News this weekend that San Jose State in California has suspended its experiment with Udacity offering low-level courses for pay and college credit and requirements:

http://www.latimes.com/news/local/la-me-0719-san-jose-online-20130719,0,4160941.story

Key point: "Initial findings suggest that students in Udacity courses performed poorly compared with students in traditional classes." Note that this is broadly in line with the prediction I made here several weeks ago, in the post titled Online Remedial Courses Considered Harmful, something that I considered to be a fairly easy and obvious call. I asked the question, "We'll see how quickly MOOCs such as UDacity, and those partnering,
paying, and linking their reputation with them, re-learn this lesson", and I'd have to say that this turnaround was faster than I would have guessed at San Jose State. Perhaps they will agree with the earlier experiment at the Philadelphia school where it was concluded, "The failure rates were so high that it seemed almost unethical to
offer the option" (see link to my earlier post above).

The last paragraph of today's news story reiterates my own views, which I've written about here on numerous occasions: "Educators elsewhere have said the purely online courses aren't a good
fit for remedial students who may lack the self-discipline, motivation
and even technical savvy to pass the classes. They say these students
may benefit from more one-on-one attention from instructors."

A few other points: "Preliminary results from a spring pilot project found student pass rates
of 20% to 44% in remedial math, college-level algebra and elementary
statistics courses." Now, it would be much better if this success rate were broken down individually for each of these several classes. I might guess that the 20% success rate is specifically for the remedial math course? That does seem marginally lower than most remedial courses where the success rate seems to be around one-quarter or one-third.

Also, the article says, "In a somewhat more promising outcome, 83% of students completed the classes." This seems unsurprising, given that students are paying $150 out-of-pocket for the course. This completion (but mostly failing) rate is about in line with the remedial courses that I teach, where students are similarly paying, meeting an absolute requirement by the college, and have no real academic penalty for failing (the course grade does not affect GPA, for example).

Perhaps charitably we might say that the $150 expense level is lower than standard college teaching costs, and perhaps someone might think it's a reasonable return on investment, even granted a lower success rate (although maybe not when accounting for student time spent). And we might also be suspicious of (a) whether this is the actual Udacity expense, or if they're operating at a loss to establish the market, and (b) the quality of the assessment at the end, when there's a clear incentive to make it easy to pass and the Udacity statistics final I've seen in the past was almost comically trivial.

Supposedly this suspension is for re-tooling and analysis of possible improvements. "The courses will be offered again next spring, [San Jose State Provost Ellen Junn] said." We shall see.

## Monday, July 22, 2013

## Monday, July 15, 2013

### Proof of Approximating Radicals to the Closest Integer

The in-house textbook that my college uses for basic algebra classes does an interesting thing -- as part of the introduction to radicals, it goes through approximating a whole-number radical by comparing it to the nearest perfect squares. An example from the book:

On the other hand, consider the cutoff between the squares of the integers in question. Any x below their average is closer to n

So you can see that there's a gap between these two cutoffs, and in fact it's exactly 1/4 in all cases, no matter what the value of n. If you pick x in the range n

In conclusion: While the claim in question is

So I think we all agree that in a case like this, the radical is clearlyExample 2: √3000 is closest to which integer?

Solution:... [after some preliminary estimates] Try between 50 and 60, (55)^{2}= 3025, still a bit too high. Try (54)^{2}= 2916, now a little too low. Thus √3000 is between 54 and 55, but closer to 55 since 3025 is closer to 3000 than is 2916.

*between*the two integers indicated (since the radical function is monotonic). But the additional step of saying which of the two it's closer to is not done in all textbooks. Here's another example (not our school's textbook). Let's clearly state the claim being made here:Above, "closest" means the minimum distance from x to any n ∈ ℕ. This claim gave me a squirrelly feeling for some time, and with good reason; itClaim: If x is closest to n^{2}, then√x is closest to n.

*isn't*true for arbitrary x ∈ ℝ.Now, let's characterize the kinds of numbers for which the claim in questionCounter-example: Consider x = 12.4. It's closest to the perfect square 3^{2}(distance 3.4 from 9, versus 3.6 from 16). But the square root is actually closest to the integer 4 (√12.4 ≈ 3.52).

