Sunday, September 8, 2013

Looking at My Inventory

I shared before about the math learning inventory I sometimes give. The greater the diversity of learners, the more inclined I am to give it. It surveys how students think they learn mathematics.

A basic break down of the learning preferences assessed follows from a Silver, Strong, Perrini and Tuculescu article, The Thoughtful Classroom:Making Students as Important as Standards.

  • Mastery learners want to learn practical information and procedures (what)
  • Interpersonal learners want to dialogue and collaborate (how)
  • Self-expressive learners want to use their imagination to explore (how)
  • Understanding learners want to learn why things work (what)
I do not think learning styles are determined for a student, but finding out what students think they prefer seems like a worthwhile preassessment to me. If I want to be a culture changer, I need to understand from what I wish to progress. The form I use is at Scribd, but I'm happy to email the Word document, too.

I thought I'd share how this semester's group of preservice elementary teachers look, and how that affects my planning. Mostly it doesn't change what I want them to experience, but it has a reasonably big impact on how I implement it. Also, with an education class, it's a chance to point out the tension between surface and core beliefs.

Yes, I put it in GeoGebra to help visualize. (I may have a problem. Here's the sketch if you want to use it to visualize your student information.)

Here's the most consistency. Students who feel like "tell me then I practice" is how to learn.  The challenge for this group is abstract or open-ended problems. Also known as all my favorite stuff in math. How I'm responding is to be explicit about what kind of problem we're working on, and support with structural suggestions. Like in working on trying to find all the pentominoes this week, I interrupted several times, solicited progress reports from groups ("They have 16?!"), and got them to share method and develop methods as a whole group.

Especially worrisome is when you compare my inventory with their composite. At the conference where I first saw this inventory I had a number of students.  They divided us up by table based on the results and I was alone! So I'm aware of my distance from students on this and how it can affect my persective.

Whoa. All over the place. This is a caution to me to not require group work always, and to look for ways to make space for some individual problem solving. Since relationship with teacher is part of this, I'm trying to be more personal to supplement for the people at the top end of this scale.

Very encouraging. Many times when learners have had negative math experiences this category can be quite low. I have to think about how to support those four students, though. So much of our class revolves around this kind of why thinking. And I feel that it is crucial for future teachers. I hope to have experiences that justify this approach to math. Maybe the way conceptual understanding furthers skill is the entry point?

Another one where the students are well spread out.  We started with a week long focus on creativity, that really got pretty good buy in from my perspective.  Again the students who do not favor this kind of learning are a challenge for me. Choice in assessment is part of the course structure, but that's probably not enough. What more can I do for them? Is it important for all teachers to be at least comfortable with this mode? 

Teacher-Student Alignment
Here's my graph compared with theirs. 
This probably comes through in the Mastery section above, but it's clear where I don't align with students. In some math classes I give choices for practicing content that can satisfy Mastery learners, but there's just not much of that for this class.  I'd welcome any suggestions here, is what I'm saying.

The view of all their regions at once also raises a concern about students who seem to have low responses across the board, as I think that's just overall discomfort with mathematics.

The goal with this post was to share what use I make of this information. If that didn't come through, or if you have ideas for more uses, please let me know!

Post Script: went through the survey with my 8th grader - very interesting discussion.  Discussing why he was answering what he did on the individual questions was enlightening. It makes me wonder about using this as an interview tool. Even for a student I know pretty well, we got to some new ground.

Post Post Script:  forgot to add the other questions' data. Even if it's mostly for fun, some illumination is to be had.

  • 16 - Show me how and let me practice.
  • 2 - I want to know why.
  • 1 - Let me play with it. (My choice)
  • 0 - Let’s talk about it and hear everyone’s ideas.
  • 10 - p. 101: 1-39 odd
  • 6 - Draw a picture that shows the ideas. (My choice)
  • 3 - Work with a group to make a math skit.
  • 0 - Report on a math controversy.
I have to work to understand the mindset of 1-39 odd. Other than comfort with the situation you know.

Thursday, September 5, 2013

Flip Flop

Jennifer Silverman made this cool motions maze the other day. (More here.) We collaborated on it a little bit, after she did all the heavy lifting. I added buttons. (I may have a button problem.) It put me in mind of these motion puzzles I used to make in Geometer's Sketchpad, and I got to thinking how much better I could make them now. So I started, with the main new feature I wanted being the ability to generate new puzzles instead of being one static dynamic puzzle.

The user moves points A and B to try to find the line of reflection between the two flip flops. When you hit the Check button, it shows you the reflection over the line you're trying. 

What follows is my GeoGebra geek out over trying to make it look right. Here's the puzzle if you want to skip that: Flip Flop.

