What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (18 page)

The Summer of Math

As I prepared to teach students in sixth and seventh grades in California classrooms, I was a little nervous—it had been some years since I had taught mathematics in schools and all of my experience had been in London. The setting was a five-week summer school class in the San Francisco Bay area. Summer school classes are not known for their serious mathematics work. The students are there for a short time, so it is difficult to establish careful and good classroom routines. The classes were also extremely mixed, combining students who loved math and wanted to spend more time with the subject at the end of the regular school year, with those who had been forced to attend because they were failing in school. Reflecting this, 40 percent of the students had gained A’s or B’s in the previous year and 40 percent had gained D’s or F’s.

Four of my graduate students—Nick, Tesha, Emily, and Jennifer—and I taught four classes of sixth and seventh graders for two hours each day, four days a week. Our teaching, and the students’ learning, was the focus of a research study—lessons were observed and students were given surveys, interviews, and assessments to monitor their learning. The classes were diverse both racially (39 percent Hispanic, 34 percent white, 11 percent
African American, 10 percent Asian, 5 percent Filipino, 1 percent Native American) and socioeconomically. I will start with a brief description of the summer school teaching and then illustrate the impact of the teaching by recalling the experiences of four very interesting young people.

One of the goals of our summer teaching was to give students opportunities to use mathematics flexibly and to learn to decompose and recompose numbers. We also wanted students to learn to ask mathematical questions, to explore patterns and relationships, and to think, generalize, and problem solve, as all of these are critical ways of working in mathematics, yet often neglected in school classrooms. As most of the students had spent the past years going through worksheets and rehearsing procedures in math classes, this was quite a change for them. We decided to focus the summer school on algebraic thinking, as we thought that would be most helpful to the students in future years, and to focus upon critical ways of working including asking questions, using mathematics flexibly, reasoning, and representing ideas. These different ways of working are all critical to success in mathematics but are often overlooked in classrooms.

It would be reasonable to assume that students, particularly those in sixth and seventh grades, know how to ask questions in a math class. Question asking is an important part of being a learner and one of the most useful things a student can do. But research has shown that student questions decline as students move through school and are surprisingly rare in classrooms.
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Indeed, such research suggests that as students progress through school, they learn
not
to ask questions but instead to keep silent, even when they don’t understand. The act of question asking has been shown to increase mathematics achievement and improve attitudes among students, and we know that students who ask a lot of questions are usually the highest achievers.
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But
while many teachers regard student questions as valuable, they do not explicitly encourage questions. In our teaching of the summer school classes we chose to encourage student questions and to teach students the qualities of a good mathematics question. We started our sessions by telling students how much we valued questions. When the students asked good questions, we would post them onto large pieces of paper that we hung up around the room. We also gave students mathematical problems and encouraged them to extend the problems by posing their own questions. When the students were interviewed during and after the summer, many of them mentioned that they had learned that question asking was a useful strategy in math classes.

A second aspect of math learning that we encouraged was that of mathematical
reasoning.
Students learn to reason through being asked, for example, to justify their mathematical claims, explain why something makes sense, or defend their answers and methods to mathematical skeptics.
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Students who learn to reason about situations and determine whether they have been correctly answered learn that mathematics is a subject that they can make sense of rather than just a list of procedures to memorize. When we talked to students (individually or in groups) or to the whole class, we would always ask them to tell us why they thought an answer made sense and to justify it to their peers. By the end of the summer, students were doing this for themselves, pushing each other to explain and justify when they talked about their mathematical ideas.

In addition to question asking and reasoning, we also highlighted the importance of mathematical representations. Proficient problem solvers frequently use representations to solve problems and communicate results. For example, they may transform a problem given in numbers into a graph or
diagram that illuminates different aspects of the problem. Or they may choose a particular representation to highlight something that helps a collaborator understand better. Although representation is a critical part of mathematical work and it is often the first thing that mathematicians do, it is rarely taught in classrooms. In our summer school teaching, we gave the students problems that were displayed in different ways, particularly visually, with manipulatives (such as cubes and beads) and diagrams, and we asked the students to produce representations as part of their work. In interviews with the students, some of them told us that they had never
seen
a mathematical idea before, and that the different representations they had seen and learned had been extremely powerful for them.

In all of our problems we encouraged and valued the flexible use of numbers. One of the best methods I know for teaching students how to use numbers flexibly is an approach called “number talks” devised by leading educator Ruth Parker.
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In number talks, the teacher asks students to work individually and without paper and pen or pencil. The teacher puts a calculation (usually an addition or multiplication problem) onto the board or overhead and asks students to work out the answer in their heads. An example of a problem we posed was 18 × 5. Teachers ask the students to signal privately when they have an answer, usually by showing one thumb, but not by raising their hand as this is too public and puts other students under pressure, also turning the activity into a speed contest. The teacher then collects all the different methods students have used. When we posed 18 × 5, four students shared their different methods for working it out.

