Arithmetic of Complex Numbers Placemat Activity - Algebra 2 + Complex Instruction (CI)

Just because you have an all-groupwork and all-Complex Instruction (CI) format doesn't mean you don't need practice activities too.

Our Algebra 2 kids were getting the concepts of complex numbers and complex conjugates, but were still kinda shaky in terms of fluency in working with them.

Based on ideas I stole borrowed a long time ago from the fabulous Kate Nowak (@k8nowak, http://function-of-time.blogspot.com) and the equally fabulous Rachel Kernodle (@rdkpickle, http://sonatamathematique.wordpress.com ), I proposed a placemat activity to my ever-game Algebra 2 teaching team and they dove right in.

Set-Up
We have typical CI four-person table teams set up in each of our rooms, with each person assigned a specific role based on where they're seated at the table. Our roles are Facilitator, Resource Manager, Recorder/Reporter, and Team Captain, although of course, your mileage may vary. Each role has specific tasks they are expected to perform; for example, only the Resource Manager may call the teacher over for a group check-in or a group question (in our program, teachers only accept and answer group questions).

Each table was given:

• two, double-sided "placemat" sheets for doing work in the center of the table
• a set of problem cards (there are four sets, one for each round of play; to simplify clean-up and organization, I printed each round of cards single-sided on a different color of paper, one set per table group. I've got 7 tables in my room, so I made seven sets of cards. I laminated them and clipped them together, but hey, that's just me)
• the sum to which all four answers for any given round should add up

The sum for each round was written on the whiteboard, though it could have been projected via document camera or Keynote/Powerpoint slide.

Objectives
We had mathematical objectives for the activity as well as CI or norms-based, group work objectives. My students in particular needed reinforcement in group work norms and collaboration. Our objectives were:

     Math Objectives

• achieve greater fluency in the arithmetic of complex numbers (including the distributive property)
• deepen understanding of and fluency with the powers of i
• deepen understanding of and fluency with complex conjugates

     Group Work Objectives

• work in the middle of the table
•  same problem, same time (no one moves on until everyone moves on)
• using table group members as resources

The next time I run this activity, I will definitely give a Participation Quiz because the group work norms are so beautifully reinforced in this activity.

How We Ran It
Recorder/Reporter writes the sum in the central oval of the first side of the placemat. Each group member gets a problem card for round 1 (problem a) and works his or her problem on his or her quadrant of the placemat.

When everybody is finished with their problem, the Facilitator facilitates the addition of all four answers. If they add up to the given sum for that round, the Resource Manager calls the teacher over for a "checkpoint" and the next set of cards for the subsequent round of work.

If their answers don't add up to the given sum, they need to work together through everybody's work on the placemat to diagnose what went wrong and where, as well as how to fix it. Then when they've fixed it, they call the teacher over for a checkpoint and the next set of cards for the subsequent round.

Group Work Benefits — Reinforcing Norms
For my classes, the greatest benefits of this activity came from the fact that it forced students to work in the middle of the table, to use each other as resources, and to talk mathematics. Getting kids to work in the middle of the table is the hardest part of CI, in my view, because it goes against the grain of most of their in-school conditioning. The placemat format makes it nearly impossible NOT to work in the middle of the table. And once they're doing that, it seemed like everything else ran pretty smoothly.

I especially liked the fact that this activity created a context in which students experienced an intellectual need for the using the rules of arithmetic for complex numbers and for the powers of i. It was situationally motivated, but extremely targeted.

Sums for Each Round 
The sums for each round are as follows (if you find an error, please speak up):

• Round 1 (problem a):   26-73i
• Round 2 (problem b):  0
• Round 3 (problem c):  165
• Round 4 (problem d):  2 – 48i

PDF Files for the activity
These are available also on the Math Teacher Wiki on the Algebra 2 page. If you haven't visited the Math Teacher Wiki, you don't know what you're missing.

