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.”

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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?

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* 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

Lakatos, the Jack Kerouac of math

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“WHEN will I need any of this in real life?” is such a common question in a math classroom that it is a cliché.

At one level it is a tried and tested way to stop a lesson. At another it is a genuine question. After all, if a student isn’t going to use the likes of the Quadratic Formula, why learn it? And spend so much time, over so many years, learning it all? And suffer so much stress¹? Why, indeed, does an average student need much more than the numeracy required to navigate life?

The question does not come so often from the student who enjoys math, which also usually means the student who is more than competent at it.

The answer used by myself, and some other teachers I know, is that math is about creative thinking skills, and, going further, the ability to think in abstract terms, to examine problems from different perspectives, the bending of minds. The other oft-used term is problem solving. Problem solving, creative and logical thinking skills are essential life skills.

Some look convinced.

Those who don’t need to ask the question (but may out of genuine, disinterested curiosity) carry on, almost entirely driven by their own motivation.

The question facing math educators is how to make math relevant — and fearless — for the rest?

Students may not need, or ever use, the Quadratic Formula, but finding the Quadratic Formula with the aid of algebra tiles, completing the square and computer manipulatives as well as some help from other students does need skills that are useful for life… the sort of math thinking skills needed in an increasingly complex technological world… an increasingly complex political, economic and social world.

Proofs and Refutations, The Logic of Mathematical Discovery by Imre Lakatos raises not just a question of how math is taught. It is also raises questions that go to the core of what education is about. Like the world, it is complex, technological, social and political in nature. There are many stakeholders, besides the students.

The pressures on teachers (and students) are contradictory.

The modern math teacher is meant to create rigorous, interesting, relevant, fun, diverse, engaging, multi-cultural, technology-based, investigative lessons that allow each student to search out their own path of discovery, at their own pace, based on their independent learning plan in co-operative group situations tailored to each student’s own learning style.

At least three times a day.

Plus relevant daily assignments with meaningful and timely, individualised feedback. Fully covering the curriculum. And preparing the student for repeated, on-the-record, high-stakes, multiple-choice computer testing… the results of which are the meat for the political-funding grinder.

So. No pressure then.

Teachers and students find themselves caught between the traditional deductivist approach and the vision of an heuristic future.

The demands of the curriculum, rigorous testing regimes, daily assignments, textbook styles and time all push the teacher back in the direction of deductivism. And as most parents are the product of deductivism, they too are fans— even while assuring the teacher they themselves hated math at school.

Competition from new interactive media, educational research and sheer desperation and frustration push the teacher to at least dream of an heuristic world where students are engaged and thinking for themselves.

Lakatos could easily have chosen a different title for this delightful read: Proofs and Refutations, the Heuristic of Mathematical Discovery.

Here Epsilon, Pi, Omega battle it out with Alpha and Beta: Adventure and the search for knowledge versus battalions of formalists, logical positivists, conceited infallibilists, dogmatists, Euclidean rigourists, monster barrers.

In the final few pages² Lakatos abandons his pseudonymous allies and speaks directly with Shakespearean passion, anger and open contempt for the deductivist approach.

Lakatos summarises this “Euclidean ritual” as painstaking lists of axioms, lemmas, unseemly definitions followed by carefully worded theorems, often loaded with heavy-going conditions. The theorem is followed by the proof.

“The axioms and definitions frequently look artificial and mystifyingly complicated.”

The student of mathematics is obliged “to attend this conjuring act without asking questions either about the background or about how this sleight-of-hand is performed.” Should a student wonder or discover by chance that mathematics could not have developed in such a fashion, “the conjuror will ostracize him (sic) for this display of mathematical immaturity.”

Lakatos complains, “mathematics is presented as an ever-increasing set of eternal, immutable truths.

“Counterexamples, refutations, criticism cannot possibly enter.” Conjectures are suppressed.

This “authoritarian” deductivist style “hides the struggle, hides the adventure.

“The whole story vanishes, the successive tentative formulations… are doomed to oblivion while the end result is exalted into sacred infallibility”.

And then the coup de grace buried in the footnote (p142, n2): “It has not yet been sufficiently realised that present mathematical and scientific education is a hotbed of authoritarianism and is the worst enemy of independent and critical thought.”

