Tomorrow, of course, are the elections here in the USA for President and other elected positions. One month ago I posted "Bungled Election Probability", where I griped about the common test-question gaffe of thinking that the preference ratio among voters will be the same as the probability of winning an election.

An excellent case study on that: Nate Silver's been getting some major attention recently with his very nice "Five Thirty Eight" blog at the New York Times, where he uses sophisticated statistical analyses to track the likely election outcome. There are numerous graphs and charts which nicely highlight the difference between the two measurements (accessed today, Nov-5):

## Monday, November 5, 2012

## Thursday, October 18, 2012

### Proofs of Distributing Exponents and Radicals

In my introductory algebra classes these days, I've switched to providing actual proofs for major principles after about the halfway point in the class. As usual, the point of this is (a) to prepare students for what real math classes are like, and (b) provide insight into why things work as they do.

What I was distressed to find yesterday is that you actually can't find any proofs for the pretty rudimentary notions of the fundamental rule (as I call it) or distribution of exponents or radicals (over multiplication or division), even for just integer powers. Not anywhere online on a Google search. Not in any math text in any of my bookcases at home.

So this in turn caused me to have weird dreams and wake up at 5am all pissed off over that fact, scribbling stuff on note paper to make up the gap, in classic MadMath style. Just in case anyone else is ever searching for the same thing, I'll present the distribution proofs below. We assume the usual properties of commuting, associating, and distributing multiplication and addition (which for integers can be proven from the Peano axioms).

[Note: Obviously I've skipped cube roots and Nth roots in the foregoing, but they're easily expanded from the above.]

Questions: Did I get those right (e.g., would it be more complete to use induction for the exponents proof)? Do you think they're illuminating for basic algebra students? And more importantly -- Can you find any place online that I missed or in any textbook that presents proofs for the preceding?

What I was distressed to find yesterday is that you actually can't find any proofs for the pretty rudimentary notions of the fundamental rule (as I call it) or distribution of exponents or radicals (over multiplication or division), even for just integer powers. Not anywhere online on a Google search. Not in any math text in any of my bookcases at home.

So this in turn caused me to have weird dreams and wake up at 5am all pissed off over that fact, scribbling stuff on note paper to make up the gap, in classic MadMath style. Just in case anyone else is ever searching for the same thing, I'll present the distribution proofs below. We assume the usual properties of commuting, associating, and distributing multiplication and addition (which for integers can be proven from the Peano axioms).

__Definition of Exponents__: For positive integers n, a^n = a*a*...a [n times]. For negative integers, a^(-n) = 1/a^n. For the zero power, a^0 = 1.__Theorem__: Exponents distribute over multiplication. That is: (ab)^n = a^n * b^n for any integer n.__Proof__: For positive powers, (ab)^n = (ab)(ab)...(ab) [n times] = (a*a*...a)(b*b*...b) [by commuting & associating] = a^n * b^n [definition of exponents]. For negative powers, (ab)^(-n) = 1/(ab)^n [definition of negative exponent] = 1/(a^n * b^n) [positive exponents distribute] = 1/a^n * 1/b^n [definition of multiplying fractions] = a^(-n) * b(-n). For the zero power, (ab)^0 = 1 = 1*1 = a^0 * b^0 [definition of zero power].__Definition of Radicals__: Square root √a means a positive number x such that x^2 = a.__Theorem__: Radicals distribute over multiplication. That is: √(ab) = √a√b for any positive a, b.__Proof__: Let x = √a, which means x^2 = a [definition of square root]. Let y = √b, which means y^2 = b [definition of square root]. Note that √(x^2 * y^2) = xy because (xy)^2 = x^2 * y^2 [distribution of exponents], satisfying the definition of square root. So √(ab) = √(x^2 * y^2) = xy = √a√b [substituting in each step].[Note: Obviously I've skipped cube roots and Nth roots in the foregoing, but they're easily expanded from the above.]

Questions: Did I get those right (e.g., would it be more complete to use induction for the exponents proof)? Do you think they're illuminating for basic algebra students? And more importantly -- Can you find any place online that I missed or in any textbook that presents proofs for the preceding?

## Tuesday, October 16, 2012

### Quaternions Anniversary

Today in 1843 William Rowan Hamilton invented quaternions (a way of using 4-dimensional numbers to concisely encode 3-dimensional positions) as he walked across Brougham Bridge in Dublin, carving them into the stone there to make sure he didn’t forget later. Begorah, that’s 13

http://en.wikipedia.org/wiki/Quaternions

^{2}years ago!http://en.wikipedia.org/wiki/Quaternions

## Friday, October 12, 2012

### Almost Homeless

One of the top study
tips many of us try to impart to our students is how mathematics (to
a degree greater than any other discipline) builds on itself, with
every day being an absolute requirement for what comes next. Much
like a metal chain (I will say), if you break any single link, then
the whole structure falls apart.

Several weeks ago, I
met a visitor at an open-house for my girlfriend's art studio. We get
to chatting, and I say that I teach college math; it's a good place
to work, my boss treats me great, and there's an enormous need for
help on the part of community-college students trying to pass
remedial courses. He agrees, saying he was one of those students, and
fortunately he did get the help he needed. I say: “For any of us,
including myself, the limit on our careers and our aspirations is
almost always how much math we were able to master in school.” He
says: “I think possibly, maybe three or four days in elementary
school, I either zoned out or something wasn't explained clearly...
and directly because of that, twenty years later, I almost became
homeless.”

Sometimes I use that
anecdote on the first day of my remedial classes now, and it does
make quite an impact.

## Tuesday, October 9, 2012

### Hope for Open Textbooks?

One of my primary
arguments against MOOC's being a revolutionary force is by comparing
them to books. In truth, while my attitude toward MOOCs is fairly
negative, I would be prone to having a distinct hope for free,
open-sourced, digital textbooks. The advantages seem multitudinous:
(1) effectively free of cost, (2) a force-multiplier for the live
classroom environment (as far as both cost and burden of carrying them),
(3) ability to actually own them on a mobile device and not be
dependent on an outside streaming service, (4) ability to read them
without any internet connection whatsoever, (5) ability to share and re-host the
work freely, (6) ease of editing for fixes, and tailoring for
individual courses and local requirements.

Compared to a suite of
video lectures, this would seem fairly

*easy*to do – and yet as far as I can tell, even*this*relatively simple project has failed to succeed to date. I've spent some time surveying open-source introductory algebra texts, for example, and found them all to be surprisingly deficient (rather reminiscent of some video lectures, in fact – frequently unplanned, technologically difficult to access, or with confusing and unprofessional jokes and puns in the text, etc.) I plan to spend some time writing up particular reviews in the future.
An argument: If making
information widely available eliminates the need for live in-person
instruction, then why didn't the printing press “tsunami” destroy
live colleges (when in fact it did the opposite)? If free MOOCs
current low quality is something easily fixed, then why aren't the
even simpler open-source textbooks yet representing high quality
offerings?

So that said, a few
news items regarding open-source text developments that do give me
cause for hope:

1. California has
passed and signed a law to fund open-source textbook development in
50 core subject areas. While there was a similar attempt under the
Schwarzenegger administration, that prior try had ambiguous definitions, weak standards, and no funding. This new law sounds like a
much stronger attempt that does give me hope.

2. Finnish researchers
and teachers engaged in a 3-day “hackathon” in which they
completed an entire open-source textbook. While I would be highly
skeptical of the quality of such an offering, it at least signals
that there is some amount of buzz and excitement for the idea, which
perhaps bodes good things to come.

## Thursday, October 4, 2012

### Bungled Election Probability

Here's a common malformed math problem that really irks me -- The idea that in a voting situation, the ratio of voters indicates the probability of one party winning. For example: a problem might say that out of a group of 50 people, 30 people favor candidate A, and 20 people candidate B, so candidate A has a 60% chance of winning an election. Obviously, this can only be the case if the election is decided by one single voter being chosen by random method, which is not remotely how any elections actually work.

I've seen this pop in one or more publisher-provided testbanks that I use. And yes, it currently also appears in the Udacity Statistics 101 course (Unit 24.4, et. al.; hopefully fixed soon?). For god's sake people, please don't do that.

I've seen this pop in one or more publisher-provided testbanks that I use. And yes, it currently also appears in the Udacity Statistics 101 course (Unit 24.4, et. al.; hopefully fixed soon?). For god's sake people, please don't do that.

## Monday, October 1, 2012

### MOOCs in the News

MIT's Technology
Review has one of the best survey/reviews I've seen of current MOOC programs,
and also pointedly asks if they might be a temporary fad. The article opens
with a fascinating comparison to the correspondence-course craze of
about a hundred years ago, which made similar promises of widespread
and personalized educational opportunities, and saw millions of
prospective students enroll – culminating in very low outcomes and
success rates, and ultimately the collapse of those programs.

American Educator
magazine has a powerful “Notebook” column assessing Khan Academy,
and pointing out the relatively poor quality of the lessons made
available there. Quoting a profile from Time magazine, “Sal Khan...
explains how he prepares each of his video lessons. He doesn't use a
script. In fact, he admits, 'I don't know what I'm going to say half
the time'... 'two minutes of research on Google'... is how Khan
describes his own pre-lesson routine.” Note that this observation
is identical to my #1 criticism of the Udacity Introduction to
Statistics course, here.

