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BOBBY’s question was simply brilliant. It was beautiful like a minimalist painting, just three square matrices sitting in the middle of the page with question marks for the missing numbers.

But despite its simplicity the question was a great test of understanding the process of matrix multiplication… but it also clearly showed that Bobby understood the process sufficiently well to create this question without any prompting.

Writing their own exam questions was the Final for my Algebra 2 classes this summer.

Each student was asked to choose a topic we’d done during the year and create a question. They also needed to list the math concepts and skills needed to answer their question as well as prepare a model answer with notes about likely common errors. Students were also encouraged to add a supplementary question aimed at probing deeper or look for ways to test a number of ideas within one question.

The students then swapped questions and attempted answers. The original authors marked the answers. The authors also explained the answers and ideas if a student got stuck on their question.

So, Cassie gave Bobby a second chance with her question built around the need to understand the Quadratic Formula, without mentioning the formula in the question.

Other questions were pleasingly complex. Becky created a log problem where the base was itself an algebraic expression which was the result of a simplifying task.

Some students also exposed errors in the model answers.

The result of the exercise was some good-humored but thoughtful mathematical discussion that continued the learning process while also revealing something of each student’s knowledge and math thinking skills.

This Final did not produce a series of right and wrong answers and a percentage at the end. But it did avoid all the damage that such traditional finals can inflict on student self-esteem and confidence. Anyone who doubts just how serious this can be should read Testing, Motivation and Learning from the Assessment Reform Group and reviewed earlier on this blog. Teachers, who almost by definition fitted well into the schooling system, frequently utterly fail to comprehend the stress repeated testing causes those not so fortunate, and indeed even many of those who can manage the system.

My own view is that traditional finals tell the teacher nothing that they should not know already. They are more ritual with their roots less in thoughtful pedagogy and more in the no-pain-no-gain vision of education and its emphasis on sorting, ranking and the motivating power of humiliation.

In my algebra 1 classes I tried a compromise… in my school the pressure is mounting to dumb down and get back to those mythical and failed so-called basics to get those damned state test scores up.

Students had the traditional exam, written by a colleague. But instead of sitting in silence and in rows, students were encouraged to try a few questions and then bring them to me. Students got to move around, chat in the small queue and get some immediate feedback plus a one-to-one lesson for a few seconds.

I got to see just what each student could do or not… and keep the teaching going targeted on individual misconceptions.

Sure, by the end of the lesson there had been considerable collaboration among some students as correct answers filtered around the room as if by osmosis. But even that can be a learning process, better than stress and the fear of inevitable failure.

The final answer must be: Do nothing to destroy hope.

Try Bobby’s question for yourself:

Find the missing numbers.

[A] x [B] = [C]

where a₁₁ = 2

a₁₂ = ?

a₂₁ = ?

a₂₂ = 5

and b₁₁ = 2

b₁₂ = 6

b₂₁ = 3

b₂₂ = 17

and c₁₁ = 13

c₁₂ = ?

c₂₁ = 23

c₂₂ = ?

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I WAS never clear what the word “rigor” meant. Mostly, it seemed to be used either by school administrators or by those teachers who seem to take pride in how hard their courses are. The no-pain-no-gain school of education.

A college professor teaching a math course I took last summer mused that he was also unsure what rigor meant, adding, “I think it means we write things down”.

Alfie Kohn tells, in his The Homework Myth, the story of a principal who was asked by a parent if his school provided a “rigorous” education. He hesitated, and added he was unsure until he’d consulted a dictionary. He returned and declared, “Good Lord, No!”

Inspired by this, I too consulted the definition of rigor even in that most traditional and quintessential US dictionary, Webster’s.

Ask me now whether or not my classes are rigorous and I would declare, “I hope not!”

Rigor dates from the early 1300s, the time of The Inquisition.

The Inquisition ruthlessly suppressed any creative or free thought, under the label of “heresy”, as well as the likes of a Galileo. Rigor in education seems to simply equate difficulty with quality. Difficulty for the sake of difficulty doesn’t seem to promote the enjoyment of free or creative inquiry.

I know teachers who boast of rigor in their courses. Some refer to other teachers as “the slacker teachers”. For them learning is about hard work, the harder the work, the better the learning. Kids get off too easy. The way to improve learning is to make it harder and then do repeatedly more of it. They point to their successes with pride but somehow seem to forget those who don’t make the grade. After all they didn’t work hard enough.

In this harder-is-better world, late or poor homework means an invitation to a mis-named after-school homework “party”. Failure to accept the invitation means a referral, detention.