In contrast Lakatos models a more open and optimistic approach, a world where imperfection is a virtue. For Lakatos it’s not the answer that counts: it’s how you get an answer, which only leads to the next question, that matters. Learning as a journey. And it’s the road that’s interesting, not so much the destination… which is only a starting point of another road.

Lakatos is the Jack Kerouac of mathematics.

“Literary criticism can exist because we can appreciate a poem without considering it to be perfect; a mathematical or scientific criticism cannot exist while we only appreciate a mathematical or scientific result if it yields perfect truth.”

Lakatos has not yielded perfect truth. But this work helps put us on a road of discovery in the classroom. For Lakatos education was about fostering independent and critical thought, and for him that would mean adopting the road of discovery and not the Euclidean ritual – “this good and evil spirit of nineteenth century mathematics”.

But Lakatos was a political man³. And much of Proofs and Refutations has the passion of a revolutionary political manifesto.

Lakatos would have recognized the tensions and politics that mire modern-day teaching.
On the one hand: open-ended investigations, discovery, problem solving and self-learning, assessment for learning. The heuristic method.

On the other: the insistent political pressure of test scores, pushing teach-to-the-test strategies. The rigidity of the curriculum map. Students trained to view education as a production line for collecting points. Assessment of learning. (Assignments means points. Tests means points. And as the most amusing, satirical show on BBC radio⁴ for many years always said, “points means prizes…” Grades, GPAs, scholarships, college, career.)

But there is a synthesis out of this Euclidean thesis and heuristic anti-thesis. Behind the closed doors of classrooms teachers are experimenting with counterexamples and stretching concepts to open up new conjectures⁵ which may yet give greater depth and breadth to real learning.

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¹ I did a Google search some time ago for “math phobia”. It returned 527,000 links. I put the same words into amazon.com and a list of 234 self-help guides was returned with names such as Overcoming Math Anxiety, Conquering Math Phobia: A Painless Primer, Danger Long Division, Overcome Your Math Phobia and Make Better Financial Decisions. And many, many more.

² Lakatos puts clues in his opening pages. Heuristics is mentioned in both the Acknowledgements and the Author’s Introduction. And in the opening pages Lakatos teasingly makes clear, in a lengthy footnote (p9), that for pre-Euclidean Greek mathematicians porisms, results that appeared by chance, springing from the proof of a theorem, were considered a windfall, or bonus. “The heuristic precedence of the result over the argument, of the theorem over the proof, has deep roots in mathematics,” says Lakatos. As the mathematicians in Lakatos’ sites often pointed back to some notion of a golden age of Greek mathematics and rigid method, this Lakatos footnote is an opening salvo.

³ Lakatos was a member of the Hungarian Communist Party and an active member of the anti-fascist armed resistance during the Second World War. After the war he was a part of the Communist administration and was involved in the reform of Hungarian higher education. He spent six weeks in solitary confinement and three years in prison. The reasons why are unclear— He was rehabilitated in 1953. After Lakatos fled Hungary and the Soviet invasion of 1956 he was supported by the Rockefeller Foundation and the London School of Economics. At LSE he remained a close friend and colleague of Georgy Lukacs, widely accredited as the father of western Marrxism. He also befriended Paul Feyerabend who formulated an anarchic theory of knowledge.

⁴ I’m sorry I haven’t a clue, BBC Radio-4, presented by legendary jazz trumpeter Humphrey Littleton. Littleton arbitrarily awarded points based on no stated criteria for games that appeared to have neither logical conclusions nor rules. No one knew what the prizes were or why points were awarded— the only real prize for all concerned being an addictive dose of hilarity that somehow commented on much of the nonsense of current events.

⁵ In my own still-mostly-deductivist classroom we have dumped the textbook as unintelligible, barred points as monsters, and have incorporated self-assessment and words such as “On your way”, “Getting it”, “Almost there”, and “Got it” instead of meaningless points, percentages and letter grades. Verbal in-class contributions count on a par with written assignments. Lower-end students say they get hope. Higher-end students are challenged to demonstrate thinking with their explanations. All students are challenged to demonstrate some learning, that they have improved their understanding. That, at least, is the aim. Some days it works. Some days not.

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.

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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.

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.

