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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Benny, Research, and The Lesson

The most recent MathAMATYC Educator (Vol 6, Number 3; May 2015) has a fascinating article “Benny Goes to College: Is the “Math Emporium” Reinventing Individually Prescribed Instruction? ” by Webel et al.  This article describes research in a emporium model using a popular text via a popular online system.  A group of students who passed the course and the final exam were interviewed; some standard word problems were presented, along with some less standard problems.

At the heart of the emporium’s approach to teaching and  learning we see the same philosophy that undergirded Benny’s IPI curriculum: the common sense idea that mathematics learning is best accomplished by practicing a skill until it is mastered.

I would phrase the last part differently, though you probably know what the authors mean … this is more of ‘the common mythology that mathematics …’ (common sense implies a reasonableness that seems lacking, given students attitudes about mathematics).

The phrase “Benny’s IPI” is a reference to a prior study by Erlwanger (1973) wherein the author looked at an individualized prescribed instruction (IPI) system; Benny was a similarly successful student who left the course with some very bothersome ideas about the types of topics that were ‘covered’ in the course.  In both studies, the primary method involved 3rd party interviews of students.

The current study had this as a primary conclusion:

We see students who successfully navigate an individualized program of instruction but who also exhibit critical misconceptions about the structure and nature of the content they supposedly had learned.

Although I am not a fan of emporium-related models, I am worried about the impact of this study.  These worries center on what the lesson is … what do we take away?  What does it mean?  The research does not compare methodologies, so there is no basis for saying that group-based or instructor-directed learning is better.  The authors make some good points about considering the goals of a course beyond skills or abilities.  However, I suspect that the typical response to this article will be one of two types:

• Emporium models, and perhaps online homework systems, are clearly inferior; the research says so.
• Emporium models, and online homework systems, just need some adjustment.

Neither of these are reasonable conclusions.

I spend quite a bit of time in my classes in short interviews with students.  Most of my teaching is done within the framework of a face-to-face class combining direct instruction with group work, with homework (online or not) done outside of class time.  Typically, I talk with each student between 5 and 15 times per semester; I get to know their thinking fairly well.  Based on my years of doing this, with a variety of homework systems (including print textbooks), I would offer the following observations:

1. Misconceptions and partial understandings are quite common, even in the presence of good ‘performance’.
2. Student understanding tends to be underestimated in an interview with an ‘expert’, at least for some students.

I have seen proposed mathematics that is equally wrong as that cited in the current study (or even worse); granted, these usually do not appear when talking to a student earning an A (as happened in the study) … though I am reluctant to generalize this to either my teaching or the homework system used.  Point 1 is basically saying that the easy assessments often miss the important ideas; a correct answer means little … even correct ‘work’ may not mean much.

Point 2 is a much more subjective conclusion.  However, I routinely see students show better understanding working alone than I hear when I talk with them; part of this would be the novice level understanding of mathematics, making it difficult to articulate what one knows … another part is a complex of expectations — social status — and instructor expectations by students.

Many of us are experiencing pressure to use “best practices”, to “follow the research”.  The problem is that good research supports a better understanding, but almost all research is used to advocate for particular ‘solutions’.  This is an old problem … it was here with “IPI”, is here now with “emporium”, and is likely to be with us for the next ‘solution’.

The “Lesson” is not “use emporium”, nor is it “do not use emporium”.  The lesson is more important than that, and involves each of us getting a more sophisticated (and more complicated) understanding of what it means to learn mathematics.  Most teachers seek this goal; the problems arise when policy makers and authorities see “research” and conclude that they’ve found the solution.  We need to be the voice for our profession, to state clearly why it is important to learn mathematics … to articulate what that means … to develop courses which help students achieve that goal … and use assessments that measure the entire spectrum of mathematical practice.

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What does ‘sin(2x)’ mean? Or, “PEMDAS kills intelligence, course 1”

My department is starting a conversation about pre-calculus and intermediate algebra.  I’m very pleased we are taking this step, and it is great that I do not know where our work will take us.

In our discussion yesterday, one of the concerns expressed was that students tend to not understand functions … including function notation.  People referred to the classic error:

sin(2x) = 2*sin(x)

The problem runs deeper than functions and function notation.

√(a²+b²) = a + b              and it’s corollary

(a + b)² = a² + b²

We cause these problems ourselves by allowing and even encouraging students to learn procedures by categorizing symbols, without needing to apply the meanings of those symbols.  In arithmetic or pre-algebra, this occurs in both order of operations and basic variables.

“Use PEMDAS to determine the correct order of operations”

“To add like terms, combine the numbers in front”

In a basic algebra course, the comparable statements for exponents and radicals might be:

“Negative exponents mean you write the reciprocal”

“Take out the perfect squares”

Much of our work in classes deals with getting students to correctly process the symbols we place before them.  When they generate mostly correct answers, we conclude ‘they understand’.  Seldom is that the case … because we seldom take the time to focus on the meanings of these objects along with the various correct choices we have in working with them.

