But it's okay, right? Starburst Candy Wrapper Prom Dress.

The Inverted Classroom

Robert Talbot
Casting Out Nines

People are talking about it here, and here. This is Robert Talbot, the first person I heard about it from. Here are some of his other articles. Let me throw out a few thoughts and attempt to nail down what I think. ("essay" from Fr. "Essayer", to try.) I'd love to hear what all of y'all have to say, too.

Essentially, Talbot (a university math professor) is reversing the order of his teaching.  He makes a video of his lecture which students are required to watch before class. Class time is then used for doing "homework" and practice, etc.  The theory is that lecture is very one-directional anyway (basically a video) and the students don't do well in lectures, so put the video online and have the kids do that portion of the class themselves. They can rewind, fast forward, review, whatever.  Then class time is reserved for face-to-face tutoring, direct and targeted instruction. Differentiated, if you will.

I'm ambivalent about this for high school. It sounds great in theory but so many great educational theories only work with college students or other adults who have different time pressures and levels of responsibility from the ones in front of me.

Some thoughts that occur to me are

Can 9th grade students learn this way? I'm not convinced. I think education at this level is more than just watching videos (and certainly NOT a lecture) and I'm not sure that the limited amount of time in the classroom the next day will suffice for re-teaching missed concepts or for correcting misunderstandings ... but they can watch the video over and over again, stop it and think. Listen to it again.  There are misunderstandings now and lots of time that homework would have gone much more smoothly with me to help. ("but my father said THIS was the way to do it.")

Really?  Students teach themselves? What am I here for? To teach the students.  They'll get more personalized attention the next day when you are working through the homework with them.

What about time? The college professors who have tried this are dealing with people who come to class every other day, people who have much more control over their lives, who can drive. I remember my college schedule: three to fours hours of class a day and then lots of free time to do this kind of thing. The expectation was one hour in class for every two hours outside of class. I can't see high school students ever fitting this pattern. When my son learned to drive, it immediately freed up a tremendous amount of time for him and for us. That's one huge difference - the amount of "loose time" and non-structured time. Time they can use to go to Office Hours. Our kids aren't allowed to come during free time, can't stay in the building without a teacher, can't meet at off times. But shouldn't they be developing that sense of responsibility? It's simple to require homework and any parent who won't support that shouldn't be surprised if the kid doesn't learn. We don't even allow them to go to Subway for lunch. that's my reality.

I can see a parent getting really annoyed that his daughter needs to use the living room TV every night for 45 minutes per class. I can also see that daughter being embarrassed to ask or simply giving in to the family's "American Idol" viewing habit. See previous comment.

My computer ate my homework.

The technological side bothers me too. Too many possible home configurations, IT issues. I had one set of parents tell me that they couldn't get their son to do a PowerPoint (I'd have accepted Keynote or Google apps, too) because their computer had a problem. For two months. True or not, it was out of my control - that kid would have had no instruction for two months. That kid had plenty of time to do the assignment, had plenty of resources with which to do it (town library, after-school, during class).  What he lacked was the desire. This will clarify matters.  Watch the video or fail.

An issue here in Vermont that the rest of the world hasn't had to deal with for a while is lack of broadband; probably a third of my students can't even get cellphone access at home. Poverty dictates that the school can't require certain things.  Require it and make it work. If there is a student who legitimately cannot, then deal with that issue at that point. Most complaints of this type are bogus. The real-world isn't stopping to wait. Lend him a computer, use a DVD, make a vidcast for the iPod the kid is carrying.

If they don't get the video done, what happens? I assume you'd let them watch it during class? What about the homework then? Does that kid get to re-invert the classroom because he simply won't cooperate? No. The classroom stays inverted.  If he refuses to do his assigned work, then that is the same as refusing to do homework.  If he refuses, he won't learn. If he doesn't learn, he'll fail. That seems a little draconian, but the students will rise to the level of your expectations.

Or not. The parents will bitch, the principal will be called and I'll be told to change. A lawsuit will be threatened. Hell, they do that anyway. Remember that kid and the PowerPoint? Give him some more time, during class, to get the work done.  That's the whole point of this - class time should be for working with a student, not talking at him.

What if YOU don't get the video done?  Well, shit.  You'd better prepare at least two weeks in advance, or just do the whole course over this coming summer.  No more of your "I've been doing this for years, let's just wing it today" bullshit.

Will students who play a sport or have another obligation get progressively farther behind? I can see them doing math problems on the bus (or on the bleachers while they wait for the JV game to finish) but not being able to do much more than placidly listen to a DVD (or podcast). Learning something new from scratch needs focused attention. Doing homework needs focused attention, too.  Would you rather they zoned out while listening to you in class, or zoned out while listening to you on their computer?  It might give them more responsibility because they know they have to get it done.  Slacking off has no upside. And you can help them with the more important part of learning, the reinforcement.

