I have struggled with teaching integer operations to students for as long as I can remember, in particular, subtraction. I have taught using rules and having the kids write the rules in a graphic organizer and apply them. I have taught using integer counters and number lines to no avail. The compression and long term understanding just hasn't been there the way I would like for it to be with my students. Why is that?
What is the difference between teaching subtraction as removal verses difference?
Mark Chubb describes it as follows:
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)
The hard part for me is to teach subtracting a negative such as 3 – (-7). I show the students the following method with integer counters (picture taken from Mark Chubb)
This is teaching subtraction as removal, however, the students don't really seem to grasp this unless the chips that need to be removed are already there.
Next I tried using a number line and teaching removal like this:
This past year, I tried using a number line and teaching difference to the students like this using 5 – (-2):
This really seemed to confuse the kids because they had to know that you start at the second number (-2) and move right to 5 which is a distance of positive 7. If the problem was -2 – 5, the answer would be -7 because they would reverse their direction.
I think in the end as I reflect on this year, I realize that in all three of my examples above, I keep saying "I show them"! Did I ever give them a chance to make senses of these problems on their own? How do they see it? For the sake of time, I try to SHOW them how to do it and get the right answer verses letting them make sense and sharing with each other. Having them make sense of the problem and listening to them will give me a better understanding of where each of them are at and I might be able to help them with their understanding better which will help them in the long run. Knowing that there are two ways to think of subtraction and having a deeper understanding myself will be invaluable to helping the students as they make sense of this. We know rules do not produce the long term results that we need. It might be worth it in the end if the kids will remember it since we use integers and rationals throughout the school year.
I have been challenging myself to gain a deeper understanding of the mathematics I teach and I saw a post of another way an elementary teacher sees and teaches students about subtracting integers. His post is below:
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).
Do our students see it this way? Would they make sense of integers this way. It is another way to think about it and a way our students might try to make sense of the problem. I would not show every student all 4 ways as that confuses them in my experience. I would make sure I understanding all 4 ways deeply myself so I can help students as they make sense of problems. Who knows, I might learn a different way of thinking about integers from the students as they share if I give them a chance to.
Should I stat this lesson with sense making and see where they are at?
How much time do I give to this? A day. Half a period?