When I say “PEMDAS kills intelligence”, I am using the pneumonic as a place-holder for the prescriptive procedures that we focus on.  As mathematicians, we are all about choices.  When we see:

3x(x – 2) + x(x – 2)

We think of two choices (combine ‘like terms’ first, or distribute first).  The PEMDAS-mentality boxes students in to the ‘one correct [sic] way’ approach to mathematics; the PEMDAS approach also encourages students to perform procedures on symbols without much regard for what that particular expression or statement meant.  We use these approaches in all kinds of math classes, from elementary classrooms to university classrooms, and it has got to stop.

In recent years, some of the reform efforts have de-emphasized symbolic work … partially as a response to this problem.  I applaud that work, and have contributed to the efforts.  However, sometimes we over-react and provide too little symbolic work.  We have course which emphasize ‘functions’ but never use basic function notation [f(x)], let alone variations such as ‘sin(x)’.  An irony is that most technologies that students use for our math courses (calculators, apps, web sites) generally use function notation.

Maintaining a strong focus on procedures and correct answers encourages a PEMDAS-mentality, causes problems for us later, and (I would suggest) limits student motivation to learn mathematics.  Think how much better it might be to have a balanced approach, where the key principles are:

1. Meaning
2. Properties
3. Choices
4. Application
5. Extension and feedback on prior steps

Some of my colleagues have said that students should “do mathematics” in math classrooms, though they are mostly talking about step 4 (application).  I also believe that students should “do mathematics” in every math class by using all levels (1 to 5 in my list) with all topics.  If we are not willing, believe that student’s can’t, or think that we do not have time … well, then we should question whether we are really committed to teaching that mathematics.

Most of our collegiate math courses are overly ‘full’, not too full of topics but too full of wasted effort.  We focus so much on “simplify” and “solve” in the basic courses that students use the PEMDAS-mentality; of course they won’t remember most of it, and of course they can’t apply ideas to other contexts — we are training them to just process the symbols.

So, if you have been wondering what I would have us do to replace “PEMDAS” for basic expressions, we should focus on four items:

• Meaning of each expression
• Inherent priority of each operation (a generally predictable list, based on level of abstraction)
• Properties for the type (meaning) of expression
• Choices for this expression

It is almost useless to know that a student can correctly calculate “8 – 3(5)”.  Value comes from knowing that there are multiple procedures to correctly calculate “6(2x) + 8(4x)”.  It is also almost useless to know that a student can correctly solve “12 – 5x = 7”.  Value comes from knowing that we have choices for that equation and for “8(x – 2) + 4x = 48”.  [I also suggest that PEMDAS itself is both incorrect and incomplete.]

Mathematics did not become so valuable because we know how to correctly arrive at an ‘answer’.  Our work is indispensable because we can present alternatives, and in some cases one of those alternatives provides great benefit to people, companies or societies.  That is ‘doing mathematics’, and is the type of experience I want for our students … whether in pre-algebra, pre-calculus, or anything else.

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Acceleration as Distraction

One of the tricks used to increase traffic on a web site is to incorporate ‘hot phrases’ in to the pages and articles.  In our field, “acceleration” is a very effective phrase to use.  Sadly, acceleration is not that important … by itself.

Like most colleges, mine has some acceleration models — two-courses in one, boot camps, and self-directed study for example.  Some acceleration work gets very high press coverage, such as the Austin CC ACCelerator program (see http://sites.austincc.edu/newsroom/accs-accelerator-and-developmental-math-course-wins-praise-of-second-lady-and-under-secretary-of-education/ )

Acceleration is better than not accelerating … or is it?

One of my friends tends to use medical analogies in our conversations.  I am envisioning him saying something like this:

A doctor knows that three lab tests being required are without any benefit to the patient (no diagnostic nor any treatment benefit).  What the patient needs is a new treatment, but the insurance will not cover the new treatment.  Is our profession better served by making the three useless tests quicker for the patient … or by working on fixing the basic problem of getting the right treatment?

Our goal should be to fix the problems.  Acceleration is not the basic problem … what needs to be changed is the mathematical treatment provided to students so that there are multiple benefits for students.  In developmental mathematics, our work needs to focus on capabilities that serve all college programs with a focus on quantitative reasoning.  In college level mathematics, our work needs to focus on empowering students for programs or groups of programs.

Acceleration tends to reinforce the current curricular system by masking a symptom (too long to complete).  An emphasis on acceleration distracts us from working on core problems.

I believe that we need fewer courses.  We can start with a course like Mathematical Literacy (or Quantway, or “FMR”), with just-in-time remediation as needed.  The next level can be a course like Algebraic Literacy (or STEM path I), again with just-in-time remediation for students who did not need an entire course before it).  We only need one course to connect that level with calculus I — a deliberate design of a pre-calculus course.

We can do better than acceleration.  With new ideas of content and course design, we can provide important mathematics for our students in an efficient manner.

“Needing acceleration” is direct evidence that the basic curricular structure is inappropriate.  Don’t worry so much about acceleration — fix the basic problem.

 
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