GL(s,R)

January 17, 2011

A multivariable calculus list

In addition to my calculus course this semester, I also get to teach a multivariable calculus course with only six students. I’ll start with the standard list for those interested in that sort of thing.

Spring 2011 Multivariable Calculus Standards List

Let me admit something, here, in between two documents—less likely to read in here—about teaching this course, now for the third time: I’m a fraud. That’s right, I’m a fake, a charlatan, an impostor. I’ve created a counterfeit course and hustle the students with a dash of hocus-pocus and a sprinkle of hoodwinking. It is only through mathematical guile that my misrepresentations, chicanery and flim-flam go unnoticed. In short, and in the passing Christmas spirit, I am a humbug. This is a physics course. It should be taught be someone proficient in physics, someone with honed intuition about the geometry of abstract mathematical notions like div, grad, curl and all that, someone who sees everything as an application of Stokes’ theorem and has strong feelings about whether it should be written Stokes’ theorem or Stokes’s theorem. About the only thing I bring to the table is that I can teach students to remember that:

\mathrm{curl}(\mathbf{F}) = \nabla \times \mathbf{F}

and

\mathrm{div}(\mathbf{F}) = \nabla \cdot \mathbf{F}

because curl and cross both start with c, while div and dot both start with d.

Here is the calendar for the course. After it, I’ll explain a little bit of what I’m trying.

Spring 2011 Multivariable Calculus Calendar

There are several big differences here from how I’ve taught this course in the past. First, I am going to try with all my might to get to Stokes’ theorem before the last week. Part of the way I plan to do this is, similar to my calculus class, to cut out most of the stuff on limits and continuity that I usually get bogged down on in the first couple of weeks—am I the only person who finds interesting the pathological examples that make Clairaut’s theorem necessary? I get to teach an extra hour a week to a subset of the class and that stuff will fit perfectly in there. For the science majors, I’m more interested in helping them figure out how to use this stuff and how to develop intuition. Second, I’m skipping Green’s theorem until the end. Yes, it changes the story I normally tell, one that progresses so nicely up the dimension chart, but the trade-off is that I get more time to show them Stokes’ theorem and more time to focus on the physical interpretation.

Speaking of interpretation, you will notice in the calendar  eleven or so ‘Group Activities’. These are stolen from an excellent guide produced by Dray and Manogue at Oregon State as part of their Bridge Project. To work within their framework, I’ve made another structural change that I’d never considered given how I think about the subject. Immediately after finishing triple integration (which, essentially, finishes the first half of the course), we start with vectors (I never start with vectors as most calculus books do) and then I want to get to line integrals and surface integrals as fast as possible. Normally, I mess around with div and curl before getting to integration of vector fields. Instead, I’m going to push out the Divergence theorem—the theorem I always cover in the last 45 minutes of the course—and use this to motivate the definition of div. Then I’ll push out Stokes’ theorem and use this to help motivate the definition of curl. This ought to give me two solid weeks to explore the physical meaning of these theorems as well as to use them to prove some of the standard cool corollaries (like Green’s theorem).

This class will also be the first of my SBG courses to incorporate a final project. If anyone has good suggestions based on experience about how best to incorporate projects into the SBGrading scheme, I would live to hear them. My current system is quite simplistic. The standards for the course are given a 90% weighting for the overall grade—did I mention that midterms and finals now are simply extended assessments whose grades are treated like an arbitrary quiz, just with a lot more standards tested?—and 10% weighting for the project.

January 12, 2011

A Calculus List

Filed under: High Effort/Low Payoff Ideas,Standard Based Grading — Adam Glesser @ 4:19 pm
Tags: ,

One of the advantages of my job is the incredible scheduling. We finished the fall semester the second week of December and my first class of the spring is next Tuesday! The downside is that this gives me way too much time to plot, scheme, doodle, dabble, think, rethink, and overthink. In the end, I usually settle on a plan that is far too ambitious, pedagogically impossible, philosophically suspect, and utterly indefensible. Thus, I bring you my plan for calculus this semester.

I read the wonderful article, Putting Differentials Back Into Calculus, which argues for using differentials in a way closer to their original creation than the way they are employed in modern textbooks.  As a huge fan of Thompson’s Calculus Made Easy, this suggestion didn’t seem half bad. Considering that only a fifth of my students are math majors and that, for the rest, using calculus outside of their physics class is unlikely, why not make things as easy as possible. I’m going to push to teach this same group next fall in calculus II, so the only teacher I can hurt is myself, right? But then I started thinking: the reason the differential approach will work so well with these students is that they will always be using differentiable functions. What reason is there to mention limits and continuity? These are technical issues that won’t help them at all in understanding calculus or how to apply it in their field of inquiry.

Oh dear, so here I am with essentially two months of material (this includes learning to differentiate any elementary function and using this to solve the standard problems). What will I do for the last month and half? I quickly remembered to add Taylor series because I love teaching that in calculus I. Then I added in the obligatory introduction to antiderivatives and integration. I even sprinkled in some partial differentiation at the end so that I could show the students the totally-awesome-implicit-differentiation-trick that would save them five minutes on the final exam. Grr…still two weeks left. These are precisely the two weeks that I usually spend on limits in the beginning. Now I remember why I always do this. It perfectly fills in the semester calendar. And then it hits me.

