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# The Importance of Additive Thinking

According to research completed by Terezinha Nunes, Peter Bryant, Rossana Barros and Kathy Sylva (2011), “Mathematical reasoning and arithmetic did make independent contributions to the prediction of mathematical achievement; mathematical reasoning was by far the stronger predictor of the two. These predictions were specific in so far as these measures were more strongly related to mathematics than to science or English.”

According to the Elementary Mathematics Professional Learning website, when students are able to think additively, “Students are able to manipulate numbers by joining, separating, and comparing while engaging in flexible mathematical reasoning.  It is

• a capacity to work flexibly with the concepts, strategies and representations of addition and subtraction as they occur in a wide range of contexts. (mathematical reasoning)
• going beyond memorization of basic arithmetic skills
• the means to communicate additive understanding effectively in a variety of ways (for example, words, diagrams, symbolic expressions, and written algorithms).”

# Misconceptions

There are several prevalent misconceptions students hold around additive thinking. As you read the misconception statements below, can you determine when and why these misconceptions

• are developed?
• are interfering with developing new understandings?

##### Misconception 2:

You can’t take a bigger number away from a smaller number.

##### Misconception 3:

Subtraction is commutative. ie. 5-4 is the same as 4-5.

##### Misconception 4:

Keywords in word problems can be used to identify the operation that should be used to solve the question. ie. “Altogether” and “more” means add. “Less” means subtract.

# Students need to understand

You can manipulative numbers in order to make solving problems easier.

Addition and Subtraction include joining, separating and comparing.

There are three essential meanings of subtraction:

• Taking away: I have 6 candies I ate 2. How many do I have left?
• Part-whole: I have 6 candies. 2 taste like strawberries and the rest taste like blueberries. How many taste like blueberries?
• Comparison: I have 6 candies and 4 chocolates. How many more candies do I have?

We spend most of our time focusing on “taking away” but we need to focus an equal amount of time on all 3 meanings.

# High Leverage Strategies

Find strategies here. These are not yet full developed but are a work in progress.

# Further Learning

Explore ARPDC’s Elementary Mathematics Professional Learning site, focusing on its “Additive Thinking” section.