Bridging the Gap
Recently at my teaching placement, I introduced the concept of negative exponents during a lesson. This concept always stumps students, and they tend to misconstrue negative exponents with multiplying by a negative number. I think of negative exponents as just notation, they denote an operation that you perform. But where do negative exponents come from? Why would we decide to denote them with a minus sign? Why not a different symbol, like the letter k or the symbol for pi? Asking these types of questions about negative exponents helps me to put myself in my students' shoes, and better empathize with them. One particular student in my class was really struggling to swallow this new idea. I pondered another way to explain the concept, and decided I could relate negative exponents to a concept this student knows well--inverse functions.
I asked the student, "Let's go back to what you know. What does it mean to square a number? What does it mean to take a number to the power of 2?" The student responded, "you multiply the number by itself." Then I asked, "how many times do you multiply it?" And they said "one time." I nodded and then asked, "Ok, so whats the opposite of multiplying?" The student paused, then said "dividing?" I nodded again. Then I said, "if I told you that a negative two in the exponent is the opposite of a positive two in the exponent, what would that mean?" The student didn't reply. I went back to teaching for a bit, then returned to the student towards the end of the period.
I started: "Earlier, you told me that to square a number means to multiply the number by itself. So if instead I took the negative two power of a number, that would mean we do the opposite of multiplying...So what is three to the negative two power equal to, do you think?" The student thought for a minute, and said "One. Its 3 divided by 3."
At this point, I thought I had misled the student. I noticed he was getting a bit frustrated, and I began to worry that I had made negative exponents even more confusing. But I gave it one more try. I said, "You're on the right track, you divide. But don't divide the number by itself. Remember, 3 to the 0 power is 1. So 3 to the negative 2 power should be not equal to 1." I wrote a sentence on the board: "A negative exponent means how many times to divide 1 by the number." The student paused, then worked out 3 to the negative 2 power on his paper. I looked over his shoulder as he worked, and he simplified the expression perfectly.
The strategy I used was really simple, I tried to bridge an academic knowledge gap by tracing back to something that the student had a solid understanding of. By linking a new concept to the old one, the student not only was able to understand what negative exponents represent, but also why we denote them that way. Just as we have positive exponents to symbolize multiplying, we need negative exponents to symbolize dividing. To have one means we must have the other.
I wish I would have explained negative exponents like this to the whole class, instead of introducing them as an exponent property that needs to be memorized. In the future, I would even like to encourage students to use other exponent properties to derive the meaning behind negative exponents. This is hard to explain verbally, so I've attached an image below of what this might look like. I think with the right guidance and prodding, students could figure out what negative exponents symbolize on their own, rather than being told what they denote right off the bat. When students discover concepts for themselves, they not only will remember them better but have more interest in them. Deriving something for yourself gives it more significance!
I asked the student, "Let's go back to what you know. What does it mean to square a number? What does it mean to take a number to the power of 2?" The student responded, "you multiply the number by itself." Then I asked, "how many times do you multiply it?" And they said "one time." I nodded and then asked, "Ok, so whats the opposite of multiplying?" The student paused, then said "dividing?" I nodded again. Then I said, "if I told you that a negative two in the exponent is the opposite of a positive two in the exponent, what would that mean?" The student didn't reply. I went back to teaching for a bit, then returned to the student towards the end of the period.
I started: "Earlier, you told me that to square a number means to multiply the number by itself. So if instead I took the negative two power of a number, that would mean we do the opposite of multiplying...So what is three to the negative two power equal to, do you think?" The student thought for a minute, and said "One. Its 3 divided by 3."
At this point, I thought I had misled the student. I noticed he was getting a bit frustrated, and I began to worry that I had made negative exponents even more confusing. But I gave it one more try. I said, "You're on the right track, you divide. But don't divide the number by itself. Remember, 3 to the 0 power is 1. So 3 to the negative 2 power should be not equal to 1." I wrote a sentence on the board: "A negative exponent means how many times to divide 1 by the number." The student paused, then worked out 3 to the negative 2 power on his paper. I looked over his shoulder as he worked, and he simplified the expression perfectly.
The strategy I used was really simple, I tried to bridge an academic knowledge gap by tracing back to something that the student had a solid understanding of. By linking a new concept to the old one, the student not only was able to understand what negative exponents represent, but also why we denote them that way. Just as we have positive exponents to symbolize multiplying, we need negative exponents to symbolize dividing. To have one means we must have the other.
I wish I would have explained negative exponents like this to the whole class, instead of introducing them as an exponent property that needs to be memorized. In the future, I would even like to encourage students to use other exponent properties to derive the meaning behind negative exponents. This is hard to explain verbally, so I've attached an image below of what this might look like. I think with the right guidance and prodding, students could figure out what negative exponents symbolize on their own, rather than being told what they denote right off the bat. When students discover concepts for themselves, they not only will remember them better but have more interest in them. Deriving something for yourself gives it more significance!
This is a great way of putting things! Negative exponents can be confusing, for sure. I like that you kept coming back to the focus student, letting him think about a solution for awhile before you came back and explicitly bridged the gap with him. So at this point students already knew that anything to the 0th power is 1, right? I assume that was proven to them.
ReplyDeleteI think it's also a good connection to use the scaffolding you found helpful with one student to teach the whole class. I bet that will be helpful, in my experience they do well with the concept of "opposite" as it relates to positive/negative numbers, multiplication/division, etc.
Dear Danielle,
ReplyDeleteThank you for sharing your thoughtful reflection of how you bridged the academic knowledge gap for your student. You were wise to start by building on your student's prior knowledge of inverse functions.
I love the way you didn't give up when your student didn't readily understand what you were trying to explain. You gave him time to think and then you came back to him with another way of looking at the problem.
I totally agree with the statement you made when you wrote, "When students discover concepts for themselves, they not only will remember them better but have more interest in them. Deriving something for yourself gives it more significance!" That is so true and so powerful!
I appreciate all of your conscientious work and your fascinating insights into teaching math, Danielle! It is such a pleasure working with you!
Sincerely, Julie Elvin