What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (10 page)

S:
We’re usually set a task first and we’re taught the skills needed to do the task, and then we get on with the task and we ask the teacher for help.

P:
Or you’re just set the task and then you go about it. . . . You explore the different things, and they help you in doing that . . . so different skills are sort of tailored to different tasks.

JB:
And do you all do the same thing?

P:
You’re all given the same task, but how you go
about it, how you do it, and what level you do it at changes, doesn’t it?

The students were given an unusual degree of freedom in maths lessons. They were usually given choices between different projects to work on and they were encouraged to decide the nature and direction of their work. Sometimes the different projects varied in difficulty and the teachers guided students toward projects that they thought were suited to their strengths. During one of my visits to the classrooms, students were working on a project called “Thirty-six Fences.” The teacher started the project by asking all of the students to gather round the board at the front of the room. There was a lot of shuffling of chairs as students made their way to the front, sitting in an arc around the board. Jim, the teacher, explained that a farmer had thirty-six individual fences, each of them one meter long, and that he wanted to put them together to enclose the biggest possible area. Jim then asked students what shapes they thought the fences could be arranged into. Students suggested a rectangle, triangle, or square. Jim asked, “How about a pentagon?” The students thought and talked about this. Jim asked them whether they wanted to make irregular shapes allowable.

After some discussion, Jim asked the students to go back to their desks and think about the biggest possible area that the fences could make. Students at Phoenix Park were allowed to choose whom they worked with, and some of them left the discussion to work alone, while most worked in pairs or groups of their choosing. Some students began by investigating different sizes of rectangles and squares, some plotted graphs to investigate how areas changed with different side lengths. Susan was working alone, investigating hexagons, and she explained to me that she was working out the area of a regular hexagon by dividing it into six triangles and she had drawn one of the
triangles separately. She said that she knew that the angle at the top of each triangle must be sixty degrees, so she could draw the triangles exactly to scale using a compass and find the area by measuring the height.

I left Susan working and moved to sit with a table of boys. Mickey had found that the biggest area for a rectangle with perimeter 36 is 9 × 9. This gave him the idea that shapes with equal sides may give bigger areas and he started to think about equilateral triangles. Mickey seemed very interested in his work and he was about to draw an equilateral triangle when he was distracted by Ahmed, who told him to forget triangles since he had found that the shape with the largest area made of 36 fences was a 36-sided shape. Ahmed told Mickey to find the area of a 36-sided shape too, and he leaned across the table excitedly, explaining how to do this. He explained that you divide the 36-sided shape into triangles and all of the triangles must have a 1-cm base. Mickey joined in, saying: “Yes. Their angles must be 10 degrees!” Ahmed said: “Yes, but you have to find the height, and to do that you need the tan button on your calculator, T-A-N. I’ll show you how. Mr. Collins has just shown me.”

Mickey and Ahmed moved closer together, using the tangent ratio to calculate the area.

As the class worked on their investigations of thirty-six fences, many of the students divided shapes into triangles. This gave the teacher the opportunity to introduce students to trigonometric ratios. The students were excited to learn about trig ratios as they enabled them to go further in their investigations.

At Phoenix Park, the teachers taught mathematical methods to help students solve problems. Students learned about statistics and probability, for example, as they worked on a set of activities called “Interpreting the World.” During that project they interpreted data on college attendance, pregnancies, football results, and other issues of interest to them. Students learned about algebra as they investigated different patterns and represented them symbolically; they learned about trigonometry in
the “Thirty-six Fences” projects and by investigating the shadows of objects. The different projects were carefully chosen by the teachers to interest the students and to provide opportunities for learning important mathematical concepts and methods. Some projects were applied, requiring that students engage with real-world situations; other activities started with a context, such as thirty-six fences, but led into abstract investigations. As students worked, they learned new methods, they chose methods they knew, and they adapted and applied both. Not surprisingly, the Phoenix Park students came to view mathematical methods as flexible problem-solving tools. When I interviewed Lindsey in the second year of the school, she described the maths approach: “Well, if you find a rule or a method, you try and adapt it to other things. When we found this rule that worked with the circles, we started to work out the percentages and then adapted it, so we just took it further and took different steps and tried to adapt it to new situations.”

