Algebraic Literacy meets Pogo (AMATYC 2016)

Pogo (by Walt Kelly) offered humor and wisdom.  That is the combination I am reaching for at my Algebraic Literacy session this week in Denver (AMATYC 2016).

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The Value of Worthless Mathematics: It’s not ALL about ME!

Within our mathematics community, much of our recent efforts have been directed at presenting students mathematics related to problems (contexts) that are likely to be important to them.  Some curricular work is limited to the mathematics for which such a context can be presented.  Although relevant context is helpful, we lose something important when the context becomes more important than ‘mathematics’.

A related movement is the ‘guided pathways’ (see http://www.aacc.nche.edu/Resources/aaccprograms/pathways/Pages/ProjectInformation.aspx) which has a goal of aligning mathematics with the intended major, a guideline based on research showing improved completion when this is done.  The guideline is being applied to both college-level and developmental course work.

In some ways, this makes sense … Mathematics has always had roots deep in practicality.

However, I see two failures resulting from these approaches:

1. Mathematics is not always practical when ideas are developed or discovered.
2. General education seeks to go beyond the parochial.

In the American culture of 2016, we seem to validate the notion that “I only have to care about things that impact me directly.”  When we honestly tell students that this mathematics is important even though we are not showing a context for it, we should be able to expect students to honor the statement.  In many ways, learning mathematics without context is a good training program for employment … I suspect that the majority of workers work in a job with little innate value to them, in which they need to honor a supervisor’s statement that doing a job a certain way is important.

The role of general education has been both integral to higher education and marginalized in higher education.  The values of ‘different perspectives’ and ‘modes of thought’ represent the building of capacity in a society to think about difficult problems without resorting to slogans and over-simplifications.  When general education works, it is a beautiful thing.  This type of rising to a higher level of problem solving can not occur when the classroom is limited to the shared current concerns of those present.

If we truly believe that students are well-served by allowing them to focus on their own interests and concerns, sure … let’s limit their mathematics to contexts that they can understand at the time.

I think that limitation is a dis-service to students (and is not respecting mathematics as a set of disciplines).  Sure, we can have lots of fun when students are enthusiastic about our work in class.   Do they have any better notion of what ‘mathematics’ is?  Did the experience result in anything more than a few concepts that are applied in concrete ways?

Our courses should always contain significant elements of what I call “beautiful and useless mathematics”.  “Beautiful” refers to the aspects of mathematics which appeal to mathematicians … which can vary from person to person, and from one domain to another.  “Useless” refers to the ideas being developed in an abstract way without knowing if there will ever be any practical use.

One example of such ‘beautiful and useless mathematics’ would be functions which have a rate of change equal to the function.  The number e is not immediately reasonable when we deal with concrete multiplicative change.  We can contrive some contexts where the base e can be used, though most of these are more accessible to students using a percent growth rate (or decay).  The use of e for the function, and for the rate of change in the function, is a thing of beauty.

In some ways, this post boils down to this statement:

Don’t sanitize any mathematics course to the point where all artistic merit is destroyed.

Although this post relates to a recent post on ‘where STEM students come from’, I think the idea is valuable for every student who walks in to a math class.  We are not mathematicians because it is practical (though it is); we are mathematicians because there was something that attracted us.  Our students deserve to see at least a small corner of the wonderful canvas called ‘mathematics’.

 
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Are STEM Students Born or Made? The STEM student paradox

A couple of things are causing me to think again about STEM students in developmental mathematics.  First, we have local data showing that over half of our pre-calculus students came from the developmental math program … about 24% start in intermediate algebra, about 23% start in beginning algebra (or math literacy), and about 5% started in a pre-algebra course.  Since we no longer have the pre-algebra course, those students will now take a math literacy course (raising the 23% to about 28%).

The other event was a student in one of my intermediate algebra classes.  One of the things we always do on the first day of the class is to have students record what their college program is (either on a class sheet or on an individual form).  This particular student recorded her program as “religious studies”.  She had taken our beginning algebra course the prior semester, so being in this class was not a surprise.

