Modern Pre-Calculus Course

Good questions are very helpful.  At a presentation recently on the Theory of Everything (Theory of Everything presentation Oct2018) one of the participants in the session asked:

What do you mean by a ‘modern math course’?  What would be in it? How would it be different?

Being a good question, I could not give that good of an answer at the time.  However, it seemed like such a good question that I should make an attempt to provide a good answer.  The initial domain for the answer is “Pre-calculus”

In order to understand how inadequate the current courses are, we need to understand ‘modern’ thinking about learning and learning mathematics in particular.  Much of the traditional college algebra and pre-calculus experience is based on the presumption that working 10000 problems with some success will prepare you for a course with a higher conceptual basis and greater cognitive demands.  One reference for modern thinking about this comes from the book “Adding it Up”, which focuses on K-12 mathematics (https://www.nap.edu/login.php?record_id=9822) which provides an image to help us visualize the learning of mathematics:

We generally understand the names for these 5 strands, and we often talk about them with our colleagues.  However, the courses currently only address two strands directly (procedural fluency and (to a lesser extent) strategic competence).  Some of us use active learning strategies which (coincidentally) provide some support for the other strands.  A basic premise of a scientific approach is that “things we want do not happen when we want if we do not plan and act intentionally”.

A core problem in college mathematics is our separation of classroom practices from content decisions.  If your instructional practices encourage conceptual understanding within a course which does not directly state ‘conceptual understanding’ as a goal, there is a mis-match between instruction and content … and this will always result in reduced student outcomes because the assessments are likely driven by the learning outcomes.  So, here is the first standard for a modern pre-calculus course:

The learning outcomes in pre-calculus represent all 5 strands of proficiency, and instructional practices support the success of students in all 5 strands throughout the course.

In general, the learning outcomes for the Dana Center “Reasoning With Functions” reflect this type of approach.  Here are the outcomes:

• DCMP_Reasoning with Functions I_Overview and Learning Outcomes
• DCMP_Reasoning with Functions II_Overview and Learning Outcomes

As an example, a traditional pre-calculus course might list this learning outcome:

Represent and recognize functions

A modern course might list this learning outcome:

Create, use and interpret functions and use them to solve meaningful problems

Hopefully, this example of outcomes is helpful in understanding what I mean by a ‘modern mathematics course’ in pre-calculus.  There is a key feature not well represented by the outcomes above — the role of numeric methods within the course.  A modern mathematics course needs to provide a balance of symbolic and numeric methods for students, whether the course is calculus I or pre-calculus.  Some of this is addressed by the ‘overview’ portion of the documents above, though I would look for an explicit statement that the course will embed technology tools for both graphing (TI, Desmos, etc) and modeling (Mathematica, etc).

In a modern mathematics course, we would see evidence of all 5 strands of proficiency on each major assessment as well as the final exam.  A modern mathematics course removes the significant amounts of current courses that fail to meet professional standards for preparation … in this case, for calculus (where we have a sound basis for identifying the nature of the preparation (MAA Calculus Readiness test https://www.maa.org/press/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness and Characteristics of Successful Programs in College Calculus https://www.maa.org/programs-and-communities/curriculum%20resources/progress-through-calculus/cspcc-publications).  In other words, we make room in the pre-calculus course(s) by dropping the unnecessary topics and problems so we can focus on the goal … helping students get prepared for calculus.

The other aspect of a modern mathematics course deals with design principles.  It is generally not wise, and often is dangerous, to create a design with implementation not including a process to collect data on the effectiveness of the design.  Conferences and journals are often well stocked with reports of the first semester or year of a ‘new’ thing; that is not what I am talking about.  I am referring to a regular process of collecting data (aggregated and disaggregated) that will show meaningful trends in a process allowing for the assessment of corrections and modifications in the ‘treatment’.  This type of work is seldom ‘fun’ in the same way that a conference presentation is.  However, serving all of our students depends upon this continual examination of the basic question:

So, how are we doing NOW?

Moving from traditional to modern mathematics courses provides an opportunity to have all students experience good mathematics reflecting current tools and applications, and we might therefore conjecture that there will be an increase in majoring in the mathematical sciences.

Hopefully, this first approximation to a good answer to that good question was helpful.  I’d like to hear from you on that!

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