Math Education: Changes, Progress and the Club

What works?  How do we help more of our students … all students … achieve goals supported by understanding mathematics in college?  To answer those questions, we need to correct our confusion between measurements and objects; we need to develop cohesive theories based on scientific knowledge.  We are far from that stage, and policy makers tend to be just as ill-equipped.

Co-requisite remediation in mathematics has been broadly implemented.  The process which led to widespread use of a non-solution illustrates some of my points.

The rush towards co-requisite models is based on two patterns in ‘data’:

• Students who place into remedial mathematics courses tend to have significantly lower completion rates in programs
• Students who pass a college math course in their first year tend to have significantly higher completion rates in programs

These patterns in the data are not in dispute.  The reason I call co-requisite remediation a non-solution, however, is based on the faulty reasoning which bases a solution on these ‘facts’.  I find it surprising that our administrators accept this treatment for a problem which never addresses the needs of the students — it’s all about responding to the data.  That approach, in fact, is a political method for resolving a disagreement where people on one side hold the power to make decisions.

Eliminating stand-alone developmental math courses will not enable more students to complete their college program, beyond the 15% to 25% who were underplaced by testing — and then only to the extent that this group failed to complete their remedial math courses.  The data on this question usually shows that the issue is not failing remedial math courses; it’s retention or attempt.

Tennessee has been the most publicized ‘innovator’, and I have been addressed with the “club” that the leader of that effort was a mathematician — with the handle of the club saying “He’s a mathematician; and he thinks this is a good solution!”  Like mathematicians are never wrong :(.  Some evidence on related work in Tennessee is showing a lack of impact (https://www.insidehighered.com/news/2018/10/29/few-achievement-gains-tennessee-remedial-education-initiative).  I would expect that evidence to show that the main benefit of corequisite remediation is an increase in the proportion of students getting credit for a college math course in either statistics or quantitative reasoning (QR)/liberal arts (LAM).

When data shows that many students can pass a college math course in statistics, QR, or LAM I don’t see any evidence of corequisite working.  Many of my trusted colleagues believe that stat, QR and LAM have very minimal mathematics required; some believe that the only real requirement is a pulse as long as there is effort.

Why do students placed in remedial mathematics have lower completion rates?  The properties of this group of students is not homogeneous.  For some of them, they have reasonably solid mathematical knowledge that is ‘rusty’ … a short treatment, or just opportunity to review, is sufficient.  A significant portion have a combination of ‘rust’ and ‘absence’; the absence usually involves core ideas of fraction, percent, and basic algebraic reasoning.  Corequisite remediation presumes to address these needs via a ‘just in time’ approach; ‘rust’ responds well to short-term treatments — absence not so much.  Another portion of remedial math students have mostly ‘absence’ of mathematical knowledge.

I’ve written previously on the question of ‘when will co-requisites work’ (Co-requisite Remediation: When it’s likely to work).  My comments here are more related to the meaning of ‘work’ — what is the problem we are addressing, and what measures show the degree to which we are solving it?

One of the primary reasons we have today’s mess in college mathematics is … ourselves.  For decades, we presented an image of mathematics where calculus was the default goal as shown in the image at the start of this post.  This image is so flawed that external attacks had an easy time imposing a political solution to an educational problem:  avoidance.  Of course, courses not preparing students for calculus are generally ‘easier’ to complete.  That is not the problem.  The problem is our lack of understanding related to the mathematical needs of students beyond what we find in guided pathways.

To clarify, the ‘our’ in that last sentence is a reference to everybody engaged with both policy and curriculum.  As an example of the lack of understanding, consider the question of the mathematics needed for biology in general and future medical doctors in particular.  We hear comments like “doctors don’t use calculus” on the job; I’m pretty confident that my doctor is not finding derivatives nor integrating a rational function when she determines appropriate treatment options.  However, mastering modern biology depends on conceptual understanding of rates of change along with decent modeling skills.  The problem is not that doctors don’t use calculus on the job; the problem is that our mathematics courses do not address their mathematical needs.

