## Posts tagged ‘Pedagogy’

### A picture is worth a thousand calculations

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BREANNA is a quiet student who patiently sits in my math class waiting for the ordeal to end. She is too polite to complain or cause a fuss or disturb any of her peers with disruptive behavior.

She does take some notes, and with encouragement will attempt some of my questions. But, to be frank, Breanna gets almost nothing from my math class, apart from sitting through a course she needs on her transcript to walk the walk.

Breanna’s passion is drawing, mainly caricatures heavily influenced by animé. She is good, very good. Tucked under the notebook for my class is her pad. As token gestures go into her math notebook, a detailed and dramatic picture builds unseen on her pad.

When we did our Pascal’s triangle and binomial theorem investigation and poster

project, Breanna got the basics, but was happier coloring the triangle. Draw me a picture of Blaise Pascal I suggested (result, left). Last year I had tried an art and geometry project with my geometry class. The task was to find an artist who uses math or geometry in their work and become an expert. For various reasons it was less than a success. But would it work for Breanna? Her eyes lit up, yes she said, it sounded interesting. She would research artworks, choose an artist or an art movement or the art of a culture (I secretly hoped she’d opt for Islamic art), become an expert on both the art and the mathematics and give a presentation to the class.Dutifully she showed me her growing list of artists and I eagerly awaited which one she would choose… Vasarely, Riley, Mondrian… hopefully not Escher.

Is it OK to look at pyramids? she asked. The proportions are interesting she explained.

Since her math notebook — not her art pad — has added calculations about the angles and proportions of the Gizza Great Pyramid and neighbors. Was it OK to look at less famous pyramids asked Breanna.

Her self-selected extension was to find some more obscure pyramids and compare the proportions.

Then came the poster, with a giant yellow pyramid, drawn to the correct proportions.

❏ For me, the value of this project was that Breanna did some mathematical thinking, where before she was doing none, other than sitting through a curriculum that had little interest, meaning or use for her.

It was also interesting to see that simply giving Breanna the freedom to pursue some mathematics of her own choice based on her own interests did indeed lead to some mathematical work.

What was surprising was that Breanna didn’t choose an overtly artistic piece of mathematics, such as a painting influenced by geometrical shapes. Though, in Breanna’s eyes a Pyramid is a piece of art.

### The elephant in the classroom

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JO BOALER’s research into what works and what doesn’t inside a math classroom has gone a lot further than just watching, literally, hundreds of math classes. She has tracked down the pupils she’s observed years later as adults and quizzed them on how their experiences in the classroom prepared them for using math in real adult life.

Her findings show not only how badly wrong the still dominant, traditional style of math education can be… but how it is possible to turn the situation around, that a growing number of schools are finding ways to engage students in deep math thinking that lasts for life. And gives pleasure.

Open-ended problem solving, mixed-ability group work and project work as well as lots of discussion apparently can unlock the hidden mathematician in every child. Of course, avoiding the superficial quest for educational silver bullets, the real implications for pedagogy in the classroom go much further.

“There is often a very large elephant standing in the corner of maths classroom… the common idea that is extremely harmful to children is the belief that success in maths is a sign of general intelligence and that some people can do it and some people can’t.” says Jo Boaler in the introduction to The Elephant in the Classroom, Helping Children Learn and Love Mathematics*.

“Even maths teachers (the not so good ones) often think that their job is to sort out those who can do maths, from those who can’t. This idea is completely wrong…

“In many maths classrooms a very narrow subject is taught to children, that is nothing like the maths of the real world or **the maths that mathematicians use** (PiFactory emphasis). This narrow subject involves copying methods that teachers demonstrate and reproducing them accurately, over and over again. Of course, very few people are good at working in such a narrow way…

“But this narrow subject is not mathematics, it is a strange mutated version of the subject that is taught in schools.

“When the real mathematics is taught instead — the whole subject that involves problem solving, creating ideas and representations, exploring puzzles, discussing methods and many different ways of working, then many more people are successful.”

