Practical Math — or Not

Last week, I spent several days with faculty who are working with the Carnegie Foundation’s Pathways — Statway and Quantway, at their National Forum (summer institute).  I continue to be impressed with the quality of these professionals; Carnegie is fortunate to have them involved.  One comment from a faculty member has been stuck in my thinking.  In the context of Quantway, this faculty member said:

Everything in this course has to be practical.  The math students see has to be practical.

I recognize that there is a high probability of head nodding and agreement with this sentiment among people reading this post.  Can we … is it reasonable or desirable … to shift from a ‘nothing in this course is practical’ to ‘everything in this course is practical’ position?

First of all, we need to recognize that ‘practical’ is a matter of perception, communication, and culture.  Our students will not see the same ‘practicality’ that we do.  For example, if we have a series of material looking at the cost of buying a car including operating and finance, many students will definitely not see this as practical.  The majority of my students are not able to consider this situation in their real life now, nor for several years; for some, they can not even imagine having a real choice to make about a car.  What we often mean is that math needs to be contextualized, not practical — context is a simpler matter to establish than practical.

Secondly, the ‘practical’ or ‘contextual’ emphasis reminds me of the old school approach to low-performing math students:  If a student was not doing well in math, put them in an applied math course (business math, shop math, personal finance), as a way of being polite about lowered expectations.  I realize that many of our students are initially happy with the lowered expectations of ‘practical math’; however, this approach does not honor their real intelligence, nor does it recognize the capacities in our students to understand good mathematics just because it is enjoyable to do so.

More important than these two points is the learning implications of ‘practical math’.  I’ve been reading theories of learning and research testing these theories … for close to 40 years now.  Nothing in the theory suggests that learning in a practical context is better than learning without the context; without deliberate steps to decontextualize the learning, the practical approach often inhibits general understanding and transfer of learning to new situations.  I do not believe that ‘all is practical’ is a desirable approach to learning mathematics.

However, context and practicality can be very motivating.  Motivation is the most elemental problem in developmental mathematics.  Therefore, it is reasonable to provide considerably more context for students than the traditional developmental math courses with its ‘train problems’.  I also would add that most students are motivated by learning mathematics with understanding when they can see the connections; true, our students need some extra support for this process, and it conflicts with the approach emphasized with them in the past (primarily memorization without understanding).

I have summarized my view on the ‘practical’ issue with this statement:

I will always include some useless and beautiful mathematics in all of my math classes.

Education is about expanding potentials and creating new capacities; practical learning is the domain of ‘training’ (which is also critical … but it is not education).  I encourage all of us to help our students learn mathematics in different ways: sometimes practical, sometimes in a context, sometimes imaginative, and sometimes logical extensions.  The mix of these ingredients might reasonably shift as a student progresses; developmental math courses might be more practical than pre-calculus.  Diverse learning is better than limited learning.  Diverse learning respects the intelligence of our students and maintains high expectations for all students.

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Change in Developmental Mathematics

Most of us involved with developmental mathematics understand that change is coming; to some extent, we welcome this — though we also have concerns.  How should we conceptualize this change?  How will we even know when this change represents progress, and not just change?

Part of this conceptualization depends on having a concise vocabulary to describe what is changing and what would be progress.  I have heard one phrase that is not helpful; I’d like to explain what I see as being so bad about “change the culture of teaching” and suggest a better vocabulary.

To change a culture, there needs to be a culture.  Culture, in the formal sense understood by anthropologists, refers to shared symbolisms and understandings by a large group of people communicated through generations.  Teaching in developmental mathematics does have some shared norms, such as developing concise work habits and reasoning in students; this does not make it a ‘culture’.  (See http://www.tamu.edu/faculty/choudhury/culture.html for some definitions of ‘culture’.)  What we have is a partially shared set of norms and values out of a larger framework of understanding our settings; we lack the ‘completeness’ of natural cultures as part of a society.  The phrase “change the culture of teaching” is an oversimplification of our problems, often meant to dismiss concerns about change.

How should we talk about change in developmental mathematics?  I suggest that we focus on some central goals and beliefs, not as cultural artifacts but as deliberate and thoughtful statements about our work.

