Plus Four — The Role of Statistics in Mathematics Education

We’ve all been reading and hearing about this … statistics is critical in modern society (in a ‘world of uncertainty’, as some people say).  Some argue that all students should take an introductory statistics course in college, while others argue that some statistics be included in a general education course.  One of the latest blog writings on this is from Kevin Carey (see http://chronicle.com/blogs/brainstorm/everyone-should-learn-statistics/46353)

There are fundamental differences between statistics and ‘other areas’ of mathematics.  [I put ‘other areas’ in quotes because I question an equivalence between statistics and mathematics as scientific endeavors.]  In most ways, statistics is a laboratory science based on logic used to support or refute claims in an argument.  Some specialties in mathematics are much like this, though those specialties do not normally show up in community college mathematics programs. 

What is unsettling about an emphasis on statistics as a ‘better math requirement’ is that statistics supports methodologies based on getting better results … not just on properties of measurement.  Ever heard of “plus 4 confidence intervals”?  The ‘plus 4’ method is used for creating proportion confidence intervals; see http://www.math.metrostate.edu/mike/Course/Fa2006/ConfidenceIntervals.pdf for some background.  Essentially, an arbitrary adjustment is made to the sample size (n+4) and to the success outcome (p+2); this adjustment provides confidence intervals that are judged to be more appropriate.  As you can see, the ‘plus four’ procedure results in a success outcome closer to 50% for use in the confidence interval; there is no scientific basis given for doing this at all … we could use ‘plus 2’ or ‘plus 6’, why ‘plus 4’ (besides the circular argument of producing the results we want)?

In developmental mathematics, we spend a lot of time (traditionally) on “percents”.  In statistics, percents are usually a proportion statement (when used as a measurement); overall, however, percents are a label (as in ‘95% confidence interval’).  A transition from the mathematical percent (calculation based) to the statistical (a label) is difficult for students.  Before you mis-interpret, let me add … I teach a course where we use both types of percents (in quantitative reasoning), and I think that this is good for students.

I have no problem with considering statistics part of quantitative reasoning; in fact, I can not envision teaching a quantitative reasoning course without significant coverage of statistics.  What I object to, or at least question, is the presumption that statistics is an alternative to mathematics in general.  Mathematics, in general, is applied theories about quantities where consistent meanings and interpretations are essential; in mathematics, our work is constrained by whether the results are consistent with established knowledge and theory.  Statistics … a wonderful field of study … is more flexible in its methodologies; competing theories with differing results are accepted and expected in statistics.  For some students, statistics (a lot) is what they need.  For most, however, not so much.

We should not use statistics as the only general education ‘math’ course for a global audience.  Even algebra and calculus have a role in general education.

 
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Modules or Clumps in Developmental Math?

A lot of people are talking about modules in developmental mathematics as part of an effort to ‘fix’ our programs.  Of course, the word ‘modules’ has implications that sometimes are not meant … so I was inspired when I saw somebody refer to them as ‘clumps or modules’. 

The word ‘module’ carries connotations, and also has a denotation.   You might be surprised to learn that ‘module’ does not have a denotation (definition) relative to the practice of clustering learning outcomes into small pieces (‘clumps’), nor with process of assigning a subset of ‘clumps’ to a given student.  Most dictionaries will not give an educational meaning to ‘module’.  What we are doing here is describing by metaphor — “this is like modules in electronics where sub-systems are replaced as a unit”. 

Which leads in to the connotations.  When we think about ‘modules’, we usually have positive images — easy, efficient, better.  “Modules” has a scientific sound, as if describing by metaphor automatically assigns a scientific basis.  I suspect many people think that ‘modules’ means that we are meeting students’ needs, and that the program is individualized.  Some people believe that ‘modules’ mean that students spend more time actively doing ‘mathematics’.

Using modules does not mean anything more than using ‘clumps’ would mean.  Of course, a particular implementation of clumps (or modules) might mean a great deal of good stuff.  Too often, using ‘modules’ means that we focus on the delivery system to the exclusion of critical analysis of the content (beyond creating clumps).

