Algebra Lament

Woe, my name is algebra.

I help people communicate about important relationships; people use me to predict future conditions.

Woe, my name is algebra.

I have been shunned and made fun of.  The fault is not mine; no, the fault is almost entirely that of ‘algebra courses’ taught without a focus on understanding, without attention to communication about the world.  The quadratic formula is not my fault!!

Woe, my name is algebra.

People think that I am another name for right answers to meaningless questions, that I am the effort to emulate some perfect series of steps to solve those meaningless questions.  I am not some worthless set of dance steps, steps being marketed in the absence of music or creativity.  Just because I can’t carry a tune doesn’t mean that I lack creativity!

Woe, my name is algebra.

I am the written language to communicate about matters quantitative.  Rejecting me is the rejection of the basic goals of education in the modern era.  For, how can people understand the world when all they can do is vaguely describe the qualitative traits … or calculate values for a few specific cases?  I may have faults, but ‘lack of clarity’ is not one of them!

Woe, my name is algebra.

My properties allow people to transition from a sum to a product, and to discover the almost magical explosion of options for working with expressions.  My properties allow people to express functions of variables in ways which uncover critical features of the relationships.  Instead of this beauty, most people are told that overly complicated trivial work is ‘algebra’.

Woe, my name is algebra.

I live in the core of science and society, despised solely for the company I’ve kept.  Did I have any say in that company?  Is it my fault that school mathematics is often taught in poor ways and with ‘outcomes’ which add no value for the learner?

Woe, my name is algebra.

My reputation has been ruined by others.  I am like a poor citizen who needs to be represented by public defenders who do not see my value.  The public defenders have good intentions about our students, but represent me in such a negative fashion that the majority of students conclude that I am worthless … and that they (the students) can never understand me.  My remote cousin with a similar name, ‘linear algebra’, has much better respect and cred.

Woe, my name is algebra.

I have been placed in two boxes.  One box is labeled “use only enough to get an answer”, perhaps to questions students might care about.  The other box is labeled “recipes for right answers to artificial questions”.  Does anybody put geometry in these boxes?  Does anybody put statistics in these boxes?  I can tell you that I seldom have any company in these boxes, and never for very long.  Let me out of the box!!

Woe, my name is algebra.

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Transitioning Learners to Calculus in Community Colleges (TLC3)

You might have heard of the MAA project “National Study of College Calculus”  (see http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus ).  That work was very broad, as it studied calculus in all 3 settings (high school, community colleges, and universities).

A recent effort is focused on community colleges  with the title “Transitioning Learners to Calculus in Community Colleges”   (info at http://occrl.illinois.edu/tlc3  )  Take a look at their web site!

One component of their research is an extensive survey being completed by administrators of mathematics at associate degree granting public community colleges, including the collection of outcomes data.  A focus is on “under represented minorities” (URM), which relates closely to a number of recent posts here (on equity in college mathematics).

I am expecting that the TLC3 data will show that very few community colleges are successful in getting significant numbers of “URM” students through calculus II (the target of this project).  The ‘outliers’, especially community colleges succeeding with numbers proportional to the local population of URM, will provide us with some ideas about what needs to change.

Further, I think the recent emphasis on ‘pathways’ has actually decreased our effectiveness at getting URM students through calculus; the primary assumption behind this (based on available data) is that minorities tend to come from under-performing K-12 systems which then results in larger portions placed in developmental mathematics.  The focus on pathways and ‘completion’ then results in more URM students being tracked into statistics or quantitative reasoning (QR) pathways — which do not prepare them for the calculus path.  [Note that the basic “New Life” curricular vision does not ‘track’ students; Math Literacy is part of the ‘STEM’ path. See http://www.devmathrevival.net/?page_id=8 ]

Some readers will respond with this thought:

Don’t you realize that the vast majority of students never intend to study calculus?

Of course I understand that; something like 80% of our remedial math students never even intend to take pre-calculus.  Nobody seems to worry about the implication of these trends.

Students are choosing (with encouragement from colleges) programs with lower probabilities of upward mobility.

The most common ‘major’ at my college is “general associates” degree.  Some of these students will transfer in order to work on a bachelor degree; most will not.  Most of the other common majors are health careers (a bit better choice) and a mix of business along with human services.  Upward mobility works when students get the education required for occupations with (1) predicted needs and (2) reasonable income levels.  Take a look at lists of jobs (such as the US News list at http://money.usnews.com/careers/best-jobs/rankings/the-100-best-jobs )  I do not expect 100% of our students to select a program requiring calculus, nor even 50%; I think the current rate (<20%) is artificially low … 30% to 40% would better reflect the occupational needs and opportunities.

Our colleges will not be successful in supporting our communities until URM students select programs for these jobs and then complete the programs (where URM students select and complete at the same rates as ‘majority’ students).  Quite a few of these ‘hot jobs’ require some calculus.  [Though I note that many of these programs are oriented towards the biological sciences, not the engineering that often drives the traditional calculus curriculum.]