New isn't always better. Old isn't always better, either.

Crap. What about the time needed? In any recording endeavor, the time to create is at least four - six times the running time. You plan, record, re-record, mix, edit, review.  In the middle of October, that pile of whatever that needs grading is not going to go away and you've got videos to make and these videos can't wait. You don't have the "research" time that Talbot does. He has 3 hours of class a day, you have 6. He has lot more free time and "office hours" than you do. Once created, they stay created.  Sure, you can be replaced next year with a much cheaper TFA teacher and we'll still show your videos, but that's the risk we all take. Did I mention that all of your work is now the property of the school, that you cannot resell it or even give it away for free? Take it off YouTube. It legally belongs to the district because you created it on school time. Thanks for your effort. Now we can give a teacher 60 kids per class and a couple of cheap aides because all the learning has been done by video. Any non-learning will still be your fault. Aren't you thrilled?

Well.  It was a nice discussion.

I think that I will take the safe route this year and wait for other people who have met AYP to do the research. In the meantime, I'll put a few videos online, increase the use of the Moodle, but keep the traditional format for now.

You can't have "improvement"
if you haven't got any data.

Changing seems too much of a crapshoot to me at this point. The most important piece to all this, and one that I'm afraid will be overlooked by almost everyone, is data. I haven't seen anything other than Talbot's anecdotal information to say this works at the high school level.  I am not willing to go through the tremendous amount of time and effort required to change over without something more than pleasant stories.

We have enough trouble in this country with educational fads. An option that MIGHT work? I have seen lots of stupendous, wonderful, groundbreaking ideas come roaring in like lions and then go slinking out embarrassed like a wet toy poodle. The money spent in the meantime is obscene.


Data Questions:

Better bone up.

1. What is your baseline? Have you got someway to measure the effectiveness of the current system before you throw it away?
2. What kind of data will convince you that this works - or doesn't?  You must develop some way to tell if this was effective or if the teacher was effective, or both, or neither.  If you can figure that out, you can sell the idea to New York.
3. When will you decide? You have to nail this point down or run the risk of picking your goalposts to suit a preconceived notion.
4. Is this tent therapy?* Are the gains simply a matter of improved attention on the teacher's part, working harder, being better because you're paying attention for the first time in a while? Is the different format making a difference or is it simply that you've made a change?  Will these gains diminish next year when the kids get used to the new method? If this goes well, will you be able to prove it?
5. Control group? Can students transfer to another class if they really don't like it? How flexible is your school? If you truly want to make this a controlled experiment - and you should - you should do it with only half of your classes and refuse to let the kids switch around.

These questions need to be answered FIRST or you are committing Egregious Research Error #1.  There's been enough of THAT already. Now for the important questions:

1. Will your principal back you up? College professor Talbot has a great deal of professional freedom. If this hits a stubborn kid or obnoxious parent in October, how much influence will be enough to shut this down? (Is 'obnoxious parent' an oxymoron or a regular moron?)
2. Do you have tenure? Is your union willing to stand by you? Do you still have a union?
3. Do you need this job? If the answer is "Yes." then you need to consider whether to take this chance. I know what the current method does. 
4. Do you have the resources to pull this off? Technology, website, bandwidth. Since you're going to be doing some of this from home, how's your home computer?
5. Do you have the time? Better start recording the lessons now.

* Tent therapy was a short-lived psychological treatment fad from early twentieth century. Patients who were brought outside and placed in tents on the lawn of the insane asylum showed improvement. In reality, the improvement came not from the tents or the fresh air but from the increased attention the staff was giving to the patients. As soon as routine settled in, the improvements disappeared.

Math Reform: Games, Practice and Psuedocontext

More interesting to students.
Probably not good at being a teacher.

Dan Meyer found Mathematics v. MTV by H. Wells Wulsin in this month's Stanford Sound Off. H. Wells, a former physics and chemistry teacher at a Washington DC private school, is making the case that math software (and by extension, teachers) must change because of competition with MTV for the eyeballs of our students. Which is kinda amusing, since none of my students watches MTV for more than a tiny percent of their information-absorbing time. Facebook, iPod, games, texting, Facebook, YouTube, cellphone, texting. The TV might be tuned to MTV in the background, but more likely it's on Spongebob.

Anyway, he does have a point when he says, "Mathematics educators now vie with a multitude of digital entertainment options to capture adolescents’ interest." "To compete more aggressively for students’ attention, mathematics software should adopt the very strategies that have made these other media so successful."

An issue I have is when he says, in the next breath, "Research shows that mathematics software can boost student learning, but these programs remain unpopular."  That's because software isn't always the answer. Teachers are primarily involved with new learning. New learning is rarely successful in a computer-learning environment because it usually needs an human explanation. If students could do this alone, you'd see a lot more kids taking summer classes online and getting credits. Repeat learning can be successful online (or by computer training) because the kids already know what the goal is, already have been taught the concept but just need repetition and practice.