One thing I love about Jennifer's sketches are her excellent images. So I tried to step it up with some nice flip flops from

It turns out the trickiest part was getting both sandals to always show up. That's why I'm writing this post. (A lot of my individual sketches I post at the tumblr.) The key to being able to do this is that in the graphics window you can put variables in for the window dimensions. Define those from the objects in the sketch, and,  voilĂ , you can see both the sandals. So I defined xmin, xmax, ymin and ymax from the two sandals. E.g.,
xmin=floor(Min[{0, x(F1'), x(F2'), x(F3'), x(F4')}]) - 1
But there's a problem then - the graphics won't be in 1:1 scale, which is always nice, but especially important for motions where the two objects should look congruent!

The Corner[ ] command is my new best friend. Corner[n] for 1, 2, 3 & 4 return the coordinates of those corners. Corner[5] returns a point with (width, height). Corner[image name, number] returns the corners of an image. This was handy for finding the corners of the reflection, F1' etc. in the command above. 
So using Corner[5] I could find out the aspect ratio of the graphics window. It took me a few minutes, but I hit on the idea of making the bottom left corner steady, and then altering the top right corner based on the aspect ratio. I defined r = y(Corner[5])/x(Corner[5]) (so, height:width) and:
  • xm = If[(ymax - ymin) / (xmax - xmin) < r, xmax, xmin + (ymax - ymin) / r]
  • ym = If[(ymax - ymin) / (xmax - xmin) < r, ymin + r (xmax - xmin), ymax]
If the sandals give a screen that's not wide enough, it uses the aspect ratio to find a suitable width. If the sandals give a screen that's not tall enough, it uses the aspect ratio to find a good height. (The If[ ] command works like If[condition, then, else], where the else is optional.

The puzzle, as it turns out, is pretty challenging. Give it a try, and let me know what you think. Or let me know an easier better way to do my GeoGebra graphics hacking.

Here's the teacher page for download or the mobile page.

EDIT: Bonus! Jennifer created an assignment to give it more structure as a lesson. (PDF in dropbox.)

Creative Pattern

So, like most semesters in most of my teacher prep classes, we started out by watching Sir Ken pose the question, "Do Schools Kill Creativity?" Especially for preservice elementary creatures, who often have trouble seeing themselves as math teachers, who often have had very negative math school experiences, and will even sometimes bust out with "I hate math" in front of their math teacher.

This semester's group got pretty into it: the story of Gillian Lynne was high impact, the idea that things need to change had traction, several recognized that they had been subject to this, and the desire to incorporate movement really resonated. (We have a drummer in class, so that might happen.) Some students wrote about creativity for their weekly work: Lauren and Kyrstin, for example.

One of the ways I'm trying to encourage creativity is a work structure (syllabus) like this:
Daily Work: I’m asking you for 1 hour per class. Document what you did somehow and keep in a binder. It is not evaluated on correctness, but on percent completed. Keep an index/table of contents for which days you have work for. This work should either be doing math or learning about the teaching of math.  It is okay to double dip - use daily time for Family Math or weekly work. Just keep track of getting in your hours. I will offer suggestions, but this is your responsibility. It’s a good opportunity to practice generating ways to meaningfully work, which will be an important part of your work as a teacher.

Creating: from our work each week I am asking you to put an additional hour or two into deeper work of your choice. Revise or extend a daily work, play or make a math game, make some math art, find and read something in an area of interest, work on a math problem of interest or create a mathematical task… there is so much different work that teachers do. If you can connect it to our course work, it’s probably okay. Each week’s work will get feedback in terms of our rubric and qualitative.  But those aren’t grades. At the end of the semester this weekly work will be evaluated ⅓ on completion (did you complete work for each week) and ⅔ on exemplars. You will pick two examplars of your doing math, and two examples of your preparing to teach math.
There's a of their weekly blogs The list helps me in finding them all for giving feedback, but I ask them to link posts to our Facebook group as well. That gets more readership amongst the class than I've ever had before. One of the purposes of blogging their work is to increase their sense of audience. So if you do take a peek, please comment!

The math content we paired with this is patterning. Our first activity (close to this previously blogged one) got us playing with the appropriately named pattern blocks, trying to get at the idea of what makes a pattern a pattern instead of a design. Our ultimate idea was that it needs to be extendable. Not necessarily predictable, but when you see what comes next it should make sense with what came before. They built and then we talked about repeating patterns and growing patterns and then sequential patterns. To emphasize the extendable idea, we built patterns, then rotated to have someone else add on. Clearly - time for pictures.
Clear to everyone
No discussion

People accepted extension,
but felt like 3rd red block
could go "anywhere"

Generated interest because
the start was in a line, and the
pattern was extended 2-dimensionally

Patter creator admitted they
didn't know what came next, but
liked the extension. Next: 3 blues
top and bottom.
Arguments! Pattern creator wanted the trapezoids
double each step, extender focused on blues
"adding one" each time.

Is this a pattern? Designer claimed it was just a design.
Extender felt like the red-blue-green were lines
extending out each direction. All agreed: lovely!
Here's the handout, if you're interested.

The next day I wanted to build on the idea of the sequential growing patterns with explicit connections to algebra. My colleague Pam Wells has the best activity I know for this, adapted from a Mathscape activity. Here's my version. (As a Word doc, if you want to edit. Wasn't displaying correctly...)

