School Mathematics can NOT be Aligned with College Mathematics

How do we help students become ready for college mathematics?  How do avoid students earning credit for learning that should have occurred before college?  Perhaps our conceptualization of these problems is flawed in fundamental ways.

As I write one of my final posts for this blog, I am pondering history and future … and the intersection called the present.  Some of this pondering has been pleasant reflection, while much of the pondering has been either professional regret or stimulating conjecture.  I hope to put some of each ‘pile’ in this post.

[Here, “common core” is a place-holder for school mathematics.]

As usual, a problem and its solutions are based on definitions.

• School mathematics is defined operationally by the curricular materials and accepted pedagogical practices.
• School mathematics is usually characterized by a closed system focused on experiencing a constrained subset of mathematics at constrained levels of learning.
• College mathematics is ill-defined with conflicting goals of historical course content and service to the discipline.
• College mathematics is characterized by a closed system serving history competing with components seeking to build mastery of modern mathematics.

The fact that one system is reasonably well-defined while the other is ill-defined suggests that any goal of alignment is unreasonable.  In other words, the reasonable-sounding effort to create a smooth transition from one level to the next is foolish.

Just as groups sought to deliberately disrupt the work of developmental education, groups using ‘alignment’ are also seeking to disrupt the world of college mathematics.  In their view, college mathematics should be more like school mathematics where the system is well-defined operationally by a limited collection of curricular objects (‘courses’).  The presumption is that the core of the college mathematics system is valid and that we can apply the school mathematics process to standardize the alignment.

All of this ignores two related and critical flaws:

1. School mathematics was (nominally) designed to prepare students for college mathematics.
2. College mathematics (as known today) is a collection of obsolete tools along with a bit of valuable mathematics.

At a CBMS meeting a few years ago, I raised the question “When are we going to question the college mathematics courses consisting of excursions into issues that we don’t care about?”  Some in attendance thanked me for saying that we should change the applications in our courses; sadly, that is not at all what I was saying.  I was suggesting that much of college mathematics presented mathematics that we no longer care about as mathematicians.

Advocating for alignment does not mean such alignment is possible; it’s not.  Advocating for alignment does not mean that people support our curricular goals; they have their own agenda (not ours).

Before we worry overmuch about ‘alignment’, we had better make basic corrections to our own system.  College mathematics could be an exciting world for our students to explore with colorful vistas combining symbolic and computational methods supported by conceptual knowledge.  Do not look to MAA and AMATYC to ‘tell us’ how and when … our organizations are too fearful of offending part of ‘us’.

Build local alliances to support experimentation in modernizing mathematics in college.  Do not let ‘alignment’ lock you in to an obsolete and harmful set of mathematics courses.

Meaningful Mathematics: That is Worth …

A few years back, my dean informed me that returning adult students wanted to know how their learning would be applied to their lives as opposed to understanding theory.  I was quite surprised by this statement, given all I’ve learned over the years; I had reached the conclusion that the more ‘seasoned’ students wanted to understand the why as well as the how and when.  This cognitive dissonance resulted in a non-discussion as the Dean would not believe my statements.  Of course, these generalizations (hers and mine) are seldom true over a broad range of situations.

This idea of ‘application’ and meaning continues to be a hot-button for me.  Somehow, academia has accepted the strange notion that learning needs to be justified by seeing how knowledge is applied to individual lives.  In the guided pathways movement, mathematics is specifically designated for ‘alignment’ with the student’s program of study.  I’ve written a bit about that; see GPS Part III: Guided Pathways to Success … Informed Choices and Equity and other posts.

There is a need for balance here, as in most things.  Traditionally, college mathematics courses were theory-driven gauntlets designed to ensure that only the fit students reached the point of seeing how mathematics is applied to significant problems and processes.  No ‘meaning’ is permitted until the student has survived entry into calculus with some sanity, and then meaning is only explored within a limited range of classic problems (‘maximize the area of …’).  The absence of meaningful uses of mathematics is only part of the problem with traditional courses.

At the other extreme are some modern courses in quantitative reasoning or statistics.  A note came out this week from Carnegie Math Pathways (the folks doing Statway™ and Quantway™) about how great it was that students could see how to apply the mathematics in their lives (“for the first time …”).  Some of my colleagues emphasize finance work in our QR course for similar reasons.    Yes, that adds some ‘fun’ to the course, and helps with motivation for some students.

A few years back I did an invited talk at a state meeting dealing with general education mathematics.  The talk was apparently well received for the wrong reasons — members of the audience thought I was advocating for a focus on applications and context that students could understand.  I left that meeting dismayed.

Why the dismay?