So, back to the two data points mentioned earlier.  The second of these dealt with an observation that students who complete a college mathematics in their first year are more likely to complete their program.  This type of data is often used as a second rationale in the political effort to impose co-requisite remediation … get more students into that college math course right away.  Of course, the reasoning is based on confusing correlation with causation; do groups of similar students have better completion with a college math course right away compared to delays?  Historically, students who are able to complete a college math course in the first year have one or more of these advantages:

• Higher quality opportunities in their K-12 experience
• Better family support
• Lack of social risk factors
• Presence of enabling economic factors

If these factors are involved, then we would expect a ‘math in first year’ effort to primarily mean that more students complete a college math course — not that more students complete their program.  This is, in fact, what the latest Tennessee data shows.  I have not been able to find much research on the question of whether ‘first year math’ produces the results I predict versus what the policy marketing teams suggest.  If you know of any research on this, please pass it along.

We need to own our problems.  These problems include an antiquated curriculum in college mathematics combined with instructional practices which tend to not serve the majority of students.  Clubs with data engraved on the handle are poor choices for the forces needed to make changes; understanding based on long-term scientific theories provides a sustainable basis for progress which will serve all of our students.

Why Do Students Have to Take Math in College?

The multiple-measures and co-requisite trends (fads, if you will) continue to gain share in the market.  Results are generally positive, and more laws are passed limiting (or eliminating) remedial mathematics in colleges.  Given the talk on these issues, I have to wonder … why do we require students to take a mathematics course in college?

Clearly, I am not raising this question relative to STEM or STEM-ish programs that some students follow; their need for mathematics is clearly logical (though that experience needs to be more modern than they usually experience).  These students normally proceed through some sequence of mathematics, whether 2 courses or 10.  No, the question is relative to programs or institutions which require one math course, usually a general education course.

Those general education math courses are often very close in rigor to high school courses common in the United States at this time; I’ll provide a specific rubric for that statement below.  “College Algebra”, the disaster that it is, happens to be pretty close to the algebra expectations in the Common Core standards; the details differ, but the level of expectations are very similar.  “Statistics”, at the intro level, is again similar to those expectations; even some of the intro stat outcomes are in the Common Core.  Liberal Arts math has topics not normally found in K-12 mathematics, but the level of rigor is generally quite low.  Quantitative Reasoning (QR) has some potential for exceeding the high school level, but most of our QR implementations are very low on rigor.  See https://dcmathpathways.org/resources/what-is-rigor-in-mathematics-really for a good discussion of ‘rigor’ as I use the word in this post.

Do we require a math course in college as a means to remediate the K-12 mathematics students “should have had”?  Or, do we require a math course in college in order to advance the student’s education beyond high school?

Those questions seem central to the process of considering those current trends.  The high school GPA, the cornerstone of most multiple measures, has a trivial correlation with mathematical abilities but a meaningful correlation to college success; if the college math course is essentially at the high school level, then using the GPA for placement is reasonable.  Co-requisite remediation can address missing skills but not a lack of rigor (in general); if the college math course is essentially at the high school level, there is little risk involved from using co-requisite remediation.

On the other hand, if we require a math course in order to extend the student’s education beyond high school, neither multiple measures nor co-requisite remediation will dramatically decrease the need for stand-alone remediation.  K-12 education does not work that effectively; prohibiting stand-alone remediation in college will punish students for a system failure.  Our ‘traditional’ math remediation involving three or more levels is also a punishment for students, and can not be justified.

I would like to believe that we are committed to a college education, not just a college credential.

Before we conclude that multiple-measures and/or co-requisite remediation “work”, we need to validate the rationale for requiring a math course in college for non-STEM students.  A key part of this rationale, in my view, is our community developing a deeper appreciation of the quantitative needs of all disciplines.  Few disciplines have been exempt from the radical increase in the use of quantitative methods, and this is a starting point for ‘why’ require a college math course — as well as the design of such courses.  Most of our current courses fail to meet the needs of our partner disciplines, which means getting more students to complete their math course will have a trivial impact on college success and on occupational success for our students.

