Addition and Subtraction Development, Research, Activities, and Assessment

Overview
• Big idea
• Research information
• Development notes
• Subtraction concepts
• Addition and subtraction related properties

Big idea (generalization) for addition and subtraction

Addition and subtraction are two ways to operate on two or more numbers to create a third number of equivalent value. The different ways these operations are represented in real life can be classified into four groups:

1. Combination of number values
2. Separation of number values
3. Part-part-whole relationships of number values
4. Comparing or equalizing number values

Research bits:

• Children who are not taught algorithms become better at mathematics. Those that are taught algorithms rarely use more efficient strategies, more appropriate for the value of the numbers to be added or subtracted. Such as:
• Making nice numbers (368 + 204 = 368 + 200 + 4 = 568 + 4=572).
• Keeping the whole (71 - 36 = {subtract 1 from both to get} 70 - 35 = 35) (342 - 37 = {add 3 to both to get} 345 - 40).
• With a good teacher students can learn a variety of strategies as well as algorithms. For example, if students are given the following problem: "I went to the store with \$32.00 and spent \$17.00, how much do I have left?" Younger children will draw 32 tallies, cross out 17, and count those left to arrive at the answer. Later, children usually decompose numbers into place value (tens and ones) and develop algorithms that they understand. 23 cards and 14 cards would be decomposed into 23 + 3; and 10 + 4; adding from left to right 20 + 10 and 3 + 4; and finally adding 30 + 7.
• When different groups of second graders were given this problems: (7 + 52 + 186) 45% of the students solved the problem without using an algorithm, 26% used part of an algorithm, and 12% used an algorithm.
• When this problem (504 - 306) was given to groups of students: A group who had been taught addition and subtraction with an algorithm and a group who were taught without an algorithm. 74% of the second graders and 80% of the third graders, taught without an algorithm, got the right answer. Where as 42% of the second graders and 35% of the third graders, taught with an algorithm, got the the problem right.
• Students who were taught relationships in which automaticity was the goal produced more correct answers to basic addition facts within three seconds (76% to 55%) than students who were taught traditionally. (Fosnot)

There are critiques who use emotional words as propaganda techniques; terms like new mathematics, fuzzy math, soft math and claim it is dumbing down students learning of mathematics. However, the mindless use of algorithms is the real dumbing down.

Development notes

Children learn addition and subtraction based on their understanding of number value. Tthey memorize the counting numbers and soon realize they are sequenced with each related in an increasing order. This order develops as an understanding of one more and then one less and the ability to put a number with a set of objects, which eventually becomes cardinality. The relationship of numbers as more or less and cardinality can be decomposed and composed in a hierarchial manner eventually is seen as the operations of additiona and subtraction.

These relationships are developed when students experience activieis such as; dot plates to subitize cardinality and learn one more and one less, begin to memorize addition facts and understand hierarchical inclusion.

Students, who are given activities to quantify groups of objects both before and after combining or separating different groups of objects will naturally compose and decompoe the numbers and invent their own algorithms. If, once they learn to count, they are are pushed away from counting, not taught to use touch points, and encouraged to use skip counting, five as an anchor, ten and more, and eventually decomposing numbers left to right.

The environment needs to include problems and activities which will enable students to naturally incorporate the following ideas when solving problem.

