# GL(s,R)

## 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.

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.

## September 6, 2010

### Cleaning Out the Gutters

Filed under: Classroom Management,High Effort/Low Payoff Ideas — Adam Glesser @ 11:26 pm

School is starting this week. Hooray!!!

No more summer
Here comes class
Time to get up
Off my chair

The syllabi are written. The schedules are finalized. The students are being informed of their impending doom. Mmmm, it’s good to be alive.

Let’s see, what can I offer you today? First, in case you aren’t checking out the awesome SBGBeginners Wikispace, here is a link to my (always in flux) standards list for precalculus. I would post my standards list for Topics in Finite Mathematics (and maybe I will still), but  I doubt there are too many people teaching a course covering precisely what this course covers. Nonetheless, I should tell you a bit about my experience with producing the grade distribution for that course.

In an effort to waste time, I collected links to videos targeting the standards for my classes (see here for precalculus and here for finite mathematics). Originally, I was going to record the videos myself, but even I can’t justify spending that much time right now on duplicating previous efforts. In the process of creating these lists, I started hanging around on the Art of Problem Solving website. This reminded me of all the time I spent taking the California Math League contests in high school. Man, those were fun. And…

Homeschooling Connection

As I’ve mentioned before, my wife and I are starting homeschooling with our five-year old this fall. Actually, since I happen to believe that

1. Summer vacation should be a time to do things you enjoy
2. School is something you should enjoy

we started a proper curriculum in April and went all summer, forgetting to tell him that he shouldn’t want to learn over the summer. (Mwoo hah hah!!! Kids are so gullible.) So, we’ve covered addition, subtraction and multiplication and are just starting with division. Incidentally, division is the first operation that I think my son finds practical. At the store today, he was constantly telling me how much each member in the family would get of the things we were buying. Anyhow, I was thinking to myself that knowing the four basic operations at his age is pretty good and just think how far I can take him over the next few years. But then, I thought to myself how my biggest complaint with my students is how they are so weak at the basics. Even those who can perform calculations efficiently, can rarely apply that math to solve real-world problems or interesting abstract problems. What if I got my son up to the level of understanding the basic vocabulary and then, instead of striving for breadth, went for depth? What if I started feeding the kid competition style problems and applied problems, not with the goal of getting him into the Math Olympiad, but rather aiming towards mastery of the foundations of arithmetic and mathematical thinking? The idea gives me goosebumps…oh, wait, the air conditioning was set to 63.

One last bit: I made up a few songs to help my son learn the multiplication tables and I think I’ll record them and post them on the blog for posterity. I searched hard for such songs and hated just about everything I found. Mine aren’t really any better, but they don’t annoy me as much and my son sings them all the time.

There once was a teacher who dared
To teach as if all students cared
But his methods all stunk
‘Til he read Think Thank Thunk
Now his class is no longer impaired

Happy New School Year!!!

## August 30, 2010

### Speaching with Toonerisms

Filed under: Classroom Management — Adam Glesser @ 9:46 pm
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Although I only recently learned that constructions like the title of this blog have a name (see Spoonerism), I have used such things in my every day repertoire for years (and you all wondered how I ended up with a such a hot wife). Given my propensity for squeaking pickly, I find it to be an ace way of ‘accidentally’ saying something off-color with a built-in ready-made excuse.

Naturally, I started trying them out on my kids while I read Dr. Seuss. They quickly picked up on my slongues of the tip and still revel in pointing out my errors, “No, Daddy. He is writing in anapestic tetrameter, not tetrapestic anameter.” By randomly throwing these (and other mistakes) into my reading, it keeps them engaged and questioning my sanity (If you aren’t a parent, you will just have to take my word for it how valuable it is for your children to believe you might be insane; we’re not talking Jack Nicholson in The Shining, but more like Richard Dreyfuss in What About Bob?).

[A caveat: I tend not to do this on the first, second or third reading of a book. I want them to have familiarity with what is right, before I start trying to trick them with what is wrong.]

I tried this during my precalculus class this summer. It wasn’t by design; it just sorta happened. As soon as I had done it, I got excited to see what they would say. Predictably, they said nothing. Even when I paused for ten seconds and gave them the patented You only think you know what I just said glare, nary a glimpse of recognition from the crowd. End of story, right? Another failed experiment, no?

The Pay-Off

I think there is something to this. Certainly, a class needs to be conditioned to listen for it, but this is simply a matter of repetition. Maybe in the first week you pump the prime a little bit—except in a course on number theory, where you will pimp the prime—and then you have a wonderful way to toss in random jokes, create inside humor and find out who is really paying attention all in one swell foop.

I don’t think one should build a lesson plan around this; it should be just another tool in your bag of tricks for keeping from jumping those on ledge staring down at the peacefulness of white noise.

Now off for some exercise.

