Acceleration as Distraction

One of the tricks used to increase traffic on a web site is to incorporate ‘hot phrases’ in to the pages and articles.  In our field, “acceleration” is a very effective phrase to use.  Sadly, acceleration is not that important … by itself.

Like most colleges, mine has some acceleration models — two-courses in one, boot camps, and self-directed study for example.  Some acceleration work gets very high press coverage, such as the Austin CC ACCelerator program (see http://sites.austincc.edu/newsroom/accs-accelerator-and-developmental-math-course-wins-praise-of-second-lady-and-under-secretary-of-education/ )

Acceleration is better than not accelerating … or is it?

One of my friends tends to use medical analogies in our conversations.  I am envisioning him saying something like this:

A doctor knows that three lab tests being required are without any benefit to the patient (no diagnostic nor any treatment benefit).  What the patient needs is a new treatment, but the insurance will not cover the new treatment.  Is our profession better served by making the three useless tests quicker for the patient … or by working on fixing the basic problem of getting the right treatment?

Our goal should be to fix the problems.  Acceleration is not the basic problem … what needs to be changed is the mathematical treatment provided to students so that there are multiple benefits for students.  In developmental mathematics, our work needs to focus on capabilities that serve all college programs with a focus on quantitative reasoning.  In college level mathematics, our work needs to focus on empowering students for programs or groups of programs.

Acceleration tends to reinforce the current curricular system by masking a symptom (too long to complete).  An emphasis on acceleration distracts us from working on core problems.

I believe that we need fewer courses.  We can start with a course like Mathematical Literacy (or Quantway, or “FMR”), with just-in-time remediation as needed.  The next level can be a course like Algebraic Literacy (or STEM path I), again with just-in-time remediation for students who did not need an entire course before it).  We only need one course to connect that level with calculus I — a deliberate design of a pre-calculus course.

We can do better than acceleration.  With new ideas of content and course design, we can provide important mathematics for our students in an efficient manner.

“Needing acceleration” is direct evidence that the basic curricular structure is inappropriate.  Don’t worry so much about acceleration — fix the basic problem.

 
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When Does Reform Succeed?

I have been thinking lately of a problem considered back when we started the AMATYC New Life project (about 2008).  The problem is not mathematical in nature, which perhaps explains why we have not solved it before.  Now, I am not saying that we found the only solution; I’m not even sure that our solution is sufficient.  I can report that our reform has grown way (way!) past any prior reform of developmental mathematics.

The problem is this:

What properties or methods enable a curriculum reform to succeed over a period of years and across regions?

Prior to our New Life work, many intelligent people had created valid reform ideas or models.  None of them survived time and space; they resulted in temporary changes (in general) and were limited to a few locations (at most).

For those interested in such things, here are my thoughts on strategies the result in successful reforms.

1. Professional organizations need to be deeply involved.

The New Life project was born in the AMATYC Developmental Mathematics committee, which had a large group of faculty willing to work on the project.  In addition, several members of the AMATYC Executive Board both supported and contributed to the work.  The involvement of the national leaders of a group enable that reform effort to connect with similar reform efforts by other groups (see below).

2. Content in the reform math curriculum created by faculty in a collaborative process, based on professional references.

If you look at the material over in the New Life ‘wiki’ (dm-live.wikispaces.com) you will notice that the learning outcomes were drawn from multiple professional sources (MAA, AMATYC, NADE, Numeracy Network, etc).  Both parts of the process were important — collaboration resulted in content that was widely accepted by math faculty, and professional resources helped create content that had external validity.

3. Avoid a focus on one issue.

In general, a reform effort built on one issue is very unlikely to succeed.  That one issue will not appeal to the general math faculty population.   For example, the NCAT redesign work tends to deal (in the curriculum) primarily with technology; as in prior calculator-based reforms, people find that this is a weak motivation for reform.   Addressing multiple issues in the reform means that most faculty will see something they like, which is a critical property for getting the reform adopted.  In the case of New Life, we addressed several content issues, classroom pedagogy, and professional development.

