Changing the Elementary Mathematics Curriculum: Obstacles and Challenges

Susan Jo Russell

TERC
2067 Massachusetts Avenue
Cambridge, Massachusetts 02140

Problem-solving in the U.S. elementary curriculum: Some history

There has been an evolution in the definition of what an appropriate problem is for students ages 6-11. For many years in the US, problems for elementary age students consisted of disembodied numbers and operations. A typical textbook would consist of pages and pages of addition, subtraction, multiplication, or division problems. The underlying assumption seemed to be that if students did many, many problems of a certain type, they would understand how to do that kind of problem. Emphasis was on following one prescribed rote procedure for the indicated operation and getting the correct answer. Students were not expected to estimate or to use what they knew about the structure of the number system and the relationships among the numbers to help them solve the problems.

Textbooks have also typically included what we call in the US "word problems" for example:

Joan and Jere went to the beach. Joan found 19 shells, and Jere found 12 shells. How many shells did they find altogether?

Some people identify such word problems as "problem solving"; However, many of these word problems are nothing more than computation practice, just like the pages of numerical problems. When the central focus in the mathematics classroom is on using a prescribed procedure to find the correct answer, we find that students ignore the context in such word problems. They extract the numbers from the problem and use some operation on them -- not necessarily the correct one. We have seen 8- and 9-year-olds who, rather than reading the problem, simply try every possible operation with the numbers in the problem until they get an answer they think is reasonable. Rather than making sense of the problem by drawing a picture or making a model, they look for key words (such as altogether in the problem above) which seem to indicate which operation is correct. Perhaps this is because they recognize that the problem is not a real-life application as it claims to be, but simply slightly disguised computation practice.

Educators who teach young children began to reassess the nature of the problems that young children encounter in their mathematics classes. Early childhood educators have long recognized that young children are engaged in the task of making sense of the world around them -- sorting, classifying, naming, sequencing, comparing. A strong movement to make mathematics more "relevant" to students' own lives began to influence the nature of the problems given to students in the elementary grades.

Problems were formulated to be more "realistic" or to use "real data" However, when these efforts did not include reassessing the goals of mathematics education, even these efforts continued to generate what young students quickly recognized as phony mathematics. Consider, for example, a problem from a textbook in which students are given the length in miles of the world's seven longest rivers and are asked to find the average length of these rivers. While the data are "real" and the length of rivers is potentially interesting to young students, the problem itself is silly. Why would we want to know the average length of these seven rivers? Students recognize that the problem is contrived -- again in order to give them practice in demonstrating that they can use a particular algorithm.

The almost exclusive focus on learning rote procedures for operations with whole numbers, fractions, and decimals in the elementary grades led to several serious and unfortunate results. First, many, many students emerged from the elementary years with a dislike for mathematics that lasted into their adult lives. Their belief that they were not "good at math" led them to avoid taking mathematics courses beyond the minimum requirements. Mathematics became a barrier which filtered out far too many students from careers that require a good grasp of mathematics. Second, many students, including those who were successful in school mathematics, never developed an appreciation for the beauty, order, and pattern found in mathematics. They saw mathematics only as a way to solve individual problems, rather than as a way of thinking that involves making conjectures, finding patterns, examining the characteristics of mathematical objects, using examples and counterexamples to test hypotheses about mathematical relationships. Third, students saw "school math" as a collection of arbitrary procedures disconnected from their own knowledge and experience. Students discarded their own sound intuitions and good number sense as they learned that their own thinking was not sought after in the mathematics classroom.

