Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Wednesday, January 27, 2010

Triangle Congruence Theorems

are so boring, and there is no nice way to teach them. A google search turns up a hojillion versions of "state the theorem and show an example." Here's what I did this year. It sucks and I'm looking for better ideas.

What makes triangles congruent? They're the same size and shape. This means all the sides and all the angles are congruent. The kids had to sit through one proof like this. What a pain, having to write 6 statements to show that two triangles are congruent.

In class the next day, I had them draw three segments with a ruler on scrap paper: one the length of their index finger, one the length of their ear, and one the width of their palm. I told them to construct a triangle with sides of these three lengths. I passed out compasses but didn't give them any instruction on how to do it. Some of them figured it out. A few of these guys showed everyone what they did with the giant compass and yardstick on the whiteboard.

We had a "discussion" about how there was only one triangle you could make with the three lengths. And when I say "discussion" I mean I said - did you notice that those three lengths would only make one triangle? That the angles were sort of 'locked in'? You couldn't just draw the other sides at any old angle? Then I did a lame little demo of how with 4 sticks, the angles are all wobbly and you can make a ton of different quadrilaterals, but with three sticks, you can only get them to make one triangle. I said something about bridges. Then I modeled and they practiced a bunch of SSS proofs.

Today was SAS. In previous years I passed out protractors, and asked them to draw a triangle with two specific side lengths and a specific included angle, then look around and notice how everyone ended up with the same triangle. I didn't feel this really sent the message that the angle has to be included. This year, I asked them to solve these two problems:

I don't honestly know how effective this was. Yeah "we" eventually made the point, but it took FOREVER, and I'm sure some kids got the point, but I'm also sure that half the class was just sitting politely waiting for the torture to be over. And some kids, of course, get downright indignant when you ask them to do something that turns out to be impossible. Which, whatever, but it's much harder to teach somebody who has concluded you are a crap teacher.

Then we practiced determining which other pair of corresponding things you would need in order to use SAS, then we did one example proof.

I know this sucks but I don't know anyone doing anything better. So let me hear it, hot shots.

Monday, January 25, 2010

Why Being Able to Look It Up is Insufficient

Eddie* came after school because he wasn't happy with his 6/10 on graphing log functions. I asked him to, without a calculator, sketch a graph of y = 2^x and y = log_2(x), then state the domain and range of each.

He did an admirable job sketching the graphs. He made a table of x vs 2^x, inverted them to make a table for the log function, and plotted points appropriately, connecting them with a smooth curve. An all around serviceable graphing effort.

Then for Domain and Range for both, he wrote "All real numbers." (Dun Dun Duuuuuun.)

Eddie, I say, your graphs are beautiful, but tell me how you decided on the domain and range. Oh, well, he says, I didn't really know. I just guessed, because sometimes it's all real numbers. OK Eddie, Domain refers to possible values of x. Look at your graph of the log function. What x's are not on the graph? Easy, he says, the negative ones.

Why should that be? Why can't you take the log of a negative number? Right now I'll give you an 8/10. You can have a 10/10 when you can tell me, in your own words, why the log of a negative is undefined. Invite its inverse function to the party. I hear he's a big help. This is not that difficult, you will just need to focus on the question for a little while. Maybe 10 minutes of concentrated effort. You are capable of figuring this out.

Eddie takes off down the hall and returns, breathless, 5 minutes later, bursting into my room Kramer-style, "Miss Nowak! You can't take the log of a negative because it's impossible! You just can't! Because it's not in the domain." Did he talk to someone in Precalc? Another teacher? I wonder. "I'm aware of that Eddie. I want you to tell me why it's not in the domain."

Eddie comes back a few minutes later, iphone in hand, and starts reading wikipedia at me. It was a drive-by Walesing. "EDDIE! No points for reading crap that you don't understand verbatim off wikipedia! Here's a secret, Ed: I ALREADY KNOW THE ANSWER! I'M NOT LOOKING FOR ENLIGHTENMENT HERE! I WANT TO KNOW THAT YOU KNOW! You have just spent 20 minutes searching for an easy way out, when in 20 minutes you could have figured it out for yourself! You can do better!"

