## Friday, November 14, 2014

### Fall 14 - Book Club

In my senior capstone history of mathematics class, I had had good luck with having students choose books to read and then sharing and swapping. (Previous post.) I thought it might be worth trying with my preservice elementary teachers, who in the past would all read one book. Currently, if I was picking just one it would be Boaler's. No more than 5 per book; choices included:
• Accessible Mathematics: Ten Instructional Shifts That Raise Student Achievement, Steven Leinwand, (Amazon) [Practical, pre-service teacher approved)] (3)
• Intentional Talk: How to Structure and Lead Productive Mathematical Discussions, Kazemi & Hintz, (Amazon) [Applies to more than math; good support for helping students learn to converse productively] (2)
• Making Sense: Teaching and Learning Mathematics with Understanding, Carpenter, Fennema, Fuson, Hiebert, Murray & Wearne (Amazon) [Writers and researchers of the best elementary math curricula around tell what they think is important.] (FULL)
• Math Exchanges: Guiding Young Mathematicians in Small Group Meetings, Kassia Omohundro Wedekind, (Amazon) [Similar to intentional talk, more strongly based in literacy routines.]
• Math for Smarty Pants, Marilyn Burns (Amazon) [Collection of entertaining problems across all kinds of math from a master math teacher.] (FULL)
• A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form, Paul Lockhart (Amazon) [Not sure about putting this on. Many readers are disappointed in the 2nd part, but the 1st part people see as a powerful argument for why math teaching has to change.] (FULL)
• Powerful Problem Solving, Max Ray (Amazon) [New book from a very deep thinker about how to teach math.]
• What's Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject, Jo Boaler (Amazon) [If I was picking one book for everybody this would be it. Dr. Boaler is doing a lot to research and shares how to make math better.] (FULL)
I was disappointed no one chose Kassia's or Max's great books - there was a bit of a follow the leader effect in choosing books.

What follows is a poor transcript of the group discussion. On book club day, I ask students to bring in enough snacks for four people. They start in small groups with people who read the same book, then we have a discussion circle where each group shares and fields questions and I try to keep my mouth shut unless asked a direct question. (Group questions in italics.)

What's Math Got to Do With It? Jo Boaler
 Anyone else hear Tina Turner when they see this book?
I rated this book as must read, because it talks about what we grew up with, and what we should do. Connected with this class really well, and I’ve never had a class like this. The other way separates you by ability, makes the kids feel dumb or come to hate math. So then in life later, they can’t even think about careers that use them. “I wasn’t good at it I didn’t like it.”
Any specific lessons? Talked about research on the effects of this.
Also had kid perspective.
Did it address grading? Not as much as they could have. Did talk about what doesn’t work with tests and encouraged smaller assessments that aren’t a big deal.

 From Mark Bennet's visual reading notes
A Mathematician's Lament, Paul Lockhart Our book talked about teaching styles, too. His big point was that math is an art. Example: triangle area and formula. By giving problem and answer in one you’re cheating the kids. 2nd part was all about how he loves math. Redundant or over my head.
Students need to get to play around with it. Example of art class, where students aren’t told how to do each brush stroke. I like that, but
It wasn’t very practical. We don’t have time to have each kid discover the area of a triangle. It really connected with me how I was taught. My high school geometry… my face got red as I read.
Any solution at all? Really, no.
Also not fair to teachers to blame them entirely. Brought up issues, just wish there was more solution. I liked the book but kinda hated it.
Was he a teacher? Sort of…

How is he teaching?
Lockhart says, 'I want to know what you think of this. If you’re a student I send my condolences. Ignore the absurdity you’re taught in your math class.'
You can’t be a teacher and tell students to ignore their teachers. I hated math, ignored my teacher, and got switched to a slower class - it didn’t work.
Kind of puts himself on a pedestal.
Talked about textbooks, and how math is stuck in the 19th century. We can’t get out of it because then schools adopt these books. You read that, get fired up, but then what do you want to do.

 Talking Heads(obviously)
Making Sense, by Fennema, Carpenter,  Fuson, Hiebert, Murray, Wearne
We want to teach all students so that they make sense of mathematics. 5 dimensions. How to communicate teacher to student.
The book was helpful, set up logically, good examples of classrooms, but then repetitive. Everyone should read the first chapter, and then read more about one dimension.
At the end, giving examples of classrooms in different modes. But just stories - a detailed description would be better. Got tedious, chapters 7-10 were repetitive. Nice, but can’t tell the difference between some of them. Did tell how each related to topics in chapters 2-6.
It was good.

