"Inception" Chain Rule

The power of Twitter compels me!

In one of my classes on Tuesday, I told the students that the chain rule for derivatives is kind of like the movie Inception. At the time, I only meant that there were nested pieces and to stand as a warning for students to pay close attention as we went through it so as not to get lost. After tweeting about it, though, and reading some of the responses there, it got me thinking about how deep that rabbit hole could go.

So, in class on Wednesday I went a bit further with the idea. I guess it would be more Dan Meyer-ish if I was able to actually get a clip of the movie, but Inception isn't on DVD yet and I don't really want to get into the illegal bits if I can avoid it. Enough students had seen the movie to provide us with details and the conceit of the film was a bit wild to those who hadn't seen it, but was generally accepted.

So, I began with a brief (spoiler free, I hope?) explanation of the relevant points of the movie. The main ones are:

1. Time works differently in dreams. I don't remember the exact numbers from the movie, but in one class the students said 1 minute passed in "awake world" corresponds to 10 minutes in "dream world." Another class said 5 minutes was meant to equal an hour, so we used a factor of 12 in that class. Another student suggested it was variable based on how long you were asleep.
2. Dreams can be set up so that you are dreaming within a dream and any time dilation (or other effects) are compounded (1 awake minute = 12 dream A minutes = 144 dream B minutes = etc.)
3. Some effects from the next higher level can be transferred to the dream worlds. While there were many parts to this (for example, person who needs to pee in the awake world can make it rain in his dream), we were mainly focused on the physics aspects. If someone in awake world were to push your bed off a cliff while you were sleeping, you would become weightless in the dream world.

So, we set off to model some of these interactions. We would start with just one dream world and one awake world. The situation we would model is: While you are sleeping, some mean person decides to push you and your bed off a cliff. Meanwhile you're having one of the most boring dreams ever wherein you are just standing still in a plain room. How would the bed motion feel to you?

Only about half of my students either are taking or have taken physics of any kind (we don't offer calculus-based physics at our school). At this point in their physics class, they have had to memorize "the kinematics equations" (with only experimental suggestions at validity--no calculus proofs). So, I appealed to those students to get us started with an equation for the bed motion.

The equation as they have memorized it is:

where is the gravitational constant (-9.8 m/s for Earth), is the initial velocity of the object, is the height at time , is the starting height, and is the time (for us, time in the awake world).

I asked the class how high we wanted our cliff to be. (I am very poor at estimating reality, so I rely heavily on the students for this.) It was settled that we would use 10,000 meters for . This seems very unrealistic to me, but they thought it was funny and the math process works out the same, so we went with it.

Also, since our bed is only being pushed off the edge, would be 0.

This leaves us with the equation:


So, the first question we began with was simple: How long until the bed hits the ground? Setting y = 0 and solving for t shows us that we have a little more than 45 seconds (awake time).

A discussion of the time dilation that was proposed in the movie led us to a factor of 12. After we set as the variable for time in dream world, we had a big discussion whether it should be or . People were on both sides pretty strongly, but we finally got to a ratio to determine a definitive answer. (It's . Plug in 1 awake-minute for t to see that it would be 12 dream-minutes to convince yourself if you're not already.)

So, how long do we have in the dream until we have to wake ourselves up and pull the cord on that parachute we always wear to bed? Multiply our previous answer by 12 to get about 542 seconds (9 minutes and 2 seconds).

Well, this is all fine for an algebra class or physics class, but this is CALCULUS! So, let's get some velocities, ok?

We're still getting used to the notation, so I asked the students what would represent. They got it pretty quickly as the velocity of the bed and after some prompting added "with respect to awake-time." They also got it quickly (having just learned the power rule last week) that . This is not surprising to the physics students since another formula they've memorized is .

But, let's see how the time dilation makes this feel in dream-world. Let be the position of the dream-Earth (whose physics we are assuming follows similar physics to our awake-bed). Then, relative to awake- and dream-times respectively.

What does represent? What is its value at various awake-times t? Well, to describe the derivative, things get even weirder than they already are. We might imagine our awake-life is actually like in a cartoon where we can stand to the side and watch the sleeper's dream in a thought-bubble over their head. In this context, the derivative would be the apparent motion of dream-Earth relative to the observer's awake time. Its value (perhaps unsurprisingly?) is the same as .

