As an acolyte of our secular object of worship, the Beloved National Council of Teachers of Mathematics Standards, I felt it was high time for one of us to search this country for the perfect Standards-based classroom. I committed to cross this country: unfettered, unannounced, unknown and on foot, a soldier in the battle on behalf of true constructivism.
I carried little to nourish me beyond the efficacy of our vision: a few nuts and berries, a package of twinkies, seeds of our vision to plant across the fruited plain. The essence of my search was to find qualitative evidence of the correctness of our weltenschaung, that quantitative data means nothing. In fact, the first commandment of our credo, is after all, any proof students know something is proof they don't!!
Why, oh why, must the shrill perpetrators of the industrial age insist upon the evil "N" word, knowledge? Why must things be known? Find them, admonish them, convert them to pay attention to the fine work of Alan Sokal who has demonstrated to us the path of the true post modern academic. Natural laws are fleeting and transitory. Apples need not always fall; the rotten apple of Newtonism is at its core pedagogical imperialism.
When will these "educators" quit insisting that something as invented and arbitrary as modern mathematics be learned in any systematic way? This is individual imperialism of the worst sort. Imperialism honed by the relentless persistence of the ruling class whose insistence on knowledge makes us intellectual serfs.
Taking my self-appointed responsibility as the math reform guru, I set out upon my journey. Wearing my new Black & Decker TI Calculator Belt, my TI-36, 82, and awesome 92 at the ready. (No more need to learn manipulation of those horrid symbols.) I had the appropriate technology for every possible problem. Technologically speaking, I was in complete accord with the NCTM Professional, Curriculum and Assessment Standards.
I was renewed, becoming the walking embodiment of math reform. A walking, talking cooperative group of one, marching our country on behalf of the NCTM.
However, I was anxious about taking this journey alone; I yearned to have this trip validated by the participation of my cooperative learning group. I yearned to jigsaw our trips to the laundry or co-op co-op our most elemental functions (one-to-one, I might say!). Lacking their physical presence, I assumed all the roles of my group. I reflected the diversity that is my group, hence I celebrated myself!
I felt powerful and rich in my mathematical knowledge even though my garb aroused the curiosity of the slack jawed proles. How I wished to reach out and shake them from their computationally induced languor. Every fiber in my being yearned to enthrall them in the rich and powerful mathematics I knew (or really didn't know, but felt!!)
Hovering before me in a state of constant vision was my authentic educator's portfolio. It reflected my self esteem and worth as a humyn and educator. It was prepared for additions. They would document the authentic assessment of the constructivism that would unfold before me.
And what reform I've witnessed!! Crisscrossing this great land, I visited schools, large and small, urban and suburban, rural views and cityscapes. Our vision of the Standards was my credo; diversity and equity my mantra.
No, this insight hasn't always been with me. The old wretched me wouldn't have known constructivism from a construction site! When I saw students' heads down on desks, I thought they might be sleeping. Now I know they were, indeed, reconstructing thousands of years of mathematics, from Archimedes to Gauss, and back! Now, I've seen the error of my ways. Yet even President Clinton now seems to agree with the former me. He stated in his speech to the educational summit in New York, "I don't believe you can succeed unless you are prepared to have an assessment system with consequences". Respectfully, Mr. President, you are wrong. It's as obvious as the nose on your face (oops, perhaps not that obvious!)
For example, take the totally discredited approach to teaching reading by phonics (not discredited by me, but, instead the reactionaries). Word decoding skills only remove the glory and beauty of literature. Bill Honig, former ringmaster of whole language, now apologizes for his past policies. Hold firm, Bill! Evidence of the efficacy of your views is on the way. The Houston study is merely a fabrication by the Luddites to maintain the status quo.(Actually, it might not be, since any tangible evidence would be evidence to the contrary. I confuse myself, yet it's authentic confusion!)