• Complex Numbers Arithmetic – Problem Cards
• Complex Numbers Arithmetic – Placemat Master

Collaboration Literacy Part 2 — DRAFT Rubric: essential skills for mathematical learning groups

I have said this before: middle schoolers are extremely concrete thinkers. This is why I find it so helpful to have a clear and concrete rubric I can use to help them to understand assessment of their work as specifically as possible. I'm reasonably happy with the rubric I've revised over the years for problem-solving, as it seems to help students diagnose and understand what went wrong in their individual work and where they need to head. But I've realized I also needed a new rubric — one for what I've been calling "collaboration literacy" in this blog. My students need help naming and understanding the various component skills that make up being a healthy and valuable collaborator.

My draft of this rubric for collaboration, which is grounded in restorative practices, can be found on the MS Math Teacher's wiki. I would very much value your input and feedback on this tool and its ideas.

I don't want to spend a lot of time talking about how and why Complex Instruction does not work for me. Suffice it to say that the rigid assignment of individual roles is a deal breaker. If CI works for you, please accept that I am happy that you have something that works well for you in your teaching practice.

This rubric incorporates a lot of great ideas from a lot of sources I admire deeply, including the restorative practices people everywhere, Dr. Fred Joseph Orr, Max Ray and The Math Forum, Malcolm Swan, Judy Kysh/CPM, Brian R. Lawler, Dan Pink's book Drive, Sam J. Shah, Kate Nowak, Jason Buell, Megan Hayes-Golding, Ashli Black, Grace A. Chen, Breedeen Murray, Avery Pickford, "Sophie Germain," and yes, also the Complex Instruction folks. I hope it is worthy of all that they have taught me.

And this is why I teach...

It was another crappy Friday in an arithmetic series of crappy Fridays that were running together and threatening to define the limit of my patience for fall trimester as x approaches a mid-sized number that is nowhere near infinity. So I have no idea what possessed me to wake up even earlier than usual to pull together an extra day's practice activity for my right-after-lunch class of rumpled and discouraged algebra students — the ones who believe to their core that California's Algebra 1 requirement is God's own punishment for unremembered karmic crimes they must have committed in previous lifetimes.

But I did it.

The topic was solving and graphing compound inequalities — a skill set that must be mastered in order to have any hope of making sense of and mastering the next topic traditional algebra curricula force-feed to our students: the dreaded topic of absolute value inequalities.

There's really nothing I can say to convince a roomful of skeptical eighth graders that compound inequalities will prove not only useful in business planning (which, after all, is simply algebra writ large across the canvas of the economy) but also amusing and possibly even interesting little puzzles to delight the mind.

To this group of students, they're simply another hoop to be jumped through.

So something in me understood that I needed to reframe the task for them, and to do so using Dan Pink's ideas about intrinsic motivation from his book Drive.

Nothing unlocks the eighth grade mind like an authentic offer of autonomy. As I explained recently to a room of educators at a mindfulness meditation training, middle school students suffer emotionally as much as adults, but they have comparatively little autonomy. A little well-targeted compassion about this can carry you for miles with them, though I usually forget this in the heat of working with them.

For this reason, I like to save practice structures such as Kate Nowak's Solve—Crumple—Toss for a moment when they are desperately needed. I have learned to withhold my Tiny Tykes basketball hoop for moments like this, when students need a little burst of wonder in the math classroom. And so even though I was tired and very crabby about the ever-increasing darkness over these mornings, I pushed myself to pull together a graduated, differentiated set of "solve and graph" practice problems to get this group of students over the hump of their own resistance and into the flow experience of practicing computation and analysis.

And oh, was it worth it, in the end.

The boys who are my most discouraged and resistant learners came alive when they understood that a little athletic silliness was to be their reward for persevering through something they considered too boring to give in to. They suddenly came alive with cries of, "Dr. X— watch this shot!" from halfway across the room. One boy who can rarely be convinced to do the minimum amount of classwork completed every problem I provided, then started tutoring other students in how to graph the solution sets and perform a proper crumpled-paper jump shot.

The girls in the class got into it too, but they seemed more excited about the possibility of using my self-inking date stamp to stamp their score sheets. So I gladly handed over the date stamp to whoever wanted to stamp their own successfully solved and graphed inequalities.

