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!

## Thursday, December 29, 2011

### Math Lesson Formula

Okay so, seven years in, and I feel I am finally cracking this nut: how do you make any math lesson work for most kids under most circumstances? Throughout the year I have been tweaking most of my lessons to follow the same basic formula. Not that we do the same boring thing every day - there are infinity variations to make it work for me or a particular group of kids. Not that I'm saying teaching doesn't require a whole mess of skills besides knowing how to set up a lesson. Anyway.

I will illustrate with the most frustrating of topics : log laws. I can't think of a topic that seems more boring and pointless to most math teachers and students. I know their virtues as well as you, but let's be honest, 99% of your kids don't really need to know them for anything they are likely to do for the rest of their lives. I posted about it last year, but there was a piece missing, and now it really sings. To believe this works, you have to believe that the one doing the work is the one learning. Nobody gets much out of Miss Nowak doing dramatic performances of math problems and proofs other than Miss Nowak learning how to do dramatic performances of math problems and proofs under the withering attention of 24 bored and irritated teenagers. I don't want to give the impression that I'm giving them a worksheet and being all like, "You're on your own, kids! Time for me to kick back and drink coffee." Because I'm running around, scanning for common questions or points of confusion or missed connections, re-capping with the whole group every five to ten minutes, encouraging and validating, etc. But if you believe "teaching" = "lecturing" then you are not going to see the validity of this approach, and I can't help you.

Phase 1: Productive Struggle
Hook the new thing to something they already know or know how to do. Then make them do it. A few times. Let them discuss and work together. No reason this has to be done in silence. Whether calculators are allowed depends on whether the calculator will let them avoid the things you want them to remember and see. (This particular lesson is no-calculators.)

Phase 2: Generalize. Make them write whatever they have been doing with letters. This is harder for most kids than you'd probably expect, especially if they've never been asked to do it before.

Phase 3: Use it. Presumably this new thing you've discovered is good for something. Even if that something is obviated by ready access to a shmancy calculator.

Phase 4: Prove it. The hardest part for kids, and the hardest part for me to figure out how to get them to be the ones doing the work. I have had some success with this approach of setting up an organizer and basically telling them what to write. But they still need lots of hand-holding. But at least they are doing more than watching/copying a dramatic performance.

Phase 5: Lots and lots of practice. I want them to understand, but I also want mental automation of relationships and procedures. Because later they are going to use this stuff to learn something new.

I would like to say Phase 6 is apply it to a novel and interesting problem, but I'll be real, I'm not there yet with log laws. Though I am there with good projects on some other topics that lend themselves to applications. Give me another seven years.

## Friday, December 16, 2011

### All I Really Need to Know about Teaching

Dave started it. Here's mine: It's been on a bulletin board at eye level at my desk for seven years. It's a little embarrassing to share because it's more than a tad hubristic. But I think it helps me be better. I don't really need to have it hanging up because I can recite it from memory. Like a mantra. But it's comforting to have it there. Like a talisman.
“I have come to the frightening conclusion that I am the decisive element. It is my personal approach that creates the climate. It is my daily mood that makes the weather. I possess tremendous power to make life miserable or joyous. I can be a tool of torture or an instrument of inspiration, I can humiliate or humor, hurt or heal. In all situations, it is my response that decides whether a crisis is escalated or de-escalated, and a person is humanized or de-humanized. If we treat people as they are, we make them worse. If we treat people as they ought to be, we help them become what they are capable of becoming.” ― Johann Wolfgang von Goethe
(There is a teacherified version floating around by Haim Ginott. I like the original better.)

I don't take away from this "it's all about me." The take away is more like, "There is a ton that is in my control, and that makes all that happens here my responsibility." Which is maybe a little oppressive and maybe a little "duh." But I still like reading it every day.

## Monday, December 12, 2011

### In Which Ben Articulates My Reasoning Better Than I Could

...and then some.

#takethetest

#passiton

I'll add, since I didn't provide much in the way of explanation. The NY Algebra 2/Trig test is a horror show. It tests many things. Notation. Vocabulary. Procedures. Graphing calculator button sequences. It is not a test of mathematical understanding. I am pretty sure any reasonably mathematically-literate adult would sit down to take it, and within twenty minutes be all like, "What the HELL is all this CRAP? And WHY are we inflicting it on our young people? Get me the Governor! Oh wait, I am the Governor!"

I just want the guy to know what his organization says is important for college-bound kids to know. Thats all. I'm not even totally anti-test. I'm anti horrible, very-bad, no-good test.

## Sunday, December 11, 2011

### "Favorite No"

Just a quick share - I have tried this a few times this year, because I was looking for ways to more frequently but still quickly assess a whole class. It works really nicely. I don't have anything to add - Lea covers it all. Just watch.

## Saturday, December 10, 2011

### This Journey to Wherever...

...is about to get slathered in chimichurri. If you keep up with me on Twitter this is old news, but everyone else: I accepted a position at an International School in Buenos Aires for 2012-2014. Reactions fall into two camps: 1) Awesome! 2) WHY IN THE WORLD WOULD YOU DO THAT? so, here goes. While a young pup Navy officer I spent great chunks of time overseas, predominantly Italy and Bahrain (which seems like an odd pairing only if you've never been a Med/Gulf Sailor,) and hearted it. I like being a stranger in a strange land. I like navigating mysterious cultural waters. I like spending twenty minutes of gesticulating to communicate the idea "I think this thing is awesome but I am not willing to pay 20 dinars for it." "Okay fine you can have it for 15." "10." "12." "Fine."

So when I started teaching in 2005 the idea of an International School really appealed, but at the time, the reputable schools wouldn't consider teachers with no experience. I hear that's not always the case these days, but at the time, it was off the table for a few years anyway. Cut to seven years later, and you'd be right to wonder what took so long. I wonder that, too.

It took me about the past three years to Get Serious about making this happen. There was comfort with the known and fear of the unknown. There were two boys who captured my attention for a time but things just didn't go that way. I had to improve my math teacher fu. I had to gain and lose fifty pounds. Would that it had all not been so painful but it was all probably necessary. There were excuses but once I Got Serious I realized my excuses were really no thing.

For instance, I was all concerned about What Would Happen with the Cat, but here is what my point of contact at my new school said when I asked if it would be reasonable for her to join me:
YES!!!! Bring Kitty! (what is his/her name....that is very important for me to know!) I came here many years ago with 2 young German Shepherds and an old grumpy cat!  So you KNOW I understand bringing your little furry friend!
So yeah, excuses loom large in your mind but sometimes go poof when exposed to daylight. And they are just that, excuses, i.e. not the real reason you are hesitating. I think my real reasons (mostly fear) were alleviated by meeting and talking to and reading the blogs of teachers currently working at International Schools. These are real people not that different from you and this is their life. I have to especially thank Mimi who spent lots of time patiently answering my questions at PCMI and afterward, and offered lots of good advice.