On the whole language front, there are technological improvements which will make this whole language vs. phonics debate moot. A high tech company is presently marketing a reading machine that will fit in a student's pocket. So why should we teach reading? Like calculators, this reading machine will free ourselves to focus on the meaning of what we study, not the minutia of words. Never more will students be brought down by the tyranny of the unpronounceable!!
So what did I see? Classrooms so close to the Standards it gave me chills. Authentic portfolio assessment replete with slumping standardized test scores (explicit proof of our success). Students so enamored of working in groups they've renounced their names, taking on the collective name of their group. Parents so devoted to our vision that they have moved in with their child's cooperative learning group! Americans working and living in alternative, authentic living situations experiencing the power and richness of mathematics. The glory of our vision!
Problem solving to the max. Minds and calculators whirling in synch. Minds so full of problem solving techniques, no place for knowledge. Nothing to think about, only problems to solve. Minds unencumbered by knowledge of the past, integrating themselves with technology. Somebody invented it; children (err, pre-adults) must use it.
Ten year olds! Discard your bicycles; to the driver license bureau! You have problems to solve, oceans to save, and planets to preserve! Fight the tyranny of withheld technology. Calculators are your birthright, computation the scourge, long division your oppressor. Aux barricades mes enfants!!
There were so many wonderful, enriching classrooms reflecting our vision. At an urban high school in the Pacific Northwest, I saw the sort of projects-based curriculum so promoted by the Standards. My AP Calculus class had a year long project on measuring the acceleration of our gym during an earthquake. Unfortunately, we had no earthquakes during this period, but the students valued their involvement.
But compared to what I saw in Seattle, I was a Standards-wimp. I watched as a group of calculus students positioned themselves on the Puget Sound, waiting for the imminent oil spill. Their teacher felt this means of experiencing a related rate problem was superior to anything they might encounter in a text. I concurred.
"Kids don't like learning like that. They hate structure; they'd rather experience things. Problem is I've got to wait for this oil spill before I teach related rates." He exhaled the sigh of the visionary constructivist.
"It's a little inefficient, but authentic. Students need to know they're improving things, even while they're in school." Then he winked. "Plus, I want the kids to like me. You know, like Mr. Holland. Americans want to be liked, especially by kids, since they're so much more authentic than adults are."
The man's a master, I thought, as I noticed a young, Cobainesque fellow sitting alone on the shore. Yes, here was the problem solver of the 21st century. I approached and asked if he preferred the essence of mathematics since the implementation of the Standards.
"Yes", he said. "I've taken quite a bit of math, too." With that he pulled back his dufflecoat to show me his collection of calculators. He pointed to them. "This one I took at Walmart, this one I jacked at K-Mart, and this TI-92 I took from my teacher's desk." He nodded at the master teacher who was involved in setting up a coffee detector on his CBL.
"You stole from your teacher?" This was even a little excessive for my now-liberated mind.
"The Assessment Standards teach us we need to value different kinds of learning. And I learn a lot from jacking these things. My dufflecoat is sort of a living portfolio of my mathematical activities." Boy, was this student authentic! And what a portfolio. This would be worth an article in the Mathematics Teacher! Authentic dufflecoat assessment!!
"Are you a mathematically promising student?" I blurted it out so quickly, he winced.
"Promise me $50 and Ill get you some math. How bout a 92?" There was an air of capitalism about the boy here that wasn't exactly in accord with the Standards. But despite that, this lad had shown me. Perhaps, there was still much I needed to know about the beauty and vision of our new ways
But all wasn't perfect on my odyssey to math perfection. At a middle school in California, I had an experience I regret sharing because it shows how far we need to go to fully implement our Standards. When I entered this class, the students were working alone on what appeared to be a test! There was obviously much angst among the students as they worked. Clearly, they were oblivious to the impending ecological disaster in the Puget Sound. No doubt they were thinking of themselves and their grades! I was angry they were diverting themselves from the salvation of our wounded planet.
