Sunday, December 26, 2010

Self-Assessment for Teachers

Merry Christmas to all!  (I offer these greetings sincerely and in peace, not trying to bludgen anyone with it.)

As my colleague David Coffey and I were planning for the winter student teachers, we were discussing ways to involve the student teachers in self-assessment.  Dave is quite the assessment guru, and we want to have them help develop the rubric, but at the same time support them with what we might develop.

Kathy Coffey, local first grade teacher par excellence, develops these amazing rubrics with her students.  If first graders can, I figure novice teachers have a shot.  (If only Kathy was teaching them...)

Of course, how to evaluate teaching is a thorny enough process that we are having a national hissy fit over it as a country. 

The best resources we've found so far are the NCTM professional standards (going on 20 years old at this point; need a membership to read the full thing, unfortunately), and the research Rebecca Walker noticed in JRME this past semester.  I wanted to share this stuff just to share, but also in hopes of feedback.  How do you evaluate other teachers, novice teachers, or support new teachers in self-evaluation?

Professional Standards: Evaluation
The Process of Evaluation

Standard 1: The Evaluation Cycle
Standard 2: Teachers as Participants in Evaluation
Standard 3: Sources of Information

The Foci of Evaluation

Standard 4: Mathematical Concepts, Procedures, and Connections
Assessment of the teaching of mathematical concepts, procedures, and connections should provide evidence that the teacher
  • demonstrates a sound knowledge of mathematical concepts and procedures;
  • represents mathematics as a network of interconnected concepts and procedures;
  • emphasizes connections between mathematics and other disciplines and connections to daily living;
  • engages students in tasks that promote the understanding of mathematical concepts, procedures, and connections;
  • engages students in mathematical discourse that extends their understanding of mathematical concepts, procedures, and connections.
Standard 5: Mathematics as Problem Solving, Reasoning, and Communication
Assessment of teaching mathematics as a process involving problem solving, reasoning, and communication should provide evidence that the teacher-
  • models and emphasizes aspects of problem solving, including formulating and posing problems, solving problems using different strategies, verifying and interpreting results, and generalizing solutions;
  • demonstrates and emphasizes the role of mathematical reasoning;
  • models and emphasizes mathematical communication using written, oral, and visual forms;
  • engages students in tasks that involve problem solving, reasoning, and communication;
  • engages students in mathematical discourse that extends their understanding of problem solving and their capacity to reason and communicate mathematically.
Standard 6: Promoting Mathematical Disposition
Assessment of a teacher's fostering of students' mathematical dispositions should provide evidence that the teacher-
  • models a disposition to do mathematics;
  • demonstrates the value of mathematics as a way of thinking and its application in other disciplines and in society;
  • promotes students' confidence, flexibility, perseverance, curiosity, and inventiveness in doing mathematics through the use of appropriate tasks and by engaging students in mathematical discourse.
Standard 7: Assessing Students' Understanding of Mathematics
Assessing the means by which a teacher assesses students' understanding of mathematics should provide evidence that the teacher-
  • uses a variety of assessment methods to determine students' understanding of mathematics;
  • matches assessment methods with the developmental level, the mathematical maturity, and the cultural background of the student;
  • aligns assessment methods with what is taught and how it is taught;
  • analyzes individual students' understanding of, and disposition to do, mathematics so that information about their mathematical development can be provided to the students, their parents, and pertinent school personnel;
  • bases instruction on information obtained from assessing students' understanding of, and disposition to do, mathematics.
Standard 8: Learning Environments
Assessment of the teacher's ability to create a learning environment that fosters the development of each students' mathematical power should provide evidence that the teacher-
  • conveys the notion that mathematics is a subject to be explored and created both individually and in collaboration with others;
  • respects students and their ideas and encourages curiosity and spontaneity;
  • encourages students to draw and validate their own conclusions;
  • selects tasks that allow students to construct new meaning by building on and extending their prior knowledge;
  • makes appropriate use of available resources;
  • respects and responds to students' diverse interests and linguistic, cultural, and socioeconomic backgrounds in designing mathematical tasks;
  • affirms and encourages full participation and continued study of mathematics by all students. 