## Thursday, September 27, 2012

### First Exercises with Variables

I'd like to share the lecture notes for the very start of my introductory algebra classes. (This occurs in the 2nd hour, after administrative procedures and a little review on the most basic terms and the order of operations.) Here's how I introduce writing with variables:

Using these, we can write precise statements about number patterns.

Ex.: Translate to math:

(a) "Any number times zero is zero" → For all x: x∙0 = 0.

(b) "Any number plus zero is the same as the original number" → For all x: x+0 = x;

(c) "Any number times one is the same as the original number" → For all x: x∙1 = x.

(d) "A negative sign is the same as multiplying by negative one" → For all x: −x = (−1)x.

Of course, there's a bit more that I say verbally, but that's what gets on the board. I prime the pump for these exercises by asking initial questions like, "What's 5 times 0? What's 8 times 0? What's 25 times 0? Who can clearly state the pattern that we're observing?", etc. We do the first two translations together, and then students do the second pair on their own.

This is a bit of an evolution on what I've done in the past. It works extremely well, and part of the reason I'm so tickled by these exercises is that they slyly manage to do at least quintuple duty. What's being accomplished here is:

**Definitions**__Variables:__Letters that stand in for numbers.__Algebraic Expression__: Series of math symbols with no equals.__Equals:__Means "is" or "is the same as"; symbol =.Using these, we can write precise statements about number patterns.

Ex.: Translate to math:

(a) "Any number times zero is zero" → For all x: x∙0 = 0.

(b) "Any number plus zero is the same as the original number" → For all x: x+0 = x;

(c) "Any number times one is the same as the original number" → For all x: x∙1 = x.

(d) "A negative sign is the same as multiplying by negative one" → For all x: −x = (−1)x.

Of course, there's a bit more that I say verbally, but that's what gets on the board. I prime the pump for these exercises by asking initial questions like, "What's 5 times 0? What's 8 times 0? What's 25 times 0? Who can clearly state the pattern that we're observing?", etc. We do the first two translations together, and then students do the second pair on their own.

This is a bit of an evolution on what I've done in the past. It works extremely well, and part of the reason I'm so tickled by these exercises is that they slyly manage to do at least quintuple duty. What's being accomplished here is:

- Getting initial practice in reading & writing variables (obviously),
- Demonstrating that math is about finding and communicating patterns,
- Generating a review of important writing rules (don't write "times one", etc.),
- Emphasizing that math notation is a language that can be translated to & from English like any other – specializing in brevity & precision, and
- Sneaky presentation of important identities, starting with the Peano-axiom definitions of zero, and then segueing to theorems about the multiplication identity, etc.

## Monday, September 24, 2012

### New Blog Subtitle

One or two readers may have noticed that this blog has a new subtitle as of a few weeks ago -- "Beauty is the Enemy of Expression." To give credit where it's due, this is a quote attributed to world-renowned solo violinist Christian Tetzlaff, speaking to a group of music students about the sometimes misplaced overemphasis on beauty-

*uber-alles*in his art-form, and how it can distract from the job of clear communication (*New Yorker*, Aug 27, 2012). I felt it had even deeper meaning for the work of mathematics, and so was perfect as a sub-header here on MadMath. More coming soon.## Monday, September 17, 2012

### Udacity Update

Interestingly, the day after I posted my review of the Udacity Statistics 101 class last week, Sebastian Thrun wrote a response on the Udacity blog site. In the response, Thrun agreed that the ST101 course "can be improved in more than one way... In the next weeks
we will majorly update the content of this class, making it more
coherent, fixing errors, and adding missing content".

First of all, I'll say kudos to Thrun for taking the time to read my blog and viewing the critique in the constructive spirit which it was intended (not everyone would do the same). If you consider the original review (here), then I'll say that this actually does address the final and overarching criticism, namely "Lack of Updates?". The fact that Udacity does intend to be fixing and re-recording lectures is undoubtedly a good sign. Personally, I think it would be ideal to have someone devoted to refining the course on a continuous basis, like a master teacher immersed in a class repeatedly every semester. Will Prof. Thrun personally have the time to make that happen (granted his other research endeavors)? Time will tell.

I'll be eager to see if the major specific content gaps are addressed in the planned updates. Among the missing topics: Probably "CLT Not Explained" (#6 on my list) is the easiest one to remedy -- taking maybe a single added lecture where it's clearly stated (and then hopefully moving the programming segment afterward so it serves as confirmation, symmetric with the other programming units). Somewhat harder would be "Normal Curve Calculations" (#5), which would require at least its own unit, alteration of how inferences are introduced, and maybe removal of the rather unhelpful derivation of the normal formula. Much harder would be "Population and Sample" (#4), which I think would require major alterations to the DNA of the course from the beginning to the end.

Now, Thrun also writes that "Statistics, to me, is a highly intuitive field... some [criticisms] are the result of... my dedication to get rid of overly formal definitions". This is a spot where we here at MadMath (i.e., this writer) will remain philosophically in a different camp. I would say that we need both good intuition AND good attention to details -- the challenge to both myself and my students being to write things as clearly and carefully as possible AND to engage in quick decision-making and communication. Each is a force-multiplier on the other (my usual analogies being that a great musician can play both very fast and very slow; or a great programmer can make use of both a high-level abstraction and low-level assembly when the need arises).

Anecdote from this past week, starting a new semester of classes: On the very first day of any of my classes, I hand out a one-page sheet of all the definitions that will be introduced in the class, and say that the first -- and sometimes the hardest -- step in learning new math is to learn the meaning of the words involved. At the end of one of those sessions last week, I had one student approach me, profusely thanking me for just such a resource. "My last teacher," she said, "I

Writing, the language-tool, is almost surely the most powerful thing humanity ever invented -- and math is fundamentally about communicating patterns in the most precise and unambiguous way possible -- so let's use the tool for the job it was intended.

First of all, I'll say kudos to Thrun for taking the time to read my blog and viewing the critique in the constructive spirit which it was intended (not everyone would do the same). If you consider the original review (here), then I'll say that this actually does address the final and overarching criticism, namely "Lack of Updates?". The fact that Udacity does intend to be fixing and re-recording lectures is undoubtedly a good sign. Personally, I think it would be ideal to have someone devoted to refining the course on a continuous basis, like a master teacher immersed in a class repeatedly every semester. Will Prof. Thrun personally have the time to make that happen (granted his other research endeavors)? Time will tell.

I'll be eager to see if the major specific content gaps are addressed in the planned updates. Among the missing topics: Probably "CLT Not Explained" (#6 on my list) is the easiest one to remedy -- taking maybe a single added lecture where it's clearly stated (and then hopefully moving the programming segment afterward so it serves as confirmation, symmetric with the other programming units). Somewhat harder would be "Normal Curve Calculations" (#5), which would require at least its own unit, alteration of how inferences are introduced, and maybe removal of the rather unhelpful derivation of the normal formula. Much harder would be "Population and Sample" (#4), which I think would require major alterations to the DNA of the course from the beginning to the end.

Now, Thrun also writes that "Statistics, to me, is a highly intuitive field... some [criticisms] are the result of... my dedication to get rid of overly formal definitions". This is a spot where we here at MadMath (i.e., this writer) will remain philosophically in a different camp. I would say that we need both good intuition AND good attention to details -- the challenge to both myself and my students being to write things as clearly and carefully as possible AND to engage in quick decision-making and communication. Each is a force-multiplier on the other (my usual analogies being that a great musician can play both very fast and very slow; or a great programmer can make use of both a high-level abstraction and low-level assembly when the need arises).

Anecdote from this past week, starting a new semester of classes: On the very first day of any of my classes, I hand out a one-page sheet of all the definitions that will be introduced in the class, and say that the first -- and sometimes the hardest -- step in learning new math is to learn the meaning of the words involved. At the end of one of those sessions last week, I had one student approach me, profusely thanking me for just such a resource. "My last teacher," she said, "I

*begged*him to define the words he was using, but he said, don't worry about that, just do the problems instead". Which just about broke my heart to hear. Although I've heard multiple voices recently call out for a reduction in formalism, I think that the end result is something like this: You (the instructor) risk putting yourself in a cul-de-sac where one day a student will ask you "why is that?" and you'll have to yell at them "just do it!" because you don't have the shared language to discuss the parts of the problem at hand.Writing, the language-tool, is almost surely the most powerful thing humanity ever invented -- and math is fundamentally about communicating patterns in the most precise and unambiguous way possible -- so let's use the tool for the job it was intended.

## Wednesday, September 12, 2012

### Sample Size of One

So in response to my review of the Udacity Statistics 101 course (earlier this week) about a hundred people all thought it was clever to say some iteration of, "your sample size is only one course, ha-ha".

The one thing I'll say on this point is that it took me several weeks of full-time work (spread over the summer) to watch, review, interact with problems sets, annotate, organize, draft, and edit my observations for the review. If someone wanted to fund a continuation of this project for other courses and institutions, then I'd be sincerely happy to do so. I would love for people to have a better capacity to identify good teaching and be able to access it, and also crap teaching and be able to avoid it. (Compare to: "Basic Teaching Motivation".)

Thanks to everyone who read and commented, particularly with their experiences in related courses; I think it was (and continues to be) a highly rewarding discussion.

The one thing I'll say on this point is that it took me several weeks of full-time work (spread over the summer) to watch, review, interact with problems sets, annotate, organize, draft, and edit my observations for the review. If someone wanted to fund a continuation of this project for other courses and institutions, then I'd be sincerely happy to do so. I would love for people to have a better capacity to identify good teaching and be able to access it, and also crap teaching and be able to avoid it. (Compare to: "Basic Teaching Motivation".)

