## Thursday, February 25, 2010

M2N2. Students will build fluency with multi-digit addition and subtraction.
a) Correctly add and subtract two whole numbers up to three digits each with regrouping.

Understanding the addition and subtraction algorithms is an important goal for Grade 2 GPS. However, what is the point in teaching algorithms? For that matter, what does it mean to teach algorithms? Some believe that we are to teach children the algorithms that are currently conventional in the United States. Others believe that teaching algorithms mean to help students make their own methods into a written procedures. This perspective doesn't mean teach children different algorithms. Many of today's textbooks include "alternative" algorithms, and some teachers will teach their students each one of them, thinking children can pick what is the easiest to them. However, the source of the algorithms is still outside of the children. Thus, although they are given some choices, algorithms are still imposed on them. Helping children make their own methods into a written procedure means the source of the algorithm is within children.

Take, for example, subtraction. Ask children to make 85 with base-10 blocks. They have no problem representing this number with 8 longs and 5 units. If you then ask them to give you 32 from what they have, they will give you 3 longs and 2 units, in that order. Ask them to make 82. Then, ask them to give you 46, they often will give you 4 longs and then pause. They see that they don't have enough units to give you. At that point some will ask if they can trade one of the remaining longs with 10 units. Once they trade the long for 10 units, they can give you 6 units, leaving them 3 longs and 6 units.

This process can be made into a written procedure like this:

This "algorithm" will work with any numbers, however long they are. If the purpose of teaching algorithms is for students to have a reliable and efficient computational method, there is nothing wrong with this algorithm. So, is that the only reason we teach algorithms?

Actually, I think there is another very important point we should help students understand when we teach addition and subtraction algorithms. That idea starts when we study simple addition and subtraction problems like 40 + 30 or 80 - 20 in Grade 1 as I alluded in the last entry. Addition problems like 40 + 30, 400 + 300, 4000 + 3000 are all related to 3 + 4 because in each case we are putting together 3 of something with 4 of the same thing. On the other hand, 300 + 40 does not relate to 3 + 4 because even though we may have 3 of 100 and 4 of 10, 3 and 4 are referring to different units. One of the important ideas of addition and subtraction is that you can only add or subtract numbers if they are referring to the same unit - we cannot add apples and oranges. To make the paper and pencil addition/subtraction easier, we arrange the numbers vertically, one on top of the other. When we do this, we also line up the place values of each number. By doing so, we know we can add or subtract the numerals in each column because they are referring to the same unit.

Although many textbooks will include the vertical notation even while studying the basic addition and subtraction facts, the importance of the notation is about this idea of lining up the place values so that we can add or subtract like numerals. With the basic facts, such an idea is rather implicit.

If we can help students make this understanding explicit, they can use it when they study addition and subtraction of decimal numbers and fractions. When we "line up the decimal points," what we are really doing is to line up the place values. Thus, we are simply following the same idea. When we add or subtract fractions, we need a common denominator, because the numerators will tell us how many while the denominators tell us what we are counting. Thus, in order to add two fractions, we have to have the two numerators referring to the same unit, or the same denominator. And, when we add, what is being counted do not change, thus the denominator stays the same. Thus, teaching of addition/subtraction algorithms is when we lay this important foundation - not just teaching them an efficient and reliable computational strategies.