Introducing Contexts for Learning Mathematics
Contexts for Learning Mathematics is a series of 24 units on the topics of number, operation, and algebra, K-6, developed by teacher educators, mathematicians, classroom teachers, and researchers from Mathematics in the City and the Freudenthal Institute. The materials can be used as either supplemental or replacement units. There are three units at each of six grade spans (K–1, 1–2, 2–3, 3–4, 4–5, and 5–6) that consist of two-week sequences of investigations, games, routines, and minilessons designed to support development on key topics such as place value, subtraction, multiplication, algebra, and fractions. There is also 7one resource unit for each grade span that includes a compilation of minilessons or games that can be used throughout the year.
The early childhood package, Investigating Number Sense, Addition, and Subtraction, was designed for grades K-3. It includes eight units that use classroom-tested, carefully crafted math situations to foster a deep conceptual understanding of the number system, place value, addition and subtraction, and early algebra. The package also includes three resource guides, consisting of a variety of minilessons and/or games that can be used throughout the year.
Mathematics in the City, located at City College of New York, is a national in-service provider for mathematics educators, K–8. The development of the center and the professional development materials that accompany this series were funded in part by the National Science Foundation. Further information on the center's professional development offerings as well as research on student achievement in Mathematics in the City classrooms is available at www.mitcccny.org.
The Freudenthal Institute is part of Utrecht University in the Netherlands. Founded in 1971 by the German/Dutch writer, pedagogue, and mathematician Hans Freudenthal (1905-1990), the institute has gained an international reputation for research and curriculum design with its theoretical approach toward the learning and teaching of mathematics known as Realistic Mathematics Education (RME). RME incorporates a specific perspective on what mathematics is, how students learn mathematics, and how mathematics should be taught. The principles that underlie this approach are strongly influenced by Hans Freudenthal's concept of "mathematics as a human activity." He felt that students should not be considered passive recipients of ready-made mathematics, but rather that education should guide students to reinvent mathematics by doing it themselves. Starting with rich contexts that can be "mathematized"at many levels, the students gradually develop mathematical tools and understanding. Subsequent contexts are carefully crafted to support progressive development. Models that emerge from the students' activities, supported by classroom interaction, are explicitly used to lead to higher levels of mathematical thinking.
Why We Wrote These Materials
Mathematics or Mathematizing?
What was math like for you as a young child? When asked that question, many of us conjure up images of students sitting in rows, with heads bent over, working diligently on worksheets, practicing procedures teachers had explained. Of course, it wasn't all drill sheets; we did do problem solving. But usually this activity came at the end of a unit to see if we could now apply what we had been practicing to a real situation. We weren't asked to do problems until the teacher had explained the procedures. If we used manipulatives, they were intended to help us understand the procedure that the teacher wanted us to know. There was also an implicit power hierarchy underlying the interaction in the classroom. The teacher asked the questions and already had answers for them that we were supposed to figure out.
I don't remember really loving mathematics until I hit geometry in high school and was finally asked to actually do mathematics—to come up with my own proofs. Wow! Now that was fun, like cracking a mystery. And the more elegant the proof, the more beauty! When I reflect on my own experiences, which I know were similar to many other students', I often think how divorced primary school mathematics was from the real world of what mathematicians do. Mathematics is a highly creative activity. Mathematicians solve problems but they also pose problems. They inquire. They explore relations, investigate interesting patterns, and craft proofs. They present their ideas to the mathematics community and those ideas hold up only when the logic of the argument is accepted. Real mathematicians don't line up before a wise one who checks their answers with a red pen!
Mathematicians are human beings who see their world through a mathematical lens. It is the problem that drives them. Humans across all cultures and through time have constructed beautiful mathematics to solve the problems around them, to mathematize their worlds, and to model their worlds mathematically. They constructed number systems to record how many, or how many more were needed, or how many altogether. They invented fractions to produce and record fair shares.
