As a class project, a student wanted to calculate the velocity of a marker being thrown into the trash. He calculated the time it took the marker to hit the trash after he released the marker and the distance to the trash. This is the horizontal average velocity, so I asked him to go further. The image below is a projectile motion still and a quick desmos graph of a parabola of best fit. Now he has an equation for height… his task is to determine the vertical velocity.
Below is a link to a video of how I created this image.
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?
The following is the abstract of a recent paper I wrote analyzing student performance data. Here is a link to the entire paper.
Measurement of student achievement is at the heart of educational policy and standardized testing has been both supported and contested as a genuine representation of student achievement. This study shows that the interpretations of standardized testing may have starkly contrasting meanings for different cohorts of students. By using quantile regression to account for conditional PSAT scores, educational factors such as gender and tracking are shown to effect students in varying ways. For the students in this study, gender was a significant indicator of performance on the PSAT test, with being male accounting for more than a five point “bump.” The conclusion from the quantile regressions is that students with extreme PSAT scores are outliers based on ability or inability, not because of their gender. Meanwhile, students that participated in the “honors” tracking system had more of an increase in their predicted score the higher their conditional PSAT score.
It’s that time of year when almost 100 million Americans go back to school. Yes, approximately 25% of the US population is enrolled at least part-time as a student at some level of kindergarten through graduate programs. My question is whether these students are learning skills applicable outside of the classroom or are they just earning credentials for their resume?
The easy answer is both, but I would argue that after initial universal skills are learned, education acts as a signal of ability to comprehend and analyze. It is important to distinguish between learning a skill, such as the ability to problem solve, versus signaling that you have the ability to problem solve. Before going too deep, an example should clarify the difference.
Consider high school math, in Algebra or Geometry a student may learn the skill of deductive reasoning and problem solving. Then a few years later in Calculus, the students signal that they have the requisite skills needed to comprehend Calculus concepts. Note that this signal (the ability to understand Calculus) may not be an applicable “life skill,” but if you ask a college admissions officer they would tell you it is quite valuable to signal to colleges that you are able to complete a Calculus course.
What skills will be acquired in schools this fall that will be applicable? Obviously basic reading, writing, and arithmetic are necessary skills learned in the lower grade levels. Once these skills are mastered, the second level of thinking comes into play. That is, using factual knowledge to solve problems. Finally third level thinking, including synthesizing (or combining) knowledge from various sources to derive original thoughts and conclusions. The last two “levels of thinking” are not necessarily taught on their own, yet students acquire these skills through high school and these attributes are the base argument for a “liberal arts education” promoted by colleges.
So, when does education stop being skill based and start becoming a signaling device? My conclusion is that this balance transfer occurs in the midst of the traditional high school years. The American high school curriculum (including calendar and schedule), include many requirements that act as hurdles. Also, many advanced graduate programs fail to provide students with necessary skills. These degrees simply act as acknowledgement that a student has read the core literature of a field and does not enable students to go out into the world ready to produce original ideas.
Below are two graphs that I believe represent traditional educational paths for American students and the subsequent returns of skill development from that education. The first graph represents a student whose returns to schooling are negative, that is each year fewer skills are acquired. Unfortunately, I believe this is the path of the majority of American students. The second graph represents a student that take a career orientated path and ends their education in a trade school or college or graduate program that gives the student real skills to be applied outside of the classroom.
If we want education to have meaning for our students, we should gear curriculum to achieve the second path shown above. Ideally, we could create a skill development graph that is a horizontal line where students are constantly acquiring applicable skills. This may seem far off, but if we want our future citizens to be equipped to face the challenges of an ever-changing economy and job market, we need to be sure that they are prepared with applicable skills, not a diploma that recognizes that they jumped through the appropriate hoops.
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.
A year ago, the LA Times published a ranking of over 5 thousand third through fifth grade teachers based on their value added scores. If you are not familiar with Value Added Ratings, they are a statistical technique that measures how much a teacher’s students improve on a standardized test over the year the teacher works with those students. These ratings are very controversial, especially as a means to evaluate teachers. To a parent reading these rankings in the LA Times, they seem like a say all end all evaluation of their child’s teacher, prompting calls to change classes and fire teachers. In May the LA Times released value-added ratings for over eleven thousand more teachers without addressing the intricacies of the data.
Last winter, I analyzed Value-Added Ratings, specifically as a way of evaluating teachers. Here is a link to my paper (see pages 3-6 for a literature review on the subject and p. 6-8 for a look at the economic model, while p. 9-17 are a critique of the method).
Here is a summary of that paper:
Economists and educational researchers studying student achievement return consistent results about teachers: they matter. While researchers agree that educational achievement depends on a quality teacher, disagreement among both economists and educational researchers occurs when considering what constitutes teacher quality. Clearly, one of a teacher’s specific duties is to improve student performance and value added statistical methods do measure student progress, but the issue arises when that progress or lack of progress is considered completely the effect of the teacher.
This paper concludes that if school administrators only evaluate teachers on student progress, then they may not be measuring the teacher’s total value to students. The review shows that many factors, both observed and unobserved, can affect student achievement and that teachers have the ability to impact their students in ways that standardized tests cannot measure.
In a 2008 New Yorker essay, Malcolm Gladwell compared hiring a new teacher to drafting an NFL quarterback. In both professions observable characteristics do not translate to production, whether on the field or in the classroom. In the NFL, coaches can examine statistics that their quarterback directly controls and that are easily observed: completions, yards, and touchdowns, but observations from college games do not translate to NFL success. Meanwhile, school administrators only evaluate teachers on observable characteristics such as experience, highest degree, undergraduate university attended, and in a recent number of cases Value Added Ratings.
Just as a quarterback’s college statistics do not directly indicate NFL performance, easily measured or observable characteristics of teachers are not always correlated with student success. Value added scores vary significantly from year to year and only measure student improvement on standardized tests, not necessarily learning. If, over a ten or fifteen year period a teacher’s Value added score was consistently higher or lower than average, then the scores can tell nus something about the teacher. But looking at one score from one year and publishing that as an all encompassing value of a teacher is unfair.
Many factors, both observed and unobserved, can affect student achievement; teachers have the ability and responsibility to impact their students in ways that standardized tests cannot measure. If school administrators only evaluate teachers on student progress on a standardized test, then they are not necessarily measuring the teacher’s total value added to their students.