These different methods all involve decomposing and recomposing numbers, changing the original calculation into other equivalent calculations that are easier. As students offered these examples, others saw a flexible use of numbers for the first time.

As the students took part in number talks, they realized that there was no pressure to finish quickly and that they could use any method they were comfortable with, and they began to enjoy them a lot. Some students learned, for the first time ever, to decompose and recompose numbers, as Gray and Tall recommended, as they saw others doing this and realized that it was extremely helpful. Many of the students mentioned the number talks as a highlight of the summer, as they particularly liked the challenge, not to mention the experience of sharing and seeing different mathematical methods. Although mathematics has a reputation for being a subject of single methods—with each problem requiring one standard method that must be remembered—nothing could be further from the truth. Part of the beauty of mathematical problems is that they can be seen and approached in different ways and, although many have one answer, they can be answered using different approaches. When we asked students about aspects of the teaching that had been helpful, learning about different methods was one of the most frequently cited aspects, second only to collaboration with classmates.

On the first day of summer school we gave students a blank journal to develop and record their mathematical thinking during the five-week course. We wanted the journals to be a space for students to play with ideas—as well as a safe place to communicate with us for those who were afraid or unwilling to share ideas publicly. We collected the journals frequently to look for mathematical thinking that we could help with and to give students feedback. Few of the students had ever been given the opportunity to write about math before, or to keep organized notes, which turned out to be very important for some of them.

In most of the lessons, students worked together in small groups or with partners. In some lessons, we allowed students to choose their seats and work groups; at other times, we chose seating assignments to help students work productively and to give them experience of working with a range of people and ideas. Our decision to allow students to sit where they wished at times was part of our general commitment to the promotion and encouragement of student choice, autonomy, and responsibility. We emphasized to students that they were in charge of their own learning, encouraging them to make changes in their behavior to improve their learning in class.

During the summer, we combined different tasks and ways of working, as variety is very important in mathematical work. There are many valuable ways to work in math classes—including through lectures, student discussions, and individual work—and there are many important types of tasks that students can work on, from long applied projects to short questions including contextual and abstract investigations. But none of these methods or tasks should be used exclusively, as there is benefit to students experiencing a range of ways of working, especially as they will need to work in different ways in their jobs and private lives. Among students who experience traditional math classes, one of the biggest complaints (and surely
the most reasonable) is that the classes are always the same. The monotony causes disaffection; it also means that students only learn to work as they have in class—using procedures that have just been shown to them. During our classes we spent some time discussing ideas as a class and some time discussing ideas in groups. We often gave the students long problems to work through with others, and sometimes gave them shorter questions on worksheets to work through alone. We gave them tasks in which they explored patterns, similar to the algebraic work given at Phoenix Park and Railside, and we also gave tasks that were applied, such as a soccer World Cup activity. In that task students needed to work out which teams would play each other and how many different games there would be, as an introduction to combinatorics. Sometimes students were given tasks to work on for a set amount of time; sometimes we allowed students to choose the tasks they worked on and to choose the amount of time they spent on them. Students were also encouraged to use their own ideas in extending problems and choosing methods that made the most sense to them. In all of our activities we encouraged students to ask questions, to represent, reason, and generalize, and to share and think about different methods. We also spent a lot of time building students’ confidence, praising them for their work and their thinking when it was good.

At the end of our classes we gave students the same algebra test that had been given them some months before in their regular classes. The students scored at significantly higher levels, even though we had not seen or taught to the content of the tests and our teaching had been significantly broader than the tests’ content. The average score before the summer was 48 percent, at the end of the summer it was 63 percent, a significant increase. In surveys, 87 percent of students reported that the summer classes were more useful to them than their regular classes and 78 percent reported that they had enjoyed the
classes a lot. In interviews, all of the students were extremely positive about the summer classes. Many had told interviewers that their regular classes were boring and frustrating, partly because they were made to work in silence. When reflecting on the summer class, the students talked about their enjoyment and learning, in particular from the collaboration with peers, from learning about multiple methods, from the different strategies they learned, and from the opportunities they received to think and reason.

In addition to the positive reports given in interviews and anonymous surveys, there was a huge improvement in the participation of the students in class during the summer. In the first class we asked students to complete a survey that asked them whose idea it had been to come to summer school and whether they wanted to be there. This revealed that 90 percent of students had been made to come and most of them said that they did not want to be there. The main reasons given by those who said they did not want to attend were that it would be boring, that they were losing their summer, that they would rather be socializing with friends, and that it would be unnecessary. The students’ initial participation in our classes reflected their lack of enthusiasm. In the first session, many either were quietly withdrawn, sitting with heads on their arms or hiding under hoods, or they socialized with friends, chatting loudly and resisting our requests to work. We were thrilled that as the summer progressed, students’ participation changed dramatically. After only a few days, students began to arrive at the door of class enthusiastic to start, they were taking the math problems very seriously, they were interested in mathematical questions; they participated in whole-class discussions, and generally their interest began to shift from social to mathematical concerns.