What Are Math Pathways? A Good Thing?

From what I see, it sure looks like “we” have decided that math pathways are a good thing.  What does the phrase refer to?  Are they usually a good thing?

The first question is more difficult than most people would expect.  Most definitions are implied … a set of objects is called a math pathway.  Digging a little deeper, the most common reason that set of objects is called a math pathway is that the identified objects form a sequence of courses which avoid algebra when possible.  As you know, there is a strong belief in the assertion that most people do not need algebra; calling something a ‘math pathway’ gives it a nice sound and appeals to this belief.

So, what mathematics remains in a curriculum (excuse me, a math pathway) if we generally avoid algebra?  We could choose to include deeper concepts from geometry, a strong background in proportionality including judging its validity in diverse situations, or other topics meant to strengthen the mathematical abilities of the students. What is the most common focus in math pathways?  “What do they need in statistics or quantitative reasoning?”  Are creating a curriculum for our students based on Lone Star’s direction after his Winnebago ran out of gas:

Take only what you NEED to survive!   [Spaceballs]

In some math pathways, content is only included if it passes this test of immediacy — We will teach it only if students really need it in basic statistics (or quantitative reasoning).

In other words, “math pathway” involves both algebra avoidance and restricting content on what is needed for one specific class.  Compare this to the traditional college/dev math program … which involved algebra obsession and restricting content to what is needed for one specific class (college algebra).  I would suggest that the vast majority of modern math pathways are just as faulty as the traditional math courses they replaced for those students.

Many of the math pathways are specifically targeted to statistics.  The role of statistics in mathematics education has been debated here before (see  Plus Four — The Role of Statistics in Mathematics Edation).  However, think about WHY statistics is being so commonly used as a general education ‘math’ course — people see it as “practical”.  [Many of the quantitative reasoning courses suffer from the same ‘usefulness’ syndrome.]  Few people seem to be questioning this love affair with statistics.  Sure, there are ‘studies’ which indicate that a number of occupations involve the use and interpretation of data.  Some of the largest occupations in this group are nursing and related programs.  Certainly, people with a long-term goal of being a high-level nurse (perhaps supervising and administering a clinic or hospital) will need to use statistics to carry out their work.  However, the vast majority of nursing graduates — especially at the associate degree level — are expected to have a different skill set, including a bit of algebra.  At the same time, the statistics class does nothing to help students deal with the mathematics they encounter in their science courses (proportionality and algebra).

It is my hope that we will awake from our current sleepy state and critically assess the proper role of statistics as a general education math course.

Some readers may have had the dubious pleasure of attending one of the various presentations I have made over the years, and some of this group my puzzle at the apparent lack of support for practical applications in this post.  We often hear more like we want to believe … I have advocated for mathematics that helps our students succeed, and — sometimes — this involves a focus on the practical.  Education is not achieved by learning only the mathematics a person can see applied at a point in time; that is a description of training.  We are mathematics educators, not occupational trainers.

On the other hand, ‘math pathways’ is beginning to be used to include all targets including calculus.  AMATYC, for example, has a grand committee on pathways.  That is fine, I suppose.

To the extent that math pathways help us improve the mathematics all students experience in our courses, math pathways are a good thing.  My motivation for the “New Life Project” New Life Project (AMATYC, et al) was based on this goal; the support of New Life for Pathways was coincidental.  Perhaps math pathways have improved the mathematics experienced by those students in the stat or “QR” pathways … I might be wrong that they haven’t.  However, why would we want to focus so much on the non-STEM students?  All students have dreams and aspirations; we should be encouraging and enabling many more of our students to see their STEM potentials.  Why should STEM students receive a second class education?

I believe that math pathways have been a net negative.  We have improved “outcomes” but not mathematics.

All of our students deserve good mathematics.

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College Algebra Must Die!

Sadly, many people look at situations with such a strong bias, born of history, that obvious problems are hidden.  Such is the case with the American “college algebra” course.  College Algebra is a glacier which has trapped college mathematics faculty for decades, causing harm to students and society.

College Algebra must die!

