This standard addresses the use of calculators, computers and manipulatives in the teaching and learning of mathematics. These tools of mathematics can and should play a vital role in the development of mathematical thought in students of all ages.

In the primary grades, manipulatives are the most natural of the three types of tools to use. Primary grade teachers have traditionally used many manipulative materials in their teaching of mathematics because they correctly perceived them to be of great value for young children. Typically, concrete materials are used to model mathematical concepts such as number or shape when those concepts are first introduced to the students.

Young children counting with lima beans, colored chips, linking cubes, smooth stones, or their fingers is a familiar sight in many New Jersey classrooms as they begin to master early counting skills and are introduced to addition and subtraction concepts. More sophisticated models should then be used, though, to begin to explore more sophisticated number concepts. Colored rods in graduated lengths give students a different sense of number than a set of discrete objects. Students should be able to see both a yellow rod and five colored chips as representative of the number five - the first being more of a measure model and the second a count model. Ice cream sticks and base ten blocks as well as chip trading activities help students begin to understand the very abstract concepts involved with place value and number base.

Attribute blocks, blocks with different shapes and colors, help students begin to classify and categorize objects and recognize their specific characteristics. Pattern blocks allow them to make patterns and geometric designs as they become familiar with the geometric properties of the shapes themselves. Geoboards allow students to explore the great variety of shapes that can be made and also to deal with issues of properties, attribute, and classification.

A great variety of different materials should be used to explore measurement. Paper clips, shoes, centimeter and decimeter rods, paper cutouts of handspans, and building blocks can all be used as non-standard units of length (even though some of them are really standard). Students place them down one after another to see how many paper clips long the desk is or how many handspans wide the doorway is. The transition can then be made to more standard measures and, following that, to rulers.

This list is, of course, not intended to be exhaustive. Many more suggestions for materials to use and ways to use them are given in the other sections of this Framework. The message in this section is a very simple one - concrete materials help children to construct mathematics that is meaningful to them.

Calculators have not been used traditionally in primary classrooms, but there are several appropriate uses for them. It is never too early for students to be introduced to the tool that most of the adults around them use whenever they deal with mathematics. In fact, many students now come to kindergarten having already played with a calculator at home or somewhere else. To ignore calculators completely at this level is to send the harmful message that the mathematics being done at school is different from the mathematics being done at home or at the grocery store.

The use of calculators at this level does not imply that students don't need to develop the arithmetic skills traditionally introduced at the primary level. They certainly do need to develop these skills. This Standard does not suggest that all traditional learning be replaced by calculator use; rather, it calls for the appropriate and effective use of calculators.

One of the most effective uses of the calculator with young children is the use of the constant feature of most calculators to count, forward or backward, or to skip count, forward or backward, by twos or threes or other numbers. This process allows children to anticipate what number will come next and then get confirmation of their guess when they see it appear in the display. Students can also greatly enhance their estimation ability through calculator use. Range-finding games ask students, for instance, to add a number to 34 that will give them an answer between 80 and 90. After the estimate is made, it is punched into the calculator to see whether or not it did the job. Calculators will prompt young students to be curious about mathematical topics that are not typically taught at their level. For example, when counting back by threes by entering 15 - 3 = = = . . . into the calculator, after the expected sequence of 12, 9, 6, 3, 0, the child will see -3, -6, -9, . . . A curious child will begin to ask questions about what those numbers are, but will also begin to develop an intuitive notion about negative numbers.

Computers are a valuable tool for primary children. As more and more computers find their way into primary classrooms, the software available for them will dramatically improve; however, there are already many good programs that can be used with kindergartners and first and second graders. MathKeys links on-screen manipulative materials to standard symbolic representations and to a writing tool for children to use. A number of different counting programs match objects on the screen to a standard symbolic representation of the number and the number is said aloud so that a young student can count along with the program. Many other new programs focus on money skills and help children recognize different coins and determine the values of sets of coins through simulated purchases.

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. Select and use calculators, software, manipulatives, and other tools based on their utility and limitations and on the problem situation.

• Students participate in races to complete a set of computation problems between some students who use calculators and others who use mental math. They try to determine what makes the calculator a useful tool in some circumstances (large numbers, harder operations) and not terribly useful in others (basic facts, easy numbers).

• Students are regularly asked to make their own decisions about what is the right type of linear measuring device for a particular situation: mental estimation, colored rods, ruler, yard or meter stick, or tape measure. Different decisions are made in different circumstances: Estimation is fine when you are deciding whether you will fit through a small doorway, but accurate ruler measurement is important if you are cutting out a frame for a picture.

