A blog for sharing my math interests on the web, to post new materials for elementary, secondary and teacher ed, and vent mathematical steam when needed.
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And then I got wondering about a tile with maximal dodecahedrons. I really liked the result of four 90^{o} rotations.

But I was having trouble extending it using the pattern blocks. So I fired up GeoGebra and starting playing with the regular polyhedron tool, and came up with a pattern I liked. If felt like 2x4 rectangles of dodecahedrons, that had the rotating 4 block in the middle, and met in a rotating four block.

That left the gap to figure out. Here is the pattern with the gap (fill in how you want). [Math Toybox is the site I used to play with the blocks. Not the satisfying click of wood, but I love the save feature!]

Here's how I filled in the gap. I had blue rhombs where the triangles matched up, but that blurred the lines of the dodecahedrons. And I wanted it to look like it was overlapping. (On Math Toybox.)

This was a little maddening, but also fun. By the end these arrangements just made a deep kind of sense and I could see the pattern and replicate it easily - definitely not the case at the beginning. So thanks/gracias/merci to Simon and Daniel.

I'm teaching College Algebra for the first time this summer, as apart of trying to revise the class. We're focusing the syllabus, and shifting some emphasis to practices from just content. I thought this problem was a good one. We're not doing proportional reasoning as an individual focus, but rate of change and difference quotient is a part of the class.

Then over the weekend, Marty & Burkard, the Australian math popularizers who collect math in the movies clips among other things, sent a link to this video from Burkard's new YouTube channel, Mathologer. It features a great math clip from Little Big League. If Joe takes 3 hours to paint a house and Sam takes 5 hours...

So much good in there, you wonder if the writer had some teaching experience.

Number mashups - check.

My uncle was a painter - check.

Trick question trope - check.

I also liked that it was a typical messed up pseudo context (paint a house in three hours), and got at what math looks like without making sense. I found a clip that had the whole scene without interruptions, and we started our math problem solving at 2:40, pausing the clip. I asked the class: what makes this a dumb question? They said the obvious, got into why you'd want to know, and discussed if two people working together would be like that. One student who is a supervisor at work countered - he would like to have an idea of how long a job should take. I added that it would be important for a bid, too. I got to paraphrase von Neumann: People think math is complicated. Math is simple. Real life is complicated. (Actual: "If people do not believe that mathematics is simple, it is only because
they do not realize how complicated life is.")

I told the story Glenda Lappan tells, about the shepherd with 132 sheep, who delivers them evenly to four fields, how old is the shepherd? They laughed, then one person gave the common student answer: 33 years old.

So then they tried to solve the problem. Some people decided 8 hours, some 4. I asked what's the most it could be: we got to less than 3. That gave one person the courage to share their answer of 2.

We showed the rest of the movie clip, but stopped at 3:22 before the math explanation started. Were they satisfied with 3x5/(3+5)? No. Earlier talking about the course (doing the Piece of Me activity) someone had asked what my teaching style was. I didn't know how to describe it, but had said I would not be up front telling them stuff, they would be working and discussing. I used this part of the clip to support that, him telling the answer did nothing for us.

I did a bit of a demonstration (more like an early 'with' in gradual release terms): what do we know for sure? Set up a timeline: at 3 hours, 1 house plus a part. (Should be more than half, students say) At 5 hours, two houses plus more than half again. At 6 hours, 3 houses... what time would make sense to think about here? 30 hours, a student said. "It's like finding a common denominator." So at 30 hours, Joe's painted 10 houses and Sam has 6. 16 houses in 30 hours. Does that tell us anything about one house? A student suggested 30/16. Why? "Because that's their average. Per house." Awesome! They invented unit rate and in hours per house not houses per hour hour. Then someone pointed out the average was the same as the answer from the clip.

Jot down what you're thinking about after solving that. Share it with your table.

So onto our next problem. I warned them that it's not the same as the painter problem because it's solved the same way, but it is the same in that we want to make sense of it. I adapted the NRICH problem for less obvious units and less information.

Work in Progress

A job needs three people to work for two weeks (10 working days).

Andi works for all 10 days.
Burt works for the first week and Claire works for the second week.
Dave works for 6 days, but then is too sick to work.
Edie takes his place for 3 days, then Fred does the last day.

When the job is finished they are all paid the same amount. At first they could not work out how much each man should have, but then Fred says: “If Edie gives $150 to Dave, then at least Dave’s got the right amount.”

