A r c h i v e d  I n f o r m a t i o n

St. Agnes School Project

Before, I taught intuitively. Now it's grounded in research. It validates my knowledge of student learning, my observations, my planning. If I design strategies based on research, I can move them along faster. I love this research.

                  --Victoria Bill, elementary mathematics teacher 

The only condition on Bill's arrangement with Resnick was that the two of them meet once a week to discuss Bill's reactions to articles and findings. Resnick, who needed a teacher's clinical insights for a strand of her NRCSL work, hoped they both would benefit from their conversations. She also felt that if she could help Bill apply research to her teaching, it would be a way of repaying St. Agnes for the school's cooperation.

"I met with Vicki Bill regularly," Resnick recalls, "and over that summer, she started building an interpretation for herself. It included some pretty radical things to do."

Bill, like many teachers in this country, had been trained to instruct her students directly in the rules and procedures of computation, to lean heavily on drill and memorization in imparting basic arithmetic skills, to teach what standardized tests would require, and to keep a quiet and orderly classroom in the process. But, Resnick explains, the research Bill was reading suggested "that if you stopped directly teaching the basic addition, subtraction, and multiplication that are the core curriculum of the first three grades of school, there was a very strong prediction that the kids would invent and use the underlying principles for themselves." This was what was "radical" about the research that Bill read. It called for a whole new approach to formal instruction, one that would build upon rather than ignore preschool children's intuitive, experience-based understanding of mathematical concepts. To Bill, it meant that conventional techniques she had been familiar with for years would have to be replaced. She concluded this from the research, but not from Resnick. "I wasn't being told I had to change, or that what I was doing in my teaching was bad," Bill recalls. "Lauren was just saying, Read this research and think about it. " The more Bill did that, the more validated she felt despite her apprehensions. "What was really a good feeling," she says, "was to read about everything that I had discovered on my own, methods I had seen children using to solve problems. I didn't have formal names for them, but I had recognized them as sophisticated kinds of thinking."

Among the techniques Bill had seen her students using were counting on, counting all, fact strategies, and MIN, which children develop sequentially in that order. "In counting all," Bill explains, "children simply count the number of objects in each of two sets in order to find the total in the two sets. Counting on is a shortcut--a device that lets children find the total number of objects in two sets without counting the first. If they see that there are three in the first set--or any number up to about five--then they just start from the three, going 'four, five, six' until they have the total for both sets." In order to count on, children must first have the concept of numerosity, which is the understanding that the number three identifies the quantity of a set with three objects in it. They must also understand that the order in which objects are counted is irrelevant to the total quantity. So when a teacher sees children move from counting all to counting on, she knows where they are conceptually. Fact strategies and the MIN strategy are further refinements of counting on, which provide additional signposts for teachers.

Despite the affirmation Bill gained by reading research, she sometimes found it "laborious" to grasp and discuss its findings. Not only were the standards and language of the research world new to her, but so were its collaborative aspects, which characterized her conversations with Resnick and occasional other researchers. "I was never part of a culture of collaboration before, where people would discuss and organize their approach to a problem together. There is nothing more stimulating," Bill declares, "but it is not part of a teacher's experience."

To Bill, the teacher's experience was to teach in isolation, work around the clock, and talk with her colleagues only in generalities about students and working conditions. "I don't think teachers have a lot of time to sit back and look at everything," she says. "Our lives are too hectic. I mean, you teach all day, and you don't stop for a minute, and then you go home to your family and do more work at night to get ready for the next day. This collaboration with researchers gave me the opportunity to sit and really think about teaching."

As a consequence of the schedule she describes, Bill had some qualms about her level of math knowledge and her ability to apply what she was learning from her reading. "I knew when I walked in here," she explains, "that there were many ways that I was not prepared, not educated." Though she agreed with researchers from the beginning that teachers needed more math, she disliked the assumptions that some made about her own background. "I taught for many years before I entered the world of research," she says. "And I considered myself good." Though she was aware of a need for a deeper grasp of mathematics, Bill knew that her clinical knowledge was at least as important. She was an expert on what she calls "the nitty-gritty details and glitches, the human chemistry of the classroom." It was because of this expertise, and Bill's determination to grapple with the instructional challenges that research presented, that Resnick had high expectations for what Bill would create out of her summer's work.

