I meet them at the door, hand everyone a slip of paper as I welcome them back. Projected are instructions to write on the slip:

1) Your name

2) Something really fun you did over break

3) Name for a quadrilateral with two parallel sides

4) Definition of a kite.

They work on it while I take roll; I collect the slips and toss them on my desk. “OK, people, it was an awesome break and now we are back. I am glad to see you all. You have amazing Geometry knowledge in your heads but it may be kind of hard to access it after all the fun time off. You may not have thought about Geometry once over break and that’s fine; in fact, maybe I didn’t either. But now we are going to blow out the cobwebs so get ready.”

We are most of the way through a unit on special quadrilaterals. I grab the deck of cards (with a kid’s name written on each card) and shuffle, and fire up the slides. We play “Name it” where I project a general quadrilateral with some extra information, draw a card, and a kid has to try to specify what kind of figure it is. They have the right to pass, but are encouraged to make educated guesses, or give a partial answer.

I do everything I can to avoid revealing whether the kid is right or not. Clever Hans is always on my mind these days. “What makes you say that?” “How sure are you?” “Who agrees or disagrees?” I encourage dissent and try to make it OK, really OK, to be wrong. I’m trying to avoid the correct answer, said once. I notice several kids are making sketches to help figure these out, which is awesome but I wish more kids were. It can take a long time for the class to come to a consensus about a difficult slide. Once they do, next slide and next card.

By the time we get through all the slides (about 20) they are worked. I have everyone get up, stretch. Go to the bathroom if you need to or get a drink, but hurry. We are about halfway through an 80-minute block.

New topic: midsegment of a trapezoid. I explain what it is, show some examples and non-examples, make sure everyone gets the definition. “Okay, what can you conjecture about these things? Consider angles and segment lengths, go!” Usually I would make small groups with the cards but today I just go with who’s already close, waving them into clumps of four.

After 10-15 minutes I call them back to order and start asking groups to share. The things they need to know about the midsegment, they’ve pretty much figured out. Every group gets parallel to the bases and corresponding angles. Most get midsegment length average of the two base lengths. One group has figured out a way to prove it by chopping the trapezoid up to make a rectangle. Nice.

Okay, what if we take this trapezoid and shrink one base down to a point? We now have the midsegment of a triangle. What can you conjecture? They get it and some kids make pretty clear, convincing arguments. I think (hope) we have dodged the bullet of the right answer, said once.

Ten minutes left. Do the homework now and you won’t have to do it at home. Work with a friend, go! I grab the stack of entry slips and thumb through. I cruise around and connect with several kids about things they did over break. “You sledded down that? Epic!” “Glad you got to see your cousins. Where were they visiting from?” “You saw the King Tut exhibit. I’m so jealous!” I also help with the homework, a little. Then the bell rings. We’re back.

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As a lead-in to the project, we did a lot of graphing on log scales this week. We also graphed on semi-log and log-log paper, and saw how they linearize an exponential/power function (respectively). This helped us discover and understand why the function through two points must be unique. It was nice to do some graphing by hand, both for the tactile experience and to see all the problem-solving opportunities that arose. Choosing an appropriate log scale was especially difficult for many kids.

To do the project, kids will need to use several mathematical tools:

- point-slope form of a linear equation (can’t hurt to review it yet again)
- point-ratio form of an exponential
- “logged” form of power function, with the slope as log ratio
- various exponent and log solving techniques

There is obviously no “real-world meaning” to the personal numbers students use during the project. During our log graphing week we’ve looked at several different data sets. I’m hoping the curve-fitting skill from this project can carry forward to future data analysis projects. At the least, it will give them some good algebra practice and might help make an interesting connection between these parent functions.

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I break my classes into groups nearly every class, if only for a short investigation. But I’ve been feeling unsatisfied, and recently returned to the book *Designing Groupwork* (Cohen) for some deeper insight. One thing the author stresses is the importance of assigning individual jobs to help break down students’ prejudged status structures. In an effort to bring the groupwork up a notch, and to be really clear and formal about what makes a mathematical proof “right,” I created both a scoring rubric for student proofs and a corresponding set of group member roles. I made up a list of simple proofs I thought the students could handle. The whole thing is a one-page (front-back) sheet, found here.

How it worked: I pitched the task to kids as their first real attempt at mathematical proof. They were actually sort of excited, and nervous. I told them their work would be scored and entered in the gradebook. ( I don’t usually do this, striving for a more righteous SGB, but sometimes it helps to raise the stakes just a tiny bit.) I provided one last model so they could see the level of clarity and justification I expected. Then I shuffled the cards and made groups of four. Each group got a sheet with one of the proof tasks highlighted. I put their names by the roles. And they got after it with the whiteboards.

All the groups were able to write a correct and clear proof, working together. One group in one section took a lot longer than the others, so kids had to do some other problems while they waited. The presentations were pretty good, though lacking in the class feedback department (which I encouraged, but didn’t require or formalize).

The whole lesson, with launch, group work, and sharing, fit into an 80-minute block just right. There are a lot of things left to prove on the list, and I plan to do the lesson again after Thanksgiving. I anticipate that they will be much faster, and more confident. Then, eventually, I’ll be asking them to come up with proofs without the group support, and I plan to use the same rubric (without the participation component) to assess them.

