Slant Wise

I caught a great talk by Eugene Peterson this week. He's a pastor and spiritual writer who gave a talk at Valparaiso University in honor of Walter Wangerin, Jr. (who was also there); the talk was "What are writers good for?" (I found a pdf of a previous iteration of the talk.) There'll be mention of God below, but really, these are my connections from his talk to math teaching. I've written one other post inspired by Peterson, Jonah the Math Teacher.

Peterson's bottom line for writers is that they can reclaim language from debasing use. For religous writers this is particularly important, because we as a society have turned our spiritual words into godtalk that is easy to ignore and not worth our time.  In other words,

Knowledge of speech, but not of silence; Knowledge of words, and ignorance of the Word... Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information? -TS Eliot, Choruses from The Rock Poems and Plays, 96

Important ideas, I think. And amazingly close to the challenges we face in mathematics teaching.

"So what are writers good for? It is our vocation to maintain and practice this core, basic, revelational, personal nature of language, living speech.    In a world in which language has been uprooted from its originating God soil and put to the use of information or propaganda, it is the vocation of writers to represent and practice language as revelation, to re-orient language into the personal world in which men and women actually live—in their families, and neighborhoods and workplaces," says Peterson.

What are math teachers called to do? To recover the debased math, practiced in schools for years, to bring it into the world in which the students live, to share math as it's truly done, to share learning that will make a difference in our students' life.

So how does a writer do it?  For Peterson, the illustrating example is the middle parable section of the gospel of Luke. A lot of Jesus most powerful teaching. Stories with no explicit mention of God, no direct lesson. A story about fertilizing a tree instead of cutting it down. Told in Samaria to people who are not interested in his religion, and not fond of his people. He might as well have been a math teacher.

Not to double up on poetry, but Peterson also went to Dickinson.

Tell all the Truth but tell it slant – Success in Circuit lies Too bright for our infirm Delight The Truth’s superb surprise As Lightning to the Children eased With explanation kind The Truth must dazzle gradually Or every man be blind -Emily Dickinson

(Fantastic art by  David Clemsha. Shows up in my Reader the next morning as a wonderful coincidence.)

How well does that capture the essence of good teaching?  Peterson notes, "A parable keeps the message at a distance, in the shadows, slows down comprehension, blocks automatic prejudicial reactions, dismantles stereotypes. A parable comes up on listener obliquely, on the slant." A writer does this by having the reader come to them, going slow, countering the impatience of the age.

This leaves teachers the challenge of knowing what is best, in a culture that wants what is lesser. When we propose that there is better, they want solutions that are immediate and rushed. How can we convince them that the answer is slow? Real stories, finding the way themselves, experiencing the superb surprise. I think we have to just persist. Write our real lessons. Participate in the community. Share our success stories that will buoy us through a dozen bad days.

"The Truth must dazzle gradually."

It's our vocation to tell it slant.

#logarithms

I had one of those great twitter moments yesterday, completely by generosity of tweeps. This captures one aspect of why I introduce twitter to our preservice teachers in hopes that they will either enter in or give it a go later.

My summer intermediate algebra class is leisurely (12 weeks instead of 6) picking its way through the summer. Linear, quadratics, exponentials with tons of technology use, simulation and experiment and a dash of art. And then comes logarithms.

My emphasis on introduction is as a way to undo exponentiation. Not a big emphasis on inverse functions, because though we're using the language, we haven't really dove into function language yet. But doing and undoing we talk about a lot.

But after that reasonably good start, we come up against the log rules. Our introduction was Kate Nowak's log law introduction (which I found through Sam Shah's Virtual Filing Cabinet; why bookmark? Let Sam do it for you.) The idea is that you see lots of examples and then try to generalize into a pattern, which is the log law. Nice pedagogy!

Yesterdays lesson included connecting the exponent rules to the log laws.





Tough going.  In terms of gradual release, we had to back up to a lot of teacher support. But the most useful law was the toughest. So I came back from class and just tweeted, to ... vent, I guess. Commiserate. (Which is definitely a purpose of twitter.)

 Wasn't expecting any real response. But then, the great feedback and constructive suggestions began immediately.

This is where we should start. We - as a teaching culture - are so steeped in general then specific, abstract before concrete, that this is a good first check.

 They did understand the multiplication to addition rule better. Look for areas where students understand and build off of those. Connections strengthen learning.