I was far more interested in reviewing their mathematics with them. One of the things I love best about practice structures like this one is that they give me an excuse to engage one on one with discouraged students under a time crunch pressure that adds a different dimension to their motivation. Suddenly they not only want to understand what they have done, but they want to understand it quickly, dammit, so they can move on to another problem, another solution, another graph, another bonus point.

Ultimately, Solve—Crumple—Toss becomes an occasion for conceptual breakthroughs in understanding.

I can't tell you why this happens. I can only tell you that it does happen — often. It makes me feel lighter, more buoyant about teaching them algebra. And it makes them feel happier too.

I wanted to write this down so I could capture it and remember this for a few weeks from now, when it stays darker even longer in the mornings and when I feel crappier and crabbier and more forgetful.

Radio Silence Does Not Mean Nothing Is Happening...

Wow, did I ever fall off the radar.

Plop. That "splat" you might have heard was me, falling off the blogging radar.

But I'm back, baby.

Last night I had the most wonderful dinner with @btwnthenumbers and @woutgeo and @mythagon, who was in town for a conference/collaborative meeting, and I tell you, it pretty much restored my faith in teaching, in mathematics, and probably in all of humanity.

I have been working at a near-frantic pace these last five weeks, prepping, teaching, grading, not grading, having parent conferences, having meetings with parents and the principal, having meetings with parents and principal and superintendent, going to IEP meetings, collaborating with my department members to write goals that will help us to align our curriculum with the Common Core, and generally dealing with all those things that go haywire as soon as you start to nail down some satisfying, finite part of your teaching.

In other words, just like you, life has been kicking my ass.

But between last night and this morning's drive to work something shifted. Something sane and healthy intervened.

That something was my connection with the Twitter- blogo-sphere.

Whenever I'm feeling exhausted and run over with skid marks across my face and body, connection with my tweeps -- any connection -- seems to be the best medicine. I don't know why this is true; I only know that it is so. Remembering this makes me think of a quote I have from von Neumann hanging in the ring of inspiring quotes that encircles my classroom: "In mathematics, you don't understand things; you just get used to them." Some days that's how I feel about things in my classroom or in my school or in my life.

I only know that five or ten minutes of venting to my tweeps about an impossible situation -- even when @woutgeo is only half-listening because (a) the Giants are sucking pretty hard against the Cardinals and because (b) my venting is both predictable and boring -- it helped just to have reconnected with the connection. In Jakobsonian structuralist linguistics, this kind of communicative connection is known as a "phatic utterance" (look it up, Riemann, I have to look up all of your crap).

By this morning, I was feeling reasonably happy driving to work for a 7:30 a.m. meeting. I was not totally thrilled about the hour or having to buy gas at that hour or the price of gas for that matter, but I felt pretty great about car-dancing in the dark to Ace of Base's "The Sign" and remembering car-dancing at #TMC12 with @mgolding and @samjshah and @jreulbach and @ bowmanimal on the way to do Exeter problem sets. And I felt great when @rdkpickle's sweet soprano voice was joined by @SweenWSweens and @jreulbach singing "Tweet Me Maybe." And I even laughed when the theme from Sesame Street came on. iPod's "shuffle" feature has a somewhat perverse sense of humor.

OK, and one other thing I have learned is that my dog always knows when it's time for me to end a blog post. Just now he jumped up on my lap and pounded the laptop keyboard with his giant panda bear paw:

34ycvzn

So that's my cue to wrap this up.

I just want to say, if you are feeling alone or frustrated or exasperated and you are reading this, then for the sake of everything we hold dear, please reach out to someone else who is of like mind. "It's hard to teach right... in isolaaaaaaaaation.... So here's some PD.... just like vacation!"

Tweet me maybe, tweeps. Over and out for now.

SOLVE - CRUMPLE - TOSS in Algebra 1: hommage à Kate Nowak

Kate Nowak creates some of the most innovative and engaging practice activities anywhere -- especially for those skill/concept areas that are more like scales and arpeggios than like discovery/inquiry lessons. Some skills, like basic math facts, simply need to be practiced. This is true not because students need to be worn down but rather because it takes the mind and body time and first-hand experience to process these as matters of technique. It takes time to get used to the new realities they represent.