Any subject can be made repulsive by presenting it bristling with difficulties.
—Silvanus P. Thompson

Limits sure confuse the heck out of students. Why in the world are we leading calculus off with limits, especially to non-math majors? For the purpose of rigor?

You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require rigid demonstrations to be expounded to beginners in the calculus.
—Silvanus P. Thompson

So I decide I just won’t do it.  No limits for us. We will just do some extra exploratory work. There is a great article on math in medicine we could read together. Okay, time for sleep.

But sleep does not come.

Toss.

Turn.

Toss.

Turn.

All right. All right. I’m up.

Why can’t the limits just die? Why do I feel the compulsion to put them back in? They’re like the tell-tale heart beating under my floor. What will become of my math majors if they don’t see limits? No, I can’t do that to them. What a cruel joke to play: send them to real analysis without having used limits. Back in they go. Of course, those biology majors are going to be completely turned off and once you lose them, they’re gone for good. Argh, out they go. On the other hand, if I get audited by the department, they are sure going to ask questions. With my review coming up, I can’t afford that kind of chatter. Put them back in…

…but later. Huh? Put them back in, but later. Yes, of course. Later. The course description says I need to cover limits, but it doesn’t say when! What if we introduced everything in a reasonable way and, only after the students know what is going on and why any of this is important, then showed them those funky limit do-hickies? Hmm. Interesting. And that is my explanation for the following calendar and skills list:

View this document on Scribd
View this document on Scribd

December 4, 2010

Evil SBG

Filed under: High Effort/Low Payoff Ideas,Standard Based Grading — Adam Glesser @ 5:40 pm

It’s been a while and I really ought to do some reflection on my first true semester of SBG. But, alas, the train ride is short and I wanted to throw something out there. I just finished proctoring the Putnam exam and had a wonderful idea for how to make SBG more evil.

You see, there are times when I really want to add weights to my topics. For instance, there is no way that finding the slope, y-intercept, etc. of a line should be worth as much in a precalculus class as graphing rational functions or using the law of cosines. It isn’t that I don’t think it is as important—it is probably more important in many cases—it is just that knowledge of lines (at that level) is considered remedial material, though for some it isn’t.

So how should I weight things? Should I do it by order of (my) perceived importance? How about by difficulty level—again from my point of view? I realized that while I may think I know the correct weightings, it is unlikely that I actually do. Consequently, I came up with this evil scheme. It isn’t really something serious, but I’m not completely joking either.

At the beginning of the semester, inform the students that each topic (or standard if you prefer) will be weighted by an amount inversely proportional—you’ll have to explain what that means—to the final average class score for that topic (or standard). The topic with the lowest class average score would then be weighted the highest. Those topics which everyone figures out would be weighted the least. This would give students (especially your top tier students) an incentive to attack those topics the whole class is failing. Potentially, students could try to game the system by learning the really hard stuff so well that it doesn’t count for so much. Hah! Now wouldn’t that be something?

September 30, 2010

You don’t know it until you know it

Filed under: Classroom Management,Standard Based Grading — Adam Glesser @ 8:36 pm

During my finite mathematics exam today, I had a little bit of time to ponder and I came up with two ideas worth exploring. Most likely, I shouldn’t implement them this semester, but I haven’t ruled it out. For posterity and potential discussion, I offer the following.

Filling the Empty Bucket

As a first approximation to the process of learning course content, imagine an empty box. This does not represent the knowledge of most of our students. Everyone has a few items in their box: some old postcards, two dead AA batteries, a dirty single sock, a couple of credit card applications, etc. The goal throughout the course is to fill that box. On occasion, a student might lose some things in their box after partying with Jack from next door and his cousin Daniels. However, with any reasonable effort, a student can keep the flow going into the box rather than going out. At the end of the course, their grade might simply be the percentage of the box that gets filled.

From a grading point of view, this suggests that my usual way of posting grades has a significant flaw. At any time, a student can see what their grade is, computed as a percentage of the possible points so far. Ah, there is the rub. A student might do well on the first quiz, regress a bit on the second, look at their grade and see 78%. Not great, but not too shabby considering they only need a C to graduate and they didn’t expect to be so close to a B-. Somehow, I’ve already lost. What if, instead of telling them how they’re doing relative to the available points, I gave them their grade in terms of the whole box? If my philosophical position is that their knowledge at the end of the course rather than their intermediate knowledge should determine their grade, then I should be telling them things like, “You have now mastered 21% of the material.”

There is an obvious problem with the original box metaphor: boxes are generally of constant size. It can’t all be about knowledge, though. I want their box to grow. This is not about content standards, but about them learning to think: the application of knowledge in logical and creative ways. How should I reflect that in the gradebook? Or is a one semester course insufficient to judge such growth? These aren’t new questions. David Cox explains things quite well here. I don’t have any new answer yet…but I’m still thinking.