Everytime I've used it the lesson has been engaging, provoking discussion, and very supportive of symbolic representation with the visual. Students wanted to work through all the letters on the front, though I only asked them to pick a couple. Many wanted to jump to building their own pattern immediately. Most glossed over the verbal description, so I pushed for that. In general with our pattern work, visual to verbal has been uncomfortable. This is a good activity for the connection between rate of change and the symbolic rule, as several students made that jump. Some students went from data to rule, and some from the visual.

A couple students extended this for their weekly work. I based my sample weekly work on the letter patterns, so I expected more, actually; but that's why we give students choice. Brett extended the letter idea to his whole name, which is actually a pretty nice context for adding functions. (File that one away!) Emily did a really interesting project, making some mathart that  had layers of patterns.

The lesson after this was dominoes - but that's clearly a story for another day. Later in the semester we'll do more patterns using ideas of perimeter, area and volume.

Monday, September 2, 2013

Sonia Sotomayor

Sonia Sotomayor at Berkeley Unified Schools
Photo: Berkeley USD @ Flickr
Out of respect for Justice, I'll eschew my usual pun post titles.

I just finished My Beloved World, a memoir by Sotomayor of her life up to her first appointment as a judge. It is a well-written story, and she will impress you with her positive attitude, perseverance and grace. It makes me extremely glad to have her on the court.

By why blog about it here? Because there were a few bits about education to share. Her experiences in poor (low SES) schools, transitioning to Princeton, the importance of role models and mentors, the impact of service work, and addressing bias throughout her life are all worth reading. Her success in overcoming adversity (family alcoholism, diabetes, etc.) are inspiring.

This story made me think about the importance of genuine assessment, and the necessity of important objectives.

... Teachers, I was finally realizing, were not the enemy.
Not most of them, anyway. There was this geometry teacher nicknamed Rigor Mortis. Word had it that she'd been at Cardinal Spellman since before the invention of the triangle ... I was shocked when she called me into her office and accused me of cheating. The basis for her accusation was my perfect score on the Regents geometry exam. No one in all of her centuries of experience had ever scored a hundred on the Regents.
"So who did I cheat from?" I asked indignantly. "Who else got a hundred that I could have copied from?"
She looked flummoxed for a moment. "But you've never scored higher than eighties or low nineties on the practice tests. How could you get a hundred?"
The truth, as I explained, was that I'd never once got an answer wrong on the practice tests; points had been deducted only because I hadn't followed the steps she had prescribed. I had reasoned out my own steps, which made sense to me, and she had never explained what was wrong with them. On the Regents exam we only had to give the answer; no one was checking the steps.
What happened next truly amazed me. She dug out my old tests and reviewed them. Acknowledging the validity of my proofs, she changed my grades. Even Rigor Mortis, it turned out, wasn't quite as rigid as all that.  (Chapter 11)
One of her high school teachers expounded the value of critical thinking, but she had never learned to do it. Justice Sotomayor's first taste of argument came in a good forensics experience. (Itself a good story.) Then, at Princeton, freshman year:
Professor Weiss told a familiar tale: although my paper was chock-full of information and even interesting ideas, there was no argumentative structure, no thesis that my litany of facts had been marshaled to support. "That's what analysis is - the framework of cause and effect," she said. Her point was a variation of what Ms. Katz had been getting at, though now it was coming across more clearly and consequentially. Obviously, I was still regurgitating information. It was dawning on me that in all my classes I was so concerned with absorbing the facts in the reading that I wasn't marshaling them into a larger argument.  By now, several people had pointed out where I needed to go, but none could show me the way. I began to despair of ever learning how to succeed at my assignments when quite unexpectedly it occurred to me: I already knew how. (Chapter 15)
This is a call to me, again to emphasize what is important in mathematics, and to support students in achieving that. There's also a lot here for me about the importance of transfer, giving learners the oppportunity to apply math in their other work and to bring in their successes in other areas into the math classroom.

And she makes a nice plug for numeracy when discussing her diabetes, as well as a nice demonstration of questioning in problem solving.
I test my blood sugar and give myself shots five or six times a day now. When deciding what I'm going to eat, I calculate the carbohydrate, fat and protein contents. I ask myself a litany of questions: How much insulin do I need? When is it going to kick in? When was my last shot? Will I walk farther than usual or exert myself in a way that might accelerate the absorption rate? If I weren't good at math, this would be difficult. (Chapter 28)
There's also stories about K-8 teachers dissuading students from their dreams (Chapter 10), an interesting exchange about a teacher listening to her students about what a Spanish course should be (Chapter 11), a good interaction with a psychology prof at Princeton over a failed experiment (Chapter 15; rats!), how learning programming influenced her thinking (Chapter 15), the law as a way of thinking (Chapter 20) and more.

Of course, I'm interested in your thoughts, if you'd care to share them on Twitter or in the comments.