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.

Modern Pre-Calculus Course

Good questions are very helpful.  At a presentation recently on the Theory of Everything (Theory of Everything presentation Oct2018) one of the participants in the session asked:

What do you mean by a ‘modern math course’?  What would be in it? How would it be different?

Being a good question, I could not give that good of an answer at the time.  However, it seemed like such a good question that I should make an attempt to provide a good answer.  The initial domain for the answer is “Pre-calculus”

In order to understand how inadequate the current courses are, we need to understand ‘modern’ thinking about learning and learning mathematics in particular.  Much of the traditional college algebra and pre-calculus experience is based on the presumption that working 10000 problems with some success will prepare you for a course with a higher conceptual basis and greater cognitive demands.  One reference for modern thinking about this comes from the book “Adding it Up”, which focuses on K-12 mathematics (https://www.nap.edu/login.php?record_id=9822) which provides an image to help us visualize the learning of mathematics:

We generally understand the names for these 5 strands, and we often talk about them with our colleagues.  However, the courses currently only address two strands directly (procedural fluency and (to a lesser extent) strategic competence).  Some of us use active learning strategies which (coincidentally) provide some support for the other strands.  A basic premise of a scientific approach is that “things we want do not happen when we want if we do not plan and act intentionally”.

A core problem in college mathematics is our separation of classroom practices from content decisions.  If your instructional practices encourage conceptual understanding within a course which does not directly state ‘conceptual understanding’ as a goal, there is a mis-match between instruction and content … and this will always result in reduced student outcomes because the assessments are likely driven by the learning outcomes.  So, here is the first standard for a modern pre-calculus course:

The learning outcomes in pre-calculus represent all 5 strands of proficiency, and instructional practices support the success of students in all 5 strands throughout the course.

In general, the learning outcomes for the Dana Center “Reasoning With Functions” reflect this type of approach.  Here are the outcomes:

• DCMP_Reasoning with Functions I_Overview and Learning Outcomes
• DCMP_Reasoning with Functions II_Overview and Learning Outcomes

As an example, a traditional pre-calculus course might list this learning outcome:

Represent and recognize functions

A modern course might list this learning outcome:

Create, use and interpret functions and use them to solve meaningful problems

Hopefully, this example of outcomes is helpful in understanding what I mean by a ‘modern mathematics course’ in pre-calculus.  There is a key feature not well represented by the outcomes above — the role of numeric methods within the course.  A modern mathematics course needs to provide a balance of symbolic and numeric methods for students, whether the course is calculus I or pre-calculus.  Some of this is addressed by the ‘overview’ portion of the documents above, though I would look for an explicit statement that the course will embed technology tools for both graphing (TI, Desmos, etc) and modeling (Mathematica, etc).

In a modern mathematics course, we would see evidence of all 5 strands of proficiency on each major assessment as well as the final exam.  A modern mathematics course removes the significant amounts of current courses that fail to meet professional standards for preparation … in this case, for calculus (where we have a sound basis for identifying the nature of the preparation (MAA Calculus Readiness test https://www.maa.org/press/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness and Characteristics of Successful Programs in College Calculus https://www.maa.org/programs-and-communities/curriculum%20resources/progress-through-calculus/cspcc-publications).  In other words, we make room in the pre-calculus course(s) by dropping the unnecessary topics and problems so we can focus on the goal … helping students get prepared for calculus.

The other aspect of a modern mathematics course deals with design principles.  It is generally not wise, and often is dangerous, to create a design with implementation not including a process to collect data on the effectiveness of the design.  Conferences and journals are often well stocked with reports of the first semester or year of a ‘new’ thing; that is not what I am talking about.  I am referring to a regular process of collecting data (aggregated and disaggregated) that will show meaningful trends in a process allowing for the assessment of corrections and modifications in the ‘treatment’.  This type of work is seldom ‘fun’ in the same way that a conference presentation is.  However, serving all of our students depends upon this continual examination of the basic question:

So, how are we doing NOW?

Moving from traditional to modern mathematics courses provides an opportunity to have all students experience good mathematics reflecting current tools and applications, and we might therefore conjecture that there will be an increase in majoring in the mathematical sciences.

Hopefully, this first approximation to a good answer to that good question was helpful.  I’d like to hear from you on that!

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The Bad Part of Dev Math

This past weekend, I was at our state affiliate conference.  MichMATYC has a long history (relatively), and we have had a number of AMATYC leaders from our state (including three AMATYC presidents).  We’ve been heavily involved with the AMATYC standards (all 3 of them).  However, you can still see some bad stuff among our practitioners.

