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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45 Years of Dev Math

These are the materials from the November 11th presentation … history and future.

The presentation slides: Forty Five Years of Dev Math in 50 minutes web

The handout: 45 years of dev math in 50 minutes AMATYC 2017 S137

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Modern Dev Math

Let’s pretend that we don’t have external groups and policy makers directing or demanding that we make fundamental changes in developmental mathematics.  Instead, let us examine the level of ‘fit’ between the traditional developmental mathematics curriculum and the majority of students arriving at our colleges this fall.

I want to start with a little bit of data.  This chart shows the typical high school math taking patterns for two cohorts of students.  [See  http://www.bls.gov/opub/ted/2012/ted_20121016.htm ]

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There has been a fundamental shift in the mathematics that our students have been exposed to, and we have reason to expect that the trends will continue.  We know that this increased level of math courses in high school does not translate directly into increased mathematical competence.  I am more interested in structural factors.

Intermediate algebra has been the capstone of developmental mathematics for fifty years.  At that time, the majority of students did not take algebra 2 in high school … so it was logical to have intermediate algebra be ‘sort of developmental’ and ‘sort of college level’.   By about 2000, this had shifted so that the majority of students had taken algebra 2 or beyond.

The first lesson is:

Intermediate algebra is remedial for the majority of our students, and should be considered developmental math in college.

This seems to be one lesson that policy makers and influencers have ignored.  We still have entire states that define intermediate algebra as ‘college math’, and a number that count intermediate algebra for general education requirements.

At the lower levels of developmental mathematics, the median of our curriculum includes a pre-algebra course … and may also include arithmetic.  Fifty years ago, some of this made sense.  When the students highest math was algebra 1 in most cases, providing remediation one level below that was appropriate.  By fifteen years ago, the majority of students had taken algebra 2 or beyond.  The second lesson is:

Providing and requiring remediation two or more levels below the highest math class taken is inappropriate given the median student experience.

At some point, this mismatch is going to be noticed by regulators and/or policy influencers.  We offer courses in arithmetic and pre-algebra without being able to demonstrate significant benefits to students, when the majority of students completed significantly higher math courses in high school.

In addition to the changes in course taking, there have also been fundamental shifts in the nature of the mathematics being learned in high school.  Our typical developmental math classes still resemble an average high school (or middle school) math class from 1970, in terms of content.  This period emphasized procedural skills and limited ‘applications’ (focusing on stylized problems requiring the use of the procedural skills).  Since then, we have had the NCTM standards and the Common Core State Standards.

Whatever we may think of those standards, the K-12 math experience has changed.  The emphasis on standardized tests creates a minor force that might shift the K-12 curriculum towards procedures … except that the standardized tests general place a higher premium on mathematical reasoning.  Our college math courses are making a similar shift towards reasoning.  Another historical lesson is:

Developmental mathematics is out-of-date with high schools, and also emphasizes the wrong things in preparing students for college mathematics.

We will never abandon procedures in our math courses.  It is clear, however, that procedural skill is insufficient.  Our traditional developmental mathematics curriculum focuses on correcting skill gaps in procedures aligned with grade levels from fifty years ago.  We appear to start with an unquestioned premise that remediation needs to walk through each grade’s math content from 5 decades ago … grade 8 before grade 9, etc.  This is a K-12 paradigm with no basis in current collegiate needs.

The 3- or 4-course sequence of remedial mathematics is, and always will be, dysfunctional as a model for college developmental education.

There is no need to spend a semester on grade 8 mathematics, nor a need to spend a semester on grade 9 mathematics.  When students lack the mathematical abilities needed for college mathematics, the needs are almost always a combination of reasoning and procedural skills.  If we can not envision a one-semester solution for this problem, connected to general education mathematics, we have not used the creativity and imagination that mathematicians are known for.  Take a look at the Mathematical Literacy course MLCS Goals and Outcomes Oct2013 cross referenced 2 by 2 .  If students are preparing for pre-calculus or college algebra, take a look at the Algebraic Literacy course  Algebraic Literacy Goals and Outcomes Oct2013 cross referenced

Pretending that the policy influencers and external forces are absent is not possible.  However, it is possible for us to advocate for a better mathematical solution that addresses the needs of our students in an efficient model reflecting the mathematics required.

