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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The Calculus River … Follow the Flow

One of the myths about developmental mathematics is that very few students take STEM courses.  Often, we hear people joke that one student makes it to calculus.

Here is some data from my college showing how many students started from various levels in mathematics (over a 3 year period).

Started in beginning algebra or lower       105 out of 937             55% of that 105 pass calculus 1

Started in intermediate algebra                  177 out of 937              58% of that 177 pass calculus 1

Started in pre-calculus                                  457 out of 937             69% of that 457 pass calculus 1

Started in calculus 1                                       162 out of 937             69% of that 162 pass calculus 1

Over 10% of our calculus 1 students began in beginning algebra or lower.  We treat intermediate algebra as a developmental math course … so we’d say that over 25% of our calculus 1 students started in a developmental math course.

Not only do we have over 25% of our calculus students starting in developmental math, their pass rate in calculus is not that much lower than students who started in calculus.  It’s true that the proportions are statistically significant.  However, given the differences in student characteristics (placed in dev math versus not), the difference is relatively small.  Of course, we would like to improve the preparation so that the proportions are not different at all.

One of the reasons to point out the false nature of this myth is that our developmental math courses need reform for ALL students … not just those in ‘non-STEM’ fields.  In the New Life model, we propose using Mathematical Literacy for all students (as needed) and Algebraic Literacy instead of Intermediate Algebra.  Algebraic Literacy has learning outcomes designed to provide some early foundational work using concepts that are critical in calculus, as well as having a stronger basis in function properties and behavior.


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Problem Solving Skills

This is the story of what a student did when confronted with a procedural problem for which she did not ‘remember’ the standard procedure.

One of the in-class assessments I used is a ‘worksheet’; it’s like an open-note quiz over a set of material.  During our last class (intermediate algebra), the worksheet is longer than usual because it ‘covers’ the entire course.  Item 6 on this worksheet is:

Rationalize the denominator 

As I said, she could not remember ‘what to do’.  However, she did a great thing … she recognized that both numbers could be written as a power of 2:




I was very pleased that she did this, but the student was frustrated … she then could not see what to do.  This is pretty typical when novices dive in to the world of ‘non-standard problems’ — problems for which we lack a remembered process.

Of course, it was pretty easy to guide her through the remainder of the work:




Obviously, the expectation (this is our traditional intermediate algebra course) was that students would apply the standard procedure (multiplying top & bottom by the cube root of 4).  Students do not like that procedure, and I tell them that the procedure itself is seldom needed.

The alternate method worked only because there was a common base between numerator and denominator, and I doubt if the student will gain any long-term benefit from this experience.  This was more of a positive thing for me, as a teacher and problem-solver: Noticing a special pattern within a problem is a critical problem solving skill.

I’m sharing this story just because I had fun with it!

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The Power of Understanding Math

One of the most rewarding experiences we can have is when a student exceeds our expectations.

This is a story of a student who initially struggled with everything and is now being successful within an individualized course structure.  In this class, I never ‘lecture’ to a group of students.  Class time is used for studying, help, consultation, and testing; we call it the “Math Lab” though it’s not what most people mean by that phase.  The common meaning of “Math Lab” is a drop-in help center open to a variety of students in a set of math classes.  Our Math Lab is a way to take a few math courses … our math help for other classes is separate from the Math Lab.

This particular student (I’ll call him Philip) was clearly having trouble on the first day.  He did not want to use the online homework system, and that was not a problem for me.  However, he opened the book to the first page of the first section and had lots of questions about the names for types of numbers, about order of numbers on a number line, and (shortly after) about adding signed numbers.  The second day brought questions about the meaning of words in statements of properties, and about the meaning of variables.

It’s not that most students “get” these things, nor that they do not need to work on them.  What was unusual was the level of the struggle (basic) along with the sheer quantity of questions.  I never tell students what my prognosis is for them (I’m sometimes wrong) but I thought this student was going to spend weeks on every chapter.

Philip did, indeed, spend weeks on chapter 1 … a chapter about real numbers in a beginning algebra course.  Following those weeks, Philip then missed several classes due to medical problems related to his PTSD and physical injuries.  With over 6 weeks gone, Philip had only tried that first chapter test.  He was about to encounter the chapter on linear equations and applications, a classic “speed bump” for students struggling to learn algebra.

Somewhere in the month after that, however, Philip began making consistent progress.  In fact, he was getting through the third chapter faster than many students.  That progress has continued, and Philip is very likely to pass the course.

The main point is that something in the way Philip dealt with the struggle made a difference in how he succeeded in the entire course.  Philip works towards understanding everything, including ideas the are relatively minor.  He writes down lists of both vocabulary to learn and problems that he needs help with.  My guess is that his turn-around from struggle to success was caused by his hard work at understanding (and not just knowing what to do).

We all have students in this level of course who interact with the material at a low level; for them, it’s more about remembering what to do than it is about understanding.  I think Philip’s intense effort at understanding provided him with a cumulative positive improvement in the ability to learn new material.

Like most of us, I strive to have all students look for that understanding in learning mathematics regardless of the specific math course.  With other students, I end up trying to pull them someplace they have no intention of going (understanding) while Philip approached the material that way without any influence from me.

As a minor point in this post, I will point out that a struggling student such as Philip will be lost prior to getting any success.  Taking several weeks on one chapter is not an option within a fixed-pace class; instead of accumulating benefits, struggling students accumulate bad grades on assessments.  Our Math Lab, with its focus on individual learning, allows a struggling student to truly become a successful student.

A fixed-pace class has a limited capacity for helping struggling students; they need to be within a relatively small range of struggle in order to succeed.  Our Math Lab expands that range considerably (though there are still limits).

Understanding … a focus on understanding … enables students to obtain power in mathematics by raising their level of functioning to a higher point.

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