Subtracting Integers – Do you see it as removal or difference???

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.

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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)

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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).

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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.

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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?

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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:

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Again, students view of subtraction is removal here (or with the case of subtracting negative numbers here, students will be adding).

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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!

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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) = 

 

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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)

Questioning the pattern of our questions

I find myself spending more and more time trying to get better at two things.  Listening and asking the right kinds of questions that will push thinking.  While I find that resources have helped me get better at asking the right questions, I have learned that listening is actually quite difficult.  The quote below is something that made me really think and reflect on my own listening skills:

More about this in a minute…

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A while ago I had the pleasure to work with a second grade teacher as we were learning how to do String mini-lessons (similar to Number Talks) to help her students reason about subtraction.  After a few weeks of getting comfortable with the routine, and her students getting comfortable with mental subtraction, I walked into the class and saw a student write this:

What would you have asked?

What would you have done?

Did she get the right answer?

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My initial instincts told me to correct her thinking and show her how to correctly subtract, however, I instead decided to ask a few questions and listen to her reasoning.  When asked how she knew the answer was 13 she quickly started explaining by drawing a number line.  Take a look at her second representation:

She explained that 58 and 78 were 20 away from each other, but 58 and 71 weren’t quite 20 away, so she needed to subtract.

I asked her a few questions to push her thinking with different numbers to see if her reasoning would always work.

Is her reasoning sound?  Will this always work?  Try a few yourself to see!

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Typically, we look at subtraction as REMOVAL (taking something away from something else), however, this student saw this subtraction question as DIFFERENCE (the space between two numbers).

I wonder what would have happened if I “corrected” her mathematics?  I wonder what would have happened if I neglected to listen to her thinking?  Would she have attempted to figure things out on her own next time, or would she have waited until she was shown the “correct” way first?

I also wonder, how often we do this as teachers?  All it takes is a few times for a student’s thinking to be dismissed before they realize their role isn’t to think… but to copy the teacher’s thinking.

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Funneling vs. Focusing Questions

As part of my own learning, I have really started to notice the types of questions I ask.  There is a really big difference here between funneling and focusing questions:

Think about this from the students’ perspective.  What happens when we start to question them?

Summarized by Annie Forest in her Blog

Please make sure you continue to read more about we can get better at paying attention to the pattern of our questions:

Questioning Our Patterns of Questioning by Herbel-Eisenmann and Breyfogle

Starting where our students are….. with THEIR thoughts

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So I leave you with some final thoughts:

• Do you tend to ask funneling questions or focusing questions?
• How do we get better at asking questions and listening to our students’ thinking?
• What barriers are there to getting better at asking questions and listening?  How can we remove these barriers?
• Is there a time for asking funneling questions?  Or is this to be avoided?
• What unintended messages are we sending our students when we funnel their thinking?  … or when we help them focus their thinking?
• What if our students’ reasoning makes sense, but WE don’t understand?

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I’d love to continue the conversation about the subtraction question above, or about questioning and listening in general.  Leave a comment here or on Twitter @MarkChubb3

What are your thoughts?