Adolescence, a time for second, third… as many chances as it takes

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.”

Assessment… it’s all in the tone of the voice

DARIO is one of my outgoing, great-fun students. Fabulous and generous sense of humor, and a little loud and a little mischievous with it. Dario is fun to work with. He also enjoys his math, works hard at it and wants to get it.

Dario was frowning as he tried to follow just why ¹/_x⁻¹⁸ is equivalent to x¹⁸.

We reviewed that x⁻¹⁸ is the same as ¹/_x¹⁸. So, ¹/_x⁻¹⁸ was the same as 1 being divided by the fraction ¹/_x¹⁸.

As earlier lessons had focused on the laws of exponents, I tried a different tack… thinking about 1 divided by ¹/₄. We drew a pizza cut into four and talked about the quarters and how four of them made up the whole pizza, and how that was like division, that ¹/₄ divided into a whole pizza four times. We talked about the pattern that seemed to be there, that the denominator of ¹/₄, became the numerator in the answer 4.

Clearly my flounderings were not helping Dario construct any meaning around all this. Everything screamed that he just couldn’t see it, did not get it. There was no light bulb flashing, or even, about to flicker. But Dario looked at the accumulated doodles and said slowly, “Oh, I get it.” I clapped him on the back and said words to the effect, “I don’t think so Dario, but don’t worry, you will.”

Dario’s friend, Joe pushes across the his work to solve 125^{2x + 4} = 25. Joe is more quiet than Dario.

Dario looks to Joe for help, but also asks probing questions. Dario doesn’t just want the answer, he does want the why behind the answer.

Joe laughs at Dario’s jokes… particularly if they’re good naturedly at the expense of the teacher.

I look at Joe’s work and start to venture that “the exponent…” But before I can get any further the paper has gone and Joe quickly says, “Oh, I get it” and smiles. The work comes straight back, the exponent corrected.

Assessment isn’t right or wrong answers in a test. It’s all in the tone of the voice.

Annah and Camilla get unstuck… and 5 million points.

ANNAH and Camilla have got themselves unstuck. And they’ve written a couple sentences each on where and why they got stuck… and how they got themselves unstuck.

They’d had trouble tackling an expression with a negative exponent, and writing it as a positive exponent. Their strategy was to look up their notes, find a similar problem and see they could take the reciprocal and change the sign of the exponent.

I can only hope they found in their notes the demonstration and informal proof using the number 2 and meant to show why expressions with negative exponents are, effectively, fractions… rather than just use the bulleted-rule “flip it and change the sign of the exponent”.

The demo takes, say, 2³ and divide it by 2⁵. By writing out the three factors and dividing them by the five factors it is clear from cancelling that the answer is ¹/_2². It is also clear from such demonstrations that the answer can be derived from subtracting the exponent 5 from the exponent 3… 2⁻².

Annah and Camilla’s brief note didn’t go in to even this limited level of detail. But at least they wrote some sentences of reflection and, indeed, had reflected on the problem. It’s a welcome start.

Student self-assessment is meant to help students focus on their own work and their own learning… to take responsibility for getting themselves unstuck. At the moment students seem reluctant to reflect, let alone to leisurely reflect and to intellectually wander and to ask what if? Getting the answer, and as quickly as possible, mostly seems preferable.

I’ve tried to break this by simply abandoning traditional assessment approaches that inevitably value the answer, rather than putting the main focus on the mathematical thinking provoked by the problem.

The closest my classes have got to an heuristic approach to questioning and thinking is when we’ve gone completely off-curriculum and thrown up a couple of problems and let the conversation and argument go where it will.

Congratulations to Annah and Camilla for stopping, looking back, thinking and working out how to solve the problem. Five million points!

Student self-assessment − the research says…

Student self-assessment… what does the research say?

Education Place: When students are collaborators in assessment, they develop the habit of self-reflection. They learn the qualities of good work, how to judge their work against these qualities, how to step back from their work to assess their own efforts and feelings of accomplishment, and how to set personal goals (Reif, 1990; Wolf, 1989). These are qualities of self-directed learners, not passive learners. As teachers model, guide, and provide practice in self-assessment, students learn that assessment is not something apart from learning or something done to them, but a collaboration between teachers and students, and an integral part of how they learn and improve.

