Since then, educators in New Jersey have been exploring ways to implement the NCTM recommendations. To support their efforts, the New Jersey Mathematics Coalition was formed in 1991. It is a partnership with representation from all sectors of the community: business and industry, K-12 and higher education, government, parents, and the general public.

The Coalition has worked with the New Jersey Department of Education and many of the most knowledgeable educators throughout the state to develop a draft of the New Jersey Mathematics Standards. Although not official policy of the Department of Education, the document endorses the NCTM recommendations and goes further to create a vision of mathematics education tailored to New Jersey needs. The development effort is a component of the New Jersey Statewide Systemic Initiative to improve mathematics, science, and technology education.

In this booklet, we share our vision with you and invite you to help make the vision a reality for your children and all of the children of New Jersey.

Students who are excited by and interested in their activities. One of our principal goals is for children to learn to enjoy mathematics. Students who are excited by what they are doing are more likely to 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 real problem-solving spirit, 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 how mathematics is used and how it works. 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 skillb>

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. Recent research shows that children often 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, children develop confidence in themselves as mathematics learners and enable teachers to better monitor their progress.

Calculators and computers being used as important tools of learning. Technology can be used as an aid to 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 at home and at school, whenever they can use them 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. They are more ambitious than current expectations for most students, but are absolutely essential if together 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 replacing traditional short-answer tests. New strategies will include tests with open-ended problems, teacher interviews, long-term projects, and self- and peer-evaluations. All of these require more time, thought, and planning than traditional problems, but more closely reflect how people in the real world actually use mathematics.

Number Sense is an intuitive feel for numbers and a common sense approach to using them. It comes from comfort with what numbers represent, how different types of numbers (fractions, decimals, and so on) are related to each other, and how they can best be used to describe a particular situation. Number sense is a quality that all successful users of mathematics possess.

Spatial Sense includes traditional geometry topics but also a whole collection of other, less formal ways of looking at two- and three-dimensional space. Geometry is all around us in art, nature, and the things we make. Understandings about properties of shapes, what happens when they are transformed are important parts of the curriculum that help children deal with the world around them.

Numerical Operations are an essential part of the math curriculum. Students must understand how to add, subtract, multiply, and divide whole numbers, fractions, and other kinds of numbers. But, with the availability of calculators which can perform these operations quickly and accurately , the instructional emphasis now must be on understanding the meanings and uses of the operations and on estimation and mental skills rather than on developing pencil-and-paper skills.

Measurement is important because it helps describe the world around us using numbers. We use numbers to describe simple things like length, weight, and temperature, but also complex things such as pressure, speed, and brightness. An understanding of how we attach numbers to those phenomena, familiarity with common measurement units like inches, liters, and miles per hour, and a practical knowledge of measurement tools are critical for children's understanding of the world around them.

Estimation is a process that is used all the time by mathematically capable adults, and it is one that can be easily mastered by children. It involves an educated guess about a quantity or a measure or an intelligent prediction of the outcome of a computation. The growing use of calculators makes it more important than ever that students know when a computed answer is reasonable; the best way to make that decision is through estimation. Equally important is an awareness of those many situations in which an approximate answer is as good as, or even preferable to, an exact answer.

Patterns, Relationships, and Functions comprise the backbone of mathematics. From the earliest age, students should be encouraged to investigate the patterns that they find in numbers, shapes, and expressions, and, by doing so, to make true mathematical discoveries. These explorations present unending opportunities for problem solving, making and verifying generalizations, and building mathematical understanding and confidence.

Probability and Statistics are the mathematics we use to understand chance and to collect, organize, and analyze numerical data. From weather reports to sophisticated studies of genetics, from election results to toothpaste preference surveys, probability and statistical language and concepts are increasingly present in the media and in everyday conversations. Our children need this mathematics to help them judge the correctness of an argument supported by data or the believability of a persuasive advertisement.

Algebra is the language of mathematics. Largely because of technology, the emphasis in algebra instruction has been evolving from a focus on rules and symbol manipulation to a view which sees algebra as a tool which all students can use to model real situations and answer questions about them. New graphing calculators allow the graphing of an equation at the touch of a button so that students can focus on what the graph means as it represents some real-life phenomenon. Formal procedures are still important, but the focus should be increasingly on the development of algebraic thought -- on how quantities are related to each other and the search for methods to express those relationships.

