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

Overview

## Overview

This page focuses on the development of addition and subtraction as operations and ways to assess it. Developmentally addition and subtraction are deeply rooted first in number value and later in place value, which are discussed in more detail in those areas. Focus here is addition and subtraction as operations and the four ways they are used on numbers to develop the addition and subtraction concepts and outcome in the mathematical knowledge base.

## Development

Children learn addition and subtraction as they develop a number sense: counting, cardinality, number relationships (more, less, equal), ordinality, and place value.

As young children subitize (one, two,& three), memorize the counting numbers, conserve numbers, and understand cardinality, they can compare numbers as more or less, equal and sequence them. As they achieve greater accuracy using them they recognize the different values can be compared and operated on by being decomposed and composed in a hierarchical manner, which they initial view as number value and number sense, and begin to connect it to the operations of addition and subtraction.

Transitional activities that connect number value to the basic operations will assist in their learning this and as they do it will also enable them to begin to learn basic addition facts along with an understanding of the operations of addition and subtraction.

Students initially understand addition as more and subtraction as less; and later as joining and separating. With many experiences and activities, that include examples of the four different ways to represent addition and subtraction operations, they become experts in solving problems creatively and effectively using both interchangeably.

As the numbers students experience get bigger, they will naturally associate different strategies to compose (join) and decompose (separate) numbers and invent their own algorithms. This is even more true if, once they learn to count, they are pushed away from solving problems by counting, not taught to use touch points, and experience activities that encourage them to subitize, use skip counting, five as an anchor, ten and more, combinations of strategies, and eventually decomposing numbers left to right. With these strategies they will become more accurate, better problem solvers, and truly math literate.

Their experiences need to include problems and activities which will enable them to naturally incorporate the following ideas with each of the four different ways to represent addition and subtraction problems in problem solving situations to develop proficiency in the following addition and subtraction outcomes:

• Recognize different values of objects (subitize) with out counting by one. 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, arrow math ...
• 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.

## Four Types of Addition and Subtraction problems

Addition and subtraction are two ways to operate on two or more numbers to create a third number of equivalent value or to compare different number values. These two operations, are represented, in real life, in 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

These are described in the following chart. Another chart with assessment tasks and a scoring sheet follow further below.

# Four Types of Addition and Subtraction problems

## 1. Join Problems (start number + change number = sum or result)

### Missing Sum or Result Unknown (start number + change number = ____________)

1. Pete had 3 apples. Ann gave Pete 5 more apples, How many apples does Pete have now?
2. Sandi had 7 dimes. Mike gave her 4 more. How many dimes does Sandi have altogether?

### Missing Change Addend Unknown (start number + ____________ = sum or result)

1. Kathy had 5 pencils. How many more pencils does she have to put with them so she has 7 pencils altogether?
2. Sandi has 7 dimes. Mike gave her some more. Now Sandi has 11 dimes. How many did Mike give her?

### Missing Start or Initial Addend Unknown (____________ + change number = sum or result)

1. Bob got 2 cookies. Now he has 5 cookies. How many cookies did Bob have in the beginning?
2. Sandi has some dimes. Mike gave her 4 more. Now Sandi has 11 dimes. How many dimes did Sandi have to begin with?

## 2. Separate Problems(start - change = difference or sum or result)

### Resulting Difference or Sum Unknown (start number + change number = ____________)

1. Sandi has 11 dimes. She gave 4 dimes to Mike. How many dimes does Sandi have now?
2. Joe had 8 marbles. Then he gave 5 marbles to Tom. How many marbles does Joe have now?

### Missing Change Addend/Subtrahend Unknown (start number + ____________ = difference or sum)

1. Sandi had 11 dimes. She gave some to Mike. Now she has 7 dimes. How many did she give to Mike?
2. Fred had 11 pieces of candy. He lost some of the pieces. Now he has 4 pieces of candy. How many pieces of candy did Fred lose?

### Initial Addend/Minuend Unknown (____________ + change number = difference or sum)

1. Sandi had some dimes. She gave 4 to Mike. Now Sandi has 7 dimes left. How many dimes did Sandi have to begin with?
2. Karen had some word problems. She solved 2 of them. She still has 3 word problems. How many word problems did she have to start with?

## 3. Part - Part - Whole Problems (part + part = whole)

### Missing Whole or Sum Unknown (part + part = ____________

1. There are 6 boys and 8 girls on the volleyball team. How many children are on the team?
2. Bobbi has 3 dimes and Azzie has 5. If they put them together how many do they have?
3. Mike has 5 pennies and 10 dimes. How many coins does he have?
4. Mike has 5 dimes and Sandi has 10 dimes. They put there dimes into a piggy bank. How many dimes did they put into the bank?
5. Sara has 6 sugar donuts and 9 plain donuts. Then she puts them all on a plate. How many donuts are there on the plate?