The upshot of all of this?  You probably have done as much of this guy's ideas as is workable.

Back to our expert ... Wulsin offers recommendations that sound good in theory:

Presenting examples in high-resolution video. "Video lets students watch the sweat beading on the athlete’s temples, see the whoosh of wind in the skydiver’s hair, hear the rev of the daredevil’s motorcycle. A photograph or cartoon cannot beat video in its fidelity and power to captivate."

Connecting to students' interests. "Monitoring a breeding bunny population would show the process of exponential growth. Baseball batting averages could introduce percentages."

Showing appealing faces. "These videos could occasionally feature famous sports or entertainment figures. What if Michael Phelps calculated the volume of an Olympic swimming pool or Beyoncé computed the time delay needed for speakers at an outdoor concert? Why not let Danica Patrick figure the monthly payment on an auto loan?"

Holding students' attention. "Make students laugh through physical comedy or corny one-liners. Introduce them to interesting people with magnetic personalities."

Sounds good. Misses the point.

To engage learning, what you need is a good question that arises out of context, asked in a way that makes sense, posed by someone who actually needs to know the answer.  While I hate to push the button marked “Praise Dan” too often, I do think that he has given us a shorthand for much of this … pseudocontext … and the suggestions fail because of it.

Show appealing Faces

“What if Michael Phelps calculated the volume of an Olympic swimming pool”

Really? When would Michael ever care? On the other hand, he might be interested in split times and speeds and the differential changes made by a new type of suit with a 5% slicker surface. Build your question from that and you'll have the class hooked. Bring in data from your pool at home and calculate the chlorine percentage. How many tabs go into this pool? In my case, it was the number of bottles of medicine I needed to add to a fishpond.

“or Beyoncé computed the time delay needed for speakers at an outdoor concert?”

Don’t make me laugh. Beyoncé’s job is to sing. When the hall was built, someone cared. Once. Then the problem was solved and everyone moved on. The audio engineer needed to know when she placed the venue's sound system. The designer needed to know where to put the reflecting panels. But not Beyonce.

And the silliest one of all: “Why not let Danica Patrick figure the monthly payment on an auto loan?”

Because she cares even less than your students do and they know it.

In a sport like NASCAR, which is run using computers, analyzed to death with computers, which has something like 200 different sensors on the cars taking measurements every 1/300th of a second generating GBs of data which is endlessly broken down before, during and after the race … and all we get is a suggestion that Danica figure the monthly payment of an auto loan? Lame. (BTW, amortization schedules? Even bankers and loan officers run freely available spreadsheet tools.)

How about spring displacement, fore and aft g-loading, pressure, O2 vs Temperature, wing downforce? Suspension settings, pitch attitude, adjusting control parameters, understeer, optimize torque during power down? Why not do this? Probably because it's more information than your kids can handle, that's why.

Going too far into the real-world is not only confusing, it's counter-productive. You spend way too much time explaining something that winds up being taken on faith rather than being understood and the math you wanted them to get is lost in the descriptions of the curve of the wheel-wells. The jargon of the job overwhelms them. I love the concert hall question but you'd have to remove a lot before it became an algebra I question.

The other suggestions are interesting but not very particularly useful. Rabbit populations are not exponential (more sinusoidal) and not very relevant to the kids unless you have a fur farm nearby, in which case you'll start up the PETA alarmists instead of the graph-makers. Fibonacci was just doing a thought experiment anyway - in practice, the numbers are fairly complicated unless none of the rabbits ever dies and they all have 2 kits per litter.

If you want to model things, you need to pick the model carefully or the idea will disappear in the myriad details. If the relationship is supposed to be linear, stay away from the exponential data, and vice-versa. If you pick a real-world problem, you need to make sure that it will work out as you expect.

Appeal to Kids' Interests

Mindlessly connecting to kid's interests is a bad idea for a couple reasons.

One, the kids aren't interested in things that are mathematical right now. They are social beings and putting them in a social situation but wanting them to be mathematical is a frustrating and pointless exercise akin to moving Mt. Fuji with a spoon.

Two, which kid's interests? Baseball and softball overlap, but the goth kid is in his anti-jock mode and deliberately tunes you out for trying and nobody can understand the Valley Girl accent you're attempting.

Baseball batting averages are great for introducing percentages. Once. After that, only baseball freaks care and only if it's "their guy." The other 96% of the class is totally bored. If the entire software package is developed around baseball, I'd scream, too.

Third, How do you Know? If a kid is interested in something that you know little about, you risk looking foolish and stupid -- which doesn't achieve the goal you are trying for. Your "real-world" question is obviously contrived and tedious. This is why the "psuedocontext" questions in the book are so discordant. They aren't written by real people.

