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.
2013-07-22
San Jose State Suspends Udacity Experiment
2013-07-15
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 n2, while any x above the average is closer to (n+1)2. This average-of-the-squares is ((n)2+(n+1)2)/2 = (n2+n2+2n+1)/2 = (2n2+2n+1)/2 = n2+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 n2+n+1/4 < x < n2+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 n2. 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) = n2+n by a value of between 1/4 and 1/2. Since n2+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.
Example 2: √3000 is closest to which integer?So I think we all agree that in a case like this, the radical is clearly 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:
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.
Claim: If x is closest to n2, then √x is closest to n.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; it isn't true for arbitrary x ∈ ℝ.
Counter-example: Consider x = 12.4. It's closest to the perfect square 32 (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).Now, let's characterize the kinds of numbers for which the claim in question 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 = n2+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 n2, while any x above the average is closer to (n+1)2. This average-of-the-squares is ((n)2+(n+1)2)/2 = (n2+n2+2n+1)/2 = (2n2+2n+1)/2 = n2+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 n2+n+1/4 < x < n2+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 n2. 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) = n2+n by a value of between 1/4 and 1/2. Since n2+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.
2013-07-08
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 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).
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).
2013-07-01
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?
2013-06-17
The War on Structure
Here is a partial list of subjects that have been seemingly expunged from primary and high school in the fairly short time since I was in those institutions:
It's particularly aggravating that several of the issues cut to the very heart of why our writing systems in numbers and words are originally designed as they were: to support simple adding, multiplying, and division procedures by hand; to easily convert spoken sounds to writing by way of phonics. Lose sight of that, and you lose the very essence of those systems of writing.
So doesn't this explain why remedial algebra is the single-most devastating course in the university curriculum, preventing about half of all community college students from ever graduating? After all, it's now the first time in their academic career that students are finally forced, inescapably, it attend to the detail and structure of things. And they've been set up for failure; when our students cannot actually parse the structure of a sentence (don't know what a verb is, what clause connects to what, how to diagram a sentence, etc.), then it's impossible for them to translate word/application problems to math (to say nothing of actually solving them). And it's very sad and heart-wrenching to watch.
What is particularly galling about all this is that this has occurred precisely at the same time as the world around has become more driven by machines, computers, technology, and an increasingly technocratic government structure. All of our young people carry computing devices at all times as a very intimate part of their lifestyle, but their understanding is at about the same level as a cargo-cult. This is why "STEM" academic careers are held out as some kind of bizarre alien life-form that the normals cannot hope or imagine crossing into. To the extent that we have removed the capacity to understand structure from our students, we are making them unavoidably victims of the highly technocratic society that controls their lives, without any hope of understanding it.
Obviously this painful disconnect between prior classes and the introductory algebra course cannot last -- and all signs are that, long term, the algebra class will likely be removed as a requirement even for a college degree (my prediction). And thus a mile-high iron wall will be put in place between the unlearned masses and the elite who are educated in the "real deal" of structure, mathematics, language, and computer skills.
- Phonics
- Multiplication tables
- Algorithms for addition, multiplication, long division, etc.
- Grammar
- Logic
It's particularly aggravating that several of the issues cut to the very heart of why our writing systems in numbers and words are originally designed as they were: to support simple adding, multiplying, and division procedures by hand; to easily convert spoken sounds to writing by way of phonics. Lose sight of that, and you lose the very essence of those systems of writing.
So doesn't this explain why remedial algebra is the single-most devastating course in the university curriculum, preventing about half of all community college students from ever graduating? After all, it's now the first time in their academic career that students are finally forced, inescapably, it attend to the detail and structure of things. And they've been set up for failure; when our students cannot actually parse the structure of a sentence (don't know what a verb is, what clause connects to what, how to diagram a sentence, etc.), then it's impossible for them to translate word/application problems to math (to say nothing of actually solving them). And it's very sad and heart-wrenching to watch.
