Addition in kindergarten and first grade. Part 1.
Goal.
A reasonable minimal goal in arithmetic skill that can be achieved by the end of first grade is that:
1. Children learn how to write and read whole numbers at least up to 20 in decimal notation, in English, and in the language they speak at home (if it is different from English).
2. They learn by heart the addition facts with sums up to 10, and can use them to compute mentally (without using manipulatives, fingers, or other aids) sums of two or more numbers at least up to 20.
3. They can use these skills to count by "grouping and adding" collections of objects at least up to 20.
Remarks.
We did not include among the goals any written addition tasks that use worksheets. We think that written column addition of two one-digit numbers and a one-digit number to a two-digit number is counter-productive and could hinder the progress of learning arithmetic.
We did not include rote counting to 20, and subitizing. Most children will learn rote counting anyway. And the skill of subitizing is not needed for reaching the goals mentioned above.
Approach.
In order to learn these skills we want children to:
(A) Learn how to "figure out" the sums of numbers using manipulatives (described below). It would provide them with a method of getting the right answer even if they forget specific combinations of addends. It would decrease the "rote learning" aspect of the process.
(B) Have enough practice to memorize facts with sums up to 10. We also think that they should be told that "doing computation in their heads" is a goal they are going to achieve.
(C) Learn the "regrouping" algorithm (described below) that allows them to compute mentally other sums using only addition facts up to 10.
"Tools" and manipulatives.
A basic tool consists of two "two-by-five boards" and twenty tokens (preferably small pebbles).
One two-by-five board is a 2 inch by 5 inch piece of an index card clearly divided (with a marker) into 10 squares.
Two such boards are used for matching groups of pebbles with numbers up to 20 (in other words representing numbers by sets of pebbles) and to find the values of sums up to 20.
Basic representation of numbers by pebbles.
(E means empty square; P means a pebble. A number can be represented differently depending on the orientation of the board)
..........................................
Numbers from 11 to twenty are represented on two boards. For example, two representations of 14 are:
Children should be able to translate both ways, namely, to tell the number of pebbles on one or two boards, and to "show" on the board a required number of pebbles.
They should learn to answer questions like "What is seven?" by answering "It is four plus three" or "It is five plus two".
Technically they give the mathematical definition of the number seven in terms of smaller numbers and addition. But of course they do not have to know anything about definitions. They need only to know the answers. Notice that we do not have definitions of 0 and 1. In
mathematics we always have to start with concepts whose meaning comes from outside of mathematics. (Greek philosophers did not call 1 a number. It was a "source of numbers" which allowed them to define "true numbers" 2, 3, and so on.)
Children and adults can easily recognize at a glance the number of objects up to three. But even very skilled adults cannot do it for bigger numbers of objects which are spaced in an irregular way. Matching numbers with pebbles arranged on the boards is much easier than if the pebbles were just kept in piles.
Addition in kindergarten and first grade. Part 2.
Addition facts with sums up to 10.
Even when children learn to count 4 and 5 objects it is important that they do it by grouping.
O O O O we have four objects because there are O O two objects and O O two objects, and NOT because the last word in the count, one, two, three, four is four. This corresponds to the fact (definition of 4) that 4 = 2 + 2.
Similarly we have five objects because they can be split into two groups, a group of three objects, and of two objects. This corresponds to the fact that 5 = 3 + 2.
Viewing bigger collections of objects as the result of joining smaller ones, and the corresponding arithmetic statements, is even more important.
A subitizing algorithm (reciting number names) is very inefficient and error prone in almost all instances.
Whether children consider 7 as 5 + 2, 4 + 3, or both, is not important but they should see bigger numbers as sums of smaller ones and not as a result of a counting process.
The "discovery" of new facts can be done by regrouping pebbles on either one or two boards, but it should be accompanied by enough mental exercises and regrouping piles of objects, so that children will not treat a board with pebbles as a "computing tool".
Addition algorithm, "split and regroup".
The principle of adding two one-digit numbers whose sum is bigger than 10 is simple: split one number and regroup to form 10, for example
7 + 6 → 7 + 3 + 3 → 10 + 3 → 13
It can be done on two boards
The fact that we need to move exactly 3 pebbles is visible, because there are 3 empty squares on the first board.
This algorithm easily generalizes for adding a one-digit number to any two-digit number: split and regroup to get one more 10.
Remarks.
• Remember that using the board is just a step to purely mental operations, so verbal practice and counting objects without using boards is important.
• Memorizing facts with sums bigger than 10 is unnecessary.But knowing them both ways, 2 + 7 = 9, and 9 = 7 + 2, is very important for regrouping.
• Adding (mentally and on boards) more than two numbers, for example, 3 + 4 + 1 + 2, is important.
• Reading and writing numbers is important, but we suggest that you do not start written algorithms until the "split and regroup" mental algorithm is mastered for sums up to 30 or 40.
• Use of a four-operation calculator can be taught simultaneously, but it should not be used for "checking" mental calculations, but rather for tasks that really require it (for example, in lessons such as "Ants’ roads", in Breaking Away from the Math Book: Creative projects for grades K to 6, which can be used in the first grade.)