It conjures up visions not so much of the 19^th century sadism of Wackford Squeers and the crushing of the pathetic Smike in Dickens’ Nicholas Nickleby, but certainly the pompous Blimber and his academy of Dombey and Son — “a great hothouse, in which there was a forcing apparatus incessantly at work.”

In the Dickensian Mr Feeder’s class, “they knew no rest from the pursuit of strong-hearted verbs, savage noun-substantives, inflexible syntactic passages, and ghosts of exercises that appeared to them in their dreams”.

Before that at Mrs Pipchin’s, Dickens describes the pedagogy as “not to encourage a child’s mind to develop and expand itself like a young flower, but to open it by force like an oyster”.

Difficult does not equal better. Difficult is a relative term. What is difficult for one student may not be so for another. What is difficult for a student this year, may not be 12-months down the line after the brain has gone through another year of development.

The question in the classroom should not be about difficult versus easy, it should be about finding what the Soviet educational psychologist Vygotsky called the zone of proximal development — that place between the actual development achieved by a child on their own and the potential development that they can achieve with the mediation of an adult or other students.

“The trick,” said Jerome Bruner (quoted by Kohn in The Schools our Children Deserve), “is to find the medium questions that can be answered and take you somewhere.”

Maximum difficulty isn’t the same as optimal difficulty adds Kohn.

Too easy and the student feels belittled, too difficult and the student feels stupid, alienated and likely to lose all interest in the subject.

Kohn adds this footnote: “One technique for finding just the right level of challenge for each student is so simple that few of us think of it: let the student choose. As long as the classroom doesn’t overemphasize performance, doesn’t lead student to think mostly about getting good grades or doing better than others, children will generally seek out tasks that are just beyond what they’re able to do easily.”

But in the rigorous classroom grades and sorting and ranking students are fetishes. Students who suffer the rigorists are never allowed to forget the grade, that’s the point. The percentages and grades are supposed to motivate, when in fact they do exactly the opposite.

Key to finding Vygotsky’s zone of proximal development is collaboration with other students or the mediation of an adult in a cultural context. This clashes with another rigorist article of faith, that a child should prove ability in the isolation of high-stakes tests, based on lessons that are taught outside of a cultural context.

Kohn quotes educationist, reformer and philosopher John Dewey; the value of what students do “resides in its connection with a stimulation of greater thoughtfulness, not in the strain it imposes.”

Parents would be better served by asking not whether a course is rigorous, but whether or not it is engaging and meaningful.

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TESTING can provide some teachable moments.

Imagine. The desks are in rows. One child per desk or sitting alternately on opposite sides. Different colored tests. “No talking,” written on the board. Adults think kids cheat is the message.

And then one student, child, asks something along the lines of I-don’t-get-this-can-you-help-me? The answer? Well, in the traditional classroom it’ll be something along the lines of No-this-is-a-test.

Let’s look at this.

If the purpose of a test is to find out if a student, child, understands an idea… then the question I-don’t-get-this-can-you-help-me? seems to provide the answer with little doubt. Incontrovertible. So, why the No? In any other lesson the same question from the same student would be seized upon, or, one would hope.

Imagine you’re the hapless student. You don’t understand, you ask for help and the teacher says No. You then have to sit there, in silence unable to do the natural thing — ask your neighbor if they can help. Children are hard-wired to talk, ask questions, communicate. That’s how it works in the real adult world too. But not in a test. Ask your neighbor, and you’re a cheat. So much for teaching co-operation, social skills and collaboration.

As the student sits there, confused as well as stuck, what does that do for their self-esteem?

Personally, I agree with the research that is conclusive — testing reveals little about a child’s knowledge and mostly does harm to the learning process. But the US education system seems wedded to testing and not to the conclusions of contemporary research data. And in my school it’s on the up-and-up. Testing is becoming an obsesssion.

Testing is what teachers talk about daily — planning the test, reviewing for the test, putting off the test and then, heads wagging in disbelief, incomprehension as to the results… they-just-don’t-get-it.

Apart from abandoning testing, is there an alternative that meets the needs of those who believe in testing and, more importantly, the confused child who needs help?

This is what I tried this week.

I took the departmentally-agreed questions and buried them inside three half-page assignments each on different colored paper. The green sheet with the word question had two other word questions — not identical questions or the same question with the numbers changed, but questions around the same idea.

The other calculation test questions on the pink and blue sheets were buried in groups of similar questions exploring much the same ideas.

Students were told to bring their working and their answers to me as they completed each question or group of questions. OK, a bit of queue formed, but I was able quickly to spot what was going on with each student and give instant feedback accordingly.

I was also able to build up a list of common mistakes, misconceptions and approaches. I was able to mark some answers to share with colleageues later. I was also able to note the inadequacies of our commonly-decided questions, the ambiguous wording, how students interpreted our questions.

