The Big Ideas inside the 'big ideas'

As I've been organizing my job search materials, I've been reflecting on some of the Big Ideas I have learned are the most important among the too-many "big ideas" our textbook and state standards emphasize for Algebra 1.

One of the problems with the state standards is that they can't let go of anything as being less important than anything else. Which is why there are 26 overall standards, plus embedded sub-standards inside the standards, and the whole thing is a nasty ball of yarn to try and untangle.

One thing I've tried this year, which seems to be working well, is to choose my emphases based on what developmental psychologists have discovered about children's mathematical development. I am particularly grateful for the work of the British psychologists Terezinha Nunes and Peter Bryant, who also do a lot of work with Anne Watson of Oxford's math education program. Nunes and Bryant's book, Children Doing Mathematics, has really blown my mind open to what they call the "generative" quality of children's mathematical development and number sense — that is to say, kids develop their sense of number and of mathematics in layers, the way some inkjet printers work, with each pass of the printhead setting down another layer that completely transforms the image that is emerging on the paper.

In Nunes and Bryant's synthesis, as well as in their summary of others' research, kids' understanding of quantity is revealed to be an extremely fluid, dynamic, and multi-faceted set of tools. As they put it, "a successfully developed understanding of number comes from four distinct developmental threads" which they summarize as:

1. the ability to COUNT discrete OBJECTS
2. a deep familiarity with a wide range of QUANTITIES OF QUALITATIVELY DIFFERENT KINDS (such as both countable and uncountable quantities)
3. the ability to COMPARE QUANTITIES OR COLLECTIONS of objects, assessing both similarities and differences regardless of their qualitative kind(s)
4. the ability to use established notation for all of these (Nunes and Bryant, pp. 1-20)

Yet, as they have observed, our current curriculum tends to address only #1 and #4.

This helped me understand something I had been struggling with for a long time — namely, the fact that there are many incomplete understandings children develop that are sufficient for them in their context, BUT that are insufficient over the long term as a foundation for mature mathematical understanding. These are the mathematical versions of ideas like "the tooth fairy" or "Santa Claus." They are enabling fictions that are developmentally appropriate in their time and place, though they are not at all what we want our young adults to rely on by the time we release them out into the grown-up world of mathematics.

One of these incomplete understandings — one that drives university-level mathematicians like Keith Devlin head-banging mad — is MIRA, or the idea that Multiplication Is Repeated Addition. Their argument is that this is such a stunted understanding of multiplicative reasoning that it threatens to undermine the very foundations of civilization, dammit.

But in truth, like the idea of the tooth fairy or Santa Claus, MIRA does have a legitimate place in a child's generative mathematical development — as long as his or her teachers understand that, like the idea of the tooth fairy, it is an incomplete understanding that is meant to be expanded upon into a much richer and more scalar concept of multiplication.

The place where I am finding MIRA to be an extremely useful tool is with Algebra 1 students making their first forays into the abstractions of algebraic reasoning, which is to say, in dealing with polynomial arithmetic. I say this because young adolescents are such intensely concrete thinkers. When I ask them to consider combining like terms such as 5 elephants and 3 elephants, they can easily understand what I am asking for. But the moment we start investigating the idea of combining 5x^2 and 3x^2, their heads explode. Things only get worse when they are asked to combine 5x^2, 3x^2, and  6x^3. It seems like they forget everything they have ever known about the combining of like terms, and they start adding or multiplying exponents and or worse things than most of you can imagine.

This is the place where some teachers find algebra tiles to be helpful. But I find that algebra tiles have a grammar and a rhetoric of their own that is not easily extensible into polynomial arithmetic beyond quadratic thinking. Also with their color-coding, they also add in moving parts I find my students are not yet ready to think about.

But guide their attention away from abstraction for a moment, and ask them what happens if they combine 5 dogs and 6 apples. They understand the logic of ConcreteLand completely. In this case, I have found, getting them to think about 5x^2 the way they think about 5 dogs and about 6x^3 the way they think about 6 apples, and their conceptual understanding shoots through the roof. I can even use Brahmagupta's idea of "fortunes, debts, and ciphers" (positive numbers, negative numbers, and zero) to help students think about what happens in a trading economy where I might "owe them" 3 dogs (or 3 x^3) as we barter our algebraic quantities away for each other's lunch components.

Students still need a lot of practice and experience with this whole crazy abstract insanity to cement their understanding in place, and they can be expected to relapse several times into believing that they need to add exponents instead of thinking about coefficients as quantifiers. But eventually the idea of quantifying (and if need be, combining) x^2s the way they count and quantify dogs or apples gives them a surer footing as they begin to construct a new and deeper understanding of multiplying variables. And that is something I can eventually build a rich and scalar concept of multiplication on top of — on that would be appropriate to eventually deliver to Professor Devlin's lecture hall.

I do this knowing that even within a few months, this conceptual framework will be revised and replaced with other incomplete understandings many times over. But I do so knowing that I am teaching my students how to learn by giving them tools for understanding how to build tools that help them understand what the heck they are doing.

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.

http://rassias.dartmouth.edu

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:

http://en.wikipedia.org/wiki/Contraction_(grammar)#Italian

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.

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.

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!