Remastering Life

I used to live with some sound engineers (did I mention I live in Nashville?). It was kinda cool to have people come record sessions upstairs and hear the process of making a record going on upstairs. The one annoying thing, though, was the, to get it just right, my roommate would listen to the same 5 second clip over and over adjusting levels and mixes. I'm sure his trained ears could hear the slight differences as he adjusted knobs, but to me it was the same thing over and over and over again...

The past few months I've been transfixed by the concept that there are some things people just have to learn on their own. The classic example, of course, is "love." No matter how much you tell someone their freshman relationship is not going to last forever, they won't believe you until the breakup happens. There seem to be enough exceptions that EVERYONE thinks they are the exception.

As a teacher, it's interesting to see these life lessons play out in front of my eyes. Year to year I teach the same subjects to new groups of people. I often start out the year playing Carnac the Magnificent:

You will be challenged in calculus for the first time in your mathematical lives--for some of you, the first time ever. Some of you will rise to the challenge, some of you will want to give up. Many will try to transfer out of the class to keep your GPA high.

Those of you who stay in the class, you will learn more about the background of the math you already think you know and expand that understanding to new material. You'll have to rethink what you think you know about math and, in some cases, life.

At the end of 9 weeks you'll cry. This may be your first B or C. You will adjust your expectations. At the end of 18 weeks, you will cry again. Some of you will be happy just to pass, some of you will be surprised to hear that you have an A and are exempt from the semester exam.

You will not want to come in for help. You will try to skate by with your old methods of learning math. You will think you're a senior and be able to rise above all this stuff I'm saying and not work hard. You are thinking it right now.

Next year, you will come back and tell me how much my class has saved you in your college class.

So sayeth the Great Carnac.


But, regardless of the warning, it plays out the same.

It's interesting to see the light bulbs flash every year over the same topics. It's funny to hear the same "aha"s or "You know..."s every year.

Making the Percentages Fit

I know a number of people are struggling with how to turn their SBG scores into report-card-type percentages and/or letter grades. Here is the solution I was excited to find this week using some software which is new to me:

I haven't taught geometry, so software programs like Geometer's Sketchpad (which our school uses) and Geogebra (which most of my twitter friends use) are new to me. They're TOTALLY FUN to play around with. In my free time, I'll just pick up a proposition from Euclid's Elements and see if I can try to prove it. The dynamic nature of it brings technology to geometry in a way that couldn't be done before. It's one of the few uses of technology that I think could not in any easy way be done before.

Anyhow, Geogebra is the one I toy with most these days, so that's what I used for this. It is free and online here.

The issue

In my SBG marking I use scores 0 through 5 to represent various levels of understanding. In my head, I equate 5 with A, 4 with B, 3 = C, 2 = D, 1 = F, 0 = 0%. I have mentioned this to students and they seemed to get that, too. The issue is, when I put it in my online gradebook, it automatically converts to a percentage. So, while the students get it when looking at the problems, they are still not out of the habit of looking at the BIG NUMBER average for the semester and freaking out about how their GPA will change.

Our school changes percentages to letter grades as:
A: 91-100
B: 81-90
C: 72-80
D: 70-71
F: Below 70

So, I wanted to adjust their percentages accordingly. I need to turn one set of numbers (0%, 20%, 40%, 60%, 80%, and 100% for the 1 through 5 scoring) into another set of numbers (0%-100% for my school's grading scale). Enter:

A Mathematical Answer

Sounds like a job for functional analysis! [insert crowd cheers]

Most models require assumptions (sadly):

1. I don't want any student's grade to decrease from the percentage that the 1-5 system gives them.
2. Other than 5 = 100% and 0 = 0%, I wanted the other numbers to fall in the middle of the grade range.

So, here are the conversions I want:
SBG = Gradebook -> Converted grade
5 = 100% -> 100%
4 = 80% -> 85%
3 = 60% -> 76%
2 = 40% -> 71%
1 = 20% -> 55%
0 = 0% -> 0%

I used geogebra to plot the points: (100,100), (80,85), (60,75), (40,71), (20,55), and (0,0). Since we have 6 points, we can fit a polynomial of up to degree 5, but for my points, a 4th degree polynomial works out to the same. So, I used geogebra to fit a 4th degree polynomial to the data. (g(x) = PolyFit[{A,B,C,D,F,G}, 4])


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


I wanted to check if that would fit my assumptions. It does indeed go through the required points (I can check by doing g(0), g(20), etc.). It also does not lower anybody's average (checked by looking at the function g(x)-x which is positive for all x between 0 and 100 as can be seen above in green).