Discrete Mathematics is a new branch of mathematics that deals with some of the most practical problems we face in life. It deals with ways to find the best route from one place to another, the best way to schedule a list of tasks to be done, the manner in which computers store data and CDs record music, and strategies for winning games. Increasingly, it is the mathematics used by decision makers in our society, from workers in government to those in health care, transportation, and telecommunications. The logical and practical emphases of discrete math problems and solutions help students see the relevance of mathematics in the real world.

Underpinnings of Calculus are usually thought of as high school topics for only the very brightest academic students. Some basic fundamentals of calculus, though, are important for everyone to understand. How quantities such as world population change, how fast they change, and what will happen if they keep changing at the same rate are questions that can be discussed by elementary school children. Another fascinating topic for all mathematics students is the concept of infinity -- the infinitely large or the infinitesimally small. Early explorations in these areas can broaden students' interest in and understanding of a very important area of applied mathematics.

• Expect your child to succeed in mathematics and be sure that he or she understands that expectation. We have good evidence that ALL children can learn the mathematics that is included in the K-12 curriculum. We have to believe in them and let them know that we do.

• Be sure that your child sees you and other adults using mathematics every day. Children are more willing to try hard to learn a subject when they believe it is very important and relevant to their lives.

• Talk with your child about mathematics and solve problems out loud together. When normal questions arise such as "How many packages of noisemakers should we buy for the 19 people who will be at the party?" or "How much longer until we get there?" or "Should we go on the turnpike, on Route 1, or on side streets?", or "Should we pay cash for the new bike or pay it off over time?," reason out the answers together with your child. Let him or her grow up understanding the value and usefulness of clear mathematical thinking.

• Play Counting Games on a Calculator. If you press this sequence of keys on an ordinary inexpensive calculator, the display will show 1, 2, 3, 4, . . and so on:

  0 + 1 = = = =

  Have your child do that, counting by ones aloud with the calculator. After awhile ask your child to say the number that will come up next, and then next, and then next, and press the equals key to verify the predictions. To make the activity a little more challenging, use the sequence above, but replace the "1" with a "2." Then, another time, try a "5" or "10" or "7."

• Estimate and Measure. Collect about seven small jars and containers of different sizes and shapes. Have your child estimate which ones hold the most and which the least and line them up in order from least to most. Is the first actually smallest? To find out, fill it with uncooked rice. If it is really the smallest, that rice should fit into the next container. Pour all of the rice into the next one. Does it all fit? Is there room for still more? Continue down the whole line of containers in that way to see if the order is correct.

• Sort Toys. Collect about ten of your child's small toys: dolls or action figures, balls, small stuffed animals, cars and trucks, and play cooking utensils. Ask your child to then sort the collection in whatever way he or she wants. Ask which things go together. Why do they go together? How are they alike? How are they different? These kinds of questions and answers will help develop logical thinking and classification skills.

• Play Target Practice with a Calculator. This game will really help your child's estimation and mental math skills. You start by entering into a calculator the first number in an addition problem and pressing the plus sign. Then challenge your child to enter a second number so that the answer will be within some range that you name. For instance:

  You enter:	23 +
  You name the range for the answer:	60 to 70
  Your child has to enter some number and hit the equals sign to make a number between 60 and 70 appear in the display. (42 would work.)

  To add to the challenge, use larger numbers or use subtraction, multiplication, or division as the operation.

  Time It. Children of this age are fascinated with how much time things take. Decide on some common tasks and let your child estimate how long they would take. He or she can then perform them while you keep time. Then let your child time other members of the family as they try the same tasks. Some tasks you might try are: tying your shoes, saying the alphabet from A to Z, or making a free-standing tower with 30 pennies. If possible, time the events with a digital stopwatch that shows tenths and hundredths of a second. Your child is probably just beginning to understand these fractions of a second.

  Hunt for Shapes. Discuss with your child three-dimensional shapes that are frequently seen in the everyday world. Use their proper geometric names: cube, sphere, cylinder, cone, and rectangular prism (a rectangular box). Over the next few days, let your child hunt for the shapes in packaging, toys, and household objects, and point them out by name when they are discovered. Which ones do you see most? Why?

