New Jersey Mathematics Curriculum Framework

STANDARD 8 - NUMERICAL OPERATIONS

 All students will understand, select, and apply various methods of performing numerical operations.

Standard 8 - Numerical Operations - Grades K-2

Overview

The wide availability of computing and calculating technology has given us the opportunity to significantly reconceive the role of computation and numerical operations in our elementary mathematics programs, but, in kindergarten through second grade, the effects will not be as evident as they will be in all of the other grade ranges. This is because the numerical operations content taught in these grades is so basic, so fundamental, and so critical to further progress in mathematics that much of it will remain the same. The approach to teaching that content, however, must still be changed to help achieve the goals expressed in the New Jersey Mathematics Standards.

Learning the meanings of addition and subtraction, gaining facility with basic facts, and mastering some computational procedures for multi-digit addition and subtraction are still the topics on which most of the instructional time in this area will be spent. There will be an increased conceptual and developmental focus to these aspects of the curriculum, though, away from a traditional drill-and-practice approach, as described in the K-12 Overview; nevertheless, students will be expected to be able to respond quickly and easily when asked to recall basic facts.

By the time they enter school, most young children can use counters to act out a mathematical story problem involving addition or subtraction and find a solution which makes sense. Their experiences in school need to build upon that ability and deepen the children's understanding of the meanings of the operations. School experiences also need to strengthen the children's sense that modeling such situations as a way to understand them is the right thing to do. It is important that they be exposed to a variety of different situations involving addition and subtraction. Researchers have separated problems into categories based on the kind of relationships involved (Van de Walle, 1990, pp. 75-6); students should be familiar with problems in all of the following categories:

Join problems

• Mary has 8 cookies. Joe gives her 2 more. How many cookies does Mary have in all?
• Mary has some cookies. Joe gives her 2 more. Now she has 8. How many cookies did Mary have to begin with? (Missing addend)
• Mary has 8 cookies. Joe gives her some more. Now Mary has 10. How many cookies did Joe give Mary? (Missing addend)

Separate problems

• Mary has 8 cookies. She eats 2. How many are left? (Take away)
• Mary has some cookies. She eats 2. She has 6 left. How many cookies did Mary have to begin with?
• Mary has 8 cookies. She eats some. She has 6 left. How many cookies did Mary eat? (Missing addend)

Part-part-whole problems

• Mary has 2 nickels and 3 pennies. How many coins does she have?
• Mary has 8 coins. Three are pennies, the rest nickels. How many nickels does Mary have?

Compare problems

• Mary has 6 books. Joe has 4. How many more books does Mary have than Joe?
• Mary has 2 more books than Joe. Mary has 6 books. How many books does Joe have?
• Joe has 2 fewer books than Mary. He has 4 books. How many books does Mary have?

Basic facts in addition and subtraction continue to be very important. Students should be able to quickly and easily recall one-digit sums and differences. The most effective way to accomplish this has been shown to be the focused and explicit use of basic fact strategies-conceptual techniques that make use of the child's understanding of number parts and relationships to help recover the appropriate sum or difference. By the end of second grade, students should not only be able to use counting on, counting back, make ten, and doubles and near doubles strategies, but also explain why these strategies work by modeling them with counters. Building on their facility with learning doubles like 7 + 7 = 14, children recast 7 + 8 as 7 + 7 + 1, which they then recognize as 15 (near doubles). Make ten involves realizing that in adding 8 + 5, you need two to make ten, and recasting the sum as 8 + 2 + 3 which is 10 + 3 or 13. Counting on involves starting with the large number and counting on the smaller number so that adding 9 + 3, for example, would involve counting on 10, 11, and then 12. Counting back is used for subtraction, so that finding 12 - 4, the child might count 11, 10, 9, and then 8.

Students must still be able to perform multi-digit addition and subtraction with paper and pencil, but the widespread availability of calculators has made the particular procedure used to perform the calculations less important. It need no longer be the single fastest, most efficient algorithm chosen without respect to the degree to which children understand it. Rather, the teaching of multi-digit computation should take on more of a problem solving approach, a more conceptual, developmental approach. Students should first use the models of multi-digit number that they are most comfortable with (base ten blocks, popsicle sticks, bean sticks) to explore the new class of problems. Students who have never formally done two-digit addition might be asked to use their materials to help figure out how many second graders there are in all in the two second grade classes in the school. Other similar real-world problems should follow, some involving regrouping and others not. After initial exploration, students share with each other all of the strategies they've developed, the best ways they've found for working with the tens and ones in the problems, and their own approaches (and names!) for regrouping. Most students can, with direction, take the results of those discussions and create their own paper-and-pencil procedures for addition and subtraction. The discussions can, of course, include the traditional approaches, but these ought not to be seen as the only right way to do these operations.