And now I have my conversion function. It's not "intuitive," so it adds a bit of mystery to the actual number grade a student will have. It's not something they can figure out without being talked through all these steps and them putting all that effort into setting it up themselves. But, I think it seems "fair" to most students.

It's also not automatic, so at various times (like now when mid-quarter grades are going out), I have to adjust them by hand. I take the grade from the gradebook, open the geogebra file and type in g(84) to get the conversion for a student who has an 84 in the gradebook.

Feel free to adapt the method to your needs if you like. I am open to suggestions for improvement.

Math Course Sequencing

I'd like to hear thoughts about the pros and cons of the two general sequences of math courses:

Currently our school does:
Algebra I, Geometry, then Algebra II.

Our admins are starting to wonder if
Algebra I, Algebra II, then Geometry
would be a better option for some students.

Thoughts?

And I Ask Myself, How Do I Work This?

Over at Dan Meyer's blog, he is proposing a lot of reforms to the way we write/use word problems in math. I agree with the direction there and the discussion of pseudo-context. But, it got me thinking: How did we get here?



I imagine, as pressures came from within and without the classroom, teachers felt pressure to continue to move forward. Maybe you didn't have enough time to spend with the slower students to bring them up to where the others in the class are. Maybe you felt that you needed to march onward, ever onward to even mention all the things your state curriculum mandate. Maybe a number of other things could come into play.

At some point, though, you get to a frustrating point and think, "What's the least you can 'get' and still be able to continue?"

Even if a student cannot get motivated or understand the background or whatever, at least he can be expected to memorize the times tables, plug in values to the quadratic formula, or use his calculator to solve an equation. And so we teach there.

Derek Bruff, quoting Mazur in a recent tweet reminded me, "A problem is when you know where you want to get but don't know how to get there. We usually assign just the opposite."

Of course, we'd love to tell our students to do what they can to build a bookshelf and let them figure out all the stuff they need to do it, find directions, and then start taking steps to make it happen. In reality, though, a large number of students will ask why we're wanting them to build it, another percentage will see the wide expanse of possibility in front of them and freak out not knowing where to start, and most of them will just ask you to tell them how to do it. So, we let them fumble for a bit, but eventually, we get to the point where we say, "Ok, if you can't do it on your own, here is box with all the pieces you'll need and here are some instructions."

We tell ourselves that the lowest common denominator should at least be able to put the pieces together when we hand them exactly the correct parts they'll need and even give them a map for how to put them together. Even if they don't understand the Swedish words, they can look at the pictures and put slot A into tab B. And if you can't even do that, then I don't know how to help you.

Plus, if students can't think on their own, they will likely end up in jobs where they are meant to just follow orders from their boss. So, we're teaching them the life skills they'll need for that level of job.

This turns our classrooms into an Idiocracy. The students get so accustomed to being handed all the pieces and the map that they don't know how to think about problems when they are left to figure it out on their own. In some cases, the teacher may not even remember how to grow plants without using Brawndo.

Diary of Infinity: Part 3

We've now seen how a lot of "different infinities" are actually the same. In this post, we explore why there's at least one different kind of infinity.

Similar to what you do in science, the best way to try to prove something is to hypothesize about the outcome first. Then, once you have a guess at the answer, you have to formulate a plan to prove or disprove your hypothesis.

In this case, I claim (hypothesize) that there are different kinds of infinities. In particular, there are "more" real numbers than whole numbers.

First, we should explore the players before we plan the game.

Whole numbers are the easy ones. They're what we're using as our room numbers: 1, 2, 3, ....

What are real numbers? Real numbers can be thought of as the location of any point on a number line. A rigorous, mathematical definition is very involved, but your intuition is generally correct on this one, so we'll stick to the number line concept.

Now for the plan. I want to lay this out first, because I know you'll want to fight against the claim as we get going and it'll start to not make sense if you're not sure where we're going. I've also found it helpful to lay out a plan for a mathematical proof so that you can know when you're done. Often you can read or follow a proof and at each step say, "Yeah. Ok, I can see that." and then the proof suddenly stops and you wonder where the magic was. So, here's the plan and if we agree on it first, then it'll make the finish a bit easier to reconcile.

