Thursday, March 26, 2015

What Is It Good For?

This is a thinking out loud post. Anyone who can push my thinking, I'd be glad for it, in comments or on Twitter. Names omitted to at least partially protect well meaning folk from my ignorance. I may be more wrong in this post than I have ever been before. (Which would take some doing.)

I was at a conference recently where I got to hear someone speak whom I respect a lot. We have our students read their work, and good things come of it. The subject of the talk was long running research on teacher preparation. This was prescient, as teacher education is being heavily questioned, there are proposals for ridiculous restrictions and regulations, and we have our first competition ever from alternative certification programs.

So how would you show teacher preparation is effective? Maybe compare the learning of the students of teachers who were prepared vs students of teachers who weren't. Oh, there aren't any of those. Many of those, anyway. Also, hard to compare different programs. OK. Evaluate your program. Hmm, but there's even a lot of variation within a school. Well, we could make everyone teach exactly the same. And always the same topics. We'll need detailed lesson plans.

That's what they did. To test effectiveness, then, they kept track of a random sample of students. (It's possible the randomness was simply which students they could track.) Then they tested them on the content and teaching of three topics from their freshman level teacher preparation course. For comparison, they also tested them on one topic not from the course. What would you expect they would find? What would you hope they would find?

They found the teachers did better on the topics that were taught. And how much better was proportional to how much emphasis they put on those topics. This was true for the students as seniors, first year of teaching, second year of teaching and so on.

Even after being tested on the un-taught topic, the teachers realizing they don't know it as well as the others, the next year found the same results. The lack of transfer, it was said, proves the effectiveness of teacher preparation. If they could transfer, all we'd have to do is teach them one thing.

What about the practices? (Which would have been the processes or proficiencies back when this research was started.) Those are taught implicitly. Because you have to do them to get at the content, you know. My experience, and that of my betters at GVSU is that you cannot teach these implicitly, or even as an add on. Front and center and even then good luck.

For the capper, it was stated that the state might only certify teacher education programs that covered all of the common core standards. This was obviously foolish to everyone in the room. There's so many! No one could cover them all! Almost no one seemed struck by the irony that one of our leaders may have just proved that it is necessary.

It was a bitter pill.

It also seemed to go with one of the themes in the Twitter week for me: teacher-proof curriculum or curriculum-proof teachers? If you have a moment, take 5 minutes to listen to Megan Taylor's (@ilovemath11) Incite.


The researchers I'm talking about, and math ed folks in general, are strongly tempted by the teacher-proof curriculum. I know curriculum is important - I was a part of an effort to move a large district to a problem solving approach when they still had a traditional curriculum. (Turns out, teachers do not have time to gather the materials themselves.) But you could hand teachers the lessons from the Teacher's Edition of The Book (stealing from Erdös*) and it won't make as much difference as what Megan is talking about.

I get to teach with David Coffey (@delta_dc) and it is a humbling experience. He's great. But one of his most frequent dictums is that he doesn't want a lot of little Dave Coffeys running around. He's not trying to turn his students into him. The wisdom of this is now something I accept. Teaching is so deeply personal, it just will not work if it is not authentic. (Belief, not research.)

I tried to suss it out with my preservice high school teachers this week.  I have 2*2*14*(55/60) hours with them in a semester. I cannot cover all of their content, nor all of their crucial content, nor all of the most difficult content to teach.

Just like they will not have enough time with their students. What are they to do? What am I to do?

Use the opportunities we do have to teach our learners how to learn on their own. The math is important, the practices are important, but this is most important. My university teaching of teachers is best when it's holistic and I am doing what I want them to do. Math is such an amazing context for doing this; we are truly blessed.

Disclaimer: I have a rather unfortunate bias against a lot of education research as it is. Maybe it's a remnant of having been a mathematician, and the culture of the hard sciences looking slightly askance at social science research. Maybe it's a function of how often education research results are abused. "There's one study that didn't find class size to matter for test scores. Let's pack 50 kids in every class!" Research is vitally important, but teaching is so hard, and there are so many variables, that I just think it's very difficult to extend results. As a consequence, I tend to value story and qualitative research more than quantitative.