I hope the TLC3 project produces some useful results; in other words, I hope that we pay attention to their results and take responsibility for correcting the inequities that may be highlighted.  We need to work with our colleges so that all societal groups select and achieve equally lofty academic goals.

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Segregation in College Mathematics: Corequisites! Pathways?

So, this post will mostly apply to those of us located in urban colleges (more than rural).  The idea is to remind of the societal costs of “separate and not equal”.

As a general statement, urban public schools have more challenges than suburban schools (and more than rural schools).  The role of poverty in this situation appears substantial, and the burden of this poverty tends to fall on ‘minority’ students more than ‘majority’.  In this post, I’m focusing on two categories — black/African American and white/Caucasian.

If you track the proportion of each course that is black, you are likely to get a chart like this one.  Note that “0” represents a college-level math course (most commonly ‘college algebra’ … more on that later).

 

 

 

 

 

 

 

 

 

 

This comes from a college where black students represent about 10% of the population; the college does not have a “-3 course” (pre-algebra).  The pattern in course enrollment is a similar pattern to the ‘placement levels’ of each group … the mean placement level for black students is about -1.4 compared to -0.6 for white students.  If all students are in a sequence (‘path’) that produces an equal chance of succeeding to all college mathematics, there is ‘equality’ (given the unequal starting points).

However, two current trends break that ‘equality’ and produce a system of separate and unequal.  In many co-requisite models, students who do not place into college mathematics are given only the option to take a non-STEM math course (statistics or quantitative reasoning aka ‘QR’).  In general, colleges using a co-requisite model find that their ‘support sections’ (ones taken by non-placing students) are predominantly minority.  I know some colleges have tried to use co-requisite models in college algebra (though more often ‘intermediate algebra’); these results are seldom published, and I think this is due to the much lower ‘results’ than statistics or QR.  The result of this type of system is an unequal result for minority students — they are discouraged (or even prevented) from pursuing a STEM or high-tech program.  A new segregation is being sold to colleges, in the name of ‘better results’; more on that later!

Some ‘pathways’ implementations also produce this same unequal pattern.  Those placing ‘lowest’ and ‘struggling students’ are strongly encouraged to take a stat or QR pathway program; some of these programs actually do allow students to select a STEM or high-tech program, but many do not.  The most common model is a side-by-side design … Math Literacy (or similar course) as an option to beginning algebra, where the Math Literacy course only leads to stat or QR.  In the K-12 world, this is called “Tracking”.  Pathways often create a segregated condition, due to the impacts of the lower-performing K-12 schools.

One argument is that the co-requisite models (and pathways) at least get students to complete a college math course, most commonly stat or QR.  The question remains … so WHAT?  There is an assumption that this stat/QR approach results in more students getting a degree (likely to be true).  But … what good is the degree?  Are there actually jobs for that program?

Obviously, the answer to that last question is ‘in some cases’.  In some regions, nursing requires either statistics or QR for their associate degrees, and the employment outlook is often good.  However, these health careers programs can be ‘selective admission’.  My experience has been that students accepted in to a nursing program tend to be ‘whiter’ than the college population in general … which likely goes back to the urban school system problems.  As a practical matter, I don’t think that a focus on stat or QR, in either co-requisite or pathways, results in ‘equal’.  We are creating separate in a deliberate strategy, without ensuring that they are equal.  [Of course, it’s also reasonable to say that we should avoid “separate” in the first place.]

Now, I’m not saying that co-requisite and pathways have no place in college mathematics.  The concern deals with the ‘scaling up’ that is often sought with them, as well as the target population.  Co-requisite remediation can be quite effective at the boundary … students who “just miss” qualifying for their college course (stat, QR, or college algebra); this system can be used to partially offset the negative impacts of lower-performing K-12 schools.  Pathways keep our focus on getting out of the way as much as possible … get them to their college course quickly; however, all pathways should preserve student options.  Any pathway that blocks student options is very likely to result in ‘unequal’ conditions.

Both of these efforts (co-requisites, pathways) remind me of the segregation caused by ‘school of choice’.  Do we really want to institutionalize segregation in these new ways?

I think the better response is to modernize the entire mathematics curriculum in colleges.  Start by replacing arithmetic and basic algebra courses with Math Literacy with an intentional design to provide students options at the next level.  Replace intermediate algebra with Algebraic Literacy with its intentional design to prepare students for modern college mathematics courses.  Replace college algebra with a course likewise designed to actually prepare students for calculus.  Reduce the calculus curriculum to fewer courses while incorporating more numeric methods (see “Common Vision”).

We do not need to create separate conditions for students, not nearly as much as we need to modernize our curriculum.

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Why We Have a Major Problem Now: Forty Five Years of Dev Math, Part V

This is the 5th post in a series on “forty five years’ of developmental mathematics, roughly coinciding with my professional experience.  The prior 4 posts took us from the early 1970’s to the late 1980’s, with the last post talking about ‘normalizing a bad curriculum’.  It’s time to move on to the early 1990s.