### Missing Part Unknown (part + ____________= whole) or( ____________ + part = whole)

1. Carlos had 8 quarters in his pocket. He reaches in and pulls out four. How many are still in his pocket?
2. Brian has 14 flowers. Eight of them are red and the rest are yellow. How many yellow flowers does Brian have?
3. Bobbi and Sandi put 12 dimes into a change purse. Sandi put in 8. How many did Bobbi put in? or Mike and Sandi put 11 dimes into a piggy bank. Mike put in 7 dimes. How many dimes did Sandi put in?
4. Mike has 10 coins. 7 of his coins are dimes, and the rest are pennies. How many are pennies?
5. Joe and Tom have 8 marbles when they put all their marbles together. Joe has 3 marbles. How many marbles does Tom have?

## 4. Compare or Equalize Problems(one value + or - difference = second value)

### Difference Unknown(one value + or - difference = second value) (one value - second value = difference)

1. Joe has 3 balloons. His sister Connie has 5 balloons. How man more balloons does Connie have than Joe?
2. Janice has 8 sticks of gum. Tom has 2 sticks of gum. Tom has how many sticks less than Janice?
3. Mike has 11 dimes and Sandi has 5. How many more dimes does Mike have than Sandi?
4. Mike has 11 dimes. Sandi has 5 dimes. How many fewer dimes does Sandi have than Mike?

### Larger Unknown (one value + difference = second value)(second value - difference = first value)

1. Luis has 6 goldfish. Carla has 2 more goldfish than Luis. How many goldfish does Carla have?
2. Dad bought 18 bottles of milk on Sunday and on Monday he brought 6 bottles less. How many bottles did he bring on Monday?
3. Mike has 4 more dimes than Sandi. Sandi has 7 dimes. How many dimes does Mike have?
4. Sandi has 4 fewer dimes than Mike. Sandi has 7 dimes. How many dimes does Mike have?
5. Jane has 7 dolls. Ann has 3 dolls. How many dolls does Jane have to lose to have as many as Ann?
6. Connie has 13 marbles. If Jim wins 5 marbles, he will have the same number of marbles as Connie. How many marbles does Jim have?

### Smaller Unknown (one value + difference = second value)(second value - difference = first value)

1. Maxine has 9 sweaters. She has 5 sweaters more than Sue. How many sweaters does Sue have?
2. Jim has 5 marbles. He has 8 fewer marbles than Connie. How many marbles does Connie have?
3. Sandi has 4 fewer dimes than Mike. Mike has 11 dimes. How many dimes does Sandi have?
4. Mike has 4 more dimes than Sandi. Mike has 11 dimes. How many dimes does Sandi have? Susan has 8 marbles.
5. Fred has 5 marbles. How many more marbles does Fred have to get to have as many marble s as Susan has?

## Where would you put these?

1. There were 6 boys on the soccer team. Two more boys joined the team. Now there is the same number of boys as girls on the team. How many girls are on the team?
2. There were 11 glasses on the table. I put 4 of them away so there would be the same number of glasses as plates on the table. How many plates were on the table?
3. There were some girls in the dancing group. Four of them sat down so each boy would have a partner. There are 7 boys in the dancing group. How many girls are in the dancing group?
4. Jim has 7 quarters. Ann has 3 quarters. How many quarters does Jim have to spend to have as many as Ann?

## Assessments

Assessments to assess the development of addition and subtraction, with suggestions, sample scripts, summary comments or outcomes, with scoring sheets. Categories include:

### Subitize, pattern recognition, or quick addition for number value -

#### Dot plates with one color of dots

Materials

Dot plates with one color of dots with dots 1-10. or Electronic dot flash
[1-5] [1-10] file.

Directions

Show students dot plates of one color 1-10. Tell them you will flash each plate for a second and they are to tell you the number of dots on each plate.

• Flash plate, wait ... repeat ...
• For numbers beyond three. Students most likely will construct the value with addition. Discussion of how they arrive at the total number of dots will provide information on their understanding of sums to 10.
• Stop any individual assessments if a child is not able to respond or responds with random answers.

#### Dot plates with two colors of dots

Materials

Dot plates with two colors of dots with dots 2+1, 3+1, 4+1, 4+2, 4+3, 5+1, 5+2, 5+3, 5+4, 5+5, 6+1, 6+2, 6+3, 7+1, 8+1, 9+1

Directions

Show students dot plates of two colors with sums 3 -10. Tell them you will flash each plate for a second and they are to tell you the number of dots on each plate.

• Flash plate, wait ... repeat ...
• Ask how they got the number they did.
• Share different ways to sum the dots.
• Stop any individual assessments if a child is not able to respond or responds with random answers.

