Sunday, May 1, 2016

Blogging to Blog

OK. I've been thinking about Anne's #MTBoS30 challenge and I'm in. It was ParkStar's post that was the tipping point somehow. But probably just for today.

I'm in the middle of grading finals, grades due Tuesday. Noon, though hopefully it won't come to that. Meant to have them done last Friday, but I've been taking every distracted path possible. It just doesn't seem as relevant. Grading with no feedback, as only .1% ever come back to see this stuff. A minority even want to see their grade breakdown. The learning is done, so why am I still here?

In particular my college algebra class that has been such a challenge. They took my lovingly crafted questions and gave them short shrift. They're showing me all the ways in which they did not learn what I wanted them to learn... and I oscillate between blaming them (which, thankfully, the #globalmath on Math Trauma by Kasi Allen helped me spot) and blaming myself.

But I know the blame cause is shared. I have to create the conditions for learning that I influence, but they have to engage and be responsible, too. And I know that I'm a better teacher for those struggles.

The struggle now is to find evidence of understanding for the standards, to fight for them to get the best grade possible. I wish we could still talk about it - so much of what they did bears discussion. I'm annoyed by people who did nothing to the last minute (although I was one of those students and am now one of those teachers; not doing nothing, but...) at the same time as I'm so happy that they didn't give up. Even if it was for a grade, and not for learning.

Here are my current learning targets for college algebra. Lots of ideas for revision, so maybe that's grist for some 2 ≤ day ≤ 30.

Friday, March 18, 2016

Block Market

How much is that number worth? It's all about location, location, location.

This is another story of impulsive teaching. I'm not recommending that, but we got to a good place, so I want to tell someone.

In my preservice elementary course, we were headed towards decimals, passing through place value, so it was time for the base 10 blocks. A wise elementary teacher taught me that new manipulatives should always start with play time. (If you can't, tell them when they will be able to play. Chris's other lesson was to use each new manipulative as a chance for the students to tell you the rules about using them. Pro tip.) The wooden Base 10 blocks we have are particularly good for building.  But playtime always ends with: 'so what did you notice about these?'

They found the 10 fit into the next one pattern, and noticed irregularity in these old, hand-cut materials.

I love when manipulatives are used for a purpose or a problem rather than a set of exercises. So I asked how many blocks were in each tub, in terms of the small cubes as the unit.   3510, 4873, 4508, 4508, 3377. Hey! That's not very fair. Ooh, I have an idea: what if each group gave the next group half of their blocks? 4191, 4691, 4003, 4508, 3443. Is that any better? Some say yes, some no.  Let's give half again! 4441, 4500, 3741, 4256, 3817. Still disagreement about what's happening.

Okay. Let's settle this like mathematicians. Make a display of the data that proves your point. We collected one more round of give half away.

 I was really impressed at the diversity of displays by happenstance. It made for a great discussion of results. As we often do, I asked for each group to get feedback from the other students: one specific thing you like about their work, and one thing that would make it stronger.
 I missed one of the graphs, but here are the other three.
The first graph shown, people liked how it made the visual comparison of the round by round numbers. Convinced people that the numbers each round were getting closer.

This display charted each group's round by round count compared to the mean.

The classic lineplot follows each group's total round by round. People agreed that this showed convergence to the mean the most strongly.

But while people were making their graphs, I noticed something, and had groups record their total for each block on the back board. What's the problem?

Our individual block distribution is out of wack. Two groups don't even have enough to compose the next unit! We had to make some trades to get a better balance. We could play...


We went table by table with people proposing trades. The whole class decided if a trade was fair. It was the most fun I've ever seen composing and decomposing by place value. Trading was heavy and fast paced. But occasionally we had to stop to check fairness. We even had one crazy three way trade. There was lots of interesting reasoning about the quantities and how they got out of whack even while the totals converged. Final count - not bad.

The idea of social relevance in math class has always been an interest. My colleague Georgi Klein made great use of Marilyn Frankenstein's algebra work. And we almost got a chance to hire Mathew Felton who looks at the political aspect of math learning. So I closed with an observation that with so many math problems about maximizing or candy, it might be nice to address big issues, and disparity is something that's going to be an issue. I got a little preachy, really. But it felt like a good day, with some real values in our place value.


