cheesemonkey wonders

cheesemonkey wonders

Wednesday, June 29, 2011

Favorite Tweets #14 - gathered on behalf of Sam Shah

Another gentle cascade of sparkly unicorns and rainbows for your time-wasting pleasure to tide you over in Sam's absence.


lost my sum school job today b/c of a post i wrote and twitter. i was asked not to post about. i'm considerin… (cont)
6/22/11 10:39 PM

@sophgermain I will totally post it. I'm Canadian, what are they going to do :P
6/22/11 10:43 PM

@park_star beat you up!
6/22/11 10:46 PM

@sophgermain Hey, if you LOST the job, there's nothing more they can do to you. So go ahead and post it.
6/22/11 10:55 PM

@sophgermain They don't have a big enough travel budget to go to SK and beat @park_star up. Unless they have access to a unicorn #doubtful
6/22/11 10:55 PM



@samjshah @sarcasymptote I am influential about math, bay area, drinks (whoo-hoo!), teaching and ...wait for it... Doritos!
6/26/11 10:56 AM

@btwnthenumbers @samjshah that is bullshit. no one has done as much to raise cultural awareness for doritos for breakfast as i have.
6/26/11 10:59 AM

@sarcasymptote You have to choose. Nutella or Doritos? Which one is it going to be. You can't have both.
6/26/11 11:01 AM

@btwnthenumbers nutella flavored doritos.
6/26/11 11:04 AM

@btwnthenumbers @samjshah @sarcasymptote I'm a 47 too. Influential about software, math, ..., and diet coke??? Specialist.
6/26/11 11:19 AM

@btwnthenumbers @samjshah @sarcasymptote can't wait for the Khan style achievements!
6/26/11 11:19 AM

@dandersod @btwnthenumbers @samjshah yeah, i saw that and had the exact same reaction. looks like @fnoschese has gotten in our heads.
6/26/11 11:29 AM

and apparently, i am a specialist! FIRST TIME IN MY LIFE! #illtakeit
6/26/11 10:49 AM

@samjshah also, nutella and puppies. so there.
6/26/11 10:52 AM

@sarcasymptote i think in this situation: greg : gary busey :: sam : marathons WTF. i love your list. especially nutella and puppies. #love
6/26/11 10:55 AM

@sarcasymptote i miss hanging out with you. i have now found my new drink: pimms cups. we must drink them together STAT. #pimmsischeap!
6/26/11 10:55 AM


according to KLOUT, my areas of influence include: math, teaching, blogging, books, guacamole, MAGIC, marathons (?!?), movies, colleges, ...
6/26/11 10:45 AM


Just used "super rad" room went silent #eastcoast
6/26/11 12:22 PM

@sophgermain "wicked" should work nicely in NH.
6/26/11 12:30 PM

@dandersod @sophgermain well, for where she is, stick with calling everything "shallow and pedantic"
6/26/11 12:41 PM

@sophgermain I know u weren't in Bay long but please use Hella in all sentences.
6/26/11 12:59 PM


chuf chuf chuf chuf chuf chuf chuf LITTERBOX TIME IS THINKING TIME chuf chuf chuf chuf A HAH SOLVED COLD FUSION chuf chuf WHOOPS FORGOT
6/26/11 4:56 AM


If @ddmeyer can fly across the country to NYC, I guess I can leave Brooklyn for the occasion.
6/27/11 5:08 PM



I would like to record a video, but the firing range near my apartment is having a busy afternoon. #republicantown
6/29/11 12:41 PM



6/29/11 12:18 PM

Sunday, June 26, 2011

"Did I answer your question...?"

One of my favorite stories about Too Much Information is the one about the kid whose parent has been so dreading the "Where did I come from?" question that, when the question finally arises, she or he responds with a lengthy disquisition about mommies and daddies (or mommies and mommies, or daddies and daddies, or any other combination thereof), sperm and eggs, etc., etc., only to be met with a brow-furrowing "Oh."

When the parent follows up, asking if there are any further details the child wanted, the kid responds, "Well -- Petunia said that she is from Trenton."