Nowhere is this more true than in tinkering with the multiple different forms and components of linear equations in Algebra 1. No sooner have students gotten the hang of finding the intercepts of a line than they're asked to find the slope. They figure out how to find the slope and the y-intercept, and they're given the slope and a non-intercept point. They figure out how to crawl toward slope-intercept form, but fall on their faces when asked to convert to standard form. Standard form, point-slope form, slope-intercept form, two points and no slope, it's a lot of abstraction to juggle. Mastery is part vocabulary work, part detective work, part scales and arpeggios, and part alchemy of different forms. It's a lot to take in.

Enter Kate's Solve - Crumple - Toss activity. I have loved this practice structure since the day I first read about it, but I have struggled with the fact that the most engaging part of the activity destroys the paper trail/evidence. This was less important with high school students, but it is really important with middle schoolers, I find, because they are so much more literal.

For today's linear equation-palooza in class, I created a basic "score sheet" for each student and I numbered each of the quarter sheets on which I glued blocks of problems (4-6 problems per mini-sheet). I also differentiated them from "Basic" level (Basic-1 through -4, Level 2-1 through -4, etc), so that students could choose their own levels. Students were also invited to work in pairs or groups of three because I find it encourages mathematical language use and increases risk-taking. It also seems to be more fun.

After the "Solve" part of the activity, students brought their solved mini-sheets to me to be checked. If they completed the problems correctly, they got a stamp on their score sheet and proceeded to the back of the classroom where I'd set up the Tiny Tykes basketball hoop over the recycling bin. There they completed the "Crumple" and "Toss" stages, awarding themselves a bonus point on the honor system if they made the shot. Then they returned to the buffet table of problems and chose a new mini-sheet.

Because my middle school students like to bank extra credit points toward a test wherever they can, I like to attach these to practice activities such as this one or Dan Meyer's math basketball. Being more literal and concrete than high school students, middle schoolers seem to find great comfort in the idea that they can earn extra credit points ahead of time in case they implode on a quiz or test. What they don't seem to realize -- or maybe they do realize and they just aren't bothered by it -- is that if they participate in the process, they win no matter what. Either they strengthen their skill/concept muscles and perform better and more confidently on the test; or they feel more confident and less pressured because they have banked a few extra-credit points for a rainy day; or both.

It was fun to hear my previously less-engaged students infused with a rush of sudden, unanticipated motivation to tease apart a tangled ball of yarn they have previously been unmotivated -- or uncurious -- to unravel. And something about the arbitrary time pressure of trying to complete as many problem sheets as possible in a short period was also fun for them. I'm feeling a little ambivalent about not having found the secret ingredient of intrinsic motivation in this required blob of material. But I am grateful that, once again, an unexpected game structure generated what the late Gillian Hatch called "an unreasonable amount of practice."

The last word goes to the one student who put it best: "The crumpling is definitely the most satisfying part."

Favorite Tweets #13 - gathered on behalf of Sam Shah

For those of you who follow Sam's favorite tweets, here are some of the highlights since June 15th, when Sam first nominated me to hold the fort while he is off developing professionally.

Sam will, of course, republish these on his own site but these should tide you over until his return.

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@sarcasymptote - My fellow teachers call me the copy machine whisperer, but really I think I'm just the only one who knows how to read.

@dandersod - @sarcasymptote and hopefully the only one who can read twitter, haha.

@sarcasymptote - @dandersod you've overestimated my tact. While fixing it, I told everyone, "all you have to do is follow the goddamn instructions."

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@sarcasymptote - @fnoschese @torquedu @calcdave @physicscarp @gotphysics there are tons of videos of eastern europeans doing stupid shit that shows physics

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@samjshah - @cheesemonkeysf would you like to create the next favorite tweets? I'm going to be busy for the next few weeks and won't be crazy tweeting.

@k8nowak - @samjshah I 2nd the nomination of @cheesemonkeysf to make the next Favorite Tweets.

@cheesemonkeysf - @samjshah You mean because I am totally wasting time instead of working on my SBG gala post for @lmhenry9 who is slaving away?

