What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (19 page)

When the students returned to their middle school classes in
the fall, we kept track of their achievement. Researchers visited the students’ classes to observe the teaching approaches they were experiencing and their participation. These observations worried us as we saw students sitting in rows, in silence, working through short, narrow questions requiring single procedures for solutions. Sadly, the students were returning to the same math environments that most had told us they hated, and students received limited or no opportunities to use the learning approaches emphasized in the summer. This did not give us much hope that the math classes we had taught over the summer, in which students had been so engaged and excited to learn mathematics, would have a long-term impact. It is very difficult for students to return to the same environment in which they have done badly before and do better, even after a summer of enjoying mathematics and learning new ways of working. This is why the home is such an important place to encourage students to work in good ways. It was pleasing that our students’ math grades did improve significantly in the fall, whereas the control group who attended the same summer school but did not attend our classes did not improve theirs significantly. Unfortunately, but not surprisingly, the students’ renewed enthusiasm and increased achievement did not last, and their grades had fallen again by the following quarter.

Some of the students who attended our classes gained higher grades in the quarter following the summer
and continued to achieve higher grades in math.
For these young people, the relatively short math intervention had given them a new approach to math that they were able to continue to use. When we interviewed Lisa, one of the students who had maintained her improved achievement, and asked her how the summer was helping her in the regular school year, she said, “When I don’t know how to solve a problem the way the teacher does
it, I have other ways to solve it.” Melissa, a student who had achieved an F before the summer and then achieved an A afterward, told interviewers that the most useful part of the summer had been learning strategies—in particular “to ask a question if you’re not sure” and to look for patterns. She also said, “I used to hate math and I thought it was boring, but in this class math was a lot of fun.” Melissa’s enjoyment and her learning of strate- gies certainly had a big impact on her achievement. I will describe the ways in which the strategies that we taught the students during the summer can be encouraged in the home in the next chapter.

The research results on the whole group of students we taught were very positive, but the stories of individuals are probably most interesting and most insightful in understanding why many children are turned off by math. Let’s look at four of the students:

Rebecca, Who Needed to Understand

Over the thirty or so years I have been a schoolteacher and researcher, I have met many students like Rebecca—and usually they are girls. Rebecca was conscientious, motivated, and high-achieving. Even though she achieved A+ grades in mathematics, she did not feel that she was good at it. Rebecca, like many other students, could follow the methods the teachers demonstrated in her regular classes and reproduce them perfectly, but she wanted to understand mathematics and she did not feel that the procedural presentations of mathematics she experienced gave her access to understanding. Rebecca described her previous math classes as always the same—the teacher would start with a warm-up and then give students worksheets to go through individually. There were no classroom discussions, and the questions they worked through were
shallow and procedural. In interviews Rebecca and her friend Alice were asked if they regarded themselves as “math people.” When Rebecca said no, Alice protested, saying that Rebecca had won an award in math. I asked Rebecca about this and why she did not regard herself as a math person. She said it was because “I can’t remember things well and there is so much to remember.”

When students tell me that there is a lot to remember in mathematics, I know that they are being taught badly and the subject is being misrepresented. I know also that they are being overwhelmed by the procedures teachers show and have come to believe that they must memorize them all, instead of understanding the concepts that link the procedures and render memorization unnecessary. Rebecca explained to me that you “have to remember” in math and that it was hard to remember because “you don’t use it in life.” All of the procedural lessons she experienced made Rebecca feel a failure even though she gained A+’s. Our math classes were very different as we taught students the strategies and ideas that connect mathematical procedures and give access to understanding. The notes that teachers had written on Rebecca in their recommendations for summer school told us that she would not, under any circumstances, talk in class as she was a “selective mute”—that is, she chose never to speak. But as Rebecca came to enjoy the math problems we presented and understand the mathematics we taught, she not only talked in class but even chose to go to the board and show her work to the whole class. We were surprised and thrilled to see Rebecca participate so publicly and knew that her participation spoke to the mathematical understanding she was gaining for the first time.

Rebecca told us that she most appreciated the summer classes because they allowed her to understand mathematics. Asked about the differences between the summer class and
previous math classes, she said: “We stretch the problems a lot more. Before we would just get answers and not really stretch them. . . . Like patterns you can really look at it and find how it grows. . . . Stretching the problems helps you understand the problem more. In this class you don’t stop when you get the answer, you keep going.”

Rebecca was clear that working on longer problems that she was able to explore and extend gave her access to an understanding she had never had before. Some readers may worry that students who are high achievers cannot learn in the same classes as students who get D’s and F’s, but Rebecca did not give any indication, in class or in interviews, that she was held back in such a mixed setting. On the contrary, she reflected in her journal that the summer school class “has been more useful because we take the time to make sure everybody understands everything and we use different methods of learning.”

When Rebecca was asked what she had learned, she talked about many aspects of the class. She said that she had learned to generalize algebraic patterns “instead of just staring at them,” that she had learned to multiply double-digit numbers in her head (from the number talks), and that she had learned strategies such as organization and question asking. Also, in her words, she had learned “to think beyond the answer to the problem.”

As Rebecca summarized her experience, “I enjoy this class more than any other math class I have had because we are learning math in a fun way, and I think we can learn as much or more math this way as in a textbook.”