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So I had to dust off the modern physics books and really get to thinking how to answer that question. My training is in math, not physics, but I took this one amazing course in college that was an intensive physics class based on the Feynman lectures. I had a vague memory that the path of least time was most probable, but I wasn’t prepared to give a full answer, by any means.

My homework led me to the videos of Feynman speaking in New Zealand on the topic of QED, which are truly marvelous. I recommend every teacher watch them, just to see how he communicates with the audience. One interesting thing I discovered is that Fermat invented the principle of least time to fit the experimental findings of Snell; of course it makes sense that the model was created to fit the data, but we obscure this when we use Fermat’s principle to derive Snell’s law!

In the end, I decided to talk with the class about the history of our understanding of light, from Newton’s particle model to Maxwell’s wave theory, and explain why neither model completely worked. I introduced the twentieth-century quantum model where “waves” of probabilities rule particles. I tried to tell them just enough to let them know there’s a big, amazing world of knowledge out there waiting for them. Then I played a clip of Feynman from the first NZ lecture (“Corpuscles of Light”); on the YouTube version it’s from about 20:00 to 26:00. In this part of the lecture he speaks eloquently about not understanding. He describes the laws of nature as hard to accept and calls them “screwy,” then announces, “If you don’t like it, go somewhere else! Another universe, where the laws are simpler!” The kids were losing it; it was so fun to watch their reactions. Then I told them that links to all the videos were posted on the class website, and invited them to explore more on their own.

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Today I projected a bunch of graphs of transformed functions, and kids sought to write an equation for each. It’s a classic Algebra 2 lesson, developing more fluency between the algebraic and graphical contexts. When I handed out the iPads and sent kids to Desmos.com, things really got going.

Desmos is so much better than a graphing calculator for this purpose. It provides instant feedback, and you see the equation and graph at the same time. The graph changes while you edit the equation. Or, you can make a copy of an equation and edit that, then compare. I got some great “What would happen if…” and “Why” questions, which I love. In both sections we had a detailed discussion of why translating then dilating a half-circle would yield a different result than dilating then translating.

The lower-achieving kids seemed to do well with this lesson. It was nice to be able to try something, see what happens, then adjust. And because everyone was engaged, I got to spend some more time working directly with them.

Downsides of Desmos: On an iPad it’s pretty easy to lose your work by deleting or navigating away. Also, the parentheses take some getting used to. The downsides were totally outweighed by how immersed the kids were in the algebra-graph environment, and how much they liked it.

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This is a big contrast from last year, when I taught at an alternative high school. There, the students were historically low achievers in math (and most other subjects) and in many cases had done some construction or other labor. Hands-on geometry was the best thing I did with those kids. For them, finding the balance point of a cardboard triangle was, like, very cool math. And what I loved was they always wanted to know why something like that worked. It was a very natural, in-the-world kind of curiosity, that felt to me like a true mathematical instinct.

My geometry classes this year are mostly composed of what we would classify as better students. They will undoubtedly be more successful on average on the state end-of-course math test than the cohort I had last year. But I’m waiting – impatiently – for their curiosity to emerge. And I’m wondering what things I can do to inspire and nurture that quality in them.

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Other courses are like this too, I know. But something about the advanced algebra content of A2, and its reliance on prior high school level math couses, makes it stand out. Here’s an example: We’ve been studying arithmetic and geometric sequences, recursive and closed-form versions. Modelling growth and decay, and some financy loan/investment type stuff. We’re headed for linear models of data and linear systems. I give a short pre-assessment to see what kids know and remember from Algebra 1 (and junior high); they have to find some slopes and solve a couple easy systems.

I get the pre-assessments back and look them over. In each class, four students have completely aced it. A couple of them even solved the systems by both substitution and elimination, and demonstrated (unasked) which name went with which solution method. Aside from these eight, everyone else pretty much bombed. But, within the bomb group there are several kids who I suspect have just forgotten how to do these problems, and some who probably never really got it the first time.

We do a one-week intensive unit on lines, slope, and solving systems, with a lot of skill practice. Kids really liked the “Arithmetic Simultaneous Linear Equations” activity found here. During this time, the students who killed on the pre-assessment work as a small group learning about matrices and operations on them. At the end of the week, I give a test (to everyone). Now I have all students meeting standard on solving basic systems (albeit with some algebra errors) except two kids in each class, whose algebra skills are seriously sketchy. I mean, they are consistently making some wild and shocking algebra blunders.

That brings us to today. After looking over the tests, we break into groups. At this point three different things are happening in class:

- The fast group is moving forward with matrices. They know that there’s a way to use the calculator to solve any linear system, but not exactly how. They are still looking for inverse matrices by hand, and I think their overall understanding is getting to where I will feel good about handing them the tech option soon.
- The bulk of the class is working in small groups on linear systems application problems, with each group presenting their solutions to the other groups. Sometimes I throw in the mistake game to keep things interesting. I am seeing signs of increasing confidence and facility from almost all of them.
- The kids who really struggled are working directly with me. We are hitting the basic algebra moves and trying to build understanding, and doing a ton of practice. Because everyone else was sort of cruising, I could work with these kids one-on-two for most of the class today (with some quick circulation to the other groups). The kids responded to this attention; I just hope they follow up with my advice to keep practicing outside class.