Relax… and watch the kids shoot some squid or crash a car

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THREE weeks from the end of the semester I decided to bring back the laptops for my three Algebra 2 classes and see where we would go. We would go further than I could possibly have hoped. And it would be me who would learn the most… the more the teacher pulls back, the further the kids go on their own.

The students would be using Geometry Expressions, a symbolic geometry system that reveals the geometry behind algebra and the the algebra behind geometry¹. To get us going we would be using problems from Geometry Expressions activities for Algebra 2 and Pre-calculus, written by high school math teachers Tim Brown an Jim Wiechmann.

For a start, I photocopied some of Tim’s and Jim’s problems and hints on parametric functions which took the students through the basic steps of using Gx and launched in. I could model the steps using Geometry Expressions on the interactive whiteboard.

First impressions were positive as the first lesson logs record:

“Positive response to the program from students (girls + boys). Most problems around familiarity with the program, but no complaints about impossibility… Several said liked the way it draws straight lines, can see numbers, etc. (p2)”

“Three girl students who normally do nothing working well on task and asking mathematical questions… Higher level of engagement than normal for this class… Girls who are rarely engage in the lesson comment, “it’s fun… we should use it more. (p4)”

But not entirely: “This class (p5) has high rate of deliberate helplessness and unwillingness to figure stuff out. Ditto here, many students get stuck, do little to get unstuck and then blame me. It’s par for the course… Least successful class to date… as expected… Need re-think for next class.” Classic grumpy teacher response. It’s the kids fault! Bad teacher!

And so it continued for several lessons. Some ups and quite a few downs. The lesson logs record:

“The first part of the investigation seems to be useful and keeps most students absorbed. (p2)”

“TAG boy resists experimentation… Explain the aim is to get him to push assignment to one side and to take over under his own initiative and inquiry. Try to explain a method of changing one thing at a time in the equations and spotting/writing the change. Look for pattern. Explain he’s used to being more spoon fed… this will seem slower, but will be more effective. He does not look convinced. (p4)”

“Better than usual focus (still not brilliant). (p5)”

But I’m still nervous. I’m off the curriculum, ignoring the upcoming finals and can see a lot of students being much better at toggling in/out of their web browser as I walk the room (we need internet access for the students to logon and the laptops to work). “Move students on… seems to indicate that need to put in more traditional lesson somewhere to check some links are established. I seem to be more phased by this than the students. (p5)”

Focus on trying to lighten up the atmosphere and take one problem very slow. Enter the powder-puff kicker trying to kick a 33-yard field goal over a 10-foot high goal.

For starters, we looked at investigating how to find one possible path for our powder-puff kicker’s shot. We decided the path would be a parabola, a quadratic.

Then by guess and check we started to use Geometry Expressions to find some possible paths.

❏ First, the parabola opens upwards, which does not fit the problem. Why?
❏ Then how to get it to start at the origin (where we had drawn a powder-puff kicker using the interactive whiteboard ability to draw over the Geometry Expressions projection).
❏ Where might the ball land if the kick had cleared the goal? Would picking a landing spot help us find a parabola and its quadratic function? Why? Is there more than one?
❏ But the highest point of the ball’s flight seemed to be at more than 2,000 feet! What would give us a reasonable curve?

Most of the classes — working in pairs — get curves that show how the ball could fly over the goal.

This gave a good review of quardratics, and particularly how we could use the kick spot and the landing spot for the x-intercepts, the zeros, and find a function in factor form.

But we still don’t know how the speed of the ball at kick-off or the angle of the kick affects the curve.

The lesson logs: “Apart from a few rough edges and reluctance of students to actively participate in much discussion, lesson feels good. Most pairs follow along and are still working using Gx at end of lesson. (p2)”

Well, most of the kids did 70 minutes of work. But, notice, they followed along, they were reluctant to participate. I ran the lesson, not the kids.

Same exercise for the next class, but with the aim: loosen up teacher-man.

“Explain we’re going to be looking at parametric equations, but we’re not likely to get there this lesson, it’ll take two. I like the idea of encouraging students to think that we’re not in a rush to get an answer, that the process is more open-ended… Second time around, much smoother. Also class more actively engaged in discussion back and forth. Keep almost all students actively engaged and on task for all lesson… I found it interesting to teach by finding the curve that fits the problem with guided guess and check approach. More students found the final curve, more quickly and more independently in this lesson.”

