A while ago I watched an interesting youtube video that dealt with student misconceptions specifically in physics. It resonated with me so much I just spent a few minutes trying to find it to share on this blog. The point of Derek Muller’s research is that students often think that they already know what they are learning. I have done many lessons where I start with a class discussion of what students know about a particular topic. Especially teaching 9th through 11th graders, many student have had Algebra I multiple times which unfortunately can reinforce certain misconceptions. Last year, I had an 11th grader who regularly announced that this was her fourth time and she was going to “pass it.”
So I just erased my 2nd paragraph, and I am starting again, so it will be shorter!
One misconception that I noticed every year with my students was subtracting integers and whether adding negative integers were positive. I have vacillated about how to deal with the misconception over the years. At this point, I usually have two responses depending on the student. If the student seems ready for the concept of subtraction, then I will use money or a vertical timeline with the student to solidify their understanding. If the student is still struggling with the concept of subtraction, then I coach the student to change subtraction to adding the inverse.
For example A) -5 -8 is the same as B) -5 + -8
I avoided the above method for a while because it doesn’t really address the concept of subtraction. However, most students can solve B) whereas many students have issues with A). I think that “converting subtracting to addition” is a shortcut that is useful for some students.
I dealt with the misconception during whole class discussions by highlighting what operation was being used when combining integers. When students were able to understand a negative and a negative is a positive was dependent on the mathematical operation, I could usually address the faulty understanding by asking them what operation they were using.
I’ve had many class discussions where students insisted that a “negative and a negative is a positive.” I try to have other students verbalize why this is sometimes true. This is a case when memorization of a key concept that is missing the condition can lead students astray.
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The simplest way to avoid any troubles with A) and B) is to use parentheses. And explain to students that there is no subtraction at all, there is just an addition of the opposite number.
I have seen some of these misconceptions, especially in students that were not ready for the abstract and still needed concrete representations. However, the catch-22 is that as the students age, they may still need the concrete, but are too ashamed to act like they need it. I think using concrete represenations like the number line can definitely help many students.
This is the 2nd blog post I’ve read today about the trouble students are having with thinking that a negative plus a negative is a positive… I too have had to keep reminding my kiddos about these concepts as we studied the rules of exponents.