Thanks to everyone who read and commented, particularly with their experiences in related courses; I think it was (and continues to be) a highly rewarding discussion.

## Monday, September 10, 2012

### Udacity Statistics 101

The prospect of massive-scale online schooling seems to be all the rage at the moment. Recent competing initiatives include Khan Academy, OpenCourseWare, Udacity, Coursera, and edX (the latter ones sponsored by top-name schools such Stanford, Harvard, or MIT, or else founded by ex-faculty members). The idea of universal and free access to college programs from top researchers has fired the imagination of many in the blogosphere, and some have predicted the imminent collapse of traditional universities in the face of this “tsunami”.

As a
college educator myself, I felt compelled to survey one of these
courses, so as to assess their general quality, advantages, and
disadvantages. (Perhaps there would be some techniques that I could
fold into my own courses.) This summer, Sebastian Thrun's Udacity
unveiled a new course, Introduction to Statistics, taught by Thrun
himself, which I felt would be ideal for my purposes – my current
job largely specializing in teaching statistics at one of the
community colleges in the City University of New York (and my
master's degree being in Mathematics & Statistics). Having
enrolled, I proceeded through the entirety of the course, watching
all of the lecture videos and taking all of the web-based quizzes and
the final exam.

In
brief, here is my overall assessment: the course is amazingly,
shockingly awful. It is poorly structured; it evidences an almost
complete lack of planning for the lectures; it routinely fails to
properly define or use standard terms or notation; it necessitates
occasional massive gaps where “magic” happens; and it results in
nonstandard computations that would not be accepted in normal
statistical work. In surveying the course, some nights I personally
got seriously depressed at the notion that this might be standard
fare for the college lectures encountered by most students during
their academic careers.

Below I
will try to pick out a “Top 10” list of problems with the course.
These are not comprehensive, but I feel that they do give a basic
sense for the issues involved.

# 1. Lack of Planning

Generally,
the lectures and the overall sequence feel like they haven't been
planned out in advance (and as a result, they don't connect together very
well). One lecture is interrupted by a visitor walking into
Thrun's office as he records it, and this is left in the video itself
(Unit 17.8). Other lectures use a data set of students' guesses about
Thrun's weight for a hypothesis test on
his actual weight – which, not being a population parameter, is
totally incorrect and “an abuse” (as he admits himself in Unit
32.1); yet this semi-accidental data set was convenient to
access, and so was apparently considered acceptable.

But probably the best example of the lack of planning is how radically off-syllabus the course went from its initial advertising. Now, I've taught courses where things didn't go entirely according to plan – maybe a lecture went a half-day long, but never in all my years of teaching has a course so massively diverged from the initial plan or course description. Below you can compare the starting advertised syllabus (before any lectures were posted) to the revised final syllabus (after the lectures were actually produced). You'll see that they are remarkably different.

Initial syllabus:

But probably the best example of the lack of planning is how radically off-syllabus the course went from its initial advertising. Now, I've taught courses where things didn't go entirely according to plan – maybe a lecture went a half-day long, but never in all my years of teaching has a course so massively diverged from the initial plan or course description. Below you can compare the starting advertised syllabus (before any lectures were posted) to the revised final syllabus (after the lectures were actually produced). You'll see that they are remarkably different.

Initial syllabus:

- Visualizing relationships in data – Seeing relationships in data and predicting based on them; dealing with noise
- Processes that generates data – Random processes; counting, computing with sample spaces; conditional probability; Bayes Rule
- Processes with a large number of events – Normal distributions; the central limit theorem; adding random variables
- Real data and distributions – Sampling distributions; confidence intervals; hypothesis tests; outliers
- Systematically understanding relationships – Least squares; residuals; inference
- Understanding more complex relationships – Transformation; smoothing; regression for two or more variables, categorical variables
- Where to go next – Statistics vs machine learning; what to study next; where statistics is used; Final exam

Corrected syllabus:

- Visualizing relationships in data – Seeing relationships in data and predicting based on them; Simpson's paradox
- Probability – Probability; Bayes Rule; Correlation vs. Causation
- Estimation – Maximum Likelihood Estimation; Mean, Mean, Mode; Standard Deviation, Variance
- Outliers and Normal Distribution – Outliers, Quartiles; Binomial Distribution; Central Limit Theorem; Manipulating Normal Distribution
- Inference – Confidence intervals; Hypothesis Testing
- Regression – Linear regression; correlation
- Final Exam

# 2. Sloppy Writing

Now,
I've become fairly “religious” about the text of mathematics –
reading the details correctly, and writing with precision, being absolutely paramount. (And I've found that for my remedial students,
this fairly simple-sounding skill is a nearly insurmountable
stumbling block.) When I saw the Udacity interface, I was initially
excited; instead of a lecturer standing in front of a chalkboard, the
frame is focused on the writing surface, which gives us the
opportunity to highlight and be careful about the writing (this
being similar to Khan Academy, etc.) But soon I became keenly
disappointed at how poor and unclear the written presentation was.

There are
at least two related issues. The first is that new terms and symbols
are almost never given written definitions. Personally, I find that
discussions and questions usually return to the definitions of
terms, so setting those out carefully is the first and most important
task. Here, new terms are casually described in the audio track, but
they are neither technically careful nor visible to the viewer. I
think this is exacerbated by the course's commitment to not
following any textbook or other written source – after the first
encounter, there is no capacity to search, index, or reference back
to terms or definitions that you might need later on (and this holds
as well for specialized symbols for sums, products, conditionals,
logical operators, etc., that tend to materialize for the first time in
the middle of a problem).

But the
second issue is that the algebraic manipulations themselves are
uniformly sloppy and disjointed; some bits of the work will be
written down, the next bit discussed verbally, then another unrelated
scrap written down, etc. There are unfixed typos in words and
equations. Statements and tables go unlabeled, so when a problem is
done you can't tell from looking at it what the point was. Notation
varies unpredictably: at different points in the course, the symbols
Î¼,
x-bar, and E(x) are all used for the sample mean without introduction
or warning. Usually formulas are absent until given in summary at the
end of a section, and then disparaged as being “confusing and
complicated” (Unit 9.10) or “really clumsy” (Unit 9.15), which
I think is a great pedagogical loss for learning to read and write
math properly. At one point you get to see the assistant instructor write that “0.1 = 0.06561” (Problem Set 2.6), which to me is an unforgivable, cardinal sin. In many cases one would have to rely on
the discussion forums for a fellow student to present a clear and
complete piece of written math for any of the example problems.

# 3. Quiz Regime

The
pattern of lectures goes like this: A video nugget of a few minutes
will be shown (perhaps 2-5 minutes), which leads to a web-based quiz
question (prompting for retries until success), and then a brief
video explanation of the answer. In general, I like this idea of frequent questioning and I do
the same thing in my own classes: regular check-ins for myself and my
students that we've successfully communicated the ideas at hand.

But a
couple of things make this wonky here. One is that, obviously, the
communication is not really two-way; neither Thrun nor the system is
really “listening” to take note of when a presentation has misfired and
needs clarification. Another is that the quiz regime timing seems
forced and frequently not at a point when there is really a
legitimate new idea to check in on. I would guess that as much as
half the time a question is actually asked

*before*students have been given the tools to answer it, being used as a means of introducing a new section. Things like, “Don't get disturbed if you don't know the answer” (Unit 1.4), or “I'd be amazed if you got this correct!” (Unit 9.13), are heard frequently. These kinds of questions seem inherently unfair and, I can only imagine, discouraging to many students.# 4. Population and Sample

Astoundingly,
the Udacity Introduction to Statistics course manages to go almost
its entire length without ever mentioning or making
any distinction between the population and sample in a study. I say
I'm “astounded” because in my classes (and any one I've surveyed
or looked at), this is

*the*key idea in introductory inferential statistics. It's the very first thing that is mentioned in my class (or the book), and it's the very last thing on the last day, too. It's the entire reason*why*inferential statistics is necessary in the first place. In fact, the very word “statistics”*means*measures for one (sample) and not the other (population) – but you'll never learn that from this class.
As a
result, Thrun goes the entire course using the symbols Î¼
and Ïƒ
to indicate the mean and standard deviation of both a random variable
(population) and a limited data set (sample), whereas normally they
indicate only the former. He'll switch between the two essentially
without notice, saying something like “the observed standard
deviation” (Unit 25.3), or “our empirical mean” (Unit 25.4).
The x-bar notation appears late in the course, mid-way through a
problem statement – and then being used to indicate the mean of a

*population*in a hypothesis test, which is exactly reversed from normal usage (Problem Set 5.5). And the customary (unbiased) formula for sample standard deviation is*entirely missing*from the course, necessitating annotated instructor comments to point out that the results you get from this class would not be acceptable in any other venue (Unit 27.3).# 5. Normal Curve Calculations

A
similar astounding absence: The entire sequence of Udacity's
Introduction to Statistics passes without ever calculating any values
for normal curves. Again, since the course is committed to being
independent of any outside resource (no textbook, no tables, no
statistical software suite), the result is that calculating
probabilities or values for normal distributions is simply impossible
and never occurs. Students don't have any opportunity to develop an
intuition for normal-curve probabilities. The Empirical Rule (the
68/95/99% rule-of-thumb for standard deviations) is never mentioned.
When the time comes to compute confidence intervals, Thrun is forced
to give the direction, “just multiply this value over here with
1.96 – the magic number!” (Unit 24.19), not having any way to
explain where this comes from, nor even mentioning at the time that
this is specific to a 95% confidence level.