The Center for Development and Learning: Self-evaluation is defined as students judging the quality of their work, based on evidence and explicit criteria, for the purpose of doing better work in the future. When we teach students how to assess their own progress, and when they do so against known and challenging quality standards, we find that there is a lot to gain. Self-evaluation is a potentially powerful technique because of its impact on student performance through enhanced self-efficacy and increased intrinsic motivation.

Testing, Motivation and Learning, Assessment Reform Group: The degree to which learners are able to regulate their own learning also appears to foster pupils’ interest and to promote focus on the intrinsic features of their work. Pupils who have some control over their work by being given choice and by being encouraged to evaluate their own work are more likely to value the learning itself rather than to focus only on whether or not it is correct.

Beyond the Black Box, Assessment for Learning Group: Current thinking about learning acknowledges that learners must ultimately be responsible for their learning since no-one else can do it for them. Thus assessment for learning must involve pupils, so as to provide them with information about how well they are doing and guide their subsequent efforts. Much of this information will come as feedback from the teacher, but some will be through their direct involvement in assessing their own work. The awareness of learning and ability of learners to direct it for themselves is of increasing importance in the context of encouraging lifelong learning… But it is important to remember that it is the pupils who will take the next steps and the more they are involved in the process, the greater will be their understanding of how to extend their learning. Thus action that is most likely to raise standards will follow when pupils are involved in decisions about their work rather than being passive recipients of teachers’ judgements of it.

More to follow…

I’m stuck! − do I get 5 million points?

THE CLASS burst into cheers and applause. Annah and Camilla looked embarrassed, but pleased.

They were stuck!

This was a great moment, they were about to enter the process of learning. But only if they were prepared to get themselves unstuck.

Getting stuck is the new joke in my classes. Students put up their hands and announce, “I’m stuck, do I get five million points?” Only if they get unstuck.

This semester the target is to get more students working on assessing their own work, student self-assessment. The argument is that if they can critically assess the quality of their own thinking, they will take more responsibility for their own learning.

I’m focussing on pushing them to work on that point when they get stuck. Why have they got stuck? Where have they got stuck? Just what is it that they don’t understand? What can they do to help themselves get unstuck?

And when they’ve got unstuck, could they get unstuck again? Could they explain it or help another student to get unstuck?

For the past couple of years I’ve used a number of rubrics to try to help students assess their own work. At the end of each semester I ask them to tell me what they think their grade should be, and why… using a rubric describing the grade levels. A tear-off form gives them the opportunity to anonymously grade me.

Basic day-to-day formative assessment in the classroom is based on student-friendly learning targets and the idea of starting through getting it to got it. The descriptions focus on talking math, to what extent students can explain or discuss or simply formulate a question about an idea.

Assignments and worksheets and working paper all contain five check-boxes with brief descriptions to help students tell me how they are getting on:

❏ 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

Lately assignments include prompt questions to help students reflect as they search for why they are stuck and then reflect on how they got unstuck. Fold the assignment from the left-hand edge to the center and read questions which help find why the student is stuck, such as:

❏ What have I done that’s OK?

❏ Is my answer reasonable?

❏ Have I missed a negative or made a calculation error?

❏ Where have I got stuck?

Fold from the right and find more reflective questions, such as

❏ What did I not understand when I got stuck?

❏ How did I get unstuck?

❏ Could I explain what I did, or help a friend to get unstuck?

For my own use, the rubrics also contain descriptions or the sort of things students say, and their behaviour, at different points in the starting-getting-it-got-it cycle. Plus how they cope with processes and concepts. That’s work in progress. Feedback and suggestions would be appreciated.

So, the big question is… will Annah and Camilla get themselves unstuck? Watch this space.

Assessment — when the numbers don’t add up

Buy the Mathematicians have Great Powers tee

ASSESSMENT and numbers don’t go together, they’re not like terms.

Yet numbers are still at the heart of most grading done by teachers, even math teachers.

Numbers in assessment do only harm.

