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

Original
  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?

EDIT:
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.

Tuesday, November 30, 2010

(How to) Get a Job

Dr. Thomas Reeder, Associate Superintendent of Wyoming Public Schools, was kind enough to come talk to our secondary mathematics student teachers about life after student-hood.  He has had an amazing career already, starting as a K-12 certified teacher and becoming a literally award-winning administrator.  He gave me permission to take notes to share.

You might here someone say, "The best thing about teaching is June, July and August", but that is nowhere near true.  The best thing is the students.  Decent pay and benefits, but it's the students that are why you teach.

This is a good time to be a math teacher because of retiring teachers (state buyouts) and Michigan's new 4 year math requirement.  There are some jobs.  Science minors are excellent, too.  As you're getting ready to apply, you want to think about what sets you apart.

There are different certifications: state certification and highly qualified.  Major or minor, plus certification test equals highly qualified.  Sciences and social sciences are content specific.  In your materials, be clear about what you are qualified to teach.  Once you're hired, the district will put you where you are needed.  Know the Michigan Merit Curriculum, which has changed even since you were in high school.  Four years of math, English and science, a world language, health, etc. 

Figure out which students you want to teach.  Figure out where you want to teach.  The state, the type of area, urban or near-urban... but evaluate where you want to be.  Rural teachers whose schools have not been able to hire new teachers for consecutive years.

Michigan teachers are paid well (top 3) with much lower cost of living than other high-paying states.  You will never be affluent, except for what you get from your students.

Resume
L Hollis @ Flickr
Succinct, no errors.

What to do to get ready: be involved in educationally relevant experiences.  What do you do outside of your teaching?  Conferences, tutoring, coaching, special events like a math night or tutoring night, etc.  The more you have those experiences the better.  I'm looking at between 100 and 1000 resumes.  If you're a rookie, what else have you done that makes you not a rookie. 

The two crucial areas of relationships are with parents of students, and with your peers.  100% of parents believe that there kids are 100% innocent... what will you do?  Get experiences with conferences.  We have a principal who can remain calm in the face of full out screaming and cursing.  You don't need to take it: "I'm not going to talk with you now if you can't speak civilly..." give a warning.  But then you can hang up.

Other work experiences can be valuable if they build up the image of who you are.  Never lie on your resume.  Never - that's a firing offense.  If you were fired from a job, you can still put it on.  When you're asked for employment history, you need to be complete.

I've only put my educationally relevant jobs - should I put the other ones?  Sure.  Maybe going back to senior year of high school.  Does your experience show the ability to balance?  Loyalty?

Should I list volunteering in high school, or is that too far back?  I would put it in.

Should I list a full education history?  Yes.  High school plus all your colleges.

You can list or not list you GPA on your resume, but I always look at transcripts.  I don't want all A's, but I want to know about retakes, how long, what kind of student you were.  Some of the teachers who have struggled for us, also struggled in college.

Personalities are more important now than before; maybe you've heard about speed interviews - we might use those as kind of a screener before more in depth interviews.  Now while you're a student, belong to the professional organizations.  It's dirt cheap and they are valuable.  Then make sure they're listed on the resume.

Get your resume down to two pages.  Include relevant personal interests.  One person got a job because they listed skiing and they needed someone for ski club.  Schools will want you to have multiple roles, coaching, clubs, drama, etc.  Plus, as a teacher, it is going to help you see your students in a different way.  And help them see you in a different way.

The number of mistakes on resumes is unbelievable.  You need to proof them and clean them up totally.  If you're not good at it, get help.  Even a professional.  The resume isn't tailored to a specific job... that's the...

Cover Letter
With internet applications we went from dozens to hundreds of applications.  The cover letter is how you tailor your application.  Be specific to the school or district.  Don't be generic.  Share what you know about the school and district.  A question we ask is:  tell us about yourself, (2) tell us what you know about Wyoming Park.  Personalize your cover letter.  If you were dating someone, you would write a personal letter.  Will that mean you write a lot of letters?  Yes.  Go there or at least do your homework.  You need to know if you want to be there anyway.  What will set you apart?

Three paragraph essay: a little about yourself, what you know about the district and your fit; an overview of yourself and what qualifies you; a wrap-up.