Intentional Talk, Hintz and Kazemi

Sets up different strategies, open discussion, targeted, gave strategies to assess and fix discussions. The examples were actual dialogues. Then they looked back and identified, here she was identifying goals, here she was introducing math ideas. Good strategies on how to make the classroom safe for discussion. The ideas and participation of all students are valued. So even wrong answers, it gave you lots of ways to use that and build from it.
The book showed how you plan discussions. Gave a template, helped you think about how to plan it before, going from what you think they are going to say.
I gave it a 5 because it was really helpful, I want to reread before I student teach. You wanted to take notes as you were reading.
Open strategy sharing. Like kids giving different ways to multiply, we need to think about different strategies so everyone has a way to learn.
That’s like at Family Math, we have 5 pizzas with 4 pepperonis. What’s a math problem like that? “Well we have 5 groups of 4…” I never thought about it that way when I was kid.
They give you talk moves, so you have ways to move it forward.

Smarty Pants, by Marilyn Burns
 On Etsy.
Our book was not about how to teach. It was just problems. Different kind of problems.  Listening to other people talk - I didn’t get that much out of this one. It was interesting, and fun to read through, but you don’t get a lot out of it except different kind of problems.
Each part has a ton of new problems and cartoons.
Parents could use it as a supplement.
My sister doesn’t know my nephews math, he’s like 14. They call me… could parents use that to practice.
This is younger, but more about abstract thinking. They got me thinking, but it doesn’t explain why it works out.
How do they take into account kids who are not ready for abstract thinking. They have specifics to do in the problem, but then it abstracts…
It is literally just a book of problems. I liked it. It was interesting. I enjoyed reading it.
Would it show them how to teach, like addition? There was a section on tricks to multiply, but it was just hints and different strategies.
It reminds me of problems of the week. I used to get my family doing them.
Even the answers were hard - upside down and backward.

Accessible Mathematics, by Steve Leinwand
 We played his Ignite on "It's Instruction, Stupid!"afterwards, so they could read in his voice.
This gives what’s wrong with math and gives different positive approaches.
I really liked it. So straightforward, 10 shifts to do. Emphasizes how to always be asking questions. I’d recommend reading it. So practical, check homework more effective, using problems of the day.

(Not much discussion, but a lot of interest in reading it afterwards.)

Their Summary My questions for them for the summary.

What would you like to read?
I’d want to read Accessible Mathematics. Like music ed, know the beats as a pattern… that’s doing math.
I’d want to read intentional Talk, because of good ways to get you to get students thinking. For teachers that are just starting out, it has very practical.
What’s Math Got to Do With It - practical, actual ways to teach.
Smarty Pants - I want to see the activities. How could you incorporate them into the classroom.
Variety or one book? Variety!
Assigned or pick your own?
Choice! Now we know multiple books, which one connects, time thing.
We get to focus on one book, get more in depth, get to borrow.

My Summary
This worked out well.  I'm going to do it again, and try it in more classes. They did a nice job with this summary day, and I was convinced of their interest and investment.

I ask them to make a reading plan once books are chosen, with the end date in mind. I ask them to keep reading notes, for accountability and retention, that include a summary and a response to each session of reading. I don't grade those, but just check when done. I emphasize this as a rough draft of doing book studies as teachers, and like that we're arming them with some different book choices for those future book clubs.

## Thursday, November 13, 2014

### Array Maker 9000

This is probably a stupid little post.

The other day I was making some math games, and I needed a rectangular array for a board. As one often does. So I made a quick GeoGebra sketch. And rather than make it so it only made one board, I made sliders so that it could make any rectangular board. Of course sometimes you'd have to zoom out to see the whole thing. A little clunky.

Then Tim Cieplowski (the BGU prof with the beautiful GeoGebra stuff) tweets:

Which makes me feel like I should do a proper job.

One of the things I love about GeoGebra is that you can make the window dynamic, so that it automatically fits what you want to show.

The problem with doing the array is that fitting it to the window will make it non-square and change the game board. So here's my workaround.

The array is just a ratio to compare a window of array plus a border of .5 to the actual graphics window. Corner[1] is the lower left corner around counter-clockwise to the upper left Corner[4]; those are helpful for making things that go right to the edge. Corner[5] gives the (width, height) in pixels of the graphics window. (You can even ask for Corner[2,5] if you have the second graphics window.) So these definitions make the window fit the height if that is taller than the width is wide, relative to the window.