I was remembering this story as I was filing away some of the nicest cards and notes I received from my students this year. More than one of them thanked me for always asking the student questioner or the whole class if I had actually answered their questions.

This made me wonder, doesn't everybody do that?

When I worked in technical support and sales, I learned that this isn't just an important part of providing a correct and accurate answer. It's important for establishing the basis of trust that underlies the relationship.

When I started teaching math, I never even thought about the fact that I always asked a question-asker this question, but now I am starting to think about its importance. It doesn't matter so much for the "telling"/"lecturing"/"demonstrating"/"modeling" part of the process, but it sure matters a lot for building my relationship with my students and for establishing myself as a trustworthy guide to the world of mathematics. And it makes me wonder if this is how a lot of our students learn their habits of discouragement. I mean, if nobody ever asks them if they received what they asked for, doesn't that reinforce the unintended curricular idea that satisfying their personal curiosity doesn't matter?

Does anybody else out there have any experiences to share around this?

WARNING: This post contains math education heresy

I am so tired of math teachers and teacher educators telling me memorization doesn't work that I am willing to take a reckless step into the fray.

My purpose in this post to demystify this dangerous misunderstanding and to say that memorization of basic facts not only can work but indeed does work and works well -- as long as several essential conditions are met.

First, a definition. When I talk about memorization, I am NOT suggesting that one can achieve mastery by rote memorization of the proof of the Fundamental Theorem of Galois theory.

Rather, I am referring to an active process of integrating certain basic, rudimentary facts into one's mind and body, both backwards and forwards. By this, I mean that, given a vocabulary word, one can produce the definition, and similarly, given the definition, one can produce the vocabulary word. Or given a basic multiplication fact, one can produce the product and similarly, given a composite number, one can produce its basic factors.

 I am talking about filling in basic math facts. Expanding critical vocabulary. And solidifying basic mathematical skills that are often missing in the high school math student. 

Facility with this kind of memorization is the sine qua non to serious foreign language study as an adult, since it is simply not practical to put yourself into the way of enough quality adult conversations to absorb all the vocabulary one will need to read, write, think, and speak another language with the basic fluency that is required if you are going to be dropped into another culture. But it IS possible to approximate that vocabulary and bring its user closer and closer to a level of acceptable fluency.

This is how I both learned and taught Italian and Latin, as well as a number of other languages as a Comparative Literature scholar. It is also how the Peace Corps trains its Corps members in preparation for their overseas, language-intensive assignments. 

I was schooled in the Rassias Method, a highly dramatic, intensive, and effective technique of drilling students in the language classroom to approximate and accelerate the contexts of listening and speaking another language. It does so through very strategic, high-energy, rapid-fire, and theatrical drilling and practice techniques.

Approximating contexts is important because, as Skemp puts it, purely instrumental learning without any relational context is just pointless. But I believe many of my colleagues and math education instructors have misunderstood this critical distinction. The way I read it, Skemp is not suggesting that there is NO room or role for instrumental learning. He is asserting that instrumental learning is insufficient without relational learning as well.

This intersection between instrumental and relational learning is where the Rassias Method really shines. One thing I used in my math classes this past year was the Rassias strategy of "flooding" students (my term, not Rassias') with productive opportunities in order to burn those facts and skills into their minds and bodies. But even more important than the drill itself is the process of breaking down student inhibition in the classroom.

This strategy is key.

Discouragement is always accompanied by inhibition. And the only way I've ever found to break down inhibition in this regard -- my own as well as that of my students --  is to insist on lots and LOTS of participation and practice -- with no chance of opting out.

The Rassias Method taught me to use a technique that blends rapid-fire drill with micro-contexts and an often humorous dramatic flair to create a heightened emotional charge in the classroom in which anyone could be called on at any moment to produce anything that is being asked for. It encourages learners to engage, to enjoy, and to stop worrying about producing the right answer because it creates dozens and dozens of chances to produce the right answer. It accomplishes this goal by flooding learners with basic language demands, all the while heightening drama, motivation, and interest in success while simultaneously lowering the stakes of failure. 

To put it another way, trying becomes more important than succeeding -- because eventual success is assumed.