So when I finally Got Serious I was looking for three things in a placement: 1. a non English speaking country so I could become fluent in another language 2. a stable, reputable school committed to supporting and developing their faculty and 3. a major urban center, for all the reasons people like living in cities. Beyond that I wasn't even particularly concerned about what continent I would end up on.

I joined Search Associates and joyjobs. Search Associates turned out to mostly be useful for demonstrating to potential employers that I was "serious" - since I interviewed and accepted an offer quickly, and never even had to go to a job fair. Joyjobs is a source of lots of good information, and frequently updated vacancy postings, and worth the small fee in my opinion.

In conclusion.... I'm ridiculously delighted with the way things turned out, and I can't wait to get there. This placement has lots to be excited about - the probability of teaching IB or AP, a block schedule, much smaller classes, no fire drills, no Regents nonsense...not to mention I won't have to scrape ice off my car any more. Considering that my new school hasn't hired a math teacher in five years, maybe all the delays make sense in the grand scheme. Now for a long six months of trying to learn Rioplatense Spanish and find new homes for all my stuff. (But not the cat. She's coming.)

## Tuesday, December 6, 2011

### f(t) is having a moment

I need you all to do something for me.

I never ask you for anything.

I need everyone.

EVERYONE.

Just go toot my buddy Governor Cuomo about how he should take the exam that decides whether my cherubs are college-ready. That's all. Here's a template you can follow. Heck you can just re-tweet me.

If you really, REALLY, TRULY want to have an impact? Repost this on your Facebook and your Twitter and etc. We are about to get exponential on his ass.

It'll just take a sec. Then you can go back to watching My Drunk Kitchen. Promise.

xoxox k8

## Saturday, December 3, 2011

### Still Relating Those Rates

First I have to express an obscene amount of gratitude to Bowman Dickson for illuminating what will be challenging for students learning related rates, and sharing how he deals with it. I basically just took his post and reorganized it into a lesson that will work for me. This post will probably make more sense if you read his first.

Second, I have been thinking this morning about what this lesson has to do with the recent discussion at dy/dan. There's probably a way to turn these into a problem we could pose without words through the cunning use of video production skills I don't have. It's really fun to think about.

Here's what I'm giving the kids. Here are relevant documents: handout for the kids, smart notebook file, ggb's that I made.

Update: Mimi Yang, a.k.a. thebomb.com, revised the cone tank ggb to reflect a constantly changing volume. That file is in there too.

---------------------------------------------------------------------------------------

SUPA Calc Lesson 4-6 : Fold this paper in half to hide the bottom half. Please don’t look at the example problem while we are doing the investigation. It will just get in the way of your learning.

1. Blow up a balloon!

2. Go here: http://bit.ly/calcballoon

3. Figure out everything you can about rates with the balloons. Record your observations below… (there is no one right way to do this. Make it make sense to you.)

4. Which one is more like inflating a real balloon and why? Write about it.

5. What is going on with the other one? Write about it.

Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?

(Here, we'll set up a solution with a diagram, givens, equation, etc, in a very structured way.)

7. The model depicts a 10-foot ladder leaning against a wall. If the bottom of the ladder slides along the floor at a constant rate, what happens at the top of the ladder? Why? How did you figure it out? Write about it below.

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?

8. Go here: http://bit.ly/calccone

9. A conical tank is filling with water. Use the slider to change the height of the water in the tank. How are the height and radius related? How are the height, radius, and volume related? Write about it.

10. Imagine you are standing in a municipal pumping station, watching this tank being filled with water. What do you think is more likely: (a) the height of the water is changing at a constant rate, (b) the radius of the water is changing at a constant rate, or (c) the volume of the water is changing at a constant rate? Why?

A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep.

## Wednesday, November 30, 2011

### Curriculum Writing for the Reluctant

I am really trying to beef up my Area, Surface Area and Volume unit for Geometry this year. It gets the job done regents-exam-wise, but it is so dissatisfying and I feel it could be so much better. Overall it basically boils down to plugging things into formula-sheet-provided formulas, and isolating variables in formula-sheet-provided formulas. There are some good things in there... we find composite areas and perimeters using aerial and other images, for example. Finding the areas of regular polygons is a good application of right triangle trig. There is an investigation of how areas change when dimensions change, which is serviceable but I suspect kids don't really see the big picture. We "do" volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres. My students tend to do very well on questions from this unit on the Regents exam, and I don't want to mess that up, but in this case I don't believe that the exam is valid for measuring understanding.

These are the kinds of things I want them to understand and/or be able to do:
• what physical property you are actually calculating when you calculate a volume or a surface area
• why the formulas are what they are
• how changing a 2D or 3D figure's dimensions affects its area or volume. for example, I think they understand that if you order a pizza that has twice the diameter, you get way more than double the amount of pizza. But I don't think that intuition has any ties to math class.
• isolate a variable in a formula. for example, solve S = lw + wh + lh for w.
I have a bunch of great resources and problems and tasks that I have collected in my Evernote over the past few years that could potentially work very nicely here.

1. Design a new label for a given tennis ball canister, oatmeal canister, or soda can. (a) Create a prototype label so that it covers the entire lateral surface of the canister with little to no overlapping paper. (b) Congratulations! The company chose your design and wants to produce 100,000 labels. Calculate how much material (paper, aluminum, whatever) you will need to order.

2. This game at NLVM is quite nice for challenging your intuition about how volumes are related to dimensions.

3. This video features people with charming accents complaining about how the volume of their chocolate bar decreased even though it appears that the surface area stayed the same or possibly increased. I've shown this in the past and found that students are unable to articulate what these people are upset about using the word "volume" (much less intelligently discuss surface area.) The word "volume" from math class is not connected in their brains to "how much stuff inside."

4. Starting with a piece of copier paper, roll it into a cylinder both the long way and the short way. Will it contain the same amount either way? If not, which way holds more? Mathematically justify your response.

5. Starting with a sheet of copier paper, cut four congruent squares out of the corners and fold up the sides to make a box. Who can make the box that holds the most? Kristen Fouss did something like this but in pre-calculus. Geometry probably doesn't need to get into deriving and optimizing a polynomial equation.

6. Starting with a sheet of copier paper, design, cut out, and assemble a right pyramid with a square base. First pass: any pyramid will do. Second pass: make the area of the square exactly ___. Third pass: make the overall height of the pyramid a specified length. Present your best-looking pyramid, including the area of its base, its overall height, its lateral surface area, its total surface area, and its volume.