• Are we only able to motivate student engagement and learning in mathematics if we can convince them with immediate applications?
• Is the value of learning mathematics constrained by specific utilitarian advantages of a constrained set of content?
• Are we so unskilled in teaching mathematics that we see a need to focus on context instead of understanding mathematics?

Some readers might see these statements as disparaging inquiry based learning (and ‘problem based learning’).  My concerns with those pedagogical approaches centers on the balance issue.  As a matter of learning and cognition, context is the classic double-edged sword — yes, context can provide an initial anchor for learning and supports motivation.  However, context also tends to constrain the learning making it difficult for students to transfer their knowledge.

At the heart of my concern is this:

If we focus on utility of mathematics, how are we to inspire the next generation of mathematicians?  Is that inspiration going to be limited to applied mathematics?

For years, I have been saying that every math course should engage students with “useless and beautiful mathematics”.  We should show students how we became inspired to be mathematicians; for most of us, this inspiration combines theory and application — not ‘or’.

Let’s keep mathematics in every math class.

Remedial College Algebra

We are all familiar with ‘predictions’ based on societal trends which are seldom validated by reality — whether it is flying cars, Facebook’s “population”, or economic stability.  Predictions are often based on a presumption of continuity within the determining forces; people attempt to apply modeling concepts to an open (or semi-open) system.  As mathematicians and mathematics educators, however, we often fail to notice the interaction between forces impacting our curriculum.

At the collegiate level, the most dramatic example of such a disconnect is the course called “college algebra”.  I’ve written before about how ill-designed this course is, considering the role it plays; see College Algebra is Not Pre-Calculus, and Neither is Pre-calc, Cooked Carrots and College Algebra, College Algebra Must Die! and also about it’s history (see College Algebra … an Archeological Study. This post, however, deals with the conditions we are operating within in approximately the year 2019.

For reference, I will be using information about the Common Core Math Outcomes.  (See http://www.corestandards.org/Math/).  I recognize that the Common Core has many detractors, as well as structural problems within (such as insufficient guidance about which outcomes have a higher priority).  However, there is no dispute with this statement:

In spite of ‘problems’ with the Common Core, the Math Outcomes listed are the only usable reference for national conversations about K-14 mathematics.

So, here is the bottom line statement: if one compares the set of Common Core Math outcomes for K-12, they exceed the outcomes normally listed for a college algebra course required prior to pre-calculus.  Even the standard pre-calculus course is repeating content described in the Common Core.  [ACT conducts regular research on ‘national curriculum; the surveys are at http://www.act.org/content/act/en/research/reports/act-publications/national-curriculum-survey.html ]

Complex numbers?  Vectors? Matrices?  Connect zeros to factors? Binomial Theorem? Polynomial functions?  Rational functions?  Those, and more, are listed as Common Core outcomes for high school mathematics for ‘all students’.  I am not trying to equate the high school courses to a college algebra course; that is not a required element for the conclusion about college algebra as a course preceding pre-calculus:

College algebra is a remedial course.

The traditional remedial mathematics courses received that designation primarily because people saw that the content was what students SHOULD HAVE HAD in K-12 mathematics.  We maintained developmental mathematics courses which taught 9th to 11th grade mathematics, and denied college credit for them because students ‘should have already learned this stuff’.  [I am not suggesting that we allow remedial math to get credit towards a degree; in particular, I don’t think intermediate algebra should meet a math requirement.]

My claim is that the college algebra course preceding pre-calculus materially meets the same conditions which resulted in the determination that our traditional ‘dev math’ courses were remedial.  Substantially, every topic in the college algebra course should have already been learned in the K-12 experience.  Certainly, not every student had that opportunity (just as before).  Certainly, not every K-12 school does a quality job in mathematics (just as before) … though this statement also applies to “us” as college math professionals.

At the college level, we often function in isolation from K-12 mathematics; in general, we also continue to work as if the client disciplines exist now as they did 50 years ago.  We have not been sensitive to the dramatic changes in intent within the K-12 curriculum, and sometimes we seem to take pride in our ignorance of school mathematics.  We presume continuity as it relates to our curriculum, in contrast to our intense efforts to improve pedagogy.   I continue to believe something I have been saying for years:

Improving our pedagogy without modernizing our curriculum is like putting a GPS on a 1973 Ford Pinto — sure, we can see a map to help us drive, but it is still a 1973 Pinto.

We teach the importance of continuity within our courses.  I find it ironic (and tragic) that we tend to make basic assumptions concerning continuity within the world around us.  College algebra is a remedial math course.

Nested and Sequential: Not in Math, or “What’s Wrong with ALEKS”?