In many elementary classrooms -- and this is still true today -- the place of the algorithm in the larger endeavor of doing mathematics became distorted. Memorization and use of particular algorithms became the whole aim and purpose in the mathematics classroom. Rather than developing a sound and deep understanding and appreciation of the number system, teachers and students believed that one was doing mathematics when reciting the chant,

"6 and 6 is 12, put down the 2 and carry the 1; 1 and 2 is 3, and 3 is 6." It is well documented that when such algorithms are taught very early, without firm grounding in the structure of the number system, young children tend to focus on the procedures of manipulating individual numerals in a prescribed way. They no longer think about the quantities 26 and 36 and the relationships between these quantities. Mistakes made through blind application of rote algorithms tend to go unnoticed by the students, and estimation is not used to predict or check results. When students learn to use algorithms in this way, we often see errors such as:

resulting from misapplied or misremembered algorithms.

When students do pages and pages of similar addition problems, following the rules they had learned for addition, they are not engaged in problem-solving, but in remembering and using an algorithm over and over. While I am certainly not making an argument here that algorithms are not useful -- they are, of course, extremely useful tools -- I am arguing that the blind, repetitive use of algorithms is not doing mathematics.

Towards a new elementary mathematics

During the last ten years, the mathematics education community in the US has been reexamining the nature of mathematical problem-solving for young children. With the publication of the National Council of Teachers of Mathematics (1989) Curriculum and Evaluation Standards, the community has come together around new objectives for the elementary classroom. The focus in the elementary classroom is shifting towards an emphasis on mathematical reasoning and problem-solving in a true sense -- thinking mathematically in order to solve a problem that you do not know how to solve. In this view, what makes a problem a problem is that it is problematic for the person engaging in trying to solve it. Further, the Standards and other current reform documents (e.g., National Research Council, 1989, 1993) emphasize that in order to solve problems, students must learn to describe, compare, and discuss their approaches to problems. Alternative strategies are valued, and multiple strategies -- rather than a single, sanctioned approach -- are encouraged. In order to learn, students must learn from each other, as well as from the teacher's questions. They must communicate about their mathematics.

Mathematics classrooms are changing. In the old style of elementary mathematics classroom, students

  • work alone
  • focus only on getting the right answer
  • record only by writing down numbers
  • complete many problems as quickly as possible
  • use a single, prescribed procedure for each type of problem

In the mathematics classrooms many educators are now striving to create, students

  • work together
  • consider their own reasoning and the reasoning of other students
  • communicate about mathematics orally, in writing, and by using pictures, diagrams, and models
  • carry out one or two problems thoughtfully during a class session
  • use more than one strategy to double-check

Many elementary school teachers are eager to change their classroom practice in order to engage their students more deeply in mathematics. However, most elementary teachers have not themselves had sound mathematics training and experience. One of the biggest tasks we face in the US is the staff development of the elementary teachers in mathematics. One of the critical needs these teachers currently have is for new curriculum materials that can help them learn mathematics content and pedagogy as they are teaching their students. 1

New curricula: Goals and philosophy

The National Science Foundation has funded about a dozen new curriculum projects to develop curricula at the elementary, middle school, and high school levels. At TERC2 , we are working on one of these projects, a curriculum for kindergarten through grade 5 called Investigations in Number, Data, and Space. The major goals of this K-5 curriculum effort are to:

  • offer students meaningful mathematical problems
  • emphasize depth in mathematical thinking rather than exposure to a series of fragmented topics
  • communicate mathematics content and pedagogy to teachers
  • serve as a tool for radically expanding the pool of mathematically literate students.

The Investigations curriculum embodies an approach radically different from a textbook-based curriculum which leads students through 50-100 separate topics, most of which involve only basic arithmetic processes. Rather, this curriculum consists of a set of six to ten units of work at each grade level. Each unit offers a set of connected investigations that focus on major mathematical ideas within the areas of number (including operations, computation, number patterns, and number theory), data collection and analysis, geometry, and the mathematics of change. Besides offering significant mathematics content, the investigations encourage students to develop flexibility and confidence in approaching mathematical problems, proficiency in evaluating solutions, a repertoire of ways to communicate about their mathematical thinking, and enjoyment and appreciation of mathematics. Because we see teachers as the primary audience for this curriculum, the materials are addressed directly to them and include notes on mathematical ideas and dialogues from classrooms designed to support teachers in learning more about mathematics and about children's mathematical thinking. The project has also developed assessment tools, videotapes for teachers, and computer environments that support this approach to mathematical investigation.