Is the cover letter and resume it?   Each district have specific information they request.  Follow their instructions exactly.  The old adage was go in person.  Don't do that anymore.  It's annoying and might even count against you.  In our district, resumes go to our HR person.  They put the piles together and vet the resumes.  Then they go to the principals.  The only thing to check with the central office is to make sure they received all your materials.

How many references?  Sometimes it's specific, some will be general.  If they don't specify, 3 or 4 is appropriate.  Make sure your references know what you're doing.

Start putting your materials together now. You can still tweak.  Two periods for hiring are a little bit in January, and then June-August or even Septemeber.  We hired 6 people at the start of school.

Preparation
Use those weeks after you graduate to sub.  Get your name out there.  Subbing is the best and worst job.  Maybe 80% of our hires come out of our sub pool.  Why?  There's safety in hiring people that you've tested out in some way.  Maybe you have another job, but is that going to lead to the teaching job you want.  You don't have to sub every day.  It will give you experiences to help you decide what you want and don't want.

What are red flags?  It's not supposed to be about your personal characteristics.  No picture, for example.  No gimmick.  Got one with a plate of chocolate chip cookies.  Some might like it, I don't.  I don't want any preconceived ideas about you as a candidate.

Some teachers don't want to teach the high level classes.  I got my first job because none of the current teachers wanted to teach AP calc.  I was an A student, but that calc class was a shock.  The kids caught on quickly, and then after a year it finally, totally clicked.  "I thought this was hard, but it is so easy."

If you want to go out of state:  check out the particular standards.  Michigan standards are pretty high, so shouldn't be too much of an issue.  Be careful about pay and benefits.  If you're young, go anywhere you want, but look into it.  There are good websites for out of state jobs.  MASB is a good source for Michigan jobs.

Resume, cover letter, portfolio (paper or DVD), put a lesson together that you could use.  You might hear on a short timeline, so you want to have things done ahead of time.

Interview
If you get called, be very courteous.  They may give you some options as to time.  What you want to do is change anything you have to do to get to that interview.  If there is a reason, work around it, share the reason and make sure it's important.  I had a conflict with a game I was coaching and they were willing to interview me on Saturday.

Bring whatever they ask.  It is okay to ask a question, "do you want me to bring..."  Don't ask anything else.  You need to show initiative to get the information you need.  Now it's time to do your homework.  You have got through the biggest hoop, you have your foot in the door.  Find out anything you can.  School Improvement Plan, classroom management system, special district programs, anything they put online.  Go visit the town and the school.  My first job I went to the town for two days, talked to people in the coffee shop, found out a ton.  Plus I found out that it was a job I wanted.  But it would have been just as good to find out if I didn't want it.

When they ask you if you have questions, have some.  Be specific with what you've found out about the school.  What about your poor grades in math, your SIP, your ...

Don't ask about salary and benefits.  With the one exception that if you're choosing between offers and need to know that information.

Dress appropriately.  Comfortable.  Suit okay? Yes.  But dress up at least equivalently of a dressed up teacher.

I will never tell you not to be yourself, but nose rings, tattoos, piercings ... be aware of the effect.  There are places where a man with an earring can not get hired.  Impressions are natural and always happening.  The interviewers will be forming impressions immediately.

Should I introduce myself to each person, shaking hands?  Absolutely.  Plus it will help you relax.  Good firm, practiced handshake.  I've interviewed people so nervous they broke out in hives.  Know yourself.  If you speak quickly when nervous, pay attention and slow down.  Don't be afraid to do what you need.  If you need to doodle, take notes, then bring a pad of paper.

Sometimes we'll ask a three part question to see if the candidate can track all three parts.  Good time to have paper.  If you bring in something extra, don't be afraid to leave it anyway.  (If it's something to be left.)  Or ask if they want to look at it before you go.

Every district interviews differently.  Short, long, test class to adults or a summer program.  The hard part of the interview is that you're looking for the best candidate, but it's insufficient to really know what kind of teacher you're going to be.

Typically, there are a dozen questions.
  • Starting with tell us about yourself.  Don't stick to the resume, complete it.  Extra training, share your personality.  
  • Tell us about the district.
  • Scenario questions:  how would you handle...
  • Philosophy: what do you believe about...
  • Management: what do you do to make a learning environment?  Not about having your thumb on the students.
  • Past experiences
  • Expertise areas
  • Maybe: content questions. 
  • Try googling teacher interview questions.

