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 grades 3 and 4, manipulatives have traditionally not been used as much as they have been in the primary grades. It is fairly common for teachers at this level to think that once initial notions of number and shape have been established with concrete materials in the lower grades, the materials are no longer necessary and a more symbolic approach is preferable. Research shows, however, that concrete materials and the modeling of mathematical operations and concepts is just as useful at these grade levels as it is for younger students. The content being modeled is, of course, different and so the models are different - but no less important.

Third- and fourth-graders can use square tiles to model one-digit multiplication arrays in a manner that makes the operation very meaningful for them, and later use base-ten blocks to model two-digit multiplication arrays. The added advantage to this kind of a model is the degree to which students who have used it can visualize what's happening with the factors in the problem and so can develop much better estimation and mental math skills than students who have simply learned the standard paper-and-pencil algorithms. The relationship between multi-digit multiplication and division is also clearly shown by such models.

Geometry models, both two- and three-dimensional, are an important part of learning about geometry and development of spatial sense in students of this age. Students should use geoboards to explore area and perimeter and to begin to develop procedures for finding the areas of irregular shapes. They can also use construction materials like pipe cleaners and straws to make three-dimensional geometric shapes like cubes and pyramids so that they can study them directly. Such models make it much easier to determine the number of faces or edges in a figure than two-dimensional drawings.

Third- and fourth-graders should also be in the habit of using a variety of materials to help them model problem situations in other areas of the mathematics curriculum. They might use different colored unifix cubes to represent all of the different double-decker ice cream cones that can be made with three different flavors of ice cream. They should be able to use a variety of measurement tools to measure and record the data in a science experiment. They might use coin tosses or dice throws to simulate real-world events that have a one-in-two chance or a one-in-six chance of happening.

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 construct mathematics that is meaningful to them.

There are several appropriate uses for calculators at these grade levels. It is never too early for students to be introduced to the tool that most of the adults around them will use whenever they deal with mathematics.

The use of calculators at this level does not imply that students don't need to develop 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 which can be continued in grade three is the use of the constant feature of most calculators to count, forward or backward, or to skip count. This process allows children to anticipate what number will come next and then get confirmation of their guess when they see it come up in the display. Students can greatly enhance their estimation ability through calculator use. Range-finding games ask students, for instance, to add a number to 342 that will give them an answer between 800 and 830. After the estimate is made, it is punched into the calculator to see whether or not it did the job.

Calculators will also prompt students to be curious about mathematical topics to which they are about to be introduced. For example, while routinely using calculators in problem solving activities, some students may notice that whenever they add, subtract, or multiply two whole numbers, they get a whole number for an answer. Sometimes that happens for division, too, but sometimes when they divide they get an answer like 3.5. What does that mean? These kinds of questions offer a great opportunity for some further exploration and investigation; for example, Which problems give you answers like those? What happens when you solve those problems using pencil-and-paper?

Computers are a valuable tool for students in third and fourth grade. As more and more computers find their way into these classrooms, the software available for them will dramatically improve; however, there are already many good programs that can be used with students of this age. MathKeys links on-screen manipulative materials to standard symbolic representations and to a writing tool for children. Logo can be used by students to explore computer programming and geometry concepts at the same time. Tesselmania! and other programs offer an opportunity to play with geometric transformations on the screen and produce striking designs. The King's Rule is a program that asks students to determine the rules that distinguish one set of numbers from another, fostering creative and inductive thinking. The World Wide Web can be an exciting and eye-opening tool for third-and fourth-graders as they retrieve and share information. Specifically, in these grades, they might look for state populations, meteorological data, and updates on current events.

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 grades 3 and 4.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

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 between some students who use calculators and others who use mental math, each working to complete a set of computation problems involving newly learned arithmetic skills. 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 work through the Tiling a Floor lesson that is described in the First Four Standards of this Framework. Third grade students test various shapes made of a variety of materials to determine which can be used to tessellate an area.

• Students choose to use a computer spreadsheet on their classroom computer as a neat way to organize tables and charts, but they also use a full-function word processor when there is a good deal of text involved or when using different fonts and text formatting.

• Students use base ten blocks rather than popsicle sticks when performing operations with large numbers because they can create models more efficiently and more quickly with them.

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

• Students use base ten blocks to demonstrate the operations of multiplication and division with multi-digit numbers using both repeated subtraction and partition methods.