The plan (Hilbert Hotel style):

As before, the hotel has infinite amount of rooms numbered 1, 2, 3, ... as the whole numbers. This time, our people who show up will have names of all the real numbers from 0 to 1. I claim that there are more people than rooms, so there is no way to match up one room per person and one person per room.

It's relatively easy to prove there is something: just find one. If I claim that time travel is possible, all I have to do is go out and do it.

On the other hand, it's really hard to prove there is NOT something. If I claim that there is no way that time travel can happen, how can I prove it?

Well, we are going to use a technique called "proof by contradiction" (sometimes called "reductio ad absurdum"). Here's how that method works:

In order to prove that X cannot happen, let's--for a minute--pretend that X can and DID happen. What would be the consequences of X having happened? We will follow some of those consequences logically and arrive at something absurd or contradictory and realize that the only flaw in our logic was the original assumption that X has happened.

For example, one reason people claim time travel cannot happen is that IF it could, we should see some time travelers around once in a while. Since we don't/haven't, the original assumption of time travel being possible must be incorrect.

Here's how it applies to our situation. I'm going to pretend for a minute that we CAN match up our people to rooms. Since we can't check all of the possible ways to match them, I won't be able to tell you exactly how we did it, but we'll pretend that I got it to work. Then, if I can find someone who doesn't have a room and I can prove he doesn't have a room, then we have arrived at a contradiction.

I'll prove that this person cannot be in any room by allowing you to think of any room number you want and I can prove he is not assigned to that room. Since you are allowed to pick any room number, he cannot be assigned to any room.

Another thing I'll use is that two numbers are not equal in their decimal representation if at least ONE of their decimal places does not match (with the one exception being instances of 0.9999... repeating DOES equal 1, but that's for another blog post and we can avoid it here).

Have I convinced you that, if I carry out this plan, then there will be more people (irrationals) than rooms (whole numbers)? Let me know in the comments if you need more convincing.

Carrying out the plan:

So, we have somehow figured out how to match our real numbered people to our whole numbered rooms so that each room has one person and each person has one room. I don't know exactly how we did it, and it doesn't even have to follow any nice pattern, but somehow we've got a list somewhere. If I need to check who is the person assigned to room #573, I can find their name (or number) by looking it up on that list.

Since I don't know the exact order of this list, I'll attempt to put some "random" numbers as examples so you can see some numbers, but any methods we use should be able to be applied to any numbers you want to put in their place.

According to our plan, I need to find someone who DOESN'T have a room and then we'll be done. So, here's how we can find the person (by "the diagonal argument"). We'll call him "x" for now and construct his name in a specific way.

Check the list to see who is in room #1. Since all of our people are real numbers between 0 and 1, room #1's person has a name that goes something like 0.32914514.... All I really care about is the first decimal place (the 3 right after the decimal in that example). I am going to pick person "x" so that his name begins 0.a where a is any number other than what is in that same spot for the guy in room #1. For the example above, I could pick person "x"'s name to start 0.1 or 0.5 or 0.7 or a bunch of others as long as it doesn't start 0.3. Let's just go with 0.1 for this example.

We continue to construct person "x"'s name by checking the list for who is in room number 2. Room #2's person has a name something like 0.42150682.... We will pick person "x"'s name so that the second decimal place is anything other than the second decimal place of the guy in room #2. Also, to avoid the 0.999... problem, let's choose that digit to be something other than 9 as well. In this example, we have to avoid the 2, so pick anything else: 0.17

Continue this way down the whole list. Now, we can continue doing this down the whole list because:

1. There is a unique decimal representation of the room occupant names (if we avoid the .9999... repeating issue and we can).
1a. For room occupants that have terminating decimals (0.5, for example), we will use 0s to fill out the infinite decimal places after the termination point (so, 0.5 = 0.50000000....)
1b. Thus each room occupant has SOME digit in the decimal place that corresponds to his room number.
2. Since each room occupant has only one digit in the required decimal place (corresponding to room number), we have 8 choices for the digit in person x's name (not the one from the room occupant and not 9 to avoid that weird quirk).

So, I claim that person x cannot be assigned to any of the rooms which is what will contradict our original assumption that everyone did have a room meaning our original assumption was wrong.

Well, person x can't be in room number 1 because the person assigned to room 1 has a different first decimal place than person x. Similarly for room 2 and 3 and all the rest by the way we picked out person x.

And we're done! See, I told you that you'd be surprised that we're done. Go back and review our plan from the beginning and see if we carried it out. I think we have, but I agree that it somehow seems unsatisfying.

"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?