Thrun
spends a surprising amount of time developing the actual

*formula*for a normal curve, but no calculations are made with it and its utility in an introductory course is highly questionable. The absence is doubly weird because at one point he asserts, “That's the purpose of the normal distribution for the sake of this class... we just do it for the normal distribution where things are relatively easy to compute”. (Unit 20.15)# 6. CLT Not Explained

Another
bizarre gap: what one would think to be the keystone to inferences
for a mean, the Central Limit Theorem (the fact that the distribution
of possible sample-mean values automatically takes on a normal shape
with large sample size) is never clearly stated, nor its importance
explained. There is an optional programming unit with the name in the
title (Unit 19), which does generate a bell-shaped histogram of a few
thousand randomized sample means, and ends by stating that how this
relates to the Central Limit Theorem will be discussed in the next
unit. The next unit is on the Normal Distribution, but it still
neglects to actually state the CLT, and instead winds up engaging in a
rather baroque discussion to wit, “it's a transition from a discrete
space of finitely many outcomes to a space of infinitely many
outcomes” (Unit 20.14). There's a later point where Thrun says,
“Remember the Central Limit Theorem? Remember what it said?”
(Unit 25.2), and weirdly, this is the first time he actually outright
(if very briefly) states it. This is cursorily tied into how
confidence intervals work (blink and you'll miss it), and also said
to relate to “1.96 the magic number” in an unverifiable way (Unit
25.2-3). It's enormously unclear, and I think a distressing misstep.

# 7. Bipolar Difficulty

Throughout
the course, lectures and exercises veer rapidly between utterly
trivial and nigh-impossible. I think this is a reflection of the
one-way communication channel, such that Thrun can't have any
awareness of what counts as easy and what counts as hard to the
students. Frequently the “problem sets” at the end of a section
will have work that is dramatically different than anything shown in
the lectures. The first half-dozen units of the class are fairly long and obvious presentations of reading different tables and charts
and linear relationships. Then at some point he switches into a
remarkably difficult “complete the proof” exercise demonstrating
that the sample mean is in fact the correct Maximum Likelihood
Estimator for the population mean (Problem Set 3.1; not that he uses
the terms sample/population) – granted that this is “optional”,
but the course hasn't had any proofs at all to that point, the
overall strategy of the proof isn't declared, and it involves
numerous calculus concepts. Even my graduate text in statistical
inference (Casella/Berger) felt compelled to present and explain that
proof in its entirety. (Later, when he revisits this same exercise
again in Unit 23, Thrun actually does finally explain the technique,
which I presume to be a response to earlier complaints in this
regard.)

Similar
whiplash will be experienced at other points in the course. For
example, one student wrote in the discussion forums for the course
(regarding a different problem), “Questions such as this one and
the one before it 'Many Flips' are counter productive. The previously
explained course material was mostly very smooth and gradual.
Reaching 'Many Flips' felt like crashing into a reinforced concrete
wall.” (Link).
That's a perfect description of what I think the experience will be for many
first-time students.

# 8. Final Exam Certification

The
course ends with a web-based final exam with 16 questions in the same
vein as the section quizzes that have appeared all along. Upon completion, the student is able to print out a PDF “certificate of
accomplishment” saying that they've taken this course from
Udacity, with one of several success levels (Highest Distinction for
all 16 questions correct, High Distinction 13/16, Accomplishment
10/16, or Completion 8/16).

Now
obviously, a somewhat delicate issue is that this is a completely
worthless, faux-certification for a number of reasons. Obvious ones
would be: (1) Udacity has no accreditation, oversight, or recognition from any
outside body, and (2) the questions are all fixed and the
answers are probably posted somewhere online in full. But even more
importantly, and what really surprised me, was: (3) the fact that you
can re-submit all of your answers as many times as you like until
they are confirmed correct (just like the quizzes; and some are even
multiple-choice). Another would be: (4) the final exam is just remarkably
easy; could this be a response to recent criticisms that only a tiny
percent of students who register for courses like these ever
complete them? If this is a PR problem for Udacity, then obviously
they can reduce the difficulty of a course to whatever level
generates a desired completion rate.

Recently, the blog “Godel's Lost Letter and P=NP” by Georgia Tech's Richard Lipton had a lengthy post considering a perceived security problem with programs like Thrun's at Udacity: namely, that a student could freely register multiple accounts and keep taking the final exam until they achieved an acceptable score. But this overlooks the rather blatant fact that no one need go to such lengths, since the system already allows you to re-submit each individual exam item as many times as you like until success. Apparently Thrun's own response to Lipton's concern was to propose tracking of IP addresses to identify duplicate students, which bizarrely suggests a complete lack of awareness of how his own final exams work. (“Well Thrun told me about it in person when I visited his company this winter. They also can track IP addresses and they can see what is going on with their students.”; “Cheating or Mastering?”, August 21, 2012)

Recently, the blog “Godel's Lost Letter and P=NP” by Georgia Tech's Richard Lipton had a lengthy post considering a perceived security problem with programs like Thrun's at Udacity: namely, that a student could freely register multiple accounts and keep taking the final exam until they achieved an acceptable score. But this overlooks the rather blatant fact that no one need go to such lengths, since the system already allows you to re-submit each individual exam item as many times as you like until success. Apparently Thrun's own response to Lipton's concern was to propose tracking of IP addresses to identify duplicate students, which bizarrely suggests a complete lack of awareness of how his own final exams work. (“Well Thrun told me about it in person when I visited his company this winter. They also can track IP addresses and they can see what is going on with their students.”; “Cheating or Mastering?”, August 21, 2012)

# 9. Hucksterism

As if
the content-based problems noted above weren't enough, running
throughout Thrun's presentations is a routine, suspiciously hard-sell
call for how stellar the class was and how much you, the viewer, have
learned. Personally, I found this to be both grating and a
thou-dost-protest-too-much
lampshading of the flaws of the course. (You might think that I'm being too harsh,
but puncturing this kind of stuff is, after all, the

*raison d'Ãªtre*of the MadMath blog). He says: “You now know a lot about scatter plots!” (Unit 3.12) (yeah, lots). “Isn't this a lot of fun? Isn't statistics really great? (Unit 6.16) (surely someone thinks otherwise). “You are a very capable statistician at this point!” (Unit 32.12) (hyperbole at best). “When people say this is a contradiction... just smile [in disagreement] and say you took Sebastian's Stats 101 and you understand.” (Unit 22.5) (yeah, I'll get right on that).# 10. Lack of Updates?

Finally,
here's a core a problem that multiplies and exacerbates all the
others. In normal college teaching, a truly dedicated instructor will
go through a never-ending process of constant refinement and
improvement for their courses, based on two-way interaction and
feedback from live students. (I know I do; I've taught my
introductory statistics course several dozen times and I still sit
down and note possible improvements after almost every single class
session.)

So in
theory, any of the problems that I've noted above could be revisited
and fixed on future pass-throughs of the course. But will that happen
at Udacity, or any other massive online academic program? I strongly
suspect not – likely, the entire attraction for someone like Thrun
(and the business case for institutions like his) is to be able to
record basic lectures once and then never have to revisit them again.
Or in other words: All the millions of students using these ventures
will be permanently experiencing the shaky, version-1.0 trial run of
a new course, when the instructor is him- or herself just barely figuring out
how to teach it for the first time, and without the benefit of
two-way feedback or any refinements.

# Summary

Based on
my review of the Udacity Introduction to Statistics course, I see
some compelling strategic advantages for live in-class teachers, that
will not be soon washed away by massive online video learning. Chief
among them are the presence of actual two-way communication between
teacher and students, such that the instructor can modify, expand,
and respond to questions when appropriate (in regards to clarity of
presentation, quiz questions, missing pieces, and rationalizing
difficulty levels); and the ability to engage in a cycle of constant
improvements and refinements every time the course is taught by a
dedicated teacher. Also, I feel that written text is ultimately more
useful than videos, being more elegant and precise, easier to search
and index key terms and examples, suffering fewer technical problems,
easier to update, and generally being truer to the form of
mathematical written presentation in the first place. In addition to
these, Thrun's lectures at Udacity have a stunning number of critical
flaws (in regards to planning, sequencing, clarity, writing, and
missing major topics) that leave me amazed if any actual intro-level
student manages to make their way through the whole class.

Perhaps
the upshot here is a restatement of the old saw: “You get what you
pay for.” (Udacity being currently free, with a mission-statement
to remain that way). Or else another: “Don't take a class from a
world-famous researcher, because they don't really have time or
interest for teaching.” Obviously, Sebastian Thrun is not just a
teacher-by-online-video; he's also a Google Vice-President and
Fellow, a Research Professor of Computer Science at Stanford, former
director of the Stanford AI Laboratory, head of teams competing in
DARPA challenges, and leads the development of Google's self-driving
car program. How much time or focus would we expect him to have for a
freshman-level introductory math course? (Not much; in one lecture he
mentions that he's recording at 3AM and compares it to his “day
job” at Google.) Some of these shortcomings may be overcome by a
more dedicated teacher. But others seem endemic to the massive-online
project as a whole, and I suspect that the industry as a whole will
turn out to be an over-inflating bubble that bursts at some point, much
like other internet sensations of the recent past.

