A “Golden Age”: Forty Five Years of Dev Math, Part II

In my continuing account of a history of developmental mathematics, we are moving from the early 1970’s to the late 1970’s.  Although ‘dev math’ existed before the 1970s (the ‘origins’), my experience started then … and this period coincides with other shifts (such as the founding of AMATYC).  This post will look at the patterns of the late 1970s and how some of them impact us in 2017.

Faculty in mathematics, and observers, might assume that developmental mathematics has always been trying to justify its existence.  However, for the early part of this story, policy makers tended to ignore both the need for developmental mathematics and out outcomes.  Budgeting in this period would reward enrollment, and developmental math classes were both easy to populate with students and economical for the institution.

These conditions resulted in larger enrollments in our courses, which contributed to one aspect of a ‘golden age’:

Dozens of publishers actively sought authors and new textbooks.  Derivatives of these textbooks still dominate the book ‘market’ today.

One of these textbooks initially begun in this period is “Keedy/Bittinger”, and the “Lial/Miller” texts also began at this time.  Previous textbooks tended to be knock-offs of high school books, and now the focus was placed directly on the needs of our courses and students.  However, the content was still organized by typical topics in chapters like one would see in high school books, and this generally continues until quite recently.  The content was quite traditional and procedural; the innovations focused on the use in a ‘college’ course by adults.  This is when “workbooks” became popular, providing instructors with homework submission before the internet.

The current environment has focused on the price of textbooks.  I think it is interesting that in the 1970s the price of textbooks was just as high (relative to the “CPI”, for example) … and that the buyer got just the book.  Today, with prices a bit above the adjustment for CPI, the buyer often gets online access.  Clearly, perception is the most important issue in an economic decision like ‘buy a textbook’.  [Students also did not have any purchase options in the 1970s.]

As the enterprise of developmental mathematics expanded, some concerns developed around ‘proper placement’.  Since this preceded most of the technology we presume today, “checking prerequisites” was an enormous undertaking for an institution.  Many colleges  had been letting students enroll for courses based on the student’s perception about what was needed.  This period pre-dated the placement tests we are accustomed to, which led to another aspect of a golden age:

Many institutions invested resources in developing their own placement instruments.

In many institutions, this meant that math departments did some analysis of what students needed to know before a given class.  Likely, a majority of these efforts produced assessments very similar to the items on Accuplacer and Compass, with a focus on one type of error … not letting a student register for a class when the test indicated a high chance of not passing.  Some of these institutions were in New Jersey, where (a few years later) the items from these original institutional placement tests were incorporated into the New Jersey Basic Skills tests, which is where many of the Accuplacer original items came from.

The emphasis on avoiding a single type of error has been at the center of mathematics placement until the present, though forces are pushing us to move beyond this concern.  We have been so focused on avoiding “over placement” that we have a strong tendency to under place students — putting them in courses for which there is little need.  That pattern has left us open to external criticism, and lies at the core of the “Complete College America” attack on remedial mathematics.

Placing students has been more about “avoiding failure” in a higher course than with the question of the “best placement” for students.

The current efforts in true ‘multiple measures’ placement are aimed at answering the better question.

I think it is important to recognize that some of the institutional efforts at placement in this era were more sophisticated in their goals and more creative in the resulting assessments.  Many of this novel approaches were shared at the first few AMATYC conferences I attended a decade later.  However, almost all of these indications of diversity were overwhelmed later during the ‘systemic years’ (another period in our history, in the 1990s).  We have generally lost the institutional placement instruments, with a few surviving as supplemental devices used by individual faculty or specific courses

Obviously, this period I am calling “a golden age” was not such a good thing.  The trends begun here caused us to under-place millions of students, and also to use textbooks which presented high-school mathematics at a low level of learning.  However, this period saw growth and large investments by both institutions and publishers.

As we move from a 1970s ‘golden age’ into the 1980s, we will be describing the impact of “back to basics” in an era prior to any content standards in the profession.

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

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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