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 6¢ 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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Never DO Mathematics in a Math Class?

Within our efforts to make major improvements for our students, both in developmental and gateway college math courses, we have been looking at our content and our methodologies.  I want to connect the over-used phrase “Do the math!” with our phrase “doing mathematics”, within this process of building a better future.

I’ve commented before that the phrase “do the math!” is frequently used as a propaganda technique, to imply that everybody would reach the same conclusion as the speaker (and often used when the ‘math’ in the statement involves a small set of numbers, far removed from meaningful evidence for any argument).  I wonder if our phrase ‘doing mathematics’ serves a similar purpose within the profession.

Part of our problem is that we assume that there is a single meaning for ‘doing mathematics’.  Historically, the phrase seems to have grown out of the view of mathematics as seeking patterns, often in a constructivist approach; this ‘doing mathematics’ just means to immerse people in a situation involving quantities with a goal to establish patterns and statements that make sense to the learners.  Within professional mathematicians, we have two contrasting meanings — occupational (actuaries, for example) using systems of mathematical knowledge, and researchers using a variety of tools to establish new knowledge or applications.

As a learning tool, the “doing mathematics” has very limited benefits for the student.  Most research I have read suggests that learners need a very directive structure for the learning to occur by discovery; this guidance takes the process out of the original meaning of ‘doing mathematics’ into the more appropriate ‘learning mathematics’.  Doing mathematics, with the goal of learning mathematics, is a very advanced process — it is what some experts can do; expecting novices to engage in this process is a bit like expecting novices to become good piano players by having them sit at the piano (without any technique, without any theory).  Doing mathematics to learn mathematics does happen, often not by design, frequently with great excitement by those involved.

Perhaps the question is:

If students can experience what we experience when we ‘do mathematics’, they will be motivated to learn more mathematics.

Now, motivating students is one of the central roles in our classrooms.  Sometimes we focus so much on content and skills that we provide no information on our world of mathematics.  If we are the ones doing mathematics, presented in a way that novices can follow, then I can see some real benefits.  I have tried to do this in all of my classes; with colleagues, I use the phrase:

Students will see and perhaps do some beautiful and useless mathematics in every math class.

I include ‘useless’ in the description, and actually focus on that.  Why?  Because it is not reasonable to expect students to understand the eventual usefulness of the mathematics which they can appreciate; to them, it will likely seem “useless” even though it is not to a mathematician.  When I do this, I am walking a little beyond the limits of what students currently understand; I am in a beautiful field on the other side of a path, and want to share this perspective with my students.  My honest answer for ‘why I do this’ is simple:  It is fun!  I also have pedagogical reasons; walking a little beyond the current level helps to create an atmosphere of respect and one where learning for learning’s sake is appreciated.

So, my advice is: never have students to mathematics in a math class.  Help them learn mathematics, and you should do some mathematics in class so they know why we are mathematicians.  We don’t do mathematicians just for the money, or the fame.  We do mathematics because we want to.

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AMATYC 2014 … The New Life Project

Here is an ‘early alert’ — among the sessions about the New Life Project at this fall’s AMATYC conference are two sessions that I am presenting.

The first session is “The Missing Link: Algebraic Literacy to Replace Intermediate Algebra“.   Here is the session description:

Does Intermediate Algebra work? Not so much! Come learn about a better model for a course that connects your students to
college mathematics, while supporting general education. Algebraic Literacy is a course from the New Life model, designed to
meet the quantitative needs of college algebra, pre-calculus, and science courses.

This session is scheduled for November 14 (Friday) at 8am.

The second session is “Accelerate and Improve Developmental Mathematics: The New Life Model“.   Here is the session description:

Acceleration is necessary, but not sufficient. Typical developmental courses deliver outdated content to our students. The
New Life Model enables many students to complete their developmental mathematics in one semester while receiving the
mathematics they really need. Learn why 40 colleges have implemented Mathematical Literacy and/or Algebraic Literacy.

That session is scheduled for November 15 (Saturday) at 2:15pm.

My goals with both sessions are to provide information about the curriculum reform work from AMATYC’s Developmental Mathematics Committee and the New Life Project subcommittee; this work is not a random branch of mathematical history — it builds on decades of professional work by other groups and teams, and uses that work to design new solutions.  The New Life courses (Mathematical Literacy, and Algebraic Literacy) can be offered side-by-side with traditional courses, and can also be a total replacement for existing courses.

I hope to see you in Nashville for the AMATYC conference.  Spread the word about these sessions!

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