Students can develop a clear sense of number from consistent ongoing experiences in classroom activities where a variety of manipulatives and technology are used. The key components of number sense, as identified in the K-12 Overview, include an awareness of numbers and their uses in the world around us, a good sense of place value concepts, approximation, estimation, and magnitude, the concept of numeration, and an understanding of comparisons and the equivalence of different representations and forms of numbers.

Kindergarten, first, and second graders are just beginning to develop their concepts of number. They have most likely come to school with some ability to count, but with differing notions of what that activity means. It is in these grades that they begin to attach meaning to the numbers that they hear about and see all around them. One useful activity that can be repeated many times throughout this age range is the keeping of a scrapbook reflecting all the uses of numbers that the children can identify. It would probably include telephone numbers, addresses, ages, page numbers, clothing sizes, room numbers, and many others. Discussions of the similarities and differences in all of these uses can provide some interesting insights.

In terms of numeration, students in these grades start by constructing meaning for one-digit numbers and build up to formal work with three-digit numbers. The regular and consistent use of concrete models for that development is essential. Kindergartners need a variety of things to count, from poker chips to marbles to beans. Both concrete and rote counting are critically important in developing a sense of number. Adequate attention to counting activities throughout these grades will help to assure both a good sense of magnitude (size) of numbers and a real readiness for all four basic operations. (See Standard 8.) Counting by ones should be followed by counting back; skip counting by twos, fives, and tens; counting from a given starting number to a given target number by ones and by other numbers; counting on by tens from non-multiples of ten like 43; and so on.

As students are able to handle larger numbers, place value and base-ten ideas are introduced through grouping activities. Many of the models with which they are comfortable for single units can translate nicely into beginning base-ten models; poker chips can be put in groups of ten into small paper cups; beans can be pasted in tens onto tongue depressors, and so on. These newly enhanced models, along with the single digit units, are then used to represent two-digit numbers. As the next step, of course, groups of ten tens can be made to create hundreds. These first models of base-ten number are the best ones to use with young children who are first encountering these notions because they can actually build larger units from smaller units. Such models are called bundle-able. Another property these have is proportionality, because the model for a ten is actually ten times as large as the model for a one. A widely used model which is both bundle-able and proportional involves popsicle sticks which are wrapped into tens and hundreds with rubber bands.

The next type of model to be used would be one which is still proportional, but no longer bundle-able. The best examples of this type are the standard base-ten blocks. They require the child to trade ten ones for a ten rather than directly constructing a ten from the ones, and, as a result, are slightly more sophisticated. The last level of sophistication in this sequence of models includes those that are neither proportional nor bundle-able. Two models of this type which are regularly used are chip trading materials and play money. With chip trading materials, there is no inherent concrete ten-to-one relationship between the red chips and the green chips; the red chips are not ten times as large as the green ones. The relationship holds only because of an external rule that is made up and followed. Similarly, there is no inherent concrete ten-to-one relationship that exists between dimes and pennies. The relationship only exists because of a rule that is external to the coins themselves. As a result, these most sophisticated models should be used after the underlying concepts are developed with the earlier models.

Children at these grade levels also begin to learn about equivalence. When youngsters find as many "names" as they can for the number 7 (such as 2 + 5, 9 - 2, and one more than 6), they are creating equivalent forms of the same number. Slightly older students should be using similar activities to generate equivalent forms of multi-digit numbers, partly in preparation for operations involving them: 67 = 6 tens and 7 ones = 5 tens and 17 ones = 4 tens and 27 ones.

Estimation should be a routine part, not only of daily mathematics lessons, but also of the entire school day. Children should be regularly engaged in estimating both quantities and the results of operations. They should respond to questions that arise naturally during the course of the day, like: About how many kids do you think there are out here in the playground? About how many pieces of construction paper will we need for this project if everyone needs three different colors? and How many of your great graphs do you think will fit on the bulletin board without overlapping? After several children have had chances to make estimates about numbers like these, they should defend their estimates by giving some rationale for thinking they are close to the actual number. These discussions can be invaluable in helping them to develop good number sense.

Technology plays an important role in number sense at these grade levels. Calculators can be wonderful teaching tools when programmed to count forward and backward by some constant. Children can do the programming easily themselves and try to anticipate the calculator display. Appropriate computer software provides environments in which students can first develop a sense of small whole numbers and then build an understanding of place-value and base-ten ideas.

