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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Looking for new Textbooks? Be an Author!

Are you looking for a math literacy book that is different from those available now?  Are you looking for any algebraic literacy textbook?

These books get written by people who want to teach the courses.  We understand the goals of the courses, the type of content that should be present, and how to present this material so that students can succeed.

Perhaps you are somebody who might be interested in writing either a Math Literacy or an Algebraic Literacy text … either by yourself or as part of a writing team.   If so, you can certainly approach any publishing company to start the process.

In particular, Pat McKeague of XYZ Textbooks is willing to work with potential authors of textbooks for our new courses.  He is excited about developing more textbook choices for us, while providing materials to students at a lower cost.  When XYZ publishes a textbook, they do some of the wrap-around work (such as videos).

I appreciate Pat’s support of our work and his willingness to work with authors.  If you are interested in learning more, contact Pat at pat@mckeague.com

The textbooks should be a close approximation to the course goals & outcomes:

• mlcs-goals-and-outcomes-oct2016-cross-referenced
• algebraic-literacy-goals-and-outcomes-oct2016-cross-referenced

Clearly, the intent is that any textbook focus on understanding and reasoning.  The level of “context” and “small group work” can vary (though always being a part of the package); some of this could be left up to the instructor using the materials.

If you have questions, feel free to contact me!

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Algebraic Literacy Presentation (AMATYC 2016)

The presentation includes some additional data related to HS math course taking trends over a 25 year period … which definitely impacts our college mathematics curriculum.

Here are the documents for the session:

Slides:  a-bridge-to-somewhere-amatycalgebraic-literacy-sample-lesson-rate-of-change-exponential-2016

References: references-bridge-to-somewhere-amatyc-2016

Algebraic Literacy Goals & Outcomes: algebraic-literacy-goals-and-outcomes-oct2016-cross-referenced

Sample Lessons:
Trig Basics: algebraic-literacy-sample-lesson-trig-functions-basics-2x
Rational Exponents:  algebraic-literacy-sample-lesson-rational-exponents-stem-boosting
Rates of Change: algebraic-literacy-sample-lesson-rate-of-change-exponential

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