4. Plan for, and support, long-term conversations with faculty.

For some reform efforts, advocates did not sustain conversations with faculty over a period of time.  Only a few faculty will accept any reform when they first hear about it; one could argue that these faculty are actually not good test cases for a reform.  For the New Life project, we sustained conversations online (email, wiki) and at many conferences, for over 4 years now; in addition, we have had people travel to put on local workshops.  In our case, these conversations often result in faculty concluding that teaching our reform course is just more fun than what they have been doing; this is a powerful force for reform.

5. Create or support multiple solutions sharing basic properties.

No matter how good one particular reform model is, some faculty will not be comfortable with it; some institutions or states involve conditions that conflict with a given solution.  Our New Life project is one of three closely related solutions:  Carnegie Foundation Pathways (Statway, Quantway), Dana Center New Mathways, and AMATYC New Life.  The three projects have collaborated, shared resources and talent, and provide faculty & institutions with choices.    The New Life project itself supported multiple solutions — we depend upon commercial textbooks, and each major publisher is creating a solution.

6. Do not depend upon “one good book”.

Prior reforms, at all levels, often involved the creation of one set of materials.  New books face several challenges both in publishing and in getting adoptions.  A single book is just not going to be good enough to result in reform long-term.  The current reform in developmental mathematics involves commercial texts, foundation developed materials, and self-published materials.

I think other areas of college mathematics need basic reform, some perhaps even more needed than developmental.  I want reforms to succeed in ‘college algebra’, pre-calculus, finite math & modeling, calculus, quantitative reasoning, and statistics.  These courses impact hundreds of thousands of students every year; the impact is not uniformly positive.

As  you look at the points above, I am hoping you reach the single biggest conclusion:  Reform is something we do together, with each other, over a period of time.

 
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College Algebra … an Archeological Study

As we make real progress on improving mathematics education in colleges, shown especially in developmental mathematics, our attention is going to focus on college algebra and the “STEM Path”.  Of course, the name “college algebra” is given to a variety of courses, some of which serve a pre-calculus purpose (and some do not).  For years, I have thought of the name (college algebra) as a statement of “not being remedial”.

Could be that I was wrong about that.  During some recent searching, I came across a paper that Jeff Suzuki gave a few years ago.  Most of that talk is available at https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxqZWZmc3V6dWtpcHJvamVjdHxneDo2MWI5YWE4YzU2MDM1MmY3; if you have trouble with that link, search for “Jeff Suzuki Project” to get a list of his presentations.

Assuming that the content of that history is essentially correct, here is a brief statement of what college algebra is today:

College algebra is a collection of mathematical topics for general education, taken in place of calculus.

Some of the information in the Suzuki paper is in the form of book references to the 19th century.  This led to a book, possibly the first, to use “College Algebra” in the title — George Wentworth’s “A College Algebra” (1888); a later edition (1902) is available at https://archive.org/stream/acollegealgebra07wentgoog#page/n12/mode/2up .  In the same period, Webster Wells authored “University Algebra” (1879) and “College Algebra” (1890); see the 1879 text at http://books.google.com/books?id=uKZXAAAAYAAJ&pg=PR7&source=gbs_selected_pages&cad=2#v=onepage&q&f=false

These courses were taught as universities (Harvard, Yale, Princeton, Bowdoin, etc) reduced their mathematics requirements.  The college algebra course was not designed to prepare students for calculus.

These early college algebra books did not contain some current topics (factoring and graphing, for example).  The addition of graphing (including properties of functions) is related to calculus preparation; factoring is generally not so related.  Overall, the current college algebra course is clearly a descendent of this earlier course.

One of my current projects is to study the math courses required before calculus in my state (Michigan); Michigan does not have a system for higher education, which results in diversity in mathematics — college algebra, precalculus, and other courses are used.  However, the overall approach (in Michigan and elsewhere) is to consider these as being an equivalence or subsets; either the college algebra course(s) equate to the pre-calculus course(s) OR the college algebra course is a prerequisite to pre-calculus (that is very rare in Michigan).