Boaler calls it a classic win-win: “teaching real mathematics, means teaching the authentic version of the subject and giving children a taste of high-level mathematical work, it also means that many more children will be successful in school and life.”

+++

Boaler followed classes in two schools in the UK for three years, and then interviewed former students almost a decade later in their mid-20s. One she calls pseudononimously Phoenix and the other Amber Hill.

At Phoenix the teachers adopted what they called “a project-based approach”. Instead of teaching mathematical procedures, students from age 13 worked every day on open-ended projects that needed mathematical methods.

When Boaler asked Phoenix students what to expect, the responses were “chaos”, “freedom…” Boaler confirms the “classrooms at Phoenix did look chaotic”. The project approach “meant a lot less order and control than in traditional approaches”.

A typical project was *Volume 216* — an object has a volume of 216, what could it be, what would be its dimensions, what would it look like?

At Amber Hill classrooms were quiet and peaceful. Teachers began lessons by lecturing from the board, followed by students working through exercises. Students worked quietly, mostly in pairs. They could check answers with each other, but they were not encouraged to discuss their mathematics.

At Phoenix a student described activity in the classroom: “You’re able to explore, there’s not many limits and that’s more interesting.”

An Amber Hill student: “In maths, there’s a certain formula to get to, say from a to b, and there’s no other way to get to it, or maybe there is, but you’ve got to remember the formula. In maths you have to remember, in other subjects to you can think about it.”

At age 16 all the students sat the UK’s major three-hour public GCSE mathematics exam. Although the Phoenix students had tested lower than the UK national average before their project-based lessons started, their GCSE grades were significantly higher than Amber Hill’s and the national average.

But it is the achievements and recollections of the students nearly a decade later that speak more powerfully. Boaler recorded her research more fully in the national book award winning Experiencing School Mathematics.

At school all the students were in similar social class levels, as defined by their parents’ jobs. Eight years later more than six out of ten of the Phoenix students had moved into jobs that were more highly-skilled or more professional than their parents. The figure for the Amber Hill students was less than one-in-four. Over half the Amber Hill students had lower-skilled jobs than their parents, the figure for Phoenix was less than one-in-six.

Looking back to his school years, Phoenix student Paul said: “I suppose there was a lot of things I can relate back to maths in school. You know, it’s about having a sort of concept, isn’t it, of space and numbers and how you can relate that back… maths is about problem-solving for me. It’s about numbers, it’s about problem-solving, it’s about being logical.”

Marcos from Amber Hill said: “It was something where you had to just remember in which order you did things, that’s it. It had no significance to me past that point at all — which is a shame. Because when you have parents like mine who keep on about maths and how important it is, and having that experience where it just seems to be not important to anything at all really. It was very abstract. As with most things that are purely theoretical, without having some kind of association with anything tangible, you kind of forget it all.”

Boaler also worked closely with in an inner-city high school in California called Railside. There teachers who had originally taught using traditional methods with classes grouped according to notions of ability focused instead on mixed-ability groups and a re-designed curriculum built around big mathematical ideas.

Instead of an approach based on isolated skills and repeated practice, the Railside students worked on themes — such as What is a linear function? — using multiple representations, the different ways maths could be communicated through words, diagrams, tables, symbols, objects and graphs.

Again Railside was monitored alongside schools adopting a more traditional approach. Although Railside students started with lower levels of achievement, after two years they were outperforming the other schools. By year 12, more than four out of ten Railside students were in advance classes of pre-calculus and calculus. The corresponding figure for the more traditional schools was fewer than on in four.

The four-year study at Railside revealed consistently higher levels of positive interest in mathematics at Railside.

At the end of the study only 5 per cent of students from the traditional schools planned a future in mathematics. At Railside the figure was 39 per cent.

Janet: “Back in middle school the only thing you worked on was your math skills. But here you work socially and you also try to learn to help people and get help. Like you improve on your socialskills, math skills and logic skills.”