• Recognize different values of objects (subitize). Dots on plates, dice, dot cards, groups of objects ...
• Respond with one more and one less for an initial value of objects. Numbered dots on a wall, dots on plates, dice, dot cards, groups of objects, ten frames, hundred chart, ...
• Respond with two more and two less for an initial value of objects. Numbered dots on a wall, dots on plates, dice, dot cards, groups of objects, ten frames, hundred chart, ...
• Count two separate sets of objects, slide them together into one group, and then count the new group to find how many altogether.
• Find out how many objects are in two separate sets of objects by counting on from the total number in one group. Rolling two dice, subitize the first number and count on from it for the value of the second die to find the total dots on the two dice. Same for dot cards, groups of objects, ten frames, numbered dots, hundred chart ...
• Find out how many objects are left in a set of objects by counting back from the total number in an initial group. Groups of objects, ten frames, numbered dots, hundred chart ...
• Decompose numbers into smaller addends, commute them and compose them (find the sums). Examples - two die with a roll of sixes, decompose them into 5, 5, 1, 1, and compose them into 5 + 5 = 10; 10 + 2 = 12.
• Decompose and compose sums less than 20.
• Decompose and compose for subtracting differences less than 20.
• Add and subtract values greater than 20 by working left to right - decomposing into tens and ones, adding or subtracting tens, then adding or subtracting ones, and then adding or subtracting the tens and ones.
• Adding on with two digits and subtracting from is the last step so that students can mentally add and subtract all sums and differences less than 100. First and second grade students will, on their own invent an algorithm for additional and subtraction by this deconstruction and construction process. For example: 46 + 23 by deconstructing 46 and 23 into 40 + 6 and 20 + 3, then adding the 40 and 20 and then the 6 and 3 and then the 60 and the 9 getting 69.
• Later, students will either discover or it can be suggested they do not need to deconstruct the initial number: 46, but can deconstruct the second 23 to 20 + 3. Then add on from: 46 + 20 to get 66 and then add on the 3 to get 69.
• Similarly, students between first and third grade will invent an algorithm for subtracting two digit numbers. First by deconstructing and constructing problems like: 47 - 23. Again decompose into 40 and 20, subtracting 20 from 40 to get 20, and then subtracting 3 from 7 and have 4 left. Recognizing that all of 23 has been subtracted and the 4 is part of the original 47, they will add back the four to the 20, therefore, taking 20 + 4 and getting 24.
• Later, students will either discover, or it can be suggested, that the first number not be deconstruct: 47, but to deconstruct the second 23 to 20 + 3. Then subtract the 20 from the 47 to get 27 and finish by subtracting 3 from the 27 to get 23. Subtraction is more difficult and if students don't have a very good understanding of number value and subtraction, then it is extremely difficult.
• Students will eventually discover that all addition and subtraction must account for the place value of each number as it is composed or decomposed to arrive at a sum or difference. Students should be encouraged to decompose numbers into expanded notation, based on place value, and then add or subtract without regrouping.
• When students are aware of the need to add and subtract according to place values in expanded notation, then continue using expanded notation with numbers that need regrouping.

At the beginning of this discussion addition and subtraction was described in four different ways they are represented in real life. To keep the categories less complicated addition was referenced as joining and subtraction as separating. However, doing this is a dangerous idea since it is important students learn all four ways addition and subtraction can be represented can be used for both addition and subtraction. In fact it is impossible sometimes to know for sure if a person is using one or the other to arrive at an answer. For example. If I ask you the difference between 18 and 15? Did you think of three before you even thought to use addition or subtraction? Thus, making it difficult if not impossible to know which operation was used. So be it.

Teachers should understand that any addition and subtraction problem can be solved with both addition and subtraction. Review these examples and think about how interchangeable addition and subtraction can be when operating on numbers.

This should raise an important question for every math teacher. Do curriculum developers or text book authors take similar short cuts? How many of the four ways and the subcategories of subtraction and addition are included in your math curriculum or text book? You can bet the ones that are not represented have been discovered as good types of problems to include in normative testing. Why? You ask, because they will efficiently sort students into different levels, whcih is the purpose of all normative tests.

If students are presented with problems and encouragement in developmentally appropriate ways to understand, they will, usually by fourth grade, invent a traditional addition and subtraction algorithm along with flexibility for selecting from a variety of ways to add and subtract efficiently.

Historically we should recognize Constance Kamii who first published ideas on how students' reinvent algorithms in her book Young Children Reinvent Arithmetic (1985. second edition 2000). She built on Piaget's development of understanding.