Therefore, I believe that this is our current method of preparing students for calculus:

After establishing that the student does not need further remediation on high school mathematics, the student enrolls in an antiquated general education math course with a few valid preparatory topics, with the unreasonable hope that this will prepare them for calculus.

Much of our apparent curricular dependency (stuff in college algebra that is needed for calculus) seems to be an artificially created dependency — we need this radical simplification because that technique is needed for a few problems in calculus, and those problems were created in calculus to show why we needed radical simplification; we need this multi-step factoring topic in college algebra because we have created a set of problems in calculus that require creative factoring.

I encourage us all to study the “Mathematical Sciences in 2025” (available at http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025 ).  Some parts of our curriculum are archeological artifacts from the 19th century, and some parts date from the mid-20th century.  Very little of our curriculum reflects either current needs of client disciplines; not much more of it reflects the needs of mathematical sciences.

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Statistical Doors Into Mathematics

That’s really a question — does statistics create a door into mathematics?  Or, is statistics (for most students) an alternative off-ramp from the mathematics highway?

The question is perhaps trivial.  In terms of the bulk of our work, we are dealing with students required to take specific courses for their program.  Every math course becomes a common off-ramp for students.  Perhaps we should be satisfied with a curriculum consisting of terminal courses for students interested in everything else but mathematics.

One of my colleagues began her higher education as a fairly typical community college student at our institution.  She reports that a turning point for her was a particular computer science course that she decided to take; after this course, she changed her major and got a degree in computer science (and later a masters in math).  There was something of beauty in that computer science course that connected with her, and changed her life.

I would be interested in any research on the question:

Do students change their career path to mathematics after taking a statistics course?

I am sure that there are students who change their path to statistics after a statistics course, though I wonder if the rate is equal to that of ‘math program after a math course’.

Like most of us, my students are just interested in passing this math course so they can get their degree or that job.  I am fine with helping them along that trail; in fact, I am happy to do so.  I teach because I find that rewarding.

However, I am also a professor is in “affirms a faith in something”.  I think I have a responsibility to show students in each course something about the beauty of mathematics; something wonderful should show in every class.  Partly, this is needed to encourage more positive attitudes about mathematics; partly, this is needed to encourage a more accurate view of the nature of mathematics, that mathematics is much more than processes to generate answers.

To me, however, the largest reason for what I try to do is “opening doors”.  A major reason for lowering expectations for a given student is mathematics; lower-skill programs are selected because they require less mathematics (or none).  Students even avoid occupations that they would love to be in … just due to mathematics.  To me, every mathematics course should be a STEM magnet drawing students towards higher skilled jobs and more security.

I do not think that statistics operates as a STEM magnet.  Of course, there are many math courses in our institutions that are not STEM magnets; however, almost all math courses could be strong attractor points drawing students towards mathematical sciences.  I think the problem with statistics is that we teach statistics as a practical discipline without a core mathematical structure.  We focus on the innate appeal of statistics, on its utility; perhaps we need to show the mathematics supporting statistical methods when possible.  If there is no mathematics supporting a method (the ‘plus 4’ rule type of thing), perhaps we should question the presence of that method in a general statistics course.

Clearly, I may be demonstrating levels of ignorance vast and wide.  I wonder, though … do we share a view that math courses in the first two years should have a property of ‘STEM magnet’?  Can a statistics course be such a magnet?

Before the reader decides that I am far too optimistic about our mathematics courses — yes, I know that we fall far short of a STEM magnet in our current courses.  We tend to cede our territory, and deliver service courses; we focus on the practical at one extreme … or the totally useless on the other.  In between is the zone needed to be a magnet for students; a magnet can not be unidimensional.

Perhaps the question is more general than statistics; my concern is with the contemporary move towards requiring statistics as the typical general education course.  Perhaps the loss is trivial.  I do wonder if there is an innate qualitative difference between statistics and mathematics that results in statistics being far less able to contribute towards larger goals such as raising student goals and drawing students towards STEM.

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