Forty Five Years of Dev Math, Part I: The Origins

I’m getting somewhat close to the end of my career, and I expect that AMATYC 2017 is the last AMATYC conference that  I will attend.  Most likely, it is natural for people to contemplate the arc of history at this point (whether this arc bends towards justice is another question).  I will be writing a series of posts on the history of (my) developmental mathematics, which might be presented at a session in San Diego.

For me, the origins lie in a coincidence:  having ‘trained’ as a high school math teacher, I was unable to find a teaching job that did not involve moving.  I applied for a part-time job at the local community college, and in a fit of inexplicable errors, was hired.  The job involved supporting the operations of the college’s “Math Lab”, where several remedial math courses were offered in a self-paced, mastery format.   The time was the early 1970s.

In general, those remedial courses were intentional copies of K-12 courses from a short period prior to this time.  We had middle-school math (basic math), beginning algebra, geometry, and intermediate algebra; soon after I started, we began offering a metric system course, a desk-top computer course, and a sequence of two statistics courses (which had a beginning algebra prerequisite … quite ahead of its time).  The faculty in charge of the courses for students were, in general, current or former high school math teachers; familiarity with K-12 math was a high priority in hiring, and support for student success was not even considered.

The core of the ‘developmental math’ curriculum was the 3 course sequence aligning with grades 8, 9, and 11 .. basic math, beginning algebra and intermediate algebra.  At this time, the mode for a student’s high school math was ‘algebra I’, with a fourth of recent HS graduates never having had any algebra course.  Those on a ‘college-prep’ track certainly had more, but the community college policies were not targeted towards the college-prep students.

This was the time period when a pattern was started that still holds in many parts of the country:

Since most of the students graduating from high school had not taken ‘algebra II’, intermediate algebra is ‘college credit’ and often meets an associate degree requirement for general education.

The rationale for this policy lost its validity within about decade, as the majority of students began to graduate with algebra II credit on their transcript.  Colleges have been slow to update their general education policies to reflect fundamental shifts in HS course taking behavior.

In terms of “hot topics” in developmental mathematics education, it was all about two systemic features:

  • Curricular materials that required little reading and provided ‘clear’ examples with lots of practice.
  • Alternative delivery methods, including self-paced and programmed learning.

The first element reflected the high-school context for  the period prior to this … school textbooks were intended to be ‘teacher-proof’ (anybody could teach math), and the content was all about procedures to calculate answers in arithmetic and algebra.  That context has changed in a basic way, as the result of the teaching standards over the past 30 years (NCTM, AMATYC, etc).  Like the general education policy, math faculty have not altered the core focus of the curriculum; most current materials still focus on clear examples and lots of practice (though there is often more reading involved).

Our focus in alternative delivery methods, though cast in naive terms, was actually critical to trends that continue through today.  Most of us find it funny that ‘programmed learning’ was a “Thing”; the central idea was to have an assessment ‘every page’ and the student was ‘branched’ to a different next page, depending upon their answer.  In more recent times, this idea has been done in a more mature fashion with adaptive computer tutor designs.

The essential transaction that was being developed in these early days was “student — does math — correct OR recycles to re-learn it, repeat”.  Faculty had a role, but this role was not seen as the most essential role for student learning.  In contrast, much of our current professional development puts great emphasis on faculty interacting with students.  Although there is an obvious and valid basis for this emphasis, I wonder if perhaps we would be better off focusing more on the student interaction with mathematics.

A subsequent post will look at the period of a few years following this ‘origins’ time of the early 1970s.

 

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The Student Quandary About Functions

For students heading in a “STEM-ward” direction, understanding functions will become critical.  Unfortunately, a combination of a prior procedural emphasis and some innate cognitive challenges tends to result in a condition where students lack some basic understandings.

For example, in my intermediate algebra class, we provide problems such as:

For f(x) shown in the graph below, (A) find the value of f(0), (B) find the value of f(1), and
(C) find x so that f(x)=0.

 

 

 

 

 

 

 

Since there is no equation stating how to calculate function values, students need to use the information in the graph.  The vast majority of students make 2 novice errors:

  • Error of x-y equivalence:  providing the same answer for (A) and (C)
  • Error of symmetry: Since the answer for (A) is x=1, stating the answer for (C) as x=1

To improve this understanding, I use the longest (time measured) group activity in the course.  This is definitely a situation where “Telling” does not correct the errors [I’ve tried that 🙁  ], and the small group process helps dismantle some of the errors.  Clearly, the correct understanding for reading function graphs is critical for success in pre-calculus and eventually in calculus.