And I’m learning from lesson to lesson:

“Spoonfed up to this point, but with input from class,” records the p5 log. “Now decide to leave and not intervene… ‘it doesn’t work’ insists best student in class… ‘we found out what we were doing wrong’… Announce remember we’re trying to do it like mathematicians have done it through history… a sort of guess and check&hellip ‘We got it… awesome’.”

Two steps forward, one step back: “Summarize where we were at… try to get responses. Little comes back… Front of room led lesson, spoon feeding all the way… but attempting to get feedback at every step.” The lesson moves on to open up the ideas of parametric functions.

It depressingly concludes: “As a didactic lesson ok. As a discovery lesson, useless. First lesson of the day always an issue with sleepiness, etc. Difficult to motivate students. (p2)”

One class goes out, the next comes in. Teacher is starting to think:

“Smoother and more straightforward explanation of trig relationship to the problem… More feedback from students and most of lesson achieved in shorter time… Tomorrow will try again, but will ponder on how to move in direction of less didactics and more student discovery. One key might be to slow the process and simply give less… the fear here is that the students simply don’t respond but just sit and talk! The issue-elephant in the room is that students who are used to being spoon-fed, see math education as getting an answer and who are not expected to be engaged on their own behalf are slow to rise to the challenge… My challenge, next year, is to work out how to build a new culture in the classroom specifically aimed at teaching the kids how to learn by themselves for their own sakes.” That’s what I thought I’d been trying to do this year.

Hands off with p5, “encouraging them to explore” using a table to record their results.

“The students in this class (p5) who are normally willing to engage, did so in this lesson. There was some engagement by those who are normally resistant to the lesson. I wandered around the room on several occasions and asked if students needed help and most said they were ok… Unfortunately could not get much for the table of results… Asked one of the more engaged students how it felt just being left to get on with it… ‘it was fun… I knew what to do and you gave us support… it was ok’.”

It’s getting close to the semester’s end and finals week. p2 “sleepy and non-responsive.” We get into a muddle with degrees and radians which throws off the results: “One of those problems that gives students the excuse to stop,” notes the lesson log.

I’m becoming resigned to spending the summer thinking about what went wrong. Last day for our seniors, the sun is out, carnival atmosphere, so try to keep things light. Let’s try to get some pirates to shoot a cannon ball at a giant squid: “What do we do first? What did we do the other day? Draw a silly picture. I want to see lots of pirates and squids… and a bit of math thinking.” A coloring-in lesson to get us through the 80 minutes!

After two weeks, some pretty drawings. “I hit enter and the point just disappeared,” says the most engaged girl in class with apparent irritation. Another frustrated girl: “The point just kept going up and down.”

Next class (p4)… “I want lots of drawings” I say with resignation. After ten minutes I decide to walk the room, for the sake of form, expecting I know not what, but not a lot. Here’s the lesson log (the numbers represent times, TAG = talented and gifted):

09:55 TAG girl trying to reconstruct a Quadratic to plot the curve. Two TAG boys appear to have drawn a giant quid in the computer (at the correct coordinate) plus a pirate ship complete with sails and a cannon (at sea level!). I go up to suggest they now… when they fire off two cannon balls complete with parametric coordinates, both of which directly hit the squid! I point out the cannon balls are traveling at different speeds… also why not put the cannon on the deck of the pirate ship?

10:00 TAG boys have shifted their cannon and can still shoot the squid.

A plethora of pirate ships drawn in Gx now appearing on laptop screens. TAG girls importing pictures of pirate ships off the internet and placing inside Gx.

10:05 TAG girls who plotted quadratic have stopped work… explain the need for them to drive a point using parametric functions as the coords.

10:17 Lots of experimentation now going on… one boy suggests putting less or more powder into the cannon to change the velocity.

TAG girls wrestling with working out the parametic functions in the coordinates… what’s theta?  Why t? “We’ve forgot”

TAG girls(2) have animated Jack Sparrow firing the cannon… alas not so much math

Remind need to find two angles.

10:20 TAG boys firing salvos from a deck full of cannons, with many seemingly many direct hits… “we’re trying to fix them, but now we’re lost” they say laughing!