I'll still read the research, looking for techniques, ideas, and inspiration. (Read a great bit from Ilana Horn and Sara Campbell just today!) But we can't start thinking it's the answer in itself. (Maybe learning is solvable?) Just because we can make something testable, doesn't mean we should. I think what I want most is integration of researchers with practicioners, working on the problem together. And then conversation with the researchers, not a lecture. Didn't someone prove those don't work?

End rant.

*Erdös: "Even at this early point in his career, Erdős had definite ideas about mathematical elegance. He believed that God, whom he affectionately called the S.F. or Supreme Fascist, had a transfinite book (“transfinite” being a mathematical concept for something larger than infinity) that contained the shortest, most beautiful proof for every conceivable mathematical problem. The highest compliment he could pay to a colleague’s work was to say, 'That’s straight from The Book.' " from the Encyclopedia Brittanica


Saturday, February 14, 2015

Skemp & Fractions


While my desert island article is the one where Brian Cambourne shares the Conditions of Learning, Richard Skemp's Relational Understanding and Instrumental Understanding” (reprinted in Mathematics Teaching in the Middle School, September 2006) is not far behind. And it may be better to discuss with preservice math teachers, since it doesn't require transfer from literacy to math. Despite being a rather difficult read, it never fails to provoke good discussion and deep thinking.

Previously on the blog I have: interviewed a baseball coach/math teacher about relational understanding, recorded student discussions, and a post about the article. So thisis only the fourth post, it's not like I'm obsessed.

I don't give a formal homework assignment too frequently, but still do for this reading as support is helpful. (Assignment.) I also have a workshop for use in class:



After time to work through the questions, Sam led the start of the discussion. She hit the ideas of relational and instrumental, and solicited examples of the contrast for fraction addition and subtraction. But as she noted - it felt like multiplication and division was where the really interesting bits would be. So I split up the groups among multiplication and division and then recorded their quick explanations.


Loved that the key question "3/2 of what?" came up here. I was fascinated by the "sometimes it works, sometimes it won't" idea. That's a real vestige of instrumental understanding, when we are given rules but often not the conditions under which they apply.


We discussed the grid here for what might confuse students, and tried to connect back to context. Students often want to draw a picture for all the quantities, even though there is not 1/6 of a whole here, but they were taking 1/6 of 1/2.


The lack of a picture was good here, and we discussed how relational doesn't mean with pictures. I tend to ask them about pictures to push their understanding because they are more likely to have rules for the numeric than the visual. Although the grid method can become very rule driven, just like the numberline for integers. This discussion was also grounds for discussing the difference between explaining why a method works and justifying that it does work. 


In the last explanation we were getting close on time, but they posed a couple good why questions to which they struggled to good answers. 

One thing about university classes is that it can be hard to get them to ask each other questions as the duck and cover principle is well learned. I try to stress that the discussion is one of our best tools for pushing understanding, and in math ed classes, I try to frame it as teacher training - you need to practice posing questions. Still tough sometimes.

I'm satisfied that they see a difference in the modes of understanding. Fractions are just such good content for this, as math majors' computational fluency is strong, but they can tell there are things they don't get. One of the gratifying parts is how much they want to get it, and take on the goal of getting their students there as well.

Bonus: as they write their next blogposts, we might see some writing on this as well. First one in is from Matt - Instrumental vs Relational.


Friday, February 13, 2015

Nova Now Notes





This is the school. Srsly.


One of the questions posed during Nova Now 15, a state conference focusing on discussions among teachers, was 'should conferences end with a reflection session instead of a keynote?' Any opportunity to encourage teacher writing.


So now I feel obligated.

As with Twitter Math Camp and EdCamp, this is a conference organized on the principle of creating teacher conversation and collaboration. It's hosted at Kent Innovation High, where, frankly, I wish I could send my kids. It's a tech friendly, open design, project based learning school. Kids attend in the morning for core classes (science, math, ELA, social studies) then return to their home schools all over the ISD for the rest of their schedule. The conference starts off with a tour and chance to see the learning happen, and then several students take the option to stay and be part of the conference, even coming back on Saturday. The single most frequently heard comment for me was about the eloquence, maturity and phenomenal perspective of these students. My favorite #KIHway quote from Peyton: "Students have to shift from doing this to make people happy to 'I'm learning how to be creative and productive'."