Rebecca continued to get A+ grades in the year after our summer class, but we hoped that she no longer believed mathematics to be a subject that had to be remembered without understanding or enjoyment.

Jorge, Who Needed a Chance and an Opportunity
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Jorge came to our math class with a history of D’s and F’s in math—and in school in general. He walked into math class on the first day with a huge smile on his face, joking with three of his friends. As on most days, he was dressed in baggy blue jeans and an Oakland Raiders cap that he only took off after much cajoling from his teacher. During that first lesson Jorge and his friends laughed and talked their way through the time, doing very little mathematics. Jorge was a social force in the classroom. Funny and charming, he could pull his friends away from work in an instant. Watching Jorge in class, it was easy to see why he wasn’t doing well in school. Jorge had all the markings of a bad boy who would try to get away with doing as little as possible in math class and also keep other students from doing good work.

Observing Jorge during the later weeks of summer school, it was still possible to catch glimpses of the bad boy behavior that we saw on the first days. He still took a while to get started on tasks, still made jokes during discussions, and still tried to whisper something to throw off other students going to the board. However, he was also taking mathematics
extremely
seriously and, as he wrote in one of the surveys, he was working harder than he had
in any other math class.
During discussions, he listened to his classmates and sometimes volunteered to present his own thinking. Over time, he presented more often and with fewer jokes to deflect attention away from his work. In both the whole class and in groups, he increasingly talked with classmates seriously about mathematics, rather than always shifting conversations into the social territory with which he was more comfortable and in which he, an underperforming mathematics student, had more status. In a particularly striking example of Jorge’s move to take math more seriously, he and
two other students spent over an hour during one lesson grappling with the generalization for a challenging pattern. They kept their focus only on the problem for almost this entire time, even moving to a different area of the room when two girls started working on one side of their table. Jorge did not just follow his peers’ conversation; he also kept the other boys on the task, asking questions when he was confused and volunteering ideas. He was deeply engaged in this task for a long period of time, a striking turnaround from the first day when he seemed to go out of his way to avoid serious mathematical work.

Jorge’s comments in journals and interviews reveal aspects of the summer environment that may have contributed to his willingness to work more seriously in the class (and, importantly for such a “cool” kid, to be
seen
working more seriously). Notably, Jorge says that he worked harder
in the summer school class than his regular classes, although he also describes the summer school classes as “funner.” He explains that he works harder in summer school because “in our [regular] class they give us, like, easy problems and all that. And in this class you give us hard problems to figure out. You have to figure out the pattern and all that.”

Asked what advice he would give math teachers to help them teach better, he says that he would tell them to “give harder problems.” These comments suggested to us that he appreciated being taken seriously as a capable mathematics student.

When Jorge discussed the hard problems he was given in the summer school class, he talked both about being able to “figure out” the solutions himself and about being given the time to do that. He explained that he enjoyed working on pattern problems because “We stay on it longer . . . so we can really get to know how to do pattern blocks and everything, and try to figure out the pattern.”

Jorge also talked about the value of working in groups.
Another piece of advice he said he would give to teachers was to put students in groups, because “You learn more from other people’s ideas.”

On the days that Jorge became deeply engaged in mathematics, he was working on problems that could be viewed in many different ways; and he was working with high-achieving boys whom he respected. Jorge left our classes proud of his math work; importantly, Jorge had experienced working with high-achieving students on challenging problems, and that had changed everything for him. He felt respected, and he could offer his own thoughts and ideas. For a student who probably had been placed into low tracks and given easy work in the past, the experience of working on hard problems and discussing ideas with others was transformative.

Alonzo, Who Needed to Use His Ideas
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Alonzo was a very popular student. He was always surrounded by other students, before and after class. Alonzo could be described as the strong, silent type because of his striking, tall, athletic build and quiet, fiercely observant ways. For the first few days, Alonzo would slip into class undetected and pull the brim of his baseball cap down, as if hiding, silently watching the activities unfold before him. As the summer progressed, Alonzo’s behavior changed and we came to realize that the mathematics he was working on was allowing his curiosity and creativity to take root and bloom. As Alonzo experienced the opportunity to use his ideas in math, he started to become a different kind of student in class.

Like many other students who attended summer school, Alonzo had gained an F in his previous math class and was forced to enroll by his math teacher. Alonzo described his previous class as one where the teacher talked and the students plodded through worksheets, and where group-based collaborations
were extremely rare. Math learning in that class was largely a repetitive, worksheet-based experience. As Alonzo described it: “In regular math class we had to go to work. We couldn’t talk or we couldn’t, like, [offer] ‘Oh, can I help?’ No. He would just give us a paper, a pencil and put us to work.”
In informal conversations during class, Alonzo described his previous class as boring and frustrating.

One activity that piqued Alonzo’s curiosity and creativity was “Staircases.” In this task students were asked to determine the total number of blocks in a staircase that grew incrementally from 1 block high, to 2 blocks high, to 3 blocks high, and so on, as a move toward predicting a 10-block–high staircase, a 100-block–high staircase, and, finally, algebraically expressing the total blocks in any staircase. Students were provided with a box full of linking cubes to build the staircases if they wished.