The Exam Enigma

I mentioned in my last post how frustrated I am by the students adherence to the typical college way of studying. I had considered changing the names of exams to something less obvious; perhaps it might fool them. But who I am kidding. If I put it on the schedule, the students will figure it out. Maybe the whole problem is that I separate things in the first place. What if instead of treating these three exams as tests of retention, I simply clumped them in with the assessable standards portion of the grade? Right now, if a student does poorly on the standards, but studies hard and pulls off a good score on the exam, they get two grades: one says they don’t know what they’re doing, the other says they do. That makes no sense to me. I suggest (to myself) that I don’t differentiate quizzes and exams in the gradebook. The 75 minute exam should count no more than a daily quiz, except that the student will see more standards than usual on the test. This remands retention to the final exam.

Does it give the student an incentive to study throughout the term? Maybe it has the opposite effect. Maybe students will think nothing of the 75 minute exam and save up their study power for the final exam. It would be suicide, of course. They certainly won’t be able to cram the material in such a short time. Moreover, under this system, the final exam would be 20% and the standards + exams longer quizzes testing standards would be 70%. Some would still put all of their eggs in the final exam basket (another basket metaphor?), but that person won’t be helped by any system of assessment I try.

A Final Thought (read “Vent”)

I was told today that one of my students was complaining about homework not being assessed. This student is having trouble because he or she isn’t doing any of the homework, nor asking about homework problems, nor writing me emails, nor attending office hours, etc. The student freely admits all of this, but, for not requiring homework, blames me. What is going on when people want to be treated like little children? Nothing is stopping the student from doing the homework problems but himself or herself. “But if I do the homework, how will I know if I got it right?” Aside from asking me, they could just look up the answers in the back of the book. I put up videos on youtube of me working the review problems. I put up links to videos of people explaining each standard. I beg for students to communicate with me outside of class. But it is my fault this student is struggling and because I don’t require him or her to do the homework? Well, sign up one more for the nanny state. Perhaps a class discussion is in order; let the kids doing the work hear from the kids not doing the work and see what they say to them.

September 21, 2010

Collecting Unlike Terms

Filed under: Classroom Management,Standard Based Grading — Adam Glesser @ 8:31 pm

The school is year is in full swing and I’ve been so busy grading that I’ve been neglecting to record my ultra-profound thoughts. As I’ve forgotten those, you’ll have to settle for my ultra-not-so-profound thoughts.

My flavor of SBG is working

One of the downsides of this new approach for me is the amount of grading I’ve thrust upon myself. Between two classes of twenty-three students taking two quizzes a week, I feel like the grading is non-stop. I apologize to all the high school teachers reading this, as they probably consider my grading load a dream; there is an acclimation process, I assume, and I’m still at the front end of it. However, I’m giving feedback and learning a heck of a lot about what the students aren’t doing well. The grade breakdowns (see here, for example) help me identify weak areas (either in my explanations or the background of the students) and I can spend extra time on things. For instance, we are currently covering circles and ellipses (this is my way of introducing function transformations) and one of the things I have the students do is convert the standard form into the center-vertex form. This uses completing the square, a skill I incorrectly assumed they had. I suppose I knew they wouldn’t have that skill (most of my math majors struggle with it), but I guess I just wanted to believe. In this case, I didn’t even have to give a quiz to know that the students needed to spend some quality time working on this skill. I broke the students into groups, threw some problems on the board and walked around giving guidance, but mostly letting them self-teach. This is quite a departure for me as I feel most comfortable lecturing and working examples.

If I didn’t need the quiz, why do I say SBG is working for me? Well, as Cornally might say, SBG ain’t all about SBG. Giving my students feedback has already helped train me. I know better what questions to ask of my students (during lecture) to gauge their comprehension. I don’t have to assess them to find out what they know. This isn’t new for me, but the approach has heightened my sensitivity. Students still aren’t sure of the system and not one has come to remediate during office hours, but I think this is because I promised that each standard will show up on at least two quizzes. However, no one is complaining to the chair or dean, so I’m good for now.

Oral Evaluation is having its ups and downs

If the few students who have commented on my system of lecturing to only three students are an indication, the idea  is a success. I’ve had at least three students volunteer to do it again and one begged me to make an exception to my rule that you can only do it once per third of the course (this rule is a practical one, not an attempt at cruelty). On the other hand, I find two things difficult with the system. First, it slows me down (probably a good thing). Second, I need a lot more practice in asking questions. Too often I ask either the wrong questions or answer the right questions myself. I may be trying to channel Michel Thomas in the classroom, but Michel Thomas I am not. Third, I haven’t found a good rubric for grading it. I take off points for tardiness and being unprepared, but I don’t really have a good feeling for how to translate their performance into a grade. In the end, I’m giving obscenely high grades and think that it is going to act as unintentional grade inflation. So, that’s not so cool.