Even before the conference started, I had a great talk with Laura Chambless. She's the K-7 math/science support for St. Clair region schools. She's got her resources organized in a protopage. (Free start pages that can also do RSS feeds.) She's a big believer in fact fluency and has been trying to find ways for teachers to get at that constructively.

#michED is our statewide Twitter chat. Wednesday evenings, 8pm ET. The first big session was a meet between the east side and westside collaborative groups, Innovation Now and the Bluewater Group, moderated by Rushton Hurley. He did a good job of guiding a discussion, and using that to also make his points. Some big ideas raised:
  • Isolation is the cancer of the teaching profession.
  • Good ed leader question: how many times have you deviated from your plans? Because that's a measure of how many cool moments you've facilitated.
  • We are not good at sharing successes. Why are you a good teacher? "I care about kids." Who doesn't? Share specific successes as individual teachers and as a school. Share them with voters!
  • Gamestorming, co-creation tools.
Derek Braman led a session on student blogging. Mostly ELA teachers - I want to hear how content teachers are using it, too.  My big takeaway was an activity he uses to introduce blogging. Have students write a list of some of their passions. Pick one to write about on paper. Then circulate and leave comments for people on stickies. I'll try that next fall and share how it goes.

'Math: are we doing it wrong?' was led by Rick Jackson, who kept good notes on resources - . Dan Meyer came up, PBL in math with the KIH teachers, SBAR, flipped classroom ... all the good stuff. Infuse Learning was recommended for BYOD formative assessment. Teachers were surprisingly reluctant to discuss, surprisingly given the context, but there were two KIH students who killed it in the conversation. 
  • "We have to still follow state standards, which have nothing to do with learning or critical thinking."
  • "The pioneering spirit is a big part of KIH, shared between teachers and students."
  •  "Students have to shift from doing this to make people happy to 'I'm learning how to be creative and productive'."
Ben Rimes led a session the next morning. (His notes, which include the cards.) Out of all the good stuff going on,  he won me with teases about the world premiere of his Keynotes for Humanity game. The game itself was a great discussion piece - I should think about how to use the Cards Against Humanity structure in math class. (Make your own.) The point about the keynote's role in ed meetings was that we need to think about it's purpose and utility.

Jeff Bush (@bushjms) and Rebecca Wildman (@RebeccaWildman) had a session on student centered classrooms. (Their ) One of the hallmarks of a meeting like this is that being discussion driven, the sessions may not go according to plan. Two of their tasks for us took over the discussion. Find a video that represents you as a teacher. The most fun:
The second task was to make an infographic of a topic. I haven't tried this yet. Rebecca assured us that there are infographics on everything, and that seems to include math topics. Good synthesis assignment. Kate Kling recommends http://piktochart.com for a free tool.

The last session I attended was Jennifer Bond's creative play. (@teambond; her Google site.) What I learned is she has the best toys. The littleBits are very curious. Another good resource is the Imagination Foundation. Best quote: "if you give them open-ended time, you'll have their attention week after week. They don't have time to play." Sad to think about students not having time for real play. Jennifer ran a creative play club for which students applied. Every single form I saw, these elementary students considered themselves creative. By the time we get them in college, few people claim that. What are we doing so wrongly?

My session was on Talking Points.  I'll write about that separately! I had an awesome group of teachers to share them with and discuss them in other disciplines.

Some other neat bits from the conference:

Sunday, February 1, 2015

Good or Bad?

Are you a good teacher or a bad one?

Image by Gillian (?)

STOP! It's a trick question. Loaded.

If you say good, you're arrogant, you don't understand how good teachers are always trying to get better, you're not aware of your faults or you're just blind to the signs.

If you say bad, you buying into some kind of talent argument, ignoring growth mindset principles, falsely humble, don't even know what formative assessment or SBAR means or, worse, you're guilty of one of the big teacher sins like low expectations, no classroom management or ...  I gasp to say it ... LECTURING.