Much of our mathematics curriculum is based on a belief in the ‘nested and sequential’ nature of our content — Topic G requires knowledge of Topics A to F; mastery of topic G therefore implies mastery of topics A to F.   A popular platform (ALEKS) takes this as a fundamental design factor; students take a linear series of n steps through the curriculum, and can only see items which the system judges they have shown readiness for.

Other disciplines do not maintain such a restricted vision of their content, whether we are talking about ‘natural’ sciences or social sciences … even foreign language curricula are not as “OCD” as math has been.  As a matter of human learning, I can make the case that learning topic G will help students master topics A to F; limiting their access to topic G will tend to cause them to struggle or not complete a math class.

Whether we are talking about a remedial topic (such as polynomial operations) or a pre-calculus topic (function analysis), the best case we can make is that the topics are connected — understanding each one relates to understanding the other.  Certainly, if a student totally lacks understanding of a more basic idea it makes sense to limit their access to the more advanced idea.  However, this is rarely the situation we face in practice:  It’s almost always a question of degree, not the total absence of knowledge.

At my institution, this actually relates to our general education approach (as it probably does at most institutions).  In our case, we established our requirements about 25 years ago; the mathematics standard (at that time) was essentially intermediate algebra.  The obvious question was “how about students who can place into pre-calculus or higher”.  One of my colleagues responded with “these courses are nested and sequential; passing pre-calculus directly implies mastery of intermediate algebra”.  My judgment is that this was incorrect, and still is incorrect.  Certainly, there is a connection between the two — we might even call it a direct correlation.  However, this correlation is far from perfect.

Learning is a process which involves forward movement as well as back-tracking.  We are constantly discovering something about an earlier topic that we did not really understand, and this is discovered when we attempt a connected topic dependent on that understanding.

Some of my colleagues are very concerned about equity, especially as it relates to race, ethnicity, and social status.  Using a controlled sequence model has the direct consequence of limiting access to more advanced topics and college-level courses for groups of concern … students in these groups have a pronounced tendency to arrive at college with ‘gaps’ in their knowledge.  A mastery approach, although a laudable goal, is not a supportive method for many students.

In some ways, co-requisite courses are designed based on this mis-conception — we ‘backwards design’ the content in the co-req class so that the specific pre-requisite topics are covered and mastered.  I don’t expect that these courses actually have much impact on student learning in the college-level courses.

Back in the ‘old days’ (the 1970s) a big thing was programmed learning, and even machine learning.  The whole approach was based on a nested and sequential view of the content domain.  My department used some of those programmed learning materials, though not for long — the learning was not very good, and the student frustration was high.

Our courses, and our software (such as ALEKS), are too often based on a nested and sequential vision of content — as opposed to a learning opportunity approach.  By using a phrase “knowledge spaces”, ALEKS attempts to sell us a set of products based on a faulty design.  Yes, I know … people “like ALEKS” and “it works”.  My questions are “do we like ALEKS because we don’t need to worry about basic decisions for learning?” and “do we think it works because students improve their procedural knowledge, or do they make any progress at all in their mathematical reasoning?”

Obviously, there if a basic fault with a suggestion to remove the progressive nature of our curriculum … there are some basic dependencies which can not be ignored.  However, that is not the same as saying that students need to have mastery of every piece or segment of the curriculum.  No, the issue is:

Do students have SUFFICIENT understanding of prerequisite knowledge so that they can learn the ‘new’ stuff?

This ‘sufficient understanding’ is the core question in course placement, which I have addressed repeatedly in prior posts.  I am suggesting that the ambiguity of that process (we can never be certain) is also valid at the level of topics within a course.  It is easy to prove by counter-example that students do not need to have mastered all of the prior mathematics before succeeding; they don’t even need to necessary have the majority of that mathematics.  Learning mathematics is way more messy — and much more exciting — than the simplistic ‘nested and sequential’ view.

There is a substantial literature based on ‘global learners’.  I definitely prefer the concept of ‘global learning’, as I think our own ‘styles’ vary with the context.  However, that literature might help you understand the ‘ambiguities’ I refer to; see https://www.vaniercollege.qc.ca/pdo/2013/11/teaching-tip-ways-of-knowing-sequential-vs-global-learners/  as a starting point.  As a side comment, ‘global learning’ is also used to describe the goal of having students gain a better understanding of ‘global’ societies, cultures, and countries; in that context, they really mean ‘world’ not ‘global’ (global refers to a physical shape, while ‘world’ refers to inhabitants).

A nested and sequential structure, by design, limits opportunities to learn.  This, in turn, ensures that we will fail to serve students who did not have good learning opportunities in their K-12 education.  Just because we can lay out a logical structure for topics and courses from a nested & sequential point of view does NOT mean that this is a workable approach for our students.

Drop as much of the sequential limitations as you can, and start having more fun with the excitement of having more learning for our students.