Sure. ;-)

@cheesemonkeysf - @k8nowak @samjshah  OK, but you realize there's going to be some insane stuff in there.

@k8nowak - @cheesemonkeysf @samjshah Natch. Kind of counting on it actually.

@calcdave - @cheesemonkeysf @k8nowak @samjshah  The insane stuff is my favorite! #favourite

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@samjshah - @Mythagon can we also make anything vegetarian from t.co/C4T53i3 ? and eat t.co/kxjsDnd again? YES! we will do these things!

@cheesemonkeysf - @samjshah @mythagon Now I'm sad. There's TONS of great vegetarian recips on smittenkitchen.com .

@samjshah - @cheesemonkeysf @mythagon why don't you just come on by utah

@cheesemonkeysf - @samjshah @mythagon Do they allow drop-ins at PCMI? Or would I just be your chef-slave? ;-)

@Mythagon - @cheesemonkeysf @samjshah I like the idea of a chef-slave. I'd be willing to hide you in my room.

@samjshah - @Mythagon @cheesemonkeysf  awesome... i think this is going to work out just fine

@park_star - @Mythagon @cheesemonkeysf @samjshah  why to to Utah and be a chef-slave when you can go to SK and be a chef-slave? Real novelty is a SK vacay

@samjshah - @Mythagon @cheesemonkeysf don't listen to @park_star! you need a passport to go there, while you just need a unicorn ponycorn to go to UT

@park_star - @samjshah @mythagon @cheesemonkeysf lies! Unicorns and their riders are admitted w/out passports. 

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@sophgermain - Post coming later. Lots of feelings.

@park_star - @sophgermain feelings? Like you need a hug? Good thing you're moving closer to @samjshah

@samjshah - @park_star @sophgermain  oh, I am weird about hugs. so ... maybe.

@k8nowak - @samjshah @park_star @sophgermain  He IS weird about hugs. You have to throw yourself at him. And have inertia. Yall are probably too skinny.

@cheesemonkeysf - Dear @k8nowak, Thanks for the tip about the need to take a running start if you want to hug @samjshah /cc @park_star @sophgermain

@k8nowak - @cheesemonkeysf don't underestimate the element of surprise, too.

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@sarcasymptote - @samjshah @sophgermain  I like the idea of that being a euphemism. "continuous everywhere but differentiable nowhere IF YOU KNOW WHAT I MEAN"

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@sarcasymptote - Holy shit, these fucking copiers are fucking pissing me the hell off.

@fnoschese - @sarcasymptote Don't hold back, Greg. Tell us how you really feel about those copiers.

@sarcasymptote - @fnoschese @btwnthenumbers @cheesemonkeysf I did unleash a tirade that included "holyf***************************tch" in the copy room, + 100 more.

@Mythagon - @sarcasymptote @fnoschese @btwnthenumbers @cheesemonkeysf have you tried threatening it with glitter?

@fnoschese - @Mythagon @sarcasymptote @btwnthenumbers @cheesemonkeysf Have you tried your Sonic Rainboom yet? bit.ly/jK8D9y  cc: @samjshah

@samjshah - @fnoschese @Mythagon @sarcasymptote @btwnthenumbers @cheesemonkeysf that's a pretty fantastic superpower

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@ddmeyer - The Two Lies of Teaching: If I say it, they will learn it. If I don't say it, they won't learn it.  bit.ly/iOLAKs

@k8nowak - @ddmeyer re 2 lies of teaching STOP STEALING MY VIRTUAL CONFERENCE POST. j/k but seriously. weird.

@sophgermain - @k8nowak you're already thinking about your virtual confgerence post? damn.

@k8nowak - @sophgermain yeah it has sparkly rainbows and everything. @sarcasymptote and @ThinkThankThunk are gonna eat it up.

@cheesemonkeysf - @k8nowak @sophgermain @sarcasymptote @thinkthankthunk Oh please. Enough with the sparkly rainbows. Like they need any more encouragement.

@k8nowak - @cheesemonkeysf @sophgermain @sarcasymptote @thinkthankthunk too late. the sparkly rainbow IS IN.

