The elephant in the classroom
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
Entry filed under: Pedagogy, Thoughts from the classroom, What's on the PiFactory blog.... Tags: enrichment, how to teach math, mixed ability groups, open-ended questions, Pedagogy, problem solving, student motivation, thinking skills.