Category: math reasoning and applications

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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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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Graphing Functions, Algebra, and Life

In our Applications for Living class, we are taking the last test before the final exam.  The primary topic for this test  is ‘functions and models’, where we cover the use of linear and exponential types — including finding the function from data and graphing functions.  What follows is a list of observations about what students seem to understand and what students tend to struggle with.

SLOPE — Everything is a linear slope to many students.  Even when the problem says specifically “Find the exponential model” [and the reference sheet includes the exponential model y=a(b^x)], a majority of students use the slope formula when we first do this.  After 3 or 4 visits to the idea in class, the majority work with the correct model — however, a third (or more) cling to the slope calculation when starting any ‘find the equation’ problem.  It’s worth noting that our beginning algebra class does not deal with any non-linear graphing.  The students who come from our Math Literacy course have an advantage — they have experience with linear and exponential graphs.

GRAPHING — Plotting points is seldom an issue … if given a pre-scaled coordinate system.  However, students struggle with the concepts of dependent and independent variables; we’ve gotten pretty good at that discrimination since the class dealt with the idea for 4 consecutive classes.  That does not mean that students know that dependent values usually are placed on the horizontal axis; I’ve seen some beautiful graphs which have the dependent variable on the vertical scale.  We talk about graphing equations as being a matter of communication, just like we did for statistical graphs; people expect the dependent to be on the horizontal axes.

GRAPHING — Scaling the axes is not easy.  We learn a routine for determining the scale size (1, 5, or whatever), and that helps.  However, many students do not see a problem with unequal intervals on their graph — especially for the independent variable.  Whether we are using graphing to communicate in a science class or in an article on global warming, equal intervals are critical.

USING MODELS — We are doing both types of problems with both types of models … we are given values of the independent and calculate the dependent, and we are given values of the dependent and solve for the independent.  In the case of exponential models, we solve for the dependent numerically via a calculator program to find the intersection.  Students seem to have a predisposition to calculating independent values; they ask if I will include the word “Intersect” in the problems where that is the correct procedure.  This is a case where the difficulty with dependent vs independent variables collides with selecting a strategy.

EXPONENTIAL FUNCTIONS — We started our work with exponential models in week 4 (12 weeks ago), when we did percent applications.  We did them again when we worked with finance models (like annual compound interest).  We did percents within probability, where we covered repeated probabilities (acting like a basic exponential model).  In the last 3 weeks, we have talked about how we discriminate between linear and exponential change based on descriptions, with ‘percent’ being the most important concept for us.  This spiral definitely improves understanding of percent problems, but students still struggle with exponential functions.  There is a tendency to use the percent as the multiplier (using .03 instead of 1.03), and some students treat the multiplier as a slope value in a linear function.  We make progress, but I would like students to be able to apply exponential equations in other classes and in life.

Here are some problems from the test we are doing:

The price of computer memory is decreasing 5% per year.  Write the exponential model for the price, and use the function to predict the price in 3 years.

The price of fresh oranges is expected to increase by 6ยข per week for the next few months.  The current price is $1.19. Write the linear function for the price, and use this to predict the price in 8 weeks.

I can purchase a motorcycle for $10,504, or I can lease it for a down payment of $750 and monthly payments of $155 per month.  Write the equation that describes the cost of the lease.  Use the equation to find how long I can lease the motorcycle before I pay more than the purchase price.

A rain forest is decreasing at a rate of 12% per year.  In 10 years, what percent of the current rain forest will remain?

A drug follows an exponential model.  After 3 hours, there are 16 mg in the body.  After 4 hours, there are 12 mg in the body.  How much will there be after 5 hours?   [Comment:  This is missed by many students.]

Twenty mg of a drug are administered at 4am, and the function y = 20(0.90^x) shows the amount of the drug in the body after x hours.  When will there be 6 mg of the drug in the body?  (nearest tenth of an hour)

I’ve made several adjustments to how I do the class to help with the struggle points described.  I can see improvements, and I can see individual students improve.  Overall, I am actually pleased with the results.

I hope you will continue to design your classes so that students understand the mathematics in a way that they can apply the ideas.

 
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