If you are thinking about how we should develop an understanding of adding and subtracting integers, it is very important to first consider how primary students should be developing adding and subtracting natural numbers (positive integers).
Of the two operations, subtraction seems to be the one that many of us struggle to know the types of experiences our students need to be successful. So let’s take a look at subtraction for a moment.
Subtraction can be thought of as removal…
We had 43 apples in a basket. The group ate 7. How many are left? (43-7 =___)
Or subtraction can be thought of as difference…
I had 43 apples in a basket this morning. Now I only have 38. How may were eaten? (43-___=38 or 38+____=43)
Each of these situations requires different thinking. Removal is the most common method of teaching subtraction (it is simpler for many of us because it follows the standard algorithm – but it is not more common nor more conceptual as difference). Typically, those who have an internal number line (can think based on understanding of closeness of numbers…) can pick which strategy they use based on the question.
For instance, removing the contexts completely, think about how you would solve these 2 problems:
Think for a moment like a primary student. The first problem is much easier for many! If the only strategy a student has is counting backwards, the second method is quite complicated! In the end, we want all of our students to be able to make sense of subtraction as both the removal and difference! (We want our students to gain a relational understanding of subtraction).
Take a minute to watch Dr. Alex Lawson discuss the power of numberlines:
How does this relate to Integers?
Subtraction being thought of as removal is often taught using integer chips (making zero pairs…). Take a look at the examples below. Can you figure out what is happening here? What do the boxes mean?
Number lines are often used with Integer operations too, but the method of using them is typically removal as well. Think about this problem for a minute:
Again, students view of subtraction is removal here (or with the case of subtracting negative numbers here, students will be adding).
However, now let’s think about how a number line could help us understand integers more deeply if we thought of subtraction as difference:
Hopefully for many students, they might be able to see how far apart the numbers (+5) and (-2) are. Without going through a bunch of procedures, many might already understand the difference between these numbers. Many students come to understand something potentially difficult as (+5) – (-2) = 7 quite easily!
I encourage you to try to create 2 different number line representations of the following question, one using removal and the other using difference:
(-4) – (-7) =
Some final thoughts:
- When is it appropriate for us to use difference? When is it appropriate for us to use removal?
- Should students explore 1 first? Which one?
- Which is easier for you? Are you sure it is also the easiest strategy for all of your students?
- The questions above have no context of any kind. I wonder if this makes this concept more or less difficult for our students?
- How do you provide experiences for your students to make sense of things, not just follow rules and procedures?
- How can we avoid the typical tricks used during this unit?
As always, I’d love to continue the conversation. Send me your number lines or share your thoughts / questions below or on Twitter (@markchubb3)