Teaching to the test is wrong…now let’s see what is on the test

My precalculus course is basically mine to do with as I please, so I don’t have to teach to the test. I can test to the teach…er test to what I taught. My class on finite mathematics, though, is one of twenty or so sections and I don’t get to write the final exam. As such, I feel an obligation to get them prepared to ace that exam, especially since I have a darned good idea what will be on it. So, I throw out meaningful topics to focus on material I know will be on the test. Am I selling out? No. Well, yes. You see, I’m not. But…I am. I don’t like it and I’m having trouble looking at myself in the mirror these days (of course, being bitten by that vampire last week may also be contributing to my mirror problems).

Words of wisdom from a biologist

My buddy and pedagogical sound board from the biology department had this little nugget for me last week:

Sometimes the hardest thing is convincing the students that I can see them.

He explained to me that students are so used to watching TV, which can’t see them back, that they forget a professor can see what they’re doing. This certainly explains why they are so quick to text message, fall asleep, pick their nose, or try to cheat while I’m standing six feet away. A girl in my class was visibly trying to read the whiteboard residue during a quiz while I stood close enough to squeeze out the holy ghost. C’mon. He suggested that I appeal to their experience in band or choir. Those who’ve been involved in such activities will surely understand looking out at the audience and seeing someone paying absolutely no attention the performance.

What’s coming

I’ve decided that I’ve put it off long enough. I’ve held back my favorite tricks of the trade post, well posts, as I didn’t want to it being read by so many people. Now that school has started and readership is down, I think it is safe to release it into the wild. One of my all-time favorite tricks involves integration by parts. I have a pretty long post on the subject that I’ll break up into three or four posts. The end result is pretty cool and as close to original as I get. I also still plan on putting up my multiplication songs, I just need to figure out when to record them (the baby now sleeps in my office and that might be problematic).

Take care all.

August 5, 2010

Giving up?

Filed under: Standard Based Grading — Adam Glesser @ 10:32 am

I can’t find the words to help you all see
Every story I tells makes sense only to me
I can’t give you reasons to learn what I teach
Why should you practice all that I preach?
It’s bothersome, I know, to sit through my blather
Your life is complete, without me, I gather
No need to make jokes or listen halfway
Just forget I was here, as I slink away.

So, sometimes I think I really know what I’m doing; then class starts. Did I mention yet how much respect I’m gaining for high school teachers this summer? I always thought I respected them; I always thought they had a hard job; I always thought they were underpaid, overworked, under-appreciated,  over-controlled, underrated and overwhelmed. Hah! I had no idea. Frankly, I still have no idea. I’m teaching one class of twenty high school students for seventy-five minutes, twice per week. I don’t know anything close to what you high school teachers go through, but I’m sure now it isn’t as easy as teaching college students.

It’ s been a rough go this summer. The administration recently asked that I require homework to be completed. Naturally, I balked and, to their credit, they listened to my arguments and, I think, even bought into some of them. They still want me to require homework to be submitted, but they are not insisting that I assess it. I told them that I doubt it will have the desired affect (to ease the load on the student mentors who try to convince the students to do their work) and that the disinterested students will, at best, do just enough to avoid getting in trouble. Probably, the administrators know I’m right, but they’re in a rough situation: they aren’t in charge of all the details of the program, but their necks are on the line if things go bad.

On the plus side, the grades are going up. Under my old system, most of the students would be failing and there would be no way for them to catch up. Their recent high scores would be outweighed by weeks of low scores. Now, as they finally are starting to get how to calculate the slope of a line or to complete the square, their past ignorance is forgotten. They are happy; I am happy. They know something they didn’t know; I know they know something they didn’t know.  It’s a win-win and that is pretty cool.

So, no. I guess I’m not giving up. It’s only math, baby!

P.S. To the loyal reader who periodically writes me personally: Please post your wonderfully insightful remarks in the comments section for all to read. To answer only one of your questions: No, I hadn’t considered taking a writing class. Do you intend your question to suggest that I need to?

August 3, 2010

5

Filed under: Standard Based Grading — Adam Glesser @ 2:21 pm

Pardon the interruption. It appears that teaching two summer classes, homeschooling my 5-year old, witnessing the birth of my third son, and trying to beat the New Super Mario Bros. Wii game was just too much for me to write the blog the last month. But, I’m not giving up on this thing. I probably would have if not for a rather nice comment a while back from Kate Nowak. No, I am finding that I need to vent a little and get some advice; this place is ideal.

The precalculus course is going quite well. There are only six students, four of whom attend every day, and while comprehension is not at the level I want, I at least feel good about the philosophical direction of the course. My students know a lot of calculus. I don’t mean things like using the product rule or chain rule, though. I mean that they can look at a curve and estimate the derivative. They understand why the derivative will be useful in finding places where a graph will turn around. I could put them in a university physics course (calculus based) and they would not struggle with the math. That is the good news. The bad news is that all of you who said I needed to cut hyperbolic functions were right. There just isn’t the time. Every topic I cover, I have to cover twice and that really eats up real estate on the class calendar. I also will need to drop sequences and series as well as Riemann sums and complex numbers.  This was all gravy material, but it is so fun I really wanted to get to it. Perhaps, next time!

