Synthesis

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. 

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.

Motions	Result-type	Know/Guess?
Translation then a translation		K/G
Rotation then a rotation		K/G
Reflection then a reflection		K/G
Glide reflection then a glide reflection		K/G

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.

Activity:

  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:

• a tale of two lessons - why can't everything work?
• Sue's MathMama Dialogue:  what's wrong with math education?

Motion Sketches

These Geogebra sketches are meant to serve as an introduction to the four Euclidean motions.  (I am a proud supporter of the Glide Reflection.)  Probably for middle school and above.  If you try them out, I am always interested in feedback.  The Motion Intro does not actually use a lot of the dynamic nature, but generated a lot of connections with my preservice teachers.  The Motion Control led to a nice discussion of what information is needed for each motion, as well as how to find that information (like the center of rotation) for a given motion.  The webpages have the questions that I asked my students.

MotionIntro: webpage and Geogebra file.

 Motion Control: webpage and Geogebra file.

Math in Action 2010

Grand Valley State University sponsors a terrific little math conference each year called Math in Action.  I think started by Jan Shroyer back in the whereupon.  It has 30ish workshops for K-12 math teachers, mostly very practical.

I'm hosting 6 wonderful preservice teachers presenting geometry games for K-8 teachers, so I'm posting the electronic versions here for people to be able to download.  If you were a participant and wanted a Word file to edit instead of a pdf, just email me.  The address is available on my workpage, linked on the right.

Anne Harkema: Rope Charades
Lauren McKee: Quadrilateral Concentration and the required Quadrilateral Cards
Emily Trybus: Area Block (link to a previous post)  
Rebecca Sochacki and Brynne O’Connell: Polygon Capture
Jill Dzierwa: Triangle Detective (link to a previous post)

As a bonus, here are the two bonus games from the Quadrilateral Concentration sheet.  Both are other uses of the Quadrilateral Cards.  Not included here is Quadrilateral Euchre, which is Euchre for a partially ordered card set.  Verrrry geeky in a midwest sort of way.

Quadrilateral Go Fish

Materials:  Deck of Quadrilateral cards. Best with 3-5 players.

Setup:  Deal 5 cards to each player.  Put the rest face down in the middle, either in a neat stack, or mixed up in a big pond.

Gameplay:  Start to the left of the dealer.  On a player’s turn they can ask a particular player for a specific property.  For example:  “Do you have a shape with opposite angles congruent?”  You can not ask for a shape by name.  (“Do you have a rectangle?”)  If the player has a card like that, they have to give it over.  If they have more than one, they get to choose which card to give away.  If you have a matched pair of the same type, you can play them down. 

Winner:  First winner is the first player to go out.  Second winner is the player with the most pairs.

Variations:
•    Instead of asking by properties, ask by name.
•    Start with 7 cards.
•    Allow players to play cards on other people’s pairs.  (If you have a pair of rectangles I can play a rectangle.)

Quadrilateral Guess Who

Materials:  Quadrilateral card deck.  2 players.

Setup:  Sort the quads by type.  Each player puts one quadrilateral of each type face up in front of them, and the others go face down in the middle.  Each player draws a card from the middle and keeps it hidden from the other player.

Gameplay:  On your turn you can ask one question about the other player’s hidden quadrilateral.  That player answers yes or no.  Turn face down the quads you have that don’t match.

Winner:  first player to guess the other player’s card.

Math Teachers at Play 23

Matht Eacher Sat Play 23 is up at Math Recreation.  Nice 23 theme and a couple cool 23 facts.  My entry is the recent money post.  But you will also find:

• Very cool post about young children learning number with a deceptive title: Infinity Plus One at republic of math.
• Sue's most recent Math Salon.   Great culture building. 
• Two related posts on binder checks as assessment.  I wish more people wrote about assessment and evaluation.  (Don't think I have either, though.)  The second is at one of my favorite blogs, the unflinchingly honest f(t).
• Lots more good stuff, of course.

More Money

My preservice elementary teachers are preparing to do some 2nd grade tutoring, and the teacher asked for money and time.  (Really, who couldn't do with more of both?)  Those are always challenge areas, in my experience, so I thought I'd share our resources.  Here is a collection of some money activities.  It includes Change for the Better and Make It Take It which were described in an old post, which are two of my favorites.   It's been constructive to use games for practice, interesting problems to help with concept development, and explore multiple representations.  One thing that's often lacking is a visual representation to help understand the relative value of coins.

A quick easy game:  Monopoly Money Madness

Materials:  play money, 2 dice.
Math content:  addition, money recognition, unitizing (grouping into new amounts.)
Game play:  Very simple – roll two dice, and take that much money.  If you can group your money into a larger bill (for example, a five and five ones into a ten dollar bill).  First player to $100 wins.
Variation:  have players “shop” a catalog or the web for something they would like.  Play until they have enough to buy the item they would like.


Money as a context offers some nice practice for developing unitizing, the understanding that allows learners to flexibly exchange between a groups and individual members, or the ability to switch what you are considering as a unit.  (I.e. switching from dollars to quarters or cents.)  It is such a key concept for 2nd grade, as students move from single digit arithmetic to multidigit arithmetic.  Lack of understanding in this will follow them for the rest of elementary school.

From Mathematics in the City: Measuring Teacher Change in Facilitating Mathematizing, Catherine Twomey Fosnot, Maarten Dolk, et al. (link goes to a pdf of the article)

Unitizing requires that children use number to count not only objects, but also
groups—and to do them both simultaneously.  The whole is thus seen as a group of a
number of objects. The parts together become the new whole, and the parts (the objects in
the group) and the whole (the group) can be considered simultaneously. For learners,
unitizing is a shift in perspective.  Children have just learned to count ten objects, one by
one. Unitizing these ten things as one thing—one group, requires almost a negating of the
original idea of number.  It is a huge shift in thinking for children, and in fact, was a huge
shift in mathematics, taking centuries to develop. Understanding that a square in a tiled
array can represent a column and a row simultaneously also involves a construction of
part/whole relations (Battista et. al., 1998), as does the relationship between
multiplication and division. There are many more. Because “big ideas” involve
part/whole relations, they require a shift in perspective by learners.

As we look at state content expectations, it's pretty clear they are focusing on skills.  Fosnot and Dolk (in Math in the City, their books on Young Mathematicians at Work, and their curriculum Contexts for Learning) have a nice way of organizing content into skills, ideas, and models, and then representing them on a landscape.  The preservice teachers took their information and tried to do the same for money.  Here's what they got: (as a pdf)

Quick Triangle Sum & Pythagorean Proof

Just a quick, pretty unoriginal sketch to help secondary/tertiary students think through the justification for the sum of the angles in a triangle.

As a dynamic webpage or the original geogebra file.

EDIT:  As my students investigated (see these sketches elsewhere), they got interested in proofs of the Pythagorean Theorem.  Since they were investigating so nicely on their own, with interesting results, I had time to draw up a familiar proof in Geogebra.  Webpage or geogebra file.  The webpage has some additional hints to help towards a proof.