Addition and Subtraction Concepts, 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 are:

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 as having four different ways to be represented, which are analyzed at this page. To keep the above list 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.

Teachers should understand that any addition and subtraction problem can be solved with both addition and subtraction. When reviewing the example on this page think about how interchangeable addition and subtraction really 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 or 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. She built on Piaget's development of understanding.

• Addition is the joining of groups (sets).
• Addition can be used to solve joining problems.
• Addition can be used to solve separation problems.
• Addition can be used to solve part - part - whole problems.
• Addition can be used to solve comparison or equivalent problems.

Scoring rubric

Subtraction concepts

• Subtraction is separation of a set from another set.
• Subtraction is the removal of a group.
• Subtraction is the joining of groups.
• Subtraction can be used to solve part-part-whole problems.
• Subtraction can be used to solve comparison and equalization problems.
• Subtraction is how much is left.
• Subtraction is how much is missing.
• Subtraction is how much more.
• A number can be represented as a difference with different groups of numbers (subtrahends and minuends)

• Addition and subtraction are related.
• Numbers can be operated on with addition and subtraction in different groups of numbers.
• Addition has the commutative property.
• Addition has the associative property.
• Addition has the identity property for 0.
• Knowing the addition facts of ten helps to use ten as an anchor for addition and subtraction.
• Addition, subtraction, and comparing operations can be thought of as measurement or using a ruler.

Scoring rubric for regrouping

Addition and subtraction algorithm related concepts

• One more and one less is the value before and after a counting sequence.
• Two more and two less is the second value before and after a counting sequence.
• All numbers in a counting sequence can be thought of a distances relative to the amounts being added or subtracted.
• Two separate sets of objects can be added together by counting each set, sliding them together into one group, and then counting the new group to find how many altogether.
• The total number of objects in two separate sets of objects can be found by using the value of one set and counting on from that total number.
• The total number of objects left in a set of objects can be found by counting back from the total number in the initial group.
• Numbers can be added by decomposing numbers into smaller addends, commuting them and composing them with the different numbers. Example - two die with a roll of sixes, decompose them into 5, 5, 1, 1, commute and compose them like 5 + 5 = 10; 10 + 2 = 12.
• Values greater than 20 can be added or subtracted from left to right by decomposing into tens and ones, adding or subtracting tens, then adding or subtracting ones, and then adding or subtracting the tens and ones. 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.
• Values greater than 20 can be added or subtracted from left to right by starting with an initial number: 46, deconstruct a second 23 to 20 + 3. Then add on from: 46 + 20 to get 66 and then add on the 3 to get 69.
• 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. This can be shown if numbers are decomposed into expanded notation, based on place value, and then each expanded number can be added or subtracted with another number with the same place value. If numbers can't be operated on, then the numbers can be regrouped into different multiples of ten as needed using expanded notation to check for accuracy.
• Addition and subtraction requires numbers be operated on with respect to their place values.

Scoring rubric for addition and subtraction algorithms