Thursday, June 28, 2012

Teaching is Not Telling

As an early proponent of Tau Day (okay, it was a circle post on June 28th) I feel justified in calling for the establishment of a new Tau Day tradition. The Sharing of Sense!

The whole motivation of Tau Day is to think about what makes sense. Pi is traditional, Tau is the product of reflection, a consideration of where we are now. There is room for both. Of course Pi looks like two Taus, but it's Tau that's twice Pi. Typical math reversism. The tradition that makes Festivus the great holiday it is is the Sharing of Grievances, and this makes a good bookend for that.

It has been a weird ten days. Dave and I filmed the now infamous video over a month ago, and it took me a while to get to the editing. He was patient with me as always. It took us a couple months to get to the filming. I expected a day of dust ups on Twitter, some people happy, some people unhappy, and that happened, though most were amused. Then Dan Meyer picked it up. Then Ed Week. Then Slate. And yesterday I was answering questions for the Chronicle of Higher Education. We've received great support, insightful criticism, boorish comments and threatening email.  Crazy, then surreal, then bizarre. There are many people with deep emotions involved.  It's like in Ken Robinson's first TED talk, people will pin you to the wall to talk about their education.

Dave has said that there is discussion among teachers, and that's a good thing.  As he's written in a couple comment threads:
Our primary purpose with this video was to get a conversation started. We realized that the satire would put some people off but many teachers have tried to engage Khan Academy in a reasonable discussion and present their case to the media about issues with this approach with little to show for it. Now that the conversation has started and Sal himself has said he is listening and looks forward to more critiques, the time has come to raise the level of the discussion. That is why several bloggers have suggested 101 Critiques and Lessons.

We are looking for at least 101 bloggers to offer a video critique of a Khan Academy video and then share an alternative lesson on that concept. The goal is for all the participants to post the critiques and lessons on August 14th (the day before the deadline for the MTT2K prize). Perhaps the sheer volume of resources will convince the media to acknowledge that while Sal Khan's approach has it's place, he could still learn something from teachers.
In the discussion with the Chronicle reporter, there were lots of questions about Mr. Khan, his academy and his videos, asked in terms of analyzing him, it or them as teachers/teaching.

That's not teaching.

Not to me, anyway.  If there's one idea I try to get at over and over (students chime in: "...and over and over") it's that teaching is more than the time with the students. Planning, assessment, and evaluation are a big part of it. And the time with the students is and should be so much more than the time when you're talking. The fundamental idea, for me, is that teaching is the practice of what we can do to support learning.  Telling can be supportive, maybe, sometimes, with some other help.

I did a project with Susan Walborn, one of my favorite teachers ever, before we went on to the Math Art festival. It was early on in my interest in math games. We got a group of 4th and 5th grade helpers who were struggling in math, to become teachers for 1st graders. We'd teach them the game for 15 minutes and then have the tutors play the games with small groups of the wee ones. It was often great.  The tutors loved being the experts in an area where they had had so much negative feedback, they often benefited from some remediation on early number ideas they had missed, and they were so supportive of the 1st graders that it was often touching.

The one teaching idea I emphasized the most with them they really got:

If you can ask, don't tell.

(This was before the similar phrase was politicized in a horrible compromise.)  The message I would want to share on Tau Day, for my bit of sense to share, is for all good citizens to understand this:

Teaching is not Telling
Telling is not Teaching

Thanks to everyone who's engaged politely in conversation about Mystery Teacher Theatre 2000, and I'm very grateful to the many who have been supportive.

Image credits: Festivus Pole by M. Keefe at Flickr. More anthropomorphic comic at the tumblr.

Sunday, June 24, 2012

Intermediate GeoGebra

So we had a small group in for GeoGebra learning this week, and I want to share what I've learned. One of the best ways to learn anything is to try and teach it, and this was true this week.  This post is a review of our approach with links to the materials, and some description of what the intermediate users were interested in: images, buttons, input boxes, and more.
First up is our revamped website. I'd really like this to be a resource for anyone learning or teaching GeoGebra to others, so please send feedback. The old website had gotten pretty clunky, as we modified and added to our materials, and it was time to start fresh. The website also includes a page with all of our handouts, and a nice Resource section.  Besides the website, we started a local GeoGebra Facebook page that we could use for links and resources.