Are some teachers better than others? Of course. I see them at work every day, in my case. Do I want to know how to get better? Of course. I think about it rather obsessively. Is it helpful to classify teachers? Of...

I don't know. I'm open to arguments, but everything I know and has experienced leads me to say no.

Part of the academic life at university is this periodic vetting of our colleagues. Who is good enough to stay?

So we have to functionally label people as good teachers or bad. Since we are so bad at measuring teaching, our main tool for this is student evaluations. This is not the purpose for which student evaluations are meant. But they have pithy comments, and numbers we can average! I can pretend it's data about the quality of teaching!

Tenure is a weird construct, and I understand why it makes people fidgety. Ultimately, my view of it is colored by my brilliant advisor, who saw it as freedom. Freedom to pursue interests instead of quantitative measures, which will produce more innovation in the long run. We want to know, what will people do with their freedom? Which is one of the fundamental problems of freedom.

I'm fortunate, in my view, to be at a university where teaching is the major factor in tenure. Getting a little less so, but it's still the case. There are not many public universities like this. Send us your children, because they are going to get a better education here than almost anywhere.

But it also brings us to these moments of passing judgment. If I'm against classifying good and bad teachers, how do I make my decisions?

Photo by VD Veksler
Is the colleague interested in teaching, working on their teaching, selecting good goals, seeking out professional development that will move them forward? If yes, then I hope they are different than I am, teaching for different goals, working on different teaching problems. Because then, as a faculty, we will be the better for it.

I don't want to judge where I think your path is going. I want to hear you talk about your journey. If you're willing to share it, let's get a coffee.

Te invito. My treat.


Wednesday, January 28, 2015

Shadow Sculpture

We just had a fun, crazy and exhausting day for Super Science Saturday, a yearly K-12 outreach by our outstanding Regional Math and Science Center. 

The project had its start when my fellow traveler Heather Harrington sent me a picture of an awesome (and, later, prize winning) installation piece from Anila Quayyam Agha at ArtPrize.  (Here’s a good article about the installation.) When I saw the the RMSC was focusing on light for Super Science Saturday the bulb went on.


Intersections, by Anila Quayyam Agha

How could  that not inspire some excellent art and math?

Our handout gave a little of the background, and some basic steps to cover.


People really gave it a go. Mostly the kids, although we did get a higher percentage of parents trying than expected. (Those who won't join in - what keeps them back? These are folks who are giving up a Saturday for their kid to do science, so...)

Over and over we tried to emphasize:
  • choose or make a structure (math)
  • predict what the shadow will be
  • test it out with a flashlight or on the overhead (science)
  • compose an image to capture  (art)
Some of the results:




Here's a few of the final images. (You can look at all of them, too.)

We saw a lot of good mathematical and artistic problem solving. People measuring struts they'd need, considering options to construct, reverse engineering things they saw in other polytopes. They sought help when they were stuck, collaborated with neighbors and could verbalize what they were doing. They experimented with shadows, trying to figure out what angles of flashlight or position on the overhead showed the structure they wanted.

It was difficult to get people to predict. The ones who did mostly found it more interesting to do the shadows and spent longer trying to match. For some reason, this was the highest level of parental involvement.  The kids who were comparing shadows to a prediction drawing often had the parents jump in and want to see. Some people were interested enough in the connection with their predictions to trace the final

Some of what people seemed to learn about light and shadow is to start thinking of it as a projection from a single source, instead of a silhouette.  People with right rectangular prisms or right pyramids were surprised at what you could see. Kids learned about the moving of shadows with the light source, and the effects of being closer or farther from the source.

Anna's gif experiment
Artistically people divided up between looking for symmetry, being representational (lots of houses) or crazy abstract. (My favorite to see, though when building I always go for symmetry.) There was a lot of pride of ownership - the vast majority of kids wanted to take their structure with them, and they were almost always excited to bring them back to show Heather, myself or a volunteer for the light work. Heather thought it was important for people to have a chance to enter into the art, to interact with the shadows, and that's why we did the overhead projector. It was a huge hit, and there was often a line of people waiting to see theirs on the big screen. Heather's other cool idea was to capture motion, as the changing shadows are visually interesting. We experimented with gif capture and Vines, but nothing was efficient enough to be able to handle hundreds of people.