For a start they are grossly inaccurate, offering only spurious objectivity. But mostly they divert students from focussing on their learning. Ditto if you replace the numbers with letters.

In short, alpha-numeric grades suck.Students focus on their learning when they no longer feel the need to focus on grades. Shift the feedback from numbers and letters to descriptive feedback, and you shift the focus from grades to learning. Focus on the learning and forget the grades.Let’s look at the accuracy.

A teacher scores assignments on a range of 0 to 4. He, or rather his percentage-based grading software, converts the number to a percentage.Say he has a student who mostly scores threes. Except on some days it is raining and our teacher is grumpy and gives a two. Or, on sunny days he is happy and feeling generous and gives a full 4.

Over, say, 12 assignments the average score would be 36. If it was always raining, the score would be 24. If always sunny the score would be 48. That’s plus or minus 12 out of a total possible score of 48, plus or minus 25 per cent.Now, let’s assume our teacher only swings 0.5 either way. If it always rains, the score would be 30, or if it shines as high as 36. That’s plus or minus six out of 48, or plus or minus 12.5 per cent.

Now let’s assume our teacher is obsessive. He scores directly as a percentage, a number out of 100. Presumably he must have have 100 criteria on which he is making the decision… 1 per cent per criteria.

Well, let’s assume the plus or minus on the average grade of 75 per cent is between say 70 per cent at its lowest on a rainy day and 80 per cent for sunny days. Then that’s plus or minus 5 per cent.

At the end of the semester is the student grade C, B or A? Well, the student could have gotten some 4s on sunny days, so on some days the student was grade A. On rainy days, the student was struggling to get a C.In percentage terms the student was probably a secure B. Or, maybe not. What if in percentage terms the student was plus or minus 5 per cent on an average of 85 per cent, or even a little higher. A or B?

But on what basis was the teacher scoring? How to define a score down to one point out of 100, or even five points out of 100? Against what is the score, percentage, defined? After all, mathematically a percentage must be out of something. What is the something? Does the student know. Indeed, does the teacher know?

Some teachers are confident they do. I’ve watched as a student, eyes full of tears (of anger or frustration?) appealed an 89 per cent and seen the teacher abdicate their professional judgement and refuse to budge from the magic computer-generated number and concede the A. She must have been confident her grading was consistently well within a margin of error of less than one in 100.I’ve seen a teacher post percentages to three decimal places… presumably the teacher had a rubric defining their grading down to 100,000 criteria!

As Alfie Kohn has pointed out in his inspirational The Schools our Children Deserve “what grades offer is spurious precision — a subjective rating masquerading as an objective evaluation.”

Research, Kohn says, has long been available confirming what all teachers know: any given assignment may well be given two different grades by two equally-qualified teachers. “It may even be given two different grades by a single teacher who reads it at two different times,” says Kohn.

Quoting Paul Dressel’s 1957 article Facts and fancy in assigning grades, Kohn says a grade “is an inadequate report of an inaccurate judgement by a biased and variable judge of the extent to which a student has attained an undefined mastery of an unknown proportion of an indefinite amount of material.”

As Kohn says: “A teacher can meticulously record scores for one test or assignment after another, eventually calculating averages down to a hundredth of a percentage point, but that doesn’t change the arbitrariness of each individual mark.”

But what if a teacher counts right answers, surely that must give an objective assessment?

What about the student who clearly understands the concept, but made a silly computational error? What about the student who gets the right answer by successfully repeating the steps of an algorithm, but who cannot explain why the algorithm works or what it means?

Does that assessment say much about either student’s learning? More to the point, does the assessment do anything to help either student achieve learning?

So, what about numbers, points, percentages, letters… and focus on learning?

As the Assessment Reform Group has concluded: “Feedback that emphasises relative performance, for example marks or grades which are formally or informally compared with those of others, encourages pupils to concentrate on getting better grades rather than on deeper understanding.”

Alan Blankstein in Failure is NOT an Option, arguing that “grades and test scores do not reflect what children are really learning,” points to an example of a child whose “intrinsic motivation to learn and do well has been replaced by an external motivator: grades”.

