Same subject + different students = different lesson

Today W was starting to learn about exponents, something I was thrilled to hear about! My favorite lesson I’ve taught was last May while subbing in a middle school, and I taught a class of 8th graders exponent rules. There was originally another teacher in the classroom and she told me to stay put and not worry about teaching the students, but then when the kids continued to ask why certain rules existed, she time after time said “that’s just the way it is.” After hearing that a few times I politely asked her if I could say a few words on the exponent rules. I then used the division rules of same base exponents to explain why anything raised to the power of zero is one. The students seemed very engaged and excited to hear somebody explain to them the “why” behind this. After that the teacher asked how I was able to come up with that proof on the spot, and after telling her I’m a double major in mathematics and secondary education, she asked if I was comfortable teaching the rest this lesson without her because she had other work she wanted to go do. The rest of the day I had the best experience working with these exponent rules in a middle level classroom, so I was overjoyed when I heard W was working on exponents.

I was excited to see how teaching W would differ from teaching the same lesson to 8th grade students. It was very interesting to show him new notation of how to express exponents and radicals. He really seemed to grasp that you multiply the base number by itself however many times based on what the exponent is. He had a tough time differentiating between something like 23 and the cubed root of 8 . These are both representing that 23 is 2*2*2, or 8, and that 2 is the cubed root of 8 or 81/3, but he wasn’t sure when a problem was asking for you to find the base, which would be a problem like cubed root of 8 asking you to evaluate that the cubed root of 8 is 2, or when you have 23 and need to find the product of 2*2*2. This was because this notation and language was all brand new to him. I took this hint and took a step back to discuss the language associated with exponents and radicals, and made him a key of which symbols correspond to which math language. This seemed to really help him grasp which process to use when.

It was intriguing to see the differences between teaching this to an ELL student compared to 8th grade students native to Vermont. The biggest factor that makes these two experiences differ so much was in the familiarity with mathematics language and notation. One thing that was helpful to W was when my placement partner made an analogy of exponent notation by drawing a picture. This is something that blew my mind because I hadn’t seen this analogy before, and the visual representation really helped our student to grasp the concept of the notation better. As seen in the picture below, the roots of the tree represents the base number, and the branches represent the exponent. This helped to show the notation and that the exponents is the number written higher than the base. The larger the exponent is, the longer the branch is, so we can think of this to show that exponents with the same base get increasing larger as the exponent increases. In addition, it is good to use the roots of the tree as the base because when you reverse this process you must take the root. For example, 8 is the base, 2 is the exponent, and it shows that 82 is 64. When you reverse this, you take 64 and use the exponent as the root, so that would be the square root 64 which gives you the base, 8. This visual helped to remind our student that in order to find the base you must take the root.