I really hope that I go into school tomorrow and can write a happy ending to this story. Everyone send calming, logical brain waves Eddie's way.

Confrontations like this make me wonder if the "They can just look it up on their iphones" crew really know what they are saying. Knowing the domain of log(x) is not tops on anyone's list of crucial life skills...that's not my point. But insisting Eddie understand the why will make him re-investigate the behavior of the exponential function that is its inverse. And exponential functions are slapping you in the face every time you turn around, whether you realize it or not. Open your credit card statement? Slap. Open up a 401k? Slap. Nuclear waste will never totally go away! Slap. Your repairables are depreciating? Slap. The effects of one-child policy in China? Modeling the adoption of disruptive technologies? (Yeah, Disrupting Class includes graphs on a log scale.)  The Richter Scale - Slap. The point is we spend our lives (incoming: barely coherent analogy) in an intellectual ocean - facts float in and float out - sometimes we can grab onto a mechanized vehicle that will move us around faster, sometimes we are provided with a guide, more often, not. We can increasingly get our hands on any discrete bit of information we want. But to do anything with all this information, we kind of need either direct experience with the ocean floor or very good maps. I'm in the underwater expedition and map-making business.

*Eddie is not his real name, of course.

Friday, January 22, 2010

Marginal Improvements

Is something that would flow like this any better? I don't know. It's better for ME, but I don't know how many kids are going to come along for the ride.

And now I'm enamored with Sue's comment on the pyramid-slide post:

Yeah, we've gotta build this crazy thing. Pharoah's given us this diagram, and wants it 432 Pharoah-feet from the center to the edge. You know it's gonna be coming in at a 53 degree angle, the way we're gonna build it. He wants us to tell him how tall it's gonna be - I think he wants to see whether it will beat the last pyramid we built for him.

Thursday, January 21, 2010

What about Students Who Don't Have a Computer at Home?

This is a faithful transcription of a conversation I had today. I am not making this up. And this is pretty much how these always go.

"Miss Nowak. The speakers on my computer at home broke. It is impossible for me to watch the videos without speakers. I mean...I've tried! But it's hard!"

(Color commentary: This kid is hoping and expecting I am going to say, that's ok, you poor unfortunate broken-speaker child, you don't have to do your assignments.)

"I agree that does pose a problem. Let's think of some other options you have for using a computer. Do you have a lunch period?"

"No. I don't have a lunch in my schedule."

"Do you have a study hall?"

"No. I don't have any free periods."


Walk nonchalantly over to my computer. Look up student's schedule. (Pro tip: Kids think they can get away with lying about their schedule even when everybody knows you can look it up. See, they think you would never suspect they're lying, because you know they know you can look it up. What they don't know you know, is that they are capable of some impressive psychological jujitsu.)

"It says here you have a Study Hall every day 7th period."

"Oh...well...I always go to the computer lab to...type essays! So, see, I don't think of it as a Study Hall."

(Color commentary: He goes to the computer lab to play Linerider and read the Blizzard forums.)

" go to the computer lab for 7th period every day."


"Does your typing take you 43 minutes every day? Or can you spare 10 minutes for math?"

"No...Yeah...I can do it then."

"You will need headphones. Do you usually carry headphones with you?"

Other kid who is not involved in the conversation but is obviously enjoying himself: "They have huge boxes of headphones in the computer lab to borrow! You can just take them!"

And, scene.

If you want to find excuses for why you can't possibly teach class differently than you have for the past n years, you will find excuses. If you want to find solutions, you will find solutions.

Edit, because I got called out, and fairly: Look...I'm just weary of that question I used to title my post. If what you're doing is working for you, and your kids are learning well what you think they need to be learning, great. Awesome. You should say this: "It's not broke. No reason to fix it." And I'm not trying to convince you to. If you feel overwhelmed, you should say "This is too much to take in all at once. Can you suggest something smaller in scale to try out?" If you think this is an ineffective way for kids to learn, argue with me on the merits. I want to have that conversation. But if you live in a place like where I live, where I have taken surveys of classes that indicate home computer and internet penetration approaches 100%, and while at school, kids are tripping over a computer every time they turn around, "What about students who don't have a computer?" is not the rhetorical trump card people seem to think it is.