2100 straws and 1000 pipe cleaners later, we were exhausted but energized. 

We found the small aperture bar straws to be best, and picked up ours at Gordon Food Service, 500 to a box. About half a length was a good basic distance for the projector and flashlights - a whole 8" straw scale was too large. The flashlights were just $1 9 LED cheapos, but they were bright enough for strong shadows in an only half unlit room. The bulk pipe cleaner pack was the best deal, all the same color so as to avoid "I wants",  cut into 5 or 6 pieces was enough to hold two straws together. If a pipe cleaner was loose in a straw, we'd hook the end of the pipe clearner to get a better grip. Originally I was going to cut the straws into Zome like proportions, but it was unnecessary as people mostly didn't make the classic solids. Flickr seemed to be the most efficient photosharing tool so that people could get the photos afterward. Take the picture from within the Flickr app, it uploads automatically, and you can set it to a public album with one touch.

People had the option to leave their sculpture and thank goodness some did because we recycled/cannibalized everything.  

When people had the choice of tons of cool science activities, what kept us so busy? Some of those had lasersI think it was the art aspect. In the other activities, though cool, you were doing what someone told you to do. What we had wasn't for everyone. The option to choose a premade sculpture was good to have (we had about 10 premade sculptures), as it allowed people to still engage in the shadow prediction and art. 

Great day, good to collaborate with an artist, and a lot of creative mathematics. Who knows what mathematics dwells in the hearts of men?


Monday, January 19, 2015

Nonviolent Teaching

Martin Luther King, Jr. Day. Seems harder to pretend things are all better this year.

Some of my favorite recent posts around the web that fit with this #MLK day...

The Principles of Nonviolence, gathered at the King Center
  1. PRINCIPLE ONE: Nonviolence is a way of life for courageous people. It is active nonviolent resistance to evil. It is aggressive spiritually, mentally and emotionally. 
  2. PRINCIPLE TWO: Nonviolence seeks to win friendship and understanding. The end result of nonviolence is redemption and reconciliation. The purpose of nonviolence is the creation of the Beloved Community.
  3. PRINCIPLE THREE: Nonviolence seeks to defeat injustice not people. Nonviolence recognizes that evildoers are also victims and are not evil people. The nonviolent resister seeks to defeat evil not people.
  4. PRINCIPLE FOUR: Nonviolence holds that suffering can educate and transform. Nonviolence accepts suffering without retaliation. Unearned suffering is redemptive and has tremendous educational and transforming possibilities. 
  5. PRINCIPLE FIVE: Nonviolence chooses love instead of hate. Nonviolence resists violence of the spirit as well as the body. Nonviolent love is spontaneous, unmotivated, unselfish and creative.  
  6. PRINCIPLE SIX: Nonviolence believes that the universe is on the side of justice. The nonviolent resister has deep faith that justice will eventually win.
I see these as connected to how I want to teach. My nonviolence training came from Sr. Liz Walters, IHM, when I was an undergrad at Michigan State. (Short bio of Liz from an award she received.) At that age, I was (so typically) filled with idea of competition and winning masquerading as justice. When I think back on the transition to peacefully campaigning for justice, and that the means matter as much or more than the ends, the training from Liz was of profound impact. Found this post with some current organizations that do nonviolence training.

So how is teaching related to these principles?

Active resistance to evil. Nonviolence is in no way nonaction. Instead, it is active pursuit of justice. Teaching is often inherently nonviolent because it is a career built on constructing relationships. Not that teaching automatically moves in this direction, because we can bring confrontational relationship strategies to the job.  Most teachers are capable of careers that earn more or are essentially easier. Even teachers that leave the field I think have sometimes just finished what they had to give. Some vocations are for a season and some are for a lifetime. (When we lose the lifetime teachers because of school injustices, though, this is a serious loss. Personally and societally.)