This may be hard to believe, but kids are okay with just learning something mathematical. Raw, pure math. Give them this just before they really are going to use it for a question THAT INTERESTS YOU and you'll have succeeded.  Forget their interests. "What use do YOU make of it" would be a better idea.

Finally, kids change. MTV is so last week. If you base your software on what was current just ten years ago, 95% of their world wouldn't exist. Are they interested in Spongebob or iCarly or Jimmy Neutron? Kid fads go out of style way too fast to try and keep up and there is nothing so dated as a teacher (or a computer tutor) trying to achieve "relevance."

Use Hi-Res Video

Hi-res video is valuable except when it’s not. The video has to show a problem. If it shows the sweat, blood and tears without a problem, then there is no point. Better to use a still picture that does than a video that doesn’t.

Make Students laugh

Making students laugh with corny one-liners that come out of nowhere only leads you nowhere. The jokes can be bad and yet still effective if they have context … like the joke that always comes up during discussion of integral of 1/x:

What’s the integral of $ \int \frac{dCABIN}{CABIN}$ ?
ln(cabin), of course. ("natural log cabin")
and when you add “C”, you get houseboat. (Better when you say it aloud.)

In the end

In the end, for me, it comes down to:
Get Real: real math, real problems.

WCYDWT - football and simultaneous equations

How DO you score this game?

 

The root of the problem

Gene Lyons has this piece in Salon:

"Except for weight loss potions, no area of American life is more prone to fads, panaceas and miracle cures than public education. Everybody agrees that schools are failing, and since everybody went to school almost everybody's an expert.

"Naturally, it's assumed that the biggest experts are those with the most money. Hence the philanthropic enterprises of what Bob Somerby calls the "Billionaire Boys Club" draw outsize attention. Whether it's New York Mayor Michael Bloomberg or Microsoft's Bill Gates, the guy with the thickest wallet is assumed to have all the answers.

Humph.

Let's make Algebra II mandatory - that'll learn them.

From Mother Jones and The Washington Post:  ed reform has gone to this:

Of all of the classes offered in high school, Algebra II is the leading predictor of college and work success, according to research that has launched a growing national movement to require it of graduates.
In recent years, 20 states and the District have moved to raise graduation requirements to include Algebra II, and its complexities are being demanded of more and more students.
....One of the key studies supporting the Algebra II focus was conducted by Anthony Carnevale and Alice Desrochers, then both at the Educational Testing Service. They used a data set that followed a group of students from 1988 to 2000, from eighth grade to a time when most were working. The study showed that of those who held top-tier jobs, 84 percent had taken Algebra II or a higher class as their last high school math course. Only 50 percent of employees in the bottom tier had taken Algebra II.
....But not everyone is convinced that Algebra II is the answer. Among the skeptics is Carnevale, one of the researchers who reported the link between Algebra II and good jobs. He warns against thinking of Algebra II as a cause of students getting good jobs merely because it is correlated with success. “The causal relationship is very, very weak,” he said. “Most people don’t use Algebra II in college, let alone in real life. The state governments need to be careful with this.”

Holy crap!  It's so damned simple!

Calculus students do really well in jobs that require mathematics.  Let's require every student to take calculus and we'll have a nation of math geniuses.

Algebra II! As a minimum requirement to hold a high school diploma! We're literally saying that if you can't factor polynomials, manipulate complex numbers, do matrix arithmetic, and understand basic trig, then you can't get a high school diploma? Really?

The push comes from Achieve, a group of idiots would really, really need to take statistics again.

Let's think here ... smart kids take algebra II ... smart kids are motivated ... smart kids are likely to maintain their motivation and work their way up the corporate ladder ... smart kids are likely to take art ... smart kids are likely to do well in literature ...  smart kids are likely to finish college ... smart kids do well in science ... smart kids work hard ... smart kids focus ... smart kids are likely to be smart ... smart kids eat healthy foods ... smart kids play a lot of video games ... smart kids read a lot ... smart kids play soccer ... smart kids volunteer their time ... smart kids eat crappy, greasy foods ... smart kids are likely to be athletic ... we could go on.

Why is anyone making the link only between algebra and success?

Why are we so stupid?

Speaking Fees

It's always been a scam. Ex-President gets $400,000 for delivering a 45-minute speech (which is equivalent to his Presidential yearly salary) or travels around the country speaking for $150,000 a pop. Schools try to garner the best people they can: diplomats, statesmen, artists, people who think and deliver the goods. The goal of landing a big speaker is usually to boost alumni donations and add to the university's endowments. The criteria usually include a lifetime of work and achievement, like what Bill Cosby has done or any number of Presidents.

And then you have Snooki, famous for a different kind of endowment, willing to donate pretty much anything to pretty much anyone.

