Tuesday, March 30, 2010

Coordinate Connect

This is a modification of a game that I can not remember where I saw. EDIT:  Yeah, remembered!  It was at NRICH, the game of the month.  They have a java implementation.  Give it a go.

Modifications by Jill, Leah, Lauren, Ashley, Megan, Nick, Jenn, Alyssa (Preservice geometry K-8 class) and myself.

GameCoordinate Connection

Materials:  10x10 grid with no axes, pen (or colored writing utensils)

Gameplay:  The idea of the game is players take turns naming a coordinate pair and then filling in the point.  The first player to make four in a row, wins.

The first point of the first player is the origin, (0,0).  It still counts as their point and can be one of the four in a row.  Later points are given by naming a coordinate pair.  For example, (2,-3), is two to the right of the origin and 3 below it.

Players take turns naming a point and then marking it.  If it's not where you thought, too bad, so sad, that's where it goes.  (Unless another point is already there - lucky.)  If players are having a tough time just verbalizing it, they can be required to write the coordinates down before drawing the point.

The winner is the first player to get four in a row, vertically, horizontally or diagonally.  The loser decides whether they want to go first or second in the next game.

  • consider blocking your opponent from getting three in a row. 
  • get and stay on the offensive
  • use a special mark
  • Play with three or four people.  (This was surprisingly fun and complex.)
  • Allow any 4 consecutive collinear points to win.  For example, (-3,-3), (-2,-1),(-1,1), and (0,3)
  • Play with younger students by giving directions from home, such as 2 left and 3 up.
  • In the original game, you marked the point then said the coordinates, losing it if you were incorrect.
Give it a go, and let us know what you think!

Paper for playing - 2 sided, 4 10x10 grids each side.

Sample game:

Friday, March 26, 2010

Throw Out Your Lesson Plans

Preamble:  the Common Core Standards for K12 Mathematics are up and available for comment.  See http://www.corestandards.org/.  My two two word reviews: too much and too little.  They just couldn't focus.  And, there's very little attention to the processes.  In related news, as Congress considers the revamping of the Elementary and Secondary Education Act, the Forum for Education and Democracy put together a pretty nice little manifesto, er, recommendations.  I glossed over it once, but Dave Coffey repointed it out to me.  Worth a look.

We were talking about lesson planning this week, and I enjoyed the think aloud enough that I thought I'd post it.

It's easy and common to confuse lesson planning with a lesson plan.  When you ask preservice teachers about planning they invariably talk about lesson plans, and most usually, particular lesson plan formats.  Then they get more classroom experience and 95/100 supervising teachers tell them that they don't use lesson plans any more and the novice teachers decide they don't them either. 

Of course, they're right.  They probably don't need lesson plans the way we often teach them.  I used to require awful things.  Huge four column Japanese style lesson plans with loads of information.  Then I started just using those to capture a lesson.  Finally, I gave up making any kind of stink about the format.  I still share those, as a way to capture a lesson.  But I make no pretense that you would use them to prepare a lesson.  As a department (okay, Dave, Rebecca and I) we are trying to get away from planning without students in mind.  Real students.  Talk about sending a bad message!

But the lesson plans are the bathwater.  Helpful.  Bubbly?  The baby is the planning.  That is absolutely essential.  And every intentional teacher I know spends time, thought and energy planning.  We may not have enough time for it, and would probably like to do more of it, but it's a crucial part of the teaching.  That's why it gets a spot of its own on the Teaching Learning Cycle.  (Adapted from the Learning Network model.)

So to try and capture the difference, this semester, I talked about both as the questions I ask myself while thinking about them.  This feels more authentic to me, because I'm constantly adding to and changing the planning questions, which feels like when I'm planning. 

Plan vs Planning:  Essential Questions

Lesson Plan
  • What will help you organize your thinking?
  • What physical record would be a good reference while teaching?  What details, sequencing or answers would be handy to have available.
  • What record will help you keep track of what was done and what you learned from it?

Lesson Planning
What do you want students to learn?
  • What do they already know about it?
  • Why do they need to know this?  Or, what’s important about this?
  • Consider big, long term goals and specific lesson objectives.
  • Consider process goals as well as content goals.
  • What do they already know about this?
  • How will you be able to tell when they’ve learned it?

What experiences will move students forward towards the objectives?
  • What lesson structure will be good for this?
  • What mode (individual, cooperative) is good for this?
  • Have you tried it?
  • What’s engaging about this?

What support would help students?
  • How will you equip diverse students?
  • What are possible student responses or questions? Your responses to that?
  • What representations will you be using?
  • Would a demonstration help?
  • Are there math or life connections to this?