TAG girls… “we’ve got it animated, but the squid is shooting back! It’s going the wrong way”. Encourage and help to examine the functions… they’ve added the distance 1000m inside the x-function

TAG(2) girls are working on the math… trying to get the cannon ball to come out of the cannon which is on the poop deck along with an animated Jack Sparrow! Discuss how they could do this.

TAG boy (loner) struggling to understand how cos and sin work and why we use them. Brief lesson on trig with some scribbled cannons and right triangles.

09:38 (lesson ends 09:40) some students still working to complete and email to me.

I am just flabbergasted! Unbelievable! I’m bouncing around the ceiling!

At the next lesson (p4): “Class keen to see its work projected from last lesson. Lots of pride. Show a range of the work.”

We move onto plotting moving cars and finding whether or not they will crash.

09:50 TAG boys solve, they don’t collide. So… how do you make theme collide? Change speed car A. What about car B? What if a car starts earlier or later? What if a car takes a different route?

TAG girl takes time to remember the variable that will change is t… we have 2-minute dialogue where discuss everything about a car journey with kids in the back seat moaning till she hit on time is what makes them impatient. She gets the coord change +45*t immediately. Changes car A coord v quick too. She checks in she has it correct… she needs lots of validation, even tho v smart.

TAG who struggled last class, gets it quick this time.

09:57 TAG boys “they crash” Now what do you want us to do? I want you to decide how to change the problem… change time, direction.

10:05 couple of pairs need support to change the coordinate. Need to emphasise idea of time driving x and y. Up till now x has always driven y, now t for time drives x and y

TAG boy (loner) completes last lesson exercise showing the two cannon balls. The target (the eye) on the squid is clearly marked in red, points labelled, explanation included. Meticulous completion of the task, with reluctance to move on till completion. Moves on to new problem. (So, why are tests timed? What are we testing? People work at different speeds to their own different standards)

Pix of cars off internet being put on screen.

10:23 TAG girl creates a car out of points and drives the car.

TAG boys make four cars travel diagonally till all hit pedestrian in the center of the screen.

TAG boy draws on-screen map, has a miss and a collision, carefully color-coded and labelled. Other students build the Starwars battle.

10:28 TAG girl now has two cars built of points that collide… suggest she changes the nature of the crash… full on, glance, 

10:32 TAG girl 2 crashes five cars… boys sneer, they’re going too slow. I can make them go faster responds girl.

Lower ability girl, who is rarely engaged, continues to work with furrowed brow trying to change the coord to get cars to meet. Ask if she’s ok and get told she’s fine.

New boy knows exact speed change. Can run the actual collision in slow-mo back and forth. Suggest he changes time of start of one of the cars. He realizes straight away that he needs to change t in some way… he multiplies it by ¹/₂… hence making the car travel at half the speed… suggest he needs to think about just making the car start one hour later… he gets (t-1)

10:40 TAG boys getting 8 points to crash into one point… no longer talking cars talking “points” More serious in tone and attitude.

All students achieve basic aims of the investigation with many going much further. Solid engagement in a fun atmosphere for 70 minutes.

TAG girl reports the exercise on her Facebook page… suggest she posts the parametric coordinates. “I will,” she says.

10:47 TAG boys now have 12 cars crashing.

The other classes seem to pick up too. The log for the p2 lesson on cars crashing records:

The concept of miles per hour times t gives a distance not much of a problem, but need a little pushing to go there. Class not prepared to ask itself questions, but will respond to questions asked.

08:45 Model on board to make sure everyone roughly at the same spot. Class engaged + feedback is flowing freely. Working in pairs v cooperatively. Brief discussion about nature of time, no wrist watches any more (kids all show their left wrist), clocks used to tick, the only sound when no cars, etc… Someone asks how did old fashioned clocks work…

Suggest consider changing the problem. How do we ensure they meet? What if one car starts at a different time? What if a car decides to drive a different direction. Talk briefly about how math developed… by changing the problem and asking what if?

09:01 class silent and completely engaged.

J (struggles with math, but tries really hard, concepts v slow to hit home) Shows me how he has got the cars to collide, what he has changed, how he has changed the coordinates + how he has adjusted the speed of the animation so that he can see the collision clearly plus he has put a point on the screen to show the point of collision.

TAG girl raises the fact the car is going in the wrong direction… girl who has not attended class for three months(!) suggests need a negative.