The only questions on which students received no immediate feedback were the magic test questions. Those I just noted, right or wrong. No student noticed as we discussed the surrounding questions.

I didn’t need to shift the desks into rows. There was no big sign saying No Talking. The students go to socialize. Each got individual and immediate feedback without having to ask for it. I got a detailed formative assessment as to the thinking, approach and understanding of each student to guide the next lesson. No one risked being accused of cheating.

And whoever is interested in the test statistics got what they need too.

Testing with teachable moments. Everybody happy.

Get this mind map from The PiFactory

The PiFactory’s student-friendly math learning targets, from the TARGETmath database, are now online as the project slowly moves towards going open source.

The several hundred learning targets, covering pre-algebra to advanced algebra, can be searched here. Many of the targets are also grouped by learning unit.

All the targets are written in the form I can…

For teachers the targets make up a handy To-Do list of stuff to be covered. For students they summarize what needs to be learned. The order of the targets roughly reflects what a student needs to understand before they can move on to later targets.

In the TARGETmath database the learning targets are at the core of a system that links nearly 2,000 questions + notes, builds assignments and gives descriptive feedback and revision advice to students. TARGETmath can be downloaded for macs.

Mindmaps, or concept maps, for many groups of targets are also available from The Pifactory to complement the targets. There’s a thumbnail of the logarithm mindmap at the top of this page.

It is expected that the textbook notes attached to the targets will be available online soon, followed by linked questions.

WHEN I was training to be a math teacher in Britain some years ago there was a popular satirical BBC radio program called I’m sorry I haven’t a clue, presented by legendary jazz trumpeter Humphrey Lyttelton.

Humph arbitrarily awarded points for no apparent reason. The rules of the games were far from clear.

And although the audience shouted “points mean prizes…” on cue, no one ever knew what the prizes were. The only real prize for all concerned being an addictive dose of hilarious nonsense that was also a biting weekly commentary on current events.

Reflecting on the issue of grading, I’m reminded of this farce.

Three years ago I decided to completely abandon all grading based on points, marks, percentages or letters. The only numbers in my classroom would be in the math my students and I were working on, and not in any assessment. I felt naked.

But since then, this simple decision has lead me radically to re-assess everything that goes on in our classroom — from testing, homework policies, seating plans, late work, make-up work, behavior management to the really important question… the very essence of what it is my students and I are trying to achieve in our thinking laboratory.

Now my assessment focuses on observation, listening, discussion, collaboration and increasingly on students’ own assessment of progress. I hope the result is a focus on what we’re trying to think about in the classroom, free of the diversion of worrying about some ill-defined percentage.

The prize is the achievement of just trying to get it.

Feedback is verbal and descriptive, striving to help the student find some way of constructing their own meaning and paths to mathematical thinking.

I changed career to be a math teacher at the age of 50. UK educationists were encouraging new math teachers to abandon the traditional mark-book full of numbers. Numbers have no place in assessment and feedback in the math classroom was the argument.

It was summed up: Students who get nines and tens out of ten, nearly always get nines and tens. It may make them — and their parents — happy, but the points don’t motivate them to seek out new challenges or how to improve. They coast, complacently.

Those who get sevens or sixes… well, they’re passing. They just need to survive the you-could-try-harder talk. But how to try harder? To do what, exactly?

Those with the fives, fours, threes… confused, disappointed, frightened, feeling an inexplicable shame, they give up and cover-up as best they can. I was given a class into which such kids had been herded: “Hi,” they said, “we’re the dumb class.” I felt ashamed.

Numbers in assessment de-motivate, set student against student and encourage assessment as a process of ranking and sorting.

Adding some words of encouragement to the numbers also seems to make little difference. Students still focus on the numbers, despite their lack of any real meaning. Changing the numbers to letter-grades, doesn’t shift the focus to learning either.

Only when you take away the numbers, and letter grades, and replace them with phrases such as, “well done, now you need to work on… the distributive property” does motivation get turned round.

A target, a goal, something to focus on finally clears the clouds of confusion for the student. Now the student can see just what to do to improve. With student self-esteem no longer punctured by the stark moral judgement of a number or letter dripping with blood-red ink, students are free to focus on what they need to do to improve… and work cooperatively together to collaborate and help each other.

BARACK OBAMA and me both spoke out to demand a conversation on the issue of race this week.

My own contribution was greeted with embarrassed, near silence.

Barack’s speech, however, was justly hailed as a “landmark” by The Washington Post.

The New York Times reported Obama “a living bridge between whites and blacks still divided by the legacy of slavery and all that came after it,” who delivered a speech that “addressed the politics of race in straightforward terms that seemed intended to keep the discussion grounded in the realities of the moment”.