Jasmine: “With math you have to interact with everybody and talk to them and answer their questions. You can’t be just like ‘oh here’s the book, look at the numbers and figure it out.’

“It’s not just one way to do it… it’s more interpretive. It’s not just one answer. There’s more than one way to get it. And then it’s like: ‘why does it work?'”

Jo Boaler concludes: “Put simply, because there were many more ways to be successful at Railside, many more students were successful.”

**Eight key questions for teachers and parents:**

❏ Is our school’s mathematics approach teaching children to think and reason and make sense of the mathematics they are learning?

❏ Is practice with skills provided in engaging, challenging and mathematically important contexts?

❏ Is persistence valued over speed?

❏ Are problem solving and the search for patterns at the core of all that children are asked to do?

❏ Is numerical reasoning emphasized?

❏ Does the mathematics approach emphasize that there is almost always more than one way to solve a mathematics problem?

❏ Does it present mathematics as relationships to be understood rather than recipes to be memorized?

❏ Are children the ones who are doing the thinking and sense making?

+++

* The Elephant in the Classroom, Helping Children Learn and Love Mathematics will be published in March in the UK. An earlier account of Boaler’s research is available in the US entitled What’s Math Got To Do With It, how parents and teachers can help children learn to love their least favorite subject, and why it is important for America

### Wizard math… day 2

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WE’D LOOKED at the sequence generated when you take a hexagon and steadily add more hexagons. It gives a linear rule. Some students got this easily, others found it challenging.

For 20 minutes the task was to work through a handful of similar problems in the textbook.

The issue was how to give an extension challenge for those who could easily do this sort of problem. Jo Boaler in her What’s Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject advises open-ended problems are one solution. She also argues that so-called low-ability students benefit from doing hard problems, or, at least listening and eventually participating in finding solutions in mixed-ability group discussion. Talking math is learning maths. The higher ability students benefit by explaining their thinking to other students.

On the board I drew a square with diagonals drawn in red, a pentagon with diagonals drawn in red, a hexagon with diagonals, a heptagon and octagon, also with diagonals drawn in red. I also put up an incomplete polygon labelled n. I numbered the polygons 1, 2, 3…

I also wrote: Challenge question, spot patterns, how many red lines?

Breanna was out of her seat immediately counting the lines. Within seconds she announced the number of lines coming from each vertex was the same as the number above the polygon. Then she sat down. So? I asked.

“I can’t do any more,” she replied. I explained she needed to write down her discovery so she had something new to look at and work on to get the next step. No, she said.

Shane was busy counting lines. Rebecca stared at the diagonals in the heptagon and octagon and said it was too complicated. “What about this one,” I said pointing at the pentagon “start here… what you find out here will work for those.”

Breanna had started to build a table. “Breanna,” I said, “tell the others about how you counted the diagonals.”

Shane spotted the number of sides of each polygon was 3 more than the number above the polygon. Breanna explained the number of diagonals was the number of vertices multiplied by the number of lines coming out of one vertex. “And the polygon labelled n?” I coaxed. “The number of sides is n + 3,” said Jonathan from the other side of the room.

Robert who had earlier struggled with the basic exercises, sat watching the board and listening intently to the discussion on Breanna’s table .

As others finished the textbook exercises I drew on the board a 4x4x4 cube made up of 64 small cubes. Next to it I wrote, “if the cube is painted, what proportion of the small cubes have paint on them?”

Stopping the class, I focused everyone on the new problem. On our wall we have a list of Polya’s problem-solving strategies. I pointed to the list with my hand next to “Make it simpler”.

Shane was walking from desk to desk debating with other students the number of cubes. Rebecca asked, “you paint the back too?”

Jonathan and Nick were back and forth at each other, Jonathan slicing out invisible cubes in the air with flattened hands. Nick drew out the net of cube and cut it out: “Look,” he announced, beaming, “I’ve made a cube.”

