Posts tagged ‘student motivation’

Engaging math for all learners

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The UK extends its revised secondary (high school) national curriculum to 15-year-olds from September. The focus is on engagement, and in particular engaging all learners — regardless of ability — with rich, varied and compelling math activities.

And to reflect the new priorities, the UK’s public examinations — the General Certificate of Secondary Education GCSE — will boost assessment of applications and problem solving from 20 per cent to 50 per cent.

“This does not mean that technical competence is no longer important, rather that just being able ‘to do’ mathematical techniques will not be sufficient,” writes Sue Pope of the UK Qualifications and Curriculum Development Agency* in Mathematics in School.

“Students will need to be able to think for themselves and decide when and how to use their mathematics to tackle problems within mathematics and in other contexts.”

The new curriculum is ambitious and defines “an entitlement of experience for all learners.

“Rather than labelling learners and restricting access, the richness of the entire progamme of study needs to be made available to all,” says Sue Pope. “Whilst this may seem daunting, particularly if you are used to teaching level by level… it can also be liberating.”

Mick Waters, director of Curriculum at te QCDA: “If we want young people to do well in mathematics, it helps if they enjoy the subject… to see that the subject is fascinating and exhilarating, to see the way it affects everyday life and helps to change the world in which we live.

“We have to strike a balance between the challenge of incremental steps in understanding, knowledge and skills, and the joy, wonder and curiosity of learning.

“It is not about ‘basics’ and ‘enrichment, all children should have a rich experience.”

For students to develop problem-solving and mathematical thinking schools “their classroom experiences need to be rich and varied”:

A rich mathematical task…

❏ Engages everyone’s interest from the start,

❏ Allows further challenges and is extendable,

❏ Invites learners to make decisions about how to tackle the activity and what mathematics to use,

❏ Involves learners in speculating, hypothesis making and testing, proving or explaining, reflecting, interpreting,

❏ Promotes discussion and communication,

❏ Encourages originality and invention,

❏ May contain and element of surprise,

❏ Is enjoyable,

❏ Allows learners to develop new mathematical understandings.

The QCDA worked with some 30 UK schools to develop programs of rich tasks aimed at “combining understanding, experiences, imagination and reasoning to construct new knwledge”.

Tasks and case studies are spelled out in the downloadable Engaging Mathematics for all Learners.

*Shortly after the election of a new Conservative government in May, the UK Department of Education announced legislation will be introduced in the autumn to close QCDA.


June 11, 2010 at 7:13 pm Leave a comment

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

November 29, 2009 at 1:43 am 1 comment

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.

challenge_diagonalsOn 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 99th 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.

November 5, 2009 at 10:42 pm Leave a comment

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.assess_screen_02
Assignments built from learning targets with student self-assessmentassess_screen_04 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 Starting through Getting 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.

December 14, 2008 at 11:14 pm 1 comment

Why I no longer set homework

I NO LONGER assign homework. That’s now a decision. I’ve told parents and guardians and I’ve a brief document that I give to them. None have complained. Some have been effusive in their thanks. I think I understand why.

The question of homework, assignments, demands some serious re-reflection. In the schooI in which I now teach, for most students and for most teachers assignments are how you get the points which get you the grade. From what I understand, my school is not unusual. Indeed, I suspect it is typical of a US public school.

The assignments are mostly written work. They are often against strict deadlines. And they’re graded with a percentage. The final percentage, calculated in a computer, defines the grade. Percentage of what exactly is not defined. Real percentages are always of something. Otherwise, they are a nonsense.

So, a student who hands in one assignment and gets an A grade score of, say, more than 90 per cent (let’s forget margins of error, rounding errors and all that math stuff), but misses a second assignment ends up getting an F according to the computer. The tough teachers who value what they call rigor say it’s an F. Softies opt for a C.

After much thought, I say such assignments are discriminatory, are more about teacher control and, most importantly,  have little if anything to do with learning. As an assessment tool they are educationally criminal.

Together with points-based grading and testing, homework assignments are about training students to focus on collecting points and jumping through hoops. Homework belongs to the school of education that believes sorting and ranking is the aim of the public school system… labelling the F students to make the As and Bs stand out more starkly.

The system is discriminatory because it sets up swathes of students to be labelled as failures, the poor students, the black or hispanic students in particular.

