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 4×4x4 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.

Wizard math

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THERE are 35 players standing in a circle. As the games wizard walks round the circle she kills every second player until only player survives. The players are numbered one through 35. Which player lives?

The guesses came fast. The first one. The last one. The one next to the last one. Then some students started to draw circles and cross off players.

Answers varied. Musbah smiled and said, “number 7, I did it in my head.” Musbah never fails to delight. But what if the number of players changes I challenged him.

Emma, Brittany and Julia worked together drawing circles and carefully working, talking and comparing results. They confirmed Musbah’s answer.

Jason, at the back of the room, asked do the dead ones get dragged away?

Emma was quick to add, “it must be a prime number.” Then Jeremiah tried a circle that gave the answer nine. Haley added, “you have to be an odd number to survive.”

Heaven, Jessica and Celeste raced as they took up the suggestion to try some smaller numbers in the circle. They quickly listed the survivor for a page full of circles, busily crossing off imaginary players. “There’s no pattern,” Heaven declared, giving me an accusatory look.

“You’ve destroyed your evidence,” I commented. “What if you write out the numbers as you cross them off,” I suggested.

Taylor was sufficiently intrigued to not need to be reminded to put away her lines for the school play… each of her players was reprsented by an open circle. As each player was killed off by the game wizard she filled in the circle. While others students were crossing off dots, Taylor’s method was simple and clear and a guard against confusion.

“Taylor, show your method at the board.” She responded, “I’ll do 35.” I resisted suggesting she did a small circle, beginning to think the pattern might reveal itself easier with larger numbers of players.

Emma and friends urged her on and soon Taylor was also listing the numbers as she filled in circles. In a stroke of pure genius Julia suggested Taylor colored the circles a different color on the second revolution by the wizard. Soon the numbers were color coded too.

By the sixth revolution we’d exhausted the supply of colors.

On the second revolution “it’s every fourth player… the next every eighth… then every 16,” Brittany pointed out.

Hadassah and Amanda, on the far side of the room, pointing to their list with numbers outlined in boxes said there was a “doubling” as the wizard circled.

As the 50 minutes ended, Brittany asked, “you do know the answer, right?” No, I confessed.

+++

Not everyone in class was as enthused by the problem. Everyone attempted to answer the straight question, but a significant number then tried to take the question further. A pleasingly large minority were still working on the problem as the lesson ended.

Significantly the students who persevered were mostly girls, and not girls who always stay so focused in class.

❏ The use of an open-ended question for a whole lesson was inspired by The Elephant in the Classroom: Helping Children Learn and Love Maths by Jo Boaler, to be published next March, and Boaler’s US classic What’s Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject

Prof Boaler, who has done research on effective math teaching involving thousands of students in schools in both the USA and UK, argues for open-ended projects, mixed-ability group work and students talking math.

❏ The problem has a name: the Josephus problem.

The UK’s Royal Institution has used it as the basis for one of its master-class series. I got the idea to use the problem from the November 2009 issue of Mathematics Teaching, the journal of the UK Association of Teachers of Mathematics (MT216).

The “best” solution is to use binary numbers.

Grading gets an F

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THE PRINCIPAL of my school has bravely initiated a discussion about whether or not D and F grades should be used by teachers. The argument goes along the lines if D and F grades are ditched then teachers will need to work with their pupils to find the success within every student.

Ds and Fs don’t motivate or promote learning. Finding success does.

Ds and Fs have enshrined the out-dated pedagogy that grading is about sorting and ranking, that students need to be judged in comparison with one another.

The result of such alpha-numeric grading is that the traditional idea of the “good” student — the students who most closely resemble the aspirations of their teachers — is the scale against which students are judged; learning is demonstrated by turning homework in on time, doing well on quizzes and tests and putting your hand up to answer questions while not talking out of turn.

The result of such behavior is turned into a mathematically nonsensical percentage inside a computer, which then spews out the grade. And for many teachers that’s it.

If a student does not fit into this rigid mould, or cannot demonstrate learning by these criteria, then the result is F for Failure.

Yet the research should be pointing us to question this approach: Not only does this traditional way of measuring learning not reveal the learning going on among many students, it is actually an obstacle to learning for all students… the achievers as well as those who appear not to be getting it.

As part of the discussion in our school I was challenged in a meeting to summarize the case against alpha-numeric grading. I mumbled a few sentences as best I could for as long as it seemed polite to do so.

Then, later, I kicked myself for forgetting the key reason grading does not work. So, I decided to summarize in short sentence bites the best case I could muster for a two-minute contribution:

❏ Grades tell students nothing about what they need to do to improve.

❏ Grades tell students nothing about what they have achieved.

❏ Grades focus students on grades and collecting points, not on what they are learning.

❏ Grades introspectively focus students on ability, or their feelings of lack of ability, not on how they can work to improve.

❏ Grades destroy intrinsic motivation.

❏ Grades don’t measure learning: grades measure obedience, compliance and how well a student can jump through a teacher’s grading-policy hoop.

❏ Grades discourage intellectual risk taking.

❏ Grades divert the attention of teachers and parents as well as students.

❏ Grades encourage rote learning, memorization not reflection.

❏ Grades pit student against student, ranking and sorting.

❏ A grades require F grades. Grades force teachers to give Fs to justify the As. Grades work against finding the success in every student.

❏ Grades increase stress. Stress is bad for learning.

❏ Grades don’t describe learning.

❏ Grades throw students off the back of the boat.

❏ Grades discourage student collaboration.

❏ Grades reward skills not valued in later life, such as memorization.

❏ Grades demoralize and demotivate.

❏ Grades label and stigmatize.

❏ Grades are part of an out-dated carrot and stick, rewards and punishment behaviorist approach to education.

❏ Grades lower the self-esteem of low achieving students and discourage risk taking among higher achieving students.

Readers will find plenty of links elsewhere in this blog on the research behind these statements. But a good start would be From degrading to de-grading by Alfie Kohn.

Take it nice and slow

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AS KIDS move into adolescence they need to become less impulsive and more reflective. So the brain’s output controls in an adolescent are pushing the teenager to take her time and not do the first thing that comes into her head.

“This is ironic,” says Mel Levine in his A Mind at a Time, “since our high schools force our kids to do everything as fast as possible.

“They have to write quickly, think fast, remember on the spot, sprint through timed tests, and meet tight deadlines,” says the professor of pediatrics and director of the Clinical Center for the Study of Development and Learning at the University of North Carolina medical school.

“This frenzied pedagogical rhythm is totally contrary to what the students’ brains are striving to become. The output controls are crying out, declaring that they exist to promote thoughtful, slowly executed work, which should be one of the principal missions of adolescence and the high school years.

“I think we should reward adolescents for taking as much time as they need to do a good job.

“The output controls are doing what they can to decelerate thinking, decision making, and output, to make kids thoughtful rather than impulsive. Secondary education, therefore, ought to incorporate as one of its principal objectives teaching kids how to work slowly.

“That’s what the developing brains are trying to tell us.”

Mel Levine MD is also the author of The Myth of Laziness and is a co-founder of All Kinds of Minds, a nonprofit institute for for the understanding of differences in learning.