7. Investigate what happens to area when dimensions change. What happens to volume when dimensions change. (Somehow.)

8. The car talk fuel-tank problem.

9. Some version of the PCMI volume/surface area problems. (If you know the perimeter and area of a rectangle, can you determine its dimensions? Are there any rectangles whose perimeter = area? If you know the surface area and volume of a rectangular prism, can you determine its dimensions? Are there any rectangular prisms whose volume = surface area?)

10. Derive the formula for the volume of a sphere without calculus. From Exeter Book 3. Would pose quite a challenge for my students. They would not be able to do it on their own. In fact, as it is written, it would completely mystify them.

What I am struggling with and probably will be for the next week or so is, how do I take any of these things and fit them into a logical, coherent unit of study of surface area and volume? NY/my district/my school does not provide us with a curriculum. We have : a list of standards, a collection of previous exams, a pacing calendar, and a kind-of crappy textbook, which are all useful in their own limited ways, but none of them tells you what to do in class. I have lessons already written that get it done, so there is no incentive to bother, other than it bothers me when I feel I could be doing a better job. Part of the dilemma is, I feel that any of this would have to be added to what I already do, not replace it. I still need them to be able to, for example, identify that the bases of a prism are the parallel sides, even if they are not on the top and the bottom. And I'm already about two weeks behind in this course.

How do you take a compelling resource and turn it into an effective lesson?

## Monday, November 21, 2011

### IVT the sensible way

I heard of a lovely activity for pseudo-discovering the Intermediate Value Theorem at a recent compulsory workshop for the calculus course I teach. It has everything i like in a thing. I do not have a record of the name of the teacher who presented it (and even if I did, I don't know if he wants to be famous on the Internet) so if you are he, please email me if you want credit.

(begin basic text of student handout/activity:)

The intermediate value theorem states: If a function y = f(x) is continuous on a closed interval [a,b], then f(x) takes on every value between f(a) and f(b).

Think about what you remember of conditional statements (from your geometry course:)

1) State the hypothesis of the IVT.

2) State the conclusion of the IVT.

3) In the following, be sure to use the endpoints (a, f(a)) and (b, f(b)).

A. Sketch a diagram where both the hypothesis and the conclusion hold true.

B. Sketch a diagram where the hypothesis is false, but the conclusion is true.

C. Sketch a diagram where the hypothesis and the conclusion are false.

D. Sketch a diagram where the hypothesis is true, but the conclusion is false.

(on to the back of the page)

4) Which one is impossible to do? Explain why.

5) Compare your diagrams with a partner. How are they similar? Different? If they are different, are they both valid?

6) Is any real number exactly 1 less than its cube?

A. Create a function whose roots satisfy the equation.

B. Find f(1) and f(2). How do you know there is a point (c, 0)? What do you know about c?

## Friday, November 11, 2011

### When the Problem Does the Teaching

This recent Geometry lesson is a good example of setting the kids in pursuit of a problem, where they have to learn the thing you want them to learn anyway in the process. (That wasn't that eloquent, sorry, I will illustrate.) On Tuesday, we developed the rule for the sum of the angles in a polygon by the chopping-into-triangles technique that many of you are probably familiar with. The next day I wanted them to be able to find the degree measure of one angle in any regular polygon, so I set them this task, which I stole from a PCMI problem set:

I did not include that first question when I did this in class, and many students stumbled over restricting their search to regular polygons. So I added it after the fact for next time I give this problem.

There are lots of these triplets to find, so all the kids met with some success pretty quickly. It is also a little like finding a pearl in an oyster, so they were rewarded and motivated to keep looking. Regular polygons are hard to draw, so with a little reminding and prodding, they started to find the degree measure of one angle in a regular pentagon, hexagon, octagon, etc (the whole, covert point of the activity, anyway! Yay!) I had them add their finds to a whiteboard everyone could see as they were discovered. They also wanted to verify by using the Smartboard to render regular polygons perfectly, and fit them together like puzzle pieces, which I was happy to allow them to do. This was actually a pretty great class - some kids conjecturing likely candidates, some kids armed with calculators cranking out angle measures, some kids organizing all their finds, some kids going up to the smartboard in groups of two or three for visual/spatial verification. And when I assessed them the next day, no one had any trouble understanding the question or coming up with correct angle measures. This problem is a keeper.

## Monday, November 7, 2011

"What is 1 Radian?" Try it. Dare ya. They'll do a little better with: "What is 1 Degree?"

## Wednesday, November 2, 2011

### Completing the Square

I made some final tweaks to Completing the Square in Algebra 2, and I find it just amazing the difference between this year and previous years, in that so much more often now, I just know what to do.

It doesn't feel like I changed all that much, but the kids just get it. I don't think it's a difference in delivery or anything. Here are the important bits.

First, I took two days instead of one. Go to hell, pacing calendar. The first day is just to see the pattern and get the idea with easy easy problems. a = 1 and b is even. The second day we work with a != 1 and odd values of b (fractions. eep. but the kids are even dealing with fractions okay.)

Tee it up: why would we want to do this? It saves us time.

Look for patterns. The kids fill out this whole table all on their own. I don't say a thing. I convince them to try and focus by telling them that if they really get how this table works, their lives will be a million times easier for the next six months. It's an exaggeration but you need them to engage here.

The bottom three rows were new this year. Hardly any students needed an assist with the * rows. I was surprised. The important part - the mathematics - was the ** row. Again I was surprised that they mostly worked this out on their own. There were some kids, I had to point at numbers, and say "Look at the 10, the 25, and the 5. Look at the 14, the 49, and the 7. How are those related? How can you write that relationship but use b?"

Once we're all on board with the table, we put the pattern together with "the genius method" from before to solve a simple quadratic in standard form:

And that is basically that. We practice a bunch of easy ones. The next day, we come back and practice a bunch of really hard ones.
Here are the smartboard files: Day 1, Day 2.

I just find it stunning that you can plan out a lesson 95% correctly and it will miss most of your kids. And you can change one little thing - add three rows to a table - and now all the kids basically get completing the square, think it's easy, prefer it to other methods of solving quadratics, and tell you why they don't get why this is such a big deal. I feel a little like I have super powers.

## Sunday, October 23, 2011

### Triangle Centers

I'm attempting to incorporate triangle centers with constructions. locus, and parts of coordinate geometry, because they all go together anyway. This is one of the things I love about Geometry - there are many topics, but they all relate to each other.

Some of my questions are exact replicas of those found in Exeter Math Book 2. I'll attach all the documents I used, and I know they are imperfect. But this is the basic progression:

Incenter
Where is a point that is equidistant from all sides? Conjecture that there is such a point on Nspire.

Figure out how to construct it by pouring salt on triangles.

Circumcenter
After doing some problems to review that there is such a thing as the Pythagorean Theorem...Where is a point that is equidistant from all vertices? Use coordinate geometry to see that there is such a locus of points on the coordinate grid.

