Should College Algebra Exist?

Once in a while, I do say something that I regret.  During a recent discussion about general education, I commented that college algebra is intermediate algebra with more fractions.  I knew that was not accurate; I wish I had said that college algebra is intermediate algebra with bigger words.  [Still not that accurate; however, the statement probably makes sense to non-mathematicians.]

“College Algebra” is one of the most common math courses on our campuses.  As a profession, we don’t agree on what this thing is … except that college algebra is one of those courses identified by the title ‘college algebra’.  In some cases, the name is given to a good math course because it will help get transfer credit when ‘general math’ would not.  In other cases, ‘college algebra’ is the title of a remedial pre-calculus course (prerequisite for pre-calc).

If the college algebra course is a STEM-focused, math intensive course, then ‘introduction to analysis’ or ‘analytical methods’ might be a better name.  Whether there is a reason to have this course as well as ‘pre-calculus’ is unclear to me.  If we mean precalculus, perhaps we should always say that.

One thing I do know — college algebra is not a good title for a general education math course.  If we have one course to develop mathematics within college students, I am sure that we can deliver more than symbolic algebraic methods and function theory.  [Some of my colleagues remedy this situation by doing modeling in gen ed math courses; I think this has limited benefits, by itself.]

When I looked for definitions of college algebra, I was pleased to find one that referenced introductory analysis.  Sadly, that source cited no references; I do not think we typically define the course this way.  I see little evidence that we have responded to the calls to modify the course to meet student needs (MAA CRAFTY, AMS Client Disciplines, AMATYC Right Stuff).

Currently, developmental mathematics is under great pressure to change … and we should.  However, the zone of the great unchanging curriculum seems to be college algebra.

No, ‘college algebra’ should not exist.  Let’s build something better in its place.

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Percents as Evil … Percents as Good (Applications of Math)

Given a percent and another number, do we multiply … divide … or something else?

A few years ago, I was at a presentation about a pre-algebra course where the presenter classified percent problems as either growth or decay.  My initial response was that these are concepts too advanced for that course; after a few minutes, I liked the idea, and my experiences since have strengthened that opinion.

Within a few days, I had a chance to work on percents in both a beginning algebra course and in our applications course.  In both courses, the percent problems are varied; one thing that was constant — students ‘wanted’ to multiply a percent and the other number in a problem, regardless of the context.  Sales tax rate and marked price … multiply and add.  Sales tax rate and final price … multiply and subtract (wrong).  Percent decrease and old amount … multiply and maybe add.  Percent decrease and new amount … multiply and add (wrong).

We seem to have reinforced overly simplistic rules about percents to the point where students are impervious to a need to change; 40% wrong answers is not enough (even if I asked ‘8 is 40% of what?’).  It’s really that 100% value that is the problem.  The connection between a growth rate of 3% and a multiplier of 1.03 is a challenge.

In the applications course, I had students work in small groups on a sequence of problems to make a transition from a simple percent value to a multiplier.  They worked hard, explained to each other, and seemed to do well.  The next day … a quiz on percents where they could use the multiplier; result — not so good.  In the applications course, we use this multiplier again — in our finance work (1 + APR) and in our exponential models [y = a(1+r)^x].  I suspect that a deliberate focus on the multiplier in 3 chapters might result in some improvement.

I actually fault our presentations on percents as the root of this ‘evil’.  We do “2 places to the left”, “is over of”, and mechanical use of “a is n% of b”; sure, we include problems where students need to find the base (divide), but the work is too superficial.  Students do not generally understand the contexts where percents are used.  An initial approach on growth or decay, which means seeing the multiplier, might just help.

The most common uses of percents in developmental courses is usually in that pre-algebra course.  Based on long-term goals of understanding, if we are not going to cover the whole story of percents (with the multiplier), we should omit percents entirely.  However, percents are one of the richest zones of overlap between math and the world students experience that we need to see percents as a good thing — and do it right.  Fewer tricks, a lot more understanding!

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Completion Agenda and Change in Mathematics Education

Today, I am at a state conference on Student Success, and there is the usual conversation about the ‘completion agenda’ with a high focus on mathematics in community colleges.  We can not hide from the completion agenda, so perhaps we should understand better how we can use this opportunity to achieve some of our basic goals.

As you probably know, the completion agenda is primarily being driven by philanthropy working through foundations and grants.  We share a goal with these stakeholders — getting more (many more) students to the achievement of their academic goals as well as employment.  However, we have some tensions and areas of disagreement.

The completion agenda and its members have released reports about developmental education — and developmental mathematics in particular.  Some reports are research studies focused on analysis of data with a slight bias towards interpreting data from the completion standpoint.  A few reports have been dramatic dismissals of any values in developmental courses.

Like many of us, I have a strong reaction to the dismissive reports.  I want to remember the audience that those reports were written for — the high-level policy makers who need to support changes.  Perhaps the actors in the completion agenda believe that something really strong needs to be used to get their attention.  Perhaps it’s an unintended consequence of these dismissive reports is that they get quoted and cited by popular media to the point that a broad spectrum of people believe the conclusions.  Our best approach might be to smile and nod — recognize the reports, smile, and don’t argue about the conclusions.

