Theory of Everything

Theory of Everything … College Mathematics in the First Two Years

 

 

 

 

 

 

 

 

 

Please spend some time with me on a quest to understand everything about college mathematics in the first two years.

 

 

 

 

 

 

 

 

 

 

In some ways, the past allows us to understand the present in a fashion which permits us to anticipate the future.  In particular, I would like to understand the present enough to envision a future based on what is important to us as mathematicians.

 

 

 

 

 

 

 

 

 

Perhaps these quotes will help us with a set of expectations about how past relates to the future.  My favorite of these is the one by Russell Brand, who (although he is certainly not a known mathematician) makes a very mathematical statement — that time is a continuous function, since it lacks discontinuity.  Of course, he does not indicate whether a derivative exists at any particular point in time :).

 

 

 

 

 

 

 

 

 

In our work, we sometimes lose focus on mathematics — we worry about how we are teaching and helping our students.  I think we need to understand that we can’t help our students without also keeping mathematics at the center.  As you will see as we ‘talk’, a core question is “WHAT is important to US?”

 

 

 

 

 

 

 

 

 

 

So, let’s start with now.  The main things are the first two points here:

• Minimize developmental mathematics (less of it, fewer students, etc).
• Avoid College Algebra

Currently, three trends are working with different models for how to achieve these ‘goals’ (which are other people’s goals). You probably understand quite a bit about all 3 trends, plus the last comment about experts (which also applies to legislators).  If you need more background on any of them, do a search … you’re likely to find some resources here on devmathrevival.net.

 

 

 

 

 

 

 

 

 

Perhaps you already have a ‘favorite’ among these sets of models.  A great thing to do … have an honest conversation with some of your colleagues, or folks from neighboring institutions.

 

 

 

 

 

 

 

 

 

 

So, let’s go into the past.  I picked 1975 because I was in the profession at that time, but the field was still young.  Our primary goal was to get students ready for college algebra even if they had not done the ‘college prep’ math courses in high school.

 

 

 

 

 

 

 

 

 

And, that actually made some sense at the time.  Only 33% of students had completed algebra II in high school, and we also had a returning adult population.  The situation is very different today, however; as of 2012, 76% of 17-year-olds had completed algebra II, and that rate is higher for graduates. The upward trend is continuing, though there is obviously a limit for this apparently-logistic function.  To see the sources for this data, look at the references: References Theory of Everything Oct2018

 

 

 

 

 

 

 

 

 

At the credit level, the 1975 courses centered on college algebra as part of the transition to calculus.  This two-semester prep for calculus also was based on the high school preparation of the day, so it usually included a course in ‘algebra’ and a course in ‘trigonometry’.  Take a look at the references and find the citation for Jeffrey Suzuki (Brooklyn College) who has traced the college algebra course back almost two hundred years — it was originally a general education course at liberal arts colleges, more as an alternative to calculus than preparation for.  The content of college algebra has obviously changed over two hundred years (graphing, functions and factoring in particular); however, very little of this has much connection to preparation for calculus.  [Some of the content is based on the symbolic techniques necessary for completing certain types of calculus problems — but that is not the same as preparation for calculus.]

 

 

 

 

 

 

 

 

 

The work of the profession in the 1970s focused on materials, with a little bit of technology.  The mathematics was considered a static product (focused on college algebra).

 

 

 

 

 

 

 

 

 

 

Here is a sample of a dev math book from the era.  Note that the slide rule was the computing technology of the day (I still have both of mine, though never used).  In dev math in particular, the ‘minimum of words’ was considered a good thing.

 

 

 

 

 

 

 

 

 

Here is one of the college algebra books from the 1970s.  This book actually acknowledge calculators as a possibly good thing, and referred to the simple calculators of the day as “electronic slide rules”.