Bonus: this is the closest I have come to doing the Border problem in real life. (Which is so well known that it comes up ahead of immigration stories, even.)

Here's the sketch on GeoGebraTube.

Can you think of a quicker or more elegant way to do this?

## Thursday, October 23, 2014

### Percy Jackson's Math Class

[I haven't written this post yet, and I know it's going to be more rambly than usual. Fair warning.]

My eldest child has been a big fan of Percy Jackson. Me, too, to be honest. I'm reading The Blood of Olympus now. The first Percy Jackson series is one of the few non-graphic novels my son read by choice. So this was a natural click for me: The Percy Jackson Problem at the New Yorker by Rebecca Mead.

It begins with a quote from Neil Gaiman (another family favorite):
“I don’t think there is such a thing as a bad book for children,” he argued, adding that it was “snobbery and … foolishness” to suggest that a certain author or particular genre might be a baleful influence upon young reading minds—be it comic books or the works of R. L. Stine. Fiction is a “gateway drug” to reading, Gaiman said. “Every child is different. They can find the stories they need to, and they bring themselves to stories. A hackneyed, worn-out idea isn’t hackneyed and worn out to them.” Well-meaning adults, he continued, can easily kill a child’s love of reading: “Stop them reading what they enjoy, or give them worthy-but-dull books that you like, the 21st-century equivalents of Victorian ‘improving’ literature. You’ll wind up with a generation convinced that reading is uncool and worse, unpleasant.”
If you're a math teacher, how can you read this and not connect? We've been feeding students, as a profession, "worthy-but-dull" math for ages. (Worthy when it was good, that is. When it was bad, we're talking Tartarus.)

The author's argument is the counter to this idea that all reading is good.
Riordan’s books prompt an uneasy interrogation of the premise underlying the “so long as they’re reading” side of the debate—at least among those of us who want to share Neil Gaiman’s optimistic view that all reading is good reading, and yet find ourselves by disposition closer to the Tim Parks end of the spectrum, worried that those books on our children’s shelves that offer easy gratification are crowding out the different pleasures that may be offered by less grabby volumes.
I don't like this argument for reading. But I have made similar arguments in math. After a steady diet of exercises, students have no interest in problems.

But I think what I mean is that students have no experience with problems. The engagement that comes from finding a really meaty one. The question is whether reading Percy Jackson is really reading. I would argue that spending time on Tumblr is not reading (a current teenager discussion), and wonder if graphic novels are reading. (Aforementioned comic-obsessed son.)

I think this is an issue K-12. In elementary, there is a danger that teachers don't believe that students can do real problems. In high school, a desire to have the students do the basics first. Working with preservice and inservice teachers I try to stress and give experience with contexts that are problematic, but accessible. If it's not a problem, it's not doing math, no matter how many numbers and operations are involved.

Just being letters and words doesn't make it reading?

The author isn't so concerned with the Percy Jackson books, as with the forthcoming book of Greek myths as told by Riordan, writing in Percy's voice.
While the D’Aulaires wrote that “Persephone grew up on Olympus and her gay laughter rang through the brilliant halls,” Percy’s introduction to the story of Demeter’s daughter reads, “I have to be honest. I never understood what made Persephone such a big deal. I mean, for a girl who almost destroyed the universe, she seems kind of meh.”
It seems to me that this is some of what the common core struggle is about. Parents don't recognize newer curricula as math. (Which, of course, really has nothing to do with the common core in most cases; the Common Core gave them something to be against.)

The author closes with this concern:
What if instead of urging them on to more challenging adventures on other, potentially perilous literary shores, it makes young readers hungry only for more of the palatable same? There’s a myth that could serve as an illustration here. I’m sure my son can remind me which one.
Ooh, clever. I'm sure she knows which myth. What if after doing Desmos and Three Acts investigations, students don't want to do hackneyed word problems from the end of the chapter? That's probably not fair. Will they not be interested in the real problems of calculus, geometry, analysis and algebra? I think if we had a Percy Jackson parallel in math, the greater numbers of young people interested in math would mean a boom in STEM fields. The Percy Jackson problem? We should all have such problems.

