Exponential Functions

[Sarah Stevens]

It seems like a major divide, among the various suggestions for arrangement, is where to put exponential functions. I find myself torn between wanting to embrace the vision for Appendix A  and accommodating for out-of-date textbooks. Some other arrangements focus only on linear equations and quadratic equations in Algebra 1 and save exponential for Algebra 2. We are making Instructional Sequences for a large district so we must consider the varying levels of learning needed to successfully implement the CCSS. Certainly, leaving exponential equations out of Algebra 1 is a softer transition and might be easier for some teachers to master. On the other hand, this may be the best time to make this significant instructional shift. I guess my question is: Do you have any additional insight (beyond Appendix A pg 1 “The critical areas deepen and extend understanding of linear and exponential relationships by contrasting them with each other…”) to help us as we wrestle with the key decision?

Thanks!

[Bill McCallum]

You are right about the divide. Whichever arrangement is chosen it’s important to have a clear organizational principle. One reason for putting exponential functions earlier is an emphasis on patterns of growth, with linear and exponential functions representing the simplest functions from that point of view. This would occur in a function-based approach to algebra. A reason for putting linear and quadratic equations together is an approach based on looking for algebraic structure and reasoning about equations. This is an approach to algebra based more on developing symbolic fluency. In the end, at the end of Algebra II, you want students to have both symbolic fluency and a good understanding of functions, so the endpoint should be the same. But it’s important to have a clear organizing principle as you arrange topics into courses, a story to tell that will be clear to teachers and students.

[Sarah Stevens]

First, thank you for expanding my question from specific to general. You raised issues I didn’t even know we had yet!  I have been researching a functions based approach to algebra and have read the NTCM book Essential Understanding- Functions, the functions progression, and a few internet sources. I still feel like this type of organization is eluding a firm seat in my knowledge base and would like to learn more. Can you direct me to any good resources? I have done google searches on “functions approach to algebra” and get back a limited number of applicable hits. Are their alternate phrases that will expand my search?

Thanks again for taking the time to answer questions and help us navigate the waters!

Sarah

[lhwalker]

Now that you mention it, the “functions based approach” is nebulous in some aspects to me as well.  If anyone finds a  complete, succinct explanation, please post.

[Bill McCallum]

I think of the functions-based approach as an approach were you introduce the concept of a function early, and use it as a springboard for a lot of the work in algebra. For example, you might think of equations in one variable as questions about when two functions (the functions defined by the expressions on either side of the equation) are equal. And you might have such equations arise in a context where both functions have some meaning related to the context and the equation of when they are equal is meaningful in the context.

[Sarah Stevens]

I continued searching and a local university professor directed me towards “Teaching Mathematics in Grades 6-12: Developing Research-Based Instrucitonal Practices” by Randall Groth (I purchased it from Amazon). I am still waiting for it to arrive but I read chapter 8 of this book and have a much clearer understanding of the functions-based approach. In the second paragraph he defines the functions based approach. “In general, a functions-based approach asks students to form their own theories about how the values of quantities depend on the values of other quantities.” The book is steeped in research and really highlights the reasons behind some of the decisions in the Common Core. For example, I found myself wondering about the standard A.REI.11. I had never considered teaching solving equations in this way. The book highlights a study that found students with multiplie strategies at their disposal (including  the functions-based approach) are the most successful while students who rely only on algerbraic manipulations are the least successful. I knew the CCSS were research based but it really helped me to read a little about instructional decisions and the studies behind them. I hope the rest of the book is as good as this one chapter!

I also purchased the Essential Understandings: Functions book from NCTM and found it filled in additional blanks and might be more digestable for teacher PD.

[Author]

Posts