 This is the connection I'm going to follow up on. Look for a way to get it across.

 Think about the broader context. How the logarithm lives as an inverse function, with a nice concrete place to start.

 Support that this is challenging, and not me just being stupid. Of course, I'm happy when people point out I'm just being stupid, too, since I don't want to be stupid.

Neat post.  It gives a lot of good thinking about introducing logs, being intentional, and paying attention to students making sense of notation. This is very close to how I introduce logs, and how it looks like Kate Nowak introduces them, too. I use to think this was inappropriate for high school but okay for college (because of their bad early experiences), but I'm beginning to think that's how we should handle bad math notation. Use constructive notation and then transition the students to traditional.

 I don't think this is accessible (YET), but I'm going to mention this, too. Part of the class is getting the students more comfortable with symbolic methods, and I like how this emphasizes the operational part of logarithms.

A reminder of where to start. This is the fundamental relationship for logarithms, and connecting back to it well and often is important for learning in both directions.

So now I have a place to go Monday, following this up as I promised the students.  Maybe I would have had some of these insights on my own - but I don't have to work and think alone about my teaching or the mathematics. It also really sparked some thinking to me about whether logs should be introduced as a function or an operation. I feel like exponentiation goes from being notation, to an operation, to a function. Logarithms could do the same. I'm thinking:

 

So then 2v2^3=3. As it should be. I'm kind of joking?

Anyway, excellent teaching advice. I am so glad to be connected to these excellent teachers. And even though I can't go to Twitter Math Camp, I still get to have the home game to play. Why don't you play along?

(Here's my brief intro to twitter for math teachers. ) Here's the activity I put together for the next class:

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.

http://bit.ly/ggbgv

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?

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.

More GeoGebra 4 Teachers

I presented two 1.5 hour workshops for teachers at the Michigan Council of Teachers of Mathematics annual conference yesterday Thursday, and thought I'd share a few quick thoughts.  Here's my session page/lesson outline. All handouts are attached.  Even in a computer lab with somewhat slow connections, running Internet Explorer, we were able to have the GeoGebra 4 beta installed and running in 10 minutes.

EDIT: Later I used those materials to share GeoGebra with the Muskegon Community College Math Tech Bootcamp and a workshop with at the Kent Intermediate School District. The materials are updated from that, and feedback from all three groups are below.

It doesn't take a lot of experience to introduce teachers to GeoGebra. I mostly just gathered resources, talked about the basics, then let them explore. It's worthwhile for me, too. In a room where many people had not even heard of it before (yet they are at the session - I love teachers' exploratory spirit) they found new corners and features for me to think about. I am no omega-class expert, but I've spent some hours with it. This is a rich program they're giving away for free.

The basics to me are:

• understanding the main areas:

• Tool bar, including pull down tools. Emphasize the selection tool, the move graphics view and zooms


• Graphics area


• Algebra view and the View menu for axes, grid and algebra view


• Input Bar

• The selection arrow


• Undo


• Object properties/Right-clicking objects

Just a few minutes and people are ready to roll. I made up a page adapted from the 2-day workshop of introductory algebra and geometry tasks, and then had options for further exploration of either. All the handouts are attached to the session page.  By the end of each workshop, most teachers had made something that impressed them. A few just thought it would be valuable for making images for tests and handouts, and I think that's a proper way to start for some people. But most were digging in deep, and several got some math learning out of what they did. "Oh, that's why..." was overheard a few times.

The teachers recorded their comments on a Google doc (benefit of a computer lab session) and they're embedded below.  If you are a GeoGebra user - share it with your fellows! If you are not, give it a try. You'll find it worthwhile within an hour.

Or I'll double your money back!

MCTM Feedback


MCC Math Tech Bootcamp Feedback


Kent ISD Feedback

GeoGebra 4 Teachers

My current logo attempt.

I had a great experience this summer co-leading with Michelle Bunton a GeoGebra workshop for teachers. We decided to do a loose structure, emphasizing algebra one day and geometry the second. People were free to register for one or both days. We got about 22 teachers, with 14 for both days. One teacher was bitterly disappointed because they wanted premade activities, and our focus was on learning the program.  Though we tried to connect people to plenty of resources and the wide-world of GeoGebra sketches, this teacher left before that.  Most of the teachers took the freedom and ran, and it reminded me of what a joy it is to be in a room full of independent, motivated learners working on stuff that matters to them.  I learned a lot, which is typical of such situations.