Students were given lots of choices as they worked. They were allowed to choose whether they worked in groups, in pairs, or alone. They were often given choices about activities to work on and they were always encouraged to take problems in directions that were of interest to them and to work at appropriate levels. Most of the students liked this mathematical freedom. Simon told me: “You’re able to explore. There’s not many limits and that’s more interesting.” Discipline was very relaxed at Phoenix Park, and students were also given a lot of freedom to work or not work.

Amber Hill School

At Amber Hill School the teachers used the traditional approach that is commonplace in England and in the United States. The teachers began lessons by lecturing from the board, introduc-
ing students to mathematical methods. Students would then work through exercises in their books. When the students at Amber Hill learned trigonometry, they were not introduced to it as a way of solving problems. Instead, they were told to remember and they practiced by working through lots of short questions. The exercises at Amber Hill were typically made up of short contextualized mathematics questions, such as:

Helen rides a bike for 1 hour at 30 km/hour and 2 hours at 15 km/hour. What is Helen’s average speed for the journey?

Classrooms were peaceful and quiet at Amber Hill and students worked quietly, on task, for almost all of their lessons. Students always sat in pairs and they were generally allowed to converse quietly—usually checking answers with each other— but not encouraged to have mathematical discussions. During the three years that I followed the students as they progressed through school, I learned that the students worked hard but that most of them disliked mathematics. The students at Amber Hill came to believe that maths was a subject that only involved memorizing rules and procedures. As Stephen described to me: “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, you’ve got to remember
it.” More worryingly, the students at Amber Hill became so convinced of the need to memorize the methods they were shown that many of them did not see any place for thought. Louise, a student in the highest group, told me: “In maths you have to remember. In other subjects you can think about it.”

Amber Hill’s approach stood in stark contrast to Phoenix Park’s. The Amber Hill students spent more time on tasks, but they thought maths was a set of rules that needed to be memorized, and few of them developed the levels of interest the Phoenix Park students showed. In lessons the Amber Hill students were often successful, getting lots of questions right in their exercises, not by understanding the mathematical ideas but by following cues. For example, the biggest cue telling students how to answer a question was the method they had just had explained on the board. The students knew that if they used the method they had just been shown, they were probably going to get the questions right. They also knew that when they moved from exercise A to exercise B, they should do something slightly more complicated. Other cues included using all the lines given to them in a diagram and all the numbers in a question; if they didn’t use them all, they thought they were doing something wrong. Unfortunately, the same cues were not present in the exams, as Gary told me, when describing why he found the exams hard: “It’s different, and like the way it’s there, like, not the same. It doesn’t, like, tell you it—the story, the question; it’s not the same as in the books, the way the teacher works it out.” Gary seemed to be suggesting, as I had seen in my observations, that the story or the question in their books often gave away what they had to do, but the exam questions didn’t. Trevor also talked about cues when he explained why his exam grade hadn’t been good: “You can get a trigger, when she says like ‘simultaneous equations’ or ‘graphs,’ or ‘graphically.’ When they say like—and you know, it pushes that trigger, tells you what to
do.” I asked him, “What happens in the exam when you haven’t got that?” He gave a clear answer: “You panic.”

In England all students take the same national examination in mathematics at age sixteen. The examination is a three-hour, traditional test made up of short mathematics questions. Despite the difference in each school’s approach, the students’ last-minute preparation for the examination was fairly similar as both schools gave students past examination papers to work through and practice. At Phoenix Park the teachers stopped the project work a few weeks before the examination and focused upon teaching any standard methods that students may not have met. They spent more time lecturing from the board, and classrooms looked similar (briefly) to those at Amber Hill.

Many people expected the Amber Hill students to do well on the examinations, as their approach was meant to be examination oriented, but it was the Phoenix Park students who achieved significantly higher examination grades. The Phoenix Park students also achieved higher grades than the national average, despite having started their school at significantly lower levels than the national average. The examination success of the students at Phoenix Park surprised people in England and the research study was reported in all of the national newspapers. People believed that a project-based approach would result in great problem solvers, but they had not thought that an approach that was relaxed and project based with no “drill and practice” could also result in higher examination grades.