First:

Developmental mathematics deals with increasing student’s capacity for dealing with quantitative situations.

Our central goal is not preparing students for pre-calculus or calculus.  We focus on basic ideas of mathematics, understood deeply, and able to be employed as needed.  We serve all mathematics, not just algebra of polynomials.

Second:

Developmental mathematics contributes to general education.

Our students are preparing for introductory college courses; therefore, specialization is not appropriate.  The design and delivery of developmental mathematics should contribute to the goals of general education, as a priority over specialization.

Third:

Developmental mathematics allows for the possibility of inspiration and discovery of mathematicians in unlikely places.

We have the opportunity to open doors, to allow students to see beauty in mathematics, whether through specific artifacts from a discipline or by the rich connections between aspects of mathematics. 

Fourth, and most importantly:

Teaching in developmental mathematics involves deep understandings of what it means to learn mathematics combined with a broad and varied collection of tools to help students learn and the professional judgment to apply appropriate methods.

Faculty who have accepted the challenge and honor of working in developmental mathematics are advanced professionals who build individual and collective expertise by sharing and learning with others.  We are not there yet, and are not even close; we achieve as much as we do now primarily due to an amazing willingness to work very hard for our students.  Faculty can not be replaced by computers, nor by Khan videos (as good as they are); we use technology as one part of our tool set, not the entire tool set. 

Up until recently, developmental mathematics has lacked a model and mission; most people used the term to describe remedial mathematics, meaning a repeat of school mathematics.  We have not articulated our goals and beliefs, distinct from the school mathematics situation.  Saying that we are doing ‘school mathematics differently and better’ is a very weak justification for our existence.  We can do much better; we can articulate positive statements about our goals and beliefs.

We need to be able to tell when we have made progress, and not just change.  A higher passing rate is only a partial measure if our design is valid; I suggest that it is not.  We need to keep our eyes on the big picture, on the strong and unique justification for developmental mathematics as part of our country’s promise of upward mobility and work ethic.

We can, and must, do a better job of maintaining a focus on mathematics in college to prepare our students for success.

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Variable Concepts … Variable Notation

Let’s talk variables.  Do we want students to develop an understanding of variable concepts (whether in a developmental math course or not)?  Is accurate use of variable notation enough?  If students can model applications and accurately determine solutions … is that enough?  Is there a role for linguistic literacy in mathematics?

A few years ago, I was able to use a sabbatical leave to explore a number of issues related to learning in developmental mathematics; the primary product of this leave was a series of short reports intended for my department though appropriate for faculty at other colleges; one of the reports dealt with variable concepts — see http://jackrotman.devmathrevival.net/sabbatical2006/1%20Variable%20Understanding%20and%20Procedural%20Skills.pdf.  The other mini reports from that sabbatical are available at http://jackrotman.devmathrevival.net/sabbatical2006/index.htm

One of the issues we face in college is dealing (or not) with prior learning.  Without intervention, prior learning (even when inaccurate) survives — often surviving in the face of conflicting information in the current learning environment.  Visualize the prior learning as being as a stable mass of ‘knowledge’ (even though it has gaps and errors); as students go through a class as adults, information that connects positively with the old reinforces the old.  When new information does not connect or conflicts with the old, the low-energy (natural) response is to build new storage … resulting in that solid core being supplemented by weak veneers of new knowledge.  This, of course, is an incomplete visualization for the actual processes in the human brain.  The suggestion is that students approach a math class with an attitude that supports old information and minimizes cognitive effort for dealing with new or incompatible information.

In my beginning algebra class this week, we did the test on exponents and polynomials.  Although the test includes some artificially difficult problems with negative exponents, most of the items deal with important ideas.  One of the most basic items on the test was this:

Evaluate a² + (3b)² for a = -3 and b = 2

Several students made this mistake with the first term:

-3² = -9

A smaller group of students made this mistake with the second term:

(32)²

Now, this is a good class — all students are actually doing homework and attending class almost every day.  We had dealt with the first situation at the start of the semester.  How could these errors survive to this point?