## August 12, 2010

### The Final

Filed under: Classroom Management — Adam Glesser @ 11:57 am

Today is the end of my summer precalculus course. In fact, my students are struggling through the final as we speak. It took some effort, but I convinced them to read through the entire test first and to start with the easiest problem. I asked them to circle the first problem on which they work, so I should have that to interest me while I’m grading. I helped them out by making the first three problems graphing problems (which students seem to abhor, at least in my classes).

Ho Hum

But I don’t want to talk about my final. I want to talk about my high school class (you remember, it is the one that drove me to write that self-serving melodramatic structured paragraph with a rhyming pattern that passes for poetry in my warped mind). Anyway, I showed up today without a quiz. I gave them a homework assignment, but there was no formal assessment.

What?!?

Yep, that was their reaction. One student blurted out, “How come every time I study, you don’t give a quiz?” I told him that this was the second time I hadn’t given a quiz (the other was the first day of class) and so I take it to mean that this was the only time he had studied.

Student: But I know this stuff, now.

AG: Great! I can’t wait for you to show me.

Student: Can I show you now?

AG: We’re going to cover some new material now, but after that you may show me, or you can wait for the next quiz on Tuesday.

Student: (mumbles for a little) But sir, I got it.

I’m silently thinking to myself during this whole conversation that I am at the doors to math teacher heaven. This guy is not one of the stronger students and has shown some, but not much, interest. Now, he is begging me to test him, to show off what he knows. He is seeking validation and I am the only one he knows that can give it. I don’t know what I did to deserve it, but I’ve got it and I don’t want to let it go.

And then…?

And then I did. I turned a boring lecture into a long boring lecture. He never did get to show me what he knows (there is always next Tuesday, but let’s face it, that might as well be next year). It is as if someone succeeds after spending hours trying to light a fire in the wilderness and I walk by moments later and blow it out, remarking, “If this got much bigger, we might have a forest fire on our hands.”

I cheated the student out of his glory, his satisfaction, his reward for a job well done: I stole his enthusiasm by making it subservient to my purpose.

Dude. That sucks! What is wrong with you?

I don’t know, man. I don’t know.

## June 10, 2010

### Story Telling

Filed under: Classroom Management,Standard Based Grading — Adam Glesser @ 2:43 pm
Tags:

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 30, 2010

### Shortcuts, Mnemonics and Why They Ain’t So Bad

Filed under: Classroom Management — Adam Glesser @ 4:20 pm
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In my previous post, a commenter bemoaned the use of shortcuts in adding fractions. This isn’t the first time I’ve read a person criticize the use of a shortcut technique or mnemonic. Very often, their point is that if a student learns only the shortcut and they don’t understand why it works or what they’re doing, what is the point? This seems reasonable to me. However, there is a difference between knowing how something works and knowing how best to do it. For instance, I spent a great deal of time studying Sudoku solving techniques (and we’re not talking about the kind of puzzle you find in a Will Shortz book; we’re talking about techniques for puzzles you would never solve with pattern-based techniques unless you were reading this). My wife, meanwhile, would work the easy and intermediate level puzzles and occasionally something harder. Put us in a competition: she would blow my socks off because while I’d be trying hard to find these nice patterns, she would be plugging in a number and working it until she solved the puzzle or found an error. The latter technique is what Sudoku enthusiasts call T&E (trial & error) and is frowned upon, except in competition where it is the method of choice for the tough puzzles because it is ridiculously fast. Dan Meyer made a similar observation in the context of solving systems of linear equations (or, more precisely, in not solving them when guess-and-check is easier).

So, a big difference between my examples and the usual shortcuts is that we aren’t usually telling our students to avoid the standard algorithm by guessing; instead, we are suggesting a different algorithm. Nonetheless, the point is the same. Why have a student use a slow method when a faster one is available? Is it just so that we can lie say to ourselves that they really understand what is going on? Anyway, my feeling is that the existence of these shortcut methods is not as important as looking for them. Consider Kate Nowak’s recent post about a student suggesting a shortcut method. Even though the method was dead wrong, the subsequent discussion is extremely valuable.

By the way, if you’re into shortcut mental arithmetic and beyond (can you tell that I am?), there is a fantastic book called Dead Reckoning that goes (far) beyond the methods of every other book on mental math I’ve ever read. We’re not just talking about multiplication and division tricks, but also ways of estimating roots, logarithms and trig functions.

What about mnemonics? These don’t give you a faster way to compute things, simply a way to remember them. Shouldn’t we get our students to remember things because they make sense that way instead of using some bizarre story, silly rhyme or reference to a non-existent indian chief? Ideally, yes. In practice, mathematics isn’t designed to facilitate this process. Our notations and naming conventions aren’t perfect and even when they’re pretty good, it doesn’t always help the student (does the word scalene really tell you anything if you don’t have a classical education that includes Greek?). No, we often need these mnemonics until the meaning becomes internalized. For students who don’t ever need to use the term again (and, thus, won’t have the required repetitions to internalize), the mnemonic may be the only thing they remember. I can’t remember much French from high school, but I do remember the verbs that are conjugated with être in the passé composé. Why? Because I learned a song to remember them.

