Making Sense of Adding

Do you have students that solve ALL problems using only the traditional algorithm? Do they stop and think about the strategy that could make the most sense for that particular problem?

Here are some strategies that support students in building understanding of what adding really means, before using “carry-on” as a set of rules to be followed to solve any problem.

It is essential for students to move from CONCRETE manipulative, to PICTORIAL strategies, to end in more ABSTRACT strategies. This progression will bring understanding and fluency, as students will have a variety of mental strategies to solve problems.

It is important to allow students to explore concrete manipulative to understand how to bundle ones into tens, and tens into hundreds when it is necessary.

When solving a problem with manipulative, students tend to start putting together the hundreds, then the tens, and the ones.

When students get more than 10 ones, they immediately re-group them and put them with the other place value that it belongs.

Then they count how many of each and find an answer. There are no rules or procedures, the concrete manipulative allows them to make sense of what they have in all.

After using these concrete experiences to make sense of addition with regrouping and making sense of place value, students can move into pictorial strategies. One strategy could be using the place value chart. Students can use real place value discs, or draw the place value discs.

Then they make a connection with the previous concrete strategy, and will know what to do with more than 10 ones.

Use the term “bundle” for students to understand that 1 ten is a bundle of ten ones.

Students can make a connection between using the manipulative and using this place value chart. It is not a new strategy, it is a connection. Same strategy different visual representation.

Now it is time to move into more abstract strategies but still based on place value, not based on a memorized set of rules. For example this expanded form strategy is another visual representation of the exact same thing they did using the place value chart.

Again, students will likely start by adding from left to right (unlike the algorithm), and will make sense of what to do with more than 10 tens, and more than 10 ones. They had the previous experience of “exchanging” the manipulative, and “bundling” the place value discs. They do not know what re-grouping even means, but they understand what the value of these numbers are in all.

Finally, students can make a connection with another visual representation. This abstract strategy is usually called partial sums.

Students can see the same values than before, they will start by adding 300 and 110, and can mentally solve using the visual representation they have built previously, and come up with 410, then mentally again add 13 by breaking it down into 10 and 3.

When students become fluent with these strategies, they will eventually make a connection with the traditional algorithm, and they will finally understand that “carry” is nothing more than “bundling”, but bundling makes sense.

Even if the math program you use in your school encourages students to solve 15+ addition problems using the standard algorithm, you could make a decision and create an environment of exploring the beauty of mathematics, and allow your students to solve 3 or 4 addition problems using CONCRETE-PICTORIAL-ABSTRACT strategies to really understand how math works, and how strategies based on place value are all connected!

“As educators we all share the goal of encouraging powerful mathematics learners who think carefully about mathematics as well as use numbers with fluency. But teachers and curriculum writers are often unable to access important research and this has meant that unproductive and counter-productive classroom practices continue.” Jo Boaler, Fluency Without Fear.