Think of things both ways

From my tutoring and teaching in a small school I have the opportunity of frequently working one on one with students. I have developed an awareness of noticing foundational misconceptions that students have. There are two that are very common, negatives and fractions. I was thinking about these a few days ago and realized that you can think of each in two very different ways.

Almost all students can see a number like -5 and know that it is negative. But sometimes we use the - as a negative and sometime we use it as a minus sign. For example when we are simplifying 8-2(x+1) is it 8 minus 2(x+1) or is it 8 plus -2(x+1). They are mathematically equivalent but sometimes we use - as a negative and times we use it as a minus sign. This can be confusing to students.

Fractions can be thought of as part of a whole and as another way to write division. For example 3/5 is three out of five equal parts and it is three divided by five. I know that these two ways of seeing fractions is mathematically equivalent but seeing it both ways can be confusing to students.

I am wondering if math teachers go back and forth between seeing things both ways without realizing it and as a result confusing our students. I will try to be more explicit about seeing things both ways and I hope this will help my students through these two very foundational and important concepts.