Look, I’m not trying to say we should avoid teaching meaning or attempting to get those messages across. However, I think we are going about it the wrong way when we fail to account for the way our students learn and remember. This next week I’ll go deeper into this topic with a look at WCYDWT and storytelling.

P.S. Thanks to my loyal subscribers for making my first week of posting a success. I appreciate all of the comments and helpful pieces of advice.

## May 25, 2010

### The Inclusion-Exclusion Principle

Filed under: Classroom Management — Adam Glesser @ 5:59 am
Tags: ,

During my first semester at Suffolk, I noticed there were several Spanish-speaking students in my classes. As my position wasn’t permanent, I figured it would be useful to add “Bilingual” to my CV and that I would have several students around who would be more than happy to feel intellectually superior to me.  The good news is that a year and a half later, I still enjoy studying Spanish; the bad news is that those students still feel intellectually superior to me.

Michel Thomas and Me

The best system I’ve used for learning Spanish (and I’ve tried a lot of them) is the Michel Thomas method. What make his tapes unique is that the recordings are of him teaching two people. You get to hear their (often incorrect) answers and how he responds to them. His style is a wonderful combination of humor, clever mnemonics and carefully planned repetition that constantly adds just a bit more to what the student already knows. With other programs, I would get bored or weary after 20 minutes, but with Michel Thomas, an hour goes by and I whisper, “Tell me more.”

The Michel Thomas Method and the Classroom

Here is the revolutionary, groundbreaking, innovative idea (that I stole from someone, but I can’t remember who) that I’ve come up with all on my own this summer: Stop teaching 25 kids at once. My plan is to take three students from the class and only teach to them. By this, I mean that they will sit at the front of the room and I will teach them. My usual style is to lecture to the class, but no more. Now, I will build the subject in tiny increments, continually asking my three students questions and assessing their understanding. A typical exchange might be:

AG (looking at all 3): Consider this plot:

(to Student 1) What is the y-coordinate of point A?

S1: 1

AG: (to Student 2) And what is y-coordinate of point B?

S2: 4

AG: (to Student 3) How much did the y-coordinate change?

S3: By 3.

AG: Good. How much did it change if I ask you first about point B and then about point A?

S3: Still by 3.

AG: Yes, but it is a curious thing here. From A to B, the y-coordinate goes up and from B to A, it goes down. We don’t want to lose that distinction. So we agree that if it goes up, we call it a positive change; if it goes down, we call it a negative change.

(to Student 1) When we go from point A to point B is the change positive or negative?

S1: Positive.

AG: (to Student 2) And from point B to point A?

S2: Negative.

AG: (to Student 2, again) How about in this picture—again from point B to point A?

S2: The change is negative.

AG: From point A to point B, the change in the y-coordinate is negative. What about from point B to point A?

S2: From point B to point A? Oh, yes, backward. It is positive.

AG: Now the way we will write the change in the y-coordinate is to use a Greek letter $\Delta$ (writes it on the board). The change in y is written $\Delta y$ (writes it on the board).

S1: But how do you know if you’re going from point A to point B or the other way around?

AG: You need to specify. Let’s make an agreement, though. If I don’t say anything, we’ll mean left-to-right, the same way we read.

(to Student 1) What, then, is $\Delta y$ in the first picture?

S1: 3

AG: (to Student 3) And what is it in the second picture?

S3: 2

AG: Are you going left-to-right or right-to-left?

S3: You didn’t say which, so I’m going left-to-right.

AG: And does it go up or down?

S3: Down.

AG: When it goes up, the change is positive; when it goes down, the change is negative.

S3: So, is it -2?

AG: Of course.

Etc…

What’s the Problem?

So, already I see some issues.

1. There are 22 other students; what if one of them wants to ask a question? The simple response is that I’ll answer it.
2. Won’t those three students get a better education than the others? Yes, but I plan to rotate the students daily.
3. What if a student doesn’t show up when it is his or her turn? Their performance in this setting is graded. If they don’t show up, their grade reflects that. Also, I can always do it with two students (it might be more effective that way!)
4. What if a student shows up under-prepared? Again, it is a part of their grade, so I expect some of them to take it seriously. But, I can imagine a student who is clearly not working at the same level as the other two students and this could be a problem. I would probably phase him or her out as the lesson grew in complexity and then discuss it with him or her outside of class. I need to take care in putting together these groups of three: if done randomly, I might end up with three slackers who could ruin the class.
5. The three at the front will pay attention, but what about the others? How do you keep them engaged when you’re ignoring them? This is the toughest question for me to answer. I hope that I can make the dialogue interesting enough to keep their attention. Ideas here would be really appreciated. One retort I have is that even if I lecture conventionally, I still can’t keep their attention the entire time. How many more would I lose this way? I don’t know.

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