Pre-Calculus, Rigor and Identities

Our department is working on some curricular projects involving both developmental algebra and pre-calculus.  This work has involved some discussion of what “rigor” means, and has increased the level of conversation about algebra in general.  I’ve posted before about pre-calculus College Algebra is Not Pre-Calculus, and Neither is Pre-calc and College Algebra is Still Not Pre-Calculus 🙁 for example, so this post will not be a repeat of that content. This post will deal with algebraic identities.

So, our faculty offices are in an “open style”; you might call them cubicles.  The walls include white board space, and we have spaces for collaboration and other work.  Next to my office is a separate table, which one of my colleagues uses routinely for grading exams and projects.  Recently, he was grading pre-calculus exams … since he is heavily invested in calculus, he was especially concerned about errors students were making in their algebra.  Whether out of frustration or creative analysis, he wrote on the white board next to the table.  Here is the ‘blog post’ he made:

This picture is not very readable, but you can probably see the title “Teach algebraic identities”, followed by “Example:  Which of the following are true for all a, b ∈ ℜ.  In our conversation, my colleague suggested that some (perhaps all) of these identities should be part of a developmental algebra course.  The mathematician part of my brain said “of course!”, and we had a great conversation about the reasons some of the non-identities on the list are so resistant to correction and learning.

Here are images of each column in the post:

When we use the word “identities” in early college mathematics, most of us expect the qualifier to be “trig” … not “algebraic”.  I think we focus way too much on trig identities in preparation for calculus and not enough on algebraic identities.  The two are, of course, connected to the extent that algebraic identities are sometimes used to prove or derive a trig identity.  We can not develop rigor in our students, including sound mathematical reasoning, without some attention to algebraic identities.

I think this work with algebraic identities begins in developmental algebra.  Within my own classes, I will frequently tell my students:

It is better for you to not do something you could … than to make the mistake of doing something ‘bad’ (erroneous reasoning).

Although I’ve not used the word identities when I say this, I could easily phrase it that way: “Avoid violating algebraic identities.”  Obviously, few students know specifically what I mean at the time I make these statements (though I try to push the conversation in class to uncover ‘bad’, and use that to help them understand what is meant).  The issue I need to deal with is “How formal should I make our work with algebraic identities?” in my class.

I hope you take a few minutes to look at the 10 ‘identities’ in those pictures.  You’ve seen them before — both the ones that are true, and the ones that students tend to use in spite of being false.  They are all forms of distributing one operation over another.  When my colleague and I were discussing this, my analysis was that these identities were related to the precedence of operations, and that students get in to trouble because they depend on “PEMDAS” instead of understanding precedence (see PEMDAS and other lies 🙂 and More on the Evils of PEMDAS!   ).  In cognitive science research on mathematics, the these non-identities are labeled “universal linearity” where the basic distributive identity (linear) is generalized to the universe of situations with two operations of different precedence.

How do we balance the theory (such as identities) with the procedural (computation)?  We certainly don’t want any mathematics course to be exclusively one or the other.  I’m envisioning a two-dimensional space, where the horizontal axis if procedural and the vertical axis is theory.  All math courses should be in quadrant one (both values positive); my worry is that some course are in quadrant IV (negative on theory).  I don’t know how we would quantify the concepts on these axes, so imagine that the ordered pairs are in the form (p, t) where p has domain [-10, 10] and t has range [-10, 10].  Recognizing that we have limited resources in classes, we might even impose a constraint on the sum … say 15.

With that in mind, here are sample ordered pairs for this curricular space:

• Developmental algebra = (8, 3)           Some rigor, but more emphasis on procedure and computation
• Pre-calculus = (6, 8)                         More rigor, with almost equal balance … slightly higher on theory
• Calculus I to III = (5, 10)                   Stronger on rigor and theory, with less emphasis on computation

Here is my assessment of traditional mathematics courses:

• Developmental algebra … (9, -2)         Exclusively procedure and computation, negative impact on theory and rigor
• Pre-calculus … (10, 1)                           Procedure and computation, ‘theory’ seen as a way to weed out ‘unprepared’ students
• Calculus I to III … (10, 3)                      A bit more rigor, often implemented to weed out students who are not yet prepared to be engineers

Don’t misunderstand me … I don’t think we need to “halve” our procedural work in calculus; perhaps this scale is logarithmic … perhaps some other non-linear scale.  I don’t intend to suggest that the measures are “ratio” (in the terminology of statistics; see  https://www.questionpro.com/blog/ratio-scale/ ).  Consider the measurement scales to be ordinal in nature.

I think it is our use of the ‘theory dimension’ that hurts students; we tend to either not help students with theory or to use theory as a way to prevent students from passing mathematics.  The tragedy is that a higher emphasis on theory could enable a larger and more diverse set of students to succeed in mathematics, as ‘rigor’ allows other cognitive strengths to help a student succeed.  The procedural emphasis favors novice students who can remember sequences of steps and appropriate clues for when to use them … a theory emphasis favors students who can think conceptually and have verbal skills; this shift towards higher levels of rigor also serves our own interests in retaining more students in the STEM pipeline.

The Rigor Unicorn

How would you define (or describe) “rigor” in college mathematics classes?  Can you define or describe “rigor” without using the words “difficult” or “challenging”?  I will share a recent definition, and counter with my own definition.

Before anything else, however, we need to recognize the lack of equivalence between rigor and difficult (and between rigor and challenging).  The basic problem with those concepts (difficult, challenging) is that they are relational — a specific thing is difficult or challenging based on how that thing interacts with a person or a group of people.  Difficult and challenging are relative concepts, not a property of the object being described.  I found the calculus of trig functions to be difficult, not because there is any rigor involved — it was difficult for me because of the heavy role of memorization of formulae in that work in the particular class with that particular professor.  Other learners find this same work easy.