Mistakes:
  • be succinct in your answers but completely answer the question.  Okay to pause and think.  If you're going on and on, catch yourself.  Not so short that you leave blanks.  
  • Be punctual.  (If something happens, call on the cell and inform.)  Be in the area early.  Don't have to go in the building, but be close.  
  • Don't be so nervous that you can't communicate who you are.  
  • Be aware of yourself.  Are you a loud talker?  Too quiet? People have probably told you already things you can work on.
  • Writing samples might happen on 2nd round.  Sometimes tech skills, although that's not an issue for novice teachers.  I had an interview where they left me with 2 questions and 2 sheets of paper and an hour to answer them.
  • Pay attention to their time if they've given you their limits.  Too fast, too slow - monitor.  Be done with questions a few minutes early to leave room for dialogue.

The most impressive things are: being confident and comfortable.  (Not too confident.  Eg the interviewee who asked when they would be signing the offer since they were sure to get it.)  Lean in towards the interviewers, make eye contact, look at everyone.  At the end, ask a good solid question.  OK to have written it ahead of time.  (Maybe have 2 in case they ask you one.)  Ask about resources that are available, ask about tech.  This would impress me: " I notice you use Prentice Hall.  Do you use..."  "I know you are on trimesters..."  "For your after-school tutoring, do teachers..."


You have worked hard to get here - don't skimp on the work for this last thing to get the job you want.
shareski @ Flickr

Profession
Can I take chances the first year?  Not sure what you mean.  But there are two grounds for immediate dismissal: (1) don't touch a kid (2) you're young; know about what is appropriate and inappropriate with a student.  I was 21 at my first job, with students who were 18 and 19.  Now we have parents that have hit on our teachers.  The third thing would be finances; be careful if you're responsible for any school funds. 

Michigan has curricula for academic content.  But some schools have policies like abstinence-only education or creationism... you should know that.  If you don't know, check with someone else.  You can always tell a student, "I will check."

You will make mistakes, how are you going to grow from them.  You might be miffed at how you see colleagues teaching or treating kids.  Classroom management is hard to develop as a student teacher.  When you start from the beginning, you'll need a way to have control. 

The four years towards tenure are for the school to evaluate your performance and your growth.  By law, every teacher needs to be evaluated every year.  You want feedback, though.  I'd rather know my struggles in November than in June.  Have people come in and watch you teach.  We have a lot of team teaching now with ELL and special education students.  You want to be the best you can be.

Looking back at my career, I would have gotten a Spanish degree in addition.  When I started, there was small need, but now, working in an urban district, it's a great need.  If you're interested in administration ever, you have to be involved in all areas of the school.  Get on a School Improvement Team (now required in each school by state law), be active in all areas of school life.  Teacher, union, coaching... when I went into administration there were 10 positions they had to fill!  I took that as a good sign that I was involved.

Monday, November 29, 2010

Math and Multimedia 5

Welcome to the Mathematics and Multimedia Blog Carnival, 5th edition.  (Hope you enjoyed the Fantastic Fourth Edition.)  Five seems like a fortunate number, since we have five senses.  Our five fingered hands are a good start to mathematics.  We all love a high five, are happy to take five or are glad when it's five o'clock (somewhere).  But my favorite five fact features 5 for the fifth Fibonacci number.  Far out!


This blog carnival seeks to promote seven principles:
1. Connection between and among different mathematical concepts


Sol Lederman at Wild About Math shares a video of an Incredible Magic Square.

Antonio Gutierrez at Go Geometry has a fascinating golden rectangle puzzle that connects with the Droste effect.




2. Connections between math and real life; use of real-life contexts to explain mathematical concepts


John D Cook at the Endeavor shares that there isn't a googol of anything.

 Grrrl Scientist suggested this article from her guardian.co.uk blog about "How the leopard got its spots" that has some literally beautiful mathematics.

Consider this beautiful film by Cristóbal Vila - Nature by Numbers.  Or this collection of Hands On Math Movies.

David Cox has just been sharing a ridiculous amount of great stuff lately.  For example, projectile motion.

It's been widely shared, but you have to check out Kate Nowak's money take on special right triangles at f(t).


