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 thetriangles 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 inthe “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
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 inthe “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
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