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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Renew the Profession!

I am at the ‘summer institute’ for Statway and Quantway, though the event is now called the ‘national forum’ for the pathways.  Hosted by the Carnegie Foundation, the meeting is being attended by over 100 faculty from across the country … some have been teaching a Statway or Quantway course this past year, some are new faculty from those colleges, and some are faculty from ‘new colleges’ who are looking to join the work.

The most satisfying aspect of the national forum is the dedication of these faculty to renew the profession.  Instead of looking for the answer, these faculty are building their understanding of the learning process for their students; they are listening to experts with theory and knowledge that applies to the issues; and they are collaborating on solutions that will help their students.

This dedication to renew the profession is part of the change process we are all facing in developmental mathematics.  Although some of us are currently dealing with a temporary ‘fix’ such as modules or mastery learning, the profession has a need to understand the learning and student needs so that we can provide courses with a purpose and a value to students. 

The Carnegie work involves phrases such as productive persistence, language & literacy, and advancing teaching.  The specifics of this work are only ‘in the network’ (the networked improvement communities).  Over the next few years, I believe that all of us are going to develop our understanding of the concepts and theories … and the efficacy of specific strategies for specific students in specific sfituations.

I hope that you will join me, and all those already working in these areas.  The time is now to renew the profession of teaching and learning in developmental mathematics.

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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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Calculators as Problem … Calculators as Resource

Earlier this year, we had a post here on teachers as a problem or a resource (see https://www.devmathrevival.net/?p=1021).  Technology — calculators in particular — presents another problem/resource discussion.  Is the use of calculators a good thing, or an evil contribution to an ignorant population of math students?

For example, an article in USA Today mentions calculators as part of a discussion on math illiteracy related to pushing too much math too soon (see http://www.usatoday.com/news/opinion/forum/story/2012-07-09/math-education-remedial-algebra/56118128/1).  I don’t usually cite a USA Today item, as the publication presents so many examples of bad statistics and mathematics.  One line in this article did resonate: Nobody in a high school math class could tell the teacher what the answer is for 8×4 was — without using a calculator.

To some extent, we are still in the “back to basics” movement (basic skills). People who complain about calculators usually mention basic skills or facts as a goal of mathematics education.  We also have colleagues who see nothing wrong with intense use of calculators in math classes; and, we have entire colleges who ban calculators from math classes.  The question, then, is why use calculators?  Why not use calculators?

We need to answer this question within our framework for education in general, and math education in particular.

Education is