On to the “final”. Ditch the common assessment and go for a Gx-based investigation: a sister/brother is using a hose to water a basket of flowers hanging above brother/sister who is sleeping in the sun underneath the hanging basket. What parametric coordinates get the water into the basket, without drenching the dozing brother/sister… or, drench the sibling and miss the basket? I add: How can you then change the problem?

p2 the first to go. We hit difficulties because I’ve misjudged some of the numbers. But now I’m not the teacher just someone in the classroom so I don’t get phased. Fixing the numbers just becomes part of the problem. The log records:

Despite the difficulties all class sticks with the problem for allotted time of 65 minutes.

Interesting… recall by students is a lot less than teachers would like to imagine. Even TAG students need a lot of reminders even though we’ve been doing this sort of problem for two weeks. Each time they catch on quicker and some can recall bits well, but few can recall all aspects. Taken back to traditional teaching this means that few students probably remembering v little in a sense that includes understanding.

The log for p5 records:

This class not able to complete this investigation without considerable amount of support, however all pairs remained on task for allotted time of 70 minutes with most completing the basic investigation. Two pairs (working together) managed to work out how to generate a stream of drops of water by adjusting the variable t… (t + 0.1).

Although this class has been working on this type of problem for several weeks using Gx, retention is an issue. The big success with this class is that students who adamantly refuse to engage with the subject, (now) do so with enthusiasm and determination when using Gx.

The log for p4:

Students willing to go further. All round good engagement. Even those who struggle, ask questions and move forward slowly.

Pair work seems to be accepted as beneficial, even where several students sit together each with a laptop!

This class v keen to draw a pic of the problem inside Gx, including placing artwork culled from the internet. All pairs manage to get to a solution where the water does not hit the  basket, but drops from above into the basket. Many plot a stream directly at the person asleep under the basket.

Explain that the parametric point represents the path taken by just one drop of water. Recount the example of the park in France where water is fired in short bursts over the pathways into barrels on either side of the pathways… bizaar effect of short isolated strings of water flying along a perfect parabola through the air. Suggest as inspiration of how to take the problem forward… how to fire a stream of drops one after the other… need to adjust t… is it t + 0.1… t – 0.1?

One student picks this up and works hard to get it to work. She does so.

One student asks why we would want to expand the problems once got an answer… explain idea that math developed through this process (heuristics).

Good end to the year.

¹ The development of Geometry Expressions is funded in part by grants from the National Science Foundation. The author sits on the NSF committee monitoring the project. The logs are being used as part of an NSF-funded research project un by Oregon State University looking into issues raised by using technology in math classroom.

We should be teaching mathematical thinking

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The use of IT in the math classroom raises implications for pedagogy in the math classroom.

This brief discusses the purpose of math education and tests the arguments for thinking more in terms of teaching creative math thinking skills instead of the current practice of teaching a series of mostly algorithmic skills. In particular the brief argues for a new form of assessment reflecting students attempts at math thinking as opposed to an ability to demonstrate facility with applying algorithms.

The brief concludes that such changes could mean more students feel success in math, grow to enjoy their math and attain more useful math skills — math thinking skills.

+++

WHAT’S the point of teaching and learning math?

For some time now, I have increasingly felt that a major issue in math education is that math teachers — plus those that create and administer the structures within which math teachers work — are not clear in giving good reasons to students about why it is so essential to study mathematics, exactly what is it we are all trying to achieve.

Current course structures, strict curriculum, standards, the emphasis on testing… all do not help teachers to reflect on or explain the purpose of what they do. Probably a majority of students, but certainly significant numbers, remain confused&hellp; and uninspired.

Part of the problem arises from the fact that just about all involved — except the students and the parents — are at least reasonably good at math, they get it, they enjoy it, they value it for its own sake, they see its value in the wider world. Few will have experienced the debilitating confusion, demoralization, despair that is the lot of substantial numbers of their students.

They see math education as turning out kids like themselves, mathematicians or one sort or another. This is fine when they encounter kids who get it, enjoy it, value it, etc.

This approach is compounded by political forces, which can include some mathematicians and their associations such as the NCTM, which view math education as little more than a utilitarian function at the service of corporate America.

For this latter group in particular, the maximum of math teaching is the delivery of a numerate workforce. And, it should be acknowledged, that is an aim of math teaching.