This post started when I discovered no way to comment on the article. Because I wanted to share my daughter's response. And I want to think of this in terms of math, too. Here's Ysabela:
If they were arguing against, like, Twilight, where the language use is bad because the author has no writing experience, AND the plot and ideas are unoriginal/problematic, then I would agree with them. When Twilight becomes people's standard for literature, they start accepting total crap without a second glance, which is bad.
But Riordan understands language? And his plots (at least in the original series) were good? I'm not saying it was Harry Potter, but Percy Jackson was quality, and saying it wasn't just because the language is accessible to people who aren't scholars is just... really elitist. Like I know a lot of people who find reading really, really hard, but were able to enjoy Percy Jackson because it actually made sense to them. Plus, the series was narrated by a teenage boy, it was realistic.
And don't even get me STARTED on the D'Aulaires, they're SO AWFUL. They watered down the myths so much they were almost unrecognizable, "for kids," and then wrote it at like a college reading level. Plus they organized it like total tools. I can't express in words how much I would have rather had a Riordan book of myths than the D'Aulaires when I was younger.
 Hello Katie @ Society 6
As she steps out the door now, she's railing against having spent two weeks on factoring. Because the last day before the mini fall break they learned the quadratic formula. "And it always works!" Do you know why it works? "He showed us from $$ax^2+bx+c$$. It's extra credit on the test." How much more dangerous is it that she believes math is that pointless and uninteresting?

So, I think I'll side with Neil Gaiman on this one.

## Saturday, October 18, 2014

### Such a Thing as Free

A friend found his new-to-him school in need of Algebra I and Geometry textbooks - for cheap or free. I took the Twitters, and people responded quickly and generously. Thought I should collect their suggestions.

Free Curricula
My first suggestion was Geoff Krall's (@emergentmath) Problem Based Learning maps. Amazing work. It's a worklife dream to develop a collaborative site like this where we could all link our best lessons and do some informal lesson study.

My second suggestion was Illustrative Mathematics, Algebra and Geometry (plus everything else).

Then #mtbos hit the gas:
• David Coffey (@delta_dc) reminded me of the Georgia Common Core Materials.
When he wasn't Khan-baiting.
This is where I would start. Very complete, a lot of excellent lessons and compatible web resources, even including 3 Act structure stuff.
•  Bridget Dunbar (@BridgetDunbar)  ·  out of Utah: Mathematics Vision Project.  They follow integrated sequence...but good materials.
• Engage NY, Algebra and Geometry (but complete K-12, math and ELA).
Inconsistent quality to me, but a lot of good stuff and assessments are there, too.
Lisa Bejarano (@lisabej_manitou) recommended one of these two.
Dan Anderson (@dandersod) noted it, but does not love it.
•  Macomb ISD Math (@MISDMath)  ·  have you looked at the EMATHS materials?
That's the online materials for Michigan's virtual schools. New to me.  Looks thorough, with PD materials, lesson plans, activities-based and assessments.
• @geonz  shared Algebra2go. Early, online algebra curriculum with videos and homework.
Other Ideas
• Peg Cagle (@pegcagle)  suggested: Visit Abebooks for old Key Curriculum Press materials-brilliant rigorous & coherent-a bargain at full price, now available for a few bucks.
I noticed some IMP stuff there. Love IMP.
• Justin Lanier (@j_lanier)  ·  There are the Exeter and Park School problem sets, which are freely available.
At Exeter they have been problem-based for a long time, in 12 student classes. Read more about their Harkness method.
The Park School curriculum is available on request, but you may have to nag them.
• Raymond Johnson (@MathEdnet)  ·  Not as cheap as they used to be, but College Prepatory Mathematics is worth considering.
Samples here; those are good stuff.
I also let him know about the single serving sites:
That list lives on my Reading Recommendations page. I also put in a plug for GeoGebra (of course).

If you have experience with any of these things, or know of other resources, please list them in the comments. And thanks to everybody who responded or retweeted.

## Friday, October 10, 2014

### Poetical Practices

 #35! Angelou's #1, but there's no way she'll make it through conference play without a loss.
I got a chance to hear Billy Collins last night, thanks to family friend Elizabeth.

I enjoy poetry a lot, but don't read as much as I would like.  I have virtually none committed to memory, despite thinking that would be very cool. I don't write it, but have been wondering this past year how you even get started.

He was charming, lovely voice, aware of the audience and built his set of poems like a jazz musician, making sure to hit the hits, but improvising based on conditions, inspirations and audience response. For example he read this poem, To My Favorite 17 Year Old High School Girl, which may be his biggest smash: (around 5:30)

We bought that book - it is a lovely retrospective with new poems including the one he and Colbert read.

In our reading, he had me right away; before his first poem he talked about how nice it was that we were there. That, in fact, it was nice that anyone liked poetry given the way most of us are introduced to it. By which, he meant, in school. Imagine if the first time we listened to music, it was someone picking a suitable piece, they played it for a whole group, and then sat us down to ask us questions about it.