Japanese Principles of Instruction
“Are There Any Places That Students Use Their Heads? Principles Of High-Quality Japanese Mathematics Instruction,” Corey, Peterson, Lewis and Bukarau, JRME, v41, n5, 2010

  1. The Intellectual Engagement Principle: High-quality mathematics instruction intellectually engages students with important mathematics.
  2. The Goal Principle: An ideal lesson is guided by an explicit and specific set of goals that address student motivation, student performance, and student understanding.
  3. The Flow Principle: The flow of an ideal lesson is built from a question or a problem that students view as being problematic. As students intellectually engage in the problem, building on their previous knowledge, the students are supported in learning the lesson’s big mathematical idea.
  4. The Unit Principle: A lesson is created in the framework of past and future lessons, particularly between lessons in a unit but also between units and grade levels. The lessons in a unit help students progress to ways of thinking, writing, and representing mathematics evident in the discipline of mathematics.
  5. Adaptive Instruction Principle: High-quality instruction adapts so that all students are engaged in mathematical work that appropriately challenges their current understanding.
  6. The Preparation Principle: High-quality instruction requires a well-thought-out, detailed lesson plan that addresses the previous five principles and interconnects them in a coherent lesson.

Draft version of this for our student teachers:
  1. The Engagement Principle: math instruction should intellectually engage students with important mathematics that is relevant to their purposes.
  2. The Goal Principle: A lesson is more effective when guided by an explicit and specific set of goals that address student motivation, performance, and understanding. Examples include: students’ progress to ways of thinking, writing, and representing mathematics the way professionals do.
  3. The Problem Principle: A lesson is more effective when built from a question that students view as being problematic. As students build on their previous knowledge by means of the problem, the students are supported in learning the lesson's big mathematical idea.
  4. The Connection Principle: Math instruction is more effective when units are a framework of connected lessons; ideally also with connections amongst units and even between grade levels.
  5. The Support Principle: High-quality instruction adapts so that all students are engaged in mathematical work that appropriately challenges their current understanding.
  6. The Assessment Principle: High-quality instruction includes gathering and analysis of data that measures student progress towards objectives, and use of that data in planning and adjusting instruction. An ultimate goal is that students will be equipped to self-assess and self-correct.
  7. The Reflection Principle: High-quality instruction requires a well-thought-out, detailed lesson that addresses the previous principles and interconnects them in a coherent lesson.

As questions for starting conversations:
  1. Why is this math content important? For what is it needed? Why is it worthwhile?
  2. What do you want students to understand from this lesson? What should they be able to do? How will it help students progress to ways of thinking, writing, and representing mathematics like mathematicians?
  3. Is this a problem or an exercise? What makes it problematic? What’s the hook of the problem?
  4. How does this connect to what they’ve done before in this unit? In this year? In previous years? How will it connect going forward?
  5. Are all students ready for this task? What kind of support do they need, as a group or as individuals? How will you tell?
  6. What do students understand? How do you know? How can they get from what they know to what they need to know?
  7. Does this lesson make sense as a whole? How would you or a student summarize it or reflect back on it?

billsophoto @ Flickr
Self Evaluation
As we think about the novice teachers self-evaluating, one of the relevant frameworks seems to be the Levels of Transfer from Joyce and Showers.  It's hard to use that in a grade situation because students want to jump to Executive Use, which is not how that works.  (Here's a nice blogpost by a teacher using them for self-evaluation; also a nice introduction to them.)

So how would this work for a self-evaluation rubric?

As teachers develop, they attend to different aspects of the profession:
  • Focus on self:  What am I doing?  How did I do? 
  • Focus on content:  What am I covering? What's a good activity or explanation for this?
  • Focus on student performance:  What was the average?  What grades are they getting?
  • Focus on student learning: What do they understand? What support do they need to do this?

What I like about this progression is that it's not from good to bad.  They are all aspects of teaching which are necessary to contemplate.  I think as teachers become more experienced, they progress through the levels more quickly and holistically.  I think novice teachers could recognize where they are at with respect to different aspects of teaching.  "In my planning I'm thinking about student learning, but in instruction I'm still concentrating on myself."

So how do you self-evaluate?  If you're a teacher educator, how do you assess your novices' teaching?  Do you have ways of supporting your teachers in self-evaluation?

Monday, December 13, 2010

Christmas Lights

aMacHan @ Flickr
Our family put up a Christmas tree this weekend, and it reminded me of one of my favorite problems ever! Students did great work on it, and developed a number of techniques.

On the weekend after Thanksgiving Karen put up a 2 meter tall Christmas tree, 170 cm wide at the base. She put an astounding 1500 lights on the tree. Write a story problem using this information.

One thing you should know about Karen's tree decorating is that she believes that lights should be distributed uniformly throughout the tree, not just on the surface. It's like the tree is full of lights. She started at the base, and put on 750 lights. Then she asked me how many will she need to finish? Now that we know she needed 1500, how far up from the ground did she get with those first 750 lights?

I saw that Abstruse Goose, an funny, edgy cartoon that's hip to math & physics also had some nice Christmas tree problems...