Centroid
After a detour into deriving the midpoint formula... Where is a point that is the center of gravity in a triangle? Use area and coordinate geometry to dissect a triangle into two equal areas.

Demonstrate center of gravity by demonstrating that the triangle will balance on that point.

Where is the intersection of the altitudes? This point is not interesting, so get through it as quickly as possible. But take the opportunity to teach what an altitude is.

Then there's an Nspire document with a summative review sheet so that they can keep straight the four different, confusing kinds of triangle centers.

Next we are going to do the famous locus scavenger hunt, after a day of basic locus notes, which will allow me to basically skip the whole pointless locus unit altogether.

And then I think I am going to have the children make their own locus-based scavenger hunt for the towns of Fayetteville and Manlius, for the fun, and for the backwards learning. Though I don't happen to have those documents prepared yet.

## Thursday, October 20, 2011

### Icebreakers

It is a little-known fact that I hate icebreakers. "Little-known" in the sense that I don't mention it every five minutes, but if you know me, you could have probably guessed.

Anyway. In previous years I have been frustrated that students in my classes, by the end of the school year, might not even know everyone's name in their class. So this year, I resolved to compel them into an icebreaker every time I changed the seating arrangement.

Well, first of all, what a coup if you have a lesson that depends on discussion and conversation and talking. Because, there is no teeth-pulling necessary to get people to talk about themselves for a couple minutes. And once they have broken the ice, so to speak, you don't have to worry about that component.

And second, how entertaining, and what a fun way to learn about your kids. Yesterday I changed all the seats and I asked the Geometry students (the youngest and traditionally most taciturn of my lot): if you were a Geometry vocabulary word, which one would you be and why? (with the caveat that in 5 minutes I was going to randomly select students to introduce their partner to the group.)

"A line. Because he just keeps going and going."

"A polygon. Because he has many sides."

"A circle. Because she is well-rounded."

It just went on and on. It was adorable.

## Saturday, October 15, 2011

### You Don't Even Know What Math Is Yet

Alex McFerron:
I tutor mathematics to kids and I can't tell you how many times I hear the words "I'm not good at math". This is from very intelligent kids who aren't out of high school. Honestly, I want to say, you don't even know what math is yet. You don't know the first thing about it or your ability to do it or not do it. I want to tell them that no one is good at math who doesn't work at it.
I think that what separates math people from non-math people in our culture is that math people continue doing math and don't spend anytime thinking they aren't math people. They just keep going on the journey. The older I get, the more it is obvious that a lot of really capable people quit the journey. I admit, this journey isn't for everyone. Its hard work. It takes focus. You have to want to do it. The financial rewards aren't really there in proportion to the work. But what really hurts me is that there are people who want to do it and quit. They change majors, go home, seek other work all because they have it in their heads that they lack talent and aren't naturals. The profession loses when this happens.
Alex has a unique perspective on Mathematics and I always learn from her insights. Professionally, she's a software engineer, but she is also a math enthusiast and evidently tutors kids as well. I am left wondering as a result of this post, what is our role in promoting the "not a math person" label? And what can we do to entice kids to not quit the journey? I have to think that choosing puzzling but accessible tasks is paramount, but hey I do that, and many of them still check out anyway. What else?

## Thursday, October 13, 2011

### The Nspire Has a Complex Mode...

...not just on the CAS version, but on the numerical version too.

...it's not disabled in test mode.

...it works well. (The TI-84 has an i button, but in my experience it's unreliable.)

...oh *&^%.

Yeah. Right.

 If there is a button I will find the button.

 There it is.

Here are some things it can do:

So after I threw out every assessment I used to use for this unit, I settled into the place of "What the *&^% do I do now?"

But it's good. Good! Good, I say.

It made me give some super-serious thought to what the complex number system is good for.

It's good for solving equations that don't have real solutions. Hello there, quadratics! We meet again. A little earlier this year.

It stops the Fundamental Theorem of Algebra from breaking.

It's full of numbers that represent two dimensions.

I can work with that.

## Wednesday, October 12, 2011

### This Batch of Children Will Do Just Fine

They didn't seem overly impressed with the origami time-lapse, but I found three (THREE!) secret-admirer origami presents on my desk throughout the day. A water bomb, a box, and a more-different box that looked kind of tulip-y. (When did they find time to make them in the midst of my dazzling instruction? It's a mystery.) (They were impressed by the ~500 views. "WHO ARE ALL THOSE VIEWS?!" "They are mostly my Mom." "Naw Miss Nowak has like 1000 Twitter followers." "WHAT?!" "Yeah she's apparently kind of a big deal." "Why?" "I have no idea.")

A child was making Vi Hart sketches of a triangle with an inscribed circle and infinity-more inscribed circles while waiting for his classmates to finish simplifying $\frac{4+\sqrt{-48}}{2}$. Me: "YOU WATCH VI HART?!?!!" Him: "Yes, I love her." Me: "ME TOO! YOU ARE MY NEW SECRET FAVORITE!"

One child keeps DOING SOMETHING to my rubik's cube in like four moves that I can't undo anywhere near that quickly and it's driving me literally insane. (Just changing the middle square color on all six faces? Wtf. I should be able to undo that THE RUBIKS CUBE HAS INVERSE FUNCTIONS FOR CRYING OUT LOUD.)

Calculus is still a very timid group, extra afraid of being wrong slash anyone knowing they are wrong, but had lots of fun looking into the "there are always two antipodal points on a great circle at the same temperature" thing today. (Which true confession I'm not totally sure is a good use of Intermediate Value Theorem day but I kind of suspect is it, so I went with it.) There was a distinct, entrenched camp arguing "no," and a diverse, poorly-organized camp valiantly arguing "yes," and the "no's" were very gracious when they realized their position was untenable.

They will do just fine indeed thank you.

This all makes up for how I got observed today during quite possibly the worst-ever Geometry class of the year. (Observation the first day back from a four-day weekend? Who does that? We spent 45 minutes recalling the mathematical differences between their assholes and a hole in the ground. Not really but it took ten minutes for someone to summon a vague recollection of having heard of the pythagorean theorem before. I wish I was kidding. Needless to say we did not even get to the dazzling lesson that was planned. Why didn't she come last Thursday? We were pouring salt on triangles. It was epic.)

## Sunday, October 9, 2011

I purposely didn't bring any school work home after I spent all day Saturday in school, planning. (We can't get into school on Sundays. Well, we can, but you have to disarm the alarm system, which I am too afraid to try.) I've had trouble this year Turning It Off, and I was trying to force myself into some down time.

Here's what my down time looks like: (or will, until it gets pulled down for copyright violations)

Also, if you're an Nspire person, here's a file with the area proof of the pythagorean theroem.

Days off are exhausting.