PISCATAWAY, N.J. - The pouf is mightier than the pen when it comes to speaking fees at New Jersey's largest university. The Rutgers University Programming Association paid Nicole "Snooki" Polizzi of the reality TV show "Jersey Shore" $32,000 Thursday to dish on her hairstyle, fist pumps, as well as the GTL — gym, tanning, laundry — lifestyle. That's $2,000 more than the $30,000 the university is paying Nobel-winning novelist Toni Morrison to deliver Rutgers' commencement address in May. Money for Polizzi's appearance came from the mandatory student activity fee.

Public University. Mandatory student activity fee. GTL lifestyle. More money than the commencement speaker, Toni Morrison. I guess it's okay if your purpose is to emphasize the role of women in today's society.

OMG indeed.

Are you teaching Calculus next year?

I've been doing it for how many years now? 27 years (I think) ... so much time, so many really cool kids. I guess one thing I really like about teaching calculus is that I can't recall a bad student.

Anyway, for those who are setting up a course for next year:

Text (by author):

• It will probably be Larson or Stewart. Stewart is the more common college text, but Larson is the more HS-friendly. I personally use Larson 7th. I've taught from this book since the third edition and I feel very comfortable with it.
• Hughes-Hallet-Gleason did things in an odd organization, IMNSHO. In a tutoring situation, I had this thing and it was oddly laid out, with integration happening before the end of differentials. You might like it, but I'm too used to the Larson / Stewart / Thomas Finney organization.
• Ostebee-Zora is dense for highschool students.
• Thomas and Finney was cool ten years ago but seems to have fallen out of favor. Don't know why.

If you want to stretch your boundaries and go 21st Century on your kids, there are many online texts.  I'd love to try this and I'm planning on putting selected sections on the Moodle. WolframAlpha will be required for some assignments.

• Neat place to start is the California site: http://www.clrn.org/home/
• http://www.whitman.edu/mathematics/calculus/
• MIT OC: http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm

Additional resources:

• http://www.math.temple.edu/~cow/
• http://archives.math.utk.edu/visual.calculus/
• and of course, Salman Khan on YouTube.

VERY IMPORTANT
AP insists you fulfill their audit process - mostly they need a syllabus, but you have to complete it if your school wants to put the "AP" label on it. Someone at your school may have already fulfilled that requirement -- make sure to check. Soon.  Here's my syllabus if you want to copy it.

The Summer Before
If you can, and can afford it or if your school with cover costs, hit up a summer AP seminar. The Summer AP Institute at St. Johnsbury VT is one of the best.
http://www.stjacademy.org/page.cfm?p=278
http://www.stjacademy.org/uploaded/documents/AP_Institute/AP_Institute_2011.pdf

The environment is as close as you can get to smart-people nirvana. One week surrounded by a bunch of AP teachers in an informal setting in the Vermont summer. Learn, work, relax, eat REALLY good food, play bocce on the lawn and drink free beer with a slew of brilliant people till dinner. Like you can get better than this? It's $1100 but worth it ... especially if your school will pony up. Skip the dorm and use the savings and a little extra of yours and take your wife -- a hotel isn't much more and it's a wonderful town.

Yeah, what the Hell, TI?

(from Randall at XKCD) And the marginally colored TI-Inspire doesn't get you off the hook, either.

Technology

• Every kid should have a TI-84 or the equivalent.  I don't require them to buy one but I'll warn them that the ones available in the room for borrowing are often changed and reset by the algebra I kids, making it difficult for the occasional user to get started. "Radians or degrees", language set to Deutsch, etc.  There is a new wrinkle, though:  The graphing calculator app for iPhone and Android devices.  It's got a better screen resolution and it only costs $3 for the phone they already own. 
• Excel is useful - Google apps works, too. Just be ready to reserve time occasionally in the computer room.
• Pencil and paper

To be continued.

Calculus Syllabus for the AP Audit.

This was enough to satisfy the review process wonks.  Feel free to use it verbatim as I can't claim to have made much up on my own. It is mostly a regurgitation of the table of contents.

I certainly don't feel that anyone should spend an iota of psychic energy on the syllabus - spend it making plans and working toward the best you can do for your students.

Case in point: "Definition by limit: epsilon-delta." was de-emphasized in AP years ago, is usually ignored by most college professors for MAT121 but emphasized by the Audit review process. I couldn't get the audit accomplished without lip service to it.

A change from this syllabus for me: I am changing from a school-hosted ePortfolio to a Moodle, blog and Wiki based on my personal server. There are several places to find free hosting for those. Additionally, I will be requiring certain homework to be done on WolframAlpha since no one can afford Mathematica.  I'd love to require an iPad but I'll have to win the lottery first!

Good luck in your setup.