In support of this assertion, let’s begin with the origins of “college algebra”.  Based on the research of Jeff Suzuki (see https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxqZWZmc3V6dWtpcHJvamVjdHxneDo2MWI5YWE4YzU2MDM1MmY3) our contemporary college algebra course is a descendant of the original ‘math for liberal arts’ course in the 19th century.  Certainly, the content has changed since the 19th century — we now cover factoring and graphing, which were not so much in the original.  However, the topics in a college algebra course indicate a ‘survey’ course orientation — not a focused preparation for success in mathematics.

This college algebra course was created before any standardized high school mathematics existed.  If we created such a course today, we would classify it is remedial — the content is primarily a subset of Common Core objectives.  Even if we live in a state or region not ‘implementing Common Core’, the local K-12 districts have an intended curriculum with similar objectives and goals.

Any effective preparation for calculus, coming from college algebra, is a coincidence of epic proportions.

Many of us combine a college algebra course with a trigonometry course as the prerequisite to calculus I.  Since this trig work — identities and memorization emphasized — is the most recent mathematics students experience before calculus, it is no wonder that our pre-calculus courses often harm the students they are supposed to help.  See the fascinating study by Sonnert & Sadler (https://www2.calstate.edu/csu-system/why-the-csu-matters/graduation-initiative-2025/academic-preparation/Documents/IJMEST-Sonnert-Sadler-Precalculus.pdf)

The other central reason for the assertion that “College Algebra Must Die” is based on the current best thoughts of both preparation for calculus as well as the improvements in calculus content.  Both of these sets of perspectives are based on long-term work of MAA and AMATYC, and have been articulated fairly consistently over the past 20 years.

Take a look at the MAA “Calculus Readiness” instrument (https://www.maa.org/press/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness) and some research on the CR (https://math.la.asu.edu/~carlson/…/CCR-Carlson,%20Madison%20&%20West.pdf).  Our college algebra courses have little to do with these outcomes and goals, even we use the politically correct title of “pre-calculus”.

You might also want to look at the current MAA CUPM report on calculus (https://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm/2015-cupm-curriculum-guide).  College Algebra is an antiquated non-preparation course for an out-of-date calculus sequence.

Many people and institutions (my own included) have worked hard to make a college algebra course a more reasonable general education course, which is ironic given the history of this course.  The result is usually a ‘modeling and functions’ course, clearly not rigorous.  Although these modeling and functions courses are very valuable, they are no longer college algebra courses.

We need to develop potential precalculus courses which have the content validity to justify an expectation that they will be effective preparation for calculus.  Much is known and understood of the basic design issues for such courses; before we can do this work, we need to escape from the College Algebra Glacier that we have been trapped within.

College Algebra Must Die!

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The Assessment Paradox … Do They Understand?

We often make the assumption that solving ‘more complicated’ problems shows a better understanding than solving ‘simpler’ problems.  This is an assumption … a logical one, with face validity.  I wonder if actual student learning refutes it.

My thoughts on this come from a class I’ve been teaching this summer, “Fast Track Algebra” for the first time.  Fast Track Algebra covers almost all beginning algebra along with all intermediate algebra; those separate courses I’ve taught for 45 years … this was my first time doing the ‘combo’ class.  In case you are wondering how we can manage it, the class meets 50% more — 6 hours per week in fall & spring, and 12 hours per week in the summer like my class did.

Our latest chapter test covered compound inequalities, absolute value equations, and absolute value inequalities.  As for most of the content, none of this is review of what students ‘know’ — any knowledge they had on these concepts is partially or fully faulty, so class time is focused on correcting deeper understanding of concepts and procedures.

The class test, in the case of absolute value inequalities, presented 3 levels of problems:

1. simplest, just need to isolate absolute value and easiest solution … like |x| – 2 < 5
2. typical, absolute value already isolated with a binomial expression … like |3w + 4| >8
3. complex, with a need to isolate absolute value and binomial expression … like 2|2k -1| + 6 < 10

The surprise was that most students did better on the ‘complex’ problems than they did the simplest problems.  On the simplest problems, they would only ‘do positive’ while they would do the correct process on the complex problems (both positive and negative).  This was a little surprising to me.

If a student does not do the simpler problems correctly, it is difficult to accept a judgment that they ‘understand’ a concept — even if they got ‘more complicated’ problems correct.  This paradox has been occupying my thoughts, though I have seen some evidence of its existence previously.