This site has a clearer direction through it. Whether you are introducing GeoGebra to people with some time - say a two or three hour block - or in a quick one hour presentation, this is how I'd recommend starting:
The program is so intuitive to use, that many teachers can go from there. GeoGebraTube is such a huge sell, that I tried starting with it, but it turns out that it's even bigger when people know how easy and free the program is to get. They can explore the sketches on the tube better with just 5 minutes on using the program first. I separated that out into three pages at the website: Start Up, GeoGebraTube, and the Quick Introduction.  I asked participants to record the Tube finds they liked the best on a Google doc, but I've also just asked them to write the Tube number on the whiteboard.

If you'd like to add to the Google doc with your favorite Tube Finds, be my guest! (Here's the link.)

We covered search, tags and user profiles, all of which enable finding cool stuff.

For the Basics, the list of intro tasks has gone through several iterations and seems to work pretty well. Getting through those gave users a fair amount of control. The first non-basic topic most people are interested in is usually is using images in sketches (often inspired by the Dan Meyer inspired Dan and the Ball sketch). If it's not images, teachers are interested in the spreadsheet view. So we've got a guide to using images and a starter for the spreadsheet.  It's also important to cover the export menu.

Once users have gotten to making some pretty impressive setches, most of what the teacher or trainer has to provide are the commands to help in what they want to do. The two that have come up the most for me are:
  • RandomBetween[ ⟨ minimum integer ⟩ , ⟨ maximum integer ⟩ ] - does just what you'd think it would. Very useful for generating targets or random polynomials using this for coefficients. You get new values by choosing Recompute All Objects from the View menu.
  • Function[ ⟨ function ⟩ , ⟨ start x-value ⟩ , ⟨ end x-value ⟩ ] - limits the domain of a function. Best technique for this is often to define a function f(x)=... then make a limited version of the function using the Function[] command.
It seems the next matter for most users is to get experience with the Action Objects Tools,  sliders, input boxes, check boxes, and buttons. (That's the intuitive order for me.)

Richard Wade shared with me a neat challenge that provoked some good work as well as some instructions on animation, etc. This is my entry for a dynamic Olympic logo. I also think they should include a purple and orange ring, but I understand the importance of tradition. I wanted an interlocking rings effect, but couldn't figure out an over/under trick.

A note about embedding animated gifs: some services are weird about it. If you're having trouble, upload it to a photo sharing site (I use the mildly annoying photobucket) and put in the image by URL.

The main feature that's come up that we don't have a guide for yet is the dynamic text. Text is next.

Whenever you get to work in a group with GeoGebra you learn something. The two tidbits I found most useful this time was the Corner[ ] command, which we found in an Anthony Or (orchiming on GGBT) sketch.   Someone asked "Well how did he make two sets of axes?" and the answer was clever use of the corners.

The other was instigated by a participant: looking into what that student option is in a GeoGebraTube collection. When you click Create version for students, you get a menu of choices.

You get a palette of sketches for students, like this.

That's going to be a great feature.

Hope this can be of use to you for your GeoGebra learning, or with fellow teachers or with your students.  I'm happy to share any materials electronically, or in person if you're within driving distance.  What would be helpful for you?

Monday, June 18, 2012

Mayim Bialik

Might as well go for Controversy Week on the blog, eh?

I like Mayim Bialik as an actor, totally think she improves Big Bang Theory, and think she has obvious science chops. I personally get pretty geared up about the immunization "controversy" (which I only think is a controversy in the same way as the climate change "controversy"), and was disappointed over her Science Friday brouhaha about immunization.

Then I got a promo from NCTM giving her top billing at a meeting.