 3. Clear and intuitive explanation of topics not discussed in textbooks, hard to understand, or difficult to teach

James Tanton has two videos explaining the principles for math genius thinking.  Hat tip: Denise at Let's Play Math.  You might also like Sue Van Hattum's interview of Dr. Tanton for the Math 2.0 interest group.

For that matter, Sue's post at Math Mama Writes about E is for Eigenvectors and Eigenvalues belongs in this category.  Has the great first sentence: "This post is about fear."

I spent some time recently looking at trigonometric function visualizations and making a couple Geogebra sketches for them and their inverses.  Seems silly to link, since it's right down there.



4. Proofs of mathematical theorems in which the difficulty of the explanation is accessible to high school students

No one nominated entries in this category, but it makes me think of work like James Tanton's explanation of Euler's proof that every even perfect number is triangular, or Alexander Bogomolny's proofs at Cut-the-Knot of the addition and subtraction formulas for sine and cosine.  (Both of these sites I've had occasion to look up recently!)



 5. Intuitive explanation of higher math topics, in which the difficulty is accessible to high school students


Derek Bruff has put together a fascinating interactive Cryptography Timeline.  I'd love to see some of these for some important math concepts.






6. Software introduction, review or tutorials

Guillermo Bautista, the founder of this here carnival, at the Math and Multimedia blog, has a roundup of essential tools for every math blogger.  Also be sure to check out his terrific Geogebra tutorials while at his site.

Maria Anderson has video of her presentation from MAA-Michigan up, Math Technology to Engage, Delight and Excite.  Also watch as her new blog, Edge of Learning, gets up and running.

Chris Betcher has some terrific Scratch (the programming language) resources and videos.

You might try one of these 15 mind-mapping tools.  (Many are free.)






 7. Integration of technology (Web 2.0, Teaching 2.0, Classroom 2.0), in teaching mathematics 

David Wees has an interesting meditation on the importance of interactivity in math teaching.

Cybraryman has a long list of math/tech integration resources and lessons.







If you're looking for Five tunes, Take 5, or try Gimme 5 from Sesame Street, High Five from They Might Be Giants, Dino 5 (who also have a great counting rap called What About 10)?, or maybe the best (in terms of math) ...







If you have ever posted a blog carnival, you know that you receive a lot of obvious spam.  But some can seem relevant, so I like to have a Best of the Spam category.  For example, the Top 40 sources for open courseware video.

And if you are mistakenly put here, or your post did not appear, please let me know and I will correct it posthaste.

Images were obtained from Creative Commons search.  Attributions are in the picture title - click on the image and you will see the source from Flickr.

If you enjoyed the carnival, please consider nominating a blogpost, your own or someone else's, for the sixth edition, to be held at Great Maths Teaching Ideas by William Emery.  It's a very active blog with many K-12 activities, so don't wait a month to check it out!

Congratulations if you recognized the five connection for each of the images above.  I tried to slip in some tricky ones.

If the carnival is done, must be time to head over to the Five Bells.  Cheers!

Monday, November 22, 2010

Trig Visualizing

Rebecca Walker and I modeled a lesson for our secondary student teachers on trigonometric equations, based on the first chapter of the Precalculus book from the very interesting CME Project curriculum.  While it has some interesting applications, this curriculum really does a good job of letting the mathematics be the context and addressing mathematical habits of mind.  The lead developer is Al Cuoco, who has a great history of interesting math and math ed work.

The lesson is a bit of a stretch, because we're just touching on one section, using a bit of information from three or four.  We did unit planning one week, lesson planning the next week, and finally the lesson.  The TAs read The Teaching Gap, so then we connected it to the idea of lesson study, and a discussion both about how to revise this lesson, and why lesson study might work as professional development.

We have two Geogebra sketches to help with visualization.

 As a sketch or a webpage.  This sketch supports visualizing sine and cosine with unit circle connections.
As a sketch or a webpage.  This sketch lets you invert trig functions using the Unit Circle representation.













This is my first attempt at a WCYDWT.  When I was making these sketches (don't worry, I disinfected them before posting) I had a bad cold, so was constantly reheating my tea.  Watching it go round and round.  Thinking, "so when do we know a position and want to know the angle, with possible multiplicities...hey, wait a second."  If I was using this, I think I would start with the video, and use that to motivate the idea of solving for information based on the circle position, as well as how periodicity relates to multiple solutions.