Bill's confidence grew the more she read and discussed. Each week, she and the researchers would meet and talk about "various topics--addition, subtraction, letting children be inventive, letting them talk with one another about math." As the fall school term approached, Bill committed herself to introducing effective reforms in her classroom instruction. The only question was, how? "The inventiveness that you want to see in children--how can you bring that in the classroom?" Bill asked herself. "How can you do that in a step-by-step way? For children, you have to be very structured. How could I structure this and not risk the kids' welfare if I made a mistake?"

Bill worked closely with Resnick on designing the reforms she would adopt. Some of her fears were eased by a guarantee that if her students did not progress, the researchers would personally tutor them until they could meet the school's requirements. Nevertheless, Bill was apprehensive. "I felt like I wanted a script," she recalls. "And the researchers were saying, We don't have anything to give you. Go ahead. Then, when I went back to the classroom, things certainly looked very different than during the summer. To be honest, everything I read did not fall into place until November, December."

But Bill began in September to change the way she taught arithmetic. She introduced many more manipulatives than she had ever used, creating opportunities for children to count, sort, match, and regroup objects in their explorations of numbers and amounts. She carefully designed her lessons "so that they were sequenced for them to make discoveries," and in doing so she was able to build deliberately upon the developmental patterns she had always seen--the progress from counting all to counting on, for example. But her first efforts "were still pretty much teacher guided. I was still telling my students what I wanted them to know, rather than helping them find it out for themselves."

Bill, in effect, had set out to test the research-based prediction that children would discover mathematical concepts on their own if teachers stopped explicitly teaching those concepts. But it was not until Resnick visited her class that she began to see how much she still was stating rather than demonstrating. "When Lauren came for the first time," says Bill, "she told me it was not what she had expected. One of her comments was, 'I expected more talking,' and here I was thinking that two kids talking together was pretty good. I was used to this silent room."

By degrees, Bill designed a set of teaching strategies that reflected many of the principles applied by the AFT's Visiting Practitioners. In fact, Bill and Resnick defined many of those principles during Bill's first year of redesigning her instruction, and they shared them with the Visiting Practitioners during that project's second summer workshop. The principles all were aimed at nurturing the habits of thought and feelings of confidence that would establish young learners, in their own and other people's minds, as mathematical reasoners. By November and December of 1987, when Bill felt her knowledge of research start to fall into place beside her clinical expertise, her incorporation of these principles had thoroughly revamped her classroom and taken her far beyond her initially cautious reforms. She did not, however, lose her concern for orderly routines of the sort Gaea Leinhardt's classroom research identified. "I had to tighten my routines," Bill says. "I developed specific rules for manipulatives--the children learned where to keep them, how to put them away, not to handle them until I gave the signal. Every move was efficient. Those things make or break your teaching, no matter how innovative it might be."

The specific principles on which Bill based her instructional changes are outlined in a paper she wrote with Resnick and others entitled, "Thinking in Arithmetic Class." They are:

1. Develop children's trust in their own knowledge. This leads to an effort to extend children's intuitive, pre-instruction knowledge through familiar learning methods, including finger counting and manipulatives, the use of everyday language to describe mathematical relationships and problems, and an emphasis on multiple procedures for solving problems. The latter, in particular, encourages children to invent numerous strategies of their own for reaching the same conclusion.

2. Draw children's informal knowledge, developed outside school, into the classroom. Specifically, children are encouraged to use counting extensively in story problems, especially about real-life situations. This helps them to quantify their intuitive knowledge, relate it directly to the use of numbers as symbols, and prepare for the connection between their informal knowledge and the formal notation they soon learn.

3. Use formal notations as a public record of discussions and conclusions. When the teacher uses standard mathematical notation to record children's conversations about math, carried out in everyday language and rooted in well-understood problem situations, the notations take on a meaning directly linked to children's mathematical intuitions and experience.

4. Introduce key mathematical structures as quickly as possible. This means explicitly laying out for children the mathematical situations that represent their initial, intuitive knowledge--that is, introducing them right away to addition and subtraction problems, the composition of large numbers, and strategies such as subtraction by regrouping, and then letting them develop mastery over time. This, as "Thinking in Arithmetic Class" notes, "constitutes a major challenge to . . . . the notion of learning hierarchies--specifically that it is necessary for learners to master simpler components before they try to learn complex skills."

5. Encourage everyday problem finding. Because children need far more practice solving math problems than they can obtain in the classroom, and because they need also to understand how ubiquitous math is in everyday life, they are encouraged to identify and solve the problems that surround them outside of school.