As Alfie Kohn concludes: “Research has found three consistent effects of traditional grades: students think less creatively, they lose interest in what they’re learning, and they try to avoid challenging tasks.

“Thus, rather than trying to improve techniques for grading, we should be looking for alternatives − and rather than complaining that too many students are getting A’s, we should be worried that too many students think that getting A’s is the point of education”

The research gives testing an F

Get math mind maps from The PiFactory

FINALS week and the school goes effectively into something akin to lock-down. Students are tense, withdrawn, depressed. So are the teachers. For me it’s the most depressing week of the semester.

For the past couple of years I’ve managed to work round it completely. Instead of a final where students work in isolation for an hour-plus to show how well they cope with stress, I’ve organized sessions where students pick a couple of challenging questions from a packet and work collaboratively in groups to show their best work.

Each question is labelled with the math skill most needed. The kids are asked to pick questions based on the skills they’ve not fully understood during the semester.

The room has usually been buzzy with chatter. If they get stuck, I help out. When everyone has had enough the kids complete a form explaining why they should get the grade they think they deserve. Then we play chess.

The final grade is based on a descriptive grades rubric which the kids were given on day one of the semester. All work throughout the semester is graded on descriptive rubrics.

The policy of other teachers varies: Some run sessions which amount to little more than giving a quiz right through to the full-blown terror thing. 

Finals brings out the simmering discussion between the Skinnerite behaviourists who sincerely believe you train students to behave like learners in the hope they become learners…and constructivists who believe students construct meaning and learning for themselves and their teachers try to create the atmosphere where that happens. 

I’m a constructivist. And I don’t believe testing helps students create their own meaning which in turn becomes learning. In fact, I believe testing can be a crippling blow to the learning hopes of many pupils.

My aspirations are summed up by this brief section from Alfie Kohn’s The Schools our Children Deserve:

“… the best teachers do not rely much on paper-and-pencil tests because they rarely need them to know how their students are doing. Teachers who base their practice on a constructivist theory of learning are always watching and listening… this kind of informal assessment is continuous, making things like quizzes very nearly superfluous.

 ”We might even say that the more a teacher needs formal tests to gauge student achievement, the more something is wrong. (With direct instruction, the teacher is talking more than listening, so traditional exams would be seen as necessary.)

“As parents, we shouldn’t be worried about teachers who rarely give tests; we should be worried about those who need to give frequent tests because they have no feel fo how their students’ minds work.”

So, unburdened by the need to spend hours with my red pen ticking and counting and working out percentages to many decimal places… I spent some time reading some research on testing.

The Assessment Reform Group, based at the UK’s Cambridge University school of education, has been in the lead in argueing for constructivist forms of assessment. Its pamphlet Testing, Motivation and Learning a classic. It reviewed 180-plus studies. It is tempting to simply quote the document in full… but here’s a representative selection of its conclusions.

Testing has a negative affect on pupil motivation.

The self-esteem of low-achieving pupils is particularly adversely affected by testing.

The testing ethos narrows the curriculum.

When tests pervade the ethos of the classroom, performance is more highly valued than what is being learned.

Repeated test practice encourage pupils to avoid responsibility and effort. It is detrimental to higher-order thinking.

Pupils should not be faced with tests in which they are unlikely to experience success.

“What emerges is strong evidence of the negative impact of testing on pupils’ motivation… Many aspects of the impact have significant consequences for pupils’ future learning. and thus are causes for concern.”

“One impact of the tests was the reduction in self esteem of those pupils who did not achieve well.”

“Pupils are aware of repeated practice tests and the narrowing of the curriculum. Only those confident of success enjoy the tests. In taking tests, high achievers are more persistent, use appropriate test taking strategies and have more positive self-perceptions than low achievers. Low achievers become overwhelmed by assessments and de-motivated by constant evidence of their low achievement. The effect is to increase the gap between low and high achieving pupils.”

“Lower achieving pupils are doubly disadvantaged by tests. Being labelled as failures has an impact on how they feel about their ability to learn. It also lowers further their already low self-esteem and reduces the chance of future effort and success. Only when low achievers have a high level of support (from school or home), which shows them how to improve, do some escape from this vicious circle.”