## Wednesday, August 29, 2012

### Fundamental Rule of Exponents

For a basic algebra class, given the rudimentary order-of-operations that looks like this:

Cases:

(1) Exponents will multiply powers, i.e., (a

(2) Radicals will divide powers, i.e.,

(3) Multiplying will add powers, i.e., a

(4) Division will subtract powers, i.e., a

We've discussed this before, but I just recently decided to apply the name shown here to the pattern. It doesn't show up on a Google search yet, so I think it's fair-game to do so. Cheers!

- Parentheses
- Exponents & Radicals
- Multiplication & Division
- Addition & Subtraction

__Operations on same-base powers shift one place down in the order of operations.__**The Fundamental Rule of Exponents:**Cases:

(1) Exponents will multiply powers, i.e., (a

^{m})^{n}= a^{m∙n}. Example: (x^{6})^{2}= x^{12}.(2) Radicals will divide powers, i.e.,

^{n}√a^{m}= a^{m/n}. Example:^{3}√x^{15}= x^{5}.(3) Multiplying will add powers, i.e., a

^{m}∙a^{n}= a^{m+n}. Example: x^{4}∙x^{7}= x^{11}.(4) Division will subtract powers, i.e., a

^{m}/a^{n}= a^{m−n}. Example: x^{9}/x^{2}= x^{7}.We've discussed this before, but I just recently decided to apply the name shown here to the pattern. It doesn't show up on a Google search yet, so I think it's fair-game to do so. Cheers!

## Wednesday, August 1, 2012

### Against Inverted Classrooms

The "inverted classroom" (or as Wikipedia calls it, "Flip Teaching") is the idea that lectures can be watched (say, by online video) prior to class meetings, and then classroom time dedicated to questions and problem sets with the teacher's coaching and assistance in trouble spots. Obviously, it's the reverse of the standard math-class process of lecturing in class and then homework after.

When I heard about this a few months ago, I was really excited and initially felt like it was a great idea. Here's why: I agree that I've independently found the moment where I can help an individual student, working on a specific problem, and identify-fix-correct-clarify the exact location where they're making a mistake, to be the most satisfying and productive use of time for both student and teacher. I try hard to get as much time for those moments of practice and error-catching in my classes as I can. So it sounded like fully devoting class time to that process would be ideal. For my summer courses, I thought very deeply how much I could go in that direction.

So at the end of that analysis, I am now very skeptical that this technique will have legs and work long-term for mathematics education. Here are some of the reasons why I say that:

In conclusion: It seems like the general student of any time period hasn't been able to learn math on their own (either from a book or a video) -- that's why they're in a classroom in the first place. It would be nice if there was an expert in the discipline with them at all times, during both initial presentation and homework. But since that's infeasible, the best we can do is some mix of presentation and troubleshooting together in the limited classroom time.

When I heard about this a few months ago, I was really excited and initially felt like it was a great idea. Here's why: I agree that I've independently found the moment where I can help an individual student, working on a specific problem, and identify-fix-correct-clarify the exact location where they're making a mistake, to be the most satisfying and productive use of time for both student and teacher. I try hard to get as much time for those moments of practice and error-catching in my classes as I can. So it sounded like fully devoting class time to that process would be ideal. For my summer courses, I thought very deeply how much I could go in that direction.

So at the end of that analysis, I am now very skeptical that this technique will have legs and work long-term for mathematics education. Here are some of the reasons why I say that:

**(1) It**It appears that online video lectures and published books are pretty obviously equivalent (in fact, I think books have the advantage in any way I can think to compare, especially for math). While other disciplines have commonly run classes with assigned reading beforehand, and critical discussion in-class (e.g., literature, history, law, etc.), math seems pretty ironclad in having avoided that in any place or time that I can detect. (Can you think of any counter-examples?) This suggests that there's something about math that demands live presentations in the first place.*could*have been done at any earlier time with books, but wasn't.**(2) Questions still need to be asked during lecture presentations.**One reason why the initial presentation has to be live: expert feedback isn't just necessary during individual problem sets, it's also necessary to clear up the initial presentation itself. Almost certainly some different level of detail will have to be presented for different audiences, and there needs to be live back-and-forth questioning in order for that initial lecture to be valuable in the first place -- and this value accrues with interest the more people are watching/present at the time. If a student simply doesn't have access to a particular necessary detail during a recorded video, no amount of rewinding or re-watching will conjure it up, and that time will be for naught. (It's been argued in the past that a fully hyper-linked presentation to arbitrary depth of detail could satisfy this need, but in practice that seems to have failed -- arbitrarily large amount of work on the part of the writer, and similar great workload and discipline needed by the student to follow all the needed links.)**(3) Nonstandard class times are particularly ill-suited for it.**The inverted classroom might work a bit better for regular one-hour, once-a-day classes (students need to catch up on one-hour chunks at a time, etc.) But like a lot of methodologies it breaks down in other cases. For example, take my summer statistics courses: they run for 6 weeks, meeting twice a week for 3 hours at a time (other classes might be 4 hour sessions at a time). There needs to be an in-class test about every 3rd class session, which will last about an hour -- note that between one-half and two-thirds of the same meeting will be spent on some other lecture topic. Experience (if not common-sense) shows that students will not have presence of mind for any new topic prior to the test, either in-class or before. So I absolutely must resign myself to presenting new information myself after the test on test days, which themselves are 1/3 of the class meetings. Work out this staggered effect (including the very first class), and I saw that there's essentially no way to make "flip teaching" work in my evening summer courses.In conclusion: It seems like the general student of any time period hasn't been able to learn math on their own (either from a book or a video) -- that's why they're in a classroom in the first place. It would be nice if there was an expert in the discipline with them at all times, during both initial presentation and homework. But since that's infeasible, the best we can do is some mix of presentation and troubleshooting together in the limited classroom time.

## Monday, July 30, 2012

### Algebra in NY Times

This weekend, the New York Times published an opinion piece by Andrew Hacker, emeritus professor of political science at Queens College in CUNY (sister college to where I teach). The piece is titled "Is Algebra Necessary?" and seems to make the argument that an algebra requirement should be waived for most students in American high schools and college.

Obviously, on a nearly daily basis, my job deals directly with the pain and death-march scenario of a majority of students coming into our community colleges and being unable to pass the equivalent of a 7th-grade algebra course. I do think that for all of us -- even myself -- proficiency at math is almost always the limiting factor in our careers. At the same time, it's almost uncanny how many old friends I have getting in contact with me to say that they've suddenly found a higher level of math really key to their professional advancement -- including psychologists, photographers, and even artists (mentioned in Hacker's article as a group that should clearly be freed from a math requirement).

Let me riff on that last point for a bit. My girlfriend is a fine artist, with an MFA in sculpting from a school here where we live in New York City. Earlier this year, she was the recipient of a month-long artist residency in Taiwan where she put together an outdoor installation in knitted recycled plastic as part of an exhibit on environmental themes. She has a fairly high proficiency at math (in fact, for about a month she was a math major in college before switching), and this gets used routinely in her career. She has to estimate volumes of complicated shapes she's planning to put together, so as to procure materials (plastic, wire, plaster, etc.) She has to do calculations with money so as to set budgets and write grant proposals. She has to estimate time for projects that might last many months. At some point she generated a calculation for people, time, and material to cover the Eiffel Tower in tiny crocheted plastic leaves (a long-term goal).

And as I know from several friends working as graphic artists (such as from my time in the video game industry) -- almost all of the work today is done on computers anyway, so they need proficiency with numbers, computer science, (x,y) coordinate systems, algorithms, etc. in order to interface with the most basic professional workflows nowadays. On the side, my girlfriend also runs a home business coding HTML and hosting websites for other artists, and last week was learning for the first time to hack in a UNIX command prompt in order to apply software patches. Such is the life of an artist in the 21st century.

This post is sort of a brain-dump in the first 5 minutes after reading Prof. Hacker's opinion piece. One thing I would like to ask is: Having criticized a bunch of other academic disciplines for, in his view, their overly-high math requirements (math departments themselves, medicine, history, etc.), what would his prescription be for his own discipline, political science? What would the bare-minimum level of math proficiency be there -- decimal arithmetic, college algebra, perhaps statistics? (He writes, "I say this as a writer and social scientist whose work relies heavily on the use of numbers.") Is it possible that the requirements set by a discipline may be there for reasons not immediately obvious to an outsider? (As one example: I wouldn't have known that statistical confidence intervals and P-value statements are a core piece of any medical literature until my own mother, a school nurse, asked for help in reading a required article for continuing education credit.)

At first blush, Prof. Hacker's criticisms don't sound entirely coherent, but I could be biased. Personally, I think I'm most worried about changing the essence of what counts as a college education in exchange for the possibly spurious idea that everyone in our society is required to have a B.A. (and yet that may be an unwinnable fight at this point). But one last thought; near the end of his article he writes,

So, what do those symbols mean? What is the proper order of applying those operations? (Noting that approximately half of my remedial algebra students end a semester unable to answer a final exam problem to apply the proper order-of-operations in several steps.) Any of these disciplines, and even a simple CPI calculation is a great example, presume that educated readers have the

Obviously, on a nearly daily basis, my job deals directly with the pain and death-march scenario of a majority of students coming into our community colleges and being unable to pass the equivalent of a 7th-grade algebra course. I do think that for all of us -- even myself -- proficiency at math is almost always the limiting factor in our careers. At the same time, it's almost uncanny how many old friends I have getting in contact with me to say that they've suddenly found a higher level of math really key to their professional advancement -- including psychologists, photographers, and even artists (mentioned in Hacker's article as a group that should clearly be freed from a math requirement).