## Saturday, September 24, 2011

### Destroy My Problem Set, Please.

Students should be able to complete this in groups without too much assistance from me. We already had a lesson on what the cube root means and simplifying cube roots to simplest form, which was also a refresher on how to simplify square roots. When it says "check on a calculator" they will have access to a CAS calculator for this lesson. I realize that if the roots don't come out rational, the calculator displays the answer with a fractional exponent. I don't know what to do about this yet. Maybe I will just put the answers on cards they can check instead of futzing with the CAS's.

The goal is for them to practice multiplying and simplifying, and investigate multiplying conjugate pairs to set us up for rationalizing denominators, both monomial and binomial.

I know the two questions at the end are kind of weird but it seems like a shame to waste the opportunity.

So...

• okay?
• Crappy?
• Suggestions for decrapifying?
• Am I missing any relevant opportunities to make connections?
• Or show multiple ways of seeing something?
• How are the kids going to noob this up in ways I'm not anticipating?

Also, I am a Latex beginner so no making fun of my typesetting. It took me four hours to make this.

## Monday, September 19, 2011

### Everything is a Parabola

Lined up the kids at the board along a big number line. Everybody picked a number : their "x." I said, "Shawn's x is at 2. However far away you are from Shawn, move that many floor tiles into the room, perpendicular to the board." They move into a lovely approximation of the graph of f(x) = |x - 2|. "Hey what kind of a shape are you guys making?" "A PARABOLA!" Practically in unison. FACEPALM.

## Saturday, September 17, 2011

### Geometry Project Follow-Up

So I think I know what I'm going to do with all these Powerpoint files submitted for the Geometry project. I'll grade them as promised, but when the kids come to class Monday, I'll...

First, show them an uber-presentation of the best possible answer to each statement. This way they will get their feathers fluffed up when they recognize their work on display, but we will also hopefully resolve any lingering doubts. I did not necessarily confirm or deny all their questions while working on the project, because it was more important they keep thinking about it, and the teacher giving the answer from on high shuts down thinking. Most often this sounded something like, "Miss Nowak, will you check my ten Always/Sometimes/Nevers and tell me if they're right?" Me: "No...but I'm happy to discuss with you any specific questions you have about what words mean or what would make a statement true or false..."

Second, each group will get 4 copies of a printout of someone's presentation slide that exhibited a misconception. The task will be, what could you do to help clear up this group's misconception? What could you say/draw/show to convince them that they misunderstand, and also help them understand?" Then, we will jigsaw to mix up the groups, and everyone will share what they learned with their new group.

I always felt weird about the flow of : Work on Project, Submit Project, Teacher Grades Project and Hands it Back, and that is The End. I think there's much to be gained from re-visiting this work and catching all those lingering misunderstandings.

## Friday, September 16, 2011

### Geometry: Points, Lines and Planes

I was going to write up a description of this project we just wrapped up in Geometry, but luckily Allison already did it! (Maybe go read that post if the rest of this doesn't make any sense.) I liked it because it gave students an opportunity to get messy with points, lines and planes. There was a whole lot of productive struggle going on in my Geometry classes.

Here are some nice examples of student work:

Here are some examples that show misconceptions! I'm going to do something with these. Not sure what precisely yet.

Some kids really did seem to be enjoying themselves while learning, but there was also an awful lot of complaining going on. Managing all of their digital photos and getting them from their phones and cameras into their accounts was a bit of a hassle, so I can appreciate the frustration there. But it was good! There were lots of conversations, which if they were blog posts or magazine articles would have titles like:
• Why You Are Making Us Do This
• It Is No Fair Making Us Think
• We Would Prefer to Just Fill Out Worksheets That Ask the Same Questions Over and Over
• Why Miss Nowak or Anyone Would Like This Job
• If You Give Miss Nowak Your Phone She Will Change Your Wallpaper to a Math Picture
However, there were also hopeful conversations like:
• I Would Rather Be in This Class Where I Actually Learn Something Even Though It's Harder
• My Friend is Jealous She Doesn't Get to Do This Project
• Holy Crap, a Three-Legged Table Can Never Be Wobbly

### Algebra 2: Graphing Absolute Value Functions

My goal with this was for students to understand "why the V shape" for absolute value functions. I think it will take three days. The TI-Nspire is used. This leads up to something very much like this, but with much more scaffolding ahead of time.

Phase 1: Learn lists and spreadsheets, data and stats skills on nspire:

1. Students watch and follow along on their handhelds "Data and Statistics: Adding and Rotating a Movable Line" tutorial.
2. Close file and do not save.
3. Send students document Squares.tns containing the case vs gray squares table from the beginning of the No Sleep Til Brookline problem set. Display these directions:
• open file
• add a Data & Statistics page
• create a scatter plot of case vs gray squares
• add a movable line, and try to get it through all the points
• remove the movable line
• type in the equation you know fits the line
Phase 2: Understand why absolute value graph has a V shape and what it means
1. Write a number line on the whiteboard from -8 to +8 or even longer if possible.
2. Students line up against wall (however many will fit - maybe a subset of the class). Students note their position along the number line written on the whiteboard. This is your "x."
3. Choose whoever is at 2 to be the vertex. Let's call him Jake.
4. Give Jake something to hold up like a flyswatter.
5. When I say go, you are going to move away from the board. The rule is, however many floor tiles you are away from Jake on the number line, you are going to step that many floor tiles into the room away from the board. You'll move perpendicular to the wall. Take a moment to decide how many tiles you will move....Go.
6. Students move into a V shape.
7. Display in succession on projector:
jump up and down if you are a solution to |x - 2| = 5
jump up and down if you are a solution to |x - 2| > 5
jump up and down if you are a solution to |x - 2| < 5
jump up and down if you are a solution to |x - 2| = 3
jump up and down if you are a solution to |x - 2| = -3
8. Let's call the distance you stepped into the room y. What is the equation of x vs y?
9. Reset students back to line up against the board. (Or get a new group up there.)
10. Get the flyswatter away from Jake.
11. Our new function is |x + 3| = y.
12. Who gets the flyswatter? (Let's call her Jill.)
14. Display a few "jump up and down" questions.
15. Note your position! Come to the smartboard and enter your name and position in the Lists & Spreadsheet.

Phase 3: Create Absolute Value Scatter Plot on TI-Nspire
16. Send everyone the lists&spreadsheet with name, position on NL
17. column 3 call it distJill
18. In formula cell, how can we calculate everyone's distance from Jill? Try difference...note problem with negatives.
19. Show how to enter absolute value function: template or abs(position - -3)
20. Show how to sort the whole spreadsheet from closest to farthest
22. Can you get the dots to arrange themselves into the V-shaped graph?
Can you add the function that goes through all the dots?
Can you add a horizontal line function that represents "4 away from Jill"?
Can you shade the region that includes people that were within 4 spaces of Jill?
Can you add a vertical line (Plot Value) that represents the average distance from Jill?