• In problem solving situations, students are regularly provided with calculators, manipulatives, and other tools so that they may choose for themselves what will be useful to help solve the problem.

2. Use physical objects and manipulatives to model problem situations, and to develop and explain mathematical concepts involving number, space, and data.

• Students use popsicle sticks to model multi-digit base-ten numbers and then use them to further model operations with the numbers.

• Students use pipe cleaners and straws to make models of two-dimensional geometric shapes. They then compare, contrast, and sort all the shapes using whatever criteria they think are important, including number of corners, straight or curvy sides, number of sides, and so on.

• Students work through the Shapetown lesson that is described in the First Four Standards of this Framework. Students in kindergarten are challenged to build towns with attribute blocks and loops based on a rule or pattern they make up.

• Kindergarten students each use a cubic inch block to represent himself or herself in a bar graph that describes the favorite flavors of ice cream of all the students in the class. On a table in the front of the room, the teacher has placed mats that say Vanilla, Chocolate, and Strawberry. One by one, the students walk past the table, dropping their blocks on one of the piles that build up on the mats. When this concrete "bar graph" is complete, the children ask questions that can be answered with the data displayed: What's the mostfavorite flavor in the class? What's the least favorite? Are there more people who like vanilla than chocolate?1150

3. Use a variety of technologies to discover number patterns, demonstrate number sense, and visualize geometric objects and concepts.

• Students use the constant function on a calculator to count by ones, twos, tens, fourteens, and other numbers, both forward and backward. As they do so, they try to keep up with the calculator by saying the numbers orally as they come up in the display, and even trying to say them before they come up.

• Students use a beginner's Logo to explore movement in two-dimensional space. They move the turtle on the computer screen forward and backward with simple commands and also turn the turtle through predetermined angles to the right and to the left with other commands. The turtle leaves a trail of where it's been on the screen so that its movements actually create a drawing of a figure. The students try to have the turtle draw a square, a different rectangle, and a triangle before progressing to harder tasks.

• Students use a geoboard to make shapes that are composed of unit squares. One challenge they are given is to find as many shapes as possible that are made up of 10 unit squares.

4. Use a variety of tools to measure mathematical and physical objects in the world around them.

• Young students develop meaning for rulers by first measuring with individual paper clips, then a paper clip chain, then taping the clip chain to a paper strip, then marking and numbering the ends of the clips on the strip, and last, removing the clip chain from the paper strip leaving just the marks and the numbers. This leaves the students with paper clip rulers with which they can measure the lengths of a variety of objects. The unit of measurement is, of course, a paper clip.

• Students use a balance scale to determine the weights of a variety of classroom objects in terms of units that are other classroom objects; for example, How many pennies does a math book weigh? How many paper clips does a pencil weigh?

• Students work through the Will a Dinosaur Fit? lesson that is described in the First Four Standards of this Framework. Second grade students measure the size of their classroom and other places in a variety of ways to determine whether dinosaurs they are studying would fit into them.

• As part of the morning calendar routine, second graders check each of two thermometers - one Fahrenheit and one Celsius - and make daily recordings of the outside temperature. They record the temperatures in a chart and look for interesting patterns. They notice that, as the school year progresses and the temperatures change, whenever one of the temperatures goes up or down, so does the other.

• Students regularly use both analog and digital stopwatches to practice timing events that are usually measured in seconds such as: the amount of time it takes a classmate to say the alphabet, how long a classmate goes without blinking, or how long the morning announcements take.

5. Use technology to gather, analyze, and display mathematical data and information.

• Students take a survey to determine every child's birth month and then use the Graph Club or Primary Graphing and Probability Workshop software to display the resulting data in graphs.

• Using a World Wide Web page that reports meteorological data (possibly http://www.rainorshine.com/weather/index/sites/njo/), students find the predicted high temperatures for a variety of cities in different regions around the country, write those numbers on a map of the United States, and then look for patterns and trends in different regions.

• Students use Table Top software to make a Venn diagram to show which of them have brothers, which have sisters and which have both (the intersection of the two sets). Students who have no siblings are shown outside the rings. Other attributes of the children are also used to make Venn diagrams.

References

Software

Graph Club. Tom Snyder Productions.

Logo. Many versions of Logo are commercially available.

Primary Graphing and Probability Workshop. Scott Foresman.

TableTop. TERC.

MathKeys. Minnesota Educational Computing Consortium (MECC).

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.