One of the sessions I attended focused on lower levels of dev math — pre-algebra and beginning algebra.  The presenter shared some strategies which had resulted in improved results for students; those improved results were (1) correct answers and (2) understanding.  That sounded good.

However, the algebra portion was pretty bad.  The context was solving simple linear equations, and the presenter showed this sequence:

• one step equations (adding/subtracting; dividing)
• two step equations (two terms on one side, one on other)
• equations with parentheses, resulting in equations already seen

All equations were designed to have integer answers; the presenter’s rationale was that students (and instructor) would know that a messy answer meant there had been a mistake.  All equations were solved with one series of steps (simplify, move terms, divide) — even if there was an easier solution in a different order.

When asked about the prescriptive nature of the work, the presenter responded that students understood that it was reversing PEMDAS (which, of course, makes it even worse for me).

The BAD PART of dev math is:

1. Locking down procedures to one sequence
2. Building on memorized incomplete information (like PEMDAS)

As soon as students move from linear equations taught in this way to any other type (quadratic, exponential, rational) they have no way to connect prior knowledge to new situations.  In other words, the student will seem to ‘not know anything’ in a subsequent class.

To the extent that this type of teaching is common practice, developmental mathematics DESERVES to be eliminated.  Causing damage is worse than not having the opportunity to help students.  When we offer a class on arithmetic (even pre-algebra), the course is very likely to suffer from the BAD PART; offering Math Literacy to meet the needs in ‘pre-algebra’ and ‘basic algebra’ will tend to avoid the problem — but is no guarantee.

All of us have course syllabi with learning outcomes.  Those outcomes need to focus on learning that helps students, not learning that harms students.  Reasoning and applying need to be emphasized, so that students seldom experience the BAD PART.

 
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The Student Quandary About Functions

For students heading in a “STEM-ward” direction, understanding functions will become critical.  Unfortunately, a combination of a prior procedural emphasis and some innate cognitive challenges tends to result in a condition where students lack some basic understandings.

For example, in my intermediate algebra class, we provide problems such as:

For f(x) shown in the graph below, (A) find the value of f(0), (B) find the value of f(1), and
(C) find x so that f(x)=0.

 

Since there is no equation stating how to calculate function values, students need to use the information in the graph.  The vast majority of students make 2 novice errors:

• Error of x-y equivalence:  providing the same answer for (A) and (C)
• Error of symmetry: Since the answer for (A) is x=1, stating the answer for (C) as x=1

To improve this understanding, I use the longest (time measured) group activity in the course.  This is definitely a situation where “Telling” does not correct the errors [I’ve tried that 🙁  ], and the small group process helps dismantle some of the errors.  Clearly, the correct understanding for reading function graphs is critical for success in pre-calculus and eventually in calculus.

Another function concept we dealt with this week is ‘domain’.  Now, once students have found a domain, there is a tendency for some students to think they should find the domain of any and all functions, regardless of the directions for the situation.  This “inertia error” (what was started … continues) is not a long-term problem.  Here is a typical problem for the long-term problem:

Find the domain for the function graphed below:

In this particular class, I provide a fair amount of scaffolding … in a small group project, we explored the behavior of rational functions (without using that label) including what the “undefined” x-value means on the graph.  We don’t use the word asymptote; rather, we talk about the fact that some x-value results in division by zero, and the graph of the function can not show any ‘point’ for such inputs.  This leads to the graphing of the function, including the behavior around the ‘gap’.

Students struggle quite a bit with this type of problem.  Sometimes, they continue the ‘function values from graph’ thinking, and latch on to x=0 or y=0 to make some statement about a ‘domain’.  Many students will correctly identify the x-values for the gaps (yay) but make illogical statements about the domain.  The typical student error is:

• (-infinity, -2) ∪ (-2, infinity)  … or even just one interval (-2, infinity)

This type of error usually follows from a process-focus, detached from the underlying meaning.  I am trying to get them to see:

• gap on graph equates to excluded values in the domain

The process focus looks at the first part of  this.  Like the function value errors, the effective treatment of this problem requires time and individual conversations.

This type of function work is not typical for an intermediate algebra course.  However, it would be typical for an algebraic literacy course.  As we transition from traditional content to modern content in our courses, I am expecting that our intermediate algebra courses will fade away … to be replaced by variations of the algebraic literacy course.

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