If we have enough information, how much was paid for the whole job, and how much does each person get? If we don’t, what’s a little bit of information that would let you figure it out?

They really gave it a go. A couple people got to a quick answer with the numbers involved, and all their tablemates couldn't dissuade them. I recorded their strategies as I heard them:

What they got from making sense of what the problem asked.

Asked a student to record her chart on the wall.

After they worked for a while, most students felt like they did not have enough information. Those who felt they had a solution could not convince the class of it.

They consolidated the chart into the center information on days worked, and grouped them into three ideal workers, who worked all ten days.

In discussion I pointed out that they hadn't used the information that $150 made Dave's pay fair. They kept losing track of the idea that they were all paid the same, and wanted to know how much Edie had left. Once they decided for sure they didn't have enough information I gave them more. (I considered just saying 'yes, you do' but they were stuck. ) So I said Fred had another idea. If he gave all his money to Andi, she would be set.

Then they were off and running. Several solutions popped up. One was more algebraic, which killed the interest that many people had.

This was revived when someone presented a just logical idea. If Andi was set, so were Burt and Claire. Because they worked half of Andi's work, and had half of Andi's money. If they were fair, then Dave had pay for 5 days, so then the $150 was one day's pay.

Once they knew one day's pay, they backward engineered the pay per person and the total.

Someone asked about the original information, and I supplied that we knew how many days of pay (person-days, like man-hours, in my head) there were - 30 - and that was split among 6 people. This didn't have a lot of traction, as I think man-hours is a weird unit. Some people connected it with work and got it.

In general, use of symbols was a barrier, not a help, which means we have our work cut out for us. On the other hand, this lesson with these two problems was packed full of the values of the class culture I want to establish, and got them discussing math for the first or one of the few times in their life.

Welcome to the 84th Math Teachers at Play Blog Carnival!

84 is a portentous number. It's the sum of twin primes (What's the previous sum of twin primes? Next?). It's thrice perfect, twice everything. It's positively Orwellian. It's even a town in Pennsylvania. It's a game (actually a variant of the domino game 42 if you play with 2 sets.) It is the #edtech that shall not be named.

(click for full size image)

84 puzzler 1:
Number the intersections of these five circles with the integers 1 to 20 so that the points on each circle sum to the same.

An exciting development this week was Jed Butler unveiling the Math Twitter Blog-o-Sphere Directory. It will be a great resource, but only if you add your information! Maybe you blog, maybe you tweet, maybe not... but you are a mathy type on the interwebs or you wouldn't be reading this!

What to Read?
It was a good month for math reading related posts.

84 Puzzler 2
84 is a side length in the smallest integer perfect tetrahedron, and is tetrahedral number to boot. Which triangular pyramid has 84 points? How many points in the nth pyramid?
(Image made in GeoGebra 3D)

Graham Fletcher introduced fractions with a tough tangram activity that highlighted misconceptions.

Mike Lawler shares the most amazing family math, often with video of the kids' insights. Here's his account of a fun ZomeTool exploration, and part 2 that digs a bit deeper.

84 Puzzler 3
What is the odd pattern that produces these multiples of 84? Highlight for hint: 7 is involved.

0, 2184, 78120, 823526, 4782960, ...

Why Do We Do the Things We Do?

Manan Shah is griping (productively) about Truth Tables. (He also regularly produces a comic, like the one at right.)

Malke Rosenfeld tried out a representation game with some of her elementary Math in Your Feet students. It helped her understand her own objectives!

Jo Morgan assembles a redonkulous amount of cool maths resources, but here are her thoughts from a workshop on a myriad of ways to teach HCF. (math & GCF in American.)

That concludes the carnival; 84 cheers! Remember, you can submit posts to the next carnival via Denise's MTaP form. If you didn't see last month's Carnival 83, it was at CavMaths. We're looking for a host for Carnival 85 - can we come over to play at your blog? Email the Founder of This Here Shindig to give the all clear. Thanks to everyone who pitched in with submissions!

This is a thinking out loud post. Anyone who can push my thinking, I'd be glad for it, in comments or on Twitter. Names omitted to at least partially protect well meaning folk from my ignorance. I may be more wrong in this post than I have ever been before. (Which would take some doing.)