Well, if we are in the dream world, how would we feel this free-fall? It wouldn't be with respect to awake-time, so we would need to discuss the whole shebang in terms of u, not t. In particular, would represent the instantaneous velocity of the dream-Earth relative to dream-time. This is what our dream-self would actually feel. Although this computation could be done easily by actually squaring the term in the equation above relating z and u, we are practicing chain rule, so we went with that method. Either way, though, it turns out that
.

Notice the large impact that the time dilation has. The dream-Earth seems to be moving at 1/144 of the speed of the awake-bed (in their appropriate time references).

This is about as far as we got in the time period of the class. We could certainly take this further to see the impact of the falling bed in a dream-within-a-dream world. You might guess--and be right--that at that level the falling bed would be almost unnoticed. *Spoiler alert* This plays out in the movie.

(I forgot to bring home the SMART-board export from the class period, so I'll add them tomorrow when I get back to school.)

Edit: Here's the link to a PDF of the SMART-board pages from class.

How do you KNOW you know you know, you know?

One thing I think math teachers struggle with (and I am one of them) is "proofs." These can be student-generated ones on the level of basic geometry (use SAS to prove these two triangles are congruent) or teacher-driven ones (where does the Power Rule come from?).

As you saw in my recent post about the card game "Mao," I tried to get students to recognize patterns by looking at various problems. They needed a lot of coaching to get things going in the right direction, but once they got the hang of it, they recognized the patterns pretty well.

The problem then became--what's my motivation for proving the rule in a general case? I usually fall back on the concept of "Yeah, but how do you KNOW it works for all functions and all numbers? Maybe you just randomly chose the right ones by luck."

You can also see this when coming from the other side. I asked my students in calculus to show that the equation of the tangent line to a given line at any point turns out to be the original line itself. A lot of them just picked a couple numbers for slope and intercept and showed that it was the same for those, then "since I chose random numbers, this should work for all of them."

Clearly that's not how math proofs work, but how do we motivate it beyond a lot of "what if"s?

I suppose I could pull out some of the less intuitive formulas or give some examples of sequences where the simple patterns don't fit the values for large indexes, but those always seem somewhat contrived.

I think there's power in working with general formulas and showing that we can say definitively that such-and-such is true. On the other hand, the students seem very used to just believing the teacher when she says, "The formula for circumference is pi times the diameter." So, they just want to take my word for it.

Any other ideas to motivate proofs?

Diary of Infinity: Part 2

As we saw in Part 1, we can tell sizes of things that are hard to count by matching the two sets item by item. Rather than checking all the items, all we really need to do is allow for the option that if any one item from the first set is chosen, we can identify its match from the second set and vice-versa. Since I am allowing you to choose ANY single item at complete random, this claim means I should be able to do it for all of them.

I use Hilbert's Hotel to give the students a basic handle on sizes of infinity. Here's my telling of it:

There is this hotel with infinite rooms numbered 1, 2, 3, 4, etc. It's a weird hotel that attracts odd people.

There's a convention for these odd people held at this hotel every year. An infinite amount of people come to the hotel who conveniently have silly names. Their names are 1, 2, 3, 4, etc. Each person likes their personal space A LOT--enough that they will do anything to keep it. This means they like to each have their own room and will pack up and move rooms in order to stay that way.

So, when the convention happens, it is easy to give everyone a room, we just tell them to go to the room whose number matches their name. Thus, every room has a person and every person has a room. (See Part 1 for why this is important)

Nothing too exciting here, we've just shown that (infinity) people = (infinity) rooms.

In the middle of the night, another guy shows up: his name is 0. The sign outside was flickering between "vacancy" and "no vacancy," so he thought he'd stop in and see if we could accommodate him. Can you think of a way to give him a room at our hotel (which doesn't include building a new room or telling him to "go to the end" and wait until he dies walking that far)?

Well, since our people have their odd quirk, they don't mind moving rooms, so the easiest way to make sure 0 gets a room is to move EVERYONE down one room and put 0 in room #1. So, how can we check that everyone has a room and every room has a person? We need a nice way to match each person with their room and each room with their person.

This one shouldn't be too hard, but here it is. If there's a lost person who forgot his room number, we tell him to add 1 to his name and find that room for his place. If there is a room that is on fire and we have to alert the person who is assigned to that room, we subtract 1 from the room number to see which person's belongings are in peril. Since we can do this for any person or room at random, we can do it for all of them.

Thus, every room has a person and every person has a room. This result is also not too surprising because really we've just shown that (infinity + 1) people = (infinity) rooms.

