Principles for a Quantitative Reasoning Course

We are seeing a large increase in the number of quantitative reasoning (QR) courses, which I think is very good for our students.  Here are some thoughts on principles that might be used to both design and teach a QR course.

PLURALITY OF MATHEMATICS
In our society, people often say that they are terrible at math.  They might be correct — because they are thinking mostly of arithmetic, which is usually a long collection of apparently arbitrary algorithms and rules for determine correct answers (all of which are more easily achieved with a calculator).  In some ways, this is our fault as we tend to have an entire course on arithmetic then an entire sequence of courses on ‘algebra’, and then courses on other domains of mathematics.  A quantitative reasoning course exists primarily for general education; therefore, the content must reflect the plural nature of mathematics.  Three domains should always be represented — proportionality (arithmetic, for unit conversions and scaling), statistics (concepts and communication), and functions (algebra, both linear and exponential models).  These 3 standard domains should be supplemented by one or two others to fit local needs and the wishes of the instructor (voting, probability & counting, networks, etc).

DISCUSSION TO SHIFT ATTITUDES
If a quantitative reasoning course leaves the students with the attitudes about math present at the start, we have missed a great opportunity.  I seldom talk directly about changing attitudes in class (except when students claim they can’t do math).  I have found it effective to immerse students in discussions about situations they can understand which involve important mathematical ideas.  The vast majority of students have negative attitudes reflecting a general experience, as opposed to trauma-triggered; I have seen most students shift in their attitudes over the semester, just from the discussion experience.  [I do these about 8 times a semester, often as initiating lessons.]

 

DISCUSSION TO BUILD REASONING
Those discussions also serve a direct instructional purpose:  Students are just not accustomed to reasoning.  Because of the prevalent attitude, they expect problems to be solved by knowing the correct procedure.  We work on understanding and connections in these discussions.  I embed a little bit of strongly guided discovery learning in the process — a deliberate series of questions for small groups to work through.  I think many students improve in their reasoning skills just by the experience of hearing themselves (and others) talk about the problems in these situations.

 

REASONING FOR NON-STANDARD PROBLEMS
I have told an administrator at my college that our QR course is the only course where students solve a number of non-standard problems.  It’s not like every problem is non-standard, but a large portion are (I’d guess about 20% to 30%).  One problem this last week was to determine the area of a lake given the volume and depth; we had not done this before the students saw the problem, though they had faced problems involving unit conversions in many situations.  We need to create a classroom environment where every student believes that they can figure out most problems that they face.  If all problems are similar to prior experience, we avoid this wonderful outcome — doing exercises is not problem solving.  [Both processes have a role in a QR course — the repetition of exercises strengthens knowledge, and problem solving strengthens the reasoning.]

ASSESSMENT: KEEP THE RIGOR
The history of QR courses connects to the much older “liberal arts math” courses.  Some liberal arts math courses are experiential and appreciation based — no particular performance required.  In a QR course, we need to keep the college-level rigor in the assessment.  Students need to know that they can not pass simply by attending every class, or even by writing a nice paper or two.  Skills and applications, including problem solving, need to be at the core of the assessments and the majority of the student’s grade.  A course with a QR focus has a different focus than the old liberal arts math class: we are not offering a course for students who don’t need math … we are offering a math course for students who need math (just not math in one domain).

BRING IN LIFE … BUT DON’T PUT THE COURSE IN THE BOX
Context is powerful.  When we cover mathematics applied to a problem that students care about, motivation is not an issue.  However, there is a fundamental difference between a QR class and an occupational math class:  a QR class is general, and contributes directly to a general education.  We need to include mathematics and problems which do not necessarily seem important to our students … we need to bring what we think is important to the class.  This might mean covering mathematics which we think is important in general, or it might mean solving problems that are more of a puzzle than a problem that students would care about.  When we put a course in the box of ‘context’, we are not helping our students.  Learning math for its own sake is part of learning for its own sake, and college should encourage all learning.  Besides, we should be showing students some of the mathematics that caused us to become mathematicians — it’s not always about what you can do with the mathematics, sometimes it is just how wonderful the mathematics is.

HAVE FUN
The typical mythology surrounding math professors is that we are not fun to be around, that we are not creative, and do not value differing points of view.  Our QR classes should be fun environments where creativity and points of view are used to learn mathematics.

 
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2 Comments

  • By Ryan Blankenship, May 15, 2014 @ 12:35 pm

    Jack:

    Having trouble finding a “Goals and Outcomes” pdf for QR that exist for MLCS and Alg Lit. Does that exist? If not, is there a time line?

  • By Jack Rotman, May 16, 2014 @ 9:07 am

    Ryan:
    The MLCS and Algebraic Lit documents are posted on the “Instant Presentations” page (https://www.devmathrevival.net/?page_id=116). Your question has me thinking about the QR outcomes as well; since QR courses are not standardized yet, there is no ‘master’ list of outcomes. However, I can post what we do in our QR course (Applications for Living).

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