A recent definition of “rigor” comes from the Dana Center (DC):

We conclude that rigor in mathematics is a set of skills that centers on the communication and use of mathematical language. Specifically, students must be able to communicate their ideas and reasoning with clarity and precision by using the appropriate mathematical symbols and terminology.

See http://dcmathpathways.org/resources/what-is-rigor-in-mathematics-really http://dcmathpathways.org/resources/what-is-rigor-in-mathematics-really

This definition avoids both ‘banned’ words (difficult, challenging), and that is good.  This definition focuses on communication of ideas and reasoning, and that seems good.  When my department discussed this definition recently at a meeting, the question was naturally raised:

Does rigor exist in the absence of communication?

The problem I have with the “DC” definition of rigor is that it suggests that rigor only exists when there is communication taking place.  In other words, rigor does not describe the learning taking place … rigor describes the communication about that learning.  Obviously, communication about mathematics is critical to all levels of learning — whether there is lots of rigor or none.  I don’t think we can equate rigor with communication.  Such an equivalence tempts us to equate rigor with how we measure rigor.

As I think about rigor, I always return to concepts relating to the strength of the learning.  I’d rather have an equivalence between rigor and strength, as that makes conceptual sense.  The rigor exists even in the absence of communication.  Rigor describes the concepts and understanding being developed within the learner, not the object being learned.

My definition:

Rigor in mathematics refers to the accuracy and strength of learning, and specifically to the completeness of the cognitive schema within the learner including appropriate connections between related or dependent ideas.

In some ways, this definition of rigor suggests that “rigor” and “like an expert” might be equivalent concepts.  I am suggesting that rigor describes the quality of learning compared to complete and perfect learning.  Rigor is not an on-off switch, rather it is a continuum of striving towards the state of being an expert about a set of concepts.

One of the reasons I approach the definition ‘differently’ is that rigor should exist in appropriate ways in every math course — from remedial through basic college through calculus an up to graduate level and research work.  Rigor is not a destination, where we can declare “this student has rigor”.  Rigor is a quality comparison between the unseen learning and the state of an expert in that particular set of content.  When we teach basic linear functions, I seek to develop rigor in which students have qualities of learning like an expert would have, concerning connections and reasoning.  When we teach calculus of trig functions, I hope we seek to develop qualities of learning similar to an expert.

I believe the development of rigor is a fundamental ingredient to making mathematics innately easier for the learner.  When the knowledge is more complete (like an expert) the use of that knowledge becomes more efficient … and the learning of further mathematics requires less energy (just like an expert).  Rigor is the core ingredient in the recipe to make mathematicians from the students who arrive in our classrooms.

Rigor does not start in college algebra, nor in calculus.  Rigor is not the same as ‘difficult’.  Rigor can exist when there is no communication about the learning.  Rigor is the fundamental goal of all learning, at all levels … rigor is a way to measure the quality of learning.  Rigor is the goal of developmental mathematics … the goal of quantitative reasoning … the goal of pre-calculus … the goal of calculus … the goal of statistics.

The “rigor unicorn” is within each of us, and within each of our students.

Can We Even Say “Developmental” Anymore?

Some of us say “remedial mathematics”, others say “developmental mathematics”.  Do you feel like you can’t say either one now?

You may have heard that “NADE” changed its name from National Association for Developmental Education to “NOSS” … National Organization for Student Success.  You can understand why this was done, with the recent attacks on all things developmental.  Being understandable, however, does not make this type of thing “right”.  As far as NADE/NOSS is concerned, I think the name change will make it difficult for the organization to articulate a clear identity … since ‘student success’ is an over-arching concept, suggesting that the group will focus on the universe of higher education.  Who will speak on the behalf of students who need advocates for over-coming weak preparation?

Clearly, this avoidance of the word developmental is a systemic problem — a symptom of massive denial — a denial being offered as a “solution”.  Obviously, remedial education (aka developmental education) has had significant problems in the past with our focus on too-many courses, and not providing enough benefit.  However, multiple measures and co-requisite courses will also be a failure in coping with the gaps in preparation that our students bring to us.  We could debate whether a high-school graduate SHOULD need coursework in college before being able to succeed in mathematics; ‘should’ is a very weak design principle for an educational system.  We must succeed in the real world.  Why should we penalize students by pretending that we have some magic that will somehow enable students with an SAT Math of 420 to succeed in a college curriculum with only added support to their ‘college math’ course?

If leaders don’t want to use labels like ‘developmental’, I encourage them to use the new replacement phrase “black magic”.  It would take some serious black magic to help students succeed in their college program with serious deficiencies in mathematics without doing some direct (prolonged) work on the problem.  In some cases, what is being done to avoid developmental math courses comes across as smoke & mirrors.  People implement grand plans, which (according to them) produce great results for all kinds of students.  Sign them up for “America’s Got Talent!” 🙂

I think we are better off using an accurate word like “remedial” and then have an honest discussion about identifying students who need one or two courses in order to be ready for success in their college program.  We need to think more about the whole college program, and less about passing a particular ‘college’ math course.  The opportunity for second chances and upward mobility are at the center of a stable democracy.

Language is important.  Not using a word (like “developmental”) does not solve the set of problems we face.  There is no magic in education; progress is made by applying deep understanding and critical thinking across a broad community committed to helping ALL students achieve their dreams.