This has to be the world's most boring video.  Enjoy!

video

Here's a slightly more polished version of the handout we used with the sketches.  There was some discusssion with the student teachers as to whether the inverse trig or the algebraic solutions part should come first.  I think they could be switched, depending on what you wanted to emphasize with the students and how strong their trig background is.  Also, the handout is written as if the teacher is demonstrating with the computer, which is what we wanted to model for them, (no lab is no reason to no have technology) but the ideal would be to have the students have access to the sketches.


Solving Trig Equations

Sunday, November 21, 2010

Images of Teaching

There was a nice (if long-windedly titled) article in the Decemeber '03 Teaching Children Mathematics called "Metaphors as a Vehicle for Exploring Preservice Teachers' Perceptions of Mathematics," by Brenda Wolodoko, Katherine Wilson and Richard Johnson.  In the article the preservice teachers made images to display themselves as teachers or learners of mathematics.  The majority of their images revealed anxieties about the content as learners, but hope for themselves as teachers.  The researchers liked the way that the images created an opportunity for dialogue and created a potential for change.  One interesting sidenote is that students used the idea of puzzles both positively and negatively, modeling both frustration and engagement.

My preservice elementary teachers recently made images of what does it mean to teach mathematics in small groups, and our secondary student teachers envisioned their future classrooms.  There were many neat ideas to share.

 The mullet came later, but I think they thought the "Business in the front, party in the back," slogan did relate to teaching.














 This and the next chart were imagined classrooms, which you'll see a lot of with the secondary teachers as well.  They value group work, technology, manipulatives and whole class time.  I often wish college classrooms had room for a carpet section for students to sit down.


 These next two charts are more like concept maps.  This group focused on the most important aspects to them.

While this group saw their concept map evolving into a hierarchy.  There's some pretty interesting connections here to look into.














Somehow missed my favorite poster here.  Clever use of Facebook, and really made me think about that page as a representation of who someone is.  I think there are lots of idealized people I'd be interested in seeing Facebook pages for.

The secondary teacher assistants made quick individual sketches at the end of a seminar.  So don't expect the artistic commitment we got from the elementary teachers.  One thing that came across in classroom images is the presence of the kind of technology to which they've been exposed.  It's becoming clear to me that we need to do a better job of teaching technological pedagogical content knowledge, primarily by explicit modeling.

Let's start off with a few of the text descriptions.  This teacher is worried about the content they will be forced to cover.







This teacher is thinking about classroom management as the start of learning.
I'm not sure if this teacher was describing life as it is or as they envision.  Somebody definitely considering the different models with which they've been presented.













Next come several visions of cooperative learning.  There seems to be a clear value on student discussion, and varying images of what the teacher's role is in relation.













And this sketch merges a vision of the classroom with a concept map of what is important to them.
































I'd be very interested in knowing what you think about the images here and what you notice, if you'd care to leave a comment or drop an email.  Thanks!

Wednesday, November 17, 2010

To Understand, Book Club 3

Ellin Oliver Keene
Today my preservice K-8 teachers are discussing Chapters 5 and 6 in To Understand.  I gave them four focus questions to try and help guide discussion:
  1. What is the author's main point in the chapter?
  2. How does she support that point?
  3. How does this idea apply in mathematics?
  4. What's your reaction or connection with this.
Chapter 5 - To Savor the Struggle
Mentor: Reynolds Price, author and poet who has written and spoken eloquently about his response to an inoperable spinal tumor, as well as just been more artistically productive than he was before.  (Sample of his poetry available at Google Books: The Collected Poems.)  What enables some people to flourish in adversity?

