## Tuesday, December 14, 2010

### 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.

## Monday, November 15, 2010

### 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:

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:
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.

## Thursday, October 14, 2010

### 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?

## Monday, October 11, 2010

### 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.

## Thursday, September 23, 2010

### "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:
$y_f - y_i = \frac{1}{2} g t^2 + v_i t$
where $\inline g$ is the gravitational constant (-9.8 m/s for Earth), $v_i$ is the initial velocity of the object, $y_f$ is the height at time $t$, $y_i$ is the starting height, and $t$ 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 $y_i$. 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, $v_i$ would be 0.

This leaves us with the equation:
$\inline y = -4.9t^2 + 10000$

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 $\inline u$ as the variable for time in dream world, we had a big discussion whether it should be $\inline 12u = t$ or $\inline 12t = u$. People were on both sides pretty strongly, but we finally got to a ratio to determine a definitive answer. (It's $\inline 12 t =u$. 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 $\inline \frac{dy}{dt}$ 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 $\inline \frac{dy}{dt} = -9.8t$. This is not surprising to the physics students since another formula they've memorized is $\inline v_f = gt + v_i$.

But, let's see how the time dilation makes this feel in dream-world. Let $\inline z$ be the position of the dream-Earth (whose physics we are assuming follows similar physics to our awake-bed). Then, $\inline z = \frac{1}{2}gt^2 + 10000 = \frac{1}{2}g(\frac{u}{12})^2 + 10000$ relative to awake- and dream-times respectively.

What does $\inline \frac{dz}{dt}$ 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 $\inline \frac{dy}{dt}$.

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, $\inline \frac{dz}{du}$ 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
$\inline \frac{dz}{du} = -\frac{9.8}{144}t = \frac{1}{144} \frac{dz}{dt}\approx -0.068 t$.

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.

## Monday, September 20, 2010

### 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?

## Monday, September 13, 2010

### 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.

## Sunday, September 12, 2010

### 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.

## Tuesday, September 7, 2010

### 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.

## Wednesday, September 1, 2010

### 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.

## Wednesday, August 25, 2010

I know Shawn Cornally says you shouldn't call it SBG unless you go all the way with it, but I'm kinda getting my feet wet with it this year. One of the things I'm trying to implement is allowing students to re-quiz or re-test by request. I'm allowing them to just redo the sections that they missed. (I'm doing some other pieces of SBG including dividing their grades among various skills and giving more feedback that I have previously. But it's the re-quizzing that I want advice on here.)

So, we're two full weeks in as of tomorrow and I think students are already trying to take advantage of this situation. I should maybe give some background that may or may not also contribute to this:

I use a SMART board to write for my notes during class and there are a couple groups of students who think they can understand the material by spacing out during class and going home to download the notes then study right before the assessments. I don't mind if students have a hard time writing notes at the same time as listening to the material, so they can just sit back and pay attention and download the notes later. It's also useful for when students miss class for whatever reason to catch up. So, I don't want to quit using that option, but students don't seem to be "learning their lesson" very quickly.

I had one of those students come to class this afternoon (on quiz day) and say, "Mr. Petersen, I'm not going to do well on this quiz. In fact, you may not want to even grade it." So, I told him not to waste his time either and not even take it. This is all with the understanding that he would come in later to redo it. I can easily see this kid pushing it back further and further and, with the hope of bringing it up later, just failing the quarter and beyond.

Many other students, as I'm passing out the quiz, before they've even looked at it ask, "So, can we redo this quiz later?" I am glad that I can take some pressure off of them by saying that they can, but I get the feeling that they're just pushing back the responsibility and not taking the whole thing seriously.

If I were a college teacher, I'd just take the attitude of "your loss," but we can't really do that at public schools.

For any who have re-quizzing as part of your assessment system, is this always an issue? Any hints on what I can do to feel better about it?

## Thursday, August 19, 2010

### Math Puzzle Questions

Moments ago, @dandersod tweeted about an interesting, but non-intuitive probability problem. There are plenty in this genre such as the Monty Hall Problem and many in other areas like the uncountability of irrationals, area/perimeter relationships (@CmonMattTHINK), etc.

I think these problems are like lifting the wolverine's lip to show the teeth. Some kids will be like, "Wow! Those are sharp! So cool! I wonder what that wolverine would do to a rabbit..." Others will just be, "AHHH! Run away! Don't ever make me go near that thing ever again!"

These kinds of problems really separate those who end up loving math from those who do not. I present one or a dozen of these throughout the school year to my various math classes. Some of the kids LOVE it and try to figure out why it is and will read up on it when they go home and have all sorts of questions and really work to pay attention to the "proof." Others will just throw their hands up and say, "See! This is why I hate math!"

