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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Saving College Mathematics

The problem with reality is that it tends to get in the way of where we want to go.

I’m thinking of two recent communications.  One, a comment in response to a post here, suggested that the Common Vision will have the same fate as Calculus Reform al a 1990 … in other words, ‘n.s.d.’ (no significant difference), no impact, nowhere.  The other, a presentation by a leader of the Common Vision work who suggested that we have reach a critical mass for modernizing college mathematics.

Both speakers are experienced professionals with a strong mathematical background.  Both can site ‘data’ to support their conclusion, and both can be wrong.  [No surprise to either of them!]

Before continuing, let us consider the three types of college mathematics courses:  Developmental; Freshman/sophomore level mathematics; and upper division mathematics.  Each of these types has a unique set of forces acting on it to either change or remain the same.  The Common Vision report is directly related to the freshman/sophomore mathematics in particular.

Attempts to revolutionize freshman/sophomore mathematics have focused on part of a system.  Both the ‘lean and lively’ calculus and college algebra ‘right stuff’ dealt with content, primarily.  The AMATYC Standards (Beyond Crossroads) maintained a focus on processes (such as instruction or assessment), though “BC” was hardly calling for revolutionizing college mathematics.

We should consider what has led to a fundamental change in developmental mathematics.  The process that is leading to long-term basic change (a good revolution) is driven by three compatible projects which focus first on the content and second on process.  [These efforts are the Carnegie Pathways, the Dana Center Mathematics Pathways, and the AMATYC New Life project.]  The three projects collaborate in basic ways, even though they could be seen as ‘competing solutions’.

For this purpose, I will ignore the co-requisite movement, which seeks to displace developmental mathematics without impacting freshman/sophomore mathematics in any significant manner.  Such an effort has a low probability of long-term survival, though it certainly will create some unintended changes.

The developmental mathematics revolution is working (though it is not yet complete) because the work appeals to mathematicians and because the modern content encourages active learning methods.  There is also a continuity with prior professional work, and the engagement of diverse stakeholders in the process.

If we seek to save college mathematics, the core of our work is the freshman/sophomore curriculum.  A variety of forces are acting on this work to make ‘revolution’ difficult; even modest reforms seem to be too much of a challenge.  However, I think the largest sets of forces in this matrix have their origins in us … the mathematics faculty of colleges.

We worry about ‘transfer’, and we sorry about ‘prerequisite material’.  The transfer worry means that we don’t change because our sister institutions might decline the transfer … the prerequisite worry means that we don’t change because it might disrupt a mythical sequence of necessary steps.  In many ways, the transfer worry feeds off of the prerequisite worry.

In many states, the transfer worry is managed by a state system.  In most of these systems, the decisions are made by ‘us’ (college math faculty).  Therefore, the transfer worry is a self-imposed set of forces to resist change.  Clearly, the solution is to develop a consensus that change is needed … which means to support colleges who are willing to begin the revolution in mathematics.

Earlier I mentioned the ‘critical mass’ comment.  This observation was based on evidence of process changes (mostly, in active learning and some social psychology) primarily in R1 institutions (research universities).  Although these changes are welcome and help students, I don’t think the long-term impact will be anywhere near large enough compared to the problems we seek to solve.

College mathematics, especially freshman/sophomore level, is defined by content and structure defined by the needs of a 1965 education for a 1955 occupation (engineers especially).  Any long-term solution has to address the known needs of today’s education for 2010 occupations (a more diverse list).  Modern teaching methods are not enough.

Saving college mathematics requires that we change the mathematics.  Sufficient information exists to develop a new system of courses.  Instead of two college algebra or pre-calculus courses followed by three semesters of symbolic calculus … perhaps we can design a system with one pre-calculus course followed by two semesters of calculus which combines symbolic and numeric methods.  People more experienced would know how to structure this work, so that both content and process are modernized.

We need to stop the pattern of ‘solving part of the problem’.  Solving part of a problem is failing to solve the problem.

It’s time to build a college mathematics system that solves problems and serves our students.  We can’t let “reality” prevent us, because often we are that reality.

 
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