I wrote:
To: NCTM Summing Up
Subject: Re: NCTM Summing Up: June 5, 2012

Do you all know that Mayim Bialik is a supporter of the anti-immunization movement? Pretty disastrous for teachers, should she have her way. I don't think we as an organization should be legitimizing her as science-related, let alone promoting her. This was in the news recently because of her appearance on Science Friday.

-John Golden
The NCTM was nice enough to write back...

Mr. Golden,

I am sending the attached letter on behalf of NCTM President Linda Gojak, who is copied on this message, in response to your message below about Mayim Bialik.

Ken Krehbiel
Associate Executive Director for Communications

Hmmm. So I wrote back,

Thanks for addressing this and for responding. I understand the tensions involved.

I do think there's a difference between having a Ph. D. and qualification in math ed. For example, the many Ph. D.s in mathematics who support Mathematically Correct would probably not be advertised by the NCTM in promotional materials.

May I share this letter with others?

John Golden
To which they quickly replied:
Hi John

Thank you for your thoughts.

Actually Mayim has been doing some work in mathematics education with a supportive stance on the importance of mathematics as fundamental to the foundation needed in STEM areas.

I think her message around mathematics is important as we move forward. Her personal stances on personal issues are exactly that...personal

I think the CCSSM provides us the opportunity to work together from a variety of viewpoints to work together to do what is best for our students. So it is no longer a divide between mathematically correct or not correct, rather it is about what prepares our students with multiples options in the workforce.

You can share the letter as you see appropriate.


Linda Gojak
Nctm president
I'm very happy to be part of a group that attends to its members thoughts, and I do understand that decisions are not going to be by consensus most of the time.  I don't want uniformity, relish diversity, and think I'm probably being over-sensitive here because of how I feel about immunization.

What do you think? Would you put her on stage at your conference?

Mystery Teacher Theatre 2000

Welcome to our first episode. Recently, in the halls of School University, some teachers attempted some selfdirected professional development, encouraged by their principal and given hope by Sal Khan, a man described as "Bill Gate's favorite teacher" (from a TED endorsement), "very popular," "extremely popular," "an educational revolutionary," and being like "like a nerdy, South Asian-American Seinfeld." (I'm not making any of those up. The last is from Wired.)

Here's what happened.

So, obviously, as comedy improv actors we're a couple of math teachers.

We're very interested in your comments. What do you think of this video? Of the teaching? What did we miss in our commentary? 

Dave was the instigator here, but is too smart to put it up at Deltascape.

Saturday, June 16, 2012

Quadratic GeoGebra Day

I'm teaching a summer intermediate algebra course, and we recently had a full day of GeoGebra for quadratic functions. With GVSU's GeoGebra training coming up, it seemed like a good time to document what all we did.

I'm using a Bring Your Own Device philosophy for the class, so students are bringing TIs (82, 83, 84, 86 so far), smart mobiles, and laptops to run Desmos or GeoGebra. When we have a particularly heavy computer day, I let them know ahead, and a few extra people bring in laptops. The university runs a pretty good wireless service accessible to all students and staff.

Here's our agenda for the day:
Agenda – Class 9
Objective: quadratic regression and connections between the vertex form and standard form.

5 start up: multiplying 2 digit numbers with place value sense.
15 HW
30 Vertex investigation (GGBT - GeoGebraTube)
50 Regression opportunities (W|A, GGB, TI)
30 Projectiles
5 Classroom concerns 
To do: project 1 – linear and/or quadratic math in life.
The way the class panned out, we left off projectiles for another day. The point of that is to make a strong connection between constant second difference and constant acceleration.

Some of what we were doing was to support them in the project. Here's the handout:

It helped engagement that they knew there was something for class that this supported.

The multiplication of 2 digit numbers was a connect forward piece to multiplication of binomials, by means of an area model that shows partial products and leads to the box method for multiplying polynomials.  There were no homework questions, so we spent a while on the multiplying, and then someone wanted to know why we were doing this, so we got into the box method right then and there.

