M118 Additive Reasoning p. 1

Chapter 9: Additive Reasoning

Additive reasoning is the first organized attempt children make to

understand the adult system of operations with numbers. Up to this point the work

children have done with number has been to try to understand and work with the

system. Early number work focuses on counting, understanding order, and

understanding quantity. With additive reasoning students begin to see some of the

interesting possibilities for joining, separating, and comparing numbers.

Additive reasoning is addition and subtraction, but conceptually it includes

far more. Researchers studying the mathematical content in addition and

subtraction, and the way young learners interact with it, have say there are four

important elements in additive relationships:

• Joining

• Separating

• Part/part whole

• Comparing (directly)

As the name suggests, joining is the act of putting quantities together to

achieve a larger sum. Problems that involve joining always have one missing piece.

Either one of the quantities to be joined is missing, or the result is missing. Here are

two examples:

Sam had 6 baseball cards. He got some more. Now he has 10. How many did he get? Maria walked 4 blocks, stopped, and then walked 6 blocks more. How far did Maria walk?

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The first problem is a missing quantity problem. This is sometimes referred to

mathematically as a "missing addend" problem. It can be solved, of course, using

subtraction. Young learners, however, seldom take that approach. This element of

additive reasoning is called "joining" because this is the conceptual strategy that

children use to solve when they ask, "What can I join to 6 to get 10?"

The second problem is a missing result problem, probably the most common

type of joining problem students encounter. Missing result problems are the

foundation for many important skills (like number facts) but they are often over

emphasized. The result of this over emphasis is that students develop a

misconception about the equal symbol. Rather than understanding it as a symbol

that means that both sides are "balanced", students take the symbol to mean "and

the answer is...". This leads to a very common error in the test question:

6 + 4 = ___ + 5

where students in the U.S. frequently put "10" in the blank space. For this reason it's

important that students working on additive reasoning get the opportunity to work

on joining problems and equations that are not always of the "result missing"

variety.

Mathematically, separating problems are quite similar to joining problems. In

a sense both of them can be characterized by the part/part/whole (or partitioning)

relationship. The thinking in separation, however, is closer to the idea of pulling

apart. This kind of thinking is conceptually related to the operation of subtraction.

Here, as with joining, there are some common misconceptions that develop as a

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result of the limited opportunities students have to explore the concept fully.

Separating (and subtraction) is often conceptualized (by students and

teachers) as "take away". This is a very useful analogy for subtraction, but it has

limitations. Separating is a much better way to think of subtraction because it covers

a much broader range of possibilities. Putting into two (or more) groups is a way

that quantities can be separated without removing anything. Consider the problems:

The soccer team had 18 players. The coach separated the team into two squads. One squad had 10 players. How many did the other have? There are 20 candies in a box. 8 of them have nuts. How many don't have nuts?

The idea of separating leads naturally into the part/part/whole relationship

which brings together joining and separating as well as addition and subtraction.

Part/part/whole thinking in additive reasoning represents a conception that

is inclusive of both joining and separating approaches to computation. It is the idea

that a larger sum is made up of two smaller quantities:

23

14 9

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The part/part/whole understanding is the recognition that partitioning a

quantity into groups creates mathematical relationships:

9 + 4 = 23

4 + 9 = 23

23 – 9 = 4

23 – 4 = 9

The relationships can be seen as joining in some instances, as separating in others.

Part/part/whole expresses the connectedness of joining and separating. It is a more

complete way of viewing the additive relationship. Unfortunately, students rarely

get enough experience viewing addition and subtraction from this perspective.

When the numbers get large enough students have difficulty seeing the relationship

at all. A first grade student, for example, who understands that 16 can be broken

into parts containing 9 and 7, will often flounder in third or fourth grade when

confronted with breaking 245 into 190 and 55. As a result, many struggling learners

see addition and subtraction as mechanical operations, rather than the expression of

numerical relationships.

Comparison is an aspect of additive reasoning that differs slightly from the

other aspects in that it asks student to judge in absolute terms between two

quantities. There are also comparisons that are made in proportional terms (more

on this later).

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Young children don't often have difficulty telling which quantity has more (or

less). Where they often get stumped is on the question: How many more?

As adults, we can see that this could be answered from either an “adding on”

or a “subtracting off” approach. For a young learner, though, the idea that the red

stack includes the quantity in the green stack and some more is a novel concept. To

develop this idea conceptually requires significant experience with both models and

problem solving. Often learners struggle because they are rushed to use a

subtraction algorithm when comparing, without understanding why that works.

What struggles do students have with additive reasoning?

Children can have a wide variety of struggles with addition and subtraction.

Three of the most common include:

• an unclear sense of magnitude.

• using numbers without grouping: 14 is a number by itself – not a group of ten and four ones • regrouping and place value

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An unclear sense of magnitude means that students are not really sure how

much greater one number is from another. When asked to put a “3” on a blank

number line between 0 and 10, for example, many students will put 3 in the middle.

?

0 3 10

Even older students have difficulty with magnitude. When told that a million

seconds is 11 ½ days, and asked how long a billion seconds would be, many

students give answers that are small multiples of 11 – like three or four months.

They do not have a sense of how much bigger a thousand times is.

Another common difficulty that young students have is not recognizing the

groups of ten that make up place value. Students may conceptualize 18 as a group of

ten and 8 ones or they may think of “18” as a symbol that means 18 individual units.

When reasoning additively, students must learn to create groups of 10 whenever

they can. They also need to make the connection between the number in the tens’

place, and its meaning as the “number of groups of ten.”