A CBS News poll reported an approval rating of seven-out-of-ten for the speech.

I, meanwhile, spoke briefly at an evening meeting at a local middle school before fewer than a dozen educators taking a school leadership course.

Howard, a colleague from the Urban League of Portland, reviewed just-released depressing data from the UL’s annual The State of Black America 2008 report, and we showed Grandma Zula’s Legacy, a documentary about the moving and difficult history of African-Americans in Portland.

The UL’s annual barometer of the collective effects of institutional racism and outright racism in the US confirmed:

❏ While the real median household income for all Americans has dropped, the poverty gap between blacks ad whites has widened with nearly three times as many African-Americans than whites living below 125% of the poverty line.

❏ The number of recent African-American high school graduates enrolling for college dropped from 63% to 56%, while the corresponding rate for whites increased from 69% to 73%.

❏ Mortgage application denial rates for blacks, already twice that of whites, increased by more than 3 percentage points.

Now the purpose of this blog is not to be sanctimonious or sniffy. But just to point up how difficult talking about the issue of race remains in America today. Finding the right words for a dialogue across the racial divide can be a struggle, most are desperate not to repeat words and associations and connotations that have caused so much damage in the past. Or, just to avoid those embarrassing gaffes you make when you’re nervous.

My own qualification for talking on the issue is tenuous. I’m white, so at best slot into the category of well-meaning white liberal. Yes… I too am a little suspicious of white liberals pontificating on the issue of race, coming as I do from various more radical -isms than liberalism. I defer to my partner who is African-American, and my children who are both black.

On embarrassment and nervousness… when my dear mother first met my beautiful black girlfriend (now my wife of 22+ years) with jewels in her braids and African silver on her wrists, she proudly announced as they watched the BBC that The Black and White Minstrel Show (this was 1985) was her favorite. My mother had never had a black person in her house before. Such cringe-making tales are the memories of every mixed-heritage relationship. My mother and her daughter-in-law now get on famously.

The UL and I had been asked to say a few words as this year we’ve worked together, along with our school principal and some other teachers, to try to support my school’s Black Student Union and facilitate black parents organizing a support group.

We chose to try to briefly point up the issue of institutional racism, how institutions can unwittingly discriminate through policy, assumptions and uncritical practice.

“Institutionally racist” was how Sir William Macpherson of the UK had branded the London Metropolitan Police in his report addressing a catalog of police incompetence and mismanagement — which many claimed was fueled by racism, corruption, bribery and deliberate sabotage — during the bungled investigation of the 1993 race-hate killing of 18-year-old student Stephen Lawrence by a white-supremacist gang well known to the police.

While the names of Stephen’s killers are in the public domain — Brothers Neil and Jamie Acourt, Gary Dobson, David Norris and Luke Knight — none has been charged with the murder. The Met continues to deny to this day that corruption or racism affected the investigation, and that action against the known five thugs is not possible.

Macpherson concluded institutional racism was “the collective failure of an organisation to provide an appropriate and professional service to people because of their colour, culture or ethnic origin”, which “can be seen or detected in processes, attitudes and behaviour which amount to discrimination through unwitting prejudice, ignorance, thoughtlessness, and racist stereotyping which disadvantages minority ethnic people”.

Stokely Carmichael had used almost exactly the same definition.

But Macpherson’s report did change British policing.

As in most other schools, I imagine, there are statistics that show we may not always be getting it right. In my school’s case, the referrals given to African-American students by a mostly white teaching workforce seem to be disproportionately high. This may indicate a problem worthy of some discussion. It does not indicate that individual teachers harbour racist attitudes, or are racists. But it does indicate that the school as a whole needs to examine its collective cultural awareness, assumptions and practices.

The fact that after these figures were announced a senior administrator felt the need to apologize to the staff for raising the issue, shows there most definitely is an issue to be addressed. The focus became one of the administrator’s manner of raising the problem… not the disproportionate number of referrals. Needless to say, neither issue has been raised since. Silence.

That’s institutional racism.

Race is not an easy subject to discuss. It often ends in a diversion rather than a discussion. The tantrum of the offended white, with the how-dare-you-call-me-a-racist type of response. Or, the counter accusation that it is the blacks who are really the racists.

Or, the the blacks are playing-the-race-card jibe.

Then there’s the chip on the black shoulder. Why don’t they take the opportunities open to them… this is a land of opportunity. Wy can’t they be more like us. Stop whining.

Even in success, there’s often little credit… often just snide pointers to affirmative action (ignoring the traditional affirmative action programs of the white middle-class which have repeatedly ensured privileged places for otherwise failures… right up to sitting in the Oval Office).