Let me riff on that last point for a bit. My girlfriend is a fine artist, with an MFA in sculpting from a school here where we live in New York City. Earlier this year, she was the recipient of a month-long artist residency in Taiwan where she put together an outdoor installation in knitted recycled plastic as part of an exhibit on environmental themes. She has a fairly high proficiency at math (in fact, for about a month she was a math major in college before switching), and this gets used routinely in her career. She has to estimate volumes of complicated shapes she's planning to put together, so as to procure materials (plastic, wire, plaster, etc.) She has to do calculations with money so as to set budgets and write grant proposals. She has to estimate time for projects that might last many months. At some point she generated a calculation for people, time, and material to cover the Eiffel Tower in tiny crocheted plastic leaves (a long-term goal).

And as I know from several friends working as graphic artists (such as from my time in the video game industry) -- almost all of the work today is done on computers anyway, so they need proficiency with numbers, computer science, (x,y) coordinate systems, algorithms, etc. in order to interface with the most basic professional workflows nowadays. On the side, my girlfriend also runs a home business coding HTML and hosting websites for other artists, and last week was learning for the first time to hack in a UNIX command prompt in order to apply software patches. Such is the life of an artist in the 21st century.

This post is sort of a brain-dump in the first 5 minutes after reading Prof. Hacker's opinion piece. One thing I would like to ask is: Having criticized a bunch of other academic disciplines for, in his view, their overly-high math requirements (math departments themselves, medicine, history, etc.), what would his prescription be for his own discipline, political science? What would the bare-minimum level of math proficiency be there -- decimal arithmetic, college algebra, perhaps statistics? (He writes, "I say this as a writer and social scientist whose work relies heavily on the use of numbers.") Is it possible that the requirements set by a discipline may be there for reasons not immediately obvious to an outsider? (As one example: I wouldn't have known that statistical confidence intervals and P-value statements are a core piece of any medical literature until my own mother, a school nurse, asked for help in reading a required article for continuing education credit.)

At first blush, Prof. Hacker's criticisms don't sound entirely coherent, but I could be biased. Personally, I think I'm most worried about changing the essence of what counts as a college education in exchange for the possibly spurious idea that everyone in our society is required to have a B.A. (and yet that may be an unwinnable fight at this point). But one last thought; near the end of his article he writes,

I WANT to end on a positive note. Mathematics, both pure and applied, is integral to our civilization, whether the realm is aesthetic or electronic... Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another.Okay -- just for argument's sake, let's say I go to the Wikipedia article on the Consumer Price Index and start reading up on that subject. What do I see there? Math, equations, written in the language of algebra:

Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives. It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.

So, what do those symbols mean? What is the proper order of applying those operations? (Noting that approximately half of my remedial algebra students end a semester unable to answer a final exam problem to apply the proper order-of-operations in several steps.) Any of these disciplines, and even a simple CPI calculation is a great example, presume that educated readers have the

*grammar*of algebra (I think that's what it's most like, really) available to converse with. The only other option is to mount a crusade to expunge this writing from professional resources -- and in so doing, bring those disciplines to a grindingly slow level of inefficiency and lack of progress.## Thursday, July 19, 2012

### Concrete Example of Confidence Intervals

**Estimating the Mean Rank of a Deck of Cards by Sampling a Hand of 4.**In my statistics class, near the end of my unit on confidence intervals (C.I.'s), my brief lecture notes say this:

**Concrete Example of C.I.'s**

Consider: Deck of cards (ranks A=1, 2-10, J=11, Q=12, K=13).

Estimate mean of all card values; obtain 80% C.I. for Î¼ & interpret.

Sample size n=4; stdev Ïƒ=3.74; population uniform.

Table: Group (A-E), sample, x', E, C.I. (later: contain Î¼ Y/N)

Interpret: There is an 80% chance that the mean of all card values is in any given interval.

We expect roughly 0.8×5 = 4 intervals to contain Î¼; check & lessons.

Hopefully you can see what I'm talking about -- time-permitting (takes about 50 min if I'm not rushing), I find this to be excellent and highly memorable way to reinforce the conceptual lessons of confidence intervals one final time. Frequently I get some students literally gasping in surprise that that population mean, Î¼, of the deck of cards is actually contained in most of the C.I.'s (as incredible as it may sound -- given that's the whole point of the procedure).

Just for a bit more detail: I'm taking a standard deck of 52 cards and declaring that we'll estimate the average rank value by a random sample of just 4 cards drawn from the deck (as above, we declare A=1, J=11, Q=12, K=13 for this purpose). I split the class up into 5 groups, somewhat as though they're five separate research facilities, each of whom will get their own sample and be asked to perform the C.I. estimation procedure (quicker students may opt to calculate all 5 C.I.'s). I may also talk about the fact that I'm actually breaking the rules for the procedure -- the population is non-normal and the sample size is small; but the procedure is robust enough that with a uniform population it tends to work out anyway. Here's an example from when I did it this last week (each of the 5 rows is a separate group's sample of 4):

As you can see, I've opted to do this at the 80% confidence level, so that we can explicitly compare our results to the number of intervals which we would expect to contain Î¼ in this case (that being revealed right at the end as a separate check). For my classes, this is one of the most insanely challenging parts of the class, so I am compelled to emphasize as often as I can -- understanding exactly what probability statements are telling you. (When I ask something like at the bottom of the picture above, the first responses are, inevitably, almost always "none of them" or "all of them". No joke!)

So as much as I'm always fighting time-management issues in my classes, this is an added demonstration that I've found to be invaluably useful at the end of this unit, basically the "crown jewel" and the hardest part of my community-college statistics class. Here are some of the many lessons that we can draw out from this one case study (esp. granted that we're doing it multiple times):

- Most intervals contain Î¼, but not all.
- The margin of error E is fixed by the chosen sampling process.
- The population mean Î¼ is a fixed number (but generally unknown).
- The probability statement is about the C.I., not about Î¼.
- The procedure is robust even when breaking the assumptions somewhat.

## Monday, July 16, 2012

### Power Rules

**A Method of Reducing Algebraic Power Rules to Just Two Major Principles**In a basic algebra class, we teach the "rules of exponents" and the "rules of radicals". Altogether, this usually appears in the form of 14 or so separate symbolic rules (and maybe more if you're careful to point out common simplifying errors to avoid).

I've taught this in a rather dramatically different way for several years, in a manner which collapses all the different rules to just

*two*major principles. This is based on some observations I made which seem pretty trivial in retrospect, but I find them to be useful -- not a panacea, but they get some traction from students, and better convey the deep global applicability of math (not a big list of disjointed rules to memorize).

At this point, I tend not to even see what I'm doing as something unusual, but I showed it to a friend of mine the other week who has a PhD in Molecular Biology, and she exclaimed, "I was never taught it that way!", and seemed quite delighted. (To my knowledge, really, no one's ever taught it this way, since it's a method I developed -- for what that's worth.) I've mentioned it in passing before but I figured I would highlight it clearly here today.

### Order of Operations (OOP)

First, I start the course by teaching a proper order-of-operations, and tell students that it's the single most important thing in the class -- the "engine that drives everything we do" -- the*only*thing that is arbitrarily made up, and from which everything else flows. It's not PEMDAS. They must memorize the following chart:

**Parentheses****Exponents & Radicals****Multiplication & Division****Addition & Subtraction**

*operations inside*parentheses" (and includes other grouping symbols like braces, brackets, fraction bars, radical vinculums, and absolute values). In any phase of calculation, we work left-to-right across the expression (just like we read), calculating any of the given operations as we encounter them. Note that after parentheses, each operation comes paired with its inverse (tied in order of operations). As we do exercises, I'm careful to verbally model the mental process: "Do we have any parentheses? No, so we don't need a written line for that. Do we have any exponents or radicals? Yes, so we'll need a written line for that..." Etc.

###
Principle #1: Operations on same-base powers shift __one place down__ in OOP.

**I now call this the "Fundamental Rule of Exponents".**

When performing any algebraic operation on powers (like x

^{2}), we have a simple mental shortcut, and that shortcut is found in the order of operations picture by shifting one place down. Some examples:

Ex. #1: Simplify (x

^{3})

^{2}. Think: I have a power (x

^{3}) and I'm

*exponentiating*it (raising it to the power 2). What is my shortcut? Find "exponent" in the order-of-operations and shift one place down and you see our shortcut:

*multiply*the powers. So (x

^{3})

^{2}= x

^{6}.

Proof: (x

^{3})

^{2}= (x∙x∙x)(x∙x∙x) = x

^{6}[finding total x's multiplied]

Ex. #2: Simplify x

^{5}/x

^{3}. Think: I have same-base powers (both base x) and they are being

*divided*. What is my shortcut? Find "divide" in the order-of-operations and shift one place down: we will

*subtract*our powers. So x

^{5}/x

^{3}= x

^{2}.

Proof: x

^{5}/x

^{3}= (x∙x∙x∙x∙x)/(x∙x∙x) = x

^{2}[cancelling x's top & bottom]

Stuff like that. You can do more initial examples based on student inquiry or interest; of course, it also works for multiplying and radicalizing same-base powers (shortcuts to add and divide, respectively). Maybe weaker students wind up having to memorize all four implied relationships anyway, but I think that's okay. It gives everyone a framework for truly understanding the relationships between operations when they need it.