6. Talk about mathematics, don't just do arithmetic. Because discussion and argument are essential to children's development of critical thought and to their ability to justify their ideas, students routinely discuss difficult problems that the teacher poses. They talk about what information the problem provides, what remains to be discovered, and what strategies might be used. Then they work together in teams to solve the problem, and they justify their solutions to the class, generating comparisons with other teams' solutions and further discussion about the nature of the problem.

These six principles, write Resnick, Bill et al., were intended to "engage children from the outset in invention, reasoning, and verbal justification of mathematical ideas . . . . Our goal was to use as little traditional school drill as possible in order to provide for children a consistent environment in which they would be socialized to think of themselves as mathematical reasoners and to behave accordingly. This meant that we needed a program in which children would successfully learn the traditional basics of arithmetic calculations as well as the more complex forms of reasoning and argumentation."

The changes in Bill's students were dramatic. Her first-grade students--nearly all of them low-income, minority children who in kindergarten had done poorly in both math and reading--entered her classroom with scarcely any formal skills. Most could not count to 100, or even across the boundaries of decades (for example, from 29 to 30). Only a half-dozen could solve simple addition problems, even with the aid of finger counting or manipulatives. But by December, nearly all could solve both addition and subtraction problems, half of them by using procedures they invented themselves. By the end of the year, all of Bill's children were performing well, and some were even able to handle multi-digit addition and subtraction problems. Their standardized test scores had risen from the 25th percentile to the 80th, with even the lowest-scoring child comfortably ensconced in the 66th percentile.

Children's confidence and enthusiasm also rose to unprecedented levels during Bill's first year of innovation. Her students turned in their homework without prodding, often asked for extra math time, showed off proudly for visitors to the school, and eagerly brought problems from home. Their parents, most of whom belonged to a population that is typically disaffected from schools, began to notice their children's delight in math and problem solving, and they asked Bill to incorporate their daughters' and sons' "found" problems into classroom lessons.

The implications of Bill's reforms, which are now in their fifth year of refinement and are being disseminated to the classrooms of 38 other teachers, have in Resnick's view begun to represent "a new theoretical direction in our thinking about the nature of development, learning, and schooling." In "Thinking in Arithmetic Class," Resnick et al. associate this new direction with "the view . . . that human mental functioning must be understood as fundamentally situation-specific and context-dependent, rather than as a collection of context-free abilities and knowledge. . . . As we developed our program, we found ourselves less and less asking what constitutes mathematics competence or ability for young schoolchildren, and more and more analyzing the features of the mathematics classroom that provide activities that exercise reasoning skills."

The importance of classroom features and of the social culture in which learning takes place is exactly why the classroom teacher, with her experience of the existing culture and her ambitions for a new one, is the key player in education reform. It is also why, as both Bill and the AFT Visiting Practitioners have discovered through their dual experience as reformers and disseminators of reform, teachers must teach each other how to introduce changes in their instruction.

Bill, like the teachers in the AFT/NRCSL collaboration, had to go through much the same process that research called for children to navigate--an integration of her intuitive, practical knowledge of the classroom with the formal, analytical findings of education research. A crucial part of a similar process for students is continuity with their familiar ways of learning and a gradual incorporation of those familiar ways with new ones. Bill's introduction to new ways was anything but gradual; she was suddenly immersed in an unfamiliar world, reading accounts of work whose supporting theories and operative hypotheses she had never before encountered explicitly. "No one could say to me, 'I know what you re going through,'" Bill recalls. "When I took what I learned into the classroom, no one could say, 'I know what it's like not to make any progress for three days.' No one could say, 'Give it a couple more days and you will be surprised at the progress.' Or 'Try this; it worked for me.'"

Bill, with her knowledge of two worlds, is now a model for the 38 teachers to whom she has introduced her new methods. "They think, 'If she can do that with her kids, who have never performed well before, I can certainly do it with mine,'" Bill says. She is also a model for the researchers, demonstrating to them the limits of their own expertise and the need for broadening it. "I show the researchers how to consider the teachers' perspective," she explains. "They now tend to watch what I do and to say, 'If Vicki did this, she had to have a reason,' whereas, before, they might have advised me to do what they thought was best. Some of them never really realized the importance of certain pedagogical moves, language, techniques. I show them how I make connections between intuitive math and formal math happen for kids."

The same recognition of the need for leadership and professionalization among teachers characterizes three other collaborative projects with ties to NRCSL. Two focus on mathematics education and draw upon the same research--conducted at NRCSL and elsewhere--that informs the AFT/NRCSL project and the St. Agnes School project.
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[Collaborative Projects Foster Teacher Professionalism]

[Thinking Through Language]