*won't*work. For some integer n, take the cutoff between it and its successor, n+1/2 (i.e., the average of n and n+1). Any*√*x below this value is closer to n, while any*√*x above it is closer to n+1. Under the squaring operation, this cutoff gets mapped to the square-of-the-average (n+1/2)^{2}= n^{2}+n+1/4.On the other hand, consider the cutoff between the squares of the integers in question. Any x below their average is closer to n

^{2}, while any x above the average is closer to (n+1)^{2}. This average-of-the-squares is ((n)^{2}+(n+1)^{2})/2 = (n^{2}+n^{2}+2n+1)/2 = (2n^{2}+2n+1)/2 = n^{2}+n+1/2.So you can see that there's a gap between these two cutoffs, and in fact it's exactly 1/4 in all cases, no matter what the value of n. If you pick x in the range n

^{2}+n+1/4 < x < n^{2}+n+1/2, then*√*x will be closer to its ceiling of n+1, but x itself will be closer to its floor-square of n^{2}. Specifically, the problem cases for x are anything a bit more than the product of two consecutive integers (also called a pronic or oblong number), exceeding n(n+1) = n^{2}+n by a value of between 1/4 and 1/2. Since n^{2}+n is itself an integer (ℕ closed under add/multiply), we see that any x in violation of the claim must be strictly between two consecutive integers, and thus cannot itself be in ℕ.In conclusion: While the claim in question is

*not*true for all real numbers, it is a trick that does happen to work for all whole-numbered values of x. How important is that? Personally, I'm pretty uncomfortable with giving our students an unverified procedure which can leave them thinking that it works for any number under a radical, when in fact that's not the case at all.## Monday, July 8, 2013

### Why Z-Scores Have Mean 0, Standard Deviation 1

This article is aimed at introductory statistics students.

Statistics, as I often say, is a "space age" branch of math --many of the key procedures like student's t-distribution weren't developed until the 20th century (and thus helped launch the revolution in science, technology, and medicine). While statistics are really critical to understanding modern society, it's somewhat unfortunate that they're built on a very high edifice of prior math work -- in the introductory stats class we're constantly "stealing" some ideas from calculus, trigonometry, measure theory, etc., without being explicit about it (the students having neither the time nor background to understand them).

One of the first areas where this pops up in my classes is the notion of z-scores: taking a data set and standardizing by means of z = (x − μ)/σ. The whole point of this, of course, is to convert the data set to a new one with mean zero and standard deviation (stdev) one -- but again, unfortunately, the majority of our students have neither the knowledge of linear transformations nor algebraic proofs to see why this is the case. Our textbook has a numerical example, but in the interest of time, my students just wind up taking this on faith (bolstered, I hope, by a single graphical check-in).

Well, for the first time in almost a decade of teaching this class at my current college, I had a student come into my office this week and express discomfort with the fact that he didn't fully understand

So I hunted around online for a couple minutes for an explanation, but I couldn't find anything really pitched at the expected level of my students (requirements: a

(Edited: Jan-9, 2015).

Statistics, as I often say, is a "space age" branch of math --many of the key procedures like student's t-distribution weren't developed until the 20th century (and thus helped launch the revolution in science, technology, and medicine). While statistics are really critical to understanding modern society, it's somewhat unfortunate that they're built on a very high edifice of prior math work -- in the introductory stats class we're constantly "stealing" some ideas from calculus, trigonometry, measure theory, etc., without being explicit about it (the students having neither the time nor background to understand them).

One of the first areas where this pops up in my classes is the notion of z-scores: taking a data set and standardizing by means of z = (x − μ)/σ. The whole point of this, of course, is to convert the data set to a new one with mean zero and standard deviation (stdev) one -- but again, unfortunately, the majority of our students have neither the knowledge of linear transformations nor algebraic proofs to see why this is the case. Our textbook has a numerical example, but in the interest of time, my students just wind up taking this on faith (bolstered, I hope, by a single graphical check-in).