I was at a conference recently where I got to hear someone speak whom I respect a lot. We have our students read their work, and good things come of it. The subject of the talk was long running research on teacher preparation. This was prescient, as teacher education is being heavily questioned, there are proposals for ridiculous restrictions and regulations, and we have our first competition ever from alternative certification programs.

So how would you show teacher preparation is effective? Maybe compare the learning of the students of teachers who were prepared vs students of teachers who weren't. Oh, there aren't any of those. Many of those, anyway. Also, hard to compare different programs. OK. Evaluate your program. Hmm, but there's even a lot of variation within a school. Well, we could make everyone teach exactly the same. And always the same topics. We'll need detailed lesson plans.

That's what they did. To test effectiveness, then, they kept track of a random sample of students. (It's possible the randomness was simply which students they could track.) Then they tested them on the content and teaching of three topics from their freshman level teacher preparation course. For comparison, they also tested them on one topic not from the course. What would you expect they would find? What would you hope they would find?

They found the teachers did better on the topics that were taught. And how much better was proportional to how much emphasis they put on those topics. This was true for the students as seniors, first year of teaching, second year of teaching and so on.

Even after being tested on the un-taught topic, the teachers realizing they don't know it as well as the others, the next year found the same results. The lack of transfer, it was said, proves the effectiveness of teacher preparation. If they could transfer, all we'd have to do is teach them one thing.

What about the practices? (Which would have been the processes or proficiencies back when this research was started.) Those are taught implicitly. Because you have to do them to get at the content, you know. My experience, and that of my betters at GVSU is that you cannot teach these implicitly, or even as an add on. Front and center and even then good luck.

For the capper, it was stated that the state might only certify teacher education programs that covered all of the common core standards. This was obviously foolish to everyone in the room. There's so many! No one could cover them all! Almost no one seemed struck by the irony that one of our leaders may have just proved that it is necessary.

It was a bitter pill.

It also seemed to go with one of the themes in the Twitter week for me: teacher-proof curriculum or curriculum-proof teachers? If you have a moment, take 5 minutes to listen to Megan Taylor's (@ilovemath11) Incite.

The researchers I'm talking about, and math ed folks in general, are strongly tempted by the teacher-proof curriculum. I know curriculum is important - I was a part of an effort to move a large district to a problem solving approach when they still had a traditional curriculum. (Turns out, teachers do not have time to gather the materials themselves.) But you could hand teachers the lessons from the Teacher's Edition of The Book (stealing from Erdös*) and it won't make as much difference as what Megan is talking about.

I get to teach with David Coffey (@delta_dc) and it is a humbling experience. He's great. But one of his most frequent dictums is that he doesn't want a lot of little Dave Coffeys running around. He's not trying to turn his students into him. The wisdom of this is now something I accept. Teaching is so deeply personal, it just will not work if it is not authentic. (Belief, not research.)

I tried to suss it out with my preservice high school teachers this week. I have 2*2*14*(55/60) hours with them in a semester. I cannot cover all of their content, nor all of their crucial content, nor all of the most difficult content to teach.

Just like they will not have enough time with their students. What are they to do? What am I to do?

Use the opportunities we do have to teach our learners how to learn on their own. The math is important, the practices are important, but this is most important. My university teaching of teachers is best when it's holistic and I am doing what I want them to do. Math is such an amazing context for doing this; we are truly blessed.

Disclaimer: I have a rather unfortunate bias against a lot of education research as it is. Maybe it's a remnant of having been a mathematician, and the culture of the hard sciences looking slightly askance at social science research. Maybe it's a function of how often education research results are abused. "There's one study that didn't find class size to matter for test scores. Let's pack 50 kids in every class!" Research is vitally important, but teaching is so hard, and there are so many variables, that I just think it's very difficult to extend results. As a consequence, I tend to value story and qualitative research more than quantitative.

I'll still read the research, looking for techniques, ideas, and inspiration. (Read a great bit from Ilana Horn and Sara Campbell just today!) But we can't start thinking it's the answer in itself. (Maybe learning is solvable?) Just because we can make something testable, doesn't mean we should. I think what I want most is integration of researchers with practicioners, working on the problem together. And then conversation with the researchers, not a lecture. Didn't someone prove those don't work?

End rant.

