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:
25-4 =
25-22=
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:
https://player.vimeo.com/video/88069524?color=a185ac&title=0&byline=0&portrait=0
Alex Lawson: The Power of the Number Line from LearnTeachLead on Vimeo.
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?
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)
Thinking Mathematically 
Lots of good thoughts here Mark. My concern with representing difference on the number line is that there is no way other than a rule for a student to make sense of the difference as positive or negative. For instance, why is (-4)-(-7) = (+3) and not (-3). To me, the idea of difference distinguishes magnitude, but not direction. After all, (-7)-(-4)=(-3) both have a magnitude of 3, but in opposite directions. I use number lines and integer chips in modelling integer operations with my students. I use addition (more) and subtraction (less) along with the quantity being subtracted to indicate direction. For instance, +(+)=more positive, -(-)=less negative with both pointing toward the positive end of the number line and -(+)=less positive and +(-)=more negative with both pointing toward the negative end of the number line. I think this is in line with the idea of difference from a slightly altered approach. (-4)-(-7) can then be interpreted as (+3) is 7 units less negative than (-4).
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I’m glad you bring this up Blair! You are right about the difficulty with the direction! I’m not sure which approach students would gravitate towards first. Actually what i’m more interested in is thinking about how we view subtraction and which might be more intuitive to our students as they begin to think about subtraction with integers. No matter what, we need to make sure we give our students multiple opportunities to make sense of things and whatever rules we decide on should come from the group as to what makes sense, not from me showing a trick.
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My experience is that students gravitate toward the idea of the subtraction as removal/taking away, but then find that difficult to model with integer chips when the number to be taken away is greater than the number of chips available or different in sign.
My hope is that in considering different models they will gain a flexibility in understanding of the different operations. I worry that sometimes this may just be confusing them more.
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I agree that giving a bunch of strategies to them can be confusing. Again, my wonder is which of these students would understand if we allowed them to reason before we taught anything.
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I have come to the conclusion recently that the sense-making about integer addition and subtraction MUST come from the students. You can offer tools like number lines or integer chips, but as soon as the teacher shows “how” to use them to find sums and differences, student work shifts from sense-making to direction-following. Almost in the same boat as providing “tricks” and “rules”.
As for why 4 – 7 is -3 even though the distance between 4 and 7 is always 3, I think its because 4 is 3 less than 7, below it. It would be interesting to hear what students think on the matter!
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Great comment!
Even though I have shared a perspective of using Difference as a way of understanding subtraction, I too wouldn’t push this thought on my students. Glad to hear you see that following these rules blindly is similar to giving tricks.
Wonder if a context might help our students make sense of things?
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I’ve tried context, too, and I think it does help them understand what to do–for that specific problem. I finally learned to have them come up with the simple situation (money, elevation, temperature) as they tended to wait for me to provide one for them. I still found myself prompting them (again and again) to make up a situation, though, which perplexes me. Maybe putting an expression into context is a strategy I am imposing rather than one they are creating? Or it is a strategy that is great for answer-getting but not so great for constructing understanding? What have your experiences been?
I’m glad you started this conversation. Helping students make sense of Integer addition and subtraction is my latest….obsession.
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