Walking the STEM Path V: Intermediate Algebra’s Role

We’ve got some problems to solve in college mathematics.  The most important problems involve the role of Intermediate Algebra.  Currently, that intermediate algebra course operates as a filter … a barrier with extremely limited benefits to students.  #STEMPath #IntermAlg #AlgLiteracy

Historically, intermediate algebra is an altered copy of a ‘typical’ “Algebra II” course from the K-12 world.  That Algebra II content was driven by a variety of forces, none of which involved STEM preparation.  For one thing, the Algebra II content was created knowing that a significant portion of those teaching the class would be non-math majors.  In addition, “easy replication” was given a higher priority than “important mathematics”; it was more important that the course could be easily delivered in most schools, than the content have benefits to the students in college.

Now, we have (sort of) the Common Core math standards.  Even in states where the Common Core is not being opposed by political forces, the impact is limited.  In my state, the opposition has been focused on the assessment using high-stakes tools; the schools can still ‘implement common core’.  However, we are seeing the results of the Common Core paradox:

Any course ever taught in K-12 exists as a subset of Common Core, even those courses clearly opposed to the outline of mathematical practices.

Algebra II is not going away in K-12 work; it’s not even changing that much.  What this means is that Algebra II, our source for Intermediate Algebra, is  just as disconnected from student needs in college.

The Intermediate Algebra focus, just as in Algebra II, is on 3 priorities:

1. Topics
2. Procedures
3. Answers

On occasion, an intermediate algebra course will coincidentally do some good mathematics on the road from ‘topic’ to ‘procedures’.  However, since this good mathematics is done in a disconnected way in a minor part of the course, the result is a huge mis-match with student needs:

1. Understanding
2. Connections
3. Reasoning

To paraphrase a student complaint at my college:

You take my money every semester, knowing that this course will not do any good.

[The actual student complaint was ‘not pass’ the course.]

In many ways, the reason for the Algebra II content is pretty similar to the reasons Intermediate Algebra has survived — we don’t require strong ‘math credentials’ to teach developmental mathematics, and replication is more important.  We have the additional force of ‘online homework systems’.

However, the intermediate algebra fiasco can not continue.  The tragedy will be ended when our best teaching mathematicians work on preparing students for STEM work (calculus-bound and others).  The New Life “Algebraic Literacy” course was created to fit this goal, and the Dana Center “Reasoning with Functions” courses also work in this direction.

So, what are YOU doing to end the tragedy named “Intermediate Algebra”?

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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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Walking the STEM Path III: Data on Intermediate Algebra

I have been getting ready for a presentation at AMATYC on the ‘bridge to somewhere’ … Algebraic Literacy.  A recent post described how to identify Algebraic Literacy, compared to Intermediate Algebra.  This post will look at some nice research on how effective intermediate algebra is, relative to preparing students for the typical kind of course to follow … college algebra, or pre-calculus.  #bridgesomewhere #AlgebraicLit #DevMath

ACT routinely does research on issues related to higher education.  In 2013, ACT published one called “A Study of the Effectiveness
of Developmental Courses for Improving Success in College” (see http://www.act.org/research/researchers/reports/pdf/ACT_RR2013-1.pdf  )  The data comes from 75 different institutions, representing well over 100000 students.  I was very interested in their results relating to intermediate algebra and college algebra.

Their methodology involves calculating the conditional probability of passing college algebra, using the ACT Math score as the input variable; this was done for two groups … those who took intermediate algebra and those who did not take intermediate algebra.   Their work involved a cutoff score of 19 for placing into college algebra (which seems low, but I trust that it was true).  Due to waivers and institutional flexibility, they had enough results below the cutoff to calculate the conditional probabilities for both groups; above the cutoff, only enough data was there for the group not taking intermediate algebra.

As an example, for ACT math score of 18: the probability of passing college algebra was .64 for those without intermediate algebra … .66 for those with intermediate algebra.  For that score, taking intermediate algebra resulted in a 2 percentage point gain in the probability of passing college algebra.  The report also calculates the probability of getting a B or better in college algebra for the two groups (as opposed to C).

Here is the overall graph:

The upper set (blue) shows the probability of passing (C or better) with the dashed line representing those who did the developmental course (intermediate algebra).  For all scores (14 to 18) the gap between the dashed & solid lines is 5 percentage points … or less.  In other words, the effectiveness of the intermediate algebra course approaches the trivial level.

The report further breaks down this data by the grade the student received in intermediate algebra; the results are not what we would like.   Receiving a C grade in intermediate algebra produces a DECREASED probability of passing college algebra (compared to not taking intermediate algebra at all).  Only those receiving an A in intermediate algebra have an increased probability of passing college algebra. [Getting a “B” is a null result … no gain.]

Our intermediate algebra course is both artificially too difficult and disconnected from a good preparation.  That’s what I will be talking about at the New Orleans AMATYC conference.

The results for intermediate algebra echo what the MAA calculus project found for college algebra/pre-calculus:  ‘below average’ students have a decreased probability of passing calculus after taking the prerequisite (when accounting for other factors).

Our current STEM path (intermediate algebra –> college algebra –> calculus) is a bramble patch.  The courses do not work, because we never did a deliberate design for any of them.  Intermediate algebra is a descendant of high school algebra II, and college algebra is a descendant of an old university course for non-math majors.