(And there's even something else that I often add later in the course: What happens if we add or subtract powers? Look below "add" and what you see is --

*nothing*. I'll literally write "No Operation" on a 5th line at this point, and even mention the analogous CPU machine language command. So it's consistent that you'll never be changing powers in an add or subtract operation; simply combine like terms and transcribe the powers.)

###
Principle #2: Operations __distribute__ over any operation __one line down__ in OOP.

**I now call this the "General Distribution Rule".**

So what I mean here is that (as I explain in the lecture) you've got parentheses, with one operation outside, and another operation inside. The observation is that we have a very nice, one-line shortcut to get rid of the parentheses by applying the outside operation to each piece inside -- so long as the inner operation is one of the items

*one line down*in the order-of-operations.

Ex. #1: Simplify 7(x+5). Think: We have parentheses. The outside operation is

*multiplication*. The inside operation is

*addition*. Since the latter is one line down in OOP, we can distribute this: namely, "distribution of multiplication over addition". (Note that the "over" in the official name echoes and recalls the relationship in the OOP picture.) So 7(x+5) = 7x+35.

Check: Ask students for a specific value for

*x*, substitute into both sides, and check to see if they are the same value. (Optional: Most students are already comfortable with this distribution, and don't need time spent on the check.)

Ex. #2: Simplify (a

^{2}b

^{3})

^{2}. Think: Once more, we have parentheses. The outside operation is

*exponentiation*. The inside operation is

*multiplication*. Again, since the latter is one line further down in OOP, we can distribute this in a one-line shortcut -- "distribution of exponents over multiplication". So, recalling the first principle for applying exponents to powers: (a

^{2}b

^{3})

^{2}= a

^{4}b

^{6}.

Proof: (a

^{2}b

^{3})

^{2}= (a∙a∙b∙b∙b)(a∙a∙b∙b∙b) = a

^{4}b

^{6}[total a's and b's multiplied]

Ex. #3: True or False: (x+4)

^{2}= x

^{2}+ 16. Think: Look at the parentheses on the left. The outside operation is an

*exponent*, while the inside operation is

*addition*. This will

__not__distribute in a one-line shortcut, since addition is two lines below exponents. Therefore the statement is False. (Note that this is one of the most common errors in basic algebra, and so it deserves special attention -- I'll write "exponents do

__not__distribute over add/subtract" on the board.)

Check: Ask students for a specific value for

*x*(not zero), substitute into both sides, and check to see if they are the same value.

Of course, this principle also works to recall any of: distributing multiplication over add/subtract, distributing division over add/subtract, distributing exponents over multiply/divide, and distributing radicals over multiple/divide. There's actually a total of a full dozen (12) relationships explained by this one single principle (including the fact that exponents/radicals do

*not*distribute over add/subtract).

### Conclusion

It pays off to do as many simplifying examples as time permits afterward, but I do think that getting these two major principles on the board as soon as possible "primes the pump" for everything that happens later. It provides an overarching organizational structure, and it also serves as a model for abstract, generalized principles being easier to remember and apply (making us more happy) than a large body of otherwise disjointed rules. Which from my perspective, may be the only thing that justifies everyone taking a basic algebra course in the first place.## Thursday, July 12, 2012

### Amendments

**On the Frequency and Probability of Amendments to the U.S. Constitution**Consider the following chart of amendments to the U.S. Constitution:

The number of times the U.S. Constitution has been amended has been very "bursty" -- there tend to be many amendments in eras of tremendous social upheaval, and then longer spans without any amendments at all. We're currently in one of the long spans basically devoid of amendments being passed. To be specific:

- Ten amendments were passed together in 1791 as the Bill of Rights soon after the country was founded, and two more in 1795 and 1804 -- but then no more for a period of 60 years.
- Three amendments were passed in the era of the Civil War (in 1865, 1868, 1870) -- but then no more for the next 40 years.
- Four amendments were passed near the end of the Progressive Era (1913, 1913, 1919, 1920).
- Two were passed in the depths of the Great Depression (both in 1933).
- Five were passed around the decade of the Sixties (1951, 1961, 1964, 1967, 1971).
- Only one was passed in the last 40 years; and it's a clear outlier, in that this 27th Amendment was first proposed in 1789 and was pending ratification for over 200 years (finally enacted in 1992).

As a possibly related matter, consider how the chance of achieving ratification changes as the country grows (i.e., adds more states). Note that after a two-thirds vote by Congress, amendments require ratification by three-quarters of the States. As a simple model, we'll use the binomial distribution and assume a fixed probability of being ratified by any given state:

Caveat: States don't have just one chance to ratify an amendment; each one can try year after year until it succeeds (so it becomes more like an "or" operator, increasing the chance over time; see the 27th Amendment above, for example).

But what we see here is that, generally speaking, more states means that a greater level of consensus is needed to pass amendments. If the chance per state is 75% (three-fourths), then the chance to ratify is basically fixed at 50%, since the expectation itself is for three-fourths of the states to approve (with some jumpiness in the percents due to granularity of rounding the three-fourths to an integer number of states). If the chance per state is lower (like 60%; a weak majority), then the chance to ratify crashes precipitously with more states; but if the chance per state is higher (like 80%; a stronger majority), then the chance to ratify increases to near-certainty.

Open Document spreadsheet with these calculations.

## Monday, July 9, 2012

### Prioritize Questions

**On Cutting Content to Prioritize Student Questions**For the last few weeks in my summer courses, I've been frustrated that I'm constantly running over time by about 5 minutes (every night in a 3-hour session). Not great, particularly when the session ends with the culminating problem of the evening, and I really want students to have time to try one while I'm present to help them out.

A related issue is that I've been entertaining more questions at the start of class than maybe I'm accustomed to in the recent past (frequently the first 30 min out of the 3 hours). So between the two -- and for a long time I've thought that this is actually the fundamental task of a college instructor -- I sit down at the end of the night and try to CUT OUT something from my presentation for the next time. (At this point, over the years, it's been cut so much to the bare bones that it feels like the only thing left is amputating formerly key concepts or problems.)

But it has to be that way. Principle: Of everything I can do during class time, answering student questions is the most important and must be prioritized. If I'm fortunate enough to have a student who's done homework, and arrives interested and asking good questions (not all questions being good ones), then the best thing I can do is to highlight and answer those questions. That's the ultimate competitive advantage we have in teaching classes with a live, expert instructor present.

## Thursday, July 5, 2012

### Backup Parachutes

I usually try to teach ways to double-check any calculation procedure (by means of initial estimation, making a graph/sketch, etc.). Frequently students resist this: they ask if they can skip the estimation step and just do the direct calculation alone. That's problematic, because it's the estimation which is perhaps the better test of actual comprehension of the overall concepts involved.

New analogy: The double-check is like a backup parachute. Most of the time the main parachute works, but no sane skydiver would take a leap out of a plane without the backup if they can avoid it. If they did, they'd be one tiny glitch away from total disaster.

New analogy: The double-check is like a backup parachute. Most of the time the main parachute works, but no sane skydiver would take a leap out of a plane without the backup if they can avoid it. If they did, they'd be one tiny glitch away from total disaster.

**Edit:**I suppose an even better analogy would be the little "drogue parachute" that is deployed for slowdown & stability prior to the main parachute (granted that estimation should occur at the beginning and only partly suggest the full answer). But not everyone knows about that, so I went with "backup parachute" instead.## Monday, July 2, 2012

### Teach Logic

Recently I've been coming to the opinion that we need to teach basic logic at a young age, as was done in classical education. Ultimately, it's the foundation for all of math and the scientific method. If the first time you study logic it is in college, then all of your education is really built on shifting sands.

A couple related thoughts: I'm coming to this largely because of how many of my students in any class (even sophomore statistics) get helplessly tangled up over something as simple as an if/then statement. Or a subset relationship (e.g., normal curves are bell-shaped, but bell-shaped is not the same thing as normal). Or an "and" statement (the z-interval procedure requires a simple random sample, known population standard deviation, and a normal sampling distribution of the mean... the last of which can be established by either a normal population or a large sample size).

I'm reminded of my first programming book in the 6th grade which introduced "and" and "or" operators and just said, "the meaning of these should be obvious", with an example of each. It may not be

Coincidentally, I just came across a delightful blog post by John Barnes on the same subject titled, "The Hobo Queen of the Sciences". Here are a few terrific highlight quotes:

Of course, to his credit John goes on to explain the vested interests that don't want fourth-graders -- or jury members -- knowing the basics of logic and reasoning.

Read it here.

A couple related thoughts: I'm coming to this largely because of how many of my students in any class (even sophomore statistics) get helplessly tangled up over something as simple as an if/then statement. Or a subset relationship (e.g., normal curves are bell-shaped, but bell-shaped is not the same thing as normal). Or an "and" statement (the z-interval procedure requires a simple random sample, known population standard deviation, and a normal sampling distribution of the mean... the last of which can be established by either a normal population or a large sample size).

I'm reminded of my first programming book in the 6th grade which introduced "and" and "or" operators and just said, "the meaning of these should be obvious", with an example of each. It may not be

*a priori*obvious to everyone, but it really shouldn't take very long, and could pay off enormous benefits later.Coincidentally, I just came across a delightful blog post by John Barnes on the same subject titled, "The Hobo Queen of the Sciences". Here are a few terrific highlight quotes:

And also:And then I got Ms. Pounding Shouter... She thumped the podium, she pointed at people and accused them of not understanding her, she ordered them to believe what she told them to... "I was totally logical. I pointed things out real loud and told people they were dumb if they didn't believe it, and I yelled so they'd get the point."