My other class is a high school level course that has been somewhat disastrous. This was expected. The level of the students is incredibly diverse (some are going into calculus and others struggle to subtract whole numbers). Somehow, I’m supposed to teach material they all can learn. This wouldn’t be a problem, except it is also supposed to essentially be precalculus related material. This is the second year I’ve done this course and I still don’t know what the right material is. I do know that I’ve lost the least advanced students and I’m staying after class to teach the most advanced students calculus. Long story short: All you high school teachers out there are simply amazing.

Is there a good side? Well, I have implemented SBG in both courses and I’m thrilled with that. It is liberating to tell a student, “You don’t need to give me an excuse for missing the quiz.” My current policy is to test every standard at least twice. I haven’t implemented the approach where I make the first iteration out of 4 and second (more difficult problem) out of 5. Instead, they are all out of 5. If a student scores lower the second time around I lower it according to the following rule: Let s_i denote the score on the i^{th} assessment and let s_c denote the composite score on assessments s_1, \ldots ,s_{i-1}. The new score for the students is s_i if this is greater or equal to s_c. If s_i is less than s_c, then I will drop their score (by 1) only if s_i is at least two less than s_c. There is one exception to this rule, namely if s_c = 5, then any lower score will drop s_c by 1. I’ll explain my rationale for these if anyone asks, and I’ll consider other options if anyone cares to suggest any.

Okay, this is a good place to stop. I am now the proud owner of a netbook computer, so I might be able to write these while on the train. Hopefully, we’re back in business now. Next time, I’ll tell you about some recent resistance to my homework optional policy.

All the best.

July 5, 2010

The Calculus Carrot

Last week in Lausanne, I was watching a talk that started with some standard material on finite groups of Lie type. There is this typical construction where you start with a connected reductive group \mathbf{G} over the field with q elements (where q is a power of a prime p) and the Frobenius endomorphism F: \mathbf{G} \to \mathbf{G}. The fixed points G = \mathbf{G}^F form a finite reductive group and this construction creates a link between algebraic groups and algebraic geometry.

Presumably, most of my readers have no idea what these things are, but don’t worry, they aren’t the point. Here is my problem: to understand algebraic groups, I need to understand enough algebraic geometry to utilize the above construction, and so I have some motivation to learn algebraic geometry. On the other hand, to motivate certain necessary things in algebraic geometry, it would be awfully nice to understand some of the theory of algebraic groups. Can you see my problem? I want to learn both and to use the knowledge of one to motivate the other. But I know neither!

Shawn discussed the use of analogies in teaching yesterday and I whole heartedly agree with him. What do you do, however, when you are trying to motivate a subject by comparing it to a subject the students will only understand once they comprehend the subject that you’re trying to teach them?

There is an easy solution to this problem: use a different analogy!

Sure. But I’m teaching precalculus and I’m pretty sure that the material is best motivated by calculus, the precise subject I’m supposed to be preparing the students for. So, I’m thinking right now that we should abolish precalculus and make calculus a two-year course. Come to think of it, let’s make calculus an n-year course (for n \approx 10) and start showing our students calculus when they are 7 or 8. Don Cohen does it! At least then we could introduce most of the mathematics we teach in context instead of pulling out of the void definitions and results whose purpose is to build up yet more definitions and results, most of which the students consider to be as enjoyable as a rattlesnake bite, but without the fun of asking your friends to suck out the poison. Yes, this is the way to do it.

Ah, but I don’t get to teach it like that. I have to teach precalculus and I only get one semester (six weeks, actually, as it is summer school). Fine.

Stream of consciousness (with apologies to James Joyce)

Maybe all is not lost what if I just start teaching it like it is calculus what if I start with the problem of describing the average velocity of an object and then build up the formulas and interpretation and then work towards instantaneous velocity then I could talk about how this motivates the need to understand lines really well or at least the point-slope form of a line I then need to transition to giving lots of examples of functions that occur naturally maybe even getting the students to provide me with examples I think I definitely need to get hyperbolic functions in they always seem to get the shaft but I love those cool things and the way they connect to exponentials is really awesome did you ever notice that awesome might mean inspiring some awe so that if you wanted something to be full of awe it should be awful hah I like that I’ll have to use that when some student does really well okay back to what I was doing so yeah hyperbolic functions they are connected to exponentials and so are trig functions I wonder if I can convince them of that and if they will believe that the exponential function is pretty important I don’t think so it is all a bit too much catnip for mathematicians we need to get back to throwing balls into garbage cans and other things that suck students in and get them invested in the answer I still like the structure here though start with the problem to be determined later and add layers that require the standards but get the students to request the layers and the standards for the second part of the course on sequences and series I’ll introduce the area problem and use that at the end of the course could it be that these students will be ready for calculus that is probably too much to ask but I can dream right

The Mind Map

I don’t understand how the mind map thing really works; I should probably read one of the books on the subject. Here, anyway, is what I did in about 20 minutes. It feels like a mess and doesn’t seem to capture what I’m trying to do.

