Category: math reasoning and applications

QR Courses: Resources to Build Good Quantitative Reasoning Courses

We had a workshop this winter on Quantitative Reasoning courses (QR) in Michigan.  The information shared at that workshop is now available on the MichMATYC web site.  Here:  http://michmatyc.org/QRCourses.html

[This workshop was sponsored by MichMATYC with operational support from the Michigan Center for Student Success.]

Take a look … information from several colleges is included, and some math path maps are available on the page.

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Even Our Puzzles Are Outdated … Mathematics for 2025 (and today)

Earlier this month, the Conference Board of Mathematical Sciences (CBMS) held a forum on mathematics in the first two years; many of the presentations are available on the web site (http://cbmsweb.org/Forum5/)

As part of one of the first plenary sessions, Eric Friedlander commented …  Students in the Biological Sciences now outnumber those in the Physical Sciences in the standard calculus 1 course.  (David Bressoud shared some specific data on those enrollment patterns.)

Historically, the developmental mathematics curriculum was all about getting students ready for pre-calculus.  Our “applications” tended to be puzzles created with physical sciences in mind — bridges, satellites, pendulums, and the like.   Few problems in our developmental courses draw the attention of those in biologically-oriented fields (including nursing).

We could include:

  • Surge functions to model drug levels
  • Functions to estimate the proportion of a population needed to be immunized to prevent epidemics (P_sub_c = 1 – R_sub_0)
  • Models for spread of cancer … and for treatments
  • Pollution prediction (simplified for closed systems)

This list is a ‘bad list’ because there is no common property (except being related to biology) … and because I do not know enough to provide a better list.  Take a look at books in applied calculus for the biological sciences; you will see applications that are perhaps better than those above.

There is a trend in the new models for developmental mathematics (AMATYC New Life, Dana Center New Mathways, and Carnegie Foundation Pathways) to include a balance of applications — including more from biology.  We need to bring in more of these applications throughout our curriculum (from the first developmental course up to calculus).

Most of us realize that the ‘applications’ in our courses and textbooks are puzzles created by somebody who knew the answer; generally, these problems do not represent the use of mathematics to solve problems and answer questions in the world around us.  Sometimes, we are not able to provide enough non-mathematical information to provide representative problems … in those cases, some reduction to the ‘puzzle state’ is acceptable.

Our puzzles should represent the diversity in the uses of mathematics, with a significant portion of applications being realistic in nature.

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Is THAT the Best You’ve Got??

A student comes to college, and needs to meet their general education requirement.  One of those is in mathematics, and this student actually has some options:

  • College Algebra (called pre-calculus at their college)
  • Introductory Statistics
  • Quantitative Reasoning

Being a typical student, this student wants to avoid the college algebra course; they thought about being an engineer but are too frightened of mathematics.  The next choice would be statistics, because everybody seems to think it is the best choice.

In looking in to the course, the student discovers that the statistics course has some nice features.  Most of the material is taught by first looking at data from the world around us, and the description says that the quantitative work is somewhat limited.  The student becomes worried when they look at the content in the text materials used — it’s got words used in a weird way (normal, deviation, inference, significance); it’s like statistics is a foreign language without any visible culture, so the student feels like much of it is arbitrary.

So, the student tries to find out what “Quantitative Reasoning” means.  The course description talks about voting, networks & paths, logic, and ‘proportionality’ (whatever that is).  Like the statistics course, it looks like the material often involves data from the world around us; however, it’s not clear how much quantitative work is actually involved.  The student is not too worried about any particular topic or phrase in the content descriptions; however, the course does not seem to have any pattern to the topics … it looks like an author’s 15 favorite lessons.

The student thinks about the basic question:

Will any of these courses help me in college courses, in my work, or in my life in general?

Basically, this student will reach the conclusion that none of these three courses will be that helpful.  As a mathematician, I would summarize the basic problem this way:

  • The college algebra course and the statistics course focus on a narrow range of mathematics.
  • This quantitative reasoning course does not focus on any particular mathematics.

There is a mythology, a story repeated so often that we believe it, that statistics is a better pathway for most students.  The rationale is something like “our world is dense with data and decision making” or “making decisions in a world of uncertainty”.  I see a basic problem, that remains in spite of what has been written: statistics is an occupational science, with few broad properties or theories.  Statistics is about getting helpful results, and for statisticians, this is great.  How does it help students when we use “n”, “n – 1″, and “n + 4″ for calculations involving sample sizes; the ‘plus 4 rule’ is a typical statistical method for producing the results we want — even when there is no mathematical property to justify the practice.  [In a field like topology, we don’t let inconsistent procedures survive.]  I think we also over-estimate the value of statistics in occupations; there are limited uses in  other college courses, and some nice uses for life in general (for those motivated).

