Remedial College Algebra

We are all familiar with ‘predictions’ based on societal trends which are seldom validated by reality — whether it is flying cars, Facebook’s “population”, or economic stability.  Predictions are often based on a presumption of continuity within the determining forces; people attempt to apply modeling concepts to an open (or semi-open) system.  As mathematicians and mathematics educators, however, we often fail to notice the interaction between forces impacting our curriculum.

At the collegiate level, the most dramatic example of such a disconnect is the course called “college algebra”.  I’ve written before about how ill-designed this course is, considering the role it plays; see College Algebra is Not Pre-Calculus, and Neither is Pre-calc, Cooked Carrots and College Algebra, College Algebra Must Die! and also about it’s history (see College Algebra … an Archeological Study. This post, however, deals with the conditions we are operating within in approximately the year 2019.

For reference, I will be using information about the Common Core Math Outcomes.  (See http://www.corestandards.org/Math/).  I recognize that the Common Core has many detractors, as well as structural problems within (such as insufficient guidance about which outcomes have a higher priority).  However, there is no dispute with this statement:

In spite of ‘problems’ with the Common Core, the Math Outcomes listed are the only usable reference for national conversations about K-14 mathematics.

So, here is the bottom line statement: if one compares the set of Common Core Math outcomes for K-12, they exceed the outcomes normally listed for a college algebra course required prior to pre-calculus.  Even the standard pre-calculus course is repeating content described in the Common Core.  [ACT conducts regular research on ‘national curriculum; the surveys are at http://www.act.org/content/act/en/research/reports/act-publications/national-curriculum-survey.html ]

Complex numbers?  Vectors? Matrices?  Connect zeros to factors? Binomial Theorem? Polynomial functions?  Rational functions?  Those, and more, are listed as Common Core outcomes for high school mathematics for ‘all students’.  I am not trying to equate the high school courses to a college algebra course; that is not a required element for the conclusion about college algebra as a course preceding pre-calculus:

College algebra is a remedial course.

The traditional remedial mathematics courses received that designation primarily because people saw that the content was what students SHOULD HAVE HAD in K-12 mathematics.  We maintained developmental mathematics courses which taught 9th to 11th grade mathematics, and denied college credit for them because students ‘should have already learned this stuff’.  [I am not suggesting that we allow remedial math to get credit towards a degree; in particular, I don’t think intermediate algebra should meet a math requirement.]

My claim is that the college algebra course preceding pre-calculus materially meets the same conditions which resulted in the determination that our traditional ‘dev math’ courses were remedial.  Substantially, every topic in the college algebra course should have already been learned in the K-12 experience.  Certainly, not every student had that opportunity (just as before).  Certainly, not every K-12 school does a quality job in mathematics (just as before) … though this statement also applies to “us” as college math professionals.

At the college level, we often function in isolation from K-12 mathematics; in general, we also continue to work as if the client disciplines exist now as they did 50 years ago.  We have not been sensitive to the dramatic changes in intent within the K-12 curriculum, and sometimes we seem to take pride in our ignorance of school mathematics.  We presume continuity as it relates to our curriculum, in contrast to our intense efforts to improve pedagogy.   I continue to believe something I have been saying for years:

Improving our pedagogy without modernizing our curriculum is like putting a GPS on a 1973 Ford Pinto — sure, we can see a map to help us drive, but it is still a 1973 Pinto.

We teach the importance of continuity within our courses.  I find it ironic (and tragic) that we tend to make basic assumptions concerning continuity within the world around us.  College algebra is a remedial math course.

Letting Go: The Final Vertical Asymptote

Shortly (like 2 months), I will be putting my professional work into the function which produces no output at all — retirement.  Perhaps a better metaphor is that the function has a final vertical asymptote at the end point of the domain.

My career has actually had several points of discontinuity, where the next function value substantially differs from the prior value.

• The first 5 years were focused on support for my college’s large and successful self-paced “Math Lab” — which initially had 13 courses in the same room with two instructors.  One of my duties was to hire and train student workers; of these workers, one of them would eventually come back to my College as an adjunct faculty.
• The longest period without a discontinuity (19 years) came next … I provided part of the faculty leadership for the courses and instruction in that Math Lab.  One of our students started in beginning algebra, and eventually came back to my College as a full-time faculty.
• The largest gap occurred next — I was loaned to the College’s registrar’s office to help implement our student software system (“Banner”), and eventually I functioned as an associate registrar.  Instead of AMATYC conferences, I attended the “Banner Summits” each year.
• After 5 years, I returned to ‘faculty’ duties though not exactly as the earlier time.  The College’s Math Lab was no longer an option seen with pride, as the administrators did not provide support and our own faculty made decisions which contributed to the downfall.  This unhappy period lasted 8 years.
• In 2010, the Math Lab officially closed.  This was the first year where all of my teaching was in ‘regular’ classrooms with larger groups of students; my initiating work with teaching was all one-on-one or pairs in the Math Lab.
• Although relatively small, another point of discontinuity occurred two years later as the department chair asked me to take over our quantitative reasoning class.  This class was the most fun to teach of any class I’ve done.  Within 5 years, this class went from 60 students per year to 400 students per semester.
• The last point of discontinuity occurred when I was declared not qualified to teach that QR class.  My final 4 years have been focused on dev math — though I spent two separate periods serving as an ‘acting academic coordinator’ for the department (planning, staffing, enrollment, etc).