We want students to:

  • develop fluency in approaching mathematical problems. Students must gradually acquire a repertoire of mathematical tools, processes, and approaches which they can use flexibly to solve problems. These include specific knowledge of number relationships, "number facts" and algorithms (these algorithms are developed by the children themselves), geometric relationships, ways of organizing and representing data using a variety of graphs, charts, tables, pictures, and concrete objects, knowledge of calculator use, and facility with mental arithmetic.
  • evaluate their solutions to mathematical problems. In order to look at the reasonableness of their results, students must develop skills in estimation and have a solid foundation in the structure of the number system and the outcomes of arithmetic processes as well as experience with spatial relationships. We want students to know that an answer is correct "not because the teacher says it is, but because its inner logic is so clear [National Research Council, 1989, p. 3" This inner logic will only be apparent to the student who is well grounded in the structure of the number system and of geometric relationships.
  • communicate about their mathematical work. In order to think about their mathematical work, students must keep track of their approaches and strategies for solving problems. "Keeping track" in mathematics is, we find, a process far from the experience of most elementary students. Without keeping track of their own work, they are unable to describe it, evaluate it, change it, or talk about it with others. We expect students to develop a large number of strategies for keeping track; through their writing, sketching, drawing, charting, graphing, students communicate with peers, with the teacher, and with their own thinking process.
  • enjoy and appreciate mathematics. If students are to use mathematics, to continue studying mathematics beyond minimum requirements, and to maintain lifelong curiosity about mathematics, they must come out of school with a sense of mastery of and appreciation for the power and beauty of mathematics. This affective component of mathematical learning in the elementary school is critical and is closely tied to the view of mathematics which is communicated to students in school.

We want to make sure that students are involved in investigations that involve number, geometry, and data. Traditional elementary curricula have included very little work with geometry or data, and we want to make sure that there is significant work in these areas at every grade level. We have also become interested in the mathematics of change -- ideas that lead to calculus but that are accessible to young students -- and are including a unit focused on change at each grade level as well.

Curriculum as teacher development

We see our curriculum as a vehicle for teacher development. The actual curriculum is not what we envision and write down, but what happens between students and teachers in the moment of teaching and learning. So while part of our responsibility is to provide the material, the actual investigations, in which students will participate -- and this in itself is no easy matter -- the other, equally critical part of our responsibility is to open up that material to teachers, to invite them in both to the mathematics and to children's mathematical thinking.

The audience, therefore, for our materials, is teachers, not students. Our units are written to the teachers with many digressions about mathematics and about children's learning of mathematics. The responsibility is absolutely on the teachers to make this material work. If they fail, the material fails. On the other hand, by not making teachers partners in the past, we have made a grievous error. By not inviting teachers in to mathematics, by attempting to make materials "teacher proof" because educators or mathematicians believed that classroom teachers were not smart enough about mathematics to teach it, not only have we denied the students a good mathematical education, but we have denied generations of elementary teachers -- largely women -- access to mathematics.

The complexity of apparently simple ideas

In order to open up mathematics to teachers of young children, our materials need to open up the complexity of apparently simple ideas to teachers.

A key issue in the elementary school for teachers -- in all subjects, not just in mathematics -- is that adults think that the ideas that are taught in these grades are simple. One of the factors that has made it so difficult for teachers to be recognized as professionals -- and this is more true at the elementary level, where teachers are not subject matter specialists -- is that everybody thinks it is easy to teach what students need to learn in these grades. After all, don't we all know how to count, how to add and subtract?