Here's where it gets a little odd. Shortly after 0 is settled in his new room and everyone else just moves down a spot, a new situation arrives. Infinitely more people show up to the hotel who also have odd names: -1, -2, -3, -4, etc. Do we have room for everyone?

Well, we can't quite have everyone "move down a few rooms" because when do we tell them they can stop? After some thinking, most clases will have at least one student who comes up with "double the room numbers." This is an option that works. We tell everyone who currently has a room (0, 1, 2, 3, etc.) they have to move again, but this time they have to double their room numbers to find their new room. Not too hard for some (0 who was in room 1 is now in room 2), but 999 who was in room 1000 now has to walk all the way down to 2000, and don't get me started with those long-namers further down. At least it's possible. This frees up all the odd numbered rooms for all of the new people.

So, it seems like it can be done, but is there an easy way to tell which person goes with each room and vice-versa? Perhaps. This is where we get into piecewise functions (or for you computer types "if" statements). IF your name is a non-negative number (0, 1, 2, 3, etc.) your room number is 2*(NAME + 1). IF your name is a negative number (-1, -2, -3, etc.) your room number is 2*abs(NAME) - 1. This matches people to rooms.

We can match rooms to people by doing: IF the room number is odd, then it contains a negative named person. So, we can do -(1/2)*(ROOM + 1) to find out who lives there. IF the room number is even, we can find the name of the person by doing (1/2)*(ROOM) - 1.

Thus, every room has a person and every person has a room. This makes sense on one level and is weird on another. On one level, we just showed that (infinity + infinity) people = (infinity) rooms. That kinda makes sense. On another level, we just showed that the number of numbers in the sequence 0, 1, 2, 3, etc is the SAME as the number of numbers in ..., -3, -2, -1, 0, 1, 2, 3, ...

Maybe you're getting it, maybe you're not. But in part 3 we'll see why not all infinities are the same. In particular, when a new group of infinity people show up (with names that represent all the decimals from 0 to 1) we CAN'T fit them in our hotel.

King Mao: Power Rule

When I was in high school (mid 90s), there was a card game that went around and was fun for some and frustrating for others, called "Mao."

Not this guy

The #1 rule of the game was that you couldn't tell anyone the rules of the game. The only way know the rules of the game were to figure it out by playing (back with the people I played against at that time, you weren't allowed to say why you were giving someone 2 extra cards for not following a certain rule and you weren't really allowed to watch a game without playing). Last year, a group of my students found this game again and were playing it all the time.

It's awesome for math class because you have to figure out the rules by noticing patterns. You're confident there are actual, logical rules out there, but you can only learn them by testing your theories in laying down certain cards or doing certain actions and then getting rewarded or punished based on the rules.

This week, I'm going to be doing a similar method with my classes in calculus. We've just "finished" learning about finding slope of tangent lines using a limit definition. I told them there was a "secret trick" so that you could just look at most of these problems and know the answer pretty quickly, but we wouldn't get to it until the next week. So, when they asked what the answer was, I could spout off the answer pretty quickly and not because I am so smart that I expanded (3+h)^4 in my head really quickly and then computed the limit, but because I knew the trick. Immediately they were intrigued. Some were guessing at what it could be and I just told them to either work it out to confirm their suspicions or wait until next week.

So, tomorrow starts "next week." It would be fun to do this with wolfram|alpha or something where they could come up with their own theories and test them individually, but we don't have the tech for that at our school. So, I'm going to be their wolfram|alpha.

I'm King Mao and I know the secret rules. I will have students throw out polynomials and points and I'll give them the slopes of the tangent lines at those points fairly quickly using the secret rule. When you think you've figured out the pattern, don't say the rule out loud. I'll throw out a problem for you to test if you're right. If you've got it right, you can sit out and the rest of the class can try to figure it out. If you're not right, keep playing until we all get it.

Diary of Infinity: Part 1

...you think you know, but you have no idea.

I'm stealing this image from Irrational Cube as a description of how students come to understand math.

I want to use it to illustrate how many students (and maybe some of you) understand infinity.

By the time they get to me in calculus, they've got step 1 down already. Infinity is a cool enough subject to have made it to somewhat mainstream life. "I love you times infinity" and "To infinity...and beyond!" and "Infinity plus one" are all phrases I expect most people to have heard at this point in their lives.

