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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2 Comments

• 

  By schremmer, July 24, 2012 @ 10:06 am

  As worthy as these goals may appear, they are hollow: Exactly what are we talking about?
  (It makes all the difference!)

  “dealing with quantitative situations” Unit prices at the supermarket (As used to be suggested)?

  “general education” In what sense? That of once upon a time’s “mathematical tourism”?

  “possibility of inspiration and discovery of mathematicians in unlikely places.”
  That one leaves me completely baffled.

  “deep understandings of what it means to learn mathematics” Shades of Liping Ma?

  “broad and varied collection of tools” Computers and Abacus?

  I would propose that the goal is to help the students learn
  “norms appropriate to rational belief formation” (Colin McGinn) i.e. acquire the ability to argue
  a case to a point where it is reasonable to accept it for true. In the spirit of, and be reference to,
  what is done, say, in courts of law.

  As Dienes used to point out, mathematics is the simplest area in which to learn that.

  Regards
  –schremmer

• 

  By Jack Rotman, July 24, 2012 @ 3:31 pm

  Wasn’t the purpose to delineate all information; general statements were intended in this post — plenty of details elsewhere on blog and wiki.
  Some people do equate ‘quantitative’ with ‘practical numbers’; I do not … I include academic situations.
  General education in the best sense (at least to me) — broad set of tools and understandings to apply to the world.
  ‘Unlikely places’ means our developmental students.
  Deep understanding & tools — like the purposes, this was a general statement with specifics available on blog and wiki.
  The reasoning, and argument, are fine; however, my current work is not seeing them as central purposes. I hope that my work does build a student’s capacity to engage in these in their college-level courses.

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