# Additive Thinking

# Stop Saying…

Sometimes we say things in order to help students **do** the math rather than **understand** the math. Those statements can create misconceptions and confusion when the statements no longer hold true at later grades.

##### “You can’t take a bigger number away from a smaller number”

##### or “When subtracting, the bigger number goes on top”

#### When does it fall apart?

This statement causes misconception at two levels.

Solve 24 – 7.

Solve 3 – 7.

#### Why?

**24 – 7: **When solving 24 – 19, they end up with 15. They start by figuring out 9 – 4 since, if you can’t take a bigger number away from a smaller number / the bigger number goes on top, you must switch the two numbers in order to make it work.

**3 – 7: **When solving 3 – 7, students will end up with 4. Students who have developed the misconception that “You can’t take a bigger number away from a smaller number” will apply this reasoning when working with subtracting integers.

Typically, teachers will use these phrases when they are trying to show students that you “put the bigger number on top” when subtracting. In younger grades, you do begin with the larger number. Beginning in grade 7, this is not always true when subtracting integers.

Take a moment to think about the number 5. What is it? Can you picture it in your head? How would you explain “5” to someone else?

Did you think about a collection of 5? 5 apples? 5 balls? What about the roman numeral V? A number is an abstract representation and can be hard to define.

A “numeral” is what we use to represent 5. A numeral can be any name or symbol used for representation. 17 is the numeral we use to represent the abstract concept of 17. XVII is also a numeral.

When you thought about the number 5, did you think about it as it relates to the numeral 51? Probably not. This is because in the numeral 51, 5 is a digit not a number…nor is 5 a numeral. There are 10 digits in the base ten system. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. When working with students, it’s important to reference the 5 as 5 tens or 50 but not as the “number” 5 when considering the number 51. If they see the 5 as the number 5 rather than a digit, it can cause misconceptions when working with addition and subtraction.

Numeral and number are often interchanged though number is the abstract concept and numeral is the symbolic representation. This seldom causes misconceptions. However, interchanging digit and numeral/number can lead to significant misconceptions.

#### Instead say:

“Our starting value/amount goes on top.” or “Our starting value/amount goes on the left.”

# “Borrow” when subtracting

#### When does it fall apart?

For some students, this causes immediate confusion and may be interfering with their understanding of subtraction.

#### Why?

Think about “borrowing” in real life. If someone borrows something from you, what is the expectation? You expect them to return it. In subtraction, you do not “give back” the 10.

#### Instead say:

Regroup, exchange or trade are reasonable replacements.

Make sure to read the next section as it is connected to this section.

##### “There aren’t enough ones so we have to regroup.”

#### When does it fall apart?

This is a statement that builds immediate misconceptions.

#### Why?

In the number 23, there are 23 ones. This is addressed in the Place Value section of this framework.

Could you solve 23 – 15 using base ten blocks without regrouping? Absolutely! Build the 23 using 23 ones. You’re not regrouping but you already have enough ones. Doing so makes it easy to remove 15 ones. We have been trained to build 23 using 2 tens and 3 ones. It takes less time to do so. However, we also need to remember that 23 doesn’t just have 2 tens and 3 ones. This will be addressed in the Additive Thinking section of this framework soon.

#### Instead say:

Say it this way.

##### rather t

# “Subtraction means take away”

#### When does it fall apart?

John has $15. Jill has $12. How much more money does John have?

#### Why?

When solving the question, did Jill lose money? Did you take away Jill’s money? Subtraction is much more than take away. This will be addressed in the Additive Thinking section of this framework.

#### Instead say:

When reading a subtraction question, say “minus” rather than “take away”. 6 minus 3 could be a “take away” question, a “part-whole” question, or a “comparison” question.

# “Line the numbers up to the right”

#### When does it fall apart?

Solve each of the questions by lining them up to the right when writing vertically:

- 15 + 23
- 73 – 15
- 4.3 + 2.1
- 21.3 + 4.25
- 3.2 + 6 + 1.47

#### Why?

“Lining up to the right” only consistently works when you are dealing with whole numbers. Students may also struggle with the phrase “line it up according to the decimal” in the last example.

#### Instead say:

Line up your numbers according to place value.