## The Student Quandary About Functions

For students heading in a “STEM-ward” direction, understanding functions will become critical.  Unfortunately, a combination of a prior procedural emphasis and some innate cognitive challenges tends to result in a condition where students lack some basic understandings.

For example, in my intermediate algebra class, we provide problems such as:

For f(x) shown in the graph below, (A) find the value of f(0), (B) find the value of f(1), and
(C) find x so that f(x)=0.

Since there is no equation stating how to calculate function values, students need to use the information in the graph.  The vast majority of students make 2 novice errors:

• Error of x-y equivalence:  providing the same answer for (A) and (C)
• Error of symmetry: Since the answer for (A) is x=1, stating the answer for (C) as x=1

To improve this understanding, I use the longest (time measured) group activity in the course.  This is definitely a situation where “Telling” does not correct the errors [I’ve tried that 🙁  ], and the small group process helps dismantle some of the errors.  Clearly, the correct understanding for reading function graphs is critical for success in pre-calculus and eventually in calculus.

Another function concept we dealt with this week is ‘domain’.  Now, once students have found a domain, there is a tendency for some students to think they should find the domain of any and all functions, regardless of the directions for the situation.  This “inertia error” (what was started … continues) is not a long-term problem.  Here is a typical problem for the long-term problem:

Find the domain for the function graphed below:

In this particular class, I provide a fair amount of scaffolding … in a small group project, we explored the behavior of rational functions (without using that label) including what the “undefined” x-value means on the graph.  We don’t use the word asymptote; rather, we talk about the fact that some x-value results in division by zero, and the graph of the function can not show any ‘point’ for such inputs.  This leads to the graphing of the function, including the behavior around the ‘gap’.

Students struggle quite a bit with this type of problem.  Sometimes, they continue the ‘function values from graph’ thinking, and latch on to x=0 or y=0 to make some statement about a ‘domain’.  Many students will correctly identify the x-values for the gaps (yay) but make illogical statements about the domain.  The typical student error is:

• (-infinity, -2) ∪ (-2, infinity)  … or even just one interval (-2, infinity)

This type of error usually follows from a process-focus, detached from the underlying meaning.  I am trying to get them to see:

• gap on graph equates to excluded values in the domain

The process focus looks at the first part of  this.  Like the function value errors, the effective treatment of this problem requires time and individual conversations.

This type of function work is not typical for an intermediate algebra course.  However, it would be typical for an algebraic literacy course.  As we transition from traditional content to modern content in our courses, I am expecting that our intermediate algebra courses will fade away … to be replaced by variations of the algebraic literacy course.

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• By schremmer, January 15, 2017 @ 12:04 am

I am not sure what your intermediate algebra class is but I do not see how such “problems” can get anybody anywhere. Things have to be built up. Otherwise, it just reinforces the students in their well grounded idea that mathematics is just a bag of tricks to be memorize. The other day I mentioned I was to give a talk on rational functions which I present to my students during the third part of Precalculus I. The text of the talk is available at My talk on Rational Functions

• By Jack Rotman, January 17, 2017 @ 2:54 pm

Clearly, you are correct in saying that “they have to be built up”. That, in a general (and vague) qualitative way, is what separates “Algebraic Literacy” from “Intermediate Algebra” … Algebraic Literacy involves the intentional design of content to develop understanding of core ideas, while Intermediate Algebra involves loosely connected sequences of topics determined by historical artifacts (without regard to long-term needs).
I disagree that “problems” such as these “can [not] get anybody anywhere”. But, then, “that’s just me” :).

• By schremmer, January 21, 2017 @ 9:37 pm

I have never been able to get my hands on an text. So, I have no idea what it is or is not. But I note your “develop understanding of core ideas”. In other words, Algebraic Literacy is not developing the contents. Just the students. Nothing new here.

• By Jack Rotman, February 11, 2017 @ 3:23 pm

Well, of course ‘nothing new’. The professional judgments about what should be in a modern mathematics course extend back 30 years or more. What we are trying to do with the Algebraic Literacy course is to encourage faculty to implement a course which is designed to work for the students. We can leave the ‘intermediate algebra’ course (along with its sad history) behind.