Fortunately, we have sufficient information about the needs of the STEM path to do better.  The content of the Algebraic Literacy course is engineered to meet the needs of a STEM path (as well as other needs).

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How to Recognize an Algebraic Literacy Course

The next AMATYC journal will have an update on the New Life Project (over 100 colleges, over 800 sections, and something like 16000 students this fall semester).  In order to prepare that update, I spent a lot of time searching various web sites and following up leads.  That detective work led me to a number of colleges using “algebraic literacy” as a course title … when the course was just ‘intermediate algebra’; I also found some courses titled ‘intermediate algebra’ that were closer to algebraic literacy. #AlgebraicLiteracy #MathLit #NewLifeMath

This post is a brief “field guide” to help us recognize an algebraic literacy course.  Algebraic Literacy (the course) is one of the New Life math courses (AMATYC Developmental Mathematics Committee) developed in 2008 to 2010, based on the professional work from the last two decades.  The material below comes from our wiki (http://dm-live.wikispaces.com/Algebraic+Literacy ).

GOALS and FOCUS:

The Algebraic Literacy (AL) course prepares students for mathematics pathways which include college algebra, pre-calculus, and other courses requiring a background beyond the Mathematical Literacy (MLCS) course.

This is similar to an intermediate algebra course … on the surface.  There is a fundamental difference, however:  intermediate algebra is a derivative of an earlier “Algebra II” course from K-12, while Algebraic Literacy is engineered to meet the mathematical needs of college mathematics (backwards designed).

The focus of the AL course is on building understanding of mathematical systems with a dual emphasis on symbolism and application. The Algebraic Literacy course includes quantitative topics from areas besides algebra, which supports the needs of both STEM (Science, Technology, Engineering, and Math) bound students and other students.

A typical intermediate algebra course is heavily symbolic, with applications playing a minor role (and often using trivial applications with little payoff for preparing students).  The Algebraic Literacy seeks a balance between procedural fluency and higher level skills.  For some Algebraic Literacy courses, the applications form the context within which the mathematics is developed; for others, the mathematics begins first with applications integrated.  In considering applications, the Algebraic Literacy course includes problems with numeric solutions which would be solved symbolically in calculus.

PREREQUISITES

Basic proportional reasoning and algebraic reasoning skills, and some function skills, are required prior to the Algebraic Literacy course.

We do list 5 specific areas of prerequisite skills following this general statement.  However, the Algebraic Literacy course is designed to allow ‘co-requisite remediation’ at the appropriate level: Building on basic algebraic reasoning skills, for example, we aim for deeper understanding and solid symbolic skills.  By contrast, the typical intermediate algebra course presents a conflicted approach: students must show higher levels of symbolic mastery before enrolling but then intermediate algebra reviews many of those skills (without directly dealing with the development of reasoning directly).

More students are able to begin an Algebraic Literacy course than a typical intermediate algebra course.

CONTENT

1. Numbers and Polynomials
2. Functions
3. Geometry and Trigonometry
4. Modeling and Statistics
The content is intended to be integrated and connected.

In the Algebraic Literacy course, we would not see a chapter on “radicals and rational exponents”; we might see a section dealing with fractional exponents in an early sequence dealing with functions, including an application in half-life models … and a later section working on radical notation focusing on domain and range, followed by a section on translating between radical and exponential forms.  Either of these sequences of topics might also include geometry and/or trigonometry, and modeling concepts such as parameters.  Almost all topics will be presented as connected to one or more other topics, both conceptually and in terms of applications.

For most intermediate algebra courses, the content is usually 9 to 12 ‘chapters’ of material arbitrarily divided up … and separated.  A minimum of connections are made to other ‘chapters’.  Overall, the intermediate algebra course does not tell any story; the intermediate algebra course is a long series of vignettes only loosely connected by ‘category’.

By contrast, the Algebraic Literacy course tells a story of mathematical reasoning with both symbolic and application dialogues.  The design of the Algebraic Literacy course is based on being the first step along a path which includes calculus and/or other significant mathematics.  We seek to build covariational reasoning, a step up from Mathematical Literacy, on the path towards a good pre-calculus experience.

This field guide would not be accurate without  emphasizing a fundamental difference: Algebraic Literacy supports other STEM fields in addition to those needing the traditional Calculus Path.  This is primarily a distinction for the two-year college situation, where our programs often include mid-skill to high-skill fields (manufacturing technology, engineering technology, health careers, electronics, computer science, etc).  This inclusive approach is why Algebraic Literacy is not just algebra … geometry, basic trig, and statistics are included.  Most intermediate algebra includes some non-trivial geometry (right triangles, for example); however, you can recognize an Algebraic Literacy course by the presence of non-trivial geometric reasoning and symbolic representations, trig functions at a basic level, and enough statistics to interpret models developed from data.

Recognizing an Algebraic Literacy course involves multiple factors — goals, prerequisites, content, and the nature of the ‘story’.    A instinctive evaluation is based on this:

As a mathematician, can I get excited about teaching this course … is the focus on good mathematics, with the goal of developing abilities as opposed to “Algebra II all over again”?

We will see colleges move in this direction; I hope that you will consider joining the work!

 
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