We decided to run the workshop using GeoGebra 4, as it's being released at the end of the summer.  Unfortunately - I was a novice on it.  I'm an enthusiastic GGB 3 user, but have been weak on spreadsheet use. Adding more novel features was a little scary. Turned out well, as it made us co-investigators with the teachers.  And the program is great. I mean it was great, but now it is greater. They've managed to add features without making it perceptibly more complex.  That's rare; I love this software and its developers. Guillermo Bautista's GGB4 sneak peek series was very helpful. (Of course!)

We structured the workshops with time to get software loaded. And I wanted the teachers on Twitter to converse on backchannel through the day.  That was hit or miss, due to Twitter's tendency to ignore new users as an anti-bot measure. I wound up having people follow me so I could follow them, then retweeting their tweets. This made some people show up but not others... mystery.  I thought it was important because I get some of my best GeoGebra support on Twitter. And, in fact, during the workshops we got some excellent input from @mike_geogebra and @lfahlberg .

So startup, a demonstration sketch to show some of the potential for classroom use, an overview of the program parts (tool bar, menus, views, etc.), specific tasks to figure out how to use, and then free explore time. The workshop website has all of our materials (see the workshop page), plus the sketches created by participants. One exciting feature is that participants spent time sorting and classifying sketches by standards strand in a Google spreadsheet with links to the sketches.

One problem were still working on is the "now what?" We're trying a Google group and want the website to morph into a place for teachers to share sketches. (If you are interested in joining the page to share your materials, just drop me a line.)  Teachers also wondered aloud what made this experience useful in comparison to some other professional development, and that's worth looking into.

As I said, I learned a lot: about professional development, workshop development with a new partner, and a lot of GeoGebra. The GeoGebra knowledge I could put into a sketch!  Note that the sketch links below use GGB4, which can download as a Beta.

Dave made a nice height vs time projectile sketch that used text box inputs. (That is my favorite new feature so far.) Cristine got me thinking about more subtle show/hide condition with her beautiful Unit Circle.  One of the participants made a face sketch for which he figured out how to limit the domain of a function. (Which he seems to have never uploaded!) @lfahlberg taught us how to make a reset button for a sketch.  Chris and Jason just pushed and pushed and explored in general. I can't remember why we figured out that sliders can have calculated min and max now.... I was seriously impressed at what good use everyone made of their time. 

After the workshop I made this projectile sketch to practice. It's a projectile in the x-y plane, where the slider advances or animates time. You put in the initial height by textbox, set the vector of your throw by dragging the vector, and can do target practice to a garbage can or by throwing at a seagull. (No actual seagulls were harmed yotta, yotta.) The button resets the targets.  This was fun to make. 

The big screen shot is an animated gif - one of GGB's new export formats. Go ahead and click on it!  The garbage can was inspired by Dan Meyer's WCYDWT ball toss, which is also a sketch.

GeoGebra is a supertool, which even has the power to get more powers, and I definitely want my students and preservice teachers skilled in its use.

A Teacher Talking

It's a little hard to share this because he speaks so nicely about our class. But my realization is that it's really his work that he's talking about.

Our end of semester assignment is one I cribbed from Maria Anderson at her college math/tech camp. She asked us each to do a Little Big. That is, a little presentation that gets at the big ideas.  I ask the teachers to make it available over the internet, as I see that as an important professional participation tool. Slideshare, Prezi, Google presentations or even a PowerPoint file shared via box.net.  Bill is the first person to share by webcam, and the result seemed pretty powerful to me.  He gave permission to share it more broadly, so you get to see what you think.

Bill is an experienced teacher who speaks his mind directly and is very free to share what he does and listen to others. Follow him on Twitter at @breenw77. 

Post a comment here or tweet him directly with your response. 

A Teacher Talking

It's a little hard to share this because he speaks so nicely about our class. But my realization is that it's really his work that he's talking about.

Our end of semester assignment is one I cribbed from Maria Anderson at her college math/tech camp. She asked us each to do a Little Big. That is, a little presentation that gets at the big ideas.  I ask the teachers to make it available over the internet, as I see that as an important professional participation tool. Slideshare, Prezi, Google presentations or even a PowerPoint file shared via box.net.  Bill is the first person to share by webcam, and the result seemed pretty powerful to me.  He gave permission to share it more broadly, so you get to see what you think.