Another function concept we dealt with this week is ‘domain’.  Now, once students have found a domain, there is a tendency for some students to think they should find the domain of any and all functions, regardless of the directions for the situation.  This “inertia error” (what was started … continues) is not a long-term problem.  Here is a typical problem for the long-term problem:

Find the domain for the function graphed below:

 

 

 

 

 

 

In this particular class, I provide a fair amount of scaffolding … in a small group project, we explored the behavior of rational functions (without using that label) including what the “undefined” x-value means on the graph.  We don’t use the word asymptote; rather, we talk about the fact that some x-value results in division by zero, and the graph of the function can not show any ‘point’ for such inputs.  This leads to the graphing of the function, including the behavior around the ‘gap’.

Students struggle quite a bit with this type of problem.  Sometimes, they continue the ‘function values from graph’ thinking, and latch on to x=0 or y=0 to make some statement about a ‘domain’.  Many students will correctly identify the x-values for the gaps (yay) but make illogical statements about the domain.  The typical student error is:

  • (-infinity, -2) ∪ (-2, infinity)  … or even just one interval (-2, infinity)

This type of error usually follows from a process-focus, detached from the underlying meaning.  I am trying to get them to see:

  • gap on graph equates to excluded values in the domain

The process focus looks at the first part of  this.  Like the function value errors, the effective treatment of this problem requires time and individual conversations.

This type of function work is not typical for an intermediate algebra course.  However, it would be typical for an algebraic literacy course.  As we transition from traditional content to modern content in our courses, I am expecting that our intermediate algebra courses will fade away … to be replaced by variations of the algebraic literacy course.

 

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The Calculus River … Follow the Flow

One of the myths about developmental mathematics is that very few students take STEM courses.  Often, we hear people joke that one student makes it to calculus.

Here is some data from my college showing how many students started from various levels in mathematics (over a 3 year period).

Started in beginning algebra or lower       105 out of 937             55% of that 105 pass calculus 1

Started in intermediate algebra                  177 out of 937              58% of that 177 pass calculus 1

Started in pre-calculus                                  457 out of 937             69% of that 457 pass calculus 1

Started in calculus 1                                       162 out of 937             69% of that 162 pass calculus 1

Over 10% of our calculus 1 students began in beginning algebra or lower.  We treat intermediate algebra as a developmental math course … so we’d say that over 25% of our calculus 1 students started in a developmental math course.

Not only do we have over 25% of our calculus students starting in developmental math, their pass rate in calculus is not that much lower than students who started in calculus.  It’s true that the proportions are statistically significant.  However, given the differences in student characteristics (placed in dev math versus not), the difference is relatively small.  Of course, we would like to improve the preparation so that the proportions are not different at all.

One of the reasons to point out the false nature of this myth is that our developmental math courses need reform for ALL students … not just those in ‘non-STEM’ fields.  In the New Life model, we propose using Mathematical Literacy for all students (as needed) and Algebraic Literacy instead of Intermediate Algebra.  Algebraic Literacy has learning outcomes designed to provide some early foundational work using concepts that are critical in calculus, as well as having a stronger basis in function properties and behavior.

 

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Problem Solving Skills

This is the story of what a student did when confronted with a procedural problem for which she did not ‘remember’ the standard procedure.

One of the in-class assessments I used is a ‘worksheet’; it’s like an open-note quiz over a set of material.  During our last class (intermediate algebra), the worksheet is longer than usual because it ‘covers’ the entire course.  Item 6 on this worksheet is:

Rationalize the denominator 

As I said, she could not remember ‘what to do’.  However, she did a great thing … she recognized that both numbers could be written as a power of 2:

 

 

 

I was very pleased that she did this, but the student was frustrated … she then could not see what to do.  This is pretty typical when novices dive in to the world of ‘non-standard problems’ — problems for which we lack a remembered process.

Of course, it was pretty easy to guide her through the remainder of the work:

=

 

 

Obviously, the expectation (this is our traditional intermediate algebra course) was that students would apply the standard procedure (multiplying top & bottom by the cube root of 4).  Students do not like that procedure, and I tell them that the procedure itself is seldom needed.

The alternate method worked only because there was a common base between numerator and denominator, and I doubt if the student will gain any long-term benefit from this experience.  This was more of a positive thing for me, as a teacher and problem-solver: Noticing a special pattern within a problem is a critical problem solving skill.

I’m sharing this story just because I had fun with it!

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