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.

HS GPA and Math Placement

In the policy world, “multiple measures” is the silver bullet for solving all issues of student placement in college.  Within the work of multiple measures, the HS GPA is presented as the most reliable measure related to student placement.  This conclusion is the result of some good research being used for disruptive purposes, where a core conclusion is generalized to mathematics when the data was directed at language (‘english’) placement.

A central reference in the multiple measures genre is the Scott-Clayton report from the CCRC ( https://ccrc.tc.columbia.edu/publications/high-stakes-placement-exams-predict.html ).  One of the key findings in that report is that placement tests have more validity in math than in english.  Other results include the fact that placement accuracy could be improved by including the HS GPA … especially in English.  However, the narrative since that time has repeated the unqualified claim — that HS GPA is a better predictor than placement tests.  Repetition of a false claim is a basic strategy in the world of propaganda.

In an earlier post, I shared a graphic on HS GPA vs ACT tests.

This data is from a large ACT study, which means that … if the GPA was a good predictor … we would see all ACT score ranges have a high probability of passing (B or better) in a college algebra course.  The fact that the two lower ACT ranges have an almost-zero rate of change contradicts that expectation.

Locally, I have looked at HS GPA versus math testing using SAT Math scores:

Although this graph does not look at ‘success’, we have plenty of other data to support the conclusion of Scott-Clayton — math placement tests have better validity than English tests.  [The horizontal reference lines in this graph represent the cutoffs for our math classes.]

One might make the argument that math tests work fine for algebraic-based math courses, and that HS GPA works better for general education math courses.  As it turns out, we have been using a HS GPA cutoff for our quantitative reasoning course (Math119) … which includes some algebra, but is predominantly numeracy.

Results:

• Students who used their HS GPA to place:  44% pass rate
• Students who placed via a math test:  77% pass rate

In fact, I am seeing indications in the data that the HS GPA should be used as a negating factor against placement tests … a score above a cutoff with a low HS GPA indicates a lack of ‘readiness’ to succeed.

In theory, a multiple-measures-formula could include negative impacts (in this case, HS GPA below 3.0).  In practice, this is not usually done.  [Another point:  multiple measures formulas are based on statistical analysis … and politics … which transforms a mathematical problem into statistics resulting in a ‘formula score’ which has no direct meaning to us or our students.  An irony within this statistical work is that the HS GPA lacks the interval quality needed to use a mean: the HS GPA itself is a bad measure, statistically.]

Regardless of formulas for multiple measures, we have sufficient data to conclude that HS GPA is well correlated with general college success as well as readiness in English but that HS GPA has little independent contribution in measuring math readiness.

Mathematics placement should be a function of inputs with established connections to mathematics.  The results should be easy to interpret for our students.  Any use of the HS GPA in mathematics placement violates principles of statistics and also contradicts research.