---

Calculus Syllabus

Instructional Materials

• Textbook: Calculus, 7th edition by Larson, Hostetler, Edwards; Houghton – Mifflin Publishing Company.
• Technology: TI-83 / TI 84 graphing Calculator 

While materials can be borrowed from the school, it is highly recommended that students purchase both the book and the calculator. The book will be useful for many years and the utility and convenience of writing notes in the margins and highlighting passages cannot be minimized.

• When appropriate, the class will make use of Geometer's Sketch Pad, excel, Google apps.
• a gmail account would be most helpful.
• E-Portfolios will be a part of the Course. See ____ for account and access information. Work will be done in groups, handed in separately (and included here). Typical Contents:
• Your two sets of school-wide showcase materials
• Chapter projects/labs assigned monthly.
• Past years FRQ questions worked out and scored by each other.
• Inter-curricular questions.

Introduction: What is Calculus About?

Functions Review

• I. Sign Graph solutions
• II. Coordinates, Slope
• III. Linear Equations: [point slope, slope intercept, general]
• A. Formulae [ distance, angle, perpendicular, bisectors]
• IV. Polynomial functions – library of Functions
• A. Domain, range, discontinuities, roots, synthetic division
• B. Odd, even, Symmetry
• C. composite functions g(f(x)), (g*f)(x), (gof)(x)
• D. Slope = derivatives, differentiability,
• E. "How to say it in Math." Introduction of proof techniques and format.

Limits

• I. x =>a, one sided limits at discontinuity, peak, break in curve, inf. disc, etc.
• II. [ graphically, algebraically, numerically]
• III. definition of derivative as lim h =>0, delta, epsilon (refer back to "More or Less")
• IV. Limits : add sub / mult / div
• V. Infinite Limits
• VI. Continuity, max-min theorem, Intermediate Value Theorem, Computer/calculator-aided exploration of situations in which lim f (x) x a D.N.E.

Derivatives of Polynomial Functions

• I. Definition by limit: epsilon-delta.
• II. Computer-aided exploration:
• A. Using the definition of the derivative as a limit of slope
• B. secants to plot f'(x)vs x. The relationship of the graph of f'(x)to f(x).
• III. Power Rule, Constant Rule, Sum Rule, Product Rule, PowerRule2, Quotient Rule
• IV. Implicit Differentiation
• V. Chain Rule: composite functions
• VI. "Explain" "Show that" "Why or Why Not" – how to justify an answer. Includes review of past free-response questions, and scoring methods, practice developing an appropriate answer with explanations. presentations by students
• VII. Trigonometry
• A. one-day review of basic functions, then Derivatives of trig functions
• VIII. Linear
• A. Tangents and Normals, Linear Approximations and Differentials – ZOOM and compare – analytic results vs. calc – relate back to epsilon/delta
• B. Newton’s Method – writing a recursive program
• C. Calculator computation of derivatives and graphs of derivatives.

Derivative Applications

• I. Sign Graph analysis recap
• II. Graphing
• A. First derivative: increase, decrease,
• B. Second derivative: concavity
• III. Symmetry
• IV. Asymptotes, limits, dominant terms, derivative of hyperbola vs parabola: behavior at extremes
• V. Extrema Theory: Relative and Absolute Maxima and Minima;
• A. 1st derivative test and the Extreme Value Theorem; Intervals,
• B. Continuity;
• 1. Why are the notions of continuity and limit crucial to the Calculus?
• 2. Informal discussion of "function niceness."
• 3. Definition of continuity theoretically compared to on the TI.
• C. 2nd derivative test for maxima and minima
• VI. Recap of "Explain" "Show that" "Why or Why Not" – justify an answer. "How does one know f(c) is minimum?" 1999 FRQ BC:4 Student Sample A and others.
• VII. Maxima and minima: Problems
• VIII. Related rates of change, physics, chemistry, Real Problems, dist-vel-acc-jerk, best solutions. Include here the instantaneous numerical calculations.
• IX. Rolle's Theorem, Mean Value Theorem, Intermediate Value Theorem
• X. Indeterminate forms and L'Hopital's Rule
• XI. Fall School-wide Showcase: Presentations and Explanations (similar to typical portfolio except this is a chance for students to present some of their work to parents, faculty and members of the community.)