*Erdös: "Even at this early point in his career, Erdős had definite ideas about mathematical elegance. He believed that God, whom he affectionately called the S.F. or Supreme Fascist, had a transfinite book (“transfinite” being a mathematical concept for something larger than infinity) that contained the shortest, most beautiful proof for every conceivable mathematical problem. The highest compliment he could pay to a colleague’s work was to say, 'That’s straight from The Book.' " from the Encyclopedia Brittanica

While my desert island article is the one where Brian Cambourne shares the Conditions of Learning, Richard Skemp's “Relational Understanding and Instrumental Understanding” (reprinted in Mathematics Teaching in the Middle School, September 2006) is not far behind. And it may be better to discuss with preservice math teachers, since it doesn't require transfer from literacy to math. Despite being a rather difficult read, it never fails to provoke good discussion and deep thinking.

Previously on the blog I have: interviewed a baseball coach/math teacher about relational understanding, recorded student discussions, and a post about the article. So thisis only the fourth post, it's not like I'm obsessed. I don't give a formal homework assignment too frequently, but still do for this reading as support is helpful. (Assignment.) I also have a workshop for use in class:

After time to work through the questions, Sam led the start of the discussion. She hit the ideas of relational and instrumental, and solicited examples of the contrast for fraction addition and subtraction. But as she noted - it felt like multiplication and division was where the really interesting bits would be. So I split up the groups among multiplication and division and then recorded their quick explanations.

Loved that the key question "3/2 of what?" came up here. I was fascinated by the "sometimes it works, sometimes it won't" idea. That's a real vestige of instrumental understanding, when we are given rules but often not the conditions under which they apply.

We discussed the grid here for what might confuse students, and tried to connect back to context. Students often want to draw a picture for all the quantities, even though there is not 1/6 of a whole here, but they were taking 1/6 of 1/2.

The lack of a picture was good here, and we discussed how relational doesn't mean with pictures. I tend to ask them about pictures to push their understanding because they are more likely to have rules for the numeric than the visual. Although the grid method can become very rule driven, just like the numberline for integers. This discussion was also grounds for discussing the difference between explaining why a method works and justifying that it does work.

In the last explanation we were getting close on time, but they posed a couple good why questions to which they struggled to good answers.

One thing about university classes is that it can be hard to get them to ask each other questions as the duck and cover principle is well learned. I try to stress that the discussion is one of our best tools for pushing understanding, and in math ed classes, I try to frame it as teacher training - you need to practice posing questions. Still tough sometimes. I'm satisfied that they see a difference in the modes of understanding. Fractions are just such good content for this, as math majors' computational fluency is strong, but they can tell there are things they don't get. One of the gratifying parts is how much they want to get it, and take on the goal of getting their students there as well.

Bonus: as they write their next blogposts, we might see some writing on this as well. First one in is from Matt -Instrumental vs Relational.

One of the questions posed during Nova Now 15, a state conference focusing on discussions among teachers, was 'should conferences end with a reflection session instead of a keynote?' Any opportunity to encourage teacher writing.

So now I feel obligated.

As with Twitter Math Camp and EdCamp, this is a conference organized on the principle of creating teacher conversation and collaboration. It's hosted at Kent Innovation High, where, frankly, I wish I could send my kids. It's a tech friendly, open design, project based learning school. Kids attend in the morning for core classes (science, math, ELA, social studies) then return to their home schools all over the ISD for the rest of their schedule. The conference starts off with a tour and chance to see the learning happen, and then several students take the option to stay and be part of the conference, even coming back on Saturday. The single most frequently heard comment for me was about the eloquence, maturity and phenomenal perspective of these students. My favorite #KIHway quote from Peyton: "Students have to shift from doing this to make people happy to 'I'm learning how to be creative and productive'."

Even before the conference started, I had a great talk with Laura Chambless. She's the K-7 math/science support for St. Clair region schools. She's got her resources organized in a protopage. (Free start pages that can also do RSS feeds.) She's a big believer in fact fluency and has been trying to find ways for teachers to get at that constructively.

#michED is our statewide Twitter chat. Wednesday evenings, 8pm ET. The first big session was a meet between the east side and westside collaborative groups, Innovation Now and the Bluewater Group, moderated by Rushton Hurley. He did a good job of guiding a discussion, and using that to also make his points. Some big ideas raised:

Isolation is the cancer of the teaching profession.

Good ed leader question: how many times have you deviated from your plans? Because that's a measure of how many cool moments you've facilitated.

We are not good at sharing successes. Why are you a good teacher? "I care about kids." Who doesn't? Share specific successes as individual teachers and as a school. Share them with voters!