@sarcasymptote - @k8nowak - @cheesemonkeysf @sophgermain @thinkthankthunk NICE! I haven't thought much about where I'm going with mine yet.

@park_star - @cheesemonkeysf @k8nowak @sophgermain @sarcasymptote @thinkthankthunk  UNICORNS!!!

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@sarcasymptote - @park_star @cheesemonkeysf in regards to eating habits? I was always really confused by the pyramid, but I GET THE PLATE NOW.

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(read from bottom up)

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@k8nowak - Subject line in email from @Sephora: "Tan for Free" Halfway expected body of email to say "Go outside, dumbass." It doesn't. Weird.

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@samjshah - "In their report, performance is described in terms of the content and process task demands of the subject matter and the nature and..."

@samjshah - "...extent of cognitive activity likely to be observed in a particular assessment situation." BORING.

@samjshah - ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ

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@jybuell - . @sophgermain just sent me a 3 word text w/ no spaces and one word spelled wrong. So yeah, sounds like they're having fun.

@jybuell - my phone was dead. also yea i was drunk.

@cheesemonkeysf - @samjshah Best line of the night goes to @btwnthenumbers: In re KA, someone asked, "What's the goal again?" BTN pipes up with, "Badges!"

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@ddmeyer - Just ran into @jybuell at the JiffyLube. Weird, but not in a weird way.

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@druinok - One thing I've learned this week - you guys have ruined me! :) I hold other teachers to a MUCH higher standard because of your excellence

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@woutgeo - Found out today that replacing the radio in a civic is a PITA. OTOH, feeling very manly right now.

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@sophgermain - dued, @woutgeo, I had dinner tonight with my parent volunteer who apparently went to your workship this year. #smallworld

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@ThinkThankThunk - Let's try to stay positive about ed reform. Conversations about feasible fixes are better than this-is-why-it's-broken. We know. Chill.

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@DianeRavitch - Teach for joy. Teach for beauty. Teach to touch students' hearts and stir their imaginations. Teach to make them care.

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@fnoschese - The only two badges I am proud to show off... #FathersDayAcademy yfrog.com/h4iqsdqj

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@sarcasymptote - @infinigons since EVERYONE loves the guy working insanely on math problems in a bar on tuesday nights

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@samjshah - great. our time was "planned from 8:30am-9pm. Now we have 3 readings and 3 written pieces to do. #waaaimtired

@samjshah - hm, still chugging away. it's 12.06. tired. #ugh #wowimtotallyawhinerwhenimtired

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@park_star - I love summer time - everyone starts blogging up a storm!

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@k8nowak - In geometry review, teacher: "distance formula? anybody?" kid: "r times t? *facepalm*

@ddmeyer - @k8nowak Feels dirty using Twitter as my gripe machine that's all. Tell me to get over myself.

@k8nowak - @ddmeyer oh at least 50% of my twitter use is "please validate my whining" so I can't talk. complain away.

@fnoschese - @ddmeyer @k8nowak Oh, c'mon. YOu know you love griping on Twitter. We all do. ;-)

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@sarcasymptote - This typo (FROM OUR GRADUATION PROGRAM) seems almost too good to be an accident: lockerz.com/s/112899279

@sophgermain - @sarcasymptote amazeballs.

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@untilnextstop - Math units should be organized around BIG IDEAS / themes in mathematics. Seems obvious but it's turning my world upside down. #klingsi11

@calcdave - @untilnextstop A month ago, I asked the district's Alg2 teachers what the main theme was for the class and they all threw up their hands.

@k8nowak - @calcdave @untilnextstop Funny, I would have said "maximize despair"

@calcdave - @k8nowak @untilnextstp Yeah, Alg2 seems to be our Franken-class, too. #everythingyouneedtoknowbutdoesntfitcoherentlyelsewhere

@jybuell - @calcdave @untilnextstop Alg 2 main theme is "things we can't put in other course so we're going to invent one"

@sarcasymptote - @jybuell @calcdave @untilnextstop  in Algebra 2 in NY, the main theme is, "cover so much shit that no one will like math in the end."