Kindergarten through second grade teachers are also responsible for setting up an atmosphere where estimation and mental math are seen as reasonable ways to do mathematics. Of course students at these grade levels do almost exclusively mental math until they reach multi-digit operations, but estimation should also comprise a good part of the activity. Students regularly involved in real-world problem solving should begin to develop a sense of when estimation is appropriate and when an exact answer is necessary.

Technology should also be an important part of the environment in primary classrooms. Calculators provide a valuable teaching tool when used to do student-programmed skip counting, to offer estimation and mental math practice with target games, and to explore operations and number types that the students have not formally encountered yet. They should also be used routinely to perform computation in problem solving situations that the students may not be able to perform otherwise. This use prevents the need to artificially contrive the numbers in real-world problems so that their answers are numbers with which the students are already comfortable.

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

Standard 8 - Numerical Operations - Grades K-2

Indicators and Activities

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. Develop meaning for the four basic arithmetic operations by modeling and discussing a variety of problems.

• Students use unifix cube towers of two colors to show all the ways to make "7" (for example: 3+4, 2+5, 0+7, and so on). This activity focuses more on developing a sense of "sevenness" than on addition concepts, but a good sense of each individual number makes the standard operations much easier to understand.

• Kindergartners and first graders use workmats depicting various settings in which activity takes place to make up and act out story problems. On a mat showing a vacant playground, for instance, students place counters to show 3 kids on the swings and 2 more in the sandbox. How many kids are there in all? How many more are on the swings than in the sandbox? What are all of the possibilities for how many are boys and how many are girls?

• Students work through the Sharing a Snack lesson that is described in the Introduction to this Framework. It challenges students to find a way to share a large number of cookies fairly among the members of the class, promoting discussion of early division, fraction, and probability ideas.

• Students learn about addition as they read Too Many Eggs by M. Christina Butler. They place eggs in different bowls as they read and then make up addition number sentences to find out how many eggs were used in all.

• Kindergarteners count animals and learn about addition as they read Adding Animals by Colin Hawkins. This book uses addends from one through four and shows the number sentences that go along with the story.

• Students are introduced to the take-away meaning for subtraction by reading Take Away Monsters by Colin Hawkins. Students see the partial number sentence (e.g., 5 - 1 = ), count to find the answer, and then pull the tab to see the result.

• Students explore subtraction involving missing addend situations as they read The Great Take-Away by Louise Mathews. This book tells the story of one lazy hog who decides to make easy money by robbing the other pigs in town. The answers to five subtraction mysteries are revealed when the thief is captured.

• Students make booklets containing original word problems that illustrate different addition or subtraction situations. These may be included in a portfolio or evaluated independently.

2. Develop proficiency with and memorize basic number facts using a variety of fact strategies (such as "counting on" and "doubles").

• Students play one more than dominoes by changing the regular rules so that a domino can be placed next to another only if it has dots showing one more than the other. Dominoes of any number can be played next to others that show 6 (or 9 in a set of double nines). One less than dominoes is also popular.

• Students work through the Elevens Alive lesson that is described in the Introduction to this Framework. It asks them to consider the parts of eleven and the natural, random, occurrence of different pairs of addends when tossing eleven two-colored counters.

• Second graders regularly use the doubles and near doubles, the make ten, and the counting on and counting back strategies for addition and subtraction. Practice sets of problems are structured so that use of all of these strategies is encouraged and the students are regularly asked to explain the procedures they are using.

• Students play games like addition war to practice their basic facts. Each of two children has half of a deck of playing cards with the face cards removed. They each turn up a card and the person who wins the trick is the first to say the sum (or difference) of the two numbers showing. Calculators may be used to check answers, if necessary.

• Students use the calculator to count one more than by pressing + 1= = =. The display will increase by one every time the student presses the = key. Any number can replace the 1 key.

• Students use two dice to play board games (Chutes and Ladders or home-made games). These situations encourage rapid recall of addition facts in a natural way. In order to extend practice to larger numbers, students may use 10-sided dice.

• Students use computer games such as Math Blaster Plus or Math Rabbit to practice basic facts.

3. Construct, use, and explain procedures for performing whole number calculations in the various methods of computation.

• Second graders use popsicle sticks bundled as tens and ones to try to find a solution to the first two-digit addition problem they have formally seen: Our class has 27 children and Mrs. Johnson's class has 26. How many cupcakes will we need for our joint party? Solution strategies are shared and discussed with diversity and originality praised. Other problems, some requiring regrouping and others not, are similarly solved using the student-developed strategies.

• Students use calculators to help with the computation involved in a first-grade class project: to see how many books are read by the students in the class in one month. Every Monday morning, student reports contribute to a weekly total which is then added to the monthly total.