Indefinite Integrals

• I. Indefinite Integrals - Primitives (antiderivatives) differential equations, Separation of variables, basic integration formulae, constant of integration.
• II. Finding the constant of integration (C)
• III. Substitution Method - U and dU
• IV. Integrals of Trigonometric Functions
• A. Basic Functions
• B. Using Trig identities to solve integrals
• V. The Area Under a Curve - Definite Integrals
• A. Rectangles & Summation (Sigma notation)
• B. Riemann Integral = Lim as number of rectangles incr. to infinity
• VI. Calculating Areas by Summation
• A. Rectangles
• B. Trapezoids
• VII. Fundamental Theorems of Calculus -
• A. First F.T. of C. -- definition of Primitive F(x): derivative F(x) is f(x)
• B. Second F.T. of C: definite integral is equal to F(b) - F(a)
• VIII. Substitution
• IX. Approximations of Definite Integrals
• A. Trapezoidal Rule A=h/2 (1 2 2 2 2 ...... 2 1)
• B. Simpson's Rule A= h/3 ( 1 4 2 4 .... 2 4 1 )

Integral Applications

• I. Distance - velocity - acceleration - jerk
• II. Areas: horizontal rectangles - dy, f(y), g(y); vertical rectangles - dx, f(x), g(x)
• III. Volumes, by revolution
• A. Creation: use potter’s wheel as example; zucchini-carving to show slicing; cardboard discs, wood tiles. Group–think: Given a lathe template for vase, find the volume
• B. Volumes of revolution around x-axis and y-axis using Discs, Washers, Shells
• IV. Known Cross Section
• V. Average Value, average rate of change
• VI. ** Optional extensions of integral theory
• A. Length of Plane curves, Area of Surface of Revolution
• B. Physics: Moment of Inertia, Center of mass, Centroids, Nonuniform mass distribution, Work done by variable forces (concurrent physics preferred)

Transcendental Functions

• I. Inverse Functions and their derivatives
• II. Inverse Trig functions
• III. Derivatives of the Inverse trig functions and associated integrals
• IV. Natural Log and Derivative
• A. INT (1/cabin) d(cabin) = log (cabin) + C = houseboat
• V. Exponential and logarithmic functions
• A. derivatives and integrals
• B. Applications [ rates of growth, decay, compound interest, Richter scale]
• VI. Spring School-wide Showcase

Integration Methodology

• I. Basic Forms
• II. Integration by parts [Berger method]
• III. Trig Functions, odd powers and even powers
• IV. Trig Substitutions
• V. Integrals involving .....
• VI. Partial Fractions [Alfonso’s Method, Matrix solutions, Millitello cover]
• VII. Numerical integration

** Post – Test Optional

• I. Conic Sections – Definitions and uses of calculus – areas, etc.
• II. Hyperbolic Functions –
• A. Definitions, Derivatives and integrals
• B. Cables, Catenaries (Gateway Arch and Power lines), and the suspension bridge.
• C. Inverse Hyperbolics
• III. Polar Coordinates
• A. Graphing and common figures,
• B. Derivatives and integrals in Polar coordinates
• II. Infinite Sequences and series
• A. e, sin, cos, pi expansions
• B. Taylor, MacLaurin etc.

Just don't call it tracking

Mike Petrilli on Education Gadfly:

It doesn’t take a rocket scientist—or even a cognitive scientist—to know that kids (and adults) learn best when presented with material that is challenging—neither too easy so as to be boring nor too hard as to be overwhelming. Like Goldilocks, we want it just right. Grouping kids so that instruction can be more closely targeted to their current ability levels helps make teaching and learning more efficient.

Apparently, it takes someone a lot more intelligent than a rocket scientist because it's news to all the guidance counselors and schedulers that I've talked with.

Why I support Unions.

The Union will block spurious, petty and mean administrators from firing the competent as well as the incompetent. This is good because the administrator usually has no real idea which is which.

The Union will ensure that teachers get a fair shake when they are dealing with someone who has no real need to be fair.  This is not the owner of a company who is watching every dollar out of his own pocket -- this is a transient employee who has too much power and not enough knowledge.

They float through once, twice a year in a choreographed display of feigned interest in what I am doing. Three weeks before, I get a note saying when they'll arrive and which class they'll be observing - get a lesson plan ready and do the pre-observation meeting. Then they'll watch the class, make a few notes. In the case of the more recent attendees at admin training workshops, they'll attempt to record my speech verbatim in the vain hope that recording every word spoken will somehow tell them more than just listening attentively. This is real?

We meet for the post-op and I'll be given a letter that specifies what they observed. Some of the more insightful comments were "Good lesson" and "knows the material." One of the less insightful was "Separates the kids into two groups by gender." I taught algebra in the chorus room -- apparently Mr. HIP didn't notice that the aisle down the middle separated the kids into two groups and that they took the same places as they normally did during chorus -- I don't do seating charts and usually pay no attention to where students sit. No, he thought I was being discriminatory.

I went 6 years before I was observed again. How was that principal supposed to know anything about me or anyone else in the building? How were they (five in six years) supposed to determine that X should go?

When an admin played games, got in the face of someone improperly, tried to ruin a teacher, there was always the Union stepping in to make sure that things were proper. That is its best role, to act as a buffer, to make sure that each i is dotted and t crossed. I was lucky (knock on wood). I had developed a CYA model and kept tons of paperwork and evidence. When parents complained or school boards took a closer interest, I always had a defense. What if I weren't paranoid?

