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 rememberit.” 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 todo.” 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
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 rememberit.” 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 todo.” 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
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