Transitioning to decimals, after work with a fixed unit, we traditionally do something like the top part of this next activity. (Probably originated with Jan Shroyer.) It starts the idea of shifting the unit for different situations. Pretty effective. This time around I added the problems at the bottom as puzzles. They were very interesting for the students to think about, and seemed to push consolidation of their decimal strategies. It really requires a lot of reunitizing. I'd love to know how middle school students thought about them. Each group made up a puzzle of their own to swap, and that also seemed beneficial.

I'd love to hear your thoughts about political values in math class, block market trading for place value, or the representation puzzles.

Sunday, February 28, 2016

Exploring the #MTBoS

My elementary preservice teachers (PST) are exploring professional development this week. The first assignment was to do a webinar or our local conference, Math in Action. Global Math was Problem Strings with Pam Harris (awesome) and Christopher Danielson is keynoting Math in Action, so fortuitous timing, say no more. The second assignment is to find a blogpost to recommend to teachers, so I thought I would pass these along. The list of leads I gave them is:

Apologies for any exclusions. These are all people who's work has come up so far in class. What elementary blogs would you add?

Their recommendations:
Dana -
Summary: This blog post is about a class of students looking at four different shapes, and trying to find the odd one out.
Review: I thought this blog was great because instead of simply telling the students the names of the different shapes, the teacher let them think and reason for themselves; she allowed them to come up with and defend their answer by themselves.

Dayna - This is a great lesson to combine English and math and get students excited to learn. My response to the lesson is that I love that as a teacher you get to see the students thought process when they are working on this problem.

Kalyn -
This post talked about how comparison problem are everywhere, even outside of school. Although they can be difficult at first, the lightbulb goes off and the problem makes sense! I really enjoyed this post and the person example that it gives of her daughter and the conversation that they had about math, but also about life. A lot of good stuff here!

Ally -
This is a great blog. It's about how is he works with "Alex", going through counting. It was a great read.

Amber - For my blog post, I chose to look at some more of Graham Fletcher's stuff. And although we aren't learning about volume right now, I thought this post was a great representation, which shows real life problem solving. It offers that children are robbed when force fed uninteresting story problems from a text book, and Graham offers an interesting 3 ACT problem as a substitute. My reaction to this concise, yet powerful read is that I would like to try a problem like the one he brings up. I bet students would be very interested in it.

Sarah - Summary: Second graders explore sorting by counting by 2's, 5's, and 10's. A similar activity is conducted with first grade students. One students has difficulty counting when there is a leftover present. Review: This post really made me think deeply and question the use of ten frames. Kristin Gray does an excellent job of questioning the thinking of students and that is something that I, personally, need some work on. 

Chris -
Summary: This blogpost is about an "artsy-mathy" activity he did with students involving creating trees out of factor trees.
Review: I absolutely loved this post because it addressed an issue with "artsy-mathy" activities, which is that they tend to be less artsy than an art lesson and less math than a math lesson. I enjoyed how he addressed the issue by making more out of the project and creating an exhibit where the students could teach to younger students.

Orina - I thought this game was really interesting and fun because students have to use number sense to try to win. A lot of students are familiar with hangman and it's a math way to play a fun game.She comments at the end about how this game is competitive and can be cooperative also. Playing games can bring more fun to the classrooms, but she comments that we must make sure there are some competitive and some that are about cooperation.

Kathleen -
this teacher was trying to get her students to understand grouping and have then work together to see what made the shapes different and what the noticed in general. a lot of then saw that there are many triangles in the rectangles.

Heather - I really enjoyed this blog post because it addressed the question we all ask ourselves, when will I ever use this? I like the way he addresses practical examples and being able to take those practical examples and use those for practice problems not "mind numbing" problems. (John says: "be sure to see Joe's follow up.")

Stephanie -
Summary: Counting circles can serve as more than one purpose of just counting; it helps you practice standards, recognize patterns, etc.
Review: I never knew you could do a counting circle in some many different ways; the more questions you ask your students about it, the more they will think about it, and deeply understand the material better.

I admire their taste in posts! 

The other thing is how much I want to thank these bloggers. By sharing your classroom you are having a profound effect on other teachers - and on the future. It takes time and vulnerability for you to write, and I want to thank you for it.

Saturday, February 20, 2016

Teacher's Block

I am struggling with my college algebra class.

There is the classic misalignment of what they think math is and what I want them to be able to do. The makeup of the class is almost entirely people who are done with math after this course. In our department, we're trying to separate this from precalc, making a new course for people moving on in math and this course will be for people finishing their math. The goal is to make this course more conceptual, and prepare students for use of mathematics in other subjects. Previously this course had so many skill objectives that teachers were put into coverage mode. Tenure track faculty teach it occasionally, full time affiliate instructors sometimes and most frequently adjuncts.