Basically, it polarizes the classroom. Those who enjoy thinking this way will get excited and encouraged to find out more. Those who already hate it will have fodder for their fears.

So, I guess what I'm asking here is: Is it a good idea to put forth these "paradoxical" problems in class?

## Friday, August 13, 2010

### Calculus Questionnaire

I gave a questionnaire to my calculus courses (one section of honors calculus, one of AP calculus AB, one of AP calculus BC) to get their minds going and preview some of the things we're doing. Below are the questions and why I asked them as well as any interesting answers given.

1. What is the furthest you've been away from home?
• Why I asked: On a personal level, I wanted to know where some of my students have been. On a mathematical level, I wanted them to think about practical distances and how we measure distance. (Nobody said it, but I mentioned things like, "If you step out your front door, that might be the furthest away if you measure around the earth the other way. But why stop at measuring around the earth? Why not measure around the sun?")
• Interesting answers: Rural China, Ethiopia, Alaska, Australia, India

2.What is the largest number you can write in the space below?
• Why I asked: Getting at the concept of infinity, but I wanted to point out that infinity is not a number (at least not in the calculus we're doing). Creative answers would've also been good like a large-font 8 or something.
• Interesting answers: ZERO (written in tall letters), "a really big number," 9^9^9^9^9, infinity, "I'm not playing this game"

3. What is the least probable (but still possible) event that you can think of? If you had to give it a percent chance of happening, what would it be?
• Why I asked: Getting at the concept of limits, specifically as x approaches 0 from the right. Also interesting to see creativity and interests.
• Interesting answers: scoring an 18 on a professional golf course, being slapped by a walrus, gingers ever being cool, a squirrel causing a 30 car pileup, someone dying because they wore a black vest and choked on the buttons (what?), failing this class (yeah!), meeting everyone in the world, everyone getting along (:(), walking through a wall because your atoms pass through the spaces and reconfiguring on the other side

4. What is the number right before 4?
• Why I asked: Another attempt to get at limits. Also pointing out that 3.9999...(repeating) is not the answer because it actually is 4 (we'll learn later in the series section). I expected a lot of them to put 3 or 3.9(repeating). Maybe "4" because on the paper it's the previous number before the "4" from the end of the question.
• Interesting answers: 3, 3.9(repeating)

5. Have you taken or are you taking physics?
• Why I asked: It's not necessary for calculus, but our school doesn't offer calculus-based physics. So, those who have taken or are taking it can see some cool stuff like where the kinematics equations come from.

6. Do you know the difference between the meaning of the words "velocity" and "speed?" What about "displacement" and "distance traveled?"
• Why I asked: Straightforward. This may not be a necessary question in the future, but it gave me an opportunity to tell them one of the basic applications of calculus.

7. Using algebra skills, can you turn this: $\frac{(x-2)^3+8}{x}$ into this: $x^2 -6x + 12$
• Why I asked: Practicing algebra using the difference quotient. We'll be coming up on these types of questions in the next week or so and I wanted to get a sense of where they were and give them an idea of what level of math we're looking at.

8. Using precalculus knowledge, write an equivalent expression for each of the following using trig identities you learned:

9. In as many steps as possible, explain how to make a bowl of cereal.
• Why I asked: I thought it would be interesting mathematically to talk about how many steps we have to show at this level (eg If you have a quadratic, can you just write the answer on the next line or do you have to show the quadratic formula and all that?). I also wanted to talk about how mathematicians think in details. How far back would you go? (Go to the store to buy cereal? Grow wheat to make the cereal? Evolve plants to make the wheat? etc.) And what level of detail would you go into? (Send a signal from your brain to your arm muscle to extend your arm towards the spoon.)
• Interesting answers: Really, there are too many to list here, but one kid wrote more than 80 steps. Another had some sort of story in his where you go to the store to choose your cereal and then a few steps about trying pickup lines on the checkout girl.

10. Find the shaded area of the shape in the picture below (Picture of a polygon that is a rectangle with triangles sticking out of each side.)
• Why I asked: I wanted them to get started on thinking about integral calculus where we take a region and find its area by dividing it into smaller regions. Also, in broader terms, just taking a large problem and breaking it down into smaller, more manageable parts (see Sam's motto "turn what you don't know into what you do know").