Redemption may be strange language for education, but when we think about caring for all our students, it is going to include those wronged by the system or suffering from circumstance. When we work to create a safe learning space, it is naturally redemptive work. When we get to really know our students, it is constructive.

Defeat injustice, not people. This can be difficult, as students act out routines and responses to which which they have been subjected. But the classroom culture building to which I respond does an excellent job of separating the student from behavior.

How does suffering relate? One of the things I try to share with my preservice teachers is to be ready for this, what I often call the heartbreak of teaching.  By caring for our students, we are volunteering to share their burdens. There are going to be students that have difficult, messy and painful lives, and we are signing up to walk part of the way with them. We are opposing the dehumanizing forces in our society that want to use them up or pass them over or sell their share for a profit.

Doesn't "spontaneous, unmotivated, unselfish and creative" describe a lot of the teachers you admire?

I also believe that teaching is inherently hopeful. We are siding with the universe on the side of justice, or our higher power, or whatever gives you faith.

So on this Martin Luther King, Jr. Day, I've taken some time to pray for teachers, pray for their students and pray for my students. I'm going to look for opportunities to stay in the struggle, and support those resisting injustice.  And know that it isn't just for this day.

Friday, January 16, 2015

Moving Negative

Preamble
Wow - I missed a blogging month. And I had so much to say about it... we did Math in Your Feet, some excellent student projects, lots of new lessons, assessment thoughts... So I was thinking about resolving to blog less big, more frequently. Then Sue Van Hattum blogs her #edustory, and I think challenges me and a few others...

I've been thinking about doing more microblogging - and maybe I'll try it. I get stopped by "nobody wants to read that" which makes me forget that I'm writing out my own understanding, so that shouldn't matter. I'm not an author who's trying to please a fan base, I'm a teacher trying to work my way to understanding.

Actual Post
But what's really on my mind is embodied cognition. Last summer I got to try a session with Malke Rosenfeld and Christopher Danielson at TMC14 on Embodied Cognition.  (My account.) Outside of their session, Malke worked with Michael Pershan and Max Ray and others on doing a life size complex plane and number line. I wasn't even a part of that but it got me wondering. Malke and Max have continued to work on the idea, and there is an MTMS article in the works. They were willing to share their writing on that so I could try things in class.

The course is for preservice middle school teachers. I start off with negative numbers (and probably end with negative numbers too if you know what I mean. Where's my Dangerfield font?) because it is a good setting for talking about operations as story and action (exposing them to the CGI structures), and rolling in some content from our preservice elementary classes on fact families and operation strategies, models and landscapes of learning.


For the Cognitively Guided Instruction stories we watched the Kindergartener uses Direct Modeling video from the Heinemann site. Then sorted these stories.  Usually students sort them by operation needed to solve them, but the video was a great focus, because they really did a great job discerning actions. The idea is that young students encountering stories before direct operation instruction classify stories by what's happening in them. Are amounts increasing or decreasing? Are we comparing separate amounts or looking at static groups of different types? They then model and invent strategies that fit the contexts. For example, students who are taught addition first but then have to use it in a decreasing context often have difficulty solving the story problems. (James gave away 3 pencils but still had 5 left. With how many did he start?)

We followed that by brainstorming contexts  for negative numbers: money, debt, bills, weights/balloons, depth/sea level, golf, football yardage, temperature… the usual suspects but a good variety. When we tried writing stories for them, it was challenging to ask the questions in a way that the answer was negative. I nudged them towards the idea that one of the strongest contexts for negative numbers is when the numbers are describing change rather than direct quantities.


Some of the questions from Max and Malke:
  • where she was compared to where she started
  • tell us how far away from someone they were, and in what direction
  • a plan for how everyone could, in a coordinated way, get from their home position to their spot that was the same distance away but in the opposite direction
  • identify if there was someone who was the same distance away from Shane, but in the opposite direction
  • give students a target result and ask them to come up with a series of moves that resulted in the given displacement



We just used stickies to make the numberline. I marked a square as 0 and asked the PSTs to place stickies for 5, 10, 15, -5, 10, -15. First discussion: are we using the squares or the edges in between? (Squares, because of placement of zero.)