@k8nowak - The six stages of NY Alg2Trig Regents grading: disgust, disbelief, amusement, apathy, ice cream sandwiches, inebriation

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@samjshah - How much do y'all miss me? I miss you! Wish I could hang out virtually with you!

@lmhenry9 - @samjshah I miss you bunches of sparkly rainbows and unicorns. Looking forward to when you can be back on Twitter.

@cheesemonkeysf - We are bereft without you. I can barely feed and dress myself any more.

Number Sense Boot Camp - Request for Feedback and Input

Maybe all of your Algebra 1 students showed up on Day 1 every year with a solid and fluent grasp of basic number sense, but mine sure didn't... and it scared the crap out of me. And then afterwards it haunted me, ALL   YEAR    LONG . . .

The stuff they didn't get was just mind-boggling to me:

• subtracting
• adding a negative number
• the basic concepts of the real number line
• fractions
• measuring
• counting
• basic ops with fractions
• absolute value (any related topic)

I mean, this is basic citizenship numeracy stuff, on the same order as basic literacy.

So since this does seem to be a general condition I am likely to encounter anywhere I am likely to teach, I decided to develop a "Number Sense Boot Camp" unit I could use to start the year off with, diagnose critical number sense deficits, use as an occasion for teaching basic classroom routines, give students a chance to dust off (or remediate) their basic arithmetic skills, and basically give us all a fighting chance of getting to some introductory algebra work.

Another thing that worked this year was stealing adopting game-like practice structures, such as those advocated by Kate Nowak in New York state and by the late Gillian Hatch in the U.K. As Gillian Hatch said, a game can provide "an intriguing context" as well as "an unreasonable amount of practice" in vocabulary, reasoning, procedural skills, generalizing, justifying, and representation than they might otherwise be inclined to do. As Hatch said, it also seems able to lead students "to work above their normal levels." As anyone who has tried any of Kate's practice structures can attest, there is something about introducing this playful element that really gets students to dive in.

IDEA #1
One thing I did this past year that worked for many individual students was to do some specific work with the real number line. I made a printable number line and gave each person their own number line (downloadable from Box.net folder) and a plastic game piece to use with it as a calculating device.

Since the rudiments and rules of board games have such wide currency in our culture, most students found this a helpful physical metaphor that gave them both conceptual understanding and procedural access to basic counting, addition, and subtraction experience that had eluded them in their previous nine to eleven years of schooling.

These had the added benefit of conferring prestige upon those who had shown up for extra help and received their very own set (though I gladly handed them out to anybody who requested one).


IDEA #2

Emboldened by my initial success, I realized could expand the idea of a number line "board game" to use as a basic structure for practice – both in using the number line and in many other basic number sense activities.

It even dawned on me that this could be made extensible by having different kinds of "task cards," depending on whether a player has landed on an even number, on an odd number, or on the origin (a decent justification for considering even- and odd-ness of negative numbers here ; go argue over there if you have a problem with this).


Players move by rolling one regular die and one six-sided pluses-and-minuses die (+ and –) (kids seem to need grounding in the positive and negative as moving forward and backward idea). Kids earn "points" in the form of game money, which could carry over and be used to purchase certain kinds of privileges (such as a "free parking" pass for a day when they don't have their homework to turn in).
Your thoughts?

UPDATE: 
Here are links to the different game boards, along with descriptions of each.
Basic Printable Number Line For Use With a Game Piece:
http://www.box.net/shared/eyy4nvhbtn5xx1qdc9j2

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 1-up version (for use with your basic at-home printer):
http://www.box.net/shared/nv0sdz65hy5p3hv8ix1x

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 3-up version (prints a 24" x 24" poster at FedEx Kinko's--costs about 2 dollars):
http://www.box.net/shared/d4uly1arl88lm3qu9vsm

I made Game Card files using Apple's Pages software (for Mac OS X) and MathType equation editor. You can use these as templates or make your own:
http://www.box.net/shared/s6ha4ol1o6tk0xltp1y1
http://www.box.net/shared/7or8g5klub7jiymshq0f
http://www.box.net/shared/5vq6cmpcd9f9lq1qhud1

Here is a link to the folder itself if you'd like to share and upload your own documents or samples:
http://www.box.net/shared/ftzkun7cvi5vxgvanvh5

Please share any experience or insights you have with them. Enjoy!