Here is an example of how I used this in teaching my Italian language classes at Stanford.

One of the biggest hurdles in learning Italian is mastering its complicated matrix of prepositional contractions. Wikipedia has a reasonable summary of this matrix here:

In Italian, a number of key basic prepositions are ALWAYS merged with the direct article preceding the noun that is the object of that preposition. So for example, to say that something is "on the table," you need to merge the preposition for "on" (in this case, "su") with the direct article "la" ("the") that precedes the noun "tavola" ("table"): in other words you need to say that something is "sulla tavola" ("on the table') instead of "su la tavola."

Practically speaking, this roughly six-by-eight matrix of prepositions and direct articles needs to be absolutely second nature for a speaker who wishes to be able to produce and recognize the right prepositional contraction for the job.

Basically, the prepositional contractions are the times tables / multiplication facts of the Italian language.

To get students using these, one Rassias technique I used involved a little plastic elephant, whom I named Signor Elefante, which I held in different positions with respect to a festive-looking cardboard box and drilled my students, asking, "Dov'è Signor Elefante?" ("Where is Signor Elefante?"). Or as we say in edu-speak, I used situational motivation (for a good discussion of situational motivation, see Wilhelm and Smith, "What Teachers Need to Know About Motivation," Voices from the Middle, Vol. 13, No. 4, May 2006).

Sometimes Signor Elefante was "nella scatola" ("in the box"), sometimes he was "sulla scatola" ("on the box"), sometimes he was "vicino alla scatola" ("near the box") or "lontano dalla scatola" ("far away from the box"). Sometimes he was "sotto la scatola" ("under the box") or "alla porta" ("at the door"). Occasionally he was "sulla lavagna" ("on the chalkboard"). He got himself into some pretty wacky prepositionally contracted situations. But after a lot of practice and inhibition-destruction -- as well as their own practice at home with flash cards -- locating Signor Elefante in time and space became more and more natural for my students. They got themselves over this major linguistic hurdle and developed their own relationship with the prepositional contractions.

They blended instrumental learning techniques with relational learning to generate understanding and fluency that was more than the sum of its parts.

In my math classes last year, I found that many of my lowest-achieving students responded well to this kind of approach to remediation. Even my higher-achieving students responded well to this approach. In fact, the most important the thing I discovered this past year is that many students have no idea how to practice basic techniques... but they get excited by the results they can achieve once someone shows them how memorize and assimilate bits of information like this.

Now since in spite of my best efforts I am bound to be misunderstood and misquoted, I'll restate this as plainly as possible: I'm not talking about using memorization with higher-level thinking and problem-solving. I am, however, talking about using these techniques as a strategic intervention to help students remediate and give them the tools and techniques they will need to fill in the gaps and potholes that riddle their elementary mathematical preparation. 

Confidence with the basics is a necessary condition to cultivating curiosity and persistence about mathematics. I speak from my own experience as well as that of my students.  Mastery of basic tools and techniques, combined with a lowering of inhibitions, is a foundation upon which confidence and curiosity can grow. And that can be the basis for a turnaround to success in high school mathematics -- regardless of where students are starting from.

Wednesday, June 22, 2011

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.


@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."


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


@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


@samjshah - @Mythagon can we also make anything vegetarian from ? and eat again? YES! we will do these things!

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

@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. 


@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.


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


@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?  cc: @samjshah

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


@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.

@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!!!


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


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



@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.



@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!"


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


@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


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


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


@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.


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


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


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


@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


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


@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. ;-)


@sarcasymptote - This typo (FROM OUR GRADUATION PROGRAM) seems almost too good to be an accident:

@sophgermain - @sarcasymptote amazeballs.


@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


@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.

Monday, June 20, 2011

On remembering to allow time and space to refill the well: lessons from Ursula Le Guin

A long time ago, in a galaxy far, far away, I studied at a women's writing workshop called "Flight of the Mind" in the forests of central Oregon.

It was an amazing experience, organized and run in ways that taught me about a lot of things in addition to honing my writing craft — insights that come from wisdom rather than knowledge, the kind of deep-rooted knowledge that comes from the soul rather than from the intellect.