What is not understood by many outside the schools, as well as by many inside them, is the complexity of these apparently simple ideas, both for the students and as they relate to mathematics. Here are three examples:

Elise, a sixth grader, was easily able to "find the average" of the grades on her spelling tests for the previous four weeks. However, when asked to answer the following question, "what would you have to score during the next four weeks to get an average of 90" she was baffled. She said, "I know how to find the average but I don't know how you find the numbers that go into an average" While Elise is right, in some sense, that you do not know the particular set of numbers that "go into an average" further questioning revealed that she had no idea how to describe any possible data sets represented by this average. As adults we, in fact, most often meet averages in this way -- we encounter the average, not the data. We must use our understanding of what an average represents to imagine what the data can be like. Elise's procedural understanding of average leaves her quite unprepared for dealing with the concept of average in the real world.

Gayle's third grade students were exploring multiplication patterns by skip-counting on a hundred number board. After students had counted by twos on the board -- coloring in 2, 4, 6, 8 . . . up to 100 -- Gayle asked her students to describe the patterns they saw. "It's the even numbers," declared one child. "The even numbers -- what can you say about even numbers?" After some further discussion about what "even" might mean, Jorge said, "the even numbers are the ones that have no middle." "No middle? Show us what you mean." Jorge came up and sketched three vertical slashes, circling the middle one. Underneath he drew two vertical slashes, not circling any. "See, three has a middle -- it's not an even number. But two doesn't have a middle. It's even." Gayle asked, "What do you all think about Jorge's conjecture? Can an even number have a middle" Students were soon busy building even and odd numbers using connecting cubes and exploring what the middles of these numbers might be. While as adults we might assume that what Ricky said was "obvious" these beginning third graders were genuinely engaged in deciding what the middle of a number like 26 might be. They were beginning an investigation of critical ideas about the structure of numbers, which might lead on to many kinds of conjectures about number relationships.

Carol's fourth grade students have certainly had many exposures to triangles during their four or five years in school. However, as Carol ran a discussion with her class which probed deeper into their knowledge, she found that many students had images of a prototypic triangle, usually equilateral and with one side parallel to the bottom of the page on which it was drawn, which restricted their views of the properties and relationships of triangles. For example, when she asked students to sketch various triangles with a perimeter of 12 centimeters, most students quickly drew an equilateral triangle with sides of 4 centimeters, but had great difficulty visualizing and sketching others. Some students sketched a triangle with sides of 6 centimeters, 4 centimeters, and 2 centimeters. Even when questioned hard about how this triangle would be constructed, many students insisted it could be done.

Illuminating critical mathematical issues in the curriculum

In all of these cases, teachers need information both about the mathematics itself and about the ways in which students grapple with the mathematics. What do we do about this in a curriculum? We can illuminate critical mathematical ideas. We can describe patterns of student learning, patterns of the ways in which students respond as they struggle with complex ideas. We can help teachers recognize ways in which we have seen many students respond, informal ideas that we have seen many students use, confusions that we have seen many students exhibit. Because we have extensive classroom data from our field tests, we are able to incorporate the experiences of many teachers and students into the final version of the materials through Teacher Notes (notes on the mathematical ideas and how students learn them) and Dialogue Boxes (examples of classroom interactions and issues that arise during them). For example, a Teacher Note called "Three Powerful Addition Strategies" is intended to help teachers (who themselves learned to add only by using the traditional "carrying" algorithm) become aware of other mathematically sound approaches to addition that their students may develop. Dialogue Boxes throughout the units gives teachers examples of discussions we have recorded in classrooms where students are encouraged to share their computation strategies. These teacher materials offer glimpses into students' mathematical thinking, highlight critical mathematical issues that are likely to arise, and provide information about mathematical content that teachers may not have encountered or thought about deeply.

