Last year as a test review, I asked my students to categorize problems. Beyond knowing the mechanics, I want my students to know the when and why. Thus I asked them to tell me what “big theme” each problem represented. With a set of review problems, I began to #hashtag the problems. Two examples are shown below:
I have no scientific research showing that this helps my students make connections and categorize concepts, but they certainly remembered and enjoyed hashtagging math problems! This is a technique I plan on continuing and is worth considering beyond just review. At minimum, hashtagging in class was unique, so it stood out as something special. Their favorite was #factoredform, so I wonder how long until its trending?
In my first (found here) of a three part series on statistical analysis, I discussed how data can inform decisions within countless industries. My research in human capital, and specifically education, provides for various usages of data in decision making. Specifically, available data can be used for predictions of when, where, and which students may have trouble, determinants of parental satisfaction, admission decisions (at both secondary and collegiate levels), and student achievement both academically, on standardized tests, and in future wages.
This post is intended to shed light on the science of data analysis, specifically conditional expectations. The mathematical approach to conditional expectation is based on heavy statistical concepts, some of which are not appropriate for this venue. That being said, my intention is to provide an accessible explanation of the techniques used in statistical analysis. For a mathematical approach, I recommend the seminal text Econometric Analysis, by William Greene, 2002; or Econometric Analysis of Cross Section and Panel Data, by Jeffrey Wooldridge.
Econometrics, and essentially any data analysis, is based on determining a prediction. In statistical terms, this is called an expectation. A conditional expectation is a prediction based on available information.
For instance, consider guessing the height of a random human being. The average human height is 5 foot 6 inches, so, this would be a logical starting point for our prediction. But if we know more information about this random person, we can improve our expectation. Specifically, if we knew the person was male, we would want to change our expectation conditional on that fact. The average height of an adult man is 5 foot 9.5 inches, so that would be our new prediction given some information.
If we also knew that the person weighed 240 pounds, we may want to increase our expected height. Note here that there is no causal assumption, just a change in our expectation, given some piece of information. This is correlation: the taller a person is the more, on average, we expect that person to weigh. We may predict 6 foot 1 inch for our random male weighing 240 pounds. Data analysis can help us with our predictions, given some imperfect information.
Next, consider that we also knew this random person’s SAT score was 1200. We probably would not consider changing our expectation of height. If we had data on a sample of people with the variables height, sex, weight, and SAT score, some variables may be good predictors of height and be statistically significant while others, SAT score in this example, would not be statistically significant and would not persuade us to change our expectation.
Multiple Regression Analysis essentially considers a sample of data and determines the predictive success of each variable. Using the information from a statistical data program (even excel can do this reasonably well), we can arrive at a predictive equation for the most logical expectation conditional on the information we have at our disposal.
The data will find the coefficients- the “B’s”- and also determine the likelihood that each “B” is a significant predictor. In this case, I suspect B3 would not be statistically significant.
In part three, I will discuss an example of student achievement data from a nation-wide sample and how conditional expectations can be used to inform decisions in many fields.
This past week Harrah’s Casino in Chester, PA held a promotion where selected patrons could come to the casino and redeem a gift card of a random amount between 20 and 100 dollars. The casino received many more patrons than expected and began to give out free slot play once demand exceeded the supply of gift cards. In fact, there was a line of over two hours to reach the promotional table.
Was the over two hour wait worth the gift card promotion? The most basic principle of economics- that if a transaction takes place then both the buyer and seller agree on the price an good- tells us that people thought the gift card promotion was worth the wait. But what exactly was the good and what exactly was the price?
Determining the good is not difficult. The specific terms of the promotion are not available to this author, but a simple expected value equation would solve that problem. If the chance of “winning” the gift card game (receiving the $100) is even and uniformly distributed then the expected value of the game is $60. I am sure that the odds were not even (this was at a casino none-the-less), so lets say that the expected value of the game was $50. So the “good” was a chance of winning $50, with a guarantee of $20.
The price is significantly more difficult to figure out. There was no explicit monetary risk for the patrons. Instead, patrons had to give up their time, which in econ 101 you learn is the “opportunity cost.” On a Saturday morning, what was their opportunity cost?
After many patrons received a $20 gift card (implying that the expected value was probably on the lower end of the spectrum), there were some grumblings that their time could have been better spent. But, doing what? Two extra hours of overtime at work may have resulted in a payoff greater than $20 or even the expected value of $50, but the opportunity to work overtime is diminishing. Production in the home is probably more valuable than a $20 gift card, but $50… thats a lot of groceries… or a nice dinner.
Ultimately, despite the grumblings that the promotion was not worth the wait- implying that the opportunity cost was too high- the ultimate economic principle proves the opposite. People came to the venue (most likely an hour total commute), waited in line for two hours, and were paid by a lottery guaranteeing $20 with a slightly higher (no more than $50) expectation value. Thus, the transaction took place so the value of the good was equal to the cost.
In my summer school PreCalculus class today we continued our discussion of Polar Coordinates. I began the class by asking the students to draw the graph of:
r = 2sin(θ)+2 by hand. Their gaph should have looked like this:
We discussed the process of translating between rectangular coordinates and polar coordinates when a student asked “can we just transform the entire function?” This led to a geeky, yet exciting, hour discussion and investigation.
First, we simplified the problem by dropping the “+2” from r = 2sin(θ)+2 (don’t worry we’ll go back to it…). So we algebraically converted the equation r = 2sin(θ) into rectangular coordinates (x and y). After substituting the conversions in (y=rsin(θ) and r2 =x2+y2), the algebra cleans up nicely by completing the square and a circle with center (0,1) and radius of 1 emerges. After half the class had already plugged the polar coordinate equation into their calculator, we all agreed that this process worked for r=2sinθ, but what about r = 2sin(θ)+2?
Here’s the input into Wolfram Alpha: graph y=2sin^2(aCos(x/(sqrt(x^2+y^2))))+2sin(aCos((x/(sqrt(x^2+y^2))))
Anyway, here was the graph Wolfram Alpha spit out:
Where’s the bottom!!!!
So, in our equation, to find the angle of the polar coordinate in terms of x and y, we had to use inverse cosine, whose range is only zero to pi. In order to get the bottom, we needed angles from pi to 2pi. To find that, we had to shift our cosine by pi; that is, we had to add pi to each of the angles:
Wolfram Alpha input: graph y=2sin^2(aCos(x/(sqrt(x^2+y^2)))+pi)+2sin(aCos(x/(sqrt(x^2+y^2)))+pi)
Wolfram Alpha output:
But how do we get both? A simple google search (remember this is the middle of class and my computer’s monitor is projected to the entire class, so a little bit of a gamble… good thing Wolfram Alpha isn’t a subsidary of GoDaddy!) landed me on this blog: Gower’s. Somewhere in the comments, I found that a simple comma would allow my to graph both equations together, so i input the following into Wolfram Alpha:
graph y=2sin^2(aCos(x/(sqrt(x^2+y^2))))+2sin(aCos(x/(sqrt(x^2+y^2)))), y=2sin^2(aCos(x/(sqrt(x^2+y^2)))+pi)+2sin(aCos(x/(sqrt(x^2+y^2)))+pi)
Wolfram Alpha thought for awhile, and returned nothing.
PLEASE HELP US if you know how to get wolfram alpha to graph two intense equations. (note: I can graph two simple graphs: input: graph y=2x,y=x^2)