The course started off on the wrong foot. Two problems that have been a smash in the past went awry. On the house painter problem they were unable to convince themselves of the answer. And on the fair pay problem a misconception was shared and caught on so that it became insoluble. When they came back, one student had a nice intuitive solution that he could not convince the class - even with my help - that it was correct. Finally someone asked me point blank if his answer was right. "Yes, but one of our goals is that you are able to decide for yourself."

Ay ay ay.

By nature, they are reticent to talk to each other. Despite my urging repeatedly, and sharing how math is best learned through discussion. Many don't engage in activities, they're waiting for me to tell them how, many are not doing the homework, and absenteeism is about 20%. Standards based grading has been a tougher sell than usual because this class, as a group, wants the math that was.

I've removed a lot of choice, I've been doing more demonstrations and spending time on the teacher half of gradual release of responsibility, and I'm super explicit about what problems show which standards on assessments. They still won't talk, and many won't engage in in-class inquiry. They will make up something rather than ask about a question they don't understand on an assessment.

Ay ay ay.

The other day I saw this image on Twitter, but stupidly didn't catch the source. Simon Gregg remembered - it was David Wees!)
Our College Algebra picks up with quadratics, but a lot of the work I do with students visualizing patterns is for linear content. (We did do the growth problems, though.) Doing some other work I realized that the students were not understanding symbolic representations as generalizing number patterns. (There are even such quadratic examples on my own blog!) They had been getting by on regression rather than representation. I though this problem would be a great introduction to this kind of generalizing.  It did seems to be helping connections form. I wanted to extend this to cubic or higher, so I built this pattern.

First we discussed what was going on, what they noticed and what they wondered about. Very few students wondered about how many cubes for the next building. More assumed that the next building was to be built with exactly that many orange blocks.  That's very different than my thinking, and emblematic of how difficult it is for me to anticipate how this class will respond to prompts.

I built a very scaffolded worksheet. I used to make stuff like this all the time, but have been moving away from it. But sometimes students need supports.

Another adjustment I'm trying to make is, instead of roaming the room to eavesdrop and do formative assessment,  to roam and ask questions, try to encourage table conversation, and hover over students doing work for other classes or just sitting. I feel a little awkward promoting engagement by (what feels like) intimidation, but students need supports sometimes.

One of the more successful areas of class so far has been the math writing. They have six assignments over the course of the semester, they can count towards SBARs, they can revise for their final exemplars. Several people are writing about this problem for their current writing. That tells me this was at a good place for them, and I'm happy to see some of the sense making.

Meg writes:

The strength of the temptation to give in, to teach them how they think they'd like to be taught is way stronger than I would have guessed. But so is my stubbornness to not give in. What I am trying to be wary of is to keep my stubbornness from stopping me giving the support that students need and teaching the students in front of me, rather than some fantasy class.

Sunday, January 24, 2016

My Favorite (Teaching): Improvisation

Sign in data. Most of the variety is size.
This is one of those things that's both a strength and a weakness of my teaching. I have a lot of ideas about things to try, but that is not especially professional. When people talk about the profession of teaching, we always seem to compare to doctors. You don't want your doctor making up treatments on the fly. Research! That's on what doctors should base treatment. Maybe it's okay because the stakes are less high - one topic in a math class vs. your health and well being, or because in teaching we are the researchers, too. Or maybe medical doctors are not the right comparison.

Regardless, I like to improvise! This is the story of two of those moments, in the same class period.

The characters, preservice elementary math teachers; the content, learning quadrilaterals with a focus on reasoning with properties; the setting, they've gone from describing quadrilaterals to thinking about their properties. Day 1 was spent describing quadrilaterals on geoboards to make, and thinking about a variety of different possibilities that still fit a type. They took home geoboard dot paper to make their own quadrilaterals, one of 11 types. (For us: square, rectangle, rhombus, parallelogram, isosceles trapezoid, right trapezoid, trapezoid, kite, chevron, convex, concave.) In general my teaching here is guided by the Van Hiele levels, in particular activities that give students reasons to transition from visual to analysis, from analysis to informal reasoning and then informal to formal, depending on the level. This is K-5 focused, so we don't push at formal reasoning too hardly.

