Walking the STEM Path IV: Content versus Preparation

Developmental mathematics has a mission to prepare students for college math courses, including those on the calculus trajectory.  Both the data I see and an analysis of the courses suggests that our current courses are not doing very well … so I want to look at this problem from a different perspective.  #STEMpath #Completion #AlgebraicLit

If your institution is like mine, the conversation about intermediate algebra preparing students for college algebra (or pre-calculus) goes something like this:

College algebra covers ‘complex fractions involving binomials and trinomials’, so intermediate algebra should cover ‘complex fractions involving monomials and simple binomials’.

We tend to obsess ‘content’, and presume that a reasonable progression of content creates a good preparation.  This approach uses procedural complexity as a proxy for reasoning at the level needed for calculus success.

Instead of looking at content at such a fine level of detail, how about starting from the target.  In calculus, students need:

• Procedural knowledge with understanding
• Flexibility
• Reasoning, especially related to multiple quantities changing in the same problem

The emphasis in intermediate algebra (and much of college algebra) is on the first half of the first item (procedural knowledge … ‘understanding optional’).  If this is true, then the results we see in the research are not surprising at all.  The question becomes: what is a more effective approach to designing the curriculum?

The ‘calculus list’ above is a list of student abilities.  We should design a sequence of courses deliberately organized to develop those abilities, building a STEM bridge from the basic algebra level to calculus I.  There is no reason to assume that one particular approach to this designing will be superior to others … should intermediate algebra develop all 3 abilities in all content areas included in the course, or should intermediate algebra focus on the first two abilities, or perhaps a mixture of levels where some content areas are done ‘deep’ (all 3 abilities) while others are done ‘shallow’ (first ability only).

We need some field testing of those ideas, but work has already begun.  In the New Life project, our outline of the Algebraic Literacy course takes the approach that we build all 3 abilities in each content area.  Curricular materials for this work are, sadly, not available at this time … I will be sharing 3 sections of material for this model at my AMATYC conference session in New Orleans.  The Dana Center “Reasoning with Functions” (RwF) materials are being developed currently; that model takes a similar approach to the abilities, from what I can see.  One difference is that the two RwF courses form a sequence, replacing both intermediate algebra and pre-calculus; the Algebraic Literacy course would replace an intermediate algebra course only … institutions would still have a pre-calculus course to follow it.

A related design question deals with pre-calculus: one semester, or two semesters (college algebra, then ‘trig’ in some form).  Our default trajectory should be one semester.  The only reasons to need two are (1) our failure to provide a good intermediate algebra course, and (2) the minority of students who MIGHT need a sequence of courses.  We often justify two semesters based on having “too much material”; I suggest that this is a fallacious argument (it’s not about the content … it’s about abilities).

Instead of our current sequence of courses copied from bygone years, we need an efficient system designed to help students move from one place (developmental) to another (calculus).  This is the most exciting work, and the most powerful opportunity, to ever face our profession.

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