Somewhere in a New Jersey elementary school:
The students in Mrs. Chaplain's fifth grade class eagerly return from recess, excited by the prospect of working on another of her famous Chaplain's Challenges. Mrs. Chaplain regularly uses a Challenge with her math class, and today she has promised the children that the problem would be a great one. She believes that all of her students will be up to the Challenge and expects that it will engage them in an exploration and discussion of the relationship between area and perimeter.
Suppose you had 64 meters of fencing to build a pen for your large pet dog. What are some of the different pens you could build if you used all of the fencing? Which pen would have the most play space? Which would give the most running space? What would be the best pen? *
As the students file into the classroom, they stand, scattered around the room, reading the Challenge from the board even before finding their seats. Then they begin to ask questions about it. What is fencing? Wouldn't the pen with the most play space be the same as the one with the most running space? What shapes are allowed? Mrs. Chaplain answers some of these questions directly (she has brought a sample of fencing to class so she could show the students what it looks like), but, for the most part, she tells the students that they can discuss their questions in their regular working groups.
The groups begin their exploration by discussing how to organize their efforts. One of the first questions to arise is what kinds of tools would help solve the problem. The students have used a variety of materials to deal with Chaplain's Challenges, and they have often found that the groups that fashioned the best models of the problem situations were the ones who found it easiest to find solutions. Today, one group decides to use the 10-by-10 geoboards, figuring that they can quickly make a lot of different "pens" out of rubber bands if they let the space between nails equal four meters. Another group decides to get some graph paper on which to draw their pens, because that gives them a lot of flexibility. Still another group, reluctant to be limited to rectangular shapes and work spaces, thinks that the geometry construction software loaded on the computers in the back of the room would let them draw a variety of shapes and even help them measure various characteristics of the shapes. One last group, striving for realism, decides to use a loop of string sixty-four inches long. As the session progresses, the groups of students make many sample pens with whatever materials they have chosen to use. Some groups switch materials as they perceive other materials to be less restrictive than the ones they are using. Keeping the perimeter a constant 64 meters, they measure the areas of the pens using some of the strategies they developed the week before. Mrs. Chaplain circulates around the room, paying careful attention to the contributions of individual students, making notes to herself about two particular children, one who seems to be having difficulty with the concept of area, and another who is doing a nice job of leading her group to a solution.
Gradually, the work becomes more symbolic and verbal and less concrete. The students begin to make tables to record the dimensions and descriptions of their pens and to look for some kind of pattern, because they have learned from experience that this frequently leads to insights. One group follows the teacher's suggestion and enters their table of values for rectangular pens into a computer, generating a broken-line graph of the length of the pen versus its area.
Toward the end of the class, the students become comfortable with their discoveries. Mrs. Chaplain reflects again on how glad she is that the faculty decided to organize the school schedule to allow for these extended class sessions. When she sees how involved and active the students are, how they try to persuade each other to follow one path or another, how their verbalizations either cement their own understandings or provide opportunities for others to point out flaws in their thinking, she realizes that only with this kind of time and this kind of effort can she do an adequate job of teaching mathematics.
The summary discussion at the end of the session allows the students an opportunity to see what their classmates have done and to evaluate their own group's results. Mrs. Chaplain learns that everyone in the class understands that if you hold the perimeter constant, you can create figures with a whole range of areas. Moreover, she feels that a majority of the class also has come to the generalization that the more compact a figure is, the greater its area, and the more stretched out it is, the smaller its area.
But the students still have very different answers to the question, What would be the best pen? That fits her plans perfectly. For homework, Mrs. Chaplain asks each student to design the pen that he or she thinks is best, draw a diagram of it, label its dimensions and its area, and write a paragraph about why that particular pen would be best for the dog. Mrs. Chaplain plans to move on from this activity to others where the students concentrate on more efficient strategies for finding the areas of some of the non-rectangular shapes they explored in the Challenge.
*This problem was adapted from one that appears in the Professional Standards for Teaching Mathematics, National Council of Teachers of Mathematics, 1991.
Somewhere in a New Jersey high school:
Ms. Diego's algebra class and Mr. Browning's physical science class are jointly investigating radioactive decay. The two teachers, with the support of the school administrators, have worked out a schedule that enables their classes to meet together this month to explore some of the mathematical aspects of the physical sciences. Both teachers regularly incorporate some content from the other's discipline in class activities, but this month was specially planned to be a kind of celebration of the relationship between the two areas. By the end of the month, they expect that the students will really appreciate the role that mathematics plays in the sciences, and the problems presented by the sciences that call for innovative mathematical solutions.