I have an old set of quadrilateral cards that has a lot of visual confusion. Looks like a rhombus but is a parallelogram, etc. They've been good for me in the past, but the longer I teach, the more I want the students involved in the manufacture of math materials. So this time, they made the cards for homework. I made a couple of extra sets in case someone hadn't had a chance.

The first activity was the same as I've done in the past: Quadrilateral Concentration. Players shuffled up their cards (I had them put initials or a symbol on their own so they could get them back, but style would have been enough for most), and dealt them out into a grid. Turn over two cards and if there's a match - two quadrilaterals of the same type - you can pick them up. Most people knew the game already. The conversations were amazing. First question, "I turned over a square and a rectangle, can I pick them up?" The table ruled no, and I supported. The best arguments are over type, though. Either while playing, "hold on, I don't think that is a rhombus," or at the end... "Why don't these have matches?"

My fair quadrilateral
To summarize, I brought up to the document camera the ones that provoked the most discussion. We also used them to talk about variety again - as there were MANY congruent examples. And we discussed the most ambiguous case, which is probably now my favorite geoboard quadrilateral. People thought it was a kite, people thought it was a right trapezoid, people argued about the length of sides, people argued about the size of angles, people compared it to a square... loverly. It was especially nice because people kept cycling back to earlier claims, which seems to prove what I was suggesting about the power of our visual processing.

Another game I've played with the old cards is quadrilateral Go Fish.  We played with the same rules as concentration, using the most specific names possible. Suddenly it occurred to me that we could play concentration as we had, but switch the Go Fish rules to allow for more mathematical subtlety and strategy. Not everyone had played Go Fish, but enough had to make the rules explaining go smoothly in each group. To play a match, they had to be the most specific type. But when your opponent asked you for a type, you could give any that fit the characteristics. BAM! This was great, and almost instantly a better game than the original Go Fish. There was the start of some strategizing, where some people weren't asking for exactly what they wanted, and the conversations were spot on. This is a trapezoid, are you sure you don't have anything that fits, etc.

That's an improvisation that paid off. Better than what I had before.

The other idea on the spot was to go farther into combining properties. I wanted to make it natural to think about what if a quadrilateral was a this and a that. So spur of the moment, sidebar into a weird movie and TV discussion. I asked, what was an adjective that described a show or movie that you liked to watch. Then I shared how my wife's favorite genre was funny + scary. "Like Krampus?" (Side discussion on Krampus, which we recommend. But only one person had seen it, so...) I wrote down the 'equation' funny+scary=Ghostbusters. (Best example is probably Buffy, though.) Then they discussed at their table until each person had one to put on the board. I was worried about = abuse, so I did mention that what we're really doing is finding examples in the intersection.

And one table really got into trying to do adjective arithmetic. We talked about the examples & shows for a bit and then I transitioned to the purpose: what if we combine the quadrilateral types this way? Each table I wanted to come up with one quadrilateral equation. Got some good ones, and I shared about the role of conjecture in mathematics. To their list of four conjectures, I added some questions.

I connected this to the homework, which is to try the very challenging problem of a Venn diagram for all the quadrilateral types. We'll discuss those and the conjectures next class.

Passed it around again, and got much more
variety of property and orientation.

This improvisation was okay. Don't think I did much harm, it was a moment of high engagement, but not necessarily in mathematics. Well, it was mathematics, but not our quadrilateral content.  The disappointing thing is that the conversation about the shows - reasonably analytical - didn't carry over to the conversation about the quadrilaterals.

I'm okay with this, however, because even a bad result is going to happen sometimes. The same activity that is a gas burner with every class that has ever tried it can crash and burn. So the improvisation increases my store of supplies, keeps my interest, and gives me things to think about for student thinking.

Tuesday, January 12, 2016

Similar Triangles

So now I'm going to blog about something that I'm just starting to think about.

For two days, I've had a tab open with a neat Futility Closet post. (So many clever bits of mathematics and reasoning there.) It has this image:

A pleasing fact from David Wells’ Archimedes Mathematics Education Newsletter
I immediately made it up in GeoGebra, but being the start of a new semester, hadn't really thought about it yet. I didn't see what parallel lines had to do with it, nor being equilateral. About to close the tab finally, I shared it on Twitter. Boom!

Matt and John jumped in. And then HenrĂ­... 

I love the cycle of generalization in math! Get rid of this restriction, and that restriction.

Get rid of the 2nd line.
Get rid of the shared vertex.