Jonathan just wanted to explain how he had worked it out, how he got the total number of cubes, how he excluded the cubes inside the large cube, how he decided to not to double count cubes with paint on more than one side… all the time his hands slicing out cubes in the air.

What are the dimensions of the cube? I asked Jonathan. 4 he responded. 4 what? 4 times 4 times 4. How would you write that? 4 to the power three… 4 cubed… Oh! he exclaimed as a giant lightbulb flashed in his brain.

As the students left, Jonathan and Shane were still telling each other about how to solve the problem.

“Do you want to see my work?” said Robert showing me the textbook problem he had completed. “I saw you watching and listening Robert,” I said. Robert smiled.

The Calculus class tries the polygon problem to relax after an intense hour wrestling with implicit differentiation. OK, I say, tell me the number of lines in the 99^{th} polygon.

Becci runs to the board closely followed by Jared. Megan is shouting how to count the lines, but Becci and Jared are engrossed in mathematical disagreement about how to move forward. Josh, Jordan, Jesse and Nicole sketch out the pentagon and hexagon.

Soon the room of nearly 20 students is loudly split between those insisting the rule includes (n −3) and those who say it is (n + 3). As agreement settles on (n + 3), good-natured boasting and mocking ensue.

But, everyone was talking math. And talking math is learning math.

### No gain from the pain of testing

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HIGH_STAKES testing on the rise since 2002 and No Child Left Behind, may have lead to more hours spent on reading and math in schools, but there have been no increases in learning.

And the curriculum, particularly in the humanities and liberal arts, has narrowed. “Today may actually be worse for poor children in the US than at any time in the last half century. This is because the lower classes are being kept from the liberal arts and humanities curricula by design,” a respected thinker on pedagogy told a key conference this summer.

“The newest difficulty in promoting the arts and humanities in the curriculum is due to the use of high-stakes testing,” Prof David Berliner told an international conference on redesigning pedagogy in Singapore.

“We need to remember that when administrators and teachers concentrate their efforts on raising only a few skills, they detract from the talent pool for individual and national success in an economy that will demand adaptability.”

In his paper the Regents’ Professor in the College Of Education at Arizona State University argues the result of high-stakes testing has been to increasingly narrow the curriculum, at a time when the challenges of the future demand the broadest possible liberal arts curriculum.

“The decrease in exposure to certain curricula is a rational response to high-stakes testing. But this decrease in exposure to a varied curriculum is of great concern as we contemplate what the 21^{st} century might have in store for our youth.

“Compared to the past, the future is likely to be more Volatile, Uncertain, Complex, and Ambiguous — A VUCA world for our children to face. I think adaptation to such a world requires a citizenry with the broadest possible curriculum, not a narrow one that constricts the skills of the youth because of a need to demonstrate accomplishments on a small set of assessments.”

“A 21st century workplace is likely to value such social skills as active and tolerant listening, helping each another to define problems and suggesting courses of action, giving and receiving constructive criticism, and managing disagreements. But in today’s high-stakes school environments, collaborative work where such skills can be developed is seen less frequently than ever because such work always means a loss of time that could be used for preparation to take high-stakes reading and mathematics tests.”

The narrowing curriculum is particularly undermining the education of the poor he argues. “America apparently has developed an apartheid-like system of education.”

“Using the argument that we must get their test scores up, we in the US are designing curriculum for poor children, often poor children of color but certainly, numerically, for poor white children, that will keep them ignorant and provide them with vocational training, at best. Their chances of entrance to college and middle class lives are being diminished, and this is all being done under the banner of “closing the gap,” a laudable goal, but one that has produced educational policies with severe and negative side effects.

Focussing on research by Hong and Youngs (2008) the response to high-stakes testing in Chicago and Texas, Prof Berliner says:

“In Chicago the researchers found that high-stakes testing seemed to narrow the curriculum and make it harder for students to acquire higher-order thinking, writing, and problem-solving skills. In Texas, it was found that schooling changed in ways that emphasized rote learning, not broad intellectual skills.”