If it is important to extend a student’s education beyond the K-12 level, then the ‘rigor’ of the learning is more important than the quantity of topics squeezed in to a given course.  The discussion of rigor cited above is helpful but a bit vague.  Take a look at this taxonomy of learning outcomes:

This grid is adapted from a document at “CELT” (Iowa State University; http://www.celt.iastate.edu/teaching/effective-teaching-practices/revised-blooms-taxonomy/), and is based on the “revised Bloom taxonomy”.  The revised taxonomy is a significant update published in 2001; one of the authors (Krathwohl) has an article explaining the update (see https://www.depauw.edu/files/resources/krathwohl.pdf ).  The verbs in each cell are meant to provide a basic understanding of what is intended.  [Note that the word “differentiate” is not the mathematical term :).]

Within the learning taxonomy, the columns represent process (as opposed to knowledge).  Those 6 categories are frequently clustered in to “Low” (Remember, Understand, Apply) and “High level” (Analyze, Evaluate, Create); the order of abstraction is clear.  For the knowledge dimension (rows), the sequence is not as clear — though we know that ‘metacognitive’ is higher than the others, and ‘factual’ is the lowest.

In both K-12 mathematics, and the college math courses listed above, most learning is clustered in the first 3 columns with an emphasis on “interpret” and “calculate”.  A direct measure of rigor (“education”) is the proportion of learning outcomes in the high level columns, with possible bonus points for outcomes in Metacognitive.   Too often, we have mistaken “problem complexity” for “rigor”; surviving 20 steps in a problem does not mean that the level of learning is any higher than simple problems.  We need to focus on a system to ‘measure’ rigor, one that can justify the requirement of passing a math course in college.

Trump Method: Complete College America

Whatever your political persuasion, I hope this comparison makes sense to you.  Most politicians use selective fact usage, and it’s normal to have candidates repeat ‘information’ that fails the fact-checking process.  Mr. Trump is just a bit more extreme in his use of these strategies.  I’m actually not saying anything against “the Donald”.

However, the Trump Method is being employed by the folks at Complete College America (CCA).  The CCA is a change agent, advocating for a select set of ‘game changers’ … which are based on a conclusion about remedial education as a useful construct.  The CCA repeats the same information that does not pass the fact-checking process, much to the detriment of developmental education and community colleges in general.

It’s not that professionals in the field believe that our traditional curriculum and methods are anywhere near what they should be.  I’ve talked with hundreds of teaching faculty over the past ten years, relative to various constructs and methods to use; though we differ on eventual solutions and how to get there, we have a strong consensus that basic changes are needed in remedial mathematics.

However, the CCA brings its anvil and hammer communication … promising simple solutions to complicated problems (just like Mr. Trump).  The recent email newsletter has this headline:

Stuck at Square One
College Students Increasingly Caught in Remedial Education Trap
[http://www.apmreports.org/story/2016/08/18/remedial-education-trap?utm_campaign=APM%20Reports%2020160902%20Weekend%20Listening&utm_medium=email&utm_source=Eloqua&utm_content=Weekend%20listening%3A%20New%20education%20documentaries]

Following up on this headline leads one to a profession-bashing ‘documentary’ about how bad things are.  Did you notice the word “increasingly”?  Things getting worse clearly calls for change … if only there was evidence of things getting worse.  Not only are the facts cited in the documentary old (some from 2004), there is no discussion of any change in the results.

Like “immigrants” for Mr. Trump, remedial education is a bad thing in the view of the CCA.  Since remedial education can not be deported or locked up, the only option is to get rid of it.  The headline says that we ‘trap’ students in our remedial courses, as if we had criminal intent to limit students.  No evidence is presented that the outcomes are a ‘trap’; the word ‘trap’ is more negative than ‘limitations’ or ‘inefficient’ … never mind the lack of accuracy.

Some people have theorized that Mr. Trump appeals to less educated voters.  Who does the CCA material appeal to?  Their intended audience is not ‘us’ … it’s policy makers and state leaders.  These policy makers and state leaders are not generally ignorant nor mean-spirited.  However, the CCA has succeeded in creating an atmosphere of panic relative to remedial education.  Because of the long-term repetition of simplistic conclusions (lacking research evidence) we have this situation at state level groups and college campuses:

Remedial education is a failure, because the CCA has data [sic].
Everybody is working on basic changes, and getting rid of stand-alone remediation.
We better get with the band-wagon, or risk looking like we don’t care (‘unpatriotic’).

This is why the CCA work is so harmful to community colleges.  Instead of academia and local needs driving changes, we have a ‘one size fits all’ mania sweeping the country.  Was this the intent of the CCA?  I doubt it; I think there intent was to destroy remediation as it’s been practiced in this country.  Under the right conditions, I could even work with the CCA on this goal: if ‘destroy’ involved a reasoned examination of all alternatives within the framework present at individual community colleges, with transparent use of data on results.

Sadly, the debate … the academic process for creating long-lasting change … has been usurped by the Trump Method of the CCA.  I can only hope that our policy makers and college leaders will discover their proven change methods; at that point, all of us can work together to create changes that both serve our students and have the stability to remain in place after the CCA is long gone.

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Alignment of Remediation with Student Programs

My college is one of the institutions in the AACC Pathways Project; we’ve got a meeting coming up, for which we were directed to read some documents … including the famous (or infamous) “Core Principles” for remediation.  [See http://www.core-principles.org/uploads/2/6/4/5/26458024/core_principles_nov4.pdf]  In that list of Core Principles, this is #4:

Students for whom the default college-level course placement is not appropriate, even with additional mandatory support, are enrolled in rigorous, streamlined remediation options that align with the knowledge and skills required for success in gateway courses in their academic or career area of interest.

What does that word “align” mean?  It seems to be a key focus of this principle … and the principle also implies that colleges are failing if they can not implement co-requisite remediation.  In early posts, I have shared data which suggests that stand-alone remediation can be effective; the issue is length-of-sequence, meaning that we can not justify a sequence of 3 or 4 developmental courses (up to and including intermediate algebra).

The general meaning of “align” simply means to put items in their proper position.  The ‘align’ in the Core Principles must mean something more than that … ‘proper position’ does not add any meaning to the statement.  [It already said ‘streamlined’ and later says ‘required or’.]  What do they really mean by ‘align’?

In the supporting narrative, the document actually talks more about co-requisite remediation than alignment.  That does not help us understand what was intended.

The policy makers and leaders I’ve heard on this issue often use this type of statement about aligning remediation:

The remediation covers skills and applications like those the student will encounter in their required math course.

In other words, what ‘align’ means is “restricted” … restricted to those mathematical concepts or procedures that the student will directly use in the required math course.  The result is that the remedial math course will consist of the same stuff included in the mandatory support course in the co-requisite model.  The authors, then, are saying that we need to do co-requisite remediation … or co-requisite remediation; the only option is concurrent versus preceding.

If the only quantitative needs a student faced were restricted to the required math course, this might be reasonable.

I again find a basic flaw in this use of co-requisite remediation in two flavors (concurrent, sequential).  We fail to serve our fundamental charge to prepare students for success in their PROGRAM … not just one math course.  As long as the student’s program requires any quantitative work in courses such as these, the ‘aligned’ remediation will fail to serve student needs:

• Chemistry
• Physiology
• Economics
• Political science
• Psychology
• Basic Physics

Dozens of non-math courses on each campus have strong quantitative components.  Should we avoid remedial math courses just to get students through one required math course … and cause them to face unnecessary challenges in several other courses in their program?

In some rare cases, the required math course actually covers most of the quantitative knowledge a student needs for their program.  However, in my experience, the required math course only partially provides that background … or has absolutely no connection to those needs.

Whom does remediation serve?  Policy makers … or students?

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