Both errors are based on variable as a symbol to be replaced by a number, which is not complete.  They might represent a visual approach, not verbal.  Variables represent quantities involved in sums and products, where products with variables are implied … and more than this.  Simplifying expressions might — or might not — uncover the incomplete understanding.  What can I do to help students with this?

I am planning on incorporating some linguistic activities around variables in the first week of the semester.  Some of the ideas are from a old book called “English Skills for Algebra” from the Center for Applied Linguistics (Joann Crandall, et al); I believe this book is out of print.  The authors wrote this book from the viewpoint of helping students with ‘limited English proficiency’, which might just apply to many of our developmental students.  Some of their activities involve listening to somebody read mathematical statements and the student writing them down.  I think I will mostly activities that deal with written statements — identifying translations and paraphrasing (both to algebra and from algebra).

I do know that just saying “that was wrong … this is right” will not help these students develop a more complete understanding.  I need to create situations where they get uncomfortable and really dig into the concepts related to variables.  Some energy needs to be created so that we don’t just place a veneer on top of that mass of prior knowledge; parts of that prior knowledge need to be broken up and put back together.  Without that process, many of these students will be limited in their mathematics and blocked from many occupations.

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The Magic Solution to Learning in Developmental Mathematics

Contextualize … discovery learning … group work … experiments … homework systems … calculators … modules … learning communities … clickers … tutoring … and smiles.

What was that a list of?  To some extent, that was a list of ‘magic solutions’ offered by somebody to improve (often ‘dramatically improve’, according to that person) the learning in our developmental mathematics classrooms.  Every single advocate of these solutions has some ‘data’ (often labeled ‘research’) to support their answer to our problems; if they don’t have this data themselves, then they are a convert or follower — often a person at a foundation or policy group.

These are not solutions, let alone magic solutions.  Solutions deal with problems; solutions make sense.  Solutions fit within the surrounding systems to enable both long-term maintenance and ‘scalability’. 

Here is the magic solution to learning in developmental mathematics:

Offer sound mathematics with academic value, supported by skilled professional educators who can help every student learn by employing a diverse set of tools, focusing on cognitive growth in students.

We currently do not have sound mathematics in the majority of our courses; there are emerging models that provide some specific alternatives (New Life, Carnegie Pathways, Dana Center Mathways).   Many of our colleagues (perhaps the vast majority in some places) have limited skills further hampered by a limited conceptualization of their profession; organizations such as AMATYC and its state affiliates provide professional development to supplement the internal opportunities.  As a profession, we have not articulated a standard set of tools necessary for faculty to meet the needs of our students.  And, far too often, we look at surface outcomes of success (completion, passing) instead of looking at measures of meaningful growth in our students.

As you can see, there is nothing simple or quick about this magic solution.  I still call it ‘magic’, because this solution creates a qualitative shift in our profession — instead of ‘avoidance’, we have a positive target; instead of a discouraged and sometimes desperate people, we can be inspired and proud (both as mathematicians and as educators).  I admit that this magic solution is not quick, nor is it easy; however, it is a real solution, not a temporary distraction like the items listed at the start of this post.  [Those items are possible tools to use, not solutions.]

Many of us are currently involved with projects that are not really a solution, whether this consists of modules or mastery learning or a temporary redesign such as emporium.  Do not worry about this work; it is part of the process … not the end.  Whether it takes 2 years or 5 years, the incomplete solution will be identified as such, and the next stage will be started.  THAT (the next stage) is what you should be concerned about. 

Behind this basic change is a more developed and refined use of research.  Much of the ‘data’ used in our profession (internally or externally) is just a little better than the charts in USA Today — they are not statistically sound, and do not fit into a body of research for our profession.  Most of this data is better left ‘ignored’.  Our work should be informed by theory and research that develops over time; fads are a distraction from basic change.

I hope that you can focus on the larger picture, on what is a ‘magic solution’; perhaps you can look at the emerging models for inspiration or encouragement.  Our success in this endeavor called developmental mathematics depends more on our internal visions of solutions than on a temporary distraction or ‘data’.

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