Yer Doin' It Wrong

Getting ready to introduce trig functions I opened the first lesson from last year and found this horror show.

Many thanks to Dan Meyer for the upgrades to my supposedly-educational crapola radar. (Don't you love how the textbook labels this "Egyptology" for us? That adds so much awesomeness to this problem. It really enhances the pixellated image.)

Now I don't really want to spend 30 minutes rehashing right triangle trig, just remind them about it before I peg a vertex to the origin and one side to the x-axis and start spinning the hypotenuse around. As of now I'm at the filtering-signal-from-noise stage of devising something better.

Saturday, January 16, 2010

Conversations with the Deficient in Attention

"Hi Miss Nowak."

"Oh! Hi ___. What's up?"

"What are you doing?"

"Getting ready to teach this class. How are you? How's your year going?"

"Oh fine. Hey is that a quiz? Can I try it?"

"Um,'s a Geometry quiz. Probably a little boring for you. Aren't you going to be late for 8th period?"

"Yeah maybe. Hey I like your nail polish."


"The answer to the first one is 80 degrees."

"I know."

"Remember that time in detention, when you let me sit at the big table with you to do my homework, and I started flipping through your notepad (pause for laughter) and found your (pause to laugh and catch breath) grocery list, and it said... tampons? That was hilarious."

"Good times."

"So I just stopped in to tell you I got into RIT."

"....What?! Wow! That's great! High five!"

"But if I want a hug, I should go to the English wing, right Miss Nowak?"

"That's right."

"Yeah I'm majoring in physics." (bell starts ringing) "Whoa, I better go."

Tuesday, January 12, 2010

English Teachers Think I'm Weird

Today in the Learning Support Center (the Holding Pen of Academic Purgatory - Keep Your Grades Up, Kid, or You Will Be Spending Your Lunch Period in Here) I got a leeeettle excited about a factoring problem. Precalc factoring always kind of scares and thrills me because sometimes I look at it and don't immediately see what to do. So this one was something like


and I won't spoil your fun, but when I got it, I let out a yelp and probably clapped my hands a little. The kid I was helping was mildly amused by my behavior, but this English teacher sitting at our table gave me this You Are So Weird look that really threw me. Whatever, dude. You wouldn't understand. I can read and write, but you probably walk around bragging about how you can't do math. Go back to your analyzing Camus for universal themes or whatever you were doing before I annoyed you with all the joy. (Edit: In hindsight, those last two sentences seem unnecessarily mean-spirited. Sorry, I didn't mean it that way.)

I probably should make this a separate post because it's going to get lost down here, but what are you all doing with your summer? Since applications are due for various summer things this week, and I haven't started them, I'm thinking that maybe I need a summer of not overscheduling myself such that in September it feels like I didn't have a break from work like I always do. I'd say "a summer of not thinking about math or teaching it" but I'm not kidding myself, that would be impossible at this point. I've been thinking about potential non-career related projects, such as

1. Learning How to Cook Meat and Fish. (Not learning how to cook - I can make lots of perfectly wonderful meals, and know enough to improvise - but was a sporadic mostly-vegetarian for ages so my milieu is veggies, grains, eggs, and sauces.) I might have to move temporarily to my parents' house for this, as they have better air conditioning and better equipment, and roasting and braising for one sounds both difficult and depressing.

2. Learning How to Bake Impressive Things like cheesecake and pie crusts.

Either of these could take on an edited Julie&Julia format where I just make something every day I don't already know how to make. Is there a canonical baking cookbook? Or maybe I will endeavor to become Alton Brown-like in my obsession over something bizarre like homemade pickles.

