1 – (-1)

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I use M&Ms (Math & Manipulatives). M side up is positive. A – sign to the left means turn it over. () means put it in a plastic bag. Another – on the left means turn it over. So what do we have? Two M&Ms, M side up. positive two.
I have also done it with stick pretzels, two stick pretzels next to each other can be slid together to make a + .
in other words, back to concrete representation.

I believe this error occurs from an oversimplifying of the operations. Students don’t fully understand how operations work,outside of whole number situations, so when they see integers or any rational numbers, they try to treat the numbers as whole numbers.

Anne

Andy, so how do you help students understand? I mean, with your students how do you explain this?

Andy

Oops, I was meaning the students oversimplify the operations, not the teacher. I don’t have a wonderful answer for how to fix this. I work a lot with what the operations mean and how they affect the numbers you have. With a problem such as this we would talk about what a difference is and what it looks like on the number line. I also try to get students to see rational numbers as just numbers ( something that should not be feared), and when teaching these concepts I try to use conceptual hands on learning slowly moving to algorithms.

Phill

On your last point about rational numbers as “just numbers.” I work with high school students and it seems they much prefer decimals to fractional forms. Do you suppose it is because it is easier to see 0.5 as a number than 1/2, because they see / as an operation? Or maybe it’s just easier to add 0.5 to 0.2 than it is to add 1/2 to 1/5, as the former makes use of the familiar adding procedure, but the latter requires a procedure that is both unfamiliar and unique?

Andy

I think it has to do with both issues. I think a lot of students see decimals working more like whole numbers, and therefore they see them as less complicated. Students try to read fractions in all different ways, but some decimals seem to make sense. One problem I see is that most of these students struggle with rounding and the true concept of decimals, making their results closer to guesses than results. On a different note, think many students see mixed numbers as three separate numbers without a fraction. These same students read 3^2 as three two; I believe more conceptual teaching and more waiting to create algorithms and abstractions help these misconceptions.