Addition with bridging(1st grade)

1. Introduction

It is hard for the 1st graders(age6-7) to learn addition with bridging. Because it includes number-concept as cardinal number and composition of number, resolution of number, and complementary number of 10, and so on that they have learned before. Students learn the unit "Calculation of 3 numbers",for example, 2+3+5, 5+3-4, after that they learn addition with bridging. That is to say, learning addition with bridging is on the base of the unit "Calculation of 3 numbers", and students have to utilize these learnings generally. But it is clear that it is so difficult for only 6-7aged students, and so some obstacles are underlying in learning this calculation.
(1) There are some students who can not learn composition and resolution of number to 10 sufficiently.
(2) There are some students who can calculate addition that either number is very close to 10, for example,9+3, 3+8, but can not calculate on the case, 6+5, so on, because they can not understand how they should make a gathering of 10, therefore, they can not calculate it.
(3) There are some students who can calculate it if they operate concrete materials(blocks or beads etc.), but if not, they can not do. (*They can not connect operating blocks with abstract calculation.)
(4) Though some students can calculate it(addition to 10) without using their fingers, however, they use their fingers when its calculation is over 10. (That is to, they do count.)
It seems that the matter of (3) is the most difficult. Because we can not have students calculate addition with bridging soon without concrete operating. Then, I will first consider it theoretically, and next search it practically.

2. Cognitive construction in order to calculate addition with bridging and its means

What cognitive process in students goes on in order to calculate addition with bridging? They can not express themselves well as they do not learn language(Japanese)enough. Therefore, teachers have to observe students' activities and guess their cognition. So, I thought cognitive construction-figure to calculate addition with bridging as framework of analyzing students' understanding. (fig.1)

Actual arrow line is showing that children's cognition would advance, on the contrary, dotted line feed back.

3. How do Japanese teachers teach students addition with bridging?

Generally speaking, Japanese teachers teach students addition with bridging as follows. Its method is called, "Boots' calculation". (fig.2)

As you see, the method is that students first resolute the addend into tow numbers(one of them is the complementary number of 10), and next add the augend. (8+2) And last thay add 10 to the remainder. (10+1) Thus, the figure formed during these calculation resembles "Boots". The method might be easy for students to understand and calculate, because they have to only calculate one by one orderly.

Further more, there are many variations about the method. For example, attaching some adjiont symbols, or writing down resolting formula. Each teacher thinks hard in order for students to able to calculate addition with bridging. And when they teach students this method, they instruct students to speak the procedure of calculation logically. (For example, "at first, to resolute the addend into x and y. Next, ....) Thus, Japanese students learn addition with bridging and practice many times.

4. Possibility of "Mapping instruction"

We know "Mapping instruction" advocated by Resnick as instruction to connect concrete operations with calculation. However, to change this instruction into effective one, we need try and error in practice. That is to say, there are underlying some problems how we should map students' concrete operations on the procedures of calculation and how we should instruct them practically. Moreover, on the case of Resnick, it was subtraction with bridging and in addition with bridging there might be other elements. By the way, let's think about 3 numbers calculation just before students learn addition with bridging. 3 numbers calculations are as 3+2+4 or 9-2-3, and students have to have learned them completely to calculate addition with bridging. But, the average score about this material over the nation is 76 points(my class, lower). From judging this fact, it is shown that students may begin to learn addition with bridging. It is very hard for the 1st graders to calculate 3 numbers calculation, I think. In fact, they have to calculate two times actually. Judging from this fact, as the instruction from concrete operation to calculation, it is suggested that we may have to only teach students the calculation, dividing the procedures into two parts, and connecting concrete operation with 3 numbers calculation. That is to say,the figure is the below.

                    The case of [8+3],

          Block-operation                       Students' Thinking
   
                   лллллллл**  ***          1) looking at the complementing part of 
                                                        10                 
                                                    2) dividing 3 into 2 and 1
                                                    
                   лллллллллл   *             3) adding 2 and 8 equals 10
                                                        8+2=10

                   лллллллллл   л             4) adding 10 and 1
                                                       10+1=11