By the same token, it's a wonder that anyone likes math, eh?

I really liked a lot of it. This poem, Aristotle, you can hear him read at the Poetry Foundation. It's about how Aristotle introduced or recognized the beginning middle and end structure for literature.
"This is the middle...
This is the bridge, the painful modulation.
This is the thick of things.
So much is crowded into the middle—
the guitars of Spain, piles of ripe avocados,
Russian uniforms, noisy parties,
lakeside kisses, arguments heard through a wall—
too much to name, too much to think about."
If there is an official poem of Three Act lessons, this is it.

Jamie Radcliffe was a young visiting prof at Penn when I was at grad school there, and an all round good guy. (Now a full prof at Nebraska-Lincoln.) In addition to telling the best ever thesis joke, he had this great line about math and poetry. Even if he had only learned enough math to write doggerel, he was glad to have learned enough math to read the classic works of poetry from the all time great mathematicians.

Sometimes I think that this is the greatest sin of school mathematics. Making people think that the worst of the doggerel is all of math, and then making the students memorize it.  Not only missing out on many of the potential future poets of mathematics, but denying most students the whole art of mathematics.

But what would be the equivalent of poetry readings in school for math? The closest I've seen, I think, is Fawn's My Math exploration of Math Munch. (See also Sam's adaptation.) In my classroom, the day they bring in their patterns they've made and share their thinking and noticing is pretty close.

And it was crucial to let them talk. Just looking, I missed a lot of their intent. Other students noticed things that even the creators hadn't. There were several comments about "what if..." that were good math thinking. I also contributed a few noticings... I think that let them know that there was some real math here.

Afterwards he did a short Q & A. One of the first questions was about his process. He said, (paraphrasing from here out) take for example "I Chop Some Parsley While Listening To 'Three Blind Mice'" I was in my kitchen, chopping parsley, listening to Art Blakey. I was thinking, who hears three blind mice and thinks it's a good jazz tune. It's hot cross buns. But then I thought, how did they become blind? Was it congenital? Think how distraught the mother would be. Maybe an accident - an explosion! Mice covering there eyes. I take the pen out of pocket and now I'm at the office. If they became blind separately, how did they find each other? I mean how hard is it for a blind mice to even find another mouse, let alone two more blind ones? And then, what, the farmer's wife?! Now they've lost their tails, too.

And I start wondering how they came to be blind.
If it was congenital, they could be brothers and sister,
and I think of the poor mother
brooding over her sightless young triplets.

Or was it a common accident, all three caught
in a searing explosion, a firework perhaps?
If not,
if each came to his or her blindness separately,

how did they ever manage to find one another?
Would it not be difficult for a blind mouse
to locate even one fellow mouse with vision
let alone two other blind ones?

And how, in their tiny darkness,
could they possibly have run after a farmer's wife
or anyone else's wife for that matter?
Not to mention why.

Just so she could cut off their tails
with a carving knife, is the cynic's answer,
but the thought of them without eyes
and now without tails to trail through the moist grass

or slip around the corner of a baseboard
has the cynic who always lounges within me
up off his couch and at the window
trying to hide the rising softness that he feels.

By now I am on to dicing an onion
which might account for the wet stinging
in my own eyes, though Freddie Hubbard's
mournful trumpet on "Blue Moon,"

which happens to be the next cut,
cannot be said to be making matters any better.
He finishes this discussion by saying, it's about curiousity. I get curious about it, and then I just have to work it out. (Here's the song; I love Blakey, and know this album well - didn't hurt when hearing the story.)

I so get that - happened to me just this week with Justin Lanier's Star Fractal pattern. I just had to work it out.

Someone asked how old he was when he started. About 10. He saw a sailboat on a drive up the Hudson River parkway that he needed to write about. He figures everyone has 50 to 300 bad poems in them; high school is good for getting through a lot of them. Someone asked if poetry was good for expressing feelings. He told her that nobody cares. You're writing to get the reader to feel things. If you're good at it, they might start caring about yours.

One of our friends with whom we went, Joanie, was a high school lit teacher among other things (see her IB thoughts). I asked her how she taught poetry. Students try, and you just read it and give them feedback. A lot of it's terrible, but you let them know if they lost their focus or what they're writing about. I do want to be a reader for my students.

So I'm still processing it, but wanted to get these thoughts down.

Do you have any thoughts on math as a liberal art? How do you teach to create an appreciation for the poetry of math, or to create a space for future mathematicians?