 

 

 

 

 

 

 

 

 

 

The rise of the “hand-held” calculators was met with a negative reaction by most of us in the profession, and we generally banned their use in math classes.  A few of us even continue that ban to this day, doing a great mis-service to our students and the profession.  Within dev math in particular, we regressed the content towards basic procedures and skills with little reasoning or communication.  [At my own institution, we never banned calculators; in fact, we created an arithmetic course based on using the calculator for learning mathematics.]

 

 

 

 

 

 

 

 

 

I think it’s a bit of fun to look at old textbooks.  These are two well-known books and authors; the publishers (of which there were many) were always looking for new authors.

 

 

 

 

 

 

 

 

 

As you might know, AMATYC was founded in the 1970s as a social network, so we were too young to develop any guidance or standards.  The MAA, of course, is much older but they did not address any content prior to calculus.  However, the 1981 CUPM document called for a standard of using numerical methods in calculus II (they are referring to computers and programming at that time).  At some institutions, this standard was actually implemented — many of us did not; most calculus content today is still based strictly on symbolic methods with technology used for visualization and calculating ‘final answers’.

 

 

 

 

 

 

 

 

 

In the early 1990s, the NCTM standards were a major influence on K-12 mathematics — but hardly a ripple was seen in college mathematics.  This is launching era for graphing calculators, and we either embraced them or avoided them.  During this time, professional authors were saying things similar to what we see currently in Common Vision and Math Sciences 2025 (see the references References Theory of Everything Oct2018).  We still focused on college algebra.  “Solutions by definitions” refers to the preparation for calculus — if a book or course said “college algebra” it was … by definition … preparation for calculus.  At the time, very few of us actually advocated for a course that would be effective preparation for calculus.

 

 

 

 

 

 

 

 

 

So, here are more books, this time from the 1990s.  The authors are not as well known, but I notice that the covers include more subtlety than prior books.  This is the last decade with many publishers involved in college mathematics.

 

 

 

 

 

 

 

 

 

In 1995, AMATYC published our first standards (“Crossroads”).  That document, combined with the NCTM standards, motivated some of us to do something — which mostly met writing books or materials, often with grants.  These people made presentations, and including on “AMATYC Right Stuff”, which most of us enjoyed but then ignored.

 

 

 

 

 

 

 

 

 

Here is a reform book in dev math from the period.

 

 

 

 

 

 

 

 

 

 

And, a college algebra ‘reform’ book — the reform property was primarily based on the integration of the graphing calculator.

 

 

 

 

 

 

 

 

 

 

As we get to the early part of the 21st century, much of the activity was based on publisher decisions — consolidation and digital as supplement.  Those choices, and economic considerations, led to a reduction in reform books and a bi-modal distribution of technology integration.  Most importantly, few of us in the profession thought of anything else besides college algebra as a target math course.  This lack of vision then opened the doors for outside influences as we saw a few years later.

 

 

 

 

 

 

 

 

 

 

So, a pair of books from the period.  These authors are more well known, though their publishing companies changed.  The content of either of these books is strikingly similar to the first books shown above — with a few minor changes, and some visibility for the graphing calculator.

 

 

 

 

 

 

 

 

 

AMATYC released a second set of standards in 2006 (“Beyond Crossroads”), which focused on process more than content.  I see this silence on content issues as an implicit acceptance of the status quo, with its obsolete content and curricular structure.

 

 

 

 

 

 

 

 

 

 

The early 2000s were a period of some outside influencers — mostly dealing with efficient delivery.  Some of us still use emporium and modular structures, though these solutions often tend to have the most conservative treatment of content (“it’s all about the procedures and the answers”).

 

 

 

 

 

 

 

 

 

 

So, here is a different metaphor for you to consider:  College algebra as a piñata.  You know the piñata party game — you put on a blindfold, grab a big stick, and try to smack the donkey hard enough that something falls out.  In the case of college algebra, the big stick is held by people outside of the profession in general; they smack college algebra, and get more quantitative reasoning courses and more statistics courses.  The in-profession work on college algebra focus on general education (quantitative reasoning), not preparation for calculus.  [Take a look at the Sonnert & Sadler analysis.]