{image is NOT a perfect match for the metaphor 🙂   }

This is my final semester of teaching mathematics.  On the other side of the last vertical asymptote, awaits other type of activities — family and (hopefully) volunteer work.

Throughout my work in AMATYC and MichMATYC as well as the Dana Center and Carnegie Foundation for the Advancement of Teaching, I have appreciated the help and support of MANY people.  For that, I thank each of you.

For the curious, this blog (DevMathRevival) will continue for another few weeks.  Some posts are likely to be reflections on my career, while other posts will be the type of commentary previously seen here.

Generalizing to Failure: “Cross Products”

The human brain naturally takes the leap between example and generalization.  We encounter one used-car-salesperson who pushes us to buy something we don’t want, and we make a generalization that all used-car-salespersons are pushy.  We encounter a method for correct answers in one fraction situation, and we make a generalization that this method works for all fraction situations.  In fact, some of us teach by taking advantage of this ‘constructive’ process.  Caveat emptor!!

Our Math Literacy course forms the basis for this specific post, though the issues with generalizing are universal.  The specific scenario is this … students have previously encountered operations with two fractions (all 4 basic operations), and now we are solving proportions.  Our proportions involve only one variable term, so students occasionally use proportional reasoning to build up or down, and this ‘works’ for now.

The problem is not that students lack any prior good knowledge about cross products.  Almost every student in my classes ‘knew’ what cross products are (in the fraction world).  The problem was that they generalized an incorrect ‘method’:  cross-products form a fraction.

Like this:

“Solve 14/12 = 126/b”

Student answer:  14/1512 = 1/108

This wrong answer becomes correct if there was a division operation instead of an equation.  The fact that students reach college with such bad knowledge is, of course, a function of the math opportunities they had K-12 … students with a good math background have normally been trained to notice such ‘trivial’ features as the symbol between two fractions.

I’m sure that this specific bad generalization comes from another process — using cross products to test for equivalence of fractions.  Those problems are often presented as a pseud-equation like this:

Test:  15/108 =?  10/76

At the micro-level, my message is “don’t let students use cross-products with fractions unless the object was a proportion complete with the “=” symbol”.  Teaching cross products for anything else causes harm to your students, just as teaching PEMDAS causes harm.

However, my main concern is not really this one situation.  In basic algebra, ‘distributing’ is a key skill.  The false generalizations involve these types of problems (resulting in the ‘answers’ shown):

• (x + 3)² = x²  + 9                <distributing an exponent>
• 3(w – 4)² = (3w – 12)²         <distributing before an exponent>
• 5y² = 25y²                           <distributing an exponent>
• (x + y)/y = x + 1                  <‘distributing’ by cancelling>

The first two types are very resistant to learning to correction.  In psychology, this faulty generalization is sometimes called ‘cognitive distortions’ or ‘hasty generalizations’, though I prefer the direct term ‘false generalizations’.

Keep in mind that “We Are the Problem” (where ‘we’ refers to people teaching mathematics at any level).  We focus on correct answers as measures of correct knowledge (see The Assessment Paradox &#8230; Do They Understand?).  Some of us avoid that paradox by requiring written explanations on assessments; that approach does help if done in moderation — having to explain in multiple situations on one assessment comes with significant overhead for us and our students, as well as the known risks of bias in grading the writing.

We have two other tools to help student correct their generalizations:

1. One-on-one (F2F) feedback
2. Problems designed to confront false generalizations

I have been using both approaches for 20 years or more.  My conclusion (hopefully not a false generalization 🙂 ) is that problems are not as effective as we think they are, in catching bad generalizations. The proportion given earlier came from a student who is very thorough in doing homework, and we had just done this problem in class the day before:

• Solve -3/(y + 4) = 2/(y – 1)

This problem is different from all of the homework, and all problems we had done in class — those binomial expressions were very confusing to students.  With suggestions and sometimes direct statements, students eventually used cross products to solve (complete with the distributing).  That experience does not help, though; the experience is short in duration, and seldom engages an emotional response that might help learning).  Prior learning complete the false generalizations is strong, compared to the experiences we control.