A study by Lipman (2004) of Chicago schools found that the more affluent students in Chicago received a much richer and more intellectually challenging curriculum than did the poor children in Chicago. Poor minority children, in particular, were required to memorize fragmented facts and information, and they were constantly taught simple test-taking techniques.

“Lipman is probably quite right when she says that this differential access to high-quality curriculum will have significant consequences in terms of the social inequalities we will observe in the future. White students who possess a great deal of the cultural capital valued by schools are going to be much more likely to get to college and thus more likely to attain higher status through higher paying jobs. But low SES and minority students in Chicago’s schools are much more likely to end up in lower-skilled and lower-paying jobs. The decisions about curriculum and instruction in Chicago and other urban districts results in access to rigorous curriculum for some, but not for others, thus allowing for the continuation of the current unequal social structure.”

What to do?

“Change the tests used for school accountability under NCLB. Currently almost all the tests used to comply with NCLB make heavy use of multiple-choice items and thus are designed to reward memory of decontextualized bits of knowledge. But we know that tests with high-stakes attached to them drive curriculum and instruction. So the construction of tests that measure things like creativity and critical thinking need to be designed so teachers have tests worth teaching to.

Simply using tests with open-ended items has also been found to change teacher’s instructional behavior. Under those conditions teachers more frequently required their students to explain their answers in the classroom, and the teachers used more open-ended tests in their own classrooms as they tried to give students experience that would help them on the end-of-year tests.”

In conclusion Prof Berliner argues: “The same politicians and business persons that want high-stakes testing to be the cornerstone of a school accountability system also want 21st century skills developed. They do not yet understand that they cannot have both at the same time. These are incompatible goals.

“It seems to me that all but the most privileged students come into public schools where the pedagogy may actually be closer to that of the 19th rather than the 21st century. In schools for the poor, Dickens’s (1854/1868) wonderfully written caricature of a teacher, Mr. Gradgrind, still lives. Gradgrind said:

**Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!**

“But it is not just pedagogy that needs improvement. Many of our students receive too limited a curriculum for dealing with what the eminent psychologist Howard Gardner (1999) reminds us are always the most important questions facing humankind: what is true, rather than false; what is beautiful, rather than ugly or Kitschy, and what is good rather than compromised, or evil.

“A broad liberal arts curriculum is needed to deal with these eternal questions. But we in the US are far from providing that now, and moving further away from that model as high-stakes testing changes what and how we teach.

“No one really knows what 21st century skills are needed to foster success for individuals and nations. But developing critical thinking, engaging in activities that require problem solving and creativity, and doing individual and collaborative projects of complexity and duration, are all good candidates for helping each child and both of our nations to thrive” told the teachers and educationalists gathered in Singapore.

Prof Berliner’s complete paper can be read at www.susanohanian.org

In this review Prof Berliner’s citations have been removed for readability.

### Assessment… a new sort of gradebook

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WHAT could authentic assessment look like in the classroom?

For four years I have been experimenting in my math classroom to replace percentage and letter grades with much more meaningful descriptive feedback.

There’s no shortage of percentage-based grade books for teachers. Punch in the numbers and out come the letter grades.

They don’t tell the student what they need to do to improve… they don’t even tell the student what it is they have achieved.

They just divert the student’s attention with a meaningless letter grade from the essential tasks of learning. The frustrated and anxious student is left vainly to plead for extra credit in a bid to get more nonsense points.

The challenge is to create a gradebook that is not based on numbers and letters, but records descriptive progress and gives the student the sort of valuable feedback that describes just what it is they need to do to improve their learning.

Well, __take a look at this powerpoint__ and let me know what you think.

The system is based on student-friendly learning targets. Assignments and questions are all linked to a learning target.

Against each assignment and learning target the teacher can record and feedback to the student a description — ranging from __Startin__g through __Gettin__g__ it__ to __Got it__ — and add a suggested revision learning target plus a customized study skill tip.

It also records the student’s own self-assessment of progress.