I designed this sketch to have just a and b on sliders, to focus attention on those. The class had previously noticed that c in the standard form was the y-intercept, so I made that literal in the sketch to emphasize it. I gave it a dynamic window (defining x- and y-min and max interms of function values) so that the parabola would be in view. (GeoGebra note: that means that the program is constantly recomputing when values are changed, so you can't really use the trace.) The sketch has these questions on GGBTube.
This sketch helps you investigate the location of the vertex for a parabola in standard form, y=ax^2+bx+c. Since the equation will tell us y if we know x, we will concentrate on the x-coordinate of the vertex.

1) There are controls for a and b, separate from c. Why doesn't c change the left-right location of the vertex?

2) Change a and b and notice the effect on whether the vertex moves left or right. What do you notice?

3) Our goal is to find a rule or formula for the vertex's x-coordinate in terms of a and b. How can you collect data that will help us determine the formula?

Extension: what else could you investigate with this sketch?

The vertex investigation led to a good bit of mathematical reasoning. They noticed that when a increased, the vertex moved to the right, unless b was negative in which case it moved to the left. Some discussion led them to decide that it moved closer to zero.  They noticed when b=0 that the vertex is always at x=0. They also noticed a cool parabola pattern that the vertex makes when you move the b slider.

I suggested that maybe we needed some hard data, and started recording on the board a and b vs. the x-coordinate of the vertex for what they had tried. No pattern was visible. So I suggested fixing a or b and varying the other. Now there's some interesting things to see.
(It wasn't this neat on the board.) They noticed that as b increased it got bigger ("Bigger?" I ask. "Well, more negative.") As a increased it got smaller. When a was steady, it was like a line. They noticed the -1/2 slope for when a=1. One student noticed that when b doubled, the x-coordinate doubled. Then another noticed that when a doubled, it went down by half. Then a student had the insight, "oh, I have it! I think... can I say?" (We've been working a bit on being a no-spoilers class.) Is it -b/2a?  Seemed to work with all the data we had.

Steve's Rule: x-coordinate of the vertex = -b/2a.

When students reflected on this investigation (jot down one thing you want to remember) many noted the formula, but some remarked on collecting the data and the trends in a and b as it affected the vertex.

Next I wanted to give them some practice on regression. Starting with the sketch Dan and the Ball,
which, of course, is from Dan Meyer's photo composite.  Students pretty naturally want to know if it goes in. Most think it does when you take a vote.

On this sketch you can use the red point to gather data. We talked about entering the data in the spreadsheet, from which you can use the two-variable data tool. 
Or make points from the list and use the FitPoly[ ] command.

You can also use Wolfram|Alpha to do the regression, or a TI. Unfortunately Desmos doesn't have regression yet. W|A is surprisingly finicky about the phrasing you use. For example it prefers "fit" over "regression."

 Students were very successful with this, and I had them move on to their choice of activity.

Several did the bridge problem, which looks at whether a bridge is a parabola or a catenary. They were bold about exploring, despite the many overheard comments about not having any idea what was going on with the catenary. Several also gave the bouncing ball sketch a try, which allowed them to see the effect of constant a on different parabolas. (OK, I'm also just pretty happy with the simulation.)

By that point many were looking for other things on their own. Someone found this great world record motocross time lapse shot, among others.

From tifr
 I made up a guide to importing and using images in GeoGebra.

One GeoGebraTube tip that I picked up from this much student use in class was to remember to change the vertical display size of a sketch if you are adding the menu bar, tool bar or input bar. About 200 pixels is enough for all three.
You get to that screen by being logged in and choosing the Edit option on the teacher page for your sketch.

This day seemed to make (by subsequent assessment) a difference in their technical ability to do regression, their understanding of the vertex formula and use of the quadratic formula, understanding of some of the features of parabolas. They also have been working with a great amount of independence on their projects.  Which I am excited to see on Monday!

Friday, June 8, 2012

Size the Day

It was time for my last game with the fifth graders, and the content was multiplication and division of fractions.  Having just been critical of a game that was very computation focused (see my Math Evolve review) I was very wary of doing the same thing.