What data will you collect?
  • What does understanding look like?
  • When and how will you observe the students?
  • What record will you make of the data?
  • How will students consolidate or reflect on their work?

What other questions do you ask yourself while planning?  What's most important to you?  I'd love to hear about it.

Wednesday, March 24, 2010

Tessellations and Geogebra

In my Geometry K-8 class we've been study transformations.  Which always leads one place for me ... my love, my joy... tessellations.  Arty, playful, deep underlying structure, corner cases that require thought even still; they're perfect.  To me.  I understand how others have dabbled and grown tired, but for me they are ever fresh.

There's a probably a few too many sketches here, but let's have a look.

First - Look at a tessellation, identify the motions, and consider what properties allow it to tile that way:
Quadrilaterals:  webpage and geogebra file

Hexagons1:  webpage and geogebra file

Hexagons 2:  webpage and geogebra file

Second - Look at a tessellation, identify the motions, and then alter the tile Escher-style!

Isosceles Triangles:  webpage and geogebra file

Quadrilaterals:  webpage and geogebra file

Third - Control the properties of the tile so that it will tessellate with the given motions:

Pentagons:  webpage and geogebra file

 (A midpoint rotation and 2 side to side rotations.)

Hexagons:  webpage and geogebra file

 (Quite challenging!  3 side to side rotations.)

Bonus - Kaleidoscopes!  What's the connection between reflectional and rotational symmetry?

Control the number of Sectors:  webpage and geogebra file

Control the Angle:  webpage and geogebra file

The kaleidoscopes were to investigate an open conjecture we have.  My one disappointment with Geogebra in comparison with sketchpad is that animation isn't as easy.  It's nice to have an animate button on your kaleidoscopes. 
The Leah-Jill Conjecture:  If a shape or design has n lines of symmetry, then it will have n-fold rotational symmetry, for n > 1.  Having rotational symmetry does not imply reflectional symmetry for any n.

I don't think the sketches helped.  I can't decide if we should tackle it another way or if we should just move on.

If you have any ideas for a dynamic tessellation sketch, please let me know.  It doesn't take much of an excuse to dive right in.

Friday, March 19, 2010

Conquer and Divide

I've been working a lot on division lately.  Long division with my son in 4th grade, with his classmates in small groups and with my preservice teachers.

Xavier's teacher sent home a note before the unit, showing the four ways they would be approaching it.  From the traditional algorithm to the "forgiving algorithm" and a couple in between steps.  My wife couldn't make too much of the note, despite being bright, living with more math than anyone would think is reasonable, and being comfortable with her own computation.  Don't know what it was like in other homes.

Xavier was making pretty good sense of it.  The curriculum sensibly starts out with dividing by 5 only, which was a nice touch.  I think I might even start with 10s, then 5s, having seen how well this worked.  He transferred this pretty well to working with other single digit numbers.  For example, 61/4.  When working, he used appropriate language and responded to questions like "how many fours go in 61?"

I did the soda machine problem (from Contexts for Learning, my favorite curriculum, Exploring Soda Machines: a context for division) with a small group the week before, and they did amazing work.  In the problem, you describe a pop machine (in Michigan, soda=pop) which has 6 flavors.  When full, it has 156 cans of pop.  How many cans of each flavor, if there's the same of each?  But wait, that's a lot of pop.  When I buy pop it's usually in a six pack.  How many six packs will that be?

Students do the most amazing work wth this problem.  Most choose to tackle the 6 flavors question first, and drew their own pop machines.  Six columns, and fill in pop cans 1 per column until they have 156.  156 is an inspired amount - not so much as to be overwhelming, but enough so that they usually have to start some kind of record keeping to keep track, often multiples of 6 as they add cans.  Sometimes drawing a line, with the total amount under that line.  Some students draw neat stacks of cans in a row, and some draw crazy piles of circles of many sizes.

Not very many students saw the connection with the two problems.  Only one realized that both were 156/6.  Others started to think about it when they realized the answer to both questions was 26, and then saw a connection. Many used their pictures of the flavors and started circling groups of 6.  Which was interesting, becuase then there are 2 left over of each flavor.  Some had 24 six packs with 12 left over, and some had 26 six packs.  (Talking they agreed on 26.)  The student who made the division connection shared, but the other students didn't really seem to hear her.

The formal language for what's going on here is that there are two different division actions.  When students can solve some division stoies but not others, sometimes this is the underlying cause.  (Quotative and partitive division, from learner.org, with kid video, too.)  When explaining this to our preservice teachers, we often use the terms fair share (how many of each flavor) and measure (how many six packs) for the different actions.