3. Going to Yoga/Meditation Camp for Grownups. There is a place I've been for a few days at a time, it's legit, not too woo-woo-make-a-donation, and they have a 30-day residential deal. Or alternatively maybe just stay home and Get Serious and Do It Every Day.

4. Take up some kind of other creative endeavor in a focused way, such as drawing or photography or picking up my violin for the first time in several years. At one time I was not-terrible at all of those things. I miss them.

I'm open to suggestions and interested in what other people do with their summers. I need a plan, or it's going to turn into the lost summer of ditzing around on the Internet.

Friday, January 8, 2010

For Your Low Tech Non-Clicker-Having, Non-Polleverywhere, Formative Assessment Needs

I came up with this my second year of teaching when our class sets of "clickers" were all the rage. I tried clickers once, and found them to be a total pain to plan around, set up, and use. Some people love them. I don't know.

Buy a pack of colored index cards and use a Sharpie to mark the yellow ones A, green B, pink C, and blue D. Arrange them in sets of the four different colors, like so:

When I'm planning to use them for a class, I set them by the door in a basket. The kids learn to grab a set of four cards as they walk in.


1. Quick checks during the lesson. Teach a concept, do an example, and then put up a new example. Make it multiple choice. Then poll the class. I usually do this in two steps: "Get your answer ready" and "3-2-1 hold em up." This sort of prevents the unsure student from waiting to see what everyone else is doing before deciding. Also for some reason, at the beginning, I always get 1 or 2 kids who think they don't have to participate. I just say, while everyone is holding up their cards, "So-and-so, I'm waiting on you..." until he relents and holds up a card.

If the answer was A, you want to see this:

2. Polling for other purposes. Examples: Can you send a text from your phone without any extra charges? Choose A for yes, B for I have a cell phone but texts cost extra, C for I don't normally carry a cell phone. I would only need to accommodate one student in this class:

3. Gauging the confidence level of the class. "How did that last example work out for you? Use the cards like a stoplight. Hold up green for "I'm good", pink for "I'm seriously confused", yellow for "somewhere in between."" This is how this class feels about the first day of the Logic unit:

I hope this is useful. Any questions, let me know!

Tuesday, January 5, 2010

Soldiers in the War on Innumeracy

I heard a version of this story from three different students on the day after winter break. "Tog" is slang for a local ski place. From a class blog comment:
This happened at tog - they have cups and creamers for .05 cents i called them out and i got yelled at...because aparently "As adults" they have it down as 5 Cents...and then i was like "but as people who r buying ur product...we have it as .05 cents, which is a 100 fold difference"
Way to go, "adults"!

Another student told me he tried to buy 100 cups for a nickel and argued with the manager for half an hour. "Then what happened?" I say. "They changed their sign."

All I did was show them this. It's not my fault.

Update: the youtube clip has since been taken down for mysterious reasons. Here it is on YTMND.

Friday, January 1, 2010

Blogs You Probably Aren't Reading But Should Be

These are some of my favorites and none of them have more than 100 GR subscribers. Get with it, secmathedublogosphere! School doesn't start until Monday, right? Clear the dead weight out of your readers and add some new voices. (fair warning, this is slanted heavily toward high school math.) If you're thinking "I/my buddy should be on this list!" please add it in a comment.

Blogarithm - Insightful and honest.

Concrete Classroom - Don your safety goggles and dig in.

Math Recreation is not for drowsy bedtime reading but holy crap I love this blog. Get yourself a cup of coffee and enjoy.

Mathspig - Has an endearing habit of posting a hungry pig grunting post in between real (knock your geeky socks off) posts.

Mrs. H writes a blog with a beating heart. If you're an all-business blog reader, you might not get her.

Megan Golding - I forgive her for stealing my hairdo because she has the best profile pic.

Meta Musings - Is asking important questions.

Mr. K doesn't post that much, but when he drops something it's always worth reading.

Stop Trying to Inspire Me - Because sometimes your inner curmudgeon needs nurturing.

Transformed Educator - Living in the intersection where I spend most of my time - tech and how we use it to improve instruction, not just make it noisier.