In discussions about race it’s usually the blacks that get the blame. Blacks may have been racism’s victims — racism as an ideology was invented in a bid to maintain the continuing support of poor whites, in particular, for a system of divide and rule at a time when many poor whites were realizing it undermined their interests as well as crushed those of the blacks — but it’s still the Blacks’ fault.

Meanwhile the insensitivities that compound the issue continue.

After the student newspaper at the majority-white Oregon State University in Corvallis carried a picture of a white student with a minstrel-like blacked up face, the paper’s editor rounded on her critics for not having a sense of humor. Students turned up for a football game wearing Afro-style wigs as well as blacked-up faces. The Corvallis campus was later the scene of a noose hanging that was allowed to stay up for days. Other nooses appeared at Central Michigan University, Columbia University, and the University of Maryland at College Park.

This was when the national headlines were dominated by the Jena six outrage of black students being accused of attempted murder and being jailed after fighting broke out after three nooses were hung from the “white tree” in their school.

In Mississippi and the south killings of civil rights workers of the ’60s are still unprosecuted and unresolved.

The lynchings have never been acknowledged as murder and the terrorism of one community against another. These cowardly mob murders have never been even contemplated as worthy of some of justice by the majority community… even though their perpetrators often posed for gloating group photographs that are still easily found in the records of numerous newspapers. And people wonder why blacks can still be angry? Many families still mourn the victims.

Little mention of the legalized discrimination in house loans, red-lining, bigotted judges, planned neglect of black neighborhoods, racism in the jobs market…

Collectively America, has been, and still appears to be, reluctant to discuss the issue. It’s the elephant in the room. And the elephant is smashing up the whole house. The whites fear the guilt and resent the potential accusations. The blacks hold in the anger and then get fed-up with what feels like a waste of time.

Perhaps there needs to be some form of Truth Commission, as there has been in both South Africa and Northern Ireland: a sort of putting all the cards on the table. A massive clearing of the air.

My partner worked for years in the UK on tackling racism and racism awareness training, particularly organized by the trade union movement. Post-Macpherson, such training is now often mandatory in the UK, in police forces for instance, and is increasingly accepted as part of the tool-kit for any self-respecting HR department.

Her recollection is clear: Little could be achieved until participants had had many hours of getting know each other, gently edging themselves forward into a safe space. Such courses rarely ran for less than three days, and more often a week. Even then the weight of personal history and the tension for many was too much. Finding the words that could express and overcome decades of fears, misunderstandings, misconceptions, accusation and counter-accusation was often too much.

Brits and Americans have been bequeathed a similar legacy of history. The wealth of the UK and the privilege of its white majority population were built on slavery in the Caribbean, the stripping of the resources of India and thieving the output of cheap labour across an empire.

The exploitation of people of color fueled an industrial revolution and provided the capital that underpinned the British economy and the life-styles of its middle and ruling classes until well after the second world war. The British flag is still reviled across much of the world with good reason.

So, it was good to be invited to speak at the local middle school this week for an hour. Me and my friend Howard stumbled in our delivery, forgot to say what we meant to say and badly expressed what we did say.

Our audience, sitting as far away as possible, engrossed and fumbling continuously with a textbook in one case, and in embarrassed, eyes-down silence in others, sat and listened… or just endured. It was the start of a difficult dialogue for us all.

So, how much more impressive was it for a mixed-heritage black American to stand up for an hour on his own without notes or prompter, in the glare of lights and TV cameras, with pundits waiting to pounce, and try to find the words to create the safe space for black and white to start to have the conversation. He did it well, brilliantly.

The least we can do is read Barack Obama’s speech. And quietly sleep on it.

Post-script: Taking the lead from me, Howard and Barack and this blog, The Oregonian also addressed the issue in an article entitled Race remains a touchy subject even in progressive Portland and carried a fact-box on entitled Why we avoid the issue

LIFE is like a business the father was telling me and his child. As he warmed to his theme he lost me, so I can’t recall whether his child’s grade was part of the product-mix, the prospectus or the marketing plan. Certainly it needed to be factored into the matrix for the overall offering… viewed helicopter, in the big picture on a blue sky.

This week was spring parent-teacher conferences at my school.

First, they work better when they’re student-parent-teacher conferences, with the student taking the lead.

In my old school in the UK it was really a big point that the conference was student-centered and student-lead. It works much better. Students almost always know how they’re doing and what they need to do in terms of effort, attitude and the like. And when they say out loud what’s needed for them to improve, then it comes much more forcefully than if the parent or teacher was to say the same thing.

You could call it a student self-assessment conference.

It’s also a chance for the child to see the adults most concerned about them, sit together and listen to their point of view. The listening is important. Not all parents, or teachers, are good at that.