In fact, the problem is that there is just way too much in there. For instance. when covering linear equations, do I really need to do more than slope and point-slope form at the beginning? Sure all the rest is useful, but only when you get to a specific problem. And why do we need all this garbage about quadratics? Other than root finding, we certainly won’t use it in calculus. On the other side of the fence, I think introducing the binomial theorem might be a waste of time. When I taught calculus, I used it just once: to prove the power rule for arbitrary natural number powers. Just a guess, but I don’t think anyone followed and only one student (hi, Ben!) probably remembers that we used it. The whole sum of n squares, cubes, etc. has to go. I better not play this game too long or there will be nothing left. It is just that I feel the need to be overly cautious: don’t include material that I find appealing; include material that is suited to the task ahead.

The Standards List

I’ve put it off long enough. Here is my first draft of a standards list. I’ve left off some things that are in the mind map because I’m trying suppress stuff they should have seen (like the distance formula) and which I’ll thus probably talk about but not assess (oh that already sounds dangerous to me). Reading this list a couple of times makes me feel rather nauseous. If you’d rather see a copy of the spreadsheet I made for this, click here.

  1. Compute the average rate of change of a function
  2. Interpret the average rate of change of a function
  3. Interpret the instantaneous rate of change of a function
  4. Describe the connection between the average rate of change and instantaneous rate of change of a function
  5. Slope of a line
  6. Point-slope form
  7. Function identification by graph: linear
  8. Function identification by graph: quadratic
  9. Function identification by graph: rational function
  10. Function identification by graph: exponential function
  11. Function identification by graph: logarithmic function
  12. Function identification by graph: trigonometric functions
  13. Sketch graph by type: linear
  14. Sketch graph by type: quadratic
  15. Sketch graph by type: exponential function
  16. Sketch graph by type: logarithmic
  17. Sketch graph by type: trigonometric
  18. Sketch piecewise defined functions
  19. Explain (intuitively) continuity
  20. Test continuity
  21. Find the effect on a function of a horizontal shift to the graph of the function
  22. Find the effect on a function of a vertical shift to the graph of the function
  23. Determine the equation of a circle given the radius and center
  24. Determine the equation of an ellipse given the center, vertex and co-vertex
  25. Describe radian measure in terms of the circumference of the unit circle
  26. Convert radians to degrees
  27. Convert degrees to radians
  28. Use the coordinates of the unit circle to find sine and cosine
  29. Use the unit circle to determine the sign of sine and cosine
  30. Use the unit circle to show that sin^2(x) + cos^2(x) = 1
  31. Use the unit circle to show sine is odd and cosine is even
  32. Compute sine and cosine for the 16 standard angles
  33. Convert rectangular coordinates to polar coordinates
  34. Convert polar coordinates to rectangular coordinates
  35. Use the law of sines
  36. Use the law of cosines
  37. Use sum formulas for sine and cosine
  38. Use product formulas for sine and cosine
  39. Use half-angle formulas for sine and cosine
  40. Solve equations with trig functions
  41. Write periodic solutions to equations with trig functions
  42. Graph sine, cosine and tangent (base functions)
  43. Graph horizontal and vertical shifts of trig functions
  44. Graph horizontal and vertical reflections of trig functions
  45. Graph horizontal and vertical stretches of trig functions
  46. Describe the effect of reflections and stretches on an arbitrary function
  47. Find asymptotes of a rational function
  48. Find the sign graph of a rational function
  49. Graph a rational function
  50. Distinguish between a sequence and a series
  51. Compute geometric series
  52. Describe a Riemann sum as a series (intuitive)
  53. Describe exponential growth in terms of exponential functions
  54. Describe the rate of change of a polynomial as x increases versus the rate of change of an exponential as x increases
  55. Define hyperbolic trig functions in terms of exponentials
  56. Graph hyperbolic trig functions
  57. Determine if a function has an inverse
  58. Compute inverse trig functions on standard values
  59. Compute logarithms
  60. Use logarithm properties to compute logarithms
  61. Perform arithmetic on complex numbers
  62. Explain connection between trig functions and exponential functions via de Moivre’s formula

HELP!!!

June 10, 2010

Story Telling

Filed under: Classroom Management,Standard Based Grading — Adam Glesser @ 2:43 pm
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I wrote, in what is now (thanks to Jason Buell) the most famous of my small collection of blog posts, that:

Our students read textbooks, refined over the years to be ruthless, efficient and deadly. The story is missing, the context is missing and the connections are missing. The textbooks are a reference, not a teacher. It is then the teacher’s responsibility to add the missing ingredients, to tell the story, to explain how experts actually think about these things and, most importantly, to teach the students how to read (or understand) a subject non-linearly.

Recently, my attention was brought to three excellent posts on story telling by Grace ChenDan Meyer and Dan Meyer (man, what are the odds those last two guys would have the same name!), all of which you should read before this post. In fact, if in the course of reading those three you get so tired or inspired as to not want to read my follow-up, then consider yourself suitably enlightened and come back later; what I’ll write certainly isn’t earth-shaking or particularly useful (nor will I be brief). However, I will tell stories and I will use bold fonted section names that arbitrarily oscillate between my voice and the voice of the reader held hostage in my head.