“When tests pervade the ethos of the classroom, test performance is more highly valued than what is being learned. When tests become the main criteria by which pupils are judged, and by which they judge themselves, those whose strengths lie outside the subjects tested have a low opinion of their capabilities.”

“The use of repeated practice tests impresses on pupils the importance of the tests. It encourages them to adopt test-taking strategies designed to avoid effort and responsibility. Repeated practice tests are, therefore, detrimental to higher order thinking.”

“The evidence suggests that teachers can be very effective in training pupils to pass tests even when the pupils do not have the understanding or higher order thinking skills that the tests are intended to measure. When test results are used for making decisions that affect the status or future of pupils, teachers or schools (‘high stakes tests’), teachers adopt a teaching style that emphasises transmission of knowledge. This favours those pupils who prefer to learn by mastering information presented sequentially. Those who prefer more active and creative learning experiences are disadvantaged and their self-esteem is lowered. External tests have a constricting effect on the curriculum, resulting in emphasis on the subjects tested at the expense of creativity and personal and social development.”

“The results of tests that are ‚”high stakes” for individual pupils have been found to have a particularly strong impact on those who receive low grades. However, tests that are high stakes for schools rather than for pupils (such as the national tests in England and state-mandated tests in the US) can have just as much impact.”

“Instead of motivation increasing with age, older pupils feel more resentment, anxiety, cynicism and mistrust of standardized achievement tests. Girls are reported as expressing more test anxiety than boys. Girls are also more likely to think that the source of success or failure lies within themselves rather than being influenced by external circumstances. This has consequences for their self-esteem, especially when they view their potential as fixed.”

“Feedback from the teacher that focuses on how to improve or build on what has been done (described as task-related) is associated with greater interest and effort. Feedback that emphasises relative performance, for example marks or grades which are formally or informally compared with those of others, encourages pupils to concentrate on getting better grades rather than on deeper understanding.”

“The degree to which learners are able to regulate their own learning also appears to foster pupils’ interest and to promote focus on the intrinsic features of their work (15). Pupils who have some control over their work by being given choice and by being encouraged to evaluate their own work are more likely to value the learning itself rather than to focus only on whether or not it is correct.”

“The research shows that the negative impact of tests can be reduced by ceasing to focus teaching on test content. It can also be reduced by ending the practice of ‘training’ pupils in how to pass the tests and by preventing the use of class time for repeated practice tests. Pupils should not be faced with tests in which they are unlikely to experience success.”

The ARG says do more of this…

q    Provide choice and help pupils to take responsibility for their learning.

q    Discuss with pupils the purpose of their learning and provide feedback that will help the learning process.

q    Encourage pupils to judge their work by how much they have learned and by the progress they have made.

q    Help pupils to understand the criteria by which their learning is assessed and to assess their own work.

q    Develop pupils’

understanding of the goals of their work in terms of what they are learning; provide feedback to pupils in relation to these goals.

q    Help pupils to understand where they are in relation to learning goals and how to make further progress.

q    Give feedback that enables pupils to know the next steps and how to succeed in taking them. Encourage pupils to value effort and a wide range of attainments.

q    Encourage collaboration among pupils and a positive view of each others’ attainments.

and do less of this …

q    Define the curriculum in terms of what is in the tests to the detriment of what is not tested.

q    Give frequent drill and practice for test taking. Teach how to answer specific test questions. Allow pupils to judge their work in terms of scores or grades.

q    Allow test anxiety to impair some pupils’ performance (particularly girls and lower performing pupils). Use tests and assessment to tell students where they are in relation to others.

q    Give feedback relating to pupils’ capabilities, implying a fixed view of each pupil’s potential.

q    Compare pupils’ grades and allow pupils to compare grades, giving status on the basis of test achievement only.

q    Emphasise competition for marks or grades among pupils.

For a constructive alternative to testing take a look at the ARG’s pamphlet  Assessment for Learning:Beyond the Black Box.

There’s lots more at The Assessment Reform Group.

And Alfie Kohn on testing is an inspiring read.

You can get my grading rubrics and some further thoughts at The PiFactory assessment page.