Students moved to various numbers on the line, called out by the teacher. Discussion: left right, direction big part of idea of negative. Distance talk, however, is naturally positive.

Then we started modeling change. If a student C walks from A to B: how far did they go? (Positive.) What is the change in their position? (Signed.) We did several iterations of moving in both directions. Discussion: PSTs started noticing how zero figures into the strategies. Frequently found change by b to 0, 0 to c.

PSTs challenged in groups to come up with a question that could be modeled on the line.
  • 1st group: stood at 8, 5 and -3. Class brought up person at 8 could be change: how big a change when walking from -3 to 5. Another said -3 could be how big a change from 8 to 5? Discussion: no way for 5 to be the change in that situation. 
  • Next group of 2 stood at 14 and -7. Their story was: Samantha climbed a 14 foot hill and jumped in the water, sinking to 7 feet below the surface. How far did she dive? Someone brought up 14, and dove 21 feet, where is she? (“Dead”)
Shared Max and Malke’s challenge to come up with a combination that resulted in a net difference. Students proposed 3, 4 and 5 move challenges to get the goal, took the challenge to mean literally standing on the line. Brought up how the challenge could lead to better strategies than counting one space at a time.

End of day 1, informal assessment: was this worth their time? All 4s and 5s on a 0 to 5 finger scale.

Homework: asked them to read one or more of the following: 
Day 2 we set the number line back up and started thinking about how to explicitly model addition and subtraction. The first group shared the idea to face positive: if addition of positive move forward, if subtraction of positive turn back, if negative turn... I raised my opposition to things that just feel like more rules and not connected to ideas. In discussion, the idea was raised to face neutral by default, turn positive to add and negative to sutract, walk forward for positive, walk backward for negative. I brought up how the class deciding this idea for themselves is probably the most valuable part.

Once we had decided on the representation, we got to some exciting stuff. We had students do a walk on the number line. (Start, turn, walk, stop) and then we wrote it down. 5 + -4 = 1. We discussed how non-threatening it was to walk for something like this, and it was a place where you were really free to experiment. When someone brought up different options, 7 - 13 or 7 + -13, we talked about how you can tell and was there really a difference. Then we hit on the idea that you could walk out equivalences. Are these two things equal? Let's try them! Someone had the idea to try commutativity. What would associativity look like? (Hard to walk.) There was an interesting side effect: some common student errors are impossible to walk. They just don't make sense in embodied cognition land



I brought a game idea, of course. Take a deck of cards, remove everything but A to 7, red is negative and black is positive.  Shuffle, deal two decks, like for War. Both players start at zero, facing each other. Flip the top card of your deck. Player with the smaller magnitude number goes first. You can add your number to yourself or subtract it from your opponent. The first player's move decides whether they are going positive or negative, and the 2nd player is going the other way. The goal is to get off the number line. (Ours went to +15 and -15.)

Sample turn:
  • player Positive is at 2 and flips red 6. Negative is at -3 and flips red 2. 
  • Smaller magnitude goes first, so Negative adds -2 to their position. 
  • Faces positive and walks back two squares to -5. Positive player doesn't want to add -6, so makes Negative subtract. Negative faces negative, then walks backward 6 squares to 1.
In the picture there are two different games going on. Other students tried the game or modeling stories with a chip model. People liked the game pretty well with the numberline, and got them a lot of practice thinking about adding and subtracting positive and negative, as well as use on the numberline.

Day 3 involved no embodied cognition. We discussed fact families and addition and subtraction strategies, and then discussed how to show a variety of strategies for integer addition and subtraction in symbolic records and in number lines. The number lines really showed the benefit of the previous classes' movement as they felt the directions really made sense.







All in all, great start to the year. I've only become more convinced of the need for more opportunities to embody mathematics, and the value of the intuition that this helps build through experience. And, of course, I'm interested in your stories about this, or ideas for what else you might have tried.

EDIT/postscript: several of the preservice teachers are blogging about this.
  • Sam - clearer explanation of the game.
  • Brittany - thinking about the human number line in terms of variety
  • Dakoda - trying it out in her tutoring
  • Kevin - finding some online number line games