One of the most important lessons I learned there was the need to stay with my own efforts and to protect my creative life — even to the point of becoming fierce in defense of my own needs. Regeneration of energies in preparation for a new beginning is what I am thinking of now, and I was blessed to have this lesson hammered home one day at lunch with the legendary writer Ursula Le Guin on the deck overlooking a bend the wild McKenzie River, in the dappled shade of ash trees.

We were talking about how many new threads and ideas we wanted to follow after the workshop, but how tired we were and afraid of running out of steam.

She told us something that has stayed with me and that seems applicable to life as a teacher as well.

She said, "You need to build in time to 'refill the well.' You've been working hard, and experiences like this one draw out even more energy and insight from you. The temptation is to go home and dive into the work with renewed intensity, but the reality is that you need to allow time for your spirit to refill the well. Be sure to give yourselves time and space to allow it to soak in. This is time that is not wasted."

Later that week, I watched her climb out onto a boulder a good way from the shore. She climbed gingerly, but with quiet confidence. When she reached the spot where she wanted to sit, she took her time settling in and hugged her knees to her chest, watching the water rushing over the clear pebbles on the river bottom.

I always have to remind myself of this when I hit this point in the process. One time is ending and the next time has not yet begun. I am tempted to allow this between-time to be gobbled up or frittered away, either in cleaning up the old year or in setting up the new year. I forget the need to just sit still and let myself rest and digest. I forget that this time is not wasted.

I am refilling the well.

Thank you, Ursula, for the gift of this memory.

Sunday, June 19, 2011

Mathematical Language Manifesto — including mathematical language skills in SBG

Yeah, I'm talking to you. But mostly I'm really writing this for myself.

One of the biggest holes I see in our SBG skills lists is the skill of using correct mathematical language. Not just the names of things, but the verbs.

Oh, the verbs, they are killing me.

Here's an example of the toxicity of mathematical babytalk.

In the math lab at lunch time, I listen in on the students trying to answer tutors' or teachers' questions. "Um, you times it by two...?" or "You minus one." or "You put three."

Which makes me say to myself, You WHAT?

What bothers me even more is hearing how other teachers respond to this. Most of them don't. They simply cringe and try to ignore it, as if overlooking the use of babytalk will improve the doing of the mathematics on paper and in the mind.

And I have not noticed magical thinking to be an especially effective intervention in the classroom.

I started listening in with regularity and I noticed two related correlations:
  • When students sound ignorant, they tend to be treated with subtle (or not-so-subtle) contempt.
  • When students sound knowledgeable, they tend to be treated treated with respect.
This gave me pause. It also made me wonder if I am doing this too. Which naturally gives rise to worry.

So since the only way I know to address this kind of blind spot is head-on, I've decided to address it head-on. No shame, no blame. Just another couple skills for students to master on the skills checklist.

I want to set my students up to sound knowledgeable and be taken seriously as math learners. And that means I have to encourage their courage by means equipping them with the language skills (both oral and written) to get themselves taken seriously in the math classroom as well as in the world outside.

So I'm adding two mathematical language skills to the skills checklist this year -- both for Algebra 1 and for Algebra 2.
  1. Use the appropriate mathematical names for arithmetic operations (noun forms) when answering short-response questions
  2. Use the appropriate verb forms of basic arithmetic operations when answering short-response questions
Same SBG rules as ever: Just show me you can do it twice perfectly and you'll be off the hook for those questions in the future — forever.

But secretly, in my heart of hearts, I'll be hoping that by that point, they will already have internalized a better set of language habits.

Thursday, June 2, 2011

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.

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 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).

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?

Here are links to the different game boards, along with descriptions of each.
Basic Printable Number Line For Use With a Game Piece:

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 1-up version (for use with your basic at-home printer):

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):

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:

Here is a link to the folder itself if you'd like to share and upload your own documents or samples:

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

Julia (@jreulbach on Twitter who blogs at 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 .

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!" :-)

Here are the game cards to use on the first day:

And here is a generic worksheet (front and back) you can print out and give to the kids to use as their template:

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. ;-)


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!