The next week I wanted to bring another context.  I brought a bunch of play money that I wanted to sort into 6 bags for my preservice teachers.  We counted up the 74 quarters together by making stacks of 10. I wanted to have enough that they were counting up in 10s, so that when we were dividing we'd see the benefit of the 10s.

Do we have enough for 10 in each bag? "Yes."  How much left? "14."  (I kept the notes on the left.  10x6=60)  How much more can we put in each bag? 10? "No!!"  "Two," one of the kids suggested.  People agreed, so in went 2 each. 2x6=12.  2 left, not enough for even one more in each bag.  "That's the remainder."  Excellent!

We then divvied up the rest of the coins.  We had the most pennies, so I asked for a volunteer team.  Then the next team chose dimes over nickels.  96 nickels, it turned out.  But then they lumped all of their neat stacks of 10 together again!  149 dimes.  And 491 pennies.  (One of the kids even noticed the anagram.)  The nickel team was done pretty quickly, 10 in each, then 5 in each and 6 left over.  "Oh, that's enough for one more in each bag."  The dime team did 10 each, then 10 again.  5 each... not quite.  Get one out of each of the five bags with 5.  Remainder 5.  The 491 team did multiple rounds of 10 each.  Then just kind of scooped that last 11 into a bag.  All the bags had lots of pennies by that point.  Then each team made a number record like on the left, and nobody saw the connection with the division algorithm they have been doing.


Okay, that's immediately a preservice teacher activity.  At GVSU we're blessed with a goodly pile of manipulatives.  So each table got 6 tubs of blocks (2 each with unifix cubes, wooden cubes, and snap cubes.)  5 (or so) minutes to play with them, because play is important.  Of course they made many mathematical designs and structures.

Then it was time for the task:

1)    How many of the object did you get?
2)    Physically divide them up into the 6 tubs evenly.  How did you do it?  How many in each tub?
3)    Show with a number record what you did.
4)    Use a sense-making method to do the associated division problem.  How would what you did make sense as physically dividing the objects?  Why does your method work?

1)    ___________ blocks in each tub.

2)    Description of method:

3)    Number record:

4)    Sensible division problem:

And then to make a poster of the connections between their number record and their division work.  Here's what they did! (Click on the images for full scale.)

And the piece of least resistance:

Those are some beautiful connections!

Sunday, March 7, 2010


So my week of thinking about teacher education started off with Sue's (Math Mama Writes) list of her Top Ten Issues in Math Education.  Then there was the preview of the NY Times weekend article on Building a Better Teacher by Elizabeth Green, and then there was Kate's (f(t)) take on that, and various other reactions to it. 

As I was thinking about Sue's list, my list quickly turned into a mini version of my teaching philosophy.  I realized that my list of issues was really just a list of ways we go away from what we know about how people learn, which is obviously related to how people learn math.  A lot of this I have learned through discussions and teaching with Dave and Kathy Coffey ( I wish they would write about teaching) and Esther Billings, and previously Jan Shroyer and Georgi Klein and, and, and... 

My list from there:
  • Learning starts with engagement.  Engagement is more likely when learners are safe, the content furthers their purposes, and they see themselves as potential doers of the content.  (Cambourne)
  • Humans learn best and retain longest when new learning is connected to prior learning.  (Piaget)
  • Reflection is essential for consolidation and extension of new learning. (Piaget/Vygotsky)
  • Learners build and consolidate understanding through discussion and communication. (Vygotsky)
  • Learners build abstract understanding by generalization of concrete experiences. (Piaget)
  • Teaching furthers learning when it is starts with what students can already do, has planned worthwhile and reachable objectives, supports students in their diversity of learning styles and preparation, and assesses the effectiveness of instruction by measuring student learning.  (Teaching-Learning Cycle)
  • Comprehension is best taught with a process focus.  (Mosaic of Thought)
  • Math should make sense and be about making sense of ...something.  The something depends on how you define math.  My current try at this is making sense of quantifiable or describable objects and their operations and properties.  Jo Boaler (and others) describe this as pattern finding activity.  (Don't know how to source this.  Seems obvious, though it took me forever to understand it.)
  • All students are capable of significant and important mathematics.  (NCTM Equity Principle)
  • The mathematical processes are at least as important as the content.  Problem Solving is central to these processes.  (NCTM, Polya)
Then, when the article came out, I was disappointed.  It was primarily about classroom management, it felt like.  Doug Lemov has compiled 49 tips (behaviors) that came from observing good teachers.  How did he find good teachers?  By test scores.  Tips include:  Stand still when giving directions, giving directions with physical miming, noticing positive behavior, be specific in what you ask, call on students randomly after asking the question, etc.  Hidden behind the show over techniques, though, is "Lemov’s taxonomy is one part of a complex training regime at Uncommon Schools that starts with new hires and continues throughout their careers."  Lemov is an administrator at the Uncommon Schools chain of charter schools.  (You can see more about the taxonomy including video samples at their site.)  The article then gets into Deborah Ball's approach to Mathematical Knowledge for Teaching. 