Well, for the first time in almost a decade of teaching this class at my current college, I had a student come into my office this week and express discomfort with the fact that he didn't fully understand

*why*that was the case, and if we'd really properly established that fact. Of course, I'd say this is the very best question that a student could ask at this juncture, and really gets at the heart of confirmation and proof that should be central to any math class. (Interesting tidbit -- the student in question is a History major, not part of any STEM or medical/biology program required to take the class.)So I hunted around online for a couple minutes for an explanation, but I couldn't find anything really pitched at the expected level of my students (requirements: a

*fully*worked out numerical example, graphical illustration without having heard of shift/stretches before, algebraic proof without first knowing that summations distribute across terms, etc.) Instead, I took some time the next day and wrote up an article myself to send to the student, which you can see linked below. Hopefully this careful and detailed treatment helps in some other cases when the question pops up again:(Edited: Jan-9, 2015).

## Monday, July 1, 2013

### Institutionalized Score Mangling

For some reason, there's been a bunch of stories of schools secretly boosting near-failing grades recently. A few that come to mind:

All of this certainly seems really bad to me in a first-pass "smell test" of credibility. It just seems like any kind of secret score-mangling is a foul wind that carries with it lack of transparency, disbelief in results, corruption, etc. Interestingly, a great many commentators at Slashdot (around the Indian story) said things like "this is done everywhere, if you don't understand it then you don't know anything about teaching", which is false in my experience. But apparently the motivation is frequently to avoid conflict and time spent around complaints over barely-failing scores. Some other institutional strategies I've seen or heard about to deal with this issue:

Now, I'm always in favor of maximal transparency, honesty, and confidence in any kind of process like this. But in some cases I've found myself to be a lone voice for this principle. Is this kind of secret score-mangling an acceptable social massaging of high-stakes testing, or is it the harbinger of corruption and non-confidence in our institutions? Do we even have any choice in the matter anymore, as educators or citizens?

- Just this weekend -- Hempstead High School on Long Island (somewhat near me) has a scandal of regular boosting failing 63 and 64 scores to passing 65's in any class from grades 6-12. Apparently this has been done for some number of decades, and the Deputy Superintendent defends it as customary at their school and others (although it was done in secret and not any documented policy). Other schools nearby deny that they engage in the same practice.
- Early last month, an Indian student attending Cornell University accessed and mined the data from the Indian national high school exams from the last year, and found that the scores being reported were very clearly manipulated in some secret way, as there were irregular gaps in the achieved scores across all subject areas. In particular -- none of the scores 32, 33, or 34 were achieved by any student for any subject in the entire country, whereas 35 is the minimum to pass.
- Less publicized (but perhaps more dramatic) is the fact that New York State Regents Examinations are in some sense getting easier, as the high school system brags about increased graduation rates at the same time as their graduates needing remedial instruction in college reaches around 80%. Someone who really ought to know told me that the scores on the exams are effectively mangled by administrators in Albany, i.e., a 45% raw performance is reported as a passing scaled score of "70" and so forth.

All of this certainly seems really bad to me in a first-pass "smell test" of credibility. It just seems like any kind of secret score-mangling is a foul wind that carries with it lack of transparency, disbelief in results, corruption, etc. Interestingly, a great many commentators at Slashdot (around the Indian story) said things like "this is done everywhere, if you don't understand it then you don't know anything about teaching", which is false in my experience. But apparently the motivation is frequently to avoid conflict and time spent around complaints over barely-failing scores. Some other institutional strategies I've seen or heard about to deal with this issue:

- Those who miss passing by 5% get to immediately take a re-test. I haven't seen this, but I've heard it said of other universities.
- Those who miss passing by 5% get a one-week refresher seminar, and can then re-test on the final. A somewhat more subtle version of the preceding which is used where I teach at CUNY for math remediation.
- Keeping both scores and the passing criteria itself secret -- reporting only pass-or-fail results for the test. This was done in the past at my college, allegedly to forestall complaints over scores. It's pretty much my least favorite option, because it just made everyone involved confused and upset over the secret criteria and unknown scores.

Now, I'm always in favor of maximal transparency, honesty, and confidence in any kind of process like this. But in some cases I've found myself to be a lone voice for this principle. Is this kind of secret score-mangling an acceptable social massaging of high-stakes testing, or is it the harbinger of corruption and non-confidence in our institutions? Do we even have any choice in the matter anymore, as educators or citizens?

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