• Plan a Trip. Let your child help plan a family trip. If you will be driving, let him or her plan the route on a map, determine the overall distance to be covered, propose a reasonable number of miles to drive each day, and figure out the amount of time it will take. Point out that different kinds of roads have different speed limits, and that rest stops need to be accounted for. You may want to add other considerations such as places that would be nice to visit along the way. You may even want to invite your child to help you figure out an appropriate budget for the trip. How much per day for souvenirs, hotels, meals, gas, and other expenses?

• Practice Mental Math and Estimation. Ask your child to determine the tip to be left at a restaurant, to figure out how much longer you'll be riding in the car if you're traveling at 55 miles per hour and have 130 miles left to go, to estimate the height of a really tall flagpole. Share with your child the strategies that you use when you solve problems like these, and be sure to ask for the reasoning involved in your child's estimates. If you have fun doing problems like these, you can make up some that will take the two of you a long time to solve. How many boards do you think are in the boardwalk?

• Find Geometry in Art. Throughout history, geometry has played a prominent part in the art of a great many different cultures. Take the time to visit an art museum or look through a collection of modern posters in a shopping mall and discuss with your child the ways in which the artists have used geometry. What shapes do you see? Are they all in the same orientation? How do they form the design or picture? You might want to do some reading together in the library about Islamic art, Egyptian art, Greek architecture, or the work of the Dutch artist M. C. Escher.

• Interpret a Graph. From news programs on TV to the sports pages in your local newspaper to your weekly magazines, the media is increasingly filled with graphs and tables and charts that summarize information. The next time you come across a data display that deals with data that would be interesting to your child, sit down together to try to interpret it. Work together to formulate answers to these kinds of questions: What conclusions can you draw from the data? What questions are raised by the data? Why is it important for you as citizens, sports fans, or consumers to understand this graph (or chart or table)?

• What's the Best Route? There are many situations where a number of alternate routes are possible. For example, in which order should you pick up the six friends you are driving to the school dance? In which order should you make the eight deliveries from the drug store where you work? In which order should you visit the seven must-see sites on your vacation trip? In each case, you want to find the "best route", the one which involves the least total distance, or least total time, or least total expense. Make up your own problems, using actual locations and the actual distances, and find the best route. For a larger project, see if you can improve the route taken by your school bus or your neighborhood's garbage truck.

• Solve Tangram Puzzles. An ancient Chinese puzzle, called a tangram, consists of 7 pieces which are cut from a square. Trace the 7 pieces shown on the back cover of this booklet onto another piece of paper and then cut them out. See if you and your child can use all of the pieces to make a rectangle, a triangle, a parallelogram, and a pentagon. How many of the designs shown on pages 8-12 can you make with all seven pieces? More tangram puzzles using the same seven pieces can be found in books at your library.

• Curriculum and Evaluation Standards for School Mathematics, 258 pages, published in 1989 by the National Council of Teachers of Mathematics. Available from NCTM, 1906 Association Drive, Reston, VA 22091-1593. $25.00 (800-235-7566). Executive Summaries are also available.

  Also available from NCTM are these pamphlets:

  • Family Math Awareness Activities
  • Help Your Child Learn Math
  • Using Calculators to Improve Your Child's Math Skills

  And this bibliography of mathematics-related children's literature: The Wonderful World of Mathematics: A Critically Annotated List of Children's Books in Mathematics edited by Diane Thiessen and Margaret Matthias. $17.00.

• Everybody Counts: A Report to the Nation on the Future of Mathematics Education, 114 pages, published in 1989 by the Mathematical Sciences Education Board. Available from National Academy Press, 2101 Constitution Ave, N.W., Washington, D.C. 20418. $13.95 (800-624-6242)

• Helping Your Child Learn Math, 58 pages, published in 1993 by the U.S. Department of Education, Office of Educational Research and Improvement. Available from Consumer Information Center, Dept. H-374A, Pueblo, CO 81009. $0.50.

• Math Matters: Kids are Counting on You, a kit for parents and elementary school children. Available from National PTA, 135 South La Salle Street, Department 1860, Chicago, IL 60674-1860 (312) 549-3253. $14.50