AND FUTHERMORE:
Julia (@jreulbach on Twitter who blogs at ispeakmath.wordpress.com) has started a Number Sense Boot Camp page on the Math Teacher Wiki where you can share and find other Number Sense Boot Camp ideas and activities. Available at http://msmathwiki.pbworks.com/w/page/42105826/Number-Sense-Boot-Camp .

UPDATE - 14-Sep-11:

It's only been one day since I introduced the tournament of "Life on the Number Line" but I am already excited about how well this is working out. It is exposing ALL kinds of misconceptions and misunderstandings about adding a negative and about interpreting negative and positive as movement along the number line. Students are playing individually as a "team," and the team with the highest number of correctly worked problems will win 10 free points (2 problems using the 5-point rubric for each person) on next Friday's unit test.

     Since they are surfacing all kinds of misunderstandings about + and - movement on the number line, this is leading to vast amounts of mathematical conversation to get it figured out. So basically, they are teaching each other about adding negatives and subtracting negatives and interpreting that as movement along the number line. 

     I can see that each day it will make sense to give some daily "notes" at the start of class on clearing up common misconceptions I've seen the previous day in students' work so they can solidify their conceptual understanding as well as their procedural fluency a little more each day.

     Best moment yesterday: a girl looked up at me beaming and said, "This is way more fun than doing math!"

     I said, "Good!" but I was thinking, "You have no idea how much math you are actually doing!" :-)


ANOTHER UPDATE:
Here are the game cards to use on the first day: http://msmathwiki.pbworks.com/w/file/45547360/1st%20batch%20of%20game%20cards.pdf


And here is a generic worksheet (front and back) you can print out and give to the kids to use as their template:
http://msmathwiki.pbworks.com/w/file/45547628/generic%20worksheet%20for%20Life%20on%20the%20Number%20Line.pdf

If you have only a ton of basic 1-6 6-sided dice, use Post-Its to make two (2) plus-and-minus dice for students to use with one (1) regular numbered die. This is a good task to give to a student helper. ;-)

FINAL UPDATE:

Four final things:

Thing #1
This unit confirmed me for that kids really do need active, multi-day practice in "living life on the number line" to gain a sense of positives and negatives as directions WHILE AT THE SAME TIME they are developing a sense of positives and negatives as additive quantities. It's not enough for us to just wave the idea of life on the number line at students. It doesn't make sense to them. They really needed experience alternating between (a) positives and negatives as indications of directional movement and (b) positives and negatives as additive or subtractive quantities in the process of deepening their additive reasoning skills.

Thing #2
Right before we started, I had the bright idea to give every group TWO +/- dice and ONE six-sided number die. If you don't mind my saying so, this ended up being a master stroke because it forced students to think about rolling (–)(–)(3) and rolling (–)(+)(3) and every possible combination thereof. This one thing alone might have done the most to deepen their sense of additive reasoning and of +/– as directions of movement.

Thing #3

Here's a link to a zip file that contains ALL of the game cards I created for this unit (on the math teacher's wiki): Game Cards- ALL

For all those who have asked and those who are thinking of asking, I'll say that my school uses the California edition of the McDougal Littell Algebra 1 textbook (by Larson, Boswell, Kanold, and Stiff). For this reason, the game cards are targeted at each of the lessons in Chapter 2. However they are not tied to that textbook and could easily be used with any curriculum or textbook (just sayin').

Thing #4

I'll have to take a photo of the final game boards our instructional aide mounted and laminated for us. They are a true work of art!

Unmediated Experience, part 2 — another use of primary texts in the math classroom

While gathering problems for my logarithms and exponential functions unit from some of my favorite textbook sources, I noticed that John Napier's birth and death dates (1550-1617) looked kind of familiar — really familiar, in fact.

In fact, they're almost the same as Shakespeare's birth and death dates (1564-1616).

And they're awfully close to Galileo's birth and death dates too (1564-1642).

So why not at least mention the historical context of logarithms?