The best impact comes from the one-on-one engagement.  Because there is another level of activity (social or emotional), our work is a bit stronger than just the problems themselves.  Some students I worked with on that unusual problem adjusted their knowledge.

My message today has two components.

1. Teach mathematics in a way that offers some control over false generalizations.

Get students engaged with problems that “don’t work” while including some problems that do work with the idea we are trying to learn.  Keep in mind that, while students helping students supports a good classroom environment, other students will tend to have similar false generalizations.  I had a team this semester where 4 of the 5 students believed the same wrong thing; the other student ‘gave in’ because the other 4 agreed.  YOU are the best resource in the classroom to control generalizations.

2. Assume that a significant proportion of your students have false generalizations about “today’s topic”

Because of the focus on correct answers, students can “go far” without having correct understanding.  Typically, this leads to a ‘crash and burn’ experience in pre-calculus/college algebra or intermediate algebra.  Since we don’t want math courses to be a filter, we need to design instruction so students are not weeded out; opportunities to correct prior learning are critical in our efforts at equity and inclusion.

There is no magic for fixing a false generalization.  Take a look at a study on correcting misinformation in health care (https://psycnet.apa.org/record/2014-41945-002).  The situation is not hopeless, but it is discouraging.  Correcting false generalizations is MUCH more difficult than learning true generalizations in the absence of faulty knowledge.  Thus, the first idea above is the most important — regardless of what you teach, or at what level, structure the learning process so that generalizations are almost always correct.  Five true generalizations with no faulty ones are more valuable than 20 true generalizations with 5 faulty ones.

False generalizations will kill your students dreams.

The Basis of Basic Algebra: PEMDAS or Order of Operations or ??

My professional work focuses on helping students who have generally completed their K-12 mathematics though they are not able to place in to a college level math course.  Based on doing this for a long time, I share the following conclusions:

• Most students (even those who can place ‘college ready’) have dismal abilities and understanding about arithmetic relationships.  However, this (perhaps surprisingly) has little impact on their success in college.
• The primary issues preventing success in college (in terms of quantitative outcomes) deals with fundamental concepts of basic algebra: expressions & simplifying; equations & solving.  The most fundamental of these issues is order of operations.

So, let’s make this concrete.  We are doing really basic expressions and equations in our Math Lit course; one of the problems for today’s group work was the following:

Solve   15 = -3(y + 2) – 3

Because we are finishing up a unit for a test, we have been doing a lot of distributing in class.  We’ve talked about concepts of order of operations as it relates to expressions like the left-hand side of that equation.  In spite of that, students claim that:

(y + 2) = 3y  (because there is a 1 in front of the y)

Now, it is very easy to tell a student that their work is incorrect; it’s easy to say “you should distribute first” (though we don’t always want to distribute).  I am more interested in diagnosis … WHY is that mistake there?  What understanding needs to change to know what to do with all problems we will see?

It is very disturbing to learn that many “bad things” students do are based on being told in the past to “use PEMDAS”.  In this problem, students honestly think that they have no choice — they MUST combine y and 2; since they know that y=1y, they add 1+2 to get 3y.  Somewhat reasonable … if the requirement to combine were true.

We need to avoid misleading (or incorrect) rules about calculating which lack a sound mathematical basis.  PEMDAS is such a rule; I have written before on this, so I won’t repeat myself (not too much anyway).  See prior posts:  PEMDAS and other lies 🙂 , More on the Evils of PEMDAS! and What does &#8216;sin(2x)&#8217; mean? Or, &#8220;PEMDAS kills intelligence, course 1&#8221;.

Our students would be better served if we focused on the relationships between operations and how that helps with ‘order’ questions — even if we don’t present such complicated (and contrived) problems.    Simple problems are sufficient for much of what we need students to learn:

• -5²  and (-5)²
• 4x²   [does the square apply to the 4?]
• 8+2(x+3)   vs 8+2(6+3)

Algebra is about properties and choices.  Students focus on what they have been told is really important, and PEMDAS is often in this category.  This conflicts with the goals of basic algebra — and with most mathematics our students will work with.  I would rather spend an hour in class exploring the 3 different ways to solve the equation 15 = -3(y + 2) – 3 than in redundant examples drilling “one way” to simplify or “one way” to solve.

Correct answers from PEMDAS are worse than worthless.  Success in basic college math and science classes is based on understanding (thoroughly) a few concepts.  Nobody should be ‘teaching’ PEMDAS, because we should never deliberately harm our students.  Understanding is what enables students to reach their dreams; quick fixes — whether in the form of PEMDAS or ‘co-requisite remediation’ — are more about correct answers than they are about student success or mathematics.

Are you so focused on ‘correct answers’ that you either limit your student’s knowledge or unintentionally cause them harm?  As I tell my students:

Correct answers themselves are almost worthless.  The value comes from our understanding.