What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (20 page)

A 4-block-tall staircase; total blocks = 4 + 3 + 2 + 1 = 10

Midway through the time allotted for the activity, Alonzo seemed to be playing with the linking cubes and not working on the problem. Drawing near, we saw that Alonzo had decided to modify his staircase so that it extended in four directions. Thus a 1-block-high staircase had a total of 5 blocks, a 2-block-high staircase had a total of 14 blocks, and so on.

Alonzo’s staircase: 5 blocks

5 + 9 = 14 blocks

5 + 9 + 13 = 27 blocks

From a distance we had thought Alonzo was off task, but he
was using his creativity and curiosity to create a problem that was more diagrammatically and algebraically involved than the one that we had given to him. Eventually other students began looking in on Alonzo’s work and, after convincing themselves they had mastered the original problem, they too tried to do the “Alonzo staircase” problem.

The teacher of Alonzo’s class was so impressed with his innovative approach to the problem, as well as his growing interest in class and willingness to push himself to work at high levels, that she decided to phone home and tell his parents of his achievements. During this call, his mother described Alonzo as a junior engineer who had designed several small projects around the house—among them was a mechanical pulley system using dental floss and a stack of pennies. He used this in his bedroom to turn the light switch off without getting out of bed. Alonzo’s mother told how he locked himself in his room while working on the project, coming out only to gather additional materials. In recounting the project to his mother, Alonzo said he’d had to experiment with different-size stacks of pennies until he figured out the exact weight it took to turn off the light switch without breaking the dental floss. Despite his creativity and apparent interest, she explained that she hadn’t heard a single good word about Alonzo’s mathematical work since he was in third grade. Alonzo’s mother was very grateful for the call and asked if it would be possible for us to keep in
touch with her after the summer to help lobby his teachers for more opportunities in which Alonzo could explore mathematics and use his creativity and resourcefulness. As we watched Alonzo work and reflected on the achievements his mother described, it was difficult to understand how he could ever achieve F grades in math classes.

Over the course of the five weeks, the open, problem-solving, group-based format of the summer program not only allowed Alonzo to play out his mathematical curiosity, but also encouraged him to take a more visible role in the classroom. During an activity called “Cowpens & Bullpens” students had to determine how many lengths of fencing were required to contain an increasing number of cows, given certain fencing parameters. At the close of the activity, student volunteers were asked to share their solutions and Alonzo was the first to volunteer. He strode to the front of the room and carefully diagrammed his fencing strategy on the overhead projector, documenting his work numerically and, eventually, algebraically.

In that moment the young man who once hid behind his baseball cap had all but disappeared and, in his place, we saw a seemingly more confident math student willing to share his ideas with the entire class. Alonzo was one of the highest scorers on the algebra test we administered at the end of the summer, with an impressive 80 percent, some 30 percentage points higher than when he took the test in the school year. But when Alonzo returned to his regular classes and was required to work on short questions in silence, he again withdrew and the grade he received for the class was, again, an F.

Tanya, Who Needed Discussion and Variety

Tanya described herself as a people person, which was easy to understand as she spent much of her time in math class talking to others. Tanya and her friend Ixchelle would talk and laugh as they worked, often whispering conspiratorially. Tanya did not seem to be aware that teachers were noticing her behavior and they would frequently ask her to be quiet. Tanya’s talkativeness must have concerned many of her teachers. Indeed, her previous math teacher recommended that she attend summer school and among her comments wrote, “Tanya has a voice that carries easily.” This not-so-subtle reference to Tanya’s assertive personality implied that her exuberance needed controlling. But Tanya’s own perspective on her social needs was interesting. In interviews Tanya described herself as needing an environment where students could work together because, as she explained: “Then I know if I’m doing wrong. If my way isn’t the only way, if my way can be easily comprehended, or if it’s a little too hard so I should try something different.”

But Tanya’s previous math classes did not allow collaborative work, as Tanya described to us: “For the past year, math was the hardest because you’re not supposed to talk, you’re not supposed to communicate. . . . [In other subjects] they let you
talk. . . . In math class, [the teacher says] just, like, ‘Okay, get to work, no talking, be quiet, shhhhh!’”

Tanya went on to describe the class and the teacher as dominated by silence: “[It is] a whole hour of silence. That’s a good class to [the teacher].”

Our summer classes were very different and, for Tanya and many other students, the collaboration they experienced in class was critical to their engagement. Tanya spoke eloquently about the opportunities afforded by the discussions, and her reasons for valuing them had nothing to do with socializing or enjoying her time but with understanding mathematics. For example: “You can do it multiple ways. . . . The way our schools [in the regular year] did it is, like, very black and white, and the way the people do it here [in summer school] it’s like very colorful, very bright. You have very different varieties you’re looking at; you can look at it one way, turn your head, and all of a sudden you see a whole different picture.”

Tanya enrolled in summer school because she was not doing well in her previous math class. In our class Tanya did extremely well and achieved one of the highest scores on the final algebra test. In interviews Tanya reported that she “learned more in five weeks than in almost a year.”
This statement reflected Tanya’s appreciation of the learning opportunities she received.

One key feature of the program that Tanya seemed to bene-fit from was the varied nature of the mathematical tasks and the ways in which they encouraged multiple methods and approaches. Tanya especially appreciated the tasks that included manipulatives (blocks, tiles, linking cubes), the pattern work, whole-class activities such as the number talks, group and pair work, journal writing, and in-class presentations. In interviews she remarked: “I’ve enjoyed finding the patterns and stuff like that and groups . . . and it really helped me. The number talk really helped me to do stuff mentally.”

Tanya seemed to revel in the plurality of ways she had to engage in math and described the tasks as more challenging, interesting, accessible, and fun: “It was much funner. You not only got to, like, just see the problem, you got to think it, you got different parts . . . you got to smell it, you got to eat it. And then after you finish the task that’s given to you, you need to have another assignment to be, like, well, what if this changed, and you did this with that, instead of, you know, that? It just opens your mind and makes it harder, a new way of thinking.”

Through interviews in the summer and fall, as well as in-class observations, Tanya revealed how the open, group-based format of summer school allowed her to express herself socially and develop mathematically. Tanya, like many of our summer school students, wanted desperately to appreciate math and to experience it on her own terms. She was, as she described herself, a “people person” who strove to see the color in math and yet found herself in situations that were monotonous and gray. Tanya’s positive experiences in the summer had an impact on her, and she tried very hard to engage with her regular math class in the fall, receiving a very respectable B grade, but by the second quarter she had dropped to a D again. Tanya’s low achievement, which did not fit her potential or enthusiasm from the summer, can probably be explained by the silent and monotonous nature of her school math class, as she aptly described: “I would say . . . the only way to describe summer school is very colorful and then this [regular school year] is just still, ugghhh, black and white. And you just wanna ask, ‘Can I have a little bit of yellow?’” Tanya’s request and that of so many other students—to experience a mathematics that is varied and interesting and colorful—is eminently reasonable.

In our summer school we gave students activities to work on and strategies to use that I will share in the next chapter. We taught the students that mathematics is a subject with many different methods and that numbers can be used flexibly. We
encouraged the students to have confidence in their own mathematical abilities, something that no doubt contributed to their significantly improved performances in the next math classes they attended. It was unfortunate that the students went back to impoverished versions of math teaching and that we could not stay with them to encourage their new ways of working and thinking, but parents
can
stay with their children and keep steering them in the right mathematical directions. In the next chapter I will set out the sorts of activities and settings that will give children the best mathematical start in life and that may encourage students of all ages to enjoy and succeed at mathematics.

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