The classes are average. Nearly every student in the high school takes these two classes at some point during their stay and, over the past few years, because of exciting real-world problems like the one on which they are working this week, the classes have become two of the most popular in the school.
Monday's class begins with a presentation by Mr. Browning about the process of carbon dating. He describes the problem that archaeologists faced in the 1940s with respect to determining the age of a fossil. They knew that all living things contained a predictable amount of radioactive carbon that began to diminish as soon as the organism died. If they could measure the amount that remained in some discovered fossil and if they knew the rate at which the carbon "decayed," they could figure out the age of the object. An American chemist named Willard Libby developed a technique that allowed them to do so. Ms. Diego explains that the classes will spend the next few days exploring the concept of radioactive decay and, toward the end of the week, they will be able to solve some of the same kinds of problems solved by those archaeologists.
On Tuesday, working at stations created by the teachers, the students begin to explore both the mathematical and scientific aspects of radioactive decay. Working in groups, the students use sets of 50 dice to simulate collections of radioactive nuclei. Each roll of the collection of dice represents the passage of one day. Any time a die lands with a "1" showing, it "spontaneously decays" and is taken out of the collection. The students plot the number of radioactive nuclei left versus the number of days passed in an effort to determine the half-life of the element - the amount of time it takes for half of the element to decay. Because the experiment is relatively well controlled, each group working on the task produces a graph that effectively illustrates the decay, but because the process is also a truly random process, each group's results are slightly different from those of other groups.
On Wednesday, in a very different kind of activity, students use graphing calculators in a guided activity to discover properties of exponential functions, and the effects on the graphs of various changes to the parameters in the functions. Working from a worksheet prepared by the teachers they start with the general form of an exponential function, y = abx. Using the values a = 1 and b = 2, they input the equation into the calculators and study the In the high school... resulting graph. Then, they systematically change the values of a and b to discover what each change does to the graph. They are directed by the worksheet to pay particular attention to the effect of changing b to a value between zero and one, because graphs of that type will be especially important for their work with radioactive decay. The culminating problem on the worksheet is a challenge to try to find the values of a and b that produce a graph that looks like the ones that resulted from the experiment with the dice. The students enjoy the problem and use their calculators to quickly check and refine their solutions, zeroing in on the critical numbers. There is a lot of discussion about why those numbers might be the correct ones.
On Thursday, the students discuss a reading that was assigned for homework the night before, focusing on carbon dating and addressing some of the mathematical processes used to determine the age of fossils. This discussion is led by the two teachers, who have brought in some fossilized samples to better acquaint the students with the kind of materials they read about. Ms. Diego then leads a session to develop the computational procedures for solving the carbon dating problems using exponential functions. The students will be given some homework problems of this type and will spend tomorrow's class discussing those problems and wrapping up the unit. The teachers are very pleased with what the classes have accomplished. The active involvement with a hands-on experiment simulating decay, the symbolic manipulations and graph explorations made possible by the graphing calculator, and the study of a particular scientific application of the mathematics have been quite productive. By working together as a team, the teachers have been able to relate the different aspects of the phenomenon to each other. The students have learned a great deal of both mathematics and science and have seen how strongly they are linked.
The vision of the New Jersey core curriculum content standards for Mathematics revolves around what takes place in classrooms like those described in the previous pages. It is focused on achieving one crucial goal:
To enable ALL of New Jersey's children to move into the twenty-first century with the mathematical skills, understandings, and attitudes that they will need to be successful in their careers and daily lives.
With this goal in mind, the following two standards explicitly address the learning environment in mathematics classrooms. They deal with concerns for fostering success in mathematics for all students, and for the linking of assessment to learning and instruction.
As more and more New Jersey teachers incorporate the recommendations of the New Jersey Mathematics Standards into their teaching, we should see the following results (as described in Mathematics to Prepare Our Children for the 21st Century: A Guide for New Jersey Parents, N.J. Mathematics Coalition, 1994):
Students who are excited by and interested in their activities. A principal goal is for children to learn to enjoy mathematics. Students who are excited by what they are doing are more likely to truly understand the material, to stay involved over a longer period of time, and to take more advanced courses voluntarily. When math is taught with a problem-solving spirit, and when children are allowed to make their own hands-on mathematical discoveries, math can be engaging for all students.
Students who are learning important mathematical concepts rather than simply memorizing and practicing procedures. Student learning should be focused on understanding when and how mathematics is used and how to apply mathematical concepts. With the availability of technology, students need no longer spend the same amount of study time practicing lengthy computational processes. More effort should be devoted to the development of number sense, spatial sense, and estimation skills.