Maybe we can look at the situation this way:  Pressures are being applied in order to create change, and the forces are now strong enough that “not changing” is not an option (if we wanted to).  The foundations and funding process tend to reward certain approaches more than others — in particular, integration of technology in some way.  We do not have to agree that these are the best approaches. And, because of the forces on our profession at this time, it is relatively easy to implement our own ideas of a better solution (or solutions).

I’m reminded of a story.  One person in a community is seen suspiciously by a few, so an investigation is begun.  The investigation uncovers some prior falsehoods by that person (minor ones at that), but no direct evidence.  However, the investigation continues.  Friends are questioned about the loyalty of the person.  Implications are made, even though no evidence is found to support them.  These implications are repeated by a small but active group over a period of days … until the community comes to believe that the person is a traitor.  In fact, the person is loyal — the falsehoods involved statements on forms.  The repetition of statements becomes ‘truth’.

We will see reports saying that “acceleration works well”, so developmental math is limited to one semester.  We see reports that “placement tests don’t work”, so all students are placed into college-level courses.  Seeing initial evidence of positive results does nothing to prove the validity of a methodology.  The scientific method is not being used by the completion agenda, even though the process is ‘data driven’.  The completion agenda works more like a business plan than science.

Our role is to keep the science in mathematics education.  “What works” needs to be understood within the context of the work (its purposes) and needs to make sense with other knowledge (such as cognitive psychology).    Piloting a methodology does not usually create any proof that we have a sound solution.

Let’s articulate what WE see as the problem being solved.  For me, it’s not about completion directly — it’s more about mathematics (each course having good mathematics) and more about not wasting students’ time.  Our view of the problem will only be heard if way communicate it.

Let’s gather and share data on our pilot programs whenever possible.  Just as importantly, when something good happens, we need to keep repeating the report of those results until more pilots are done … and until we understand how the pilot works (or not).  [The completion agenda has forgotten the value of experiments that fail.]

We can not match the communication capabilities of foundations, though we can come close.  The people working on ‘completion’ want to improve education, but they need the profession to be involved with solutions.  We can keep all voices in the conversation.  We can provide additional results that might help explain patterns in data.  Collectively, we need to understand the problem and why a ‘solution’ works in order to bring progress to our profession.

We each have a role to play.  What’s yours?

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Math — What is so good about THAT??

I was thinking that I could post a “What we see … What they see” item on math.  After all, we see many good things but our students see very little worth their time.  Then I thought some more about the ‘we see many good things’ part, and realized that this is not as obvious as we might think.

You’ve probably heard the phrases ‘intended curriculum’ and ‘received curriculum’ (or similar phrases).  When we advocate for a mathematics curriculum we are suggesting that the intended curriculum has sufficient value for students.  In mathematics, I think we confuse daily topics with the curriculum more than most disciplines do.  Instead of saying that a course deals with properties and relationships involving symbolic expressions of certain types, we say that the course covers factoring – rational expressions – radical expressions – and quadratic functions.  One problem is that these are internal code words that mean almost nothing outside of mathematics; the bigger problem is that these statements do not communicate any mathematics (in most cases).

Here is another concept about curriculum:  A course should have a basic story to tell.  Think about asking students who have completed a math course with a good grade:  “What was that course about?”  or “What are some good things to come from that course?”  Given that we do so much of our work in a symbolic world without a strong narrative, students will have great difficulty answering these types of questions compared to non-math courses.

So, if we want students to see the ‘good stuff’ about mathematics, we better increase our use of narrative forms.  This is not easy for us, since mathematics encourages brevity and non-repetitiveness.  However, this lack of narrative is one reason students find math ‘different’ … and ‘difficult’.

Which brings up a related point:  Can a person reason using only symbols?

We have a reputation for focusing on procedures and answers.  We often justify what we do by saying that we are building the reasoning skills of students by requiring that solutions involve a specified level of detail in symbolic form.  Instead of thinking about the reasoning, many students respond by mimicking our solution steps — which is not what we want (for most of us).

My favorite course to teach avoids some of the problems I’m talking about.  The Math – Applications for Living describes the content in phrases that many students can understand — quantities, geometry (a tougher one), finance, etc.  We also employ more narrative in the course (which frustrates some students, but usually helps overall).  We talk about good steps and solutions, but we talk more about how to figure something out — reasoning.  It’s not that we ‘cover Polya’ or anything like that, but students know that we are learning how to solve problems.

I think the curricular problems in math are causally related to the calculus-fixation that still drives much of our work:  It’s like everything prior to Calculus I is considered remediation, so we don’t do any real problem solving before then.  In the old days, this remediation approach was stated explicitly at the larger universities; I have not heard this recently, but our courses before calculus have not changed very much overall.

Developmental mathematics is changing.  We are developing courses that tell a story, we emphasize reasoning, and use narrative more than previously done.  It’s time for courses after the developmental level to look at these issues; it’s time for US to look at these issues and what they mean for our courses and students.

What would you want students to say was “good about that math class”?

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