The topics that should comprise the number sense focus of the kindergarten through second grade mathematics program are:

whole number meanings through three digits

place value and number base

counting and grouping

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in kindergarten and grades 1 and 2.

Experiences will be such that all students in grades K-2:

1. Use real-life experiences, physical materials, and technology to construct meanings for whole numbers, commonly used fractions, and decimals.

• Young students make and use a variety of models for "number" ranging from poker chips to dot patterns on a paper plate, to Cuisenaire Rods, to tally marks, to domino and dice combinations. A large component of their early work with number focuses on the various parts into which any given number can be broken.

• Students play the Broken Key game on their calculators. Kindergartners try to get the calculator display to show 7 while pretending that the 7 key is broken and cannot be pressed. Second graders might try to get the display to show 45 without pressing the 4 or the 5 key.

• Students use 5-frames and 10-frames to help develop initial ideas of small numbers. By filling up a 5-cell grid with counters first and then putting out 2 more while trying to show "7 in all," the child not only learns about "7" but also about its relationship to "5."

• Students use numbers throughout the school day as they discuss the date, attendance, time, snacks, money, etc.

• Students investigate fractions by listening to the story Gator Pie by Louise Mathews and by discussing how Alvin and Alice can share their pie with more and more alligators.

• Second-graders record prices as decimals ($0.39) and use this notation to find totals over $1 on a calculator.

• Students find half of a sheet of paper by folding horizontally, by folding vertically, and by folding diagonally. They compare the results and discuss how they are alike and how they are different.

• Students use Balancing Bear software to find combinations of numbered weights that will balance a seesaw or that will be greater or less than a given weight.

2. Develop an understanding of place value concepts and numeration in relationship to counting and grouping.

• Calendar activities at the beginning of the school day incorporate a Daily Count feature where each day another popsicle stick is added to a collection representing all of the days of school to date. Whenever 10 single sticks are available, they are bundled with a rubber band and are thereafter counted as a ten. On the hundredth day of school, the ten tens are wrapped together to make a hundred, and the class celebrates the event with a party.

• Students progress from a proportional and bundle-able base ten model like popsicle sticks to a proportional but not bundle-able model like base-ten blocks to a model that is neither proportional nor bundle-able like pennies and dimes. (See K-2 Overview)

• Pairs of students play Race to One Hundred with base ten blocks. Each, in turn, rolls one or two dice and takes that many unit cubes. Whenever there are ten unit cubes in a player's collection, the player must trade for a ten block. The first player able to trade ten ten blocks for a hundred block is the winner.

• Students have 3 dimes and 4 pennies to spend on a variety of items that are displayed in a classroom store. The items have tags ranging from 3 cents to 56 cents and the children are asked: Which of these items can be bought for exactly the amount of money that you have (requiring no change)? Which items can you buy and have some money left over? Which of these items cannot be bought because you do not have enough money? What items are left?

• Student understanding of place value for two-digit numbers is assessed by asking each student to represent a different number using popsicle sticks or base 10 blocks.

3. See patterns in number sequences, and use pattern-based thinking to understand extensions of the number system.

• Students find patterns in a hundred number chart. When asked to describe patterns that they see, some children see a counting by ones pattern horizontally, others see the tens digit increasing and the ones digit staying the same as they move down the chart vertically, and still others see in the last column the numbers that they use to count by tens.

• Students use the constant function feature of their calculators to program a skip count. They press + 2 === to watch the display count by twos, try to anticipate what number comes next and make predictions to each other. Any number can replace the "2" to add difficulty to the activity.

• Students play Find the Number on a hundred number chart located at the front of the room, with each of the numbers covered by a Post-it or a small tag. One child calls out a number, like 45, and a volunteer tries to identify where it is on the chart. The indicated Post-it is then lifted to check the guess.

4. Develop a sense of the magnitudes of whole numbers, commonly used fractions, and decimals.

• Children are presented with four jars of jelly beans - one with 3 beans in it, one with 19 beans in it, one with 52 beans in it, and one with 156. The teacher then asks Which of these jars do you think has about 50 beans in it? The students discuss their reasons for believing as they do.

• Second graders are challenged to guess how many sheets of paper are in the ream of paper on the front table. After everyone has made a guess, one student counts out 25 sheets from the top of the pile and places them next to the rest of the pile. Everyone is offered a chance to change their estimates and to discuss the reason for their change. Then students agree on a way to verify their guesses before trying to guess how many such reams it would take to reach the ceiling!