Students who are posing and solving meaningful problems. When students are challenged to use mathematics in meaningful ways, they develop their reasoning and problem-solving skills and come to realize the potential usefulness of mathematics in their lives.
Students who are working together to learn mathematics. Children learn mathematics well in cooperative settings, where they can share ideas and approaches with their classmates.
Students who write and talk about math topics every day. Putting thoughts into words helps to clarify and solidify thinking. By sharing their mathematical understandings in written and oral form with their classmates, teachers, and parents, students develop confidence in themselves as mathematical learners; this practice also enables teachers to better monitor student progress.
Calculators and computers being used as important tools of learning. Technology can be used to aid teaching and learning, as new concepts are presented through explorations with calculators or computers. But technology can also be used to assist students in solving problems, as it is used by adults in our society. Students should have access to these tools, both in school and after school, whenever they can use technology to do more powerful mathematics than they would otherwise be able to do.
Teachers who have high expectations for ALL of their students. This vision includes a set of achievable, high-level expectations for the mathematical understanding and performance of all students. Although more ambitious than current expectations for most students, these standards are absolutely essential if we are to reach our goal. Those students who can achieve more than this set of expectations must be afforded the opportunity and encouraged to do so.
A variety of assessment strategies rather than sole reliance on traditional short-answer tests. Strategies including open-ended problems, teacher interviews, portfolios of best work, and projects, in combination with traditional methods, will provide a more complete picture of students' performance and progress.
Learning environments like this should and can become the reality in virtually all New Jersey classrooms before the turn of the century. Making this vision a reality is both necessary and achievable.
Perhaps the most compelling reason for this vision of mathematics education is that our children will be better served by higher expectations, by curricula which go far beyond basic skills and include a variety of mathematical models, and by programs which devote a greater percentage of instructional time to problem-solving and active learning. Many students respond to the current curriculum with boredom and discouragement, develop the perception that success in mathematics depends on some innate ability which they simply do not have, and feel that, in any case, mathematics will never be useful in their lives. Learning environments like the one conceived in these standards will help students to enjoy and appreciate the value of mathematics, to continue to develop the tools they need for varied educational and career options, and to function effectively as citizens and consumers.
Preparing our students for careers in the twenty-first century requires that we make this vision a reality. Our curricula are often preoccupied with what national reports call "shopkeeper mathematics," competency in the basic operations that were needed to run a small store several generations ago; yet very few of our students will have careers as shopkeepers. To compete in today's global, information-based economy, students must also be able to solve real problems, reason effectively, and make logical connections. Jobs requiring mathematical knowledge and skills in areas such as data analysis, problem-solving, pattern recognition, statistics, and probability are growing at nearly twice the rate of growth of overall employment. To prepare students for such careers, the mathematics curriculum must change.
This vision of excellent mathematical education is based on the twin premises that all students can learn mathematics and that all students need to learn mathematics. In order to take seriously the goal of preparing all students for twenty-first century careers, we must overcome the all too common perception among students that they simply lack mathematical ability:
Only in the United States do people believe that learning mathematics depends on special ability. In other countries, students, parents, and teachers all expect that most students can master mathematics if only they work hard enough. The record of accomplishment in these countries - and in some intervention programs in the United States - shows that most students can learn much more mathematics than is commonly assumed in this country.
-- Mathematical Sciences Education Board, 1989
Curricula that assume student failure are bound to fail; we need to develop curricula that assume student success.
A first step in curriculum development is the formulation of goals. For that purpose, sixteen mathematics content standards have been delineated to translate the vision into specific goals. The term standard here encompasses both goals and expectations, but is also meant to convey the older meaning of banner or rallying point. These mathematics standards are intended as a definition of excellent practice, of what can be achieved if all New Jersey communities rally behind the standards so that this excellent practice will become common practice. The sixteen standards in this section were not designed as minimum standards, but rather as world-class standards which will enable all of our students to compete in the global marketplace of the twenty-first century.
These sixteen standards, along with the two additional learning environment standards listed above, define the critical goals of mathematics education today. In addition to more familiar content are many topics which have not been part of the traditional curriculum. Included are new emphases on the whys and hows of mathematics learning: posing and solving real world problems, effectively communicating mathematical ideas, making connections within mathematics and between mathematics and other areas, active student involvement, the uses of technology, and the relationship between assessment and instruction.
Although philosophically aligned with the Curriculum and Evaluation Standards for School Mathematics of the National Council of Teachers of Mathematics (1989), the New Jersey Mathematics Standards are designed to reflect conditions specific to New Jersey, as well as national changes in mathematics education since the NCTM document was written.