It’s also a great opportunity to get beyond just the disciplinary stuff that parents and many teachers seem obsessed with… talking in class, not listening to the teacher and other stuff like that.

Are you happy? is a good question to ask a kid at a student conference. Is it working for you? is another. Is there anything we can do to help? What do you think? And then listen.

Paul, the head of my old maths (with an ’s’ in the UK) department once told me one of those secrets-of-being-a-good-teacher… once the kids know you’re on their side, then it’ll work. And he was right.

The fall parent conference, in particular, is crucial to telling a kid you’re on their side. The are-you-happy? question can go a long way towards changing the student-teacher relationship overnight. If before you were adversaries, then after teacher and student can start to be a team working together to help the kid.

Here in the US the parent-teacher conference can be obsessively grade focused.

Armed with pencil and list of courses, parents go from one teacher to the next to ask the single question: “What’s the grade?” This can then be followed by versions of the good-cop/bad-cop threat and bribe speech on the need to get/keep those grades up. The kid rarely wins. Even a good grade often ends in a speech along the lines of well-we-need-to-keep-this-up-no-slacking-off-now.

Last year I had a father and daughter. He had taken an evening class in business leadership I believe. “Right,” he barked in opening, “I’m taking control of this meeting…” Clearly, it was not expected that I would respond. “Grade!?” he shouted.

She sat next to her father in terror, her face quivering with emotion. Overwhelming fear, but with a look in her eyes that appealed, screamed at me, “what you now say could condemn me to absolute misery and pain for the rest of my life… and certainly as soon as I leave this conference.”

Normally, the grade question is my cue to deliver my I’m-not-so-concerned-about-the-grade-I-want-the-student-to-be-focused-on-what-we’re-learning speech. This time it didn’t seem as if it would work.

The student in question was an absolute pain to try to teach. She was funny, cocky, self-confident, mouthy… and completely focused on destroying any progress in my class till she’d had the last word. The secret, of course, was to work out how to let her have the last word… and move on. Not always so easy. She often just got chucked out of class. Somehow all our battles, tussles every other day now made sense.

“B,” I invented. “She could get a B if…” I tried to continue, but by then the father was delivering his own speech that B wasn’t good enough. The lie had not really saved her, and alas it did not in the end do much to save or help us build a new relationship in class. In the pantheon of win some, lose some… I lost her. So, I suspect, did her father.

But back to my own don’t-focus-on-the-grade speech. Well… it’s three paragraphs above, and then I give the parent a rubric describing what it is to be an A, B or C student. I ask them to give it to their child and ask the child to tell them what their grade is. The kids are usually far more tough on themselves than I. And the rubric is like a To-Do-List of how to improve. Most parents are intrigued by the descriptions and seem to like the idea.

There are formulae describing this:

Grades = stress

Stress = no learning

The point is, judgemental letter grades (destructive, and certainly meaningless in the absence of any descriptive definition) do nothing for learning. Non-judgemental descriptions provide a learning-centered focus. And so, do help learning.

One last aside before I conclude: Grades can only serve a useful purpose if they help motivate a student to build meaning for themselves. Grades, if they must be used, need to be divorced from the destructive rank-and-sort mentality that so obsesses much of the US corporate education system.

Such grading guarantees Fs… it is a system of success and failure. The fear of the F is what motivates the successes going. You can’t have the fear, if you have no Fs. The system depends on the Fs for its seeming success. Except, it is not successful. The Fs are branded, and the successes often have little clue how to think creatively or critically for themselves… only how to collect points and jump through repeated hoops.

Many teachers feel most comfortable with a handful of As, lots of Bs and Cs and some Fs. The teacher who gets this out of balance is held in suspicion. Ranking and sorting is part of the job. Except it isn’t. The job is to help every student find their success within. Every class should be a classroom of As… or, rather, a classroom of positive and individual and unique successes.

Grades should serve the learning needs of each individual student as if they were in a bubble. The grade should have no role in comparisons with other students. Which means, of course, grades are redundant.

Far better would be to replace letter grades with descriptions… as some more progressive colleges and schools are now doing.

Finale. Of course, it is easy to be sniffy about grades and grade-obsessed parents. But there are real reasons driving very real concerns.

Grades are linked to numbers. The numbers build up an average, the Grade Point Average (GPA).

Forget the learning. The difference between an A or a B or a C could mean the difference in a GPA that qualifies for getting a scholarship or not getting a scholarship. It could mean the difference between affording a respected college or less-respected college or no college.

The obsession goes on. It could mean the difference between a good job… or even no job. A house, a health plan.