What is particularly galling about all this is that this has occurred precisely at the same time as the world around has become more driven by machines, computers, technology, and an increasingly technocratic government structure. All of our young people carry computing devices at all times as a very intimate part of their lifestyle, but their understanding is at about the same level as a cargo-cult. This is why "STEM" academic careers are held out as some kind of bizarre alien life-form that the normals cannot hope or imagine crossing into. To the extent that we have removed the capacity to understand structure from our students, we are making them unavoidably victims of the highly technocratic society that controls their lives, without any hope of understanding it.
Obviously this painful disconnect between prior classes and the introductory algebra course cannot last -- and all signs are that, long term, the algebra class will likely be removed as a requirement even for a college degree (my prediction). And thus a mile-high iron wall will be put in place between the unlearned masses and the elite who are educated in the "real deal" of structure, mathematics, language, and computer skills.
2013-06-14
2013-06-10
Online Remedial Courses Considered Harmful
Online remedial courses are inherently absurd. Even though this is UDacity's (for example) great-white-hope moment, as it has begun offering elementary algebra courses for college acceptance in California (link), it practically beggars the mind that this will help the crushing wave of need that students in such courses evidence.
The reason why is that everything that online courses do well is precisely the opposite of what remedial students need. We know that online courses require a higher level of discipline, dedication, and self-starter initiative than in-person courses do. Online courses are inherently tougher to follow than live courses. You also need a certain technical proficiency just to interface with the platform (and occasionally troubleshoot problems). This is all well and good for high-functioning academic-types who fundamentally love to learn on their own.
But our remedial students have none of that. One of the first overwhelming problems is that they don't have self-discipline in schedule or study habits -- frequently helping with this is itself part of the remedial math course. And they don't like the subject; surveys routinely show an overwhelming and long-seated hatred for the discipline; often a large proportion of a remedial class doesn't show up for the very first class. Minor technical problems will routinely frustrate them and throw them off completely.
Frankly, what the remedial student needs is clear -- if we were serious about getting these students educated, then they would need more individual, one-on-one interaction to address their deep level of need (not less). They need a personal touch to get them over their often pathological resistance for the technical subject matter. They need personal tutors -- but the cultural structure is not one that is interested in paying for that.
Here is a quote, based on the reform experience at a community college near Philadelphia, as they focused effort on the classes with the highest failure rates throughout their college, and in many cases improved their statistics by as much as half (based on interventions such monitoring no-shows on the first day, requiring early evidence of class participation, academic probation communication procedures, etc.):
The reason why is that everything that online courses do well is precisely the opposite of what remedial students need. We know that online courses require a higher level of discipline, dedication, and self-starter initiative than in-person courses do. Online courses are inherently tougher to follow than live courses. You also need a certain technical proficiency just to interface with the platform (and occasionally troubleshoot problems). This is all well and good for high-functioning academic-types who fundamentally love to learn on their own.
But our remedial students have none of that. One of the first overwhelming problems is that they don't have self-discipline in schedule or study habits -- frequently helping with this is itself part of the remedial math course. And they don't like the subject; surveys routinely show an overwhelming and long-seated hatred for the discipline; often a large proportion of a remedial class doesn't show up for the very first class. Minor technical problems will routinely frustrate them and throw them off completely.
Frankly, what the remedial student needs is clear -- if we were serious about getting these students educated, then they would need more individual, one-on-one interaction to address their deep level of need (not less). They need a personal touch to get them over their often pathological resistance for the technical subject matter. They need personal tutors -- but the cultural structure is not one that is interested in paying for that.
Here is a quote, based on the reform experience at a community college near Philadelphia, as they focused effort on the classes with the highest failure rates throughout their college, and in many cases improved their statistics by as much as half (based on interventions such monitoring no-shows on the first day, requiring early evidence of class participation, academic probation communication procedures, etc.):
In some cases, Hayden said, the college's analysis has led officials to believe that some courses were being offered in inappropriate formats. For instance, several of the highest failure rates were in online developmental courses (around 60 percent) -- and various reforms didn't budge those numbers. So the college has ended online remedial education. "The failure rates were so high that it seemed almost unethical to offer the option," Hayden said. (link)
This quote inspired me to write this post tonight. "Unethical to offer the option" (for online remedial education) seems about right. We'll see how quickly MOOCs such as UDacity, and those partnering, paying, and linking their reputation with them, re-learn this lesson.
Subscribe to:
Posts (Atom)