In practice, there usually is one difference between using ‘clumps’ and using ‘modules’.  With ‘modules’, there is not (normally) any summative assessment at the end of a ‘course’.  This means that there is no need, from the student’s point of view, to integrate knowledge and understand how parts fit together.  “Connections”, in a modular math environment, is limited to those that can be developed within a single module.  “Clumps” might share this property, but ‘modules’ almost always do.

As mathematicians, we have shared values — reasoning, application, relationships, representations, and even creativity.  Whether you call them ‘modules’ or ‘clumps’, do our values come through?  This really is important; imagine a freshman writing class where students learned about components of writing in isolation, and never had the opportunity to develop a position or argument.  Like writing, the purpose of mathematics is centered on communication.  Let’s build courses where our values are accessible to students.

 
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Treisman & Rotman Webinar – June 6 (AMATYC)

Uri Treisman and I have been involved with efforts to systematically reform developmental mathematics, such as New Life, Carnegie Pathways, and the Dana Center New Mathways.  Uri has been very supportive of our AMATYC work, including the New Life project.

On June 6 (4pm Eastern), we will be doing a joint Webinar on Issues in Implementing Reform in Developmental and Gateway Mathematics as part of the AMATYC webinar series.  The goals of this webinar are to present some general concepts to guide our work in reform, and to share some practical means to implement those concepts.

Here is the way the AMATYC webinars work — AMATYC members can register for a webinar (at http://www.amatyc.org/publications/webinars/index.html).  Registration usually begins about two weeks before the event (so you won’t see this one listed in April!).  AMATYC members who register will receive an email with directions (the day before the webinar). 

One thing to point out — people can watch the webinar as a group!  One person needs to be an AMATYC member and register; you can include non-members in the viewing process.  (The directions you receive will even tell you how to make the group process work better.)

I hope you can join us for this webinar.

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Math on the Other Side

A recent post here dealt with the metaphor of developmental mathematics as a bridge, designed to help students reach the other side.  The ‘other side’ is not just mathematics (would we really want that?), with a diverse collection of courses … some of which are called ‘gateway courses’, while others are ‘just’ college courses.  So, the question today is “What about the math on the other side”?

Is the math ‘on the other side’ the good stuff (important mathematics)?  Do courses ‘on the other side’ place a high priority on student success?  If we reform developmental mathematics in to a program which makes a difference in the mathematical learning of students, will their ‘college math courses’ have the same vitality?

These are questions which I can not answer; I am not immersed in the world of college-credit math classes (just parts of it).  However, I do know that our profession is rather silent on this component of our curriculum … we are talking a great deal about developmental mathematics, and I hear quite a bit about STEM and calculus.  Not so much about college algebra … pre-calculus … liberal arts math … or math for elementary education majors.

The easy target in this list is college algebra.  Pre-calculus … at least we know what the goal is (calculus), and students taking pre-calculus can be assumed to have that goal (even if incorrectly assumed).  However, we have absolutely no agreement on what ‘college algebra’ is.  For some of us, college algebra is what we happen to call our pre-calculus course; for this group, I would say “Hey, be honest … call if pre-calculus!”   For others, college algebra is actually a prerequisite to pre-calculus; on this … “how much time is needed getting ready for calculus?”  [Perhaps we place additional steps in between to make sure that only the best survive; I hope not.]  For still others, college algebra is a course outside of the pre-calculus sequence, perhaps used as a preparation for symbolic-based science courses; this is a good reason to have a course … though I question whether ‘algebra’ is the majority of what the students need.  Some use ‘college algebra’ as a general education course; I suggest to you that a course could be either college algebra OR general education … but not both.  One of the problems with the ‘college algebra’ label is that the traditional developmental math courses generally have ‘algebra’ in the titles; is ‘college algebra’ more of that developmental stuff?

Perhaps my worries here are just due to my extensive ignorance of some aspects of our curriculum.  Perhaps, outside of the college algebra mess … perhaps we have generally sound mathematics and important ideas in our curriculum.   Perhaps my problem is that I look at textbooks.  If most of my colleagues who specialize in these courses tell me that ‘things are okay’ on the other side, I would certainly be relieved.  However, with all of the current focus on developmental mathematics, it is possible that we are ignoring something equally important.

In our bridge metaphor, are we working on improving the bridge … just so that students can be delivered to a great wasteland of college mathematics on ‘the other side’?

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