To close, I'll include one more of his poems. And ask if maybe we should be commiserating with poets more often.
Introduction to Poetry

I ask them to take a poem
and hold it up to the light
like a color slide

or press an ear against its hive.

I say drop a mouse into a poem
and watch him probe his way out,

or walk inside the poem's room
and feel the walls for a light switch.

I want them to waterski
across the surface of a poem
waving at the author's name on the shore.

But all they want to do
is tie the poem to a chair with rope
and torture a confession out of it.

They begin beating it with a hose
to find out what it really means.

## Friday, September 5, 2014

### What's a Problem?

We had a fun class in the elementary math course today. Introducing SMP1 - problem solving, we got to an interesting question: what's a problem?

Here's the story as told by the residue on my whiteboards.

Schema Activation: jot down about a time you solved a real problem in your life. You won't have to share with anyone if you don't want to - this can be private.

After a few minutes to write, I shared how one of the big justifications for teaching and prioritizing math in school, other than the jobs to which it gives access, is that it teaches problem solving.

Actually more yes than I expected!

People asked if I meant did math class help with the problem that they journaled about or in general. I said we want to know about problem solving in general, but they could use their instance as a specific.

Next question: is it possible that math class could help teach problem solving? Short time to discuss at table.

These were definite yesses, with a lot of confidence.

One of the reasons I like teaching teaching math is teaching is so much like math to begin with. So rephrasing: our problem is how to go from the current situation to get what we want.

So the next prompt was to quickly brainstorm. Ideas for making this happen.

I shared that I liked how many of these were things that were up to the teacher. And I paraphrased Marzano, about how there is bad news: a small percentage of factors affecting student learning are under the teachers control; good news: that percent still makes a big difference.

What did they like?
• The emphasis on real life. This brought out uniform hatred for unlikely impractical story problems.
• Logic problems: one of the students shared how engaging and powerful these were for here. I asked about the contrast with real world, but people were comfortable with the crazy logic problems because the emphasis was on how did you do it.
• Teachers can ask for multiple ways and have students compare them.
Okay. So if we're going to teach problem solving, we need problems. I asked a question, then asked them to think about if that was a problem or not. There was a tub of square tiles on each set of two tables.

After they all had answers, I asked them to think about the second part. After a short time to discuss, we went to +cheesemonkeysf 's talking points structure for the statement.
This is a problem.

We're still working on the structure, so some of my feedback was about that. The no comments idea is hard.

Pretty strong agreement. People were willing to share their reasoning with the whole class ...  even the small minority. Maybe especially the small minority?

I was really happy to see the "depends on what the teacher does" idea come up. And I added the "depends on students" too. This is not a problem for me. For first graders, a serious problem - there aren't enough tiles in the tub to cover! For them... well, they talked about methods, chose how to do it, discussed results; these are problem indicators to me.

Then we went back to the question. What answers did they find?

There was shock at the diversity.

One question I ask a lot is: is this a question where different answers could all be right?

They discussed and...

There was concern about the largest answer being too big, but that table and an adjacent table figured out the first group's table is actually larger.

I shared how measurement is one of my favorite content areas, exactly for this reason that a diversity of answers is to be expected. It can be culture setting.

So, with all this, definitely time for a problem. One of my favorites. How many pentominoes are there?

We played with domino patterns last class, so that's a natural starting point. We agreed that there was only one way to make a domino with two squares. If they touch, they have to share an edge.

With three squares, the issue came up about putting them "in the middle." That's solved by the edge rule. What about the elbows in a different direction? No, the class agrees, if you can turn it to match, it's the same shape.

So what I want to know: is how many pentominoes?

People got to work with tiles and graph paper. No group found all the tetrominoes as a step, which I was trying to suggest. A couple of times I had tables report on how many they had and were they expecting more. 8 more, 9 more, 12 probably, etc. Next time, 12, 14, 15 and a mix of have them all and think there are more. When our time was up, they sent emissaries to the board to draw what they had:

Now the best part: math fight! Are flips the same or not?

They divided up into groups based on their answer: 14 flips matter, 9 flips are the same. They shared reasoning. Flips matter because these are flat things and a flip requires another dimension. And if you try to match them up they do not line up. Flips don't matter because you can get them to line up, and what's the difference between a flip and a turn, really?

I refuse to settle the issue, and ask them to make a complete set before Monday.

If you care to comment or tweet a response, I'll share your answers with them:
• How do you recognize what is a problem for your students?
• Are flipped pentominoes the same or different? Why?