Bill is an experienced teacher who speaks his mind directly and is very free to share what he does and listen to others. Follow him on Twitter at @breenw77. 

Post a comment here or tweet him directly with your response. 

MCC Math and Tech Workshop

This week I'm at Maria Anderson's Math and Tech Workshop.  I'll try to update this post as we go through the week.  There's a large Explorers group and I'm in the Adventurer's group.

Monday Morning

Adventurers brainstormed the following for discussion.  So look for more on this this week.

• Wolfram Alpha
• Attention-Getters
• Textbooks: cost, open source
• Geogebra
• Vyew
• Student interaction
• Twitter/texting
• Math Ed Research on Tech

Resources:

• Textbook cost: article in NYT 
• Math Ed Tech Research: CITE Journal, which has different topic areas, including math.

Social Media:

• Use Facebook pages for students.  They like the page and get those updates rather than your updates.
• Go over privacy settings with your students.  They are often ignorant of the reality. Check your own.
• Linked In: limited use, like a modern rolodex.
• Twitter: internet equivalent of drinking from a firehose.
• Twitter is more collegial.  Follow people who have worthwhile info to you, get nearly instantaneous feedback, hashtags to organize.
• Twapper Keeper (love that) to archive tweets.
• Twitter is a completely public space.
• Math Blogs: a node on Maria's MindMap of Spicing Up Mathematics

Monday Afternoon

Finally got time to play with Scratch.  I'm wondering about it as an activity directly for students, for preservice teachers, or as an instructional design tool to make activities.  Here's my first - kind of like pidgin Logo.




Learn more about this project

The end of the afternoon is working on Mathematica. It has developed quite a bit in the 15 years since I've last used it!

Jonah the Math Teacher

If he thought being a prophet was a tough gig...

OK, prophet's probably still tougher.  Pastor, too.  But math teacher is up there.

Warning:  This is probably over-synthesis on my part.

I just finished rereading Under the Unpredictable Plant by Eugene Peterson for a book group.  (Link to Google Books, which has the first 40 pages.)  It's a book about vocational holiness.  For many of the teachers I know, teaching is a vocation.  It's literally what we were called to do.  So the book applies on that level.  On another level, pastoral work and teaching have lots of connections, so it applies there.  But on a metacognitive level, Mr. Peterson wrote a 200 page think aloud about how he worked his way towards living his vocationally meaningfully.  He parallels this process with the book of Jonah, from the Old Testament, a story about a guy who is definitely struggling with his vocation.  It's an interesting book regardless, possibly essential for Christians or Jews and maybe Muslims (Jonah is in the Qur'an), and should definitely be required reading for vocational workers.

Mr. Peterson breaks Jonah's story up into four parts.

The first is the journey to Tarshish.  God tells Jonah to go to Nineveh, an ancient and future enemy of the Israelites, and he does not want to go.  So he heads in the opposite direction, to a mystery destination full of romance.  Mr. Peterson connects this with careerism.  Before we begin, or in our fantasy moments, or before we leave to take another job, we may indulge in the fantasy of teaching.  Stand and Deliver, or Glee, or a classroom full of attentive, willing students, or the couple students whose lives are turned around by our near miraculous intervention.  I have those days and years, for sure.  The solution to this is to stay.  Teach my students that I actually have.  These guys who were complete duds and still haven't read the article/done the homework/even tried it.

The second part is the storm and the belly of the whale.  Jonah is asleep on the ship, despite it being in a life and death battle with a storm.  Somehow, this is when jonah recovers his vocation, and he tells the sailors to throw him into the sea.  This is when the big fish swallows him up.  In the whale, Jonah amazingly prays a song of praise, made up completely of lines from the psalms.  (Maybe a whale, maybe not, depending on your view of many things.)  Waking up is seeing our students for who they are and claiming them.  Mr. Peterson connects this with the need for askesis.  (This is a Greek word for athletic training - cf. Wikipedia - he uses because the idea of ascetic has some really strange stuff attached to it now.)