GPA is sorting, ranking and competition in an unforgiving society committed to competition… with no safety nets. This is a society that lives in fear.

For parents, it’s the stuff of sleepless nights. And for students too.

After all in America, education, as well as life, is a business.

BRODY got stuck. And did exactly as I’d requested: he turned over the worksheet and completed the questions.

Where am I stuck?… Why am I stuck?… How am I going to get unstuck?

He confessed. Where he was stuck was right at the start.

Why was he stuck?… He’d got stuck because he hadn’t read the question carefully.

How was he planning to get unstuck?… read the question carefully.

It was the first time I’d used these particular student self-assessment prompts on the back of an assignment. Brody’s cooperation was helpful… to us both.

For him, it was not a bad lesson to realize that it helps to read the question carefully before you start. For me, it gave me the opportunity to talk again about student self-assessment, and the advantages of self-reflection and students’ taking responsibility for their own learning… as well as reading the question.

Soon, much of the rest of the class was submitting the assignment with notes on the back. Some even noted math difficulties. Mick was smiling broadly when he gave me his sheet that said he was frightened of exponents! But if he remained calm he could overcome his fear and learn to love them. This smacked of the discussion on how even negative numbers need someone to go to the prom with.

And then there was Anya who wrote “Nowhere” to the question where. “I’m not stuck” to the question why. And “Nothing” to the question of what she intended doing about getting unstuck. Seems reasonable.

Five million points for all those who completed the self-reflection.

Other assignment designs call for an explanation of how students have tackled a problem. They all contain a box along the lines of…

❏ Got it!… I can do this without help + I can help others

❏ Almost got it… I can figure it and sometimes help others

❏ Getting it… I’ve got the idea + I can do it with a little help from my friends

❏ Good start… I can do it with help step-by-step

❏ Starting… I can start problems and ask questions to get help

Self-assessment rubrics can be downloaded at The PiFactory assessment page

JJ’s question was a bit of a knock-out blow. Not just because it left me stumped for an answer — nor because I didn’t have the self confidence to let a class discussion find the answer — but because… well, to be frank, I hadn’t thought up till then that JJ could ask this level of question.

My mistake. My assumptions. My ignorance. Sorry JJ.

We’d been looking at writing an explicit equation for an arithmetic sequence. We’d done the recursive routine, now we were finding the equation based on the n, the position of the n^th term of the sequence. Arithmetic sequences involve repeated addition — or subtraction — of a constant, the common difference.

Over several terms of the sequence I’d tried to demonstrate, in true traditionalist didactic style, that this repeated addition was the same as multiplication.

Out of the blue JJ asked, “does that mean repeated subtraction is division?”

Wow! We’d bashed away at inverses. The inverse of addition is subtraction, the inverse of multiplication is division… I could see where JJ was coming from. But I was thrown, I didn’t even feel confident enough to say that the answer was yes.

But if JJ asked that level of question, he didn’t want a yes, he wanted the explanation. He wanted to see it built up on the board. And he deserved his explanation.

This is what is called a teachable moment. Teachable moments are supposed to be gifts for teachers. They are also tests of teachers.

Well, I failed the test.

But I was determined to try to seize the moment, even if the moment was to go through a time shift… and a couple of weeks later JJ did get his explanation (see the end of this post).

It wasn’t so much about the question or JJ’s mathematical curiosity. It was about JJ’s belief that he could build some meaning for himself in a subject he would happily tell you isn’t his favorite.

I’d first taught JJ in Algebra 1. He wanted to get it. And he’d try, he really would, but getting three or four consecutive steps correct and then getting the right answer… discouragement was always near at hand.

It was the first year I’d started to think about and experiment with throwing grades — damned confidence and learning killers — out of the window. JJ so desperately didn’t want to get the F, but he just didn’t know how not to get the F he expected. It came up again and again… his mom. It wasn’t the fear of parental anger, but he just didn’t want to disappoint his mom. JJ felt unable to deliver.

That year, I announced: if you turn up, try, don’t smash up the classroom… there’d be no Fs. I can’t remember JJ’s grade that year, but it wasn’t an F.

First lesson this year, JJ was quick to re-check the rules hadn’t changed. Smiles, thumbs up… telling the class it was going to be cool.

Since then JJ has never had the facial expression of early in his first year with me… that look of overwhelming worry, the furrowed brow, has gone. Sometimes, too often, JJ the football star talks too much about football or basketball or something to do with balls. But some work comes in, and increasingly frequently the attempts to make contributions in class discussion.

And then came the question. JJ was thinking mathematically. And he felt sufficiently confident to form the thought and say it in front of a class.