Last and far from least, in a related course where I used to teach listening for logic as a way of improving listening comprehension and retention, one student asked me at the end of the class, "Why wasn't I taught this in fourth grade?"

Of course, to his credit John goes on to explain the vested interests that don't want fourth-graders -- or jury members -- knowing the basics of logic and reasoning.

Read it here.

## Thursday, June 28, 2012

### Homework on the Board

Where I teach in New York, I know that other instructors commonly do the following -- Prior to the start of a class session, require a few (2 or 3) students to write the results of a homework problem on the chalkboard. From what I can tell, these problems are not assessed or discussed in any way (the instructor just starts the class with some other lecture topic). My impression is that students are checked off for meeting this requirement a few times through the semester (but it's not attendance; that's done separately). This is not something I ever encountered in my schooling, nor have I ever read about as a suggested technique in any report or study.

Can someone explain to me, or link for me, what the rationale of this practice is?

Can someone explain to me, or link for me, what the rationale of this practice is?

## Monday, June 25, 2012

### Look Closely at This Rectangle

And how it's labelled.

Shaking my fist at you, CVS, dummies!!

(Big thanks to BostonQuad for pointing this out.)

Shaking my fist at you, CVS, dummies!!

(Big thanks to BostonQuad for pointing this out.)

## Thursday, June 21, 2012

### A Very Personal War

A good friend pointed me to the start of a recent

*New York Times*article:A MadMath salute to G.H. Hardy!In the early years of the 20th century, the great British mathematician Godfrey Harold Hardy used to take out a peculiar form of travel insurance before boarding a boat to cross the North Sea. If the weather looked threatening he would send a postcard on which he announced the solution of the Riemann hypothesis. Hardy, wrote his biographer, Constance Reid, was convinced ''that God -- with whom he waged a very personal war -- would not let Hardy die with such glory.''

## Monday, June 18, 2012

### Tragic Examples

**Can Our Real-World Statistical Examples Be More Optimistic?**At one point in the recent math conference, one of the better speakers was analyzing the famous graph of Napoleon's march into Russia (which some argue is the greatest statistical graph ever). And then this speaker ruminated:

"It's remarkable that some of our best examples of quantitative reasoning are based on tragedies... I'm looking for more optimistic examples."

The MadMath response would be: You won't find them. (None that are so intense and urgent, at any rate.) As a say near the start of my stats courses: many of our examples will be dealing with violent acts, or deaths from disease, or drug abuse, or other unpleasant circumstances.

**You wouldn't bother with this kind of math unless you could save someone's life with it.**Math isn't a pleasant serenade; it's a battle of necessity.

## Thursday, June 14, 2012

### Remedial Classes in the News

**Remedial Classes in the News; Possible End-Games**The AP article from this week, "Experts: Remedial college classes need fixing" is worth reading. The statistics are consistent with what I've seen lots of other places (including my own university's research publications). Let me focus on one line:

Probably the most heart-breaking part of teaching remedial college math are the very many students who tell me that they've completed every course they need for a degree, except for one remedial (non-credit) algebra course, which they may take and re-take without success. (You might ask, "Isn't passing remedial math required before taking, say, a science course?" -- the answer is yes, but someone is incented to keep giving waivers in that regard). How to avoid this trap?Legislation passed earlier this month in Kansas prohibits four-year universities from using state funds to provide remedial courses.

In the past, I thought it was a financial-aid issue; funding is given (as I understand it) as long as a full-time course load is taken, which means students are required to take credit-bearing courses at the same time as they attempt remediation. Supposedly NY will soon start enforcing a rule to not pay if remediation is not completed after the first year.

But either way you go on that issue, students wind up committed (sunk cost) to a program that appears "mostly done" except for the math requirement. And they'll wind up in the same cycle of re-taking remedial math, now at their own out-of-pocket expense, with that being close to the

*last*class they have to take. Even if you go the Kansas route and don't pay for remediation, you can pay for everything else first and wind up in the same sunk-cost situation (students paying for repeated remediation near the end -- exacerbating debt which is the other whipping-boy of the linked article).

I'd like to suggest that clear, up-front communications (perhaps mandatory reporting requirements) on passing and graduation rates would do the trick. But then you've got the Dunning-Kruger Effect (the weakest students overestimate their abilities/chances), and frankly the very weakest, in remedial arithmetic, are there precisely because they don't understand percentages (et. al.) enough to parse information like that.

So it seems like only two ultimate solutions remain: (a) Bar students with certain deficiency levels from college -- i.e., end "open admissions", or (b) Void these requirements and allow people to get associate's degrees without ever mastering basic algebra (at least). My guess is that in some form or other, the latter is nigh-inevitable.

## Monday, June 11, 2012

### Reading, Writing, and Video-Watching

**Personal Milestones in Learning & Teaching Math (via Programming and Reading); Struggle With Students; Superfluousness of Video Lectures**I think that there were two key, major developmental leaps in my own learning of math:

**(1)**

**Writing Computer Programs**-- When I learned to program in BASIC on a TRS-80 color computer circa 6th-7th grade, working mostly on my own at home from a book that came with the computer, this increased my precision enormously. When writing code, the fact that the computer complained with a "syntax error" if any single character was malformed caused me to pay attention to the details, and likewise attend to the fact that every single symbol has important, condensed meaning in math/computers. You can have the top-level concepts basically down, but if your details aren't also correct, then your work will collapse into nonsense. As a free and added bonus, it simultaneously got me dealing with logical issues like if/then, not, or, and (I think the book just said "the meaning of these should be obvious"), as well as variables. In junior high school I was writing programs to spit out the results of numerous math homework assignments automatically, so the application was obvious and actually time-saving.

**(2)**

**Reading Math Textbooks**-- When I was a senior in high school, I started taking a calculus course via experimental technology; namely, a course at the state college that was being delivered by closed-circuit video feed to the high school. In theory, we had microphones on our desks with which we could ask the college instructors questions and participate. In practice, it didn't work well at all: (a) we felt disconnected and intruding when we asked a question (which would hit the instructor by surprise, he'd appear startled, and have to ask where it was coming from), (b) the one TV in our classroom made it hard to see the instructor and what he was writing, (c) the lights in our room were turned off for visibility, as I recall, and (d) I don't think the instructor was terribly good in the first place. It was the first math class I took where I was routinely falling asleep at my desk. In desperation, the only way for me to learn calculus was on the weekend, go down to the basement alone, and read it from the book. And the transformative realization was that it was fundamentally

*readable*, all the information was right there, if I just read it slowly and carefully enough. So while this wasn't how I always approached math classes from then on, it was always my backup plan -- you could learn all of math just from a book if you really wanted to.

Questions I ruminate on: Are these special skills? Can everyone pick up these abilities, or is there evidence that is not the case? Why is it so overwhelmingly difficult to get students to read a math book? Does anyone teach or assess reading a math book? Does anyone teach or assess reading

*anything*in careful detail? Would math instructors everywhere be out of a job if everyone realized that the information is just sitting in a book that you could read on your own time? (What's that

*Good Will Hunting*quote -- "... an education you could have got for a $1.50 in late charges at the public library"?) Part of me very much wants good, open source textbooks to conserve student money and resources -- but is text already dead?

Over the years, I've tried to share these developmental leaps with my students. When I was a graduate student, I tried incorporating some BASIC programming into the algebra course I was teaching (actually, it was included in the book at that time, I think). About 10 years ago I was using MathXL online homework software. A few years ago I was assigning carefully-written algebra homeworks in the standard book format, and grading every symbol/character carefully (which got

*enormous*resistance and hostility).

Again, it seems like the primary struggle we have, with any technique, is the attempt to get students to actually put in the study/ reading/ exercise time required for math (whether in-class, out-of-class, online, etc.), and I've seen all of these initiatives at some point fail when students simply gave up on them. Research seems to show that the more learning students do, the less they like it, because the more work they're doing for it. Perhaps we just have to admit that the primary factor is just how well students are situated in life to actually spend time studying. (Perhaps.)

The point I'm trying to get to is this -- There's a lot of scuttlebutt these days about Khan Academy, other online teaching initiatives, the "inverted classroom" where a video lecture watched before class (we hope) and exercises worked and coached in-class. But I honestly don't see any intuitive advantage to video lectures over having a textbook (like we've had for hundreds of years) and expecting that it be read prior to class. All the same information is there -- Just in a far more efficient format (textbooks). In fact, math notation is fundamentally symbolic manipulation created for the written page -- to me, video seems largely beside the point. If we just taught students to

*read*properly (and truly, my math classes always seem to transmogrify into language-arts classes), wouldn't it be hundreds of times more efficient to just give them a (possibly digital) book? All the reputed advantages of self-pacing, being able to pause and rewind -- is that not inherently possessed by books as well, and more elegantly? Is the idea just that "video" is all the rage and kids are trained to respond better to it?

**I just flat-out don't see any advantage to video over books -- video lectures go too slowly and make me impatient and irritable -- am I crazy?**

Amusingly, one of the recent CUNY research studies on video-lectures had the same problem with their videos as I did in my old calculus class; what the video-instructor was writing on the board behind them was apparently difficult to see, and students complained that they then had to turn to the textbook to understand what was going on (according to the speaker at the recent conference). This being approximately 25 years after my identical experience with closed-circuit video. Written language is still the

*uber-tool*of humankind -- and mathematical writing the most intensely condensed and powerful -- and I'm not seeing any way to avoid embracing it and teaching it as such.

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