By Friday I’m hoping to see everyone meet standard on basic systems, so we can move on with confidence to more complex material.

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I teach two sections of Geometry and they meet on alternating days. Monday, the lab went pretty darn great. I did a launch using the classroom projector, and we headed out. In the computer lab, kids were engaged, there was a good mix of sharing and independent work, several kids had breakthroughs, and the lesson differentiated itself fairly well. I was happy. One item on the list is a triangle with a segment from each vertex to the midpoint of the opposite side. We had sketched this during the bellwork and wondered whether those segments always had to meet in a single point. Opinions differed. Now they were seeing that the medians always did meet, no matter how they morphed the triangle. I intended this as a setup for introducing inductive and deductive reasoning, and it seemed to be working. We had a a little discussion about how you can really know something. Joy!

Tuesday was a whole different story. The class is a little larger and has a lot more attitude. Still, I felt ready, emboldened by my success on the day before. Well. First off, I got a glitchy launch of GGB in the classroom, it kept freezing, and I had to relaunch twice. Kids were like, “um, this is lame.” I soldiered on. We got through it and headed to the lab.

Things didn’t exactly soar. Several kids were way not into it, and let everyone know. The toughest thing about this group is the intelligent, high status kids with shitty attitudes. One of them, H., was clearly feeling way too cool for this. She has the potential to be a great math student but has already let me know all she cares about is the grade. Finally I got them going and most kids were making progress.

We get pretty far along and I notice several kids have made the triangle with medians. I get a discussion going about how they always seem to meet in one point, no matter how the triangle changes. “So we can see how that always seems true, but is that the same as **knowing** that it’s true? As we go forward we will gain the tools to **prove** that it’s true, which is the same as understanding **why** it is true.” Pretty nice lead-in to reasoning and proof, right? Lockhart-esque, even? Wrong. Says A-student H.: “Things just are. I don’t really care why.” General agreement from her peers in 5th block Geometry.

Which brings me to my joke.

Jesus was out walking when he came upon a man crying. “What is wrong, my son?” “Once I was a great violinist, Lord. But last week I crushed my hand, and now I am unable to play.” And Jesus laid his hand upon the man, and Lo, he was healed and able to make music once again.

Later Jesus came upon a woman who sat crying. “What is wrong, my child?” “Once I was a painter, Lord, but now I am blind and all art is lost to me.” And Jesus laid his hand upon the woman, and Lo, she was healed and went on to paint once again.

Still later Jesus came upon yet another man who was crying. “What is the matter, my son?” “Lord, I am a high school math teacher.” And Lo, Jesus sat down and wept with him.

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On the other hand, I knew that a certain percentage of kids would work the system however they could, including to their own detriment. No homework grading? Cool! No homework!

Then there are some kids who probably will not do the homework whether you grade it or not. Some of them have chaotic home lives. Homework inherently discriminates against these kids. Many of these kids are the ones we will work ourselves to the bone to prevent from failing.

So, what to do? I feel I need to encourage practice, and responsibility. I do not want to grade 135 kids’ homework. I don’t want to “check in” homework, unable to tell whether it is correct, or copied. I want everyone focused on learning, as much as possible.

I told kids the first week that I’d be giving a weekly “Homework Quiz.” These would be designed so that, if you did the homework, you should ace the quiz. Quizzes are given a full class later than the homework is due, so we have a chance to go over the work together and find any misconceptions. I’ve been making the quizzes short (half a page) and only allowing about 15 minutes of classtime. They are quick and easy to grade, and have the tested skills listed at the top. They get entered into the gradebook, divided by skill, under Formative Assessment, which altogether will be 20% of their grade (schoolwide SBG-hybrid policy… long story).

This week, in an attempt to see how things were working (and to keep everyone on their toes) I checked homework. Students were, needless to say, shocked. Three weeks in, and they had already decided I would never check the homework! “I thought you didn’t do that!!” I love teenagers.

Does it work? Yes and no. Here are some gross generalizations based on the first few weeks:

- Kids who are quick learners either did or didn’t do the homework, and did fine on the quizzes. Honestly, if they could have nailed the assessment just from paying attention in class, do they need homework?
- Compliant kids – who probably would have done the homework without the quizes – did the homework, but seemed a little clearer on what they should be getting out of it.
- System-working kids who would do anything to avoid the homework did a minimal amount of the work and often did poorly on the quizzes. I am still hoping that as the weeks go by they will really swallow the connection between practice and success.
- Kids who lack home support have mostly not done the homework. I suspect that would also be the case in a graded-homework setting. They have done poorly on many parts of the quizes, but not all. The only advantage of this system, for them, is that we have some data on what they know, and what they don’t, so I can get some targeted help headed their way.

There is one other thing about this system that I like: it’s a low-stakes test. Kids get more practice in a testing environment, and another opportunity to recall and build memory.

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