And then the what ifs. What if we restricted a vertex in the preimage?


Lines and circles to lines and circles... must be complex. But John had already gotten there!

Then Simon found circles another way!

Now I'll go find someone to talk with locally about the complex transformations here. I could do it alone, but I prefer math in dialogue! I think I want to see someone else get excited the math, too.  I also enjoyed this coming up so soon after the post on Willingham's 4C's of story. Great illustration of the causality and complications inherent in an interesting mathematics situation.

It's available in GeoGebra if you want to play, too.

Sunday, January 10, 2016

Story Teaching

Quick post thinking about Dan Willingham's post on The Privileged Status of Story, which I got to via Dan Meyer's post Study: Implicit Instruction Rated More Interesting Than Explicit Instruction

What constitutes a story?
"The first C is Causality. Events in stories are related because one event causes or initiates another. For example, "The King died and then the Queen died" presents two events chronologically, but "The King died and the Queen died of grief" links the events with causal information. The second C is Conflict. In every story, a central character has a goal and obstacles that prevent the goal from being met. "Scarlett O'Hara loved Ashley Wilkes, so she married him" has causality, but it's not much of story (and would make a five-minute movie). A story moves forward as the character takes action to remove the obstacle. In Gone With the Wind, the first obstacle Scarlett faces is that Ashley doesn't love her. The third C is Complications. If a story were just a series of episodes in which the character hammers away at her goal, it would be dull. Rather, the character's efforts to remove the obstacle typically create complications—new problems that she must try to solve. When Scarlett learns that Ashley doesn't love her, she tries to make him jealous by agreeing to marry Charles Hamilton, an action that, indeed, poses new complications for her. The fourth C is Character."

At the end, Willingham challenges us to incorporate these C's into lessons. In particular, the most important C, Conflict. "Teachers might consider using 10 or 15 minutes of class time to generate interest in a problem (i.e., conflict), the solution of which is the material to be learned."

I think this is compatible with several MTBoS approaches, in particular & obviouly, 3-Act lessons.

Character - my biggest question after my first read was who are the characters? Not in a heavy handed Life of Fred way, but in the story. I think it must be teacher and students for us. We resolve the conflict, after all.  Probably one of the inherent advantages of inquiry teaching is making the students the central characters. Not that we teachers can't be involved - I think we have to be ready to jump in, too. But we can't be Deus Ex Machina everytime, and let the students know there's always an out.

Math lessons are well set up for storytelling otherwise, I think.

Causality - why does this work is a great basis for an investigation. Add up the digits - if that's divisible by three the original number was, too. What? How could that work? Look - these three centers of a triangle are always on the same line. Why on earth...? Of course, if we make it out that knowing the fact is more important, we're killing the story. This is historically a great spark for mathematical developments as well. While I was writing this Sam Shah posted this image which got my mind wandering, making me go off and do some GeoGebra.

Conflict - I have no idea if this is unusual, but I try to get good math arguments going every chance I get. I usually refuse to be the authority. ("Is this right?" What do you think? "I think so..." Well, let's ask the class!) Plus anytime I ask for an answer, I always ask if there are any other answers. And when the students propose answers, there's a chance for a math argument. It also makes me think of Chris Luzniak and his Math Debates.  Even whether a particular topic is math can be a great argument. There's a course I start with Sudoku, and the last question is, were we doing mathematics? I have never had a class agree on this answer.

Complications - is there anything more mathematical than this? Oh, that worked. What if we added this? Could we do it still if we didn't know that? Messing around with conditions is prime mathematical behavior. And if the problem is problematic enough, this happens by itself. I could solve it if I knew that, now how do I find that out? Or you're trying all the cases and get to one where the freak out lives. Or you're practicing the very mathematical habit of mind of trying to find counter-examples to your own idea.

Where I think these C's might be helpful to me is in being more intentional about the type of math the students are working on, and using this structure to help design how I'm going to try to get my lead characters to find the problem.

For my first math for elementary teachers tomorrow, I want to create the conflict for my students between what they know about elementary mathematics and what they need to know. I loved Graham Fletcher's progression of multiplication, so I'm going to try to use that in contrast with their native ideas about teaching multiplication. (Also such a nice synthesis of understanding to model for them.) In the past I've mirrored mathematics development in children and schools, starting with number concept and building up. This will essentially be going in reverse, but will hopefully be a more obvious need to know that will motivate the deconstruction on supposedly simpler topics to follow. Wish me luck!