Can ANY Sequence of Math Courses Succeed??

We share a commitment to student success … we work hard to help students reach their goals.  Sometimes, it just seems like that success is very rare.  We’re told by many sources that most students fail to reach their goal, especially if they are placed into the “dead end” (developmental mathematics).  What’s the problem?  #CCA  #pathways  #collegemath

Mathematics in college consists of sequences of courses.  Historically, the primary variable was the ‘exit point’ (the student’s last math class); recent work has created a more subtle solution where the prerequisites are variable … not all college-level courses require intermediate algebra.  We still have sequences.

The extremes of a sequence design are easily seen as failures.  At the one extreme, we might have a sequence of 4 prerequisite courses prior to the ‘one that counts’; even if we have an astounding pass rate (80%) and perfect retention (100% to next course), the net result is 41% start the college course.  The more reasonable pass rate (70%) and retention rate (80%) mean that about 12% start the college course.

At the other extreme, we have no prior courses … it’s a sequence of one course, the college level one.  That’s what the radical “co-requisite remediation” advocates suggest (and some states try to implement).  In this approach, 100% start the college math course, so even if just 30% pass it’s a gain over the long sequence.  Most of us do not support this type of policy.

So, the question is this:

Can ANY Sequence of Math Courses Succeed?

As for many human endeavors, it is far easier to create something that does not work than something that does.  Here are some principles for designing for success.

• Courses copied from another context will not support success in the sequence.

Developmental mathematics is full of course copies … basic math copied from 8th grade, beginning algebra copied from algebra I, intermediate algebra copied from algebra II, etc.  These remedial courses are part of a different tradition: students should be ‘college ready’ so we provide courses to remediate high school.  College algebra, erroneously seen as pre-calculus, is also a copy of a course.  These courses have nothing to do with success in the sequence.  We create some coincidental features for success in the algebra courses, but the entire package is doomed.

• Arithmetic is too difficult for college

Learning arithmetic involves applying properties of real numbers, standardized rates (percents), and solving fractional equations (proportions).  These advanced topics might make sense after a good algebra course, but certainly not before.  Saying that arithmetic is a prerequisite to algebra is like saying that running is a prerequisite to walking.  A course on arithmetic is doomed; either the content is too advanced … or we take all of the arithmetic out and just deal with correct answers.

• One course at the developmental level should be enough for at least 80% of the students.

Too often we think about “what the student does not know”, instead of “what is needed for success”.  We get trapped in to a process that tries to fill in all ‘holes’.  Being ready for success in a college level math course does not involve ‘everything’.

• Capabilities are just as important as skills.

In the traditional sequence, we do exclusively skill work; sure, we include ‘applications’ with the thought that these will improve something (though we are not sure what).  Our courses often delay intense work on reasoning until the calculus I course.  General reasoning is one of the ‘capabilities’ required for success; we might even focus on the 5 strands of mathematical proficiency.  Other capabilities are number sense, proportional reasoning and algebraic reasoning.

• Good mathematics should start from the first day of every class.

The traditional sequence directly says “you are not ready for the good stuff yet; let’s see if you make it though this n-course sequence and then start good mathematics”.  I leave “good mathematics” as an undefined term.  If you look forward to teaching it and are proud of the mathematics, you are probably doing good mathematics.  One trait of good mathematics is likely to be that connections are built for each idea.

The AMATYC New Life Project has advocated a curricular model consistent with these principles.  We’re not alone in doing so; the Dana Center New Mathways work also does a good job.  The old courses (developmental and college algebra) need to be replaced by courses designed to succeed.  The New Life  math courses emphasize important mathematics with a plan to efficiently get students ready:  Mathematical Literacy, and Algebraic Literacy.   [I’m doing a session at this year’s AMATYC conference on the Algebraic Literacy course.]

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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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