So, here is what I think is happening.  As you know, ‘learning’ is a process of connecting the stimulus (such as a problem) with the stored information about what to do.  Most of the homework, and the latest work in class, deals with the ‘complex’ problem types.  The paradox seems to be caused by responding to the surface features of the problem and retrieving a process memorized in spite of a weak conceptual basis.  In the case of absolute value inequalities, they ‘memorized’ the correct process for complicated problems but failed to connect the concept to the simplest problems because they ‘looked different’.

If valid, this assessment paradox raises fundamental questions about assessment across the entire curriculum.  As you know, the standard complaint in course ‘n+1’ is that students can not apply the content of course ‘n’.  Within course ‘n’, the typical response to this complaint is to emphasize ‘applying’ content to more complicated problems.  Perhaps students can perform the correct procedure on complicated problems without understanding and without being able to apply procedures in simple settings.

I see this paradox in other parts of the algebra curriculum.  Students routinely simplify rational expressions with trinomials correctly, but fail miserably when presented with binomials (or even monomials).

Some of us avoid this paradox by emphasizing applications — known as ‘context’, and focusing on representations of problems more than procedural fluency.  With that contextual focus, we will seldom see the assessment paradox.  The challenge on the STEM path is that we need BOTH context with representation AND procedural fluency.

I’m sure most faculty have been aware of this ‘paradox’, and that this post does not have novel ideas for many of us.  I wonder, though, whether we continue to believe that students ‘understand’ because they correctly solved problems with more complexity.

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How to Impact Student Success

College leaders (presidents, trustees, chancellors, etc) have discovered “student success” as an issue, and they promptly implement systemic changes which impede student success.

In some ways, their errors are understandable.  We’ve got plenty of data which shows …

• Traditional remediation in mathematics most often functions as a barrier to students
• Students who complete college math in their first year are more likely to complete their program/degree
• Placement by single-measure tests tends to underplace 20% to 30% of the students

Leaders have also accepted the surface logic of “alignment” (At the Altar of Alignment  ), just like some folks accept the logic of ‘trickle-down-economics’.  Alignment takes many forms … from aligning K-12 and college expectations to selecting a math course for a student’s program.  Little data exists to show that alignment improves student success; like tax cuts, alignment is difficult to argue against — even though we should.

When I talk about student success, I am referring to the important measures of student success — learning, preparation, and a liberating education.  Passing my math course is not a measure of student success … being able to deal with mathematics in other situations IS.  Curiously, I asked by college president about measuring student learning as a component of student success; the response was that we should drop course grades and move to a portfolio.

So, here is the type of thing I mean by student success.

In a conversation with a small group of science faculty, they shared their frustration with student’s inability to apply math — algebra in particular — to scientific contexts.  A low level example was a simple temperature conversion:  T[sub C] = (5/9)(T[sub F] – 32), given temperature of 40 degrees C, convert to degrees F.

Many students treat this as a calculation problem (5/9)(40 – 32), instead of algebraic.  It seems to make no difference if subscripts are used or the letters C and F instead.

Student success is being able to reason (algebraically) in this case to get the job done.

In this case, we have ‘alignment’. The math course students took before the specific science course included replacements for both independent and dependent variables.  Alignment is a very (VERY) weak estimate of preparation for student success.

My goal of student success is not especially lofty.  In a nutshell, this is it:

Given a situation involving application of concepts and skills easily within the mathematical reach of the students, they will formulate a reasonable solution method and execute this solution with reasonable precision.

This goal is quite a bit above the useless definition of student success seen by college leaders: course completion one-at-a-time.  Student success means that my colleagues in other disciplines would be pleasantly surprised by how well our students apply mathematical concepts and relationships which arise in that discipline.  Those faculty would not need to dilute the scientific rigor of their course (in whatever discipline) just because the students we send to them lack quantitative understanding.

We live in an era of ‘completion obsession’.  It’s not that program completion is bad … completion is a great thing; the best day of my year is getting to see some of my students walk across the stage to get their degree.  The problem is that the obsession with completion devalues the education we are supposed to be providing to our students.  In the completion fixation, we watch students on the marathon course to make sure that they pass each critical point — without noticing that many students are running without understanding strategy or skill.  It’s like perseverance is the only trait we value.

Our job is to keep education in mathematics.  Student success means that we’ve made a difference in how our students are able to deal with quantitative situations; mathematics is an enabler of multiple career options for all students, not a subject to be gotten-done-with.

 
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