Hans Freudenthal believed that mathematics should be thought of as a human activity of "mathematizing"—not as a discipline of structures to be transmitted, discovered, or even constructed, but as schematizing, structuring, and modeling the world mathematically. When I first became a math educator twenty years ago, my passion for the field came from wondering if we were teaching mathematics as a dead discipline, like Latin, rather than the living, rich discipline that it is. Were we perhaps only teaching the history of mathematics—the results that mathematicians constructed centuries ago? How could I work to change this situation? Could children, even young children, engage right away in the activity of doing mathematics, finding ways to mathematize their lived worlds? If so, what would that look like?
The Role of Contexts
Math classes today usually look very different than they did when I was a child, thank goodness. The National Council of Teachers of Mathematics has published new standards and principles (NCTM 1989, 2000). Manipulatives abound. Children work on problems using a variety of strategies. In whole class discussions, they are asked to justify and explain their thinking to their peers. The emphasis is on conceptual understanding, not just procedures and practice of them, and many new curricula now exist with these goals in mind. But are they sufficient? Are they good enough? Are children really developing increasingly sophisticated ways to mathematize?
Over the last several years, my colleagues and I have been working with many talented, wonderful teachers across the United States and Canada. Because our in-service model is based on co-teaching, we have worked with children in many classrooms over time using a variety of curricula. No matter what the curriculum, we found ourselves constantly supplementing, deleting, and replacing the curriculum's isolated (and often trivialized) activities with richer investigations designed to ensure mathematical development. Because of our collaboration with the Freudenthal Institute, where the RME didactic was developed, we often used contexts from children's lives as starting points and crafted them to support learning. We built potentially realizable suggestions and constraints into the contexts and carefully chose numbers that would foster and support the development of specific strategies. We juxtaposed related problems so children would have to grapple with the resulting patterns in the data and wonder: Why is this pattern occurring? Will this always happen? Could I prove it?
This didactical use of context is not prevalent in North America. Instead, many curricula use an activity-driven approach with isolated "hands-on" activities and trivialized "school-type" word problems. In the hands of very talented teachers, these activities sometimes generated very interesting discussions, but more often they went nowhere. And even in the best of cases when good discussions did occur, the next day's activity might be unrelated or, even if related, fail to support progressive development.
The Importance of Emergent Modeling
In an attempt to focus on conceptual understanding, many of the child-centered curricula do make use of models like number lines and the rectangular array. But they do little to develop the models. The development of modeling is also part of the RME approach. In our work, we have found it important to develop models progressively: first, as models of a realistic situation, then as models to represent computation strategies, and only later as mathematical tools to think with (Gravemeijer 1999). Math ideas do not exist in the model; they exist in a learner's head as he or she makes meaning (Resnick and Omanson 1987). Learning how to make jumps on a number line, as a way to do addition, can become just a rote, meaningless activity if the learner does not fully understand what a number line is. It is not just a line with numerals on it; it is a distance of iterated units. Once these models are developed, they can be used as powerful tools for thinking—for generalizing, proving, and even doing algebra.
From Numeracy to Algebra
Numeracy and algebra are critical issues in today's world. Children need a strong understanding of number and operation, solid mental arithmetic strategies, and a deep enough understanding of operations that the transition to algebra is easy. From our perspective, none of the curricula we were working with treated computation sufficiently and none really pushed children to generalize or transition to algebra.
To strengthen computation, we began to design minilessons with strings of related problems to enhance students' development of numeracy and to expand their repertoires of strategies for mental arithmetic. Our goal in designing these strings was to encourage children to look to the numbers first before they decided on a strategy and to develop a sense of landmark numbers and a toolbox of strategies in order to calculate efficiently and elegantly—like mathematicians who employ a deep understanding of number and operation.
For three years we ran a think tank on the emergence of algebra. We field-tested sequences using the open number line to develop an understanding of equations and strategies for solving for unknowns. We encouraged children to develop conjectures and proofs.We helped teachers find the moments to push for generalization—to extend the work on number and operation to algebra.
Over the years, many of our teachers pleaded with us to write up the sequences we were developing into curriculum units. They asked for resources to engender investigations; suggestions for developing models, games, and ideas from our algebra work; and resource books full of our minilessons. Finally, with the nudging and support of Harcourt School
Publishers and Heinemann, we began to write. The result is Contexts for Learning Mathematics.