# Addition and Subtraction Concepts, Development, Research, Activities, and Assessment

- Big idea
- Research information
- Development notes
- Addition concepts
- Subtraction concepts
- Addition and subtraction related properties
- Addition and subtraction algorithms

## 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, and 4. comparing or equalizing number values. See examples for each and analysis of the differences.

## Research bits:

- Children who are not taught algorithms become better at mathematics. Those that are taught algorithms rarely use strategies 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). Or other strategies that would be more efficient, depending on the value of the numbers, to be added or subtracted.
- With a good teacher students can learn a variety of strategies as well as an algorithm. 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?" Most will draw tallies and cross out to arrive at the answer. Later, as children are given the opportunity to invent algorithms they usually split the numbers into place value and develop not only usable efficient algorithms but an understanding to go with them.
- The following problem (7 + 52 + 186) was given to different groups of second graders and 45% of the students solved the problem without using an algorithm, 26% used part of an algorithm, and 12% used an algorithm. The following problem (504 - 306) was given to different groups of students, some that were taught addition and subtraction with an algorithm and some that were taught without an algorithm. 74% of the second and 80% of the third graders taught without algorithms got the right answer compared to 42% of the second and 35% of the third graders taught with an algorithm.
- How do you get students to develop efficient strategies? Students who were taught relationships in which automaticity was the goal significantly outperformed the traditionally taught students in being able to produce correct answers to basic addition facts within three seconds. 76% to 55%. (Research from Fosnot…)
- Many critiques of this type of mathematics have used emotional propaganda techniques by saying that the new mathematics is soft or is dumbing down mathematics instruction. 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. For example: as students learn number value, they will develop the understanding of one more, one less, and the sequence of numbers. For example on activity to develop this is the use of dot plates to subitize cardinality and learn one more and one less, begin to memorize addition facts and understand hierarchical inclusion. Once these are achieved students will be better able to construct and deconstruct numbers with an understanding of how different number values are related so they can accurately use the operations of addition and subtraction.

Students, who are discouraged from counting or using touch points, will naturally begin to decompose and compose numbers. Students will naturally use decomposition and composition to invent their own algorithms - if they are provided an enriched environment and plenty of encouragement.

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 is represented. Additionally teachers should know any addition and subtraction problem can be solved by by either adding or subtracting. Review the example on this page and think about how interchangeable addition and subtraction really are 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 to include in normative testing, because they will efficiently sort students into different levels.

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 and flexibility in 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 concepts

- 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, subtraction and related properties

- 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 algorithms

- 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