• Students work through the Will a Dinosaur Fit? lesson that is described in the First Four Standards of this Framework. They discuss how many dinosaurs of different types might fit into the classroom.

• Students fold paper circles into halves, fourths, and eighths and are asked questions like: Which would you rather have, a half of a cherry pie or a fourth of the pie? How about three-eighths of a pizza or one-fourth?

• Students read or listen to a piece of children's literature that has fractions as its theme, such as Eating Fractions by Bruce McMillan.

5. Understand the various uses of numbers including counting, measuring, labeling, and indicating location.

• A kindergarten teacher announces to her class: Boys and Girls! Great News! The principal told me that our class has just won FIVE! A discussion then ensues regarding the need for that number to exist in some context, to have some unit or label before it makes sense.

• Second graders are given a stack of old magazines. They cut out any information which uses numbers and sort them according to how they are used: as page numbers, as prices, as dates, as addresses, and so on.

• The class takes a walk around the school or neighborhood pointing out to each other the numbers they see, and discuss how they are used.

6. Count and perform simple computations with money.

• Students use play money to show different combinations of coins that can be used to "buy" an object. For example, an 11 cents pencil can be bought with 11 pennies, a dime and a penny, one nickel and six pennies, or two nickels and a penny.

• Students earn 2 cents each day for attendance and 1 cent for good behavior. They keep their play money in a bank, count it regularly and use it to buy objects from a treasure chest.

• Students play Spend a Dollar. They each start with $1 (either as a bill or in change) and then roll one or two dice to find out how much they "spend" on that turn. They trade coins as needed. The student who spends all of her money first wins.

• Students play a shopping board game. They each begin with a given amount of money in coins. They roll two dice to determine how far they move each turn. As they land on a space, they must buy whatever is shown. Some spaces may provide refunds. The winner is the first person to go around the board and still have money left.

• Students' abilities to recognize coins and find the value of a group of coins are assessed by having each student select three objects to "buy," identify and name the coins needed to purchase each object, and find the total amount of money required to purchase all three.

7. Use models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number.

• When modeling 2-digit numbers with base-ten models such as popsicle sticks, base-ten blocks, or pennies and dimes, students are frequently asked to show all the ways they can make a given number. Children then begin to see that 3 tens and 7 ones, 2 tens and 17 ones, 1 ten and 27 ones, and 37 ones all represent the same number 37.

• Students each develop questions whose answers are all equivalent to some target number. For example, if the target is 8, students may ask the following questions: What is 4+4? What is 9-1? What is 8+0? How many hands do four children have? How many days is one more than a week? or How much is a nickel and three pennies?

• Students use geoboards, pattern blocks, Cuisenaire Rods, paper folding, and tangrams to explore simple common fractions like halves, thirds, and fourths. For instance, they may be challenged to model 1/2 with all of the different models.

8. Compare and order whole numbers, commonly used fractions, and decimals.

• Young students use dot pattern cards or dominoes to practice more, less, and same. For example, given a card with 6 dots on it, students use counters to make a set that is more, another that is less, and one that is the same. They can then label the sets with cards that show the appropriate words. With dominoes, students work in pairs to compare the dots on the two halves and state which is more and by how much.

• Students play the old favorite card game war with either dot cards or with a deck of regular playing cards minus the face cards. Every now and then, the rule changes so that the student with the card that is less wins the play.

• Students play Guess the Point. A long number line with endpoints of 20 and 75, for example, is drawn on the board where all of the intermediary points are labeled above the line. The labels are then covered by a long piece of paper that can be lifted to reveal them. A student places a finger somewhere on the line and others must estimate the numerical label of the point chosen. The paper is then lifted to check the accuracy of their responses.

9. Explore real-life settings which give rise to negative numbers.

• Primary classrooms are equipped with Celsius thermometers, in addition to Fahrenheit ones, so that "below zero" outdoor temperatures can be recorded. Temperature reports, possibly in both scales, become a part of the everyday calendar routine.

References

Baratta-Lorton, Mary. Mathematics Their Way. Menlo Park, CA: Addison-Wesley, 1995.

Mathews, Louise. Gator Pie. New York: Dodd, Mead, and Co., 1979.

McMillan, Bruce. Eating Fractions. New York: Scholastic, 1991.

National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, VA, 1989.

Software

Balancing Bear. Sunburst Communication.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.