The Union negotiates a contract. The contract is held up as a object of derision by many on the right - "Look at this egregious waste of taxpayer money. Look at this contract. We shouldn't allow teachers to quote the contract when they might get into trouble!" What an incredible thought, that a contract shouldn't be followed by both sides. "Gullible Boards sign outrageous contracts!" Actually, they sign a contract that both sides agreed to.

Show me any industry that breaks contracts at a whim. I'll wait.

Everybody complains about the LIFO, salary schedule, no merit pay clauses in the contract. When the dust settles, these clauses are seen as the most manageable. Since the admins are rarely around long enough to figure out the names of the teachers in the building, how are they supposed to know which one is worth keeping? The one who has been there for ten years or the kid just out of college who has no idea of what she's getting into? If you eliminate the salary schedule, then you'll have people like me getting $15,000 signing bonuses and thousands more than the English teachers. Really? Same job. Same experience. Think there won't be any bad feelings? Think the school will run just fine anyway? If you don't have a contract, then I'll demand more money and different working conditions because I know the school is desperate. Maybe I'll train for six weeks and be a dilettante TFA who does the school a favor by my presence and leaves when a real job is available.

The Union works for that contract and makes sure that both sides abide by it.

As for merit pay, this is a fantasy dreamed up by a billionaire. If I get a bonus every year because I suck up to the principal, how long do you think there will cooperation in the department? If I don't trust him to know his ass from a hole in the ground when it comes to hiring and firing, I sure don't want a large fraction of my income held over my head - "maybe you will and maybe you won't."

Far from being the leech on the blood veins of education, unions are the force that makes education work as smoothly as it does.

Bill Gates and Teacher Spaghetti Sauce

Bill Gates is at it again.  For such a smart guy, it is amazing how dense he can be when it comes to education reform. To wit:

We can “flip the curve,” raising performance “without spending a lot more,” if we “measure, develop and reward excellent teaching." ... of all the variables under a school’s control, the single most decisive factor in student achievement is excellent teaching. To flip the curve, we have to identify great teachers, find out what makes them so effective and transfer those skills to others so more students can enjoy top teachers and high achievement.

SO, Billy, we have to identify great teachers and find out what makes them so effective. Then we have to transfer that to other teachers. That will flip the curve.

Hey !  "Find out what makes them so effective"???

If you don't know WHY the great teachers are great, how in Bloomberg's Bloomers can you say definitively that

1. You can identify these great teachers and find out what makes them so effective.
2. These unknowns are transferable.
3. This transference will raise performance “without spending a lot more."

Sometimes life is like judging pasta sauce.

There might not be a best, only a best for you. That teacher might not be the BEST because you can't define best for more than one type of student. There are more types of teachers than there are types of spaghetti sauce. Go watch the Malcolm Gladwell talk at TED.

Academic Xenophobia.

NPR is reporting on school districts who are cracking down on residency issues.

Why?

1. Those parents who are going to go out of their way to "work" the system and exert extra effort to get their kids into your school are statistically more likely to have better motivated students.That's the KIPP difference, the charter difference.  Why not use it? Helping your AYP is nice.
2. It's not like the kid is free-loading. You get the money for every kid who attends regularly and these kids are already going out of their way to get there, so they'll probably have better attendance.
3. The kids are more likely to behave because calling the parents would expose the scam. 
4. District lines are very often arbitrary at best. Gerrymandering the lines doesn't make the bus ride shorter, the school better or the environment safer.

In my mind, I think the best thing would be to make noise publicly. "We're cracking down in the best interests of our community" while making sure that only the troublemakers who try this get caught at it.

Besides, I never quite figured out why it was so important.  We have to educate every kid somewhere and isn't it a compliment that family after family wants to be at YOUR school? I'd be rolling out the red carpet for kids who wanted to come and bragging about it if I could, but maybe that's the private school experience coming out of me.

How do I square this with my stand against vouchers? Because it is a switch between public schools. Most often, vouchers are touted as a transfer of funds from the public school system to a private school and that's the part I don't like.

I can be a member of the local school board (actually have been). I can pick up the phone and directly effect what is going on at the public school in my town. Even as a private citizen, I can walk into the principal's office and ask a question directly - and get an answer (okay, yes, make an appointment and wait a little but you get what I mean). If I am calm, dispassionate, and right, the school will make a change to its policy because I am a taxpaying member of the town and the school is being run by the town for the benefit of any citizen who wishes to take advantage of it.

The Catholic Schools and the private schools won't do that. Why should they?

I do not have a problem with the public - public voucher system. It works well here in Vermont.  I really have a problem when money goes to private institutions which don't follow the same testing and accountability requirements that the rest of the schools do.