On his end-of-quarter progress report, I can remember JJ got a B. But more important were the sentences on his report card describing that he’d demonstrated mathematical thinking skills. He came up and shook my hand and thanked me for the words… his mom had been delighted he said with a huge grin.

❏ Start at 0… repeatedly add 5

0 + 5 + 5 + 5 + 5 → 20

0 → +5 → (x 4) → 20

0 ← −5 − 5 − 5 − 5 ← 20

0 ← − 5 ← 5 ← (÷ 4) ← 20

Of course, repeated subtractions are multiplications. But division is also multiplication… multiplication by the reciprocal.

My colleague, Roscoe, said that multiplication counts the number of additions from zero… and division counts the subtractions back to zero.

Any other explanations out there?

EIGHTY-PLUS adolescent brains are walked into my classroom each day by their legs. The following 80 minutes working with each brain can be interesting, or just plain confusing, or maddeningly frustrating. On the whole it’s fun.

Then each evening plus weekends there are the two teenage brains that sometimes I find wandering my home. They slump by and crash through a door complete with a surgically-sprouting phone, or, they rush up and give me a hug and lift me off the floor. I’m ok with that.

So, when I found a copy of Why are they so Weird? What’s really going in a teenager’s brain in Portland’s famed Powells I was intrigued.

Its theme is that it is not just raging hormones that make many adolescents such joyful hard work to be around… new neurological research indicates that major changes in the growth of the brain during the teenage years may also be having a much bigger affect than previously thought.

As author Barbara Strauch reports, “most scientists working in this area today think that changes taking place in the brain during adolescence are so profound that they may rival early childhood as a critical period of development”.

She concludes: “Indeed, the remodelling of the adolescent brain — a brain that science had considered largely finished — spreads over such a wide range of systems we should rethink how we think of teenagers altogether.”

Since the late ’90s the advent of the MRI scanner has helped scientists peer into the adolescent brain, for the first time in terms of serious research. The pioneer of this research, Dr Jay Giedd, concludes, “we shouldn’t give up any teenager, there is hope.”

There are serious implications for education in this research. Perhaps the teenage years are too early for us to be labelling our young people as successes and failures. Probably these are the years for second, third… as many chances as are needed.

Jay Giedd, a neuroscientist at the National Institutes for Health, started scanning teenage brains in the late 1990s. Hundreds of them. He detected major changes in areas of the brain associated with logic, spatial reasoning, problem solving, language… and most importantly in the frontal lobes behind our foreheads.

It is the frontal lobes that guide our reasoning, decision-making, our judgement, our motivation, that help us plan ahead, that say stop that impulse isn’t the best idea. The frontal lobes are the chief executive and the police wrapped up in one. Developed frontal lobes are what make us grown up. As neuroscientist Chuck Nelson summed it up: “This is the part that tells you to count to ten before you call your mother old and stupid.”

Giedd found the frontal lobes — the very area that makes adults do the right thing — “are one of the last areas of the brain to reach a stable grown-up state”. He adds:”… perhaps not reaching full development and refinement until well past age 20″. It can be as late as 25 in boys. 25!

The teenage brain remains far from finished, says Strauch, “it remains a teeming ball of possibilities, raw material waiting to be synaptically shaped.”

Says Giedd: “If that teenage brain is still changing so much, we have to think about what kinds of experiences we want that growing brain to have.”

Yet the teenage brain seems to be growing in an ever-confined space. One Washington DC school counsellor told author Strauch: “We’ve simply made schools impossible for the regular kid. There are not enough options for how to be a successful teenager.”

A 16-year-old told Strauch the world was too black-and-white, with academic success the only barometer for success. “It seems you have to go to Harvard — or you will be a druggie and drop out.

“There don’t seem to be any in-between choices. People just talk about getting into a good college all the time; they pound that into you. They never talk about being a nice person or having a good marriage or a nice family.

“It’s all about grades. And there isn’t any room for mistakes.”

Giedd is unsure what influence we can have on the developing brain. “We may find out that all we can do is tinker around the edges…

“But we might find out that there are things we can do to improve things. My guess is that, if that is so, it’s going to turn out to be something we already know about.

“And we could find out that the way to make a better brain is not through four hours of homework.” Knowing what he does about the teenage brain, Giedd often lets his own four children decide for themselves how to use their own free time.

“What we might find out, in the end, what the brain wants is play… what if the brain grows best when it’s allowed to play?”

Bob Blum, professor at of the University of Minnesota, who has analyzed much of the data in the National Longitudinal Study of Adolescent Health concludes of the neural development of the teenage brain: “Ten years ago there was nothing. Now, I think it will be the frontier of the field for the next ten years. It will change the whole debate about adolescents. It will have huge implications for policy, for laws.

“It will change the whole way we think about kids. Forever.”