• Students recognize the following patterns or sums of dots two seconds or less.
• 2+1, 3+1, 4+1, 4+2, 4+3, 5+1, 5+2, 5+3, 5+4, 5+5, 6+1, 6+2, 6+3, 7+1, 8+1, 9+1

### Hierarchical inclusion for five

Materials

Five objects, cup,

Directions

Put five objects in a cup. Ask the student if they could use the objects in the cup to show someone what four objects would look like.

• What other numbers could you show a person using the objects in the cup.
• Dump the objects out and place the empty cup in front of the student.
• Stop any individual assessments if a child is not able to respond or responds with random answers.

• Shows that knows hierarchical inclusion: Knows that a number of objects can make a set of objects from zero to and including the number of objects.
• Can show objects of five, four, three, two, one, zero

### Five as anchor for addition and subtraction

Materials

nine objects

Directions

• Put nine OBJECTS on the table. Arrange five into a pattern that the students would recognize as five.
• If the student does not recognize a pattern of five have the student count out five and set them aside.
• Cover the five OBJECTS with your hand or a bowl and ask the student how many objects there are all together.
• Stop any individual assessments if a child is not able to respond or responds with random answers.

Responses

• Can’t do the task Moves hand or bowl and counts by ones Counts on to nine

Summary

Uses counting on strategy

Continue

• Put eight OBJECTS on the table. Ask the student how many objects there are on the table.
• Cover five OBJECTS with your hand or a bowl and ask the student how many objects you have under your hand or bowl.

Responses

• Moves hand or bowl and counts by ones
• Counts back from eight to five
• Subtracts three from eight

Summary

Uses counting back strategy

Continue

• Pick up a number of OBJECTS, with out the students seeing how many.
• Hide them in your hand.
• Put four or five on a plate.
• Tell the student that there are NUMBER of OBJECTS in your hand.
• Ask. How many are there in your hand and on the plate?

Response

Student can count on for

Summary

Uses counting on strategy

### Ten plus leftovers with Ten frames and anchor of ten

Give students a ten frames plus cards or a complete ten frame and partial ten frame (10-20) and ask them to sequence them from least to most. Stop any individual assessments if a child is not able to respond or responds with random answers.
Response
Student counts most cards and has inaccuracies. Student counts some, uses visual pattern for some, and sequences cards. Student mostly uses visual pattern and sequences cards quickly and accurately.
Other
Summary
Assessment for Number Value (Inclusiveness)

### Combinations of addends to 12: hierarchical inclusion

Directions

• Start with a practice session with combinations of numbers 3 and 4.
• Use the number of beans as the sum.
• Count that number of beans into your hand.
• Hide the beans behind your back and distribute them into both hands.
• Show the child one hand and ask how many are in the other hand.
• Repeat for all possible combinations of whole numbers (6, 7, 8, 9, 10, 11, & 12) in a random order. For example: for five beans (2 + 3, 0 + 5, 1 + 4, 5 + 0, 3 + 2, 4 + 1).
• If a student misses one, you can try it again and if they get it right the second time, then it can count as right with a prompt.
• Stop any individual assessments if a child is not able to respond or responds with random answers.

How many beans are in this hand (hold up the other hand).

• Repeat for all possible combinations of whole numbers (6, 7, 8, 9, 10, 11, & 12) in a random order. For example: for five beans (2 + 3, 0 + 5, 1 + 4, 5 + 0, 3 + 2, 4 + 1).
• If a student misses one, you can try it again and if they get it right the second time, then it can count as right with a prompt.
• Record on scoring sheet

Yes means all right on first or second attempt.

• Combinations of 5
• Combinations of 6
• Combinations of 7
• Combinations of 8
• Combinations of 9
• Combinations of 10
• Combinations of 11
• Combinations of 12

### Name of Evaluator:

Assessors note The child can either do it or not do it. It's important to give prompts to help the child feel successful, but if the task is completed correctly, the appropriate column must be recorded. The child must be able to complete the task without a prompt to receive a √.

Comments might include: used join, separate, part-part whole, compare, count on, count all...

