Common Core, Common Vision, and Math in the First Two Years

I’ve been thinking about these ideas anyway.  However, a recent comment on a blog post here got me ready to make a post about predicting the future of mathematics in the first two years.  I’d like to be optimistic … past experiences would cause considerable pessimism.   The truth likely lies between.

One of the “45 years of dev math” posts resulted in this comment from Eric:

If Back2Basics is what drifted up to CC Dev Math programs back then, what do you see the impact of CommonCore being on CC Dev Math now?

This post was about the early 1980s, when we had an opportunity to go beyond the grade level approach of the existing dev math courses (one course per grade, replicating content).  Instead of progress, we retrenched … resulting in courses which were subsets of outdated K-12 courses.  Much of the current criticism of dev math is based on these obsolete dev math courses.

We again have an opportunity to advance our curriculum.  This time, the opportunity exists for all mathematics in the first two years.

• The K-12 math world is changing in response to the Common Core State Standards.  Even if politics takes away the assessments for that content, many states and districts have already implemented a curriculum in response to the Common Core.  (see http://www.corestandards.org/Math/)
• The college math world is responding to the Common Vision (see http://www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf) which is beginning the process of articulating a set of standards for curriculum and instruction in the first two years.  AMATYC is developing a document providing guidance to faculty & colleges on implementing these standards.  [I’m on the writing team for the AMATYC document.]

The two sets of forces share quite a bit in terms of the nature of the standards.  For example, both K-12 and college standards call for significant increases in numeric methods (statistics and modeling) along with a more advanced framework for what it means to ‘learn mathematics’.

These consistent parallels in the two sets of forces would suggest that the future of college mathematics is bright, that we are on the verge of a new age of outstanding mathematics taught by skilled faculty resulting in the majority of students achieving their dreams.  This is the optimistic prediction mentioned at the start.

On the other hand, we have some prior experiences with basic change.  One example is the ‘lean and lively calculus’ movement (conference and publications in 1986 & 1989).  It is very sad that we had to modify ‘calculus’ with something suggesting ‘good’ (lean & lively) … the very nature of calculus deals with coping with change and determining solutions for problems over time.  As you know, this movement had very little long-term impact on the field (outside of some boutique programs) while the “Thomas Calculus” continues to be taught much like it has been for the past 50 years.

Here are some factors in why we find it so difficult to change college mathematics (the levels beyond developmental mathematics).

1. Professional isolation:  membership in professional organizations is low among faculty teaching in the first two years.  The vast majority of us lead isolated professional lives with limited opportunities to interact with the professional standards.
2. Adjunct faculty as worker bees: especially in community colleges, adjunct faculty teach a large portion of our classes … but are separated from the curriculum change processes.  The existing curriculum tends to be limited by these artificial asymptotes  created by our perceptions and the desire to save money by the institution.
3. Autonomy and pride:  especially full-time faculty tend to place too high an emphasis on autonomy & academic freedom, with the false belief that there is something inherently ‘good’ about opposing all efforts to change the courses the person teaches.  Although most prevalent at universities, this ‘pride’ malady is also a serious infection at community colleges.

I’ve certainly missed some other factors.  These three factors represent strong and difficult to control forces within a complex system of higher education.  Thus, I consider the pessimistic view that ‘nothing will change, really’.

I think there is a force strong enough to overcome these forces restraining progress in our field.  You’d like to know the nature of this strong force?

The attraction of teaching ‘good mathematics’ is fundamental in the make up of mathematicians teaching in college.  If faculty can see a clear path to having more ‘good mathematics’, nothing will stop them from following this path.

If the Common Core, the Common Vision, and the AMATYC new standards can connect with this desire to teach ‘good mathematics’, we will achieve something closer to the optimistic prediction.  The New Life Project has experienced some of this type of inspiration of faculty.  Perhaps AMATYC will create a new project to bring that inspiration to a larger group of faculty teaching in the first two years.

One thing we know for certain about the future:  the future will look very much like the present and the past unless a group of people work together to create something better.  I would like to think that our profession is ready for this challenge.

Are you ready to become engaged with the process of creating a better future for college mathematics?

 
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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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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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