The quantitative reasoning (QR) course has a different problem — we don’t have a shared idea of what this course should accomplish.  For some, it’s an update to a liberal arts course (like the example above).  For others, QR means applying proportionality and some statistics to life.  Still other examples exist.

Is that the best we’ve got?  We are giving students options now (a nice thing), but the options are really not that good for the student.  For the student above, they really should take the college algebra course — perhaps they will find that mathematics is not their enemy after all; they might become an engineer, an outcome not likely at all with the other two choices described.

As mathematicians, we need to claim the problem and be part of the solution.  That college algebra course?  Modernize the content and methods so that it actually helps students prepare for further mathematics without becoming a filter that stops students.  That QR course?  We need professional conversations around this course; MAA and AMATYC should jointly develop a curricular model of some kind.  In my view, the QR course is the ideal general education math course; we should include significant mathematics from multiple domains, done in a way that students can discover that they could consider further mathematics.  The statistics course?  Let’s keep a realistic view of the value of this course; it’s not for everybody, and we tend to think of statistics as the option for people who never need anything else.

No, THAT is NOT the best we have.  We have some basic curricular work to do; together we can create better ideas, and help our profession as well as millions of students.

 
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Outcomes for a Quantitative Reasoning Course

When we look at reports summarizing enrollment trends in college mathematics (like CBMS; see http://www.ams.org/profession/data/cbms-survey/cbms2010-Report.pdf) the entry “Quantitative Reasoning” (QR) does not appear — which will likely change, given the increase in the number of colleges offering QR.  As a profession, we have not settled on the general nature of the learning outcomes for a college level QR course.  As a supplement to the entry of Principles for a QR course, I will list our QR outcomes; these are from our Math119 Math – Applications for Living course.

So, here is the list of outcomes:

  1. Use mathematical principles, concepts, processes, and rules to investigate, formulate, and solve problems in disciplinary and career contexts.
  2. Work with others in teamed situations using mathematical principles, concepts, processes, and rules to investigate, formulate, and solve problems in disciplinary and career contexts.
  3. Use appropriate tools and equipment, including graphing calculators, in investigating, and solving problems in disciplinary and career contexts.
  4. Use standard references and resources, both print and electronic, from disciplinary and career areas as resources in investigation, formulating, and solving problems in disciplinary and career contexts.
  5. Use measurable attributes of objects and the units, systems, and processes of measurement in disciplinary and career contexts.
  6. Apply appropriate techniques, tools, and formulas to determine measurements in disciplinary and career contexts.
  7. Use and develop formulas for applied situations in disciplinary and career contexts.
  8. Use proportions, ratios, and percents in disciplinary and career contexts.
  9. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships as they apply in disciplinary and career contexts.
  10. Specify locations and describe spatial relationships using coordinate geometry and other representational systems in disciplinary and career contexts.
  11. Apply transformations and use symmetry to analyze situations in disciplinary and career contexts.
  12. Formulate questions in disciplinary and career contexts that can be addressed with data and collect, organize, and display relevant data to answer them.
  13. Select and use appropriate statistical methods to analyze data in disciplinary and career contexts.
  14. Develop and evaluate inferences and predictions that are based on data in disciplinary and career contexts.
  15. Understand and apply basic concepts of probability in disciplinary and career contexts.

We blend occupational and academic contexts in this class, as you can see from these outcomes.  As you would expect, some outcomes are emphasized more than others.  Proportionality and percents are very important in the class; functions are emphasized using a variety of representations.

When I teach this course, I organize the content in these units:

  1. Quantities and Geometry
    Converting units (linear, area and volume) and dimensional analysis; significant digits; scientific notation; geometry (2D and 3D) applied to objects, including compound objects (2D).
  2. Percents and Finance
    Growth and decay to algebraic statements; relative change; interest; savings plan balance; savings plan payment; loan payment.
  3. Statistics
    Concepts (population, sample, bias, hypotheses, significance); confidence interval; measures of center; distributions (concepts — symmetry, variation); communicating statistical information (frequency tables, bar graphs, histograms, line charts, 5-number summary).
  4. Probability
    Calculating outcomes; basic probability; sequences of events (independent and dependent); at least once probability; counting formulas (sequences, permutations, combinations)
  5. Functions and Models
    Linear and exponential models; writing models from verbal statements; solving for parameters (finding slope and y-intercept in context, finding multiplier and starting value in context); doubling time and half life; logistic growth; solving exponential equations numerically; graphing linear and exponential functions (including creating scales for axes).

This is not an easy class.  Regardless of background, many students have difficulty with the transitions from verbal information to mathematical symbolism.  We blend presentations and workshop activities in class, and — due to student effort — usually get a pass rate about 70%.

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