Sunday, December 12, 2010

Change the Channel

Yeah! The K-8 geometry video is ready for release! The students did a great job getting footage, and had lots of creative ideas that used their talents. I hope you enjoy it!  (The students would love it if you share the link.)

The video started back at the midsemester when students at our school were making a lipdub.  During my class.  "Can we go?" (GVSU lipdub - came out pretty well.)

"Not now - you had to be involved before."

"What's a lipdub?"

So I showed them what I consider the classic of the genre, Shorewood High School's reverse lipdub.  We were just starting our unit on teaching (following doing, preceding learning).  So we talked about how many teachers find it hard to get students to do any work, and yet here's a whole school working their butts off to make a video.  They had a great discussion about it, bringing up choice, student interest, engagement and other factors.  Then... "can we make one?"

Any reasonably bright teacher would have seen that coming, but not me.  "I'll think about it."

Starting our last unit, I brought it up.  If we're going to do it, it's time.  Discussion led them to believe that the reverse lipdub was right out, followed shortly by a lipdub.  Some students were really into the idea, most were in favor, and a few were dead against it.  There's a fair number of choice workshops in my courses, and I said most of the prep work would be choice.  They wouldn't be graded on the success or not.  The class voted on it with most in favor.  I started freaking out.  It was worth doing to me because:
  • Student interest was high, 
  • The idea of how to capture and communicate math is relevant to a math ed class,
  • By the end of the semester it could be connected to review, and
  • The preservice teachers wanted to be able to show it to their students to answer why they should be engaged.
But as the last couple weeks went by, only a few students were contributing.  (We had a google doc for students to add their ideas and development to.) My freak out got freakier.  Was it even going to be worth trying?  Part of this was also I wanted them to see a teacher giving something a go, taking a risk.  That's better if I'm uncomfortable, right?

We used our last class period to do the filming.  One of the cameras failed completely.  I just wheeled in all the manipulatives with which they had had the most fun.  Objective: get some good footage.

One student had written the Math, Math, Baby rap, and found someone to rap, so they started choreographing it.  Other students got building and drawing.  One student had come up with an I Love Charts style demonstration (on Jeggings, which you can buy but she proves do not exist), and another was ready to demonstrate our amazing rubber band enlarger.  The atmosphere got charged and they really got into the spirit.  In hindsight, we should have gotten this footage earlier, and then students could film and add to it.  Ultimately I had to do the first pass editing, but it was inspiring to do it because they got such good footage.  The Geometry song came from me noticing that it would fit to Adam Sandler's Hanukkah song, the guitarist learning it in 5 minutes from youtube, and the singer coming up with the lyrics (with some crowdsourcing) on the spot.

Even if there was no video result, I would have been happy to see the students so engaged in making math visible and engaging.  But it's better with the video!

Resources: Jamendo was a great place to find cc 3.0 music, and I think the songs from Antony Raijekov (jazz) and Josh Woodward (pop/folk) really help make it. They're not math songs, but nobody's perfect.

Friday, December 3, 2010

Triangle Puzzle

Have you ever had a nice problem that you just thought about at odd moments?  Boring meeting, stuck waiting somewhere, few surprise extra minutes in a day?

For a while now, my favorite problem like that has been finding a nice way to divide up a square into the seven triangle types.  I love tangrams, and I like Pierre Van Hiele's mosaic puzzle even better.  If you do too, stop reading right now and try this problem.  It's fun and worth a surprising amount of thought.  (For me, anyway.)  Then suddenly this week, one of my little thumbnail sketches worked out.  I don't know whether to be happy or sad.  Being a geogebra nerd, I wanted to make a sketch of it, and that led to making a puzzle out of it.

You can print this picture of the pieces to try in real life, or try it with the Geogebra file or as a webpage.    (A solution is an option on the file or webpage.)

But... now I'm left wondering what to think about in those rare extra moments.  Then on Twitter, Justin Lanier (@j_lanier) tweets:
Had an insight in the shower this morning. Example: .717171... = .717171.../1 = .717171.../.999999... = 71/99 (!)
 Hmmm.  Really?  Maybe it's a coincidence, because 100 times .717171... minus the original leaves you 99... hmm.  Would it work for .717171.../.6666... ?  It does.  Tweet back:
@ cool. So is .a_1 a_2...a_n repeating / .xxx... =a_1...a_n/xx...x (n times) for any x? Or divided by .b_1 b_2... b_m repeating ...
Which connects to another problem (from Dave Coffey) I like thinking about: how many digits does it take 1/17 to repeat and how can you tell?  In general?

OK.  Deep breath.  There's always more problems.