And rather than add up meaningless percentages, the system summarizes to what extent the teacher and student agree on their assessment of progress.

It can show a matrix of an individual student’s progress. Or, it can show, color-coded, the progress of an entire class.

Input screens allow the teacher to add questions (+ pictures, diagrams and equations), new targets, target notes and assignments — and link them together in a host of assignment designs.

### Competitive grading still sabotages good teaching

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TWELVE years ago a professor and a postdoctoral fellow at Stanford wrote a paper entitled Competitive Grading Sabotages Good Teaching. I found the paper this weekend and read a depressingly accurate description of how my school works today.

Assigning competitive grades “skews teachers’ values” and affects teachers’ behavior in five basic ways argued Prof John Krumboltz and Christine Yeh in their 1996 paper:

❏ it turns teachers into students’ opponents,

❏ it justifies inadequate teaching methods and styles,

❏ it trivializes course content,

❏ it encourages methods of evaluation that misdirect and inhibit student learning, and

❏ it rewards teachers for punishing students.

Below are some quotes from their damning indictment:

__Teachers become o__pp__onents__

“To assign grades, teachers must become critics whose focus is negative, always seeking errors and finding fault with students’ work. Moreover, students must be compared with one another, because there is no accepted standard for a given letter grade. A performance that earns an A in one classroom could earn a C in another classroom because of differences in the teachers’ standards or in the composition of the two classes.

“When judging the relative merit of students’ performances takes precedence over improving their skills, few students can feel good about their accomplishments. Only one student can be the best; the rest are clearly identified as less able. Comparative grading ensures that, unlike children in Lake Wobegon, half of the students will be below average.”

__Gradin__g__ __j__ustifies inade__q__uate methods of teachin__g

“When students fail to achieve course objectives, whose responsibility is it – the teachers’ or the students’? Current grading practices put the onus squarely on the students. Teachers can use the most slipshod of teaching methods, discover that many students do not understand the material, and then assign grades accordingly.

“Current grading practices do not encourage teachers to help students improve, because only the students are blamed when they fail to learn.

“If every student achieved all the objectives of a given course, every student would earn an A – an unacceptable state of affairs in the current view. Thus teachers are reinforced for using methods that ensure that some students will not succeed.”

__Grades trivialize course content__

Which of the following questions is more challenging to a student?

❏ When was the Declaration of Independence signed?

❏ Would you have signed the Declaration of Independence if you had lived in 1776? Why or why not?

“The answer seems clear. The first question requires students to memorize a date. The second question requires them to think — to imagine themselves in another time and place and then to justify an action that would profoundly affect their own lives and the lives of others. However, many teachers might hesitate to include such thought-provoking questions on a test.

“If assigning grades were not required, teachers might opt for the second question. Thus course content is determined, at least partly, by the need to grade students. Teachers would be liberated to teach toward more consequential goals if they were not obligated to assign grades.”

__Gradin__g__ inhibits constructive evaluation__

“Ideally, the evaluation process would help students discover how to improve their achievement of important goals. Grading defeats this purpose by discouraging the vast majority of students, who receive below-average grades, and by not challenging students who could improve on what they have already learned.

“Pressure to perform well often causes students to attend only to ‘material that will be on the final’.

“Students develop learning styles that they expect to yield good grades. They quickly learn that the operational definition of a course objective is ‘what appears on the final exam’.”

__Teachers can take pride in failure__

“Some teachers feel proud when a high percentage of their students fail. They want others to believe that a high failure rate signifies a difficult course and an intelligent teacher. To a large extent, they succeed.

“There is a common assumption that taking a ‘tough’ course is more prestigious than taking a ‘Mickey Mouse’ course. Some teachers believe that giving students low grades adds luster to their own reputations. Such teachers may choose to include excessively difficult material in their courses simply to enhance their own self-importance.

“One way of guaranteeing a high failure rate is to present material that is too difficult for most of the class to comprehend. But the inclusion of material for this purpose stands education on its head. Teachers deserve shame, not praise, if their students fail to achieve.