The question for us:  do we want to enable a STEM path for many students, or do we want to have the path restricted so that only the privileged few ‘survive’?

 

 

 

 

 

 

 

 

 

In our focus on mathematics, we need to see the current uses.  This image shows 3 — one from social sciences about income inequity (algebra and statistics), one from climate studies (calculus) and one from health sciences (calculus).  The sources are on the references.  References Theory of Everything Oct2018

 

 

 

 

 

 

 

 

 

 

The year 2010 was a ‘turning point’, with 3 dev math reforms working together to help move away from the traditional curriculum.  The most important development might have been that we generally began thinking about other targets besides college algebra.

 

 

 

 

 

 

 

 

 

 

However, because of our lack of earlier action, policy influencers had an easy target — with a goal of disrupting the profession, they sold specific solutions to decision makers (not us, generally).  Much of the power in their messages was based on avoiding college algebra.

 

 

 

 

 

 

 

 

 

 

The movements in college math, circa 2010, focused on dev math.  We still don’t have much professional guidance on the zone between dev math and calculus.  I encourage you to read the 2015 CUPM document chapter on calculus; I think the ideas related there will give you some powerful ideas for reforming pre-calculus curriculum.

 

 

 

 

 

 

 

 

 

So, now let’s look at the 3 core movements in college mathematics of 2018.

 

 

 

 

 

 

 

 

I think the ‘co-requisite’ message has been accepted the most broadly by those outside our profession; the folks at Complete College America have done an efficient job of selling this solution to our decision makers, and some of us have heard these messages so often that we are beginning to believe them.  I don’t think the way co-reqs have been done are the methodology that serves us and our students the best.

 

 

 

 

 

 

 

 

 

The word “pathways” has been so heavily used that it nearly lacks any meaning.  In this context, “pathways” means having distinct developmental math courses for students based on their target credit math course.  Frequently, this is something like Math Literacy (or Quantway or “FMR” [Dana Center]).  Now, it’s nice that these students do not experience the out-dated traditional courses.  Similar to co-requisites, however, there is an assumption that all quantitative needs are defined by a specific credit math course; Pathways silences our partner disciplines (science in particular).

In addition, the normal implementation of pathways is to leave the traditional curriculum intact for “STEM” students.  Thus, my comment:  The ‘best’ math students get the worst math courses. Is this what we really want?

As for co-requisite methodologies, pathways devalues algebra and the STEM path.

 

 

 

 

 

 

 

 

 

 

Hidden within the work of “2010” (see above) is a fantastic finding — when we look at the mathematical needs of students (whether for statistics, QR, STEM, or science), there is considerable convergence of need at the Math Literacy level.  The fundamental proportional reasoning, algebraic reasoning, and other goals serve all students.  Another treasure in this work: There are actually very few prerequisite abilities that need to exist in students prior to starting a Math Literacy course.  Therefore, we can start with Math Lit; drop all arithmetic and pre-algebra courses; replace the basic algebra course with Math Lit (or similar).

After the Math Lit course, we clearly can not connect to the traditional intermediate algebra course, which is antiquated and non-functional anyway.  Develop a new algebra course, perhaps based on the design of Algebraic Literacy (see Algebraic Literacy (A Bridge to Somewhere) ) .  Both Math Literacy and Algebraic Literacy have appropriate content for today’s students, and allow us to employ current pedagogy and technology to provide powerful preparation for ALL students.  We can support upward mobility, and help all students avoid the traps of poverty.

 

 

 

 

 

 

 

 

 

 

I think those of us who teach the traditional credit math courses have falsely concluded that the curriculum beyond dev math is generally okay as is.  No, that is not the case.  The content of the calculus path is even more obsolete than the traditional dev math sequence (see notes above and the references shown here).