Part 1

8-Bit Education

At some point, my wife and I decided we would homeschool our children. Thankfully, we’ve had a good deal of time to prepare; deciding between the plethora of homeschooling structures and then choosing the specific curricula is a daunting task. We finally settled on the classical method popularized recently in The Well Trained Mind. Part of this program is to break up the child’s education into three phases: grammar, logic and rhetoric, roughly corresponding to 1st-4th, 5th-8th and 9th-12th grades. Within each phase, you study world history from the ancients to the present. So as to be practical, you probably don’t start at the beginning of history, but at around 5000 B.C.

While my oldest son is only four and so a little young to start the world history curriculum, he is old enough to begin his video game curriculum (and if you doubt the importance of this aspect of learning, please do watch Jane McGonigal’s informative TED talk). Sure, I could start him on modern gaming systems (he does enjoy Wii Sports quite a bit), perhaps spend $50 on Super Mario Galaxy 2, but it really wouldn’t give him a full picture of the gamer universe, now would it?

The answer: start at the beginning. Well, not really. Much like ancient history, we don’t want to start too early. Although I did enjoy playing games on the Commodore 64, the TRS-80 Color Computer and the Atari 2600, to me gaming as a civilized hobby begins with the Nintendo Entertainment System. Naturally, we started with the original Super Mario Bros. The exploits of plumber brothers Mario and Luigi quickly enamored my son, Alex. Soon, we were watching episodes of the Super Mario Bros.  Super Show! and now both kids want a Mario themed birthday party. Good work, Dad. During one episode, there is a trailer for the Legend of Zelda animated series. My kids were hooked and Alex wanted to play the game.

Your 4-year old is playing Legend of Zelda?

No, not really. He sits on my lap while I play. So here we are, a few weeks later, just one level away from entering Death Mountain to fight the evil wizard Ganon for the Triforce of Power and control of Hyrule. I remember playing it as a kid and it took months for me to unlock all of the hidden loot, destroy all of the bosses, get the magic sword, figure out the NWSW maze trick, etc. It is a wonderful game that kept my friends and I occupied for a long time. Ah, but now I am older and the age of the interweb is upon us! I don’t have the time to search through Hyrule for everything again. No, I simply googl’d “Legend of Zelda walk-through” and found this delightful site which not only explains an efficient way to work through the game, but also has videos of the author doing it.

Isn’t that cheating?

Well, of course it is cheating.

So, you’re teaching your son to cheat?

Wha-buh-guh!!! Yes, that is the point. If studying math has taught me anything, it is why struggle to learn things when others have struggled before? Stand on the shoulders of giants and pick the (now) low-hanging fruit. It took people thousands of years to come up with modern mathematics; why reinvent the wheel?

…seriously?

Blow the Whistle

Part 2

Is it better to know a few things really well or a lot of things reasonably well? I suppose it depends on what you want to do with your time. However, either is preferable to not knowing anything at all. Here is an experiment:

  1. Repeat the number 8 one hundred times in a row to somebody. At the end (assuming they didn’t leave or punch you in the nose) ask them what number you were saying. Chances of success: extremely high.
  2. Now ask (for your sake) a different person after repeating 38502 twenty times in a row (still 100 numbers spoken), what five numbers you were saying. Chances of success: high.
  3. Find a third person and tell them any random string of one hundred numbers and ask them to repeat it. Chances of success: essentially zero.
  4. Ask them about the first number of the sequence. Chances of success: low.
  5. Ask them about the first five numbers of the sequence. Chances of success: essentially zero.

So what?

So what? This is what our math curriculum feels like sometimes: a seemingly random ordered sequence of topics with often arbitrary repetitions that leave an unmotivated student with, essentially, zero chance of success. Our smarter students can see through the games and learn to hate math; our weaker students just learn to hate themselves.

Standard Based Grading to the rescue!

No! SBG isn’t a panacea. It is like watching sports in high-def: yes, you get to see Tom Brady’s nose hair, but a 42-10 rout still isn’t much fun to watch. The problem is with the game, itself. At several points, standards have come down from various organizations as if they were commandments from Mount Sinai. At best, I gather that someone took a reasonable modern calculus text and backtracked to decide what and when students should learn. Of course, one can’t simply do that. This wouldn’t take into account current practice, teaching training, etc., so you would have to create standards that compromised a bit to deal with reality. Does this make a good story? Hardly. The student is told: follow these instructions to the promised land. The student hears: Wa wa wa wa. The student learns: Nada Nada Nada.

A Tale of Two Stories

Much like the Star Wars Trilogy (x2) where there is an overarching storyline as well as individual episode with subplots, teachers also need to be aware of the overall story of mathematical education along with the day-to-day practice of teaching concepts, methods and algorithms. While the posts of Grace and Dan are geared at addressing the second part, the first part is equally important.

Currently, we teach from a walk-through. There is no sense of exploration and no sense of importance. Everything is deemed important except for those things that are difficult to test. While I think of mathematics as a subject that should bring order, we teach nothing but chaos. Ask a high-school student what precalculus is about. Hah! You might as well ask them what the dictionary is about. It seems we’re afraid to make the difficult decisions, afraid to cut down the quantity for fear that they’ll need it somehow, somewhere, sometime.