Through opening up the complexity of early mathematical ideas to teachers, we hope to engage teachers as researchers in their own classrooms. We hope our curriculum will help teachers to pay closer attention to what their students say and do as they are engaged in solving mathematical problems. For a classroom teacher, this most often means asking questions designed to illuminate the way in which students are thinking about a mathematical idea. By asking the question, "what can you say about even numbers?," Gayle showed interest in the deeper thoughts of her students. She wondered what the meaning underneath their words really was, and did not take for granted that all her students meant the same thing as she did when they used mathematical terms. Opening themselves up to the complexity of students' thinking can be disconcerting for many elementary school teachers. When they ask their students to think mathematically rather than simply repeat what they have been told, it often becomes clear that students know a lot less than the teacher thought they did. The teacher who conducted the conversation about triangles was appalled at how little her fifth graders knew. Further, the teacher begins to understand how truly heterogeneous her students are and how difficult it is to tailor learning experiences to meet all the needs in her classroom. Reading accounts drawn from other teachers' experiences and beginning to become familiar with patterns of student responses help not to make every student an isolated case.

What new curricula provide for students

Besides providing new models for teachers, curriculum must, of course, provide substantive mathematical experiences for students. There are two needs in developing elementary curriculum. One is to find appropriate, engaging problems for children at this age. The other is to develop a pedagogy in which the emphasis is on the development of a mathematical frame of mind. The focus for young children, as in later mathematics, must be on thinking and reasoning mathematically.

Redefining work with number. If this is to be the case, work with number must be redefined and refocused in these new curricula. First, much more emphasis must be placed on developing a sound understanding of the structure of the number system and reasoning about number relationships based on this knowledge. For example, many children who have learned rote procedural approaches to solving problems solve the problem 1000 - 3 using the cumbersome method of "borrowing." We want children to reason from their knowledge of the place of 1000 as an important landmark in the number system. When students envision where 1000 is placed in the number system, they can easily count backwards from 1000 to 997.

Similarly, current research on young children developing their own strategies for addition and subtraction shows that children naturally add from left to right, dealing with the larger portions of the numbers first, rather than adding from right to left as we do when using the traditionally taught algorithm. Adding from left to right, students more readily retain a sense of the magnitudes of the numbers involved and are more likely to consider what a reasonable result might be.

Students make more use of prediction and estimation when they are encouraged to reason about numbers. For example, 8-year-old Anna reasoned in the following way as she solved the problem, "how many dollars do I need to give to the supermarket clerk in order to pay for potatoes that cost $3.45 and ice cream that costs $3.69?" Anna reasons, "Three dollars and three dollars, that's already $6.00. Then I round 69 cents to 70 cents. I know that 70 cents and 45 cents is already over a dollar. 40 cents and 60 cents is another dollar, so that's seven dollars, and five and nine go over, so I'll give her $8.00."

Finally, work with number should not be limited to work with operations. We want students to understand number as a way of describing relationships in the real world, but we also want them to encounter the purely mathematical (what patterns can you find in a 10X10 array of the numbers 1 to 100? how can you find the sum of all the counting numbers between 1 and n without adding each number? what is the tenth row in Pascal's triangle? what is the relationship between two consecutive square numbers?). Number relationships are in themselves fascinating objects of study. We want students to experience, appreciate, and be fascinated by the patterns of number and, on occasion, to catch glimpses of the quite surprising ways in which the purely mathematical turns out to fit reality (e.g., the occurrence of the Fibonacci sequence in nature).

Number theory offers a rich and accessible domain for exploration by young children. For example, second and third graders can become immersed in the study of the characteristics of odd and even numbers. In traditional mathematics classrooms, students learned to define even and odd numbers, but never spent time exploring how these numbers behave. In a third grade classroom working with our curriculum materials, students studied what happens when two even numbers are added together, or two odd numbers are added together. They drew pictures and used their knowledge of the structure of these numbers to develop statements that two even or two odd numbers would always result in an even number. When the question was posed, what would happen if you add an odd number to an even number?, students were eager to generate their own examples and develop informal proofs of their conclusions by referring to the structure of odd and even numbers. Most students reasoned from their knowledge that when the odd number is added to the even number, there is always "one extra" that is not paired, so that the result is an odd number. However, one student had a more unusual solution: "When we added two even numbers, the answer is even. When we added two odd numbers, the answer is even, too. So, when we add an odd and an even, the answers have to be odd, or else there wouldn't be any odd numbers."