Phase 4: Apply skills to novel problem
For the past week, I left this sitting out on a desk in my room:

Something like 100 students entered their guess. I was able to copy the column containing their guesses from Google Docs into an Nspire lists and spreadsheets page. They'll get this Nspire file and these directions, and have the period to do what they can with it.

Phase 5: Transformations on the Absolute Value Function
Students will spend time playing with this Nspire file with this investigation, to understand how changes to the parent function transform the graph. To assess, they will try to match the pictures with a function.

## Thursday, September 15, 2011

### Algebra 2: Solving Absolute Value Equations

You know how you can show them this way, all justified and with lots of practice untilblueintheface:

But then a couple days later half of them will do this

and the other half will do this:

So, I stopped teaching it that way. I'm starting with something much like what most of us probably do:

Allison lives at 15 Sycamore Drive, and Sarah lives 8 houses away. Where does Sarah live?

But then, I'm sticking with that model for all kinds of problems.

and

Earlier in the lesson I made them write it out in words, i.e., "the distance from 200 to 3x is 896."

It was more of a pain initially, and not the most effervescent lesson I have ever delivered, but MAN, it did the trick. No more of that autopilot, forget to write two equations, forget that absolute value can't equal a negative nonsense.

This is an idea I stole wholesale from the article "A Conceptual Approach to Absolute Value Equations and Inequalities" by Mark W. Ellis and Janet L. Bryson, Mathematics Teacher April 2011, Volume 104, Issue 8, Page 592.

Inequalities are a natural extension of this concept. Where on the number line are all the values that are more than 896 away? That are less than 896 away?

## Monday, September 12, 2011

### My First Days of School

My first day schtick has changed quite a bit over the years. The first year or possibly two I drank the Wong kool-aid because I was terrified and I had no idea what else to do, and also my school gave us all the Wong book for free. That didn't work. It probably works for the Wongs but it didn't work for me. I felt like a pretend drill instructor. (Wong. Wong-wah-wah-wong-wong.)

Then for a while I got kids to fill out a sheet about themselves that I stole from Dan Meyer. I had hopes these pieces of paper would capture the essence of each child, and had every intention of perusing them leisurely with loving tenderness. In reality, I just scanned the "Anything going on you want me to know about?" section so I'd have a heads up about all the imminent divorces and cancers and then I threw them in a drawer. Since that only took about ten minutes of day 1, I would just start in on a lesson for the rest of the period. Giddy up!

This year is the first one I feel I started out semi-competently. The cluebird is circling and coming in for a landing. My only measure for this is the proportion of students that will make eye contact when they talk to me.  I considered what I actually wanted to achieve from the interaction (imagine that): I want them to know they can be successful. I do want to start getting to know them, but just as importantly, I want them to know that I want to know them. I want to know who has the tech in their pocket that we can exploit for the learning. I want them to think about what they want to improve about themselves as students, and I want it to dawn on them that it's in their power to change whatever that is. I want them to feel comfortable in my room, to not feel trapped and helpless, at least know some of their classmates' names, know that I expect them to work hard but I'm pretty darn reasonable, and to tell me where funny things are on the Internet. Okay I'm going to stop with this paragraph now because it's getting out of control and was obviously was too ambitious for 43 minutes with 121 strange kids. Ahem.

Phase 1: Snowball Icebreaker
(I saw this on somebody's blog but I can't find where. Speak up and I'll happily give you credit.) Everyone gets 1/2 sheet of paper and writes three distinctive (you will have to give examples of distinctive vs non-distinctive) characteristics about themselves, but not their name. Everyone crumples up the paper, and we toss them around the room (suggested verbiage: "My biggest rule is that you show respect for each other and for me. If you can do that we will get along fine. So there will please be no whipping your snowball at anyone's face. Now, all together, let's pick a direction and gently fling our snowballs.") Everyone picks up a snowball and uncrumples it. Their job is to find its owner and write his/her name on it, and be prepared to introduce this person to the class. Their job is also to facilitate being found. As soon as they do both things, they sit down in the closest empty seat. (Suggested verbiage/modeling: "This is what I don't want to see: (hold up sheet in someone's face) 'IS THIS YER SHEET?!' (pause for giggling.) This is what I do want to see: 'Hi! My name is Kate. What's yours? Audrey? Nice to meet you! Tell me, Audrey, are you allergic to wheat? No? That's too bad. Although fortunate for you, I suppose. Ok, is there anything you'd like to ask me?")

Once the dust settles, start asking kids to introduce each other. This was great fun, because I kept asking for more details that would let them show off a little and/or amuse us all.
"This is James. He works at Wegmans and plays guitar."
Me: "Cool! I wish I could play guitar. What is your best song?"

"This is Angelina. Her brother is a pilot who lives in London."
Me: "How old is your brother?"
Angelina: "30"
Me: "Is he single?"

So that all took 15-20 minutes. I liked it because it fit one of my goals for this year: everything we are doing is for a reason, and we will follow up on things and not drop them without processing them and assessing you and making sure we all got the point (more on my lofty '11-12 goals in a later post.) It also hit several of my goals for day one: some kids learned some other kids' names, it was low-stress and all the chatter made the room feel inviting, I got to learn a little about them and they saw me being interested.

Phase 2: Distribute Books and Collect Data
Then I passed out my heavily modified Who I Am sheets, and the kids worked on them while I passed out textbooks and made smalltalk. This exercise felt different than previous years, because they weren't filling them out all scared in stony silence, but there was productive chatter and informal sharing and it just felt nicer.

Since I asked them questions I actually cared about, it was no drudgery to take all their filled in sheets and read them thoroughly and enter them in a spreadsheet. I sort of had the idea that analysis would yield some interesting things but I don't know what I was thinking. There are six days a year where three of my students have a birthday at the same time...that's kind of neat. I got a roughly even mix of math-likers and -haters:

And we have mixed opinions on whether Mathematics is invented or discovered:

And I suspect that some of the kids who claim they like math, don't know what it is, and like it for all the wrong reasons:

But we will see what we can do to change that.

Phase 3: Some Blah Blah
I spent the last 5-10 minutes telling them about supplies they need and what to do if they have to use the bathroom, that sort of thing. We didn't do any math. I don't feel bad about it. I feel better about the way this year started than I ever did. I'm excited about how we're learning, too. More soon.

## Wednesday, August 31, 2011

### Pre School Year Jitters

I don't think I've ever posted at this point in the school year before. To recap, I'm coming up on year seven of this delightful and terrifying profession.

I appreciate that I get the opportunity to totally start over every year. And I don't even have to go to the trouble to press RESET and pay a new quarter.