Derek Braman led a session on student blogging. Mostly ELA teachers - I want to hear how content teachers are using it, too. My big takeaway was an activity he uses to introduce blogging. Have students write a list of some of their passions. Pick one to write about on paper. Then circulate and leave comments for people on stickies. I'll try that next fall and share how it goes.

'Math: are we doing it wrong?' was led by Rick Jackson, who kept good notes on resources - http://bit.ly/1zWbae1. Dan Meyer came up, PBL in math with the KIH teachers, SBAR, flipped classroom ... all the good stuff. Infuse Learning was recommended for BYOD formative assessment. Teachers were surprisingly reluctant to discuss, surprisingly given the context, but there were two KIH students who killed it in the conversation.

"We have to still follow state standards, which have nothing to do with learning or critical thinking."

"The pioneering spirit is a big part of KIH, shared between teachers and students."

"Students have to shift from doing this to make people happy to 'I'm learning how to be creative and productive'."

Ben Rimes led a session the next morning. (His notes, which include the cards.) Out of all the good stuff going on, he won me with teases about the world premiere of his Keynotes for Humanity game. The game itself was a great discussion piece - I should think about how to use the Cards Against Humanity structure in math class. (Make your own.) The point about the keynote's role in ed meetings was that we need to think about it's purpose and utility.

Jeff Bush (@bushjms) and Rebecca Wildman (@RebeccaWildman) had a session on student centered classrooms. (Their ) One of the hallmarks of a meeting like this is that being discussion driven, the sessions may not go according to plan. Two of their tasks for us took over the discussion. Find a video that represents you as a teacher. The most fun:

The second task was to make an infographic of a topic. I haven't tried this yet. Rebecca assured us that there are infographics on everything, and that seems to include math topics. Good synthesis assignment. Kate Kling recommends http://piktochart.com for a free tool.

The last session I attended was Jennifer Bond's creative play. (@teambond; her Google site.) What I learned is she has the best toys. The littleBits are very curious. Another good resource is the Imagination Foundation. Best quote: "if you give them open-ended time, you'll have their attention week after week. They don't have time to play." Sad to think about students not having time for real play. Jennifer ran a creative play club for which students applied. Every single form I saw, these elementary students considered themselves creative. By the time we get them in college, few people claim that. What are we doing so wrongly?

My session was on Talking Points. I'll write about that separately! I had an awesome group of teachers to share them with and discuss them in other disciplines.

If you say good, you're arrogant, you don't understand how good teachers are always trying to get better, you're not aware of your faults or you're just blind to the signs.

If you say bad, you buying into some kind of talent argument, ignoring growth mindset principles, falsely humble, don't even know what formative assessment or SBAR means or, worse, you're guilty of one of the big teacher sins like low expectations, no classroom management or ... I gasp to say it ... LECTURING.

Are some teachers better than others? Of course. I see them at work every day, in my case. Do I want to know how to get better? Of course. I think about it rather obsessively. Is it helpful to classify teachers? Of...

I don't know. I'm open to arguments, but everything I know and has experienced leads me to say no.

Part of the academic life at university is this periodic vetting of our colleagues. Who is good enough to stay?

So we have to functionally label people as good teachers or bad. Since we are so bad at measuring teaching, our main tool for this is student evaluations. This is not the purpose for which student evaluations are meant. But they have pithy comments, and numbers we can average! I can pretend it's data about the quality of teaching!

Tenure is a weird construct, and I understand why it makes people fidgety. Ultimately, my view of it is colored by my brilliant advisor, who saw it as freedom. Freedom to pursue interests instead of quantitative measures, which will produce more innovation in the long run. We want to know, what will people do with their freedom? Which is one of the fundamental problems of freedom.

I'm fortunate, in my view, to be at a university where teaching is the major factor in tenure. Getting a little less so, but it's still the case. There are not many public universities like this. Send us your children, because they are going to get a better education here than almost anywhere.

But it also brings us to these moments of passing judgment. If I'm against classifying good and bad teachers, how do I make my decisions?

Is the colleague interested in teaching, working on their teaching, selecting good goals, seeking out professional development that will move them forward? If yes, then I hope they are different than I am, teaching for different goals, working on different teaching problems. Because then, as a faculty, we will be the better for it.

I don't want to judge where I think your path is going. I want to hear you talk about your journey. If you're willing to share it, let's get a coffee.