By working with numbers in this way, students do real mathematical thinking -- developing and testing mathematical conjectures, exploring the relationships among mathematical objects, using examples and counterexamples -- not just solving a problem given by the teacher and coming up with the right answer.

Expanding the domain of mathematics to include work with data and geometry. While number has been the traditional center of the elementary curriculum, we believe that young students should also be doing substantial work in data analysis and geometry. Students in the elementary grades have experienced a very restricted view of mathematics as a discipline. To the elementary student, mathematics is arithmetic. Students in this age group can also be fruitfully engaged in collecting, representing, and describing data and in manipulating, visualizing, and reasoning about geometric objects.

Geometry, as the study of spatial objects, relationships, and transformations, is an essential component of mathematics. Geometric representations are essential for understanding such topics as functions and calculus (Balomenos, Ferrini-Mundy, & Dick, 1987) and, through measurement, geometry serves as a major source of practical applications of numerical concepts. As important as geometry proper is spatial thinking. Hadamard argued that much of the thinking that is required in higher mathematics is spatial in nature, and Einstein's comments on thinking in images are well known. Investigations in geometry and measurement provide opportunities for students to mathematically analyze their spatial environment, to describe characteristics and relationships of geometric objects, and to use number concepts in a geometric context.

Data collection and analysis is a critical skill in an information-rich society. From the earliest grades, students can collect, display, describe, and interpret real data so that they learn to become critical users of data and graphs. Students need to pose their own questions, collect data, critique and refine their own data collection methods, compare different ways of displaying their data, and, in the later elementary grades, learn to use appropriate statistical measures (Russell & Corwin, 1989; Russell & Friel, 1989). Research on students' understanding of statistical ideas in the elementary grades indicates that, just as in work with number, premature focus on memorization of definitions and algorithms (such as the algorithm for calculating the mean) undermines students' learning to make sense of a set of data. Just as students pull numbers out of "word problems" and manipulate them blindly, they pull numbers out of data sets and carry out calculations that no longer have meaning to them in terms of the data themselves (Mokros & Russell, 1995). The elementary curriculum must include many opportunities for students to describe, analyze, and interpret a variety of data sets so that they begin to understand how data analysis can provide important information about a variety of populations.

Teacher education: A central issue for reform

In order for this reform of the elementary math curriculum to work, we have a massive job to do in reeducating and supporting teachers as they attempt to expand and deepen the content of their mathematics program and to develop a pedagogy in which students are challenged to think mathematically (Ball, 1991; Russell & Corwin, 1993; Simon & Schifter, 1991). There is no question that we would prefer to see our curriculum used in the context of a strong, long-term staff development program. In fact, we know that a curriculum cannot provide all the necessary elements of such a program. One in particular -- the opportunity for teachers to do mathematics together, at an adult level, on a regular basis, and to reflect with peers about their own learning of mathematics and its implications for their teaching -- cannot easily be included in a curriculum. However, new curricula can be one of the tools that support teachers as they rethink their mathematics teaching. Insofar as these materials can invite teachers into mathematics and into the world of student thinking about mathematics, we believe they can help teachers open the doors of their classrooms to serious mathematics.

Footnotes

1 The work reported in this paper was supported in part by the National Science Foundation, Grant No. MDR-9050210. Opinions expressed are those of the author and not necessarily those of the foundation. This paper is based on an earlier paper prepared for the University of Chicago Mathematics Education Conference. A version of this paper was also presented as part of the China-U.S.-Japan Seminar on Mathematical Problem-Solving, October 1993. [Back to text]

2 TERC is a nonprofit company located in Cambridge, Massachusetts that works to improve mathematics and science education. TERC's projects include research on children's understanding of mathematics or science, research on teacher development, and development and implementation of curriculum materials, staff development materials, and new software and technological tools for use in educational settings. The letters T-E-R-C no longer stand for anything. [Back to text]

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