However…

There are swathes of young people, children, who do not get it, do not enjoy it and do not see any value in their math education. Yet many of these young people may, indeed, have math thinking abilities. They will certainly need math thinking skills, math understanding, in the fast-changing world in which they will live.

Crudely put, these children are not always well served by mathematician math teachers. Or, are not well served by mathematician math teachers who do not reflect wider on issues of pedagogy or wider (social, political and philosophical) concerns about education in its widest sense.

One dire result is that math education in the public schools is often confined and restricted to training children to be numerate, with little more.

The emphasis on standards, state testing and curricula driven by textbook adoptions, militate against wider reflection. Math teachers simply have too much on their plates to reflect… or they do reflect, and then knuckle down.

Students themselves give the clue to solving the dilemma when they ask — and they always do — “when am I going to use this in real life?”

The frank answer is most are not going to use much of it at all, and certainly not the math that gets tested in multiple-choice computerized testing. One realistic answer to the question is that math education is attempting to teach abstract thinking skills, or problem-solving skills.

But that doesn’t go far enough. It’s not just about what and why students are learning&heliip; but also how they are learning. Indeed, the how can really be the embodiment of the what and the why.

Focussing on how children learn math may be the answer to the student-question. If the experience in the classroom is totally focussed on math thinking — with the student feeling in meaningful control of the progress, mentored Vygotsky-style within the zone of proximal development by adult guidance and peer collaboration — then the teaching pedagogy itself may give real meaning, be itself the explanation of what math education is all about.

And if the assessment supports this approach, helps guide it forward, focussing on helping the student to find their own thinking skills — rather than seeking to reward or punish — the student will not so much be learning math thinking skills but experiencing using math thinking skills.

The how becomes the what and the why.

+++

The how does not include didactics, the pressure of tests and quizzes, points, grades.

The how does include peer collaboration, teacher as mentor, student control… and time for the child to play, think and work out their own solutions.

That can be done without technology. But there is no doubt that technology can really aid the approach, providing the classroom atmosphere and activities allow the child to get lost in their thought, explorations and discussions. Indeed, the creative use of technology almost demands a new approach to pedagogy in the math classroom.

It also demands a new approach to assessment…. still working on this!

Avis foregoes the restroom and expresses some geometry

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AVIS was in my Geometry class this year. Or, occasionally he was in my Geometry class. For a few minutes at a time.

Avis is not disruptive in class, as much as he simply leaves class. He’s frequently found just wandering the halls.

He has numerous accommodations to account for his behavior. Some teachers respond with referrals, but I’ve decided little is achieved by repeated referrals when the aim is learning.

In my class Avis asks — insistently — to go to the bathroom. Again and again. On his return it can be minutes before the demand is repeated. If I explain he has just been, he disappears from the room the moment my attention is elsewhere.

And when he is in class, it is hard for him to sit in one place. Avis doesn’t do much geometry, though he will turn in some work if he can sit next to an accommodating pretty girl willing to help.

Earlier in the year the class had used laptops to do various constructions and investigations using Geometer’s Sketchpad. The exercises had not been as successful as hoped, more likely because of the didactic approach I had adopted, making the investigations quite formal.

But Avis’s attention had been grabbed, and the frequency of his bathroom visits slowed. Avis had spent much of his time playing, using the program to draw pictures. As play is a positive start to learning, this seemed a welcome step.

At the end of the year I decided try again with the laptops, using a different program, Geometry Expressions, and adopting a looser teaching style.

Geometry Expressions ¹ is a geometry-algebra system. Algebraic notation is used to “constrain”, or fix, the drawings. The outcomes can also be expressed as algebra. The focus of the mathematical thinking involved is different to that used in attempting to do traditional straight-edge and compass geometric constructions.

If you constrain a triangle side-angle-side, and then try to add a second triangle using the third, unconstrained, side Gx forces some interesting thinking if you try to impose constraints on the second triangle.

Teachers and academics who have worked with Gx readily acknowledge its power for older or higher-level students — my own AP Calc class produced some lovely work based on animating hypocycloids after their exam.

The question has been could the progam help younger or lower-ability children with their mathematical thinking? Avis was not the student I had most in mind when we tried our first exercises using Geometry Expressions.