  • Greatest gift to a student is the experience of struggle leading to understanding.  We don't do it in younger grades; students may not experience struggle till college.  
  • Intentional struggle for students requires time and modeling.
  • Teaching kids how to learn to think more critically requires struggle, versusthe students just doing without reflection.  In math, struggle leads you to know why you're solving, and where the answer comes from.
  • Part of conceptual understanding is problem solving.  If it's too easy there's no problem solving.  Struggle requires problem solving.
  • How do you motivate students to struggle?  They'll sit there until someone else does it.  One student, or the teacher, or.. (several stories to back this up).
  • Once you get them willing to struggle, then they can gain understanding.  They will know why they got that answer.
  • A lot will blow off the work b/c of the struggle.  They didn't have a need to struggle until college.
  • When you conquer through struggle you're more likely to remember how, remember your process, and be able to use it again.
  • Does more struggling now mean less struggling later?  It means you will be less frustrated with struggle later.  Got through it, rather than going through it.  
  • Frustration is a problem.  You have to continue to struggle to add new understanding.  My elementary struggles make me able to struggle for a long-time now.  
  • With struggling comes a work ethic.  It was a shock in college to have to learn to do the work.
  • Struggle motivates me to want to figure it out.  
  • How much is too much struggle?  Different for every student.  Know your students and differentiating.
  • Mixed emotions.  Why don't we want to find a way to prevent struggle, for them to get it without struggle.  What happens in real life?  Remember waiting for others to solve and just get this lab done; later on, on my own, I had to do it for myself.  Want our students to get content, but more so to be able to solve their real problems in life.  They will have struggles, I want first struggles to be in the safety of my classroom.  Help them now or help them later?
  • Isn't a struggle enough now? Some of the kids struggling will ask, but some just shut down.  Do nothing.  How motivated is each student? 
  • Why don't we have leveled math problems like leveled readers?
  • How does it happen - when a student moves from resisting to engaging struggle? 
  • If you create struggle, what about that 10% that cannot even begin.  Everyone's different. 

She discusses these systems in literacy instruction, and I think it would be worth investigating in math.
  • Semantic System - word meaning and use
  • Schematic System - process focus on what it means to read
  • Pragmatic System - what do you do with the content

Leonardo's horse at Meijer Gardens, Grand Rapids

Chapter 6 - A Renaissance of Understanding
Mentor: thinkers and artists of the renaissance.  Considering the idea of intellectual liberation.  What conditions are necessary for a renaissance?
  • Allow the students to go beyond your expectations.  Challenge my students.  A teacher who designed tests so no one could get an A - that's not challenge.
  • Be interesting to see reading levels transferred to math.  A math corner... and then science.  Time and effort from teachers too much?  Maybe they don't have the background to do it.
  • Choice is a part of the answer to this.
  • The title renaissance: something new.  The testing is a problem because it's based on a norm.  Students should be awarded for original thinking.  Writing you can imagine this.  What is it like in math - original thinking?
  • Grading is fear inducing, restricts creativity.  Especially when it rewards conformity.
  • Are you going to take the classes where you struggle, or where you can get an A.  It looks good on your GPA.  Could grade technical difficulty and performance.

In this chapter she considers the importance of text and genre in literacy instruction.  What are these things in mathematics?  Is it our content areas?  Problem types?  Different processes?  Different uses like investigation, homework, tests?

The Chapter 5 discussion was energetic and purposeful.  Definitely one of the reasons to use this book.

Photo by cliff1066 @ Flickr

Saturday, November 13, 2010

Scratch and 8th Grade Geometry

http://welovetypography.com
From twitter I found out about the K12 Online Conference. In particular about a Scratch session by Chris Betcher (@betchaboy on Twitter).  The session has a nice 22 min video about using the Scratch programming language with Year 5 students.  Very worthwhile info in a sitcom sized bite.  Chris also has a Scratch wiki devoted to getting students going with Scratch, and all the resources you need to get going.  Scratch is perpetually one of those things I'm going to investigate when I have obtained some sparemomentium.

I got a chance to work with a local 8th grade teacher who's looking into Geogebra.  He sent me some of the state standards he was interested in exploring, and I made up a couple sketches to play around.

The first is just a demonstration of the area formula for a parallelogram.  It seems so unreasonable that all parallelograms with the same base and height have the same area.  That connects with one of Geogebra's strengths to me - providing essentially infinite examples so that students can notice.


The next is probably pointless.  I was considering how to make a dynamic area measuring sketch.  I thought the advantage would be being able to change the figure, and easily check your answer regardless of how you've changed the shape.  I used sliders to build the shapes so that the distances would be nice.
Webpage and geogebra file

The third sketch is for similarity - I'll post that later this week with the world's easiest activity.