11. Find the following:
• The slope of a line passing through the points (-1,2) and (3,-1).
• An equation of the line that goes through the points (2,4) and (5,2).
• An equation of a line with slope 0.
• Something in the school that has a positive slope

• Why I asked: Basic review. I also pointed out that the last part can be tricky, because if you go to the other side of the "up staircase," then it's a "down staircase" and has negative slope.
• Interesting answers: Knowledge gained, age of a student

12. What do you think calculus is?
• Why I asked: I thought it'd be interesting to know how many students actually knew what they were getting into. (Very few did.)
• Interesting answers: A difficult class that can only be understood when taught, the next class after precalculus, something to do with curves, I have no idea, hard math, the highest math class

13. Write the word "minimum" below in cursive.
• Why I asked: I always thought it was a fun word to write. You kinda scribble up and down a lot and then go back and dot some i's.

14. Haha! Wasn't that fun? Now, why are you taking this class?
• Why I asked: I thought it would be interesting to read the different reasons they thought of--especially in light of #12 on this questionnaire.
• Interesting answers: because it looks good on college apps, because it's next after precal, I wanted to take a math my senior year and I don't like statistics, I like punishing myself with challenging math classes, good question

Overall, I think it was a fun experience for them. It was interesting to note the difference in answers from the "creative" people from the "beaten-down-by-the-system" people. A couple kids in my BC class said, "You're going to hate me when you read my answers, Mr. P." When I asked why, they'd say, "Well, like for the 'biggest number' question, I just wrote a huge number 2." I tried not to smile too big.

## Monday, August 9, 2010

### SBG: Tiger Woods

I wouldn't necessarily call myself a sports fanatic, but I do have a few teams that I follow and I usually listen to sports radio on my 45 minute drive to school each morning. This morning, the topic of discussion (at least for the part where I was driving) centered mostly on Tiger Woods and his current streak of poor play.

For those of you who don't follow golf, basically, you know Tiger Woods was a pretty awesome golfer. Then there was a bit of scandal about his personal life and he took a long break from the sport. He is working his way back into playing tournaments now, but has been nowhere near his previous level on the course. While there have been "glimpses of his former self" on certain holes, the general consensus is that he is not only playing poorly, but his body language seems to sometimes say that he doesn't even want to be there anymore. This is especially distressing for Tiger Woods because one of his best features previously was his mental focus and toughness.

Anyhow, here's why I'm writing about it here. For those of you with experience in Standards Based Grading (or some other similar system where current knowledge is more heavily weighted than previous), how would this sort of situation play out in your classroom? By the way the PGA ranks its players, Tiger Woods is still the #1 golfer (even after his long absence and recent poor play--that's how dominant he was previously), but anyone watching his current performances would know that he doesn't deserve that title currently.

I guess I'm asking it in two parts. One, do we take into account that his poor performance may be largely due to personal issues from off the course? If so, how can we help the student (and our gradebook) separate those issues from the performances? And if we don't do any sort of "averaging" system with the numerical grades, what's the best way to assess this student?

Two, if we disregard the personal issues (or don't believe they're the cause), what do you do with a student who has performed exceptionally for such a long time and then drops off dramatically (maybe "senioritis" or some other non-catastrophic issue comes into play)?

## Thursday, August 5, 2010

### Experimental Classroom

I have an ideal situation with my Honors Calculus course for experimenting in the classroom. In Tennessee, there are basically 3 standards: Derivatives, Integrals, Applications. They are very vague, so I tend to just follow the AP Calculus AB curriculum and go at a slightly slower pace or less in depth. In addition, most students in Honors Calc are seniors, so it's pretty much a capstone course--they don't get college credit, so they'll have to take calculus again at college and I don't have to prepare them for another math course in high school either. So, I get a lot of leeway on what direction the class goes.

Just to keep my head above water (I've had 5 preps every year I've been teaching), I've not really taken advantage of the above very much. I've had the class follow behind the AB course and adjusted on the fly. Now that I feel more comfortable with the other courses I'm teaching, I'll be able to try some new/different things in there and move them into the other courses I teach if they work.

The two things I think I'd like to try at some point are: "Flipping the classroom" (record lectures and put them online for students to watch as homework, then in class work through the problem solving together) and "Standards-Based Grading" (tying assessment questions to specific skills and reporting student achievement based on level of mastery of each standard).

Since there are many blogs about SBG out there, I thought I'd try flipping the classroom and report my experiences of it, but the tech limitations (as mentioned in a previous blog entry) in our system are making me wonder if I can get it done. Among other things, youtube is blocked at our school. They do give us some web space to use, but they complain when you put too many files on there and it takes up too much memory. We're not really "allowed" to use outside resources. So, I could skirt the system and upload to youtube from my house and have the students watch the videos from home as well, but it would be going against the board policy.

I'll keep things updated here if/when I think I'm doing something interesting.

