What to do: Intermediate Algebra Dies

What do we do when we terminate our intermediate algebra course?  A new course is necessary, with a focus on reasoning and communication — a more rigorous course (see The Rigor Unicorn).  What to do?

I’ve written before about the necessary demise of intermediate algebra as a college course (see Intermediate Algebra Must Die!! and Intermediate Algebra … the Barrier Preventing Progress).

The traditional narrative is that algebra is a barrier to college success.  Actually, the barrier is obsolete algebra courses (developmental and pre-calculus) which focus on drill more than understanding, and focus on artificial applications rather than fundamental relationships and concepts.  Mathematical Literacy forms a great starting point for a modern curriculum.  When we ‘kill’ intermediate algebra, the solution is to offer an algebraic literacy course (see Algebraic Literacy Presentation (AMATYC 2016).

My colleagues continue to show a dedication to a modern curriculum.  Within the past 5 years, we have dropped both pre-algebra and beginning algebra courses, and replaced them with a Math Lit class in two formats — regular and ‘with review’ (for students with especially weak numeracy skills).  Last month, we made the decision to eliminate our intermediate algebra course.

Temporarily, we will use a revised “fast track algebra” course.  That fast-track course has existed for 3 years, side-by-side with intermediate algebra.  However, the fast track algebra course still uses out-dated content and lower expectations.  Why?  Because there are not available algebraic literacy materials.  Actually, there aren’t any materials dealing with algebra focusing on communication and reasoning.  It’s like the books (and HW systems) are stuck in 1995 in terms of content.  [It’s only 1995 instead of 1975 because of a little bit of technology that some books incorporate.]

Our obituary sadly reads as follows:

After a long life, perhaps too long, the intermediate algebra course at ____ will be removed from life support on December 31, 2019, surrounded by many family and friends.  Intermediate algebra was preceded on the path to the math after life by basic math, pre-algebra, and beginning algebra.  Surviving are a temporary Fast Algebra course, a Math Lit course, and several college level math courses which are also on life support (although unaware of that fact).  A memorial service will  be held at some time in the future when a modern (current) algebra course can take it’s place to serve our students.

We face this change without a sense of closure.  There is some grief at the old course going away (though that was deserved), but there is a dissatisfaction with continuing the same type of algebraic work.  We are generally pleased with the learning occurring in Math Lit, and want a similar course to follow it … which might be called algebraic literacy or algebraic reasoning.

This is “our” problem, where “our” refers to faculty involved with mathematics in the first two years.  We have not written book materials to support a modern (algebraic literacy or reasoning) course.  Publishers have not pursued this, partially because of huge transitions in their “business model”.  That is, however, no excuse for our lack of movement.  There are smaller publishing companies that could undertake this work (XYZ for example) and we also have options with “OER”.

Is the lonely death of our intermediate algebra due to our disinterest?  A lack of understanding?  Are the enthusiasts for a modern course all too ‘seasoned’ (ie, old) to have the energy to write stuff?  Are the younger professionals only thinking about what they have to teach next week and next semester?  There are issues of professional involvement and responsibility behind this lack of newer materials; that is “on us”.

If you have seen value in an algebraic literacy type course, consider developing materials.  Network to find like-minded colleagues.  Collaboration and technology make the work of developing materials much easier.  Where are the people who will create the next level of new materials in developmental and pre-calculus?  Are you one of them?

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.

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.