“Teachers who take pride in giving low grades blame the students, not themselves, when course material is not mastered as quickly as it is presented.

“The students who fail are blamed undeservedly, and the teachers who fail them are esteemed undeservedly — but the real culprit is the grading system.

“Competitive grades turn educational priorities on their head. Classes in which most of the students master the material are perceived as unchallenging. High grades are often dismissed as “grade inflation,” not as a sign that the teacher and the students have successfully achieved their mutual objectives. Meanwhile, prestige is accorded to teachers who are unable to help most of their students learn the material.

“The situation is ridiculous.”

And in conclusion they added “…under the competitive grading system, teachers are not required to help every student learn, but they are required to judge every student. Judgment is mandatory; improvement is optional.

“Competitive grading de-emphasizes learning in favor of judging. Learning becomes a secondary goal of education. Clearly, then, the need to grade students undermines the motive — to help students learn — that brought most of us into the profession.”

Click to read the full article by Prof John Krumboltz and Christine Yeh.

Similar points were also made in From Degrading to De-grading by Alfie Kohn… nine years ago.

### Win, win not fail, fail

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GRADING day, end of the first quarter. After a long day writing individual assessments, I have some envy for my colleague teachers who have been punching in percentages into their computers for the past few weeks.

As the email reminded us at the beginning of the day: hit the yellow button and the magic software will turn all the numbers into a grade and get the report cards ready for dispatch. Bingo! Easier than punching in 180+ grades and descriptive assessments by one-by-one.

I note one of my colleagues’ students has a percentage of 92.827. Well, that’s an A, then. To three decimal places!

Clearly the assessments that built this grade had not just 100 criteria, not 1,000, not 10,000… but 100,000. Some grading!

But what is this percentage of..? What exactly has our student achieved 92.827% of..? What, exactly, is being measured?

Percentages are a mathematical nonsense, unless they are of something. Just what did our student fail to do to miss out on the last 7.133%? Nothing on the report card gives an explanation.

Even being more sensible about the three decimal places (the very expensive software used across school district spewed those out, not I), just what would, say, a rounded 90% actually mean?

More important what about the student who got a 65% and got an F? What practical advice does the 65% contain to tell the poor unfortunate who has been branded a failure need to do to become a success?

The A, B, C, D or dreaded F may contain no help in specifically describing what a student has or has not learned… and certainly contains no help in telling a student what they need to do to improve. But it does label the student.

This might be ok for the (albeit stressed-out) student labelled an A or B… but it’s not so hot for the student labelled C, D or F. Labelled a failure… but given no clue as to what to do.

And, believe me, where these percentage-based letter grading systems are used with enthusiasm, then these numbers have been pinned up fresh every week in classrooms… raising stress levels in all the students weekly and forcing them to focus and re-focus… not on the joys of learning, but on the terrors of the grade.

You might get an A one week on your assignment. The next, you miss it. That means you’re at best 50% and failing wildly. There’s plenty of teachers who practice, and defend this as a perfect reflection of their students’ learning. You get a perfect 100% A the following week… that still does not lift you back up to passing!

You might be well on top of the learning… but you can’t meet deadlines = F! What’s important here?

So, what’s on your mind? The beauties of that Shakespearean sonnet or doing something desperate to scrabble together a few more percentage points? Or, just call it quits? You can’t win.

What if you are the kid who doesn’t get As? And you miss an assignment? And you work nights?

Is this education… or just a confusing nightmare? Life was much more fair in Catch-22. Welcome to school.

❏ So, what is to be done?

Drop the As, Bs down to Fs. Learn the lessons of pre-algebra and accept that percentages without a definable “of” are a mathematical nonsense.

Instead, describe in student-friendly phrases just what it is they need to learn, what they have learned, and what is the next achievable step they need to take to improve.

And give them as many chances as they need to do it, to learn. It’s the learning that’s important, whenever and wherever it finally happens. Not the grade.

That’s a win, win. Not fail, fail.

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