### Hipster Teacher

Not me

You know how hipsters really like the indie band that nobody's ever heard of? Good or not, an unknown band will beat out a famous artist any day. Then they complain when that band gets famous or "sells out" by having songs featured on TV shows or movies or whatever. At least they can still say, "I knew them before..."

Part of me feels that way with you guys (anyone reading this). I get a lot out of discussing teaching, math, and everything else through blog comments and twitter and everything else. I get why I should share my experiences here with my companion teachers (although I don't think many of them would actually do it), but then the hipster in me comes out and I don't want to give up my trade-secrets!

Maybe one day I'll get around to giving a professional-development talk to the teachers in my area about the usefulness of a PLN and the online component of that, but in the meantime, I'm enjoying the spoils for myself!

## Sunday, August 1, 2010

### My New Blog (About Me)

For those who don't know me well yet, I'm sure you'll come to learn a few things about me as this goes on. Here are a few things to get you started:

I'm a math teacher in middle Tennessee at a public school in a suburb of Nashville. I have all the calculus courses offered at our school (Honors Calculus, AP Calc AB, and AP Calc BC) and then usually a couple extra courses to round out my schedule (in the past, Algebra II, but PreCalculus this coming year).

I think of myself as a "helper." That's partly why I'm only now starting this blog, but am somewhat well known in certain circles for my comments on others' blogs. If you look through my twitter feed, I bet you'll find that most of them are responses to others. It's also why I've become a teacher.

I am somewhat of a wolverine wrangler. I find myself mentally correcting grammar/spelling mistakes I see online (and elsewhere) all the time, I have trouble with "real world" problems because it gets really complicated really fast and I get overwhelmed very quickly, I love math because you can set the rules however you want and then go nuts within those boundaries while safely knowing the boundaries are there (later blog post on this maybe?).

The 2010-11 school year will start my experience at 4.5 years. I left graduate school (seeking a PhD in math at Vanderbilt) mid-year because it wasn't getting me where I wanted to go and I didn't really need it to teach high school math like I actually wanted. So, I took a position at an inner-city school in Memphis for the spring semester before making my way back to Nashville and starting at a new school that had just opened and have been there since.

The school where I am now opened in 2004 with just freshmen and sophomores and added a new class each year until 2006 when I showed up to teach their first calculus classes.

The good: Being a new school, the facilities are pretty great. The area for the school is mostly made up of students whose parents work at the auto plants nearby (formerly Saturn, now Nissan) and these parents tend to value education and keep an eye on their child's progress. Being in the rural South, most of our students are very polite and respectful. The Williamson County School District is one of the best in the state.

The bad: During the 6 years that the school has been open, there have been at least 3 instances of teachers and students having inappropriate relationships. This has led to severe limitations on our tech access (many sites are blocked at school for both teachers and students including youtube, google image searches, twitter, and many other useful sites). At this school more than most others I've seen, "senioritis" is a huge issue. Being in the rural South, there is a small amount of racism that persists in our school culture--two years ago we had to ban all students from wearing the "rebel flag" after some students found a noose in a kid's pick-up truck in the parking lot.

I'm sure you'll pick up quite a bit more about me and my teaching philosophy and whatnot as I let more of myself out through this blog (and comments on others').

Both of my parents are teachers (dad teaches Chemistry at a university; mom used to teach English and History on the middle school level until she moved into the library), so it's something I've grown up with and I owe them a lot for my getting this far.

As a blog, I'm not too interested in building up numbers of followers or becoming popular. I plan on just throwing idea-noodles out there and if something sticks to you, it's ready. Good luck finding something useful!

I'm a dog person.

### Teaching Method: Dissonance/Randomness

One way that can really connect with students is adding a bit of "randomness" to the classroom. Especially in the math classroom where things are often kept very logical and dry, adding something different or even weird can make it memorable.

Ask students what they remember best from their previous math class and I bet it's something more along the lines of, "That time Jimmy fell out of his chair" or "When we sang that song" rather than some topic they were actually meant to learn.

Think back through your own past and the things that jump out at you were probably a little bit "off." But, this is especially true in today's culture. Consider the following cartoon that many of my students love:

It's just random enough to be funny. There are probably a few lines in there that stick out at you because they are just odd enough to make you laugh out loud.

Anyhow, in the classroom it doesn't have to be quite that odd, but something out of the ordinary can make something memorable. Here are some things I've done to make things stand out:

• Discuss something in a funny accent (real or imagined).
• Zis ees za unit circle. YOU VILL LEARN EET!

• Make up a word or name for a topic.
• The answer is 45 degrees, or as I call it 'Jimmy Blue Eyes.' (Remember--it doesn't have to make sense.)

• If your school dress code allows it, wear something crazy.

• Anything to mix it up a little and make it memorable!