• Students look forward to the hundredth day of school, on which there will be a big celebration. On each day preceding it, the students use a variety of procedures to determine how many days are left before day 100.

• As part of their assessment, students explain how to find the answer to an addition or subtraction problem (such as 18 + 17) using pictures and words.

• Students find the answer to an addition or subtraction problem in as many different ways as they can. For example, they might solve 28 + 35 in the following ways:

8 + 5 = 13 and 20 + 30 = 50, so 13 + 50 = 63
28 + 30 = 58. Two more is 60, and 3 more is 63
25 + 35 = 60 and 3 more is 63.

• Students use estimation to find out whether a package of 40 balloons is enough for everyone in the class of 26 to have two balloons. They discuss the strategies they use to solve this problem and decide if they should buy more packages.

4. Use models to explore operations with fractions and decimals.

• Kindergartners explore part/whole relations with pattern blocks by seeing which shapes can be created using other blocks. You might ask: Can you make a shape that is the same as the yellow hexagon with 2 blocks of some other color? with 3 blocks of some other color? with 6 blocks of some other color? and so on.

• Students use paper folding to begin to identify and name common fractions. You might ask: If you fold this rectangular piece of paper in half and then again and then again, how many equal parts are there when you open it up? Similarly folded papers, each representing a different unit fraction, allow for early comparison activities.

• Second graders use fraction circles to model situations involving fractions of a pizza. For example: A pizza is divided into six pieces. Mary eats two pieces. What fraction of the pizza did Mary eat? What fraction is left?

• Students use manipulatives such as pattern blocks or Cuisenaire rods to model fractions. For example: If the red rod is one whole, then what number is represented by the yellow rod?

5. Use a variety of mental computation and estimation techniques.

• Students regularly practice a variety of oral counting skills, both forward and backward, by various steps. For instance, you might instruct your students to: Count by ones - start at 1, at 6, at 12, from 16 to 23; Count by tens - start at 10, at 30, at 110, at 43, at 67, from 54 to 84, and so on.

• Students estimate sums and differences both before doing either paper-and-pencil computation or calculator computation and after so doing to confirm the reasonableness of their answers.

• Students are given a set of index cards on each of which is printed a two-digit addition pair (23+45, 54+76, 12+87, and so on). As quickly as they can they sort the set into three piles: more than 100, less than 100, and equal to 100.

• Students play "Target 50" with their calculator. One student enters a two-digit number and the other must add a number that will get as close as possible to 50.

6. Select and use appropriate computational methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results.

• The daily calendar routine provides the students with many opportunities for computation. Questions like these arise almost every day: There are 27 children in our class. Twenty-four are here today. How many are absent? Fourteen are buying lunch; how many brought their lunch? or It's now 9:12. How long until we go to gym at 10:30? The students are encouraged to choose a computation method with which they feel comfortable; they are frequently asked why they chose their method and whether it was important to get an exact answer. Different solutions are acknowledged and praised.

• Students regularly have human vs. calculator races. Given a list of addition and subtraction basic facts, one student uses mental math strategies and another uses a calculator. They quickly come to realize that the human has the advantage.

• Students regularly answer multiple choice questions like these with their best guesses of the most reasonable answer: A regular school bus can hold: 20 people, 60 people, 120 people? The classroom is: 5 feet high, 7 feet high, 10 feet high?

• As part of an assessment, students tell how they would solve a particular problem and why. They might circle a picture of a calculator, a head (for mental math), or paper-and-pencil for each problem.

7. Understand and use relationships among operations and properties of operations.

• Students explore three-addend problems like 4 + 5 + 6 =. First they check to see if adding the numbers in different orders produces different results and, later, they look for pairs of compatible addends (like 4 and 6) to make the addition easier.

• Students make up humorous stories about adding and subtracting zero. I had 27 cookies. My mean brother took away zero. How many did I have left?

• Second graders, exploring multiplication arrays, make a 4 x 5 array of counters on a piece of construction paper and label it: 4 rows, 5 in each row = 20. Then they rotate the array 90 degrees and label the new array, 5 rows, 4 in each row = 20. Discussions follow which lead to intuitive understandings of commutativity.

References

Butler, M. Christina. Too Many Eggs. Boston: David R. Godine Publisher, 1988.

Hawkins, Colin. Adding Animals. New York: G. P. Putnam's Sons, 1984.

Hawkins, Colin. Take Away Monsters. New York: G. P. Putnam's Sons, l984.

Mathews, Louise. The Great Take-Away. New York: Dodd, Mead, & Co., 1980.

Van de Walle, J. A. Elementary School Mathematics: Teaching Developmentally. New York: Longman, 1990.

Software

Math Blaster Plus. Davidson.

Math Rabbit. The Learning Company.

On-Line Resources

http://dimacs.rutgers.edu/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.