I approached the teacher who was obviously unaware of the severe damage he was doing to these youngsters. "What is your vision, sir?"
"My vision?" He pondered the question. "About 20-40, I suppose. What's yours?"
"Not what I perceive in this classroom!" I tried to repress my indignation. "What is your philosophy of teaching, if I might ask? Are you currently in line with the three volumes on the teaching of mathematics published by the NCTM?"
"Yes, I'm a member. Occasionally, they actually have some math in the Mathematics Teacher. But reading those books, especially the Assessment Standards gave me indigestion. There's too much talk about ethnicity differences and things like that. We sound like South Africans!"
I paused considering this notion. But no, clearly the teacher had not considered our intentions. I continued,"But, sir, you know you are doing these students damage by not engaging in mathematics from a problem solving perspective?"
He smiled at me. "Well, sir, let me tell you what I know. First, understanding is not exactly learning. We can help the kids understand and you guys have come up with some good things to help that happen. In fact, I use them sometimes. But long term retention is based on consistent practice of basic skills and concepts. And while I like some of your problem solving activities, long term problem solving is primarily a function of recognizing well known embedded principles in new situations."
I made a quick adieu then headed out the door. My goodness! The man knew about our vision and had rejected it. I retched in the bushes outside.
It took me the rest of the day to recover from this visitation. To lift my flagging spirits, I stopped by a bar. Instead of drinking, I engaged several of the bar patrons in some EQUALS cooperative learning problems. I left the bar refreshed; the bar patrons were drunk, but authentically educated in math for the first time.
And so my trip went for many months until a summer day several weeks ago. In a Midwestern state, knapsack on my back, the fields of corn engulfed me. I half expected to see ghosts of great math educators, coming forward to facilitate again.
And there, suddenly, it was. An elementary school sitting alone in a field of math dreams. It shimmered in the morning air. There was an instantaneous sense of anticipation as I walked towards the building.
As was my habit when visiting a school unannounced, I went into the school's office. To my surprise, the school secretaries were involved in a cooperative learning exercise. I was amazed, "what are you working on?" I asked.
"We're doing the bell schedule, like we always do", the woman seemed surprised I was asking.
"You mean you figure it out anew everyday?" I was astonished.
"Every hour. Otherwise, we'd need to type it out and post it, then our ringing of the bells wouldn't be authentic." I had to admit they had a point. Authentic bell ringing assessment. Everything in its proper contextual and conceptual place. This was a special place, a rich and powerful place, perhaps.
A student led me to a class of pre-adults. I had asked to visit a fourth grade class, but he didn't seem to understand. As I entered the class, I was immediately struck by something unusual. Children weren't dressed individually; they were dressed by cooperative learning group. Now that's what I call an ensemble!
I asked the Large Humyn (LH, once known as "teacher") who had been ordered by her principal not to speak at all to the children as a preventative to lecturing. I agreed with this measure as did the LH. Our "silence for the lambs" was a mere inconvenience towards promoting true constructivism.
When I asked my usual introductory questions about standardized scores, hoping to find the preferred collapse, the LH smiled. She wrote on a pad of paper supplied by the NCTM for silenced teachers, "Our scores are zero across the board".
"Absolute zero?" Yes, she nodded. Holy cow, they've achieved absolute zero! This was a strange and wonderful place. "Then can I see your portfolios?" My mind literally ached to see the authentic evidence of what was going on here.
"We don't use portfolios either", she scribbled. I was perplexed and disheartened.
"Then how do you evaluate?" I still had this bourgeois notion I needed to expunge from my memory bank. (I wish I could link my TI with my brain and transport many of these mistaken ideas to a real computative device.)
"Just observe, you'll see."
And so I did. For days I sat there, watching these kids. I learned there was no fourth grade at this school, because numbers weren't known. The students had decided number systems were too arbitrary and must be reinvented everyday for every person. There was even supremely reformist groups who chose to not acknowledge mathematics at all. We valued their decision.