Askesis involves finding a mentor (or mentors) and devoting yourself to training.  That maintenance is the big thing.  Regular participation.  Peterson found he couldn't get that from the institution or the congregation, and started reading Doesteyevsky, and other classics.  He makes quite a point out of not doing what the congregation asks for, as that leads to golden calf moments.  (Not good, wrath of God situations.)  But the worst barrier by far is ego.  When I abstract teaching too much, I'm forgetting the students.  If I'm thinking about teaching in the abstract, or even focusing on what I'm doing instead of what they are learning, it gets in the way of teaching.  Sometimes terminally.

For me, askesis is the colleagues who challenge me, the reading I do, and, maybe, this online community.  Dave and Esther seriously help me monitor and think through my practice.  Reading Debbie Miller and Ellin Oliver Keene is seriously challenging.  Trying to find teaching connections to other reading (hence this post!) is also helpful.  And blogging and twitter... well, I think it helps against the incredibly strong pull of isolationaism that teachers face.  It is so easy, on most days, to close our classroom door.  Especially the weirder I get teaching.  At the end of the semester, when students are working mostly independently, and I'm sitting and watching them... I get some looks.

Peterson describes that a practical askesis has a rule, or a structure, that is supported by disciplines.  For him, his rule was a cycle:  worship with the congregation -> praying the psalms -> individual recollected prayer -> praying the psalms -> worship with the community.  It's a real question to me what a practical rule or structure would be for teachers.  The workshop structure is central to my class; is it some variation of that?  For him, the pslams are where you learn how to pray.  Where do we learn how to teach?  Maybe that's our teaching frameworks?  So for me the rule could be:

The rule would be supported by teaching disciplines.  What do you see as the disciplines which improve teaching practice? 

The third part of the Jonah story is Nineveh.  He finally goes to meet the people he's supposed to reach.  He's obedient.  He walks a day's march into the city, not even just shouting the message in through the gate.  Peterson connects this with getting to really know his congregation for whom they are, and differentiating his work with them based on where they are.  He describes being pulled into being either a messiah or a program director.  The messiah is looking to help, and never allows struggle.  The program director recognizes strengths, and plugs people into service whether that would be good for them or not.  It was easy to recognize both of these traits in my own teaching.  The solution to this is what he calls eschatology.  This is usually used to refer to study of the end times, but Peterson uses it to talk about being goal-oriented.  As teachers, if we keep the learning goals in mind and combine it with authentic assessment, we'll see our real congregation and stay as their teacher.

The final part is the unpredictable plant.  In what is the least known part of the story, Jonah has finished his message and goes up to a hill above the city.  To watch it be destroyed.  God even grows him a shade plant for his comfort.  When God takes away the shade plant, Jonah loses it.  Argues with God, complains repeatedly, and whines that he is angry enough to die.  Jonah rejects the mess:  the city has repented, Jonah doesn't accept this and doesn't know what to do.  Peterson points out that pastoring is essentially a creative activity, and that requires mess.  Easy to apply that to teaching.  "In any creative enterprise there are risks, mistakes, false starts, failures, frustrations, embarassments, but out of this mess - when we stay with it long enough, enter it deeply enough - there slowly emerges love or beauty or peace."  It helps in this to have a simple statement of what it is that you are called to do.  For me, it's something like "Support students to learn how to learn."  Math or math teaching is the context, but not the mission.

The Jonah story ends open to interpretation - the Inception of its day.  Will Jonah see his mistake or die from his anger in rejection of his calling?  I'd love to imagine him descending back into the city, meeting his class where they are.

PS> I'm always interested in comments, but this really felt like going out on a limb for me, and I would especially be interested in if it was worth your reading or not and why.

Stigler on training teachers

Michael Goldenberg over at Rational Math Ed pointed this out on his blog and it's worth spreading out.

See this thread at the Math Forum for a transcript. Dr. Stigler is the co-author of The Teaching Gap and The Learning Gap, which distill research from The International Math and Science Study (TIMSS) into fresh and relevant information for any teacher, but especially math and science teachers.

The big points to me:
-Teaching is cultural => really hard to change
-Teaching needs to change <= student achievement data
-Teaching can only change gradually, but it can change, by focusing on teaching instead of teachers, and evaluating based on students' thinking and reasoning.


A relevant quote: "A review of research by Hiebert and Grouws identifies two features of classroom instruction that are associated with students' understanding of mathematics:
- CONNECTIONS: Making mathematical relationships-among concepts, procedures, ideas-explicit in the lesson
- STRUGGLE: Students spend at least some time struggling with important mathematics."

Check it out.