Students were invited to draw some triangles and then constrain — fix — side lengths, angles, to create right triangles.

The class then had a variety of problems involving Pythagoras. They were asked to solve the problems using Gx to recreate the problem. One particularly interesting task was to find all the possible sides of a right-triangle if two of the sides are lengths 10 and 15.

Below are extracts from the lesson log written as the the lesson progressed (TAG = “talented and gifted”, RR = “rest room”, AA = Avis):

30 mins in, most students able to change sides a and b for numbers + observe how the formula calculates the answer.

Some students use the program to draw pictures not related to the exercises. (This also happened when students introduced to GSP).

40 minutes in… higher level of engagement with activity/focus better than normal for this class. Students interested in confirming answers in discussion.

TAG-type student complains the exercises can be done more quickly without the computer… explain am training him to use the program with aim of using it as a thinking tool + more difficult problems later. Agrees to give it a second go on a more complex problem.

45 mins in… one of most difficult students (usually can’t focus at all, frequent trips to RR, etc) still engaged and asking questions… and has not visited RR! Call this student AA. AA working with girl BB who does not find subect easy, but who tries.

50 mins in… some signs of disengagement among some students.

60 mins in… TAG student happier… still feels he’s working slower, but is seeing the program can aid his thinking.

Two students raise calculation that gives pi/2. Leads to brief discussion that this is 90 degrees (this class not done radians).

65 mins… engagement now down to about one in three students. This is an improvement for this class. Class also much more quiet than normal.

Allow internet access for last 20 minutes… some students still continue with Gx.

Girl BB + boy AA still v engaged and 70 mins in are now trying to work out how to constrain a side using a radical. Still no visit to the RR by AA!

TAG student comments that Gx good “verification tool”, useful for checking answers you’re not sure of.

Pack away 80 mins in.

AA cleans board, still quiet (!!!) and no visit to RR. Wow! AA smiles genuinely brightly when I congratulate him on not going to the RR.

Two days later the same class works on using Gx to solve problems involving the equations of circles. This time Avis is very much the focus of my attentions.

Below are extracts from the log written as the class progressed:

blitz start to lesson… doors locked, etc. Computers eventually arrive from another classroom.

Students specifically told to work in pairs.

13:30 all students working. Enough computers.

AA + BB work together and ask frequent questions (on task).

Questions from students make it v easy to answer with a question focusing on why?

14:48 Class engaged. One TAG student (girl) asked for help, but figured answer while I worked my way round. Questions focus on basic use of Gx… partic how the animation works. These are easily solved questions and students pick up quick.

Talk in room is almost exclusively on the task.

pair work v successful.

13:54 Deep discussion between two TAG students on equations of circles.

Respond to earlier request from BR for help… “I got it! it’s ok…” Doesn’t look up, stays completely focussed.

LI, BB, AA have discussion about where the “variable”s go… these students never talk about “variables”!

14:03 Five students pulled out of class for OAKS state testing… they’re really upset! Includes LI who is working really hard on task.

14:12 AA + BB want to discuss what are the two things on which they must agree for them both to draw the same circle. BB using words like “segment”. AA has not been to the RR!

Two students working in Gx, but drawing pictures. Only occasionally are students caught on the internet.

Easy to get students moving.

Three TAG students now working on the extension activity.

14:12 BB and AA in deep discussion looking at the calculated equation of a circle.

14:22 AA + BB still on task. HT ok writing out definition of circle with help. BB points to an equation and asks “does this make sense to you?” We spot that she had not replaced r for radius with a specific value.

14:24 Tell students to start to shut down… BB shouts out to AA “quickly, let’s do this one.” And they do.

Sadly, departmental decisions about the need to deliver aspects of the curriculum before the end-of-semester test meant an end to the experiments with Geometry Expressions and a return to textbook-based didactics for Avis’s class. At the following lessons Avis mostly left the classroom, though he did frequently ask whether or not the class would be working on the computers.

But the brief experiment did demonstrate that Geometry Expressions can motivate and aid the mathematical thinking of younger and lower-ability children. This was confirmed by similar reactions by lower-ability students in Algebra 2 classes where it was possible to use the technology over a much longer period.

¹ The development of Geometry Expressions is funded in part by grants from the National Science Foundation. The author sits on the NSF committee monitoring the project.