Eerily, at first, not even realizing what I was observing, it came over me. The smiles. Yes, they all smiled so beautifully! And when they were engaged in the topic we once called mathematics, the smiles increased.
"It's the smiling, isn't it?" I asked the LH , now bound to her chair, gagged a la Bobby Seal. Facing the wall, she nodded. Eureka, I had discovered it!!
As an observer, it was up to me to quantify (ooops, assess) what I had observed. I worked day after day, going through one idea after another. Testing, we all know shows the opposite of what is truly known by students. Teachers in California not able to pass the CBEST were actually the best mathematicians of all. I knew that from the Assessment Standards. Even portfolios were too restrictive for the beauty and power of our vision. I needed to bring this news back to the NCTM discussion group. They would want to know; they would ache to know!
Then the perfect means of evaluating true educational reform came over me. It was the smiles; so the smiles must be measured. A respected LAUSD administrator had once spoken of a "smile gauge". The true means of evaluating students had been directly before me, yet I'd not had the sense to see it.
The Hart Smile Gauge () is based on a simple mathematical (or what we formerly called mathematical) principle. The first prototype was a hand held device to measure the eccentricity of the lower lip. The shape of the lower lip is generally elliptical (except among tight lipped traditionalists), approaching a circle as a Standards-based pedagogy is implemented. In other words, in a Standards based classroom the lower lip of the students will be more circular than in "drill, kill, smash, destroy and pillage" classrooms.
I had several problems with my early designs. Younger children have a knack of slobbering all over the Hart Smile Gauge. Therefore, constructing it out of paper or even cardboard is not recommended. My current design is an electronic version which plugs into the Sonic port of the TI-CBL (Calculator Based Laboratory).
But here is the rub. Students in this class insisted they do the measurements of their cooperative group colleagues. This brings up a potentially dangerous situation that you must all be aware of before attempting this on your own. The very act of performing a smile measurement using this device is, in fact, an activity aligned with the Curriculum and Assessment Standards. Therefore, we have inadvertently set up a self feeding recursive process that produces ever more circular and circular smiles, indicating more and more authentic education reform.
This phenomenon which I have dubbed Accelerated Authentic Assessment (AAA) was apparent upon my initial experiment. The measurement was going beautifully, Smile Gauge attached to CBL, attached to a TI-82 which was being projected onto a screen. At first, I heard a slight buzzing like a swarm of bees passing through the field. It became louder and louder, sounding like a tornado! Only then did I realize what was happening. This lad who was achieving the most effective math education reform in existence was setting up a vortex of math reform.
Alarmed that this might, indeed, cause some harm, I immediately discontinued the experiment. The last thing we need is the NCTM to be responsible for a black hole of math reform sucking up all of America. This would not be good for publicizing our vision even though kids would be learning for once. Since then, I have installed a Child Vortex Preventative Device which will come built in by the time this product reaches calculator distributors.
Again, my word of warning. Watch your students carefully. High school students, in particular, have been known to "joy ride" with this device. Make sure they do not attempt to disable the Child Vortex Preventative Device. The results might, indeed, be catastrophic, although preferable to traditional classrooms which have been known to produce brain damage.
As I left this apex of math education reform, I felt like those surfers in "The Endless Summer" who'd finally found the perfect wave. Walking away, I heard the lilt of young voices raised in song.
[Sung to the tune of Smile by Charlie Chaplin]
Smile, when your calculator's breaking,
Smile, when you don't know what mistake you're making,
When there's a use for pi,
You'll wonder why
If you smile, no need to borrow.
Smile, a wrong answer's no sorrow.
You'll see our NCTM vision come true,
The Standards are you.
Fill up your portfolio with gladness,
Don't abide mathematical nonsense,
Although a sneer may destroy your math veneer,
That's the time you must keep from computing,
Smile, what's the use in tutoring.
You'll be assessed to be just fine,
If you just smile.