Chapter 11 K-2

New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition

STANDARD 11: NUMERICAL OPERATIONS

All students will develop their understanding of numerical operations through experiences which enable them to construct, explain, select, and apply various methods of computation including mental math, estimation, and the use of calculators, with a reduced role for pencil-and-paper techniques.

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 rote memory approach.

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 childrens understanding of the meanings of the operations. They also need to strengthen the childrens sense that modeling such situations as a way to understand them is the right thing to do. It is important that the variety of situations to which they are exposed include the full gamut of addition and subtraction. There are several slightly different taxonomies of the types of addition and subtraction, but one describes change problems, part-part-whole problems, equalize problems, and compare problems. Students need to recognize and model each for each operation.

Basic facts in addition and subtraction also continue to be very important. Students should be able to quickly and easily recall one-digit sums and differences. The most effective approach to enabling them to acquire this ability has been shown to be the focused and explicit use of basic fact strategies - conceptual techniques that make use of the childs 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 and back, make ten, and doubles and near doubles strategies, but also explain why they work by modeling them with counters.

Students must still be able to perform multi-digit addition and subtraction with pencil and paper, 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 follow, some involving regrouping and others not.

After initial exploration, students share with each other all of the strategies theyve developed, the best ways theyve found for working with the tens and ones in the problem, and their own approaches (and names!) for regrouping. Most students can, with direction, take the results of those discussions and create their own pencil-and-paper 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 involved in a good deal of 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 real-world problems so that the numbers come out friendly.

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

addition and subtraction basic facts

mutli-digit addition and subtraction

STANDARD 11: NUMERICAL OPERATIONS

K-2 Expectations and Activities

The expectations for these grade levels appear below in boldface type. Each expectation is followed by activities which illustrate how the expectation can be addressed in the classroom.

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

A. develop meaning for the four basic 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 good strong number concepts make 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 vignette on page 44 of the New Jersey Mathematics Standards. It challenges students to find a way to fairly share a large number of cookies among the members of the class, promoting discussion of early division, fraction, and probability ideas.

B. develop proficiency with basic facts through the use of a variety of strategies.

Students play "One More Than" Dominoes by changing the regular rules such 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 vignette on page 39 of the New Jersey Mathematics Standards. It asks them to consider the parts of eleven and the natural, random, occurrence of different addend pairs 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 strategy use 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.

C. 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 can be 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.

D. use models to explore operations with fractions and decimals.

Kindergartners explore part/whole relations with Pattern Blocks by seeing which blocks can be replicated with other blocks. For example: 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. 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.

E. select and use an appropriate method for computing from among mental math, estimation, paper-and-pencil, and calculator methods and check the reasonableness of results.

The daily Calendar Routine provides the students with plenty of opportunities to do computation. Questions like these arise almost every day: There are 27 children in our class. 24 are here today. How many are absent? Of the 24, 14 are buying lunch. How many brought their lunch? It's now 9:12. How long until we go to gym at 10:30? The students always choose a computation method with which they feel comfortable. They are frequently asked why they chose the method they chose and whether an exact answer to the question was important.
Students regularly have Human versus 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?

F. use a variety of mental computation and estimation techniques.

Students regularly practice a whole variety of oral counting skills, both forward and backward, by various steps. For instance: 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 are used to estimating sums and differences both before doing either pencil-and-paper 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.

G. understand and use relationships among operations and properties of operations.

Students explore three-addend problems like 4 + 5 + 6 =, looking first to see if adding the numbers in different orders produces different results and, later, looking 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 = 20. Then they rotate the array 90 degrees and label the new array, 5 rows, 4 in each = 20. Discussions follow which lead to intuitive understandings of commutativity.