• Students work through the Sharing Cookies lesson that is described in the First Four Standards of this Framework. Fourth grade students use manipulatives to determine how to divide 8 cookies equally among 5 people.

• Students use a variety of devices such as dice, coin flips, spinners, and decks of cards for generating random numbers and understand the essential equivalence of these devices.

• Students use pipe cleaners and straws to build and study three-dimensional objects, finding it easier to discuss things like numbers of edges, faces, and vertices and the relationships among them if they have a physical model with which to work.

• Students use geoboards to solve Farmer Brown's problem. She has 16 meters of fencing and wants to fence in the largest rectangular area possible for her dog to romp around in.

• Students use colored rods or pattern blocks to develop early notions of fractions, using different rods or blocks as the unit and discovering by trial-and-error the resulting fractional values of all of the other pieces.

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

• Students play the game target practice in the New Jersey Calculator Handbook. In it, one student enters a number into a calculator to be used as an operand, enters an operation (addition, subtraction, multiplication, or division) into the calculator by pressing the appropriate sign, and then specifies a "target range" for the answer. For instance, the student may enter: 82 x and specify the range as 2000-3000. A second student must then enter a second operand into the calculator and press the equals key. If the answer is within the specified target range, the shot was a bull's eye.

• Students play The Biggest Product, also from The New Jersey Calculator Handbook. In it, four cards are dealt face up from a shuffled deck of cards containing only the cards from ace to nine. The students who are playing then use their calculators to try to compose the multiplication problem that uses only the digits on the cards, each only once, that has the largest possible product. After several rounds, the students begin to notice a pattern in their answers and become much more efficient at finding the correct problems.

• Students begin to use Logo to create geometric figures on the computer screen. They write routines that have the turtle's path describe a square, a rectangle, a triangle, and other standard polygons. As a challenge, they write a routine to have the turtle draw a simple house with windows and a roof.

• Students solve the problems posed in Logical Journey of the Zoombinis by using logic and classification and categorization skills. In it, they create Zoombinis, little creatures that have specific characteristics that allow them to accomplish specified tasks.

• After reading Counting on Frank by Rod Clement, students practice their estimation skills by using software of the same title.

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

• Students regularly use both analog and digital stopwatches to practice timing events that happen in short time periods such as: the amount of time it takes a classmate to recite the Pledge of Allegiance or count to 60, how long a classmate takes to run a 50 meter dash, or how long the morning announcements take. They begin to record the elapsed time in decimals that include tenths or hundredths of a second.

• Students first estimate and then use a metric trundle wheel to measure long distances such as the distance from the cafeteria doors to the sandbox, the distance from the classroom door to the principal's office door, or the distance all the way around the school on the sidewalk.

• Students read Counting on Frank by Rod Clement and repeat some of the estimates made by the boy in the book. How many peas would it take to fill up the room? How long a line can a pen write? They make up their own silly things to estimate, and devise ways to make the appropriate measures and estimates.

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

• Students use the New Jersey State homepage http://www.state.nj.us on the World Wide Web to gather data about the latest reported populations for each of the municipalities in their county. They then enter the collected data into a simple spreadsheet and use its graphing function to produce a bar graph of all of the populations of the towns and cities. They highlight their own town to show where it stands in relationship to the others.

• Students use the Graphing and Probability Workshop or similar software to generate large amounts of random data. This software simulates a variety of probability experiments including up to 300 coin tosses, spinner spins, and dice rolls. Discussions focus on whether the simulated outcomes were as expected or were different from what was expected.

• There is always math help available at the Dr. Math World Wide Web site (dr.math@forum.swarthmore.edu). In Dr. Math's words, "Tell us what you know about your problem, and where you're stuck and think we might be able to help you. Dr. Math will reply to you via e-mail, so please be sure to send us the right address. K-12 questions usually include what people learn in the U.S. from the time they're five years old through when they're about eighteen."

References

Association of Mathematics Teachers of New Jersey. The New Jersey Calculator Handbook. 1993.

Clement, Rod. Counting on Frank. Milwaukee, WI: Gareth Stevens Children's Books, 1991.

Software

Counting on Frank. EA Kids Software.

Graphing and Probability Workshop. Scott Foresman.

Logical Journey of the Zoombinis. Broderbund.

Logo. Many versions of Logo are commercially available.

MathKeys. Minnesota Educational Computing Consortium (MECC).

Tesselmania! Minnesota Educational Computing Consortium (MECC).

The King's Rule. Sunburst Communications.

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.