The standards rest on the notion that an appropriate mathematics curriculum results from a series of critical decisions about three inseparably linked components: content, instruction, and assessment. All students must have the opportunity to develop an understanding and a command of mathematics in an environment that provides for both affective and intellectual growth. The New Jersey Mathematics Standards place high value on both process and product, describe appropriate learning environments, and challenge us to formulate effective assessment strategies which both inform instruction and improve learning. But these standards will only promote substantial and systemic improvement in mathematics education if the what of the content being learned, the how of the problem-solving orientation, and the where of the active, equitable, involving learning environment are synergistically woven together in every classroom.
These standards also contain a strong focus on the use of technology as a regular, integral part of school mathematics curricula at every grade level. The state mandate for the use of calculators in statewide assessment is but one indication of the strong movement that has already begun in this direction. Teachers and students who adopt these standards will understand, and develop the abilities to use, powerful, up-to-date mathematics and technology.
Implicit in the vision and these standards is the notion that there should be a core curriculum for grades K-12. What does a "core curriculum" mean? It means that every student will be involved in experiences addressing all of the expectations of each of the sixteen content standards. It also means that all courses of study should have a common goal of completing this core curriculum, no matter how students are grouped or separated by needs and/or interests.
A core curriculum does not mean that all students will be enrolled in the same courses. Students have different aptitudes, interests, educational and professional plans, learning habits, and learning styles. Different groups of students should address the core curriculum at different levels of depth, and should complete the core curriculum according to different timetables. Nevertheless, all students should complete all elements of the core curriculum recommended in the New Jersey Mathematics Standards.
A curriculum framework has been developed to assist teachers and districts in implementing these mathematics standards. The document includes not only an elaboration and clarification of the content standards, but also an expanded discussion of the two additional learning environment standards listed earlier in this introduction. The result of a collaborative effort of the New Jersey Mathematics Coalition and the New Jersey Department of Education, with Eisenhower funding from the United States Department of Education, a preliminary version was published early in 1995 and a revised version will be available in spring 1996.
The New Jersey core curriculum content standards for Mathematics presented in this chapter offer a powerful challenge to all teachers, schools, and districts: to enable all of our students to step forward into the twenty-first century with the mathematical skills, understandings, and attitudes that they will need to be successful in their careers and daily lives. The standards are also a powerful tool to help us meet that challenge, providing a vision and standards which both inform us and rally us in our efforts. It will not be easy to meet this challenge, nor can it happen overnight. But it can happen if all of us together decide to make it happen.
Mathematical Sciences Education Board. (1989). Everybody counts. Washington, DC: National Academy Press.
National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA.
New Jersey Mathematics Coalition. (1994). Mathematics to prepare our children for the 21st century: a guide for New Jersey parents.
New Jersey Mathematics Coalition. (In process, preliminary version published 1995). New Jersey mathematics curriculum framework.
4.1 All students will develop the ability to pose and solve mathematical problems in mathematics, other disciplines, and everyday experiences.
4.2 All students will communicate mathematically through written, oral, symbolic, and visual forms of expression.
4.3 All students will connect mathematics to other learning by understanding the interrelationships of mathematical ideas and the roles that mathematics and mathematical modeling play in other disciplines and in life.
4.4 All students will develop reasoning ability and will become self-reliant, independent mathematical thinkers.
4.5 All students will regularly and routinely use calculators, computers, manipulatives, and other mathematical tools to enhance mathematical thinking, understanding, and power.
4.6 All students will develop number sense and an ability to represent numbers in a variety of forms and use numbers in diverse situations.
4.7 All students will develop spatial sense and an ability to use geometric properties and relationships to solve problems in mathematics and in everyday life.
4.8 All students will understand, select, and apply various methods of performing numerical operations.
4.9 All students will develop an understanding of and will use measurement to describe and analyze phenomena.
4.10 All students will use a variety of estimation strategies and recognize situations in which estimation is appropriate.
4.11 All students will develop an understanding of patterns, relationships, and functions and will use them to represent and explain real-world phenomena.
4.12 All students will develop an understanding of statistics and probability and will use them to describe sets of data, model situations, and support appropriate inferences and arguments.
4.13 All students will develop an understanding of algebraic concepts and processes and will use them to represent and analyze relationships among variable quantities and to solve problems.
4.14 All students will apply the concepts and methods of discrete mathematics to model and explore a variety of practical situations.
4.15 All students will develop an understanding of the conceptual building blocks of calculus and will use them to model and analyze natural phenomena.
4.16 All students will demonstrate high levels of mathematical thought through experiences which extend beyond traditional computation, algebra, and geometry.