Materials 20 pennies, beans, or small objects

Addition and subtraction Do with no prompts
Do with one prompt
1
Can or can not do with one or less prompts
-
1. Join
Sandra had 8 pennies. George gave her 4 more. How many pennies does Sandra have altogether? (8 + 4 = ?)
Sandra had 7 pennies. George gave her some more. Now Sandra has 13 pennies. How many did George give her? (7 + ? = 13)
Sandra had some pennies. George gave her 6 more. Now Sandra has 15 pennies. How many did Sandra have to start? (? + 6 = 15)
2. Separate
Sandra had 11pennies. She gave 5 pennies to George. How many pennies does Sandra have now? (11 - 5 = ?)
Sandra had 12 pennies. She gave some to George. Now she has 9 pennies. How many did she give to George?
Sandra had some pennies. She gave 5 to George. Now Sandra has 9 pennies left. How many pennies did Sandra have to begin with?
3. Part-part whole
George has 5 pennies and 10 pennies. How many coins does he have? (5 + 10 = ?)
George has 13 pennies. Five pennies are in his right hand and the rest are in his left. How many pennies are in his left hand?
4. Compare
George have 11pennies and Sandra has 8 pennies. How many more pennies does George have than Sandra? (11 - 8 = ? or 8 + ? = 11)
George have 13 pennies. Sandra has 9 pennies. How many fewer pennies does Sandra have than George? (13 - 9 = ? or 9 + ? = 13)

## Hierarchical inclusion scoring sheet

Combinations of addends to 12: hierarchical inclusion Completed
√, *, -
Task: Use the number of beans as the sum. Count the beans into your hand. Hide the beans behind your back and distribute them into both hands. Show the child one hand and ask how many are in the other hand. Repeat for all possible combinations of whole numbers in a random order.
(2 + 3, 0 + 5, 1 + 4, 5 + 0, 3 + 2, 4 + 1)

1. Combinations of 5
2. Combinations of 6
3. Combinations of 7
4. Combinations of 8
5. Combinations of 9
6. Combinations of 10
7. Combinations of 11
8. Combinations of 12
Summary

## Addition and Subtraction Whole Number Calculation Algorithmic Scoring Guide

1
• Counts on for values to 20
• Uses five and ten as anchors

2
• Can tell one more and one less.
• Can tell two more and two less.
• Compose and decompose numbers to 20
• Mentally Adds and subtracts values to 20

3
• Add and subtract from left to right by decomposing double digit numbers to tens and ones.
• Works with tens, then works with ones, and then adds tens and ones.

4
• Adds on or subtracts from a two digit number
• Mentally Adds and subtracts values to 100 including regrouping for addition and subtraction
• Adds and subtracts money values by composing and decomposing into coin values.

5
• Constructs and deconstructs addition and subtraction problems to 10 000.
• Regroup with addition fairly easily
• Regrouping with subtraction is cumbersome with problems requiring multiple regroupings

6
• Constructs and deconstructs addition and subtraction problems of any size with and without regrouping until confidence in the accuracy of the solution is attained.

## Research:

1. Be aware of emotional propaganda techniques that claim any instruction, but traditional memorization of procedures, is soft or is dumbing down mathematics instruction. However, the mindless use of algorithms is the real dumbing down.
2. 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).
3. 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. Like, 23 cards and 14 cards would be decomposed into 20 + 3; and 10 + 4; adding from left to right 20 + 10 and 3 + 4; and finally adding 30 + 7.
4. A study of Dutch second graders found two left to right strategies more effective than traditional algorithms and enhanced student's mental math flexibility .
• Strategy 1 left to right or split method or 1010 strategy. For example: 46 + 23 is solved by decomposing 40 + 6 and 20 + 3, then adding 40 + 20 to get 60 and then adding 6 + 3 to get 9 and adding 60 + 9 to get 69.
• Strategy 2 left to right or jump method or N10 strategy. For example: 46 +23 is solved by decomposing the smaller number to 20 + 3, then adding 20 to the larger number 46 to get 66 and then adding 3 to get 69.
Source: Mental Strategies and Materials or Models for Addition and Subtraction up to 100 in Dutch Second Grades. Meindert Beishuizen. Journal for Research in Mathematics Education. 1991, Vol 24, No. 4, p. 294-323.
5. 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.
6. When this problem (504 - 306) was given to groups of second and third grade students: Groups taught addition and subtraction with an algorithm and groups 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.
7. 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)

## Last words

Algorithms vs mathematizing: People who criticize the way mathematics is taught with broad general terms like: new mathematics, fuzzy math, soft math, or claim it is dumbing down students learning of mathematics are not aware of the many possible ways a person can connect mathematical ideas to other mathematical ideas and to life. They are most likely good at memorizing facts and procedures and limiting their use of mathematics to the use of algorithms, which is the real dumbing down.

At the beginning of this article addition and subtraction is described by the four different ways we represent these operations in our world. Teachers often try to help students by consistently referencing addition as joining and subtraction as separating. However, doing this is a dangerous idea, because to be mathematically literate students must understand all four ways addition and subtraction can be represented to solve addition and subtraction problems.

In fact sometimes, it is impossible to know for sure if a person is adding or subtracting when solving a problem. 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 a particular 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, which 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, using many addition and subtraction strategies. 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.