In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission again comes from my former student Austin DeLoach. His topic, from Algebra: negative and zero exponents.
B1. How can this topic be used in your students’ future courses in mathematics or science?
The topic of negative and zero exponents is very important when or if the students get to calculus. Although that will be several years down the line, having a solid fundamental grasp on the idea of negative and zero exponents will help them understand derivatives a lot better. Because derivatives of “simple” functions just multiply the coefficient by the exponent and then subtract one from the exponent, it is important for the students to have a good understanding of what negative and zero exponents are. If they do not understand already, they will be confused about why, for example, the derivative of 3x is just 3. It also greatly simplifies derivatives of things like 4/x2, as the students will simply be able to recognize that that is the same thing as 4x-2 and follow standard rules instead of needing to think about the quotient rule and waste time with that. It will also help them in the more near future when they work with simplifying expressions with the exponents written in different terms (i.e. with a positive exponent or with a negative exponent in the denominator), as it will help them recognize what simplifications mean the same thing. Explaining that understanding negative exponents will thoroughly help them in the future may be enough for some students to want to solidify their grasp on the topic.
D2. How was this topic adopted by the mathematical community?
Although this is not about the early adoption of negative and zero exponents in the mathematical community, Geoffrey D. Dietz points out more recent bias for or against the use of negative exponents in textbooks in his Journal of Humanistic Mathematics (linked at the bottom of this answer). Dietz brings up the idea of what is considered “simplified” when it comes to negative exponents vs exponents in denominators. He rated over 20 mathematics textbooks from 1825 to 2012 from “very tolerant” of negative denominators in simplified answers to “very intolerant”. Interestingly, his first encounter with an “intolerant” textbook was not until the 20th century, and textbooks began getting more polarized as very tolerant or very intolerant closer to the end of the 20th century and getting closer to today. This is interesting when it comes to adoption by the mathematical community, as there is a significant inconsistency, even today, about whether negative exponents can be considered “simplified” or not. It will be important to point this out to your students so they can be prepared for their future teachers who may have different preferences on simplification from you, as that will help them understand the polarity in the mathematical community on this topic, as well as hopefully make them want to understand what negative exponents really mean. Dietz recommends giving your students practice with not only converting negative exponents to positive exponents, but also from positive to negative, in order to make sure they are prepared for whatever preferences come up as well as solidifying their understanding of what negative exponents mean.
E1. How can technology be used to effectively engage students with this topic?
This video from Khan Academy does a good job at explaining why negative and zero exponents are what they are. Although Khan Academy videos will likely not be the most engaging for all students, this video is short enough to maintain the attention of the class, and it the logic in it is helpful for the students who don’t understand how the definition of negative and zero exponents was decided on. The presenter does well explaining the idea of “going backwards” and dividing by the number when you decrease the exponent. It’s a good way to explain the “why” for students who ask about it, and it also is a good way to change up the pace for students, as playing videos during class could prevent it from becoming stale for the students, keeping them engaged for longer.