When the  NCTM released it’s standards in 1989, people teaching developmental mathematics could immediately see the implications for our work.  The conceptualization of the dev math curriculum was a one-to-one mapping to 9th to 11th grade mathematics (in ‘the old days’).  The NCTM calls for increased attention and decreased attention had a lot of appeal.

During this same time period, the graphing calculator became a reasonable tool for mathematics classrooms; both TI and Casio had good machines, with some design elements driven by what math teachers wanted.  [The HP calculators of the time were designed for engineer use, so they seldom had much traction in schools.]  These graphing calculators provided a tool that would help teachers implement the NCTM standards.

In the Spring of 1992, Ed Laughbaum had an opinion piece published in the AMATYC journal (called “The AMATYC Review” at that time), with the title A Time for Change in Remedial Mathematics.  One of Ed’s main points was:

To change the current pattern of instruction, I propose that teaching methods be changed to support implementation of the graphing calculator into the remedial sequence.

Ed’s article is primarily an agreement with Lynn Arthur Steen that most mathematics remediation is a failure.  You might notice that this is the same message being sent in the last 5 years by change agents such as Complete College America.  Twenty-five years ago, we were saying it.  What happened?

One thing that happened was that I wrote a response, which appeared in the AMATYC Review a year later, with the title “Time, Indeed, for a Change in Developmental Mathematics“.  http://files.eric.ed.gov/fulltext/ED373817.pdf   This was written just as I was about to become chair of the AMATYC Dev Math Committee (the first time, 1993 to 1997).  My response was a little too soft in terms of critiquing our work at the time; I regret that now.

This response was the first time I used the phrase “mathematical literacy”, written in the general sense (not course specific).  Sadly, one of the things I said was that graphing calculators should not be used in courses at the beginning algebra level.  My position on this changed over the subsequent five years, but my comments resulted in a number of AMATYC members thanking me … they felt supported in doing their traditional courses (which was not my intent).

My conclusion in that article had this:

The basic issue facing mathematics educators today is how to integrate the various forces attempting to drive our mathematics curriculum. The solution involves dialogue and consensus building. Institutions such as AMATYC provide a needed forum and structure for this work. As we work together, our theories and standards will converge, resulting in changes in our curriculum which will certainly integrate technology in many ways.

It’s clear that this change process did not occur … in spite of the NCTM standards resonating with our own interests as shown in the first AMATYC Standards (“Crossroads”).  What happened?

Our collective resistance to the graphing calculator is the primary reason that we did not make any progress when there was another opportunity.  Partially, this was due to the overwhelming resistance to calculators at the college math level (college algebra, pre-calculus) … and much of this still exists today.  The fact that students could not use numeric methods in the next course meant that our use of those methods in developmental mathematics was a possible risk to our students.

In some ways, the content of our courses became ‘locked in’ by 1990.  We resisted professional calls for numeric methods, we collectively ignored the NCTM standards; we even ignored most of our own AMATYC standards (which were being written during the early 1990s).  From 1995 to 2010, fewer natural opportunities for change would arise.  Our default support for an antiquated curriculum is exactly why dev math was an easy target for policy makers and change agents in 2012 … 20 years after the early 1990s.

We are facing a similar call for change today.  The Common Vision suggests that our courses emphasize numeric methods alongside symbolic ones, as well as suggesting that our teaching methods change.  This is the danger of ‘pathways’ … that only non-STEM students get a modern course with numeric & symbolic methods; STEM students are required to survive a series of courses overly focused on symbolic methods with little emphasis on reasoning, and far too little emphasis on connections between concepts.  “Right Answer” still is the goal in these courses, which is the wrong answer for students.

I am hopeful that we individually and collectively will respond today with “let us build better courses for ALL students”.   No student should be required to take a course known to be defective.  In particular, I am hoping that AMATYC will develop a project that links the Math Intensive committee with the New Life Project to work on revitalizing the courses which follow developmental mathematics.

If our profession fails to seize the current opportunity for creating our own modern curriculum, external change agents will control the primary playing field: the initial college level math course(s) such as college algebra, pre-calculus, and similar courses.  These courses suffer the same defects as the traditional developmental mathematics curriculum — antiquated topics delivered inefficiently and with harm to the overwhelming majority of college students who will never take a calculus course.  [Our calculus courses are just as antiquated and inefficient; external change agents just don’t care about calculus very much.  They should!]

We have a problem NOW (2017) because we did not have sufficient motivation to make systemic changes 25 years ago.  The profession let a few visionaries create boutique programs which were locally successful but totally isolated from the mainstream of our work.  Today’s boutique program is “Pathways”.  We need systemic change to create modern mathematics courses for ALL students.  Do we really think that non-STEM students deserve a modern course while STEM students  slog thru disfunctional artifacts clustered as pre-calculus & calculus courses?

It really is “Time for a Change” … not just in remedial mathematics, but in all college mathematics.

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