Recognize that sequences are functions…for example the Fibonacci sequence, F.IF.3.
Evidenced by the number of times I have seen sequence equations recently,
f(n)=f(1)+d(n-1) +d, or recursively f(n) = f(n-1)+d
I am assured I am not the only one that missed that shift the Common Core is making. Students are learning to read and write function notation like f(3) = 8, so it makes sense to build onto that instead of having them also learn to write sequences as equations with subscripts. I see it now, and am helping my students to develop a thorough understanding of function notation to meet the Common Core shift toward narrower, deeper standards.
When I asked Dr. McCallum about subscript vs. function notation, he was clear that students should see subscripts in high school. And they do see those in the slope formula. The progressions document for the Functions progression states, “In courses that address material corresponding to the plus standards, students may use subscript notation for sequences.” The reason the CCSS call for function notation in Algebra I, is to to build a coherent foundation for functions, to understand sequences as functions instead of teaching students a compartmentalized method for finding the value of the nth term. If we want to have time in our Algebra classrooms to explore rich problems, we need to work with the nuances in the standards to keep our procedural fluency expectations succinct.
Understanding sequences as functions was new for me. I didn’t realize how shallow my understanding was until I made that connection. I never really understood how Fibonacci fit before that point. (Okay, I know I’m slow; but when I get it, I run with it.) It clicked when I was having my students work on Khan Academy. (How has Khan changed?) Khan did not stop with f(n) = f(n-1) + 5 but progressed to f(n) = f(n-1) + f(n-2). Ah ha! So the point is not to memorize particular forms for arithmetic and geometric sequences, rather understand how to write sequences in general. Of course arithmetic and geometric are important, but to be able to generalize sequence writing is even more important. In a high-tech society, we need to understand processes and relationships so that we can adapt them to real contexts. In this way we have a natural and subtle opportunity to integrate 21st Century (STEM) topics.
I have heard the arguments that subscript notation for sequences would carry better to upper levels of math but my response to that is to recall how important function notation in general is in upper levels of math:
- Cos x or Cos (x+3): How many times have we seen a student divide off x or subtract 3?
- f(x,0) = 0
- f(x, S(y)) = g(f(x,y),x)
- g(x, S(y))=s(g(x,y))
In any discussion, it would be important for ACT, SAT, and AP to weigh in as we would not want to put our students at a disadvantage on those tests by inappropriately favoring equations over functions. As it stands, both are important. The College Board (AP) notes the subscript notation is a plus standard recommended for fourth year courses, which would include pre-calculus. The ACT College and Career readiness standards mention “function” 44 times and “equation” 22 times. If a student has a firm grasp of function notation, converting a function to a subscript equation should be fairly seamless. This would be especially true if students have already been exposed to subscripts in formulas such as the slope formula. I got some affirmation of that today when I accidentally gave my students a problem asking for a13. They asked, “Do they want the value of the 13th term?” Yep.
I’m running with it.