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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Walking the STEM Path 2: One Course, or “APL Design”

In the early days of personal computing, it was clear that digital storage was very limited; initial on-board memory was often measured in kilobytes (great by those standards in the 1970s).  The computer speed was decent for that time; as a result, programming languages faced issues and constraints.

As a mathematician, the most beautiful programming language was “APL” … the acronym for the obvious name “A Programming Language”.  You say you’ve never seen this  language?  Well, take a look at the stuff over at http://en.wikipedia.org/wiki/APL_%28programming_language%29 .

APL used an applied mathematics approach to programming.  Need a matrix invert operation?  One symbol did that.  Need a row operation?  One symbol.  Each symbol in APL was a wonderful contraction of a big idea, just like mathematics.  Of course, you needed a special keyboard to use APL.  Small price to pay.

Here is the theme song for the person who ran the local training for APL back in the day:

If your program does not fit on one line, you have not thought about it enough!

In other words, if you have not analyzed the problem intelligently and with insight, your program becomes multi-line and shows that you have more work to do.  Of course, programming has gone in a totally different direction, where we worry about ‘time’ more than lines of code.

In the STEM path, we are talking about connecting developmental-level mathematics with Calculus I. Think about this path as a problem to solve.  If we can not write this program for one semester, we have not thought about it enough.

Over the years, we have developed several ‘solutions’ for this path. Some involve a two course sequence of ‘college algebra’ and trigonometry.  Others involve ‘college algebra’ then pre-calculus.  Some have 3 courses — college algebra, trig, and pre-calculus.  Some institutions have a one-semester option (often called ‘pre-calculus’ or ‘college algebra and trig’).  A few other combinations exist.

We often allow content inflation in these courses by focusing on procedures rather than capabilities.  A well-prepared student can either figure out a needed procedure, or look it up once.  On the other hand, a student who has experienced the “100 most important tricks before calculus I” will not be able to figure out much, and will lose most of these tricks quickly.

What are the capabilities needed for calculus I?  We have a very good starting point for that conversation.  Take a look at the MAA Calculus Concepts Readiness test (http://www.maa.org/publications/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness).  The first item on that web page shows this problem:

Suppose you have a ladder leaning against a wall. Now suppose that you adjust the slant of the ladder so that it reaches exactly twice as high on the wall.  The slope of the ladder [now] is:  a. Less than twice what it was   b. Exactly twice what it was …

A student knowing how to handle that problem is likely to be better prepared than a student who can correctly evaluate a difference quotient for some arbitrary function.

If your pre-calculus path has more than one course between developmental and calculus I, you have not thought about the problem enough.

This “one semester … if not, finish solving the problem so it is” approach has been a recent trend at the developmental level.  Many of us are replacing 3 (or 4) procedural courses with 2 courses which provide both skills and reasoning.

We need national leadership from MAA and AMATYC on these issues; those organizations are ready.  We need many of us involved with an effort to upgrade and reform the STEM path.  Are YOU ready?

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Walking the STEM Path I: Take Time to Smell the Functions

As we engage in a conversation and discussion about “pre-calculus” (or ‘college algebra’ to some), I am thinking of our curricular goals and how we emphasize what is apparently important.  When the two align, we have potential for success; when our goals differ from what we emphasize, non-success is guaranteed.

Our work in pre-calculus deals primarily with functions (of all kinds).  That makes sense.  However, take the case of ‘inverse function’; whether we are talking about a specific relationship (exponents and logarithms) or the general concept, the idea is important on the STEM path.  The emphasis for most of our courses is on the following:

1. Replace y with x (once), and x with y (all times).
2. Solve for y
3. This is the inverse function, called f^-1

We often feel good about this when combined with the identification of one-to-one functions.  Once we practice finding the inverse, we sometimes explore what the inverse does … sometimes, we present this in terms of composite functions.

This procedural emphasis on ‘finding the inverse’ hides the purpose:  All inverse functions are a matter of undoing.  Algebra starts with inverse operations to solve equations of limited types, where we almost always emphasize the WHY.  In pre-calculus, we take a remedial approach:

• The ‘why’ is too difficult, and we wait until calculus to deal with it.
• Correct answers are an accepted proxy for understanding mathematics.

The procedural approach submerges and prevents understanding; transfer of learning will not occur in most cases.  We can do better: Inverse functions can be approached from the ‘undoing’ perspective, in two senses:  We undo the operations in the function in the appropriate order, and the output for f, when substituted into f^-1 results in the original input.  [We should really create a more reasonable notation for inverse functions.]

Another example is ‘end behavior’ of rational functions.  Our typical approach is:

• If the leading term of the numerator is a higher degree than the leading term of the denominator, the function approaches positive or negative infinity as indicated by the coefficient of the numerator’s leading term.
• If the leading term of the numerator is a lower degree than the leading term of the denominator, the function approaches zero.
• If the leading terms have equal degrees, the function approaches the value of the quotient of the coefficients of those two terms.

Some textbooks do base this end-behavior topic on a discussion of limits (a good idea).  Seldom do we approach end-behavior with an understanding base, which might go something like this:

• End behavior analysis has nothing to do with reducing a fraction.
• Terms never ‘reduce’; factors do.
• End behavior is based on analyzing the terms with the greatest influence on the values of the numerator and denominator.

Our complaint in calculus is that students do not know algebra; however, many pre-algebra topics are approached in a way that avoids dealing with those algebraic struggles — like ‘when does a fraction reduce’.

The pre-calculus experience must involve deep work with functions, combined with a focus on fundamental algebraic ideas.  Procedures can help students become efficient; when presented without that deeper understanding of functions and basic algebra, we create our own potholes and ditches in calculus.

Unless your calculus students never struggle with function ideas, your pre-calculus course deserves a critical analysis — does the course provide a good sense (feeling, smell, vision, etc) for functions and covariation?  Unless your calculus students never make algebraic faux pas, your pre-calculus course deserves a critical analysis — does an emphasis on procedures avoid dealing with basic algebraic ideas?

 
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Equity and Participation; our Response

This was one of those news reports that really got my attention as an educator.  There are many, of course, which should get my attention as a person and citizen; this report talked right to the ‘teacher’ in my brain.

The role of AP courses has been debated; in the mathematics community, we have some concern about how much benefit actually accrues to the majority of students in AP calculus.  However, as long as AP courses are offered … our goal needs to be equity: no group of students should be participating or not-participating at significantly different rates.

My college serves a blended district — a small-to-medium city (Lansing) and a surrounding area made up of suburbs and rural communities.  AP courses are offered in both the city and suburbs.  Here is a quick breakdown for two of the local districts for student population and AP population.

The school data came from a tool at marketplace.org; see http://www.marketplace.org/topics/education/learning-curve/spending-100-million-break-down-ap-class-barriers.  The city population data came from the 2010 Census.

One way to look at this data:  The Lansing high schools are about 73% ‘minority’; the AP classes in Lansing are about 60% minority.  Another view: the AP participation rate for white students is 50% higher than their proportion of the population would indicate, while the black student participation is 25% lower.

We might conclude that this discrepancy is a Lansing school problem; that is not the case.  The same pattern is present in Holt … just not quite as extreme, due to the smaller minority student population.

In case you are wondering, the national figures are:

We’ve known that minority students are over-represented in developmental math courses in college.  This recent data suggests that the equity problem extends through the whole range of abilities.  I respect the difficult work that our K-12 colleagues are doing, often without support or respect; this equity problem is not about the teachers … it’s about society and us.

We can, and must, do better.  How will we respond?  I think we recognize signs of a problem; what actions can be taken?

 
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