I'm at the point now where I've designed more games than I can remember easily, so one of my first steps in game design is to search my own stuff. Failing finding a game to use, maybe there's one to revise. Failing that, maybe one to revise. I did stumble across my post on multiplying fractions. Oh.

I have many lessons that get at the meaning of the operations, can be used to start discovering, exploring and justifying the rules... but only the Product Game (adapted for fractions) for practice. But that post closes with an Ant Man and Wasp cartoon, and we'd been talking a lot about them because of the Avengers movie.  My son is a comic book fan (sounds better than monomaniacally obsessed, right?) so this was quite a debate. He loved the movie maximally, but would have loved it even more with the original comic book Avengers crew. (Joss Whedon wanted characters to whom non-super powered audiences could relate as well as the super powerful ones.)

By Xavier Golden, Super Hero Squad style.
"The hexagons are Pym Particles."

So I was struck with the idea of a size-changing game. But why would our heroes have to constantly change size?  To get past obstacles! Sometimes they'd have to grow, sometimes to shrink. I tried to think of a battle game because there were several boys always interested in that, but for battle it seemed like you'd almost always want to be giant-sized instead of ant-sized. So a kind of maze... so it could be a race game.

I tried to think of a way to turn dice rolling into fractions for multiplying or dividing, or to roll three and choose two, but that didn't feel appropriate for such new content. I wanted actual fractions to see and think about. 

If you have transparent spinners, this is a good place to use them; I just used bobby pins, which make excellent spinner needles.  I experimented with the spinner entries and maze heights to find settings that were not too immediate but not too difficult either. Thinking about the framework I've been using...
  1. Goal(s) - solid. I wanted students to get the understanding of the effect of multiplying and dividing by fractions, so contrary to their expectations. I wanted to get some sense of estimation, and some experience with calculations that would lead to support the symbolic rule they'll learn later. I'd also noticed that they were very interested in calculators, but had little experience with using them. This put everyone on an equal footing, as the numbers were messy and required a calculator.
  2. Structure - like the stretching/shrinking as a context for multiplication/division. The spinners allowed a lot of flexibility in getting values to be used. Makes the game highly adaptable. And the intention of having to choose a multiplying or dividing spinner helped get across the stretching or shrinking effect.
  3. Strategy - weak as it was.
  4. Interaction - typically weak in race games, though .
  5. Surprise - spinners help here and with...
  6. Catch-Up .
  7. Inertia - not meant to be a game that requires a lot of replay.
  8. Rules - basic premise, spin and change your height. Move forward when you fit.
  9. Context - thought this was strong, plus pop culture tie ins to a heavily advertised movie. Kids were interested and engaged, though I sold it a bit explaining about Ant Man and the Avengers. There's a little suspension of disbelief, as Wasp could just shrink and fly through all the obstacles, and it's rare that she grows in the book.

I added the Spin Again option to help with catch-up, strategy and interaction. But most students were so immersed in their own spins that they rarely used them! The other idea that I like quite a bit was the customizable board. Most of the fifth graders were happy to use the board as printed, but a few experimented with rearranging the board.  Maybe with middle school students, more would be interested in giving it a go. Designing a board for your opponents is a great opportunity for some open-ended problem solving.  I picked up a couple packs of mini post-its, and they were perfect for keeping track of the players' heights.

It was a good last game of the year. In the debrief, they definitely got the point that multiplying and dividing by fractions did not just have the same effect as multiplying by whole numbers, and a few kids were noticing that dividing by unit fractions was like multiplying by the denominator. I also saw considerably increased skill with the calculators, and some sensible rounding of the decimals involved.  (Parentheses were almost entirely new to them.) They asked me to leave the supplies so they could play later, and it got almost 100% thumbs up for keep or dump - both good signs.

Hopefully you can get a chance to give this one a try. It has some interesting features, and I think the choice of spinners and rearrangeable board will show up again - good game mechanic features. I'm always interested in your feedback, if you have any ideas or get a chance to use it with students. One dramatic need: the name is a terrible joke, and of absolutely no use with middle school.  Ideas?