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.

Using my students to improve my teaching

It's those moments that I think "I never would have thought of that" that become some of the most powerful lessons. One example is an experience I had student teaching. If I haven't mentioned this before, I had an awesome student teaching experience. I remember struggling for several weeks with knowing how to determine if I should move on or not. I would often ask my cooperating teacher what he thought. He taught me that I need to find out from my students if I should move on or not. He also taught me that I should use my students to improve my teaching. There is a principle here that I know would have taken me years to find on my own, and I am so glad that I learned it.

The reason that I haven't posted anything for a few days is because of TAKS testing. Last week Tuesday through Friday the students were taking different benchmark exams. Yesterday I decided to spend some time talking about their preparation for the TAKS test. I handed out a half sheet of paper with two questions: "What are three things that helped you prepare for the TAKS test?" and "What are three things that you wish we did differently?" The responses were very enlightening. For my discussion here, I am not going to go into depth about how we reviewed although from my past 7 years of tutoring I have learned how to prepare students for tests. The point is my students were telling me my strengths and my weaknesses. I will continue doing the strengths and work on the weaknesses.

I do these kinds of activities occasionally in my classroom. At first my students weren't sure what I was doing. One student thought that I was trying to "save my job." I just explained that I want to be a better teacher and that this would help me know how.

This is a powerful principle that has helped me improve as a teacher and I thought I would share that with other readers.