asking the questions. Looking at this in a “competitive” frame of mind (as I did then), I felt like I was losing the game. This same moment though, is where my thinking was really changed. There was one thing, and one thing only that I would consider a failure and that was not finishing. I view everything else as a matter of swallowing my pride and readjusting my thought process to help get to that point. Unfortunately though, many others do not get to this point. Many get lost in the scramble of the competition and do the equivalent of folding your hand in poker. They realize that at their current pace there is little to no chance that they’d win and so they just leave the game. And this is a real risk that we’re running with competition, particularly as STEM fields are becoming more and more important and we’re trying to encourage students to focus on these areas. What may be necessary to bring this about is a more cooperative approach to these things.
I’ve added a script which helps to understand geometric sequences. Suppose you were to draw an equilateral triangle on a sheet of paper. It might look something like this: Now suppose that you draw lines connecting the midpoints of each of the edges of this triangle. This will dissect the larger triangle into four smaller triangles, each of which are equilateral. Three of these smaller triangles will be oriented in the same direction as the original triangle, whereas one will not. Consider the second image below, with the three triangles with the same orientation as the original triangle numbered. We can continue to draw lines connecting the midpoints of the edges of the marked triangles and counting the resulting triangles that have the same orientation as the original triangle and we see that a pattern emerges. What one notices is that each time we draw a new triangle by connecting the midpoints of the marked edges, we wind up with three times the number of triangles that were in the previous picture. So (assuming we had enough space) we could draw out the figure that would be the result of doing any number of these dissections. However, if we are only interested in knowing the number of triangles that each image will contain, we can take advantage of the fact that this pattern represents a geometric sequence. A geometric sequence is a sequence with an initial term, a1 and a common ration, r, where each term after the initial term is obtained by multiplying the previous term by the ratio (a1 cannot be zero, and r cannot be zero or one). In a geometric sequence, if we know the first term and the ratio, we can determine the nth term by the formula an = a1*rn – 1 Similarly, if we know the first term and the ratio, we can determine the sum of the first n terms in a geometric sequence by the formula:
For the previous example with the triangles pointed in the same direction, we can show the results in the following table:
The script is available at http://www.learninglover.com/examples.php?id=34
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|Drawing Number||Number of Triangles||ratio||sum number||sum value|
I’ve added a script which helps to understand arithmetic sequences. At a previous job of mine, there was a policy of holding a dinner party for the company each time we hired a new employee. At these dinners, each employee was treated to a $20 dinner at the expense of the company. There was also a manager responsible for keeping track of the costs of these dinners. In computing the costs, the manager noticed that each time there is a new dinner, it was $20 more expensive than the last one. So if we let a1 represent the cost of the first dinner, and let ai represent the cost of the ith dinner, then we see that ai = ai-1 + 20. Sequences like this, where t arise quite often in practice and are called arithmetic sequences. An arithmetic sequence is a list of numbers where the difference between any two consecutive numbers is constant. For the example above, the term an will represent the cost of dinner after the nth employee has joined the company (assuming that no employees have left the company over this time period). Also the term Sn will represent the total cost the company has paid towards these dinners. Before we continue with this example, consider the following table which lists the first five terms of an arithmetic sequence as well as the common difference and the first five sums of this sequence.
One of the beauties of arithmetic sequences is that if we know the first term (a1) and the common difference (d), then we can easily calculate the terms an and Sn for any n with the following formulas:
an = a1 + d*(n – 1), where d is the common difference.
Sn = n*(a1 + an)/2
We can use these formulas to derive more information about the sequence. For example, if my manager wanted to estimate the cost of dinners once we had added 30 new employees, this would be term a30 of the sequence, which we can evaluate with the above formula by a30 = a1 + d*(n – 1) = 0 + 20*(30 – 1) = 0 + 20 * 29 = 580.
The script is available at http://www.learninglover.com/examples.php?id=33.
Other Blogs that have covered this topic:
Study Math Online
|term number||term value||diff||sum number||sum value|