Today I looked around my classroom and realized it is TOTALLY TRICKED OUT. Smartboard. Document camera. TI Navigator. (Really, Nowak, you have no excuses. None.) Gigantic Darth Vader poster. (Fun Miss Nowak fact: my childhood dog was an all-black ChouChou/shephard mix, with a black tongue. Named Darth Vader. We just called him Darth. He was great.)

I have things to complain about. My Regents classes that are supposed to be a healthy mix of accelerated and not kids are pretty much totally NOT. Out of 54 Geometry kids, I have one 9th grader. ONE. Out of 42 Trig kids, I have four 10th graders. FOUR. I don't know if this is some kind of conspiracy or scheduling fluke or what, but this year is not going to be a walk in the park from the classroom management or instructional perspective. NOT. But they're in my computer scheduling thingie and I can see their names and pictures. And I love them a little already. And I can't bring myself to object although I realize Guidance is probably trying to see how far they can push me. Now, when I can only speculate about them. By reputation. Much like they're looking at my name on their schedule, and making predictions about me, by reputation. She's hard. She's easy. She's a bitch. She's awesome. Just make her laugh. Just be yourself. You're doomed. There is hope.

I posed an inservice class to my department that was basically "us hanging out and working collaboratively on difficult math with maybe food" and they were totally on board. I wrote up a proposal and sent it in. That class is going to be amazing if it gets approved. And maybe if it works and I'm a little bit lucky it will change the way we teach and change the nature of what the children learn.

Since something like 90% of our faculty are new in the last five years, our new-ish principal wants to open discussions of just about everything, including scheduling and grading. We talked about it in small cross-discipline faculty groups this morning, and that experience surprisingly did not leave me in abject despair. I gingerly broached my lunatic-academic-fringe stance on grading and they did not treat me as if I were radioactive.

Shocking!

You're doomed!

There is hope!

## Wednesday, August 24, 2011

### It's Amazing What a Good Night Sleep will Get You

Linear review lesson for Algebra 2 - version 2 - I like this much more better.
Here it is as a word document if you want to steal parts.

## Tuesday, August 23, 2011

### Linear Equations Review Lesson for Algebra 2

So, I just spent two days making one lesson for one class! Yeah, this does not bode well for this year. This pays serious homage to the PCMI problem sets by Bowen Kerins and Darryl Yong, who I already know are way funnier than I will ever be.

The goals for the lesson are the students remembering and being able to... (NB, they should already "know" all this from their previous Algebra 1 and Geometry courses)
• Explain the meaning of all the terms in slope-intercept form
• Write equations of horizontal and vertical lines and know how their slopes work
• Sketch the graph of a line given various kinds of information about the line
• Interpret point-slope form
• Write the equation of a line in point-slope form given its slope and a point on it
• Find the slope of a line given two points on the line, or its graph, or its equation in either form
• Know how slopes of parallel and perpendicular lines work
Open questions
• Is this too ambitious and going to scare the bejeezus out of the poor summer-addled adolescent brains?
• How am I going to assess who knows what as the students are working?
• What's the best way to organize the kidlets so that they might benefit from some cooperation? I'm thinking groups of three or four with minimal guidance about how they "should" work together.
• Aside from the lame jokes in the marginal notes, how can I bring more joy into this exercise?
• Are there better ways to ask any of these questions that make them more tangible?

As always, I welcome your thoughts.

## Monday, August 15, 2011

### New Blogs You Should Read

The authors in this list have one thing in common - I have met them all in person! I know, weird, right? So I feel utterly qualified to endorse them as smart, interesting, nice people. I will try to tell you something about them that is compelling and not readily apparent. Their blogs are relatively new, but all shaping up nicely. Check it out:

Tina (not sure if she wants her last name used) was at PCMI '11. She is SMIZZ-ART, yo, and one of those earnest, wholesome, authentic people who you suspect might not own a television and might spend her weekends hiking and canning seasonal produce. She could also fit in your pocket.

Bill Thill is one of the most thoughtful educators I have ever met. He will push back against all your assumptions and you can count on him to ask the most laser-like, insightful questions. Seriously, your bullshit is not safe within 50 yards of him. Also does a mean Chloe Sevigny impression.

Allison Krasnow, in the first conversation I had with her, gave me a brilliant way to manage homework to make it much more useful as a self-checking tool for the kiddos, but no more work for me. She's warm, genuine, and wears very cool earrings. Her new blog has four posts so far and I want to hug every one of them.

I met Paul Salomon at a School of Math session where we worked on a super-fun problem together. Paul teaches at Saint Ann's School, where they have no grades and the loosest of a math curriculum a.k.a. heaven. He writes a lot about the way math should be taught but he has a bit of authority in this arena, as he gets to teach math the way it should be taught. He's also a demon on Twitter and has been stirring the pot lately on the "how much paper/pencil computation is too much" front.

Chris Luzniak has really just dipped his toe into blogging about teaching math and running his school's speech and debate team, and I am hoping he sticks with it and starts writing some more. But this pattern fits with his persona - he mostly keeps is own counsel when it comes to teaching math and how to do it, but when he does weigh in, it knocks you over, and you wonder just what is going on in there the rest of the time. A real tour de force.

## Friday, August 12, 2011

### Good Problems: Follow That Diagonal

This is a sweet little problem because it is simple to state and understand. It seems like anybody should be able to make progress investigating it, but it won't be obvious to your smartypants kids.

Draw a 9 by 3 rectangle on a square grid. Draw one diagonal. How many squares does the diagonal pass through? Draw some non-similar rectangles with one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size.

I think I'm going to keep it in my back pocket for a day when I need to kill half a period. It might be nice for the first day of school if you like that sort of thing. I don't think I've seen it before. It was sent to me by Øistein Gjøvik - he has a post about it that includes access to a Geogebra file. (One benefit of blogging I would have never predicted: a cool Norwegian sends awesome math problems to my inbox.)

I have been on a bit of a Sketchpad bender since we used it at PCMI, so here's a sketch I made.

I am torn about giving guidance about posting solutions in the comments. I have one way to think about it that works, but I'm sure there are more and I really want to hear them. On the other hand, I don't want to spoil anyone's fun. So maybe if you want to work on it, resist looking at comments?

Another thing I'd like to hear about is, do you see this fitting into a curriculum? Or is it just a nice problem that doesn't have a home in a unit of study?

## Monday, August 1, 2011

### Summer Learning, PCMI Edition: Odds and Ends

This is a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program.

This is a catch-all for things I want to remember and post that aren't big enough for their own post.