I have students who go to work a second day as soon as the school day ends — not for pocket money, but as key income earners for the family. Many live in a culture of poverty which has profound respect for learning, but cannot deliver any of the requirements for quiet study. 

But it also discriminates against students whose learning styles don’t fit a narrow vision of training based on written work, reproduction of algorithms, deadlines and drill-and-kill. The majority of high school drop-outs are not the supposed least able, but among the most discriminating and thoughtful students.

It discriminates against young people struggling with hormonal fluctuations, sleep patterns over which they have no control, erratic front lobal development and all the hard-wired tribulations of teenagerdom.

For many the label of F for failure is branded before they even get started.

Such an education system suits students with access to the largely white middle-class, second-chance affirmative action programs of private tutors, articulate parental advocates and their vision of education as grade-based stepping stones to the safe gated-community of college and entry to the corporate careers of laargely white America. And that requires sorting and ranking, narrowly-defined successes and failures… precisely what points, testing and homework assignments deliver.

Sadly, some parents see homework assignments as some sort of symbol of the worth of a school. A school that does not set or value homework must be a school of slackers. It could not possibly be a place where the hard work gets done inside a vibrant classroom, and which values and respects the home and the private time of students and families alike.

Yet many parents will acknowledge the nightly hell of homework, the nagging, the denials, the tears, the slamming of doors, the simmering rows and breakdown in communication between tired and frustrated parents and their tired and frustrated children as conversation and human contact is narrowed to irritable interrogations about homework completion.

Instead of a stress-free chat about what interesting stuff may have been learned, the end-of-the-day communion between parent and child becomes an exercise in rewards, punishments and time management.

Many parents are angry at the nightly and weekend intrusions into family time.

They are angry that another worksheet and more busy work dictates the time of a tired family needing time to build and maintain the fleeting relationship of child and parent.

They are angry that with no say in the matter, a stranger can impose arbitrary demands in their home and effectively pronounce judgement on the moral worth of the family with a simple letter.

They are angry at how meaningless or simply impossible are almost all of the tasks set.

And they’re the lucky ones… for the students most ill-served by the points-testing-homework trinity of tyranny there is no end-of-day solace and comfort as parents are absent at second jobs or through poverty-driven family breakdown.

If any homework should be set, it should be for parents and childen alike to recall that four innocent people were hanged to death for demonstrating for the eight-hour work day.

Yet, there is no evidence that homework achieves any positive educational gain whatsoever. Indeed, if the evidence shows anything, it is that homework is destructive of learning.

Certainly, there are plenty of pundits — and politicians — who insist on the value of the discipline of homework. Their concern seems to focus on discipline. Here we must be grateful to Mr Alfie Kohn and his excellent The Homework Myth — Too Much of a Bad Thing for exposing the chasm between the conclusions of these charlatans and the actual evidence of their data.

At best homework seems to be set because it seems common-sense to assume that it must be good. But that homework is then used as a prime basis for assigning ill-defined labels on a child’s worth is scandalous.

My homework policy now reads as follows:

“Research indicates it does little to promote real learning, and frequently is a source of conflict in the home between tired and frustrated students and their equally tired and frustrated adult supporters.

“Occasionally I do assign a project to be done in school and at home over a period of time.

“Students are welcome to work on what we are doing in school… if that gives them pleasure or satisfaction, but not if it is a source of stress or worry.

“However, I do understand that parents are anxious to help their students… and may feel homework is a discipline that will help.

“If you want to help your student at home, the most effective way, in my view, is to ask your student to explain the ideas we are tackling in class. Talking math is learning math. Exploring the ideas, is learning math.

“Try to listen and not intervene. Ask a few mildly probing questions. If the explanation does not help you understand the idea, suggest your student raises the matter in class to seek a better explanation.

“Please, do not push the conversation to a confrontation or anything that is anything other than a pleasant conversation in the evening in the home. Please do not try to teach your student the point under question. Just suggest the student seeks a better explanation in school.”

And grading? I don’t use use points in any form. Instead I rely on a description of the grades, and the students’ own self assessment based on descriptions of what it is to be on top of a subject (the descriptions can be found on the assessment page at assessment at The PiFactory).

A factory worker works in the factory. An office worker in an office. My students work in the classroom… not at home.

For a more cogent argument against homework see re-thinking homework by Alfie Kohn. 


January 1, 2008 at 12:17 am 1 comment


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