However, this week, as we talked about a test in the course she told me that she was thinking of changing her major to mathematics.  Of course, we shared a “how cool is that!” moment; we then talked about what math course she would take next semester.  That was a good day!

Since then, I’ve been thinking about what led to the student’s statement about changing majors.  This particular class uses a “Lab” approach … class time is used for doing some of the homework, getting help, and taking tests individually.  We’ve had this format for about 50 years; although the method has been through many changes, the basic concepts have remained.  One of my mottoes for the method is “get out of the student’s way!”   We have pass rates that are just below that of traditional ‘lecture’ classes.

My impression of this student is that she got to really like the process of working through problems on her own.  If she had to listen to me lecture … or if she had to work in a group to deal with math problems … I don’t think she would have had the meaningful experience which led to a ‘change major’ state.

Here is the STEM student paradox:

A focus on getting more students through a math course can lead to conditions that never inspire students to make a commitment to a STEM major.

Now, I am not saying that continuous lecturing will inspire a student.  Continuous lecturing has no defense, and can be considered educational malpractice.

The issue here is that many of the processes we are using, combined with a limited symbolic formality based on contextualizing most topics (especially in developmental mathematics), tends to create a social focus for the learning while minimizing the symbolic complexity of the problems.  More students might learn the course outcomes at the cost of seldom inspiring students to select a STEM major.

Of course, like pretty much any generality, this one has plenty of exceptions.  I’m talking about the directionality of math classes, not about absolute location.

I would like to have a conversation with my student to see if she can articulate a reason, or even a description of the experience that led to a change.  I might get some feedback concerning my assessment, which might support the hypotheses stated her (or might not).

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What’s in that Fraction?

Sometimes students conceptualize math problems in ways that are mysterious to experts, but make sense to them.  On occasion, a bad conceptualization seems to be reinforced by features in the technology they are using.

I was helping a student work with rational expressions in our intermediate algebra course.  This particular student finds the material difficult, and often puts off dealing with the course.  Today, he was starting the first section which includes this problem:

If f(x) = 10/(x+1), find f(½)        [Presented in typical rational expression format.]

I think the student conceptualized fractions as two connected buckets (one for numerator, one for denominator) without seeing any particular meaning for the buckets together.

This student was doing most of their work on an older Casio graphing calculator, which shows fractions like this:

In other words, the calculator has a “a b/c” key used to enter fractions.  The student was trying to type in “10_½+1” so the calculator was showing ’21’ for an answer (which was a mystery answer for this student).  When I suggested using the division symbol instead of the fraction key, there was a resistance … until he discovered that it gave the correct answer for the online homework system.

I think it is pretty common to have students missing concepts in the meaning of fractions.  Frequently, they have trouble connecting a fraction with both one division AND with a combined product and quotient … where this last meaning allows for most of our algebraic work on rational expressions. Our instructional materials frequently emphasize the first concept (a single division), and never make explicit that a fraction also means multiplying and dividing … that “(3x)/(x²+2x)” means multiplying by 3x and dividing by (x²+2x).  Result: memorized rules for how we reduce a fraction.  It’s so much easier to focus on ‘multiplying and dividing by the same factor results in one’ as a concept … rather than ‘cancel common factors’ alone.

We might blame such misconceptions on an over-use of technology, or on a given calculator providing the ‘a b/c’ fraction key.  I think students have the misconception independent of the technology, and that the technology my student was using made it easier for me to identify the issue.

When a person looks for either research on learning fractions, or for suggested instructional sequences, there is agreement that a flexible and more complete set of concepts is critical for the diverse settings where fractions are used.  Our course materials (especially in developmental math, both in pre-requisite and co-requisite models) tend to focus so much on procedures that we never develop any further concepts about fractions.  That is really a shame, since students will forget the procedures; the concepts have a longer shelf life in the human brain.

We should always start with meanings and concepts … especially with fractions.

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