One thing Kate points out about the article is that it posits that teaching is not just an innate gift, that we can get better at it through work.  I agree wholeheartedly - it is no better to say some are teachers and some are not than it is to say some can do math and some can not.  The main attributes needed to both are desire and willingness to learn.

Several of my student teacher assistants are in struggling schools this semester as student teachers.  They are very frustrated.  It is clear that what is being done is not working, but teachers and administration are afraid to change anything as that might make it worse.  The student teachers are forbidden to change anything, and that leads to the frustration as the problems continue and grow.

Jim Knight is an expert at Kansas on instructional coaching.  (Knight has a blog covering coaching and various sundry topics.)  His big questions for coaching are: (my paraphrases)
1.    Is classroom behavior under control? 
2.    Are you comfortable with all the content?  The pedagogy of the content?
3.    Do all students master the necessary content?
4.    Do you know and how do you know if students are mastering the content? 

He starts with classroom management.  I, unfortunately I think, have traditionally left this issue to the College of Education.  I have seen my role in covering 2-4.  But everything continues to point to first things first.  The push to make teacher education more clinical, supports this - IF we support novice teachers in the same ways we know all novice learners need support.  It is a whole cloth.

I do believe that engagement is key to real classroom management, and for math to be engaging it has to be authentic, and that means teachers need to have experienced it authentically... these things cannot be separated.  But first, the class needs to be open to teaching and learning.  And there are places where that's counter-cultural.  Especially in math class.  It has been proven to students that it is not a safe place to learn, that they are not potential doers and it is irrelevant to their purposes.  It is natural and even wise in a way that they have closed it off.

Then classroom management is justifiably number one, except even the name is misleading.  Environment construction, conditions setting... hosting?  Whatever it's called, it's got to be part of the process of teaching as we teach it.  I think placing student teachers in groups or pairs in classrooms is part of this, too.  Students learn best when they can discuss and dialogue.  Most of this stuff has been known since Dewey... why is it so difficult?

Teaching hard.

So I'll give Lemov's book a try, and pass it on to preservice teachers if there's even a hope of it helping. 

Tuesday, March 2, 2010

More Motion Sketches

These sketches are to investigate the composition of motions, starting with reflections.  For a schema activation, I asked my preservice teachers to think about compositions.

Schema Activation:  What happens when you do two of a motion?  (Same type, not necessarily the same motion.)  Please guess if you don’t know.

Translation then a translation   K/G

Rotation then a rotation

Reflection then a reflection

Glide reflection then a glide reflection

This brings up the idea of orientation both in terms of turning and in terms of face-up/face-down.

For a focus we have the following:
Focus:  Today we’re just going to concentrate on reflections and their compositions.  We have three different sketches to consider, and will also consider the questions that could be asked about each.  A composition of motions is when you make one movement and then another.  The combination is still a motion, as the original and image are still congruent.

We're also going to consider using questions to move us forward.  The types of questions described by literacy instructors are:
  1. Literal - factual answer available or quickly available by recall, or can be found directly in the text.
  2. Application - answer found by applying known method or looking up with slight modification.  The method of getting the answer is known.
  3. Inference - answer requires prediction or extension from known information.  Can be an outright prediction or come from reading between the lines.
  4. Analysis/synthesis - answer requires combination or deduction from other known information, possibly requiring a method not currently known by the respondent.
In the three sketches, the students are asked to take more and more responsibility for the questions they are answering.

  Two Reflecting Lines:  webpage or geogebra file 

  Two Skew Lines:  webpage or geogebra file

   Two Parallel Lines:  webpage or geogebra file

We discussed these sketches together.  They asked about finding the center of rotation and saw a neat connection with the reflecting lines.  They also saw a neat connection between finding the center of a rotation and finding the center of a circle, but couldn't remember or figure out how.  They found a cool relationship between the direction and distance between parallel lines and the resulting direction and distance of the translation. 

Reflection:  What did you do during this workshop?  So what did you learn?  Now what would you want to consider next about motions or questioning?

Bonus: (or... extension)

Two Glide Reflections: webpage or geogebra file

Coming Soon: