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.

Learning, Success and Mathematics: 100%??

A few years ago, the chief academic officer (aka “Provost”) at my institution proposed that we adopt an “Operation 100%” which involved committing ourselves every student passing each course and every student completing their program of study.  Faculty reaction was more negative than positive, especially about a goal of 100% success rate in every course.

Eventually, the 100% success rate was dropped and the 100% program completion goal was kept.  This was driven, in large part, by the faculty reaction; we correctly pointed out that the 100% success rate was not a reasonable goal, especially in a community college setting.  Although it was a relief to not have the 100% success rate goal, I have to admit … we should have taken the challenge.

In most cases, we design our courses with the assumption that a significant proportion of students will not succeed.  More specifically, courses are designed based on a ‘definition’ that some students will be unable to learn the material in the allowed time frame.  Sometimes, we say “this group of students were not ready for my course” or “that group of students has trouble understanding, and they try to memorize”.  We tell ourselves that many of our students have challenges in their lives which make success in a course unlikely.

And, in terms of data, each of those statements can be shown to be ‘true’.  However, that is simply proving a point of view which justifies the acceptance of low pass rates as ‘normal’.  Another point of view, equally justified by data, is that faculty don’t know how to help students learn and succeed if the student actually needed their help in doing so.

So, let me frame the issue more scientifically:

Instead of designing a course assuming that some students won’t learn or pass, we should consider designing our courses so that we help all students learn and succeed.

You are probably thinking that this is exactly what we do right now.  Read the statement again … it does not say that we “try to help students learn and succeed”; it says “we help all students learn and succeed”.  Of course, not all students will succeed … not all students will learn.  However, 100% success (and learning) should be our fundamental design principle.  Accepting failure, and taking ‘lack of learning’ as a given, is an exceptionally weak design goal.

Imagine a surgeon who says “Well, I know some patients will not survive heart surgery so I am not going to stress myself out with worries about whether this patient survives.”

What does “design for learning and success” look like?  I am working on a complete answer to that question.  In the meantime, here are some implications I see in “100% success” as a goal:

• Every class is an opportunity to help every student learn more mathematics
• Every student knows some mathematics, though some ‘knowledge’ is faulty
• Readiness to learn a topic is part of the class where we ‘teach’ the topic
• Every student is active, all of the time: engaging with work sequenced to proceed from readiness to learning to knowing

I’m using this design for learning and success in algebra courses.  If a class is primarily about learning to solve quadratic equations using square roots, the initiating activity makes sure that every student reviews basic concepts of radicals and the symmetry of square roots.  Teams of 4 or 5 are used, so that every student has a reasonable opportunity to contribute and participate in the process.  “Faulty knowledge” is caught by team members, or the instructor, or both — starting with the prerequisite knowledge.  The activity proceeds to explore the primary concept in the new material, often starting with a simple example to solve followed by a ‘cloze’ type statement (fill in blanks) to complete a summary of the concept.  Next, the activity involves the application of this concept to a more complex situation.

I have been engaged with the profession for a long time.  As you probably know, the fundamental ingredient for student success is MOTIVATION … it’s hard for a student to learn if they are not motivated to attend class.  Some of us use tricks to improve motivation — we have students play games in class, or we find some application using mathematics in a context that the student might care about.

What I am observing is that students find this intentional design innately motivating — especially the struggling student.  For example, in one class this semester I have 8 students with mathematical challenges that are significant enough that I might normally ‘expect’ them to fail.  In fact, prior to my current design, they all would have failed. [These challenges were obvious within the first week.]  However, all 8 of the students continued to attend class; they found it motivating that every class was designed so that they would learn some mathematics.  Initially, they did not learn enough mathematics … partly because these 8 students had a larger amount of faulty knowledge.  For 2 of these students, they eventually stopped attending class when it became clear to them that their test scores were too low for them to pass the course.    The other 6 are successful; none of these 6 struggling but succeeding students will receive a 4.0 grade; on the other hand, they are not all headed towards ‘barely passing’ either.

Do you want your students to succeed?  Well, you better start by designing a course which provides 100% of the students an opportunity to learn every day.  We can not afford to leave learning to unknown or random processes.  Some patients do not survive … some students will not succeed.  We should plan — and design — our classes for what we want to see, rather than what would happen without effective intervention on our part.

So, I am all in on “100% success in our courses”.  I realize that some readers are in states where they are subject to some arbitrary minimum pass rate within their courses.  That is not what I am talking about — I am talking about designing courses so that every student learns and can succeed.  The last thing we need is some uninformed arbitrary ‘standard’ being inflicted on us and our students; this can not help but cause harm to learning and to students.  We should focus on what we care about … learning mathematics, for every student.

If you want to base your career on failure being normal, go in to politics.  Education should be based on learning to success as the goal for everybody in our classes.

Sharing Impedes Progress

Join me on a trip to a time and world in which sharing was expected, and done almost universally.  We watch how the medical profession used sharing to improve patient care.

This world, known to inhabitants as “Flux Major”, is quite small but the technology is wonderful.  The doctors have both Snype™ and IntEx™ which are used to collaborate with colleagues (and occasionally work with patients).  Now, even though small, the planet has about 10,000 doctors; we’d like to observe all of them — even with great technology that is not practical.  So, we will use three doctors as representative specimens.  Below are partial transcripts from a recent sharing session between Dr. Newton, Dr. Turing, and Dr. Zhang; please forgive the inexact translation.

Dr. Newton:

Twenty patients recently presented symptoms associated with viral infections apparently flu-like.  All 20 agreed to be treated with my mixture of cinnamon bark and amoxicillin over a 10 day period.  At the end of the 10 days, all had improvements in their symptoms and 6 were symptom free.  In addition, there was a marked improvement in their attitude towards medical care.

Dr. Zhang response: That is interesting, and sounds helpful.  I will try a similar treatment with my patients.

Dr. Turing: Yes, it does sound good.  Were you able to quantify the improvements?  [I am also working on this computing device, so I like data. :)]

Dr. Newton:  As it turns out, yes.  We measured the basal body temperature over the 10 days.  The mean basal temperature was 99.8 degrees (F) at the start, and 99.0 degrees at the conclusion.

Dr. Zhang:

My colleagues and I are focusing on psychological factors in illness.  We recently treated 30 patients with generally shared symptomology centered on digestive issues.  Twenty-nine of the patients agreed to a course of treatment involving the use of our copyrighted “Positive Music” CD (available for purchase at zhangpositive.com), specifically 3 treatments daily for at least 30 minutes each time.  Twenty patients completed at least 5 days of treatment, and all of them reported partial or total remission of symptoms.

Dr. Turing response: Thanks for the data.  I can see this being implemented in my own practice.

Dr. Newton: Can you share some technical data on the Positive Music CD?

Dr. Zhang:  Absolutely.  The CD contains 20 tracks which vary in length from 30 seconds to 6 minutes.  The tempo of all tracks is between 90 and 120 beats per minute, with a preponderance of major keys and limited use of sharps and flats.

Dr. Turing:

My partner and I are focusing on social factors in wellness. Currently, we have 25 patients in our care with the most common condition being recovery from conversion therapy.  Treatment is completed within a residential setting where we focus on developing security and acceptance followed by increasing confidence in advocating for personal needs.  Of the 25 patients, 15 have been with us for over 3 months; they each report marked improvement in personal health along with self-efficacy.

Dr. Zhang response: This is very interesting, though I don’t see how I an apply it to my non-residential practice.

Dr. Newton: So, you are saying that patients will observe an improved modal health condition if they feel safe and accepted regardless of personal differences?

Dr. Turing: Yes, absolutely.  We see those as the primary ingredients in personal health.

Obviously, the medical practice on Flux Major is not similar to Earth, in most cultures.  In developed countries, improvements in health care are informed by research (especially ‘gold standard’ research) combined with professional standards which have been developing over decades (if not centuries).  Most of us are relieved that our health care provider does not base their treatments on what they heard from other providers.

However, “sharing” is the prevalent form of professional development among teachers.  I am especially concerned with mathematics in the first two years.  Our conferences, publications, and ‘standards’ place the highest priority on sharing of practical ideas.

So, what’s the problem?  The problem is that sharing is both non-scientific and non-intellectual.  Somebody does a session describing their ‘flipped’ classroom and shows good results … what is the scientific basis for the results?  We have no idea, in general; we just know ‘it works’.  How about co-requisite classes?  We see an anecdote about what works and what did not, but we do not gain any significant understanding of the process or problem involved.  How often have you tried a promising practice that was shared, only to find that it did not ‘work’ for you?

This is an issue of professional development.  Most of us experience professional development (PD) as a sharing process.  Not only do we not develop a framework for understanding what we are doing, we do not generally do any critical thinking about what we hear.  When people share their best practices, they don’t expect an intellectual exchange which might involve classic tools (like classification, compare/contrast) nor general theories of learning.  As a result, we treat buzz phrases as if they were theory — student centered, active learning, and so on.  Improving the practice of mathematics education at the college level moves at a snails pace, when it moves at all; we likely have as much regression as we have progression.

I’ve been thinking about these issues for a while.  Recently, I was directed towards a book — from 1991.  The book was titled “The Academic Crisis of the Community College” by McGrath and Spear (see https://books.google.com/books?id=eiyJL7z0nAgC). One of the quotes from this book is shown here:

Sharing a commitment to teaching, but without a shared notion of what effective teaching might be, with strong affective ties to one another, but without the intellectual guidance and constraint provided by disciplinary cultures, faculties take on the aspect of a “practitioners’ culture.”  They come to undervalue intellectual exchange and mutual criticism, and to overvalue “sharing” as sources of professional and organizational development.”  (pg 147-148)

As a group, we who teach college mathematics have a much larger toolbox compared to 30 years ago but with little additional understanding of the process of selecting tools to achieve our purposes.  We march forward, not knowing whether we approach a precipice or a grand vista of success.  We congratulate ourselves on sharing our work, without developing a framework to understand that work.  Feeling good about the sharing does not change the fact that we don’t understand, sufficiently, the work we are engaged in.

Sharing, in the absence of a framework for understanding, impedes progress.

Controlled Burns in the Forest of Developmental Mathematics

Are there connections, or parallel conditions, between the worsening wildfires and developmental mathematics?  The destruction of a wildfire is terrible, and this post is not meant to minimize the problems experienced in that process.  However, it occurs to me that we can learn some lessons from fire management techniques.

Specifically, the overall danger from wildfires can be managed do some extent by setting controlled burns — fires deliberately set, with an expected path and amount of burn.  The process of a controlled burn is intended to reduce both the amount of flammable material AND the risk of fires spreading quickly in a region.  The forest, in effect, is made more healthy by intentionally burning some of it.

Now, the big problem with wildfires is that conditions have created larger and more aggressive fires, especially in regions of the American West.  The effects of climate change have increased the mean temperature in the areas as well as reduced the annual precipitation.

Some of us might  see developmental mathematics as being consumed by uncontrolled wildfires.  These wildfires come with catchy phrases — “corequisite remediation” and “multiple measures” being two of the most common fires.  We focus our attention on the wildfire; we fail to see the conditions which required some type of fire in developmental mathematics.

Between 1970 and 2010, enrollment in developmental mathematics grew … and grew.  We also tended to create additional non-credit math courses in dev math.  Given the poor results guaranteed by a long sequence, a correction is necessary.  Since we (in the profession) did not manage to create a controlled burn to limit the danger, outside forces released the wildfires of co-requisite remediation and multiple measures.

Eventually, the co-requisite remediation and multiple measures “wild fires” will burn up all of the readily available fuel.  Quite a bit of this destruction was necessary given the climate and conditions in developmental mathematics.  Some of the destruction was not necessary, like the areas of a forest that did not need to burn but the wildfire could not be controlled.  Many of us are dealing with both types of destruction in developmental mathematics.

The necessary destruction includes:

• sequences of length greater than 2 (prior to college level math, including ‘college algebra’)
• content based on an obsolete K-12 structure
• teaching methodologies based on low-level learning of unimportant mathematics

The valuable parts of developmental mathematics can still be saved from the wildfire.  These valuable parts include:

1. college-prep math courses focused on mathematical reasoning for adults
2. a balance between general education and math for specific programs or target courses
3. mathematics faculty skilled in delivering courses which dramatically increase the abilities of the students

These properties of future prep mathematics represent our commitment to support the success of all students, in future mathematics … in science courses … and in academia in general.

We, in the profession, will need to play the role of fire fighters who work to change a wildfire into a controlled burn.  A good result from a wildfire is improbable without intense effort by a committed group of people.  We can work to create a fire break to limit the continued burning from “co-requisite remediation” and “multiple measures”.  The total destruction of developmental mathematics is possible if we are not willing to do the hard work of stopping the wildfire.

This is about us, not about the people who started these wildfires. Are we willing to do what it takes to be able to continue to provide developmental mathematics that makes a difference to our students?  Do we see equal access and upward mobility as worthy goals?

I hope you will stand against the wildfires and work with me for the future of developmental mathematics.