Livescribe
In a 5-minute short, Cal Armstrong presented his use of Livescribe smart pens. I had a little "holy cow" moment during his presentation, because I've long dreamed of kids' recording their problem-solving process, but there's only one smartboard in the room, and writing with a mouse is hard. Enter the Livescribe pen which records both your writing as you write, and audio along with it. And they are only like \$100 a pop. I could ask kids to record a livescribe of them solving a problem as their reassessment, or record a tutoring session of them teaching it to someone else. We could put them on blackboard and build up a little library of these, or upload them to voicethread for feedback.

Google Forms for Recording Small-Group Discussion
I am pretty good at incorporating small-group or partner discussion, but I don't often have an efficient way for groups to share their thinking. One technique I noticed frequently deployed at PCMI was to give groups a link to a google form, so that each group could send in a summary of their discussion or response to a prompt. We aren't a 1:1 school, but it would be sufficient for each group to have one laptop for this purpose, and I'm pretty sure I could secure 5-6 laptops to keep in my room. Then again, I am supposed to have a TI-navigator system next year, so maybe I could just use it for this purpose.

Do you ever get stuck in a problem-writing rut? I do. Throughout, I was keeping track of all the tasks I saw that were something other than "find the missing value:"
• write an equivalent expression
• give an example
• show that two expressions are equivalent
• interpret expressions/equations in writing
• interpret a graph in writing
Metacognition: See How I Think
We spent a few days talking about what is metacognition, and ways for students to "do" metacognition. We participated in an exercise that I think could be adapted for students to use. In a group of three, students take on three roles: problem solver, listener, notetaker. The listener is NOT HELPING solve the problem, just asking the problem-solver to clarify their process and state it out loud. Meanwhile, the notetaker is writing down any evidence of metacognition or "thinking about thinking" that she hears. I think this could be very beneficial in helping students see how the same thought processes (making use of structure, considering extreme cases, organizing data, etc) cut across mathematical content, but I wonder at designing it in such a way that they can see the point. I need to spend some more time thinking about this.

The Vampire Animations
I worked on a lesson as part of our working group, and I don't think I'm supposed to disclose all the inner-workings of the lesson because it may be reviewed for publication as part of a larger project, but I do want to share this super-fun simulation we made. If you can use it, steal away.

Here is a "question" video of an infection spreading up to 64 victims:

And here is an "answer" video up to 512 victims:

Arts and Crafts
Finally, what is camp without crafts?

### Summer Learning, PCMI Edition: Formative Assessment

This is a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program.

Our middle session every day was called Reflecting on Practice, and it was basically a mini ed-school class. The focus this year was on formative assessment or assessment for learning. These types of classes are not usually my favorite (make a fake assignment! watch a video of someone teaching! talk about your feelings!), but in this case they were exceptionally well planned and executed so I didn't have much time to feel sorry for myself.

Biggest takeaway - there needs to be deliberate feedback, not attached to a numerical grade, built in to classes. Because when there's a number there, kids don't pay attention to anything else. (On the flip side, in the absence of a grade you run into kids not taking the work seriously, so giving feedback on their marginal efforts feels like a waste of time.) At least some of the time, the attitude toward assessment should be less "judgment day" than "a conversation about learning and understanding." I was influenced especially by two articles we read: Classroom Assessment: Minute by Minute, Day by Day and Working Inside the Black Box: Assessment for Learning in the Classroom (which does not appear to be available online.)

These simple ideas lead me to rethink the whole process for "level 1" quizzes - the kids' first stab at a concept on an SBG quiz. I spent way too much time re-designing what a quiz paper should look like:

That's super-helpful for you, right? Sorry. After I try this in class I'll have more to say about it with a nicely-typed up version. But the idea is, there's half a page for the student to do his work, and predict his score. The bottom half of the page is set up for structured teacher, self, and peer feedback. I want the message for level 1 to be "I want to help you figure out what you still don't understand" instead of "Fear my red pen!"

I asked the cherubs to weigh in on Facebook and got mixed responses.

So anyway, the idea for a process will look like this for Level 1 questions:
• Students take the quiz, and predict a score.
• I collect the quizzes and write feedback in sentence form, like "right idea but computational errors" or "a more careful and accurate diagram would be helpful"
• Next day, students give feedback to each other. One idea, so that they are working to understand and not to just get the right thing on the paper without understanding, is to not let them use pens or pencils, but communicate with mini-whiteboards. While they are doing this, I can be assessing/rewarding/publicizing helpful dialog that I hear.
• Then, students have an opportunity to re-work the problem, or possibly a new but similar problem.
• Then...what, grade the quality of their feedback to each other? Or never grade this part of the process? I am thinking that if I want the focus to be on the learning and not a numerical grade, I can't give a grade to this part ever.
Obviously there are questions here that will take time to sort out. But this seems like a step in the right direction.

## Saturday, July 30, 2011

### Summer Learning, PCMI Edition: Deeper Criteria

This is a series of posts that are reflections from the Park City Mathematics Institute Secondary School Teachers Program.

One afternoon we listened to a lecture/powerpoint by Douglas Corey of Brigham Young University about comparisons of effective teachers at home and abroad. Toward the end of his talk, he seemed to have partaken of a generous serving of the edureformer kool aid and came across as anti-teacher or at least teacher-concern-dismissive, which obviously turned many people off. However I took away some notes about his research that struck me as important.

Based on classroom-level comparisons between different countries from the TIMSS video study, researchers found
• there is no single effective teaching method
• all high-achieving countries teach quite differently
• we can not judge a lesson's effectiveness by methods used, but rather
• effective lessons have deeper criteria in common he called "instructional principles"
He asked us to predict the instructional features which must be present for students to learn with understanding. These were the guesses that my group brainstormed. If you want to play the home version of the game, take a moment to jot down what you think they are too.
• teacher content knowledge
• problem-solving
• students have to be working and thinking
• deeper explorations
• students making connections
• continuous assessment that informs instruction
• deliberate metacognition is part of instruction
• teacher believes all students can learn rigorous, conceptual mathematics
• students need to spend time thinking about math outside of class
However researchers only found two:
• "intellectual engagement" - the teacher has to get the kids thinking about a problem. Students have to struggle. "Struggle" means students expend effort to make sense of math, to figure something out that is not immediately apparent. It does not mean needless frustration.
• "connection-making" - the focus of the teaching has to be on making connections. Connections don't come by accident but must be an explicit focus of planned instruction.
The struggle thing rang true for me. At some level I internalized that idea long ago. I'm still coming to terms with the connection-making point. The same concept was approached earlier by Gail Burrill with respect to the Common Core standards. She pointed out that in American classrooms, teachers can plan and ask connection-making questions and activities, but students mostly still end up doing procedures. A big question I am still grappling with is how to design and deliver instruction so that the students are doing connections. I have only vague notions about what that would even look like. I don't really know what to do with this yet beyond hang a sign on the bulletin board next to my desk at school that says "make the students do connections."