We also tend to see that our work at this level is trapped by one word: transfer!  Of course, we want these courses to transfer to other institutions.  If you are bound by a state system which defines your courses and content, then you have a defined avenue in which to advocate for change — a modernization of these courses to provide a more efficient system.  [I show a potential model below.]  If you work in ‘non-system’ state, you have networks of professionals which can be used to create a new curriculum which would be shared among several institutions — and would transfer.

Earlier, I commented that external forces sought to disrupt our profession.  Here is a thought:

We can use intentional disruption to begin the process of modernizing our courses.
Sometimes, we need to break something in order to build a better future.

Please note that I am not advocating anarchy.  I am just saying that the box ‘transfer’ is a poor excuse to do nothing for our students.  They deserve better.

 

 

 

 

 

 

 

 

 

The traditional dev math model is based on the premise that more courses is better.

 

 

 

 

 

 

 

 

 

 

Most of the people who make the big decisions, and certainly all of the policy influencers named here, see the work in a very different way.

 

 

 

 

 

 

 

 

 

 

The external view of our work is that there is an exponential decay function involved, while we see a piece-wise linear function.  Our view can not win this debate.  The solution is to change the debate — stop using the traditional words for our work, specifically ‘remedial’, ‘developmental’, and ‘college algebra’.  We can focus on good mathematics for all students and articulate a positive message about effective preparation.  I like to think of it as “one (or none) and done” for dev math.

 

 

 

 

 

 

 

 

 

 

The initial value of this logistic function represents the rate of placing into credit math courses (with our without support or ‘co-requisites’).

 

 

 

 

 

 

 

 

 

 

My institution has been making curricular changes over the past 5 years which creates a good first approximation to the “one (or none) and done” vision.  We have Math Literacy for the first level, regardless of incoming level of abilities; if students needs QR or Stat, those students are ‘done’ along with all those who can place into those credit courses directly.  The other level of prep math is ‘algebra’; we are working on updating this algebra piece to be modern.

 

 

 

 

 

 

 

 

 

 

Just by making the curricular changes, we have doubled our credit course enrollment rate in 5 years — without making any changes to the placement process, nor sacrificing rigor in the credit math courses.  I am guessing that we are at about 75% compared to the dream of 90% (to a credit math course with one or zero prep courses).

 

 

 

 

 

 

 

 

 

 

As I said … that algebra course (prep) needs work.  The college courses in the STEM path needs a lot of work.  I think co-requisites work better within the primary math courses — not the terminal QR and stat courses they’ve been used with.

 

 

 

 

 

 

 

 

 

So, this version shows an approximation for the co-requisite populations for each mainline math course — called ‘co-support’ on the chart, and also has some annotations for the calculus courses.  The fact that we have 3 (or more) basic calculus courses seems to be due to a design meant for engineering programs 50 or 60 years ago.  Not only do we have a very different mix of students in calculus today, but the needs within the discipline have changed greatly.  We should never teach calculus as just a symbolic enterprise — numeric methods are just as important in the world around us.  And, numeric methods should not just be ‘graphing calculator’; we have a variety of modeling tools available (Mathematica, MatLab, etc).  In the current curriculum, students learn the theory and concepts about numeric methods within their discipline — if they ever encounter them before the workplace.

 

 

 

 

 

 

 

 

 

There is my first prediction.  “Developmental Mathematics” will not exist within a few years, as there are multiple strong forces pointing in that direction.  I believe that ‘college algebra’ is in even worse condition, and we should replace it immediately.  Recognizing that the college algebra course has connections, especially in transfer, we can take a little longer … How about in the year 2019??

I’m trusting that many of us believe that a good preparation for calculus is important; it’s still about the mathematics, whatever the final mathematics course happens to be for a student.

 

 

 

 

 

 

 

 

 

Currently, we might be leaving the STEM path in mathematics open to forces which do not necessarily want it to be successful.  We don’t need to surrender the STEM path — for many students, this is the path which will lead them to social and economic stability, which is great for them and very good for our society and the world around us.