We use the concept of infinity in calculus a LOT, so I try to give students a glimpse into some of the interesting facts about infinity. In particular that there are different KINDS of infinity (countable and uncountable, to those "in the know") and "all infinities are created equal, but some infinities are more equal than others."

In Part 1 here, I'm going to just talk about how to compare sizes of two groups of objects mathematically.

When you're talking about infinities, how can you compare their sizes? In particular, how can you compare things that you don't want to (or can't) count?

Imagine two piles of sand about waist high. Which has more sand? How could we figure it out without counting? Well, one option would be to turn to physics and assume that all sand grains have the same density/size, so we could weigh the piles. Which is great, but physics doesn't apply very well to numbers and infinities.

So, let's go in another direction. Imagine two piles of wooden blocks (like for babies): one pile has 19 blocks and the other has 20. How could a baby tell which pile has more blocks? The piles probably look about the same size to him and he can't count, so how could he figure it out?

Well, an option that works is to match them up in some way. Grab one block from each pile and move them behind you. Keep doing that until you get to the end when you go to reach for another pair and have only one left in one hand and none in the other.

This process also seems impossible with infinity (because there won't be an "end"), but it's a process we can exploit. Imagine those baby blocks were each labeled. The pile with 19 blocks had letters on them: A, B, C, ..., S. The pile with 20 blocks had numbers on them: 1, 2, 3, ..., 20. Now, if we can make up a function that tells us which numbers can match up with which letters, then we don't ever have to even touch the blocks and notice that one is left over.

As we move to the math, we have to make sure that our processes fit our intuition with the blocks. So, what does it mean to match the blocks with a function? It means that we have some list or maybe a process so that a person can give us any block (letter or number) and we can tell them which other block it is matched with. There may be more than one way to match them. One option could be: A = 1, B = 2, C = 3, ..., S = 19. Another might be A = 19, B = 18, C = 17, ..., S = 1. There are lots of others that could work, but in order to say the two piles are the same size, you only need one that works. What we have to make sure we avoid, though, is "matching" a block twice so that if A = 1, then no other letter can also be matched with 1 and no other number can be matched with A.

To say that one pile of blocks is bigger seems like a much harder chore. We have to show that there is NO WAY to match the blocks. How can you show that you can't do something? We'll address that when we attempt to do it later in another "diary" entry.

SBG: You Really Have to Want It

In the previous post, you may have read how my version of SBG is set up logistically this year. Today I wanted to point out some "issues" I'm noticing so far with my own classes.

I love the idea "in theory" of standards-based grading, but the practice and logistics is what was scaring me before this year. A few weeks into the year and trying it, it's still the logistics and work that are making it difficult for me.

Those of you reading this may cringe at the next bit, but it's what I did and what many teachers do, I think. Last year, I would go through teaching, give the students practice problems to work in class and as homework, then give a quiz or test over that material. I'd take those home and grade them for a few hours and then hand them back. We'd spend the next day or two going over any common mistakes and clearing up any issues and then move on to the next topic. I'd warn the students that the material built on itself and they would need to learn from their mistakes on the prior test and I would take the information from the previous and help out individual students and the group as a whole as we moved on and the students in my classes mostly seemed to understand that, but off we'd go into new topics.

I would spend my afternoons/evenings planning the next lesson on most days and grading heavily on the days with quizzes and tests. I'd spend my mornings setting up any equipment for the lessons and getting my head in the right mode for the next class (I have 5 different courses, so between each period I have to get in the right mindset for the next batch of students).

This year, I'm trying to allow for later-learning by allowing retakes of tests/quizzes. You've read most of how that works in the classroom in the previous post, but here's how it's affecting me, personally:

I now have to spend my afternoons/evenings as before, but IN ADDITION, I'm writing new questions for the students who signed up to retake quizzes and I'm constantly grading all the retakes that students are taking throughout the day. In the mornings, I have to do all the things as previously AS WELL AS printing out all the questions I've written the previous night, running those questions to the appropriate places throughout the building for students to take, administering the retakes to students who came in early for that, and helping students who missed the information on the previous quiz/test.

I feel at least a dozen times busier each day because of this "student-centered" program.

I also don't know if this system is actually working. The students (and parents) seem to like the safety-net of retakes, but still seem to be point-grubbers who aren't really changing their habits.

There are a LOT of factors here, mostly including the idea that it's a new year and a new system to me. So, am I not doing it right? Are the kids just not used to it yet? Are the kids not used to me yet? Is it this group of students?