Assessment: Is this “what is wrong” with math education?

I have been thinking about a problem.  This problem is seen in too many of our students … after passing a test (or a course) the proficiency level is still low and understanding fragile.  Even accepting the fact that not all students achieve high levels of learning, the results are disappointing to us and sometimes tragic for students.

Few concepts are more basic to mathematics than ‘order of operations’, so we “cover” this topic in all developmental math classes … just like it’s covered in most K-12 math classes.  In spite of this, college students fail items such as:

• Simplify    12 – 9(2 – 7) ÷(5)(3)
• Write  3x²y without exponents

I could blame these difficulties on the inaccurate crutch called “PEMDAS”, and it’s true that somebody’s aunt sally is causing problems.  I might explore that angle (again).

However, I think the basic fault is fundamental to math education at all levels.  This fault deals with the purpose of assessment.  Our courses are driven by outcomes and measurable objectives.  What does it mean to “correctly use exponential notation”?  Does such an outcome have an implication of “know when this does not apply?”  Or, are we only interested in completion of tasks following explicit directions with no need for analysis?

Some of my colleagues consider the order of operations question above to be ‘tricky’, due to the parentheses showing a product.  Some of my colleagues also do not like multiple choice questions.  However, I think we often miss the greatest opportunities in our math classes.

Students completing a math course successfully should have fundamentally different capabilities than they had at the start.

In other words, if all we do is add a bunch of specific skills, we have failed.  Students completing mathematics are going to be asked to either apply that knowledge to novel situations OR show conceptual reasoning.  [This will happen in further college courses and/or on most jobs above minimum wage.]  The vast majority of mathematical needs are not just procedural, rather involve deeper understanding and reasoning.

Our assessments often do not reach for any discrimination among levels of knowledge.  We have a series of problems on ‘solving’ equations … all of which can be solved with the same basic three moves (often in the same order).  Do we ask students ‘which of these problems can be solved by the addition or multiplication properties of equality?’  Do we ask students to ‘write an equation that can not be solved just by adding, subtracting, multiplying or dividing?’

For order of operations, we miss opportunities by not asking:

Identify at least two DIFFERENT ways to do this problem that will all result in the same (correct) answer.

When I teach beginning algebra, the first important thing I say is this:

Algebra is all about meaning and choices.

If all students can do is follow directions, we should not be surprised when their learning is weak or when they struggle in the next course.  When our courses are primarily densely packed sequences of topics requiring a rush to finish, students gain little of value … those procedures they ‘learn’ [sic] during the course have little to no staying power, and are not generally important anyway.

The solution to these problems is a basic change in assessment practices.  Analysis and communication, at a level appropriate for the course outcomes, should be a significant part of assessment.  My own assessments are not good enough yet for the courses I am generally teaching; the ‘rush to complete’ is a challenge.

Which is better:  100 objectives learned at a rote level OR 60 objectives learned at some level of analysis?

This is a big challenge.  The Common Core math standards describe a K-12 experience that will always be a rush to complete; the best performing students will be fine (as always) … others, not so much.  Our college courses (especially developmental) are so focused on ‘procedural’ topics that we generally fail to assess properly.  We often avoid strong types of assessment items (such as well-crafted multiple-choice items, or matching) with the false belief that correct steps show understanding.

We need conversations about which capabilities are most important for course levels, followed by a re-focusing of the courses with deep assessment.  The New Life courses (Math Literacy, Algebraic Literacy) were developed with these ideas in mind … so they might form a good starting point at the developmental level.  The risk with these courses is that we might not emphasize procedures enough; we need both understanding and procedures as core goals of any math class.

Students should be fundamentally different at the end of the course.

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