But here is the funny thing…

At some point, a student will ask their teacher: what is this all good for? One variation of the answer is to say: while this specific topic may not ever be relevant to you, the skills you learn from the process are. Okay, so some of us think that mathematics helps develop critical thinking skills and that the material is simply a catalyst. On the one hand, we’re almost willing to concede that some specific things they learn probably aren’t relevant to their lives, but, on the other hand, are terrified that they might worry about how awful it would be if we didn’t teach it. But, if mathematics really does help develop critical thinking skills, then it really shouldn’t matter if we skip some material: they can learn it later if they need it.

Part 3

I play a lot of board games and one of my favorites is Race for the Galaxy. Really, I love hand-management type games. There is nothing more frustrating than having to choose between two awesome cards, knowing you’ll have discard the one you don’t choose. But, the decisions must be made and the game will go on. Educators in charge of curriculum design might first play these games and get used to making these types of choices. Imagine if you could only teach a student three things. What would they be? Obviously, your subject and level dictates this choice a bit. You wouldn’t teach multiplication to students who already know calculus, for instance. As someone who trains mathematics majors (many of whom will become mathematics teachers), I choose the following three facts (along with the appropriate discussion of what those facts mean):

  1. 1 is not a prime number
  2. 0 is an even number
  3. Derivatives describe rates of change

If you let me teach math majors about just the first two, I think I could put together a pretty solid class; the third is gravy.

Where is the story?

The story begins with a choice: what do my students need to learn. Right now, there are too many characters and not enough character development; too many storylines and too few stories; too much mathematics and too little opportunity to appreciate mathematics. Throw out what you would like them to learn or what you think future teachers would want them to know. Focus on the essentials. Now teach it, tell it, break it down, build it up, grade it, shake it, bake it, use it, love it, leave it, come back to it and, finally, they will know it. Let the students explore and let the students struggle. Don’t help them, help them, undermine them, create doubt, create certainty, destroy the certainty, add characters, conflict, irresolution, resolution, chaos, and, then, order. Every class should have an answer to the question: What is this class about? Help them answer this question.

But, I can’t throw out the national standards.

Yeah, I know. This is where Standard Based Grading comes to the rescue. You see, SBG is a panacea…

Hold on a sec…

Wait! I’m on a roll here. Set your own standards, find the most important things you could teach a student. Teach it; teach it well. And then when they come and ask why your students don’t know about topic xyz, tell them, “I forgot.”

???

It’s Steve Martin, damn it.

???

All I’m saying is that once we teach our students the core ideas about how to think about the world in a mathematical way (read “logically creative way” not “creatively logical way”), then and only then does it make sense to start adding things back in. The standards are the problem. But you have to teach them the standards. But the standards are making it harder to teach and harder to learn. But you have to teach them the standards. But teaching them the standards is, in most cases, tantamount to making sure they will not learn any real mathematics. But you have to teach them the standards. I give up. Teach the standards. The story sucks, but so do most movies.

Dénouement

I apologize for wasting your time. I don’t have any practical ideas. I’m lucky: I can get away, in many of my classes, with doing my own thing. If you’re teaching middle or high school, you are accountable to so many people that trying something like what I’m suggesting is probably career suicide. Don’t do it. If you’re reading this, you’re already too good a teacher to waste, especially on this rubbish. Maybe I’ll get back to reading Dan Meyer instead of trying to write like Dan Meyer.

Man, I started this post feeling so enlightened. Now, I feel…defeated.


May 27, 2010

Stopped Before It Started?

Filed under: Standard Based Grading — Adam Glesser @ 5:13 am

A couple of weeks ago, the course coordinator for finite mathematics (one of the courses I’m teaching in the fall and the first class for which I’m preparing standards) came to discuss my ideas. The concern is that I would, potentially, be giving so many make-up quizzes. Aside from the extra work and stress it would put on me, the main issue is that it is fairly standard here for professors to have a “No make-up quiz” policy (independent of other faculty, this has always been my policy, as well) and mightn’t students in other sections of the course become hostile to their instructors upon learning that they can’t re-assess content while their peers in my course can?

The Unspoken Bit

This next year will be my third year at Suffolk and that marks a key point in my academic career—the 3-year tenure review. The last thing I want is to jeopardize tenure by irritating faculty, especially those faculty who have supported me and are simply trying to avoid disasters.

The Spoken Bit

Another faculty member supported my system, arguing to the course coordinator that as long as I didn’t use class time for re-assessment, then my system isn’t a problem. I argued that there is still the issue of other faculty being affected by my cavalier ways, but he was not impressed.

The Tricky Bit

I think they’re both right. If other students react by challenging their teachers, it might turn those classrooms into nightmares to teach. On the other hand, some of those classrooms may already be nightmares to teach; I should have the authority to design my course around demonstrably solid academic principles and it isn’t clear to me that the other students will respond so negatively to their teachers.

“…and to make a long story short—TOO LATE”

I’m not sure what to do. I want the best for me and my students, but not at the cost of making the lives of twenty or so adjunct faculty extremely difficult. Ideas, anyone?

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