There's a lot of unlearning to be done, and I'm not sure if/when it will get through to them how this is meant to work. I'm trying to emphasize the learning and understanding at every point I can, but I think their brains equate "understanding" with "grade on a test" (which is how it SHOULD be, I guess, but is most often not the case--and not in their heads, I'm afraid).

I'm sure a lot of this is just "change is hard" stuff. Eventually I think the students will come around and study more for the retakes rather than just come in right after the test and think they know how to do it now that they've got it back in their hands. Eventually I think I'll have a larger base of problems to pull from so I don't have to constantly be writing new questions for students all the time. Eventually I think it'll all be worth it. But right now, it seems to be a lot of work for little gain.

So, for all you readers who may be considering trying this method, know that the theory is great and will probably be wonderful in practice if you do it "right," but you really have to want it because it's a lot of hard work.

Then again, you're a teacher, so the entire job fits that description.

The Way I Work It (no diggity)

Well, I don't know if I'm doing an official "SBG" thing, but this stuff I'm doing now is finally starting to work for me.

Here's what I'm doing (chronologically): On the test/quiz (the only difference to me is that tests reach back a bit further and are more comprehensive) I list the topics that each question covers. Students take the test/quiz. I grade each question individually (2/2 for #1, 4/6 for #2, etc. depending on the number of parts that each question has or level of difficulty) and then combine skills in the gradebook (so, if questions 1 and 2 are the same in the previous parenthetical, then that student would have a 6/8 for skill 1). I write feedback where appropriate and return to the students. So far, I have been able to return the assessments within a day or two.

Students then have a few minutes to talk with their group (of up to 4--so far, student chosen) about what they missed and how to correct their mistakes for the future. Before today, I would have them mention to me when they would like to come back to retest or requiz over certain questions and I'd stay with them after or before school. Well, due to outside-of-school-circumstances, my wife and I were a one-car-family for a couple weeks and I had to run out before it was convenient for many of the students. So, today I've implemented a slightly new system.

I have a sign-up sheet in the back of the room with columns for Student Name, Test/Quiz to Retake, Topic(s) and Question Number(s), and Date to Retake. Students may go back there at any time and sign up to retake questions within the topic (in the above example, again, I would make them redo both questions 1 and 2). I grab the list at the end of the day and write any appropriate questions for the individuals on the list.

Our school has a room called "the learning lab" (LL) where teachers can drop off materials and students can go during their study hall to make-up any missed work. Since almost every student has a study hall at our school, it is most convenient for them (and me) to have them go to the LL to take their retest/requiz. They are also allowed to come in before or after school to my room to make up the test or go over any content questions they have (and now that we are back to having individual transport, I don't have to stress about my wife waiting for me to come get her).

This is week 4 of the new year and students still seem to be getting a feel for the kinds of questions I ask and the level of understanding I expect from them (much more difficult and and much deeper than they are used to, it seems).

Many still get their tests back and struggle with the idea of, "Is a 7/8 worth it to keep on this topic or should I come back in and retest?" When asked directly, though, I tell them, "You should always come in and retest. If you got a 7/8 the first time, you pretty much get it, so it'll be easy to run through those the next time and you can get that one part you missed right and you'll have a 100 in here." Win-win-win. Generally they seem to agree with me and probably have in the back of their mind that even if a stupid mistake comes out the 2nd time, they'll have a third time to come back as well.

At the moment I'm limiting it to 2 retests because I don't want them to come in every day and just hope they get an easy question that they get by luck (and also because our online gradebook "Gradespeed" has only 2 spots for retakes). As they're getting used to the system, I'm sure a lot of them would try something like that. Two retakes gives them an option to realize that they didn't just forget the negative, but missed the entire concept and focus to come in a third time and get it right.

At first it was a bit of a hassle trying to accommodate students on-the-spot by either writing out a second quiz and just telling them to do the ones they wanted or making up questions while they stood there (they were impressed by that, and I don't mind doing it, but when there's a line of students, it takes a bit to write them all down). With the sign-up sheet, though, I feel much better about taking my time to write out individualized retests for the student requests and bring them in the following morning for the LL to file away until the student shows up.

So, I'm starting to get excited about the method. I don't think students are quite at the point of being "excited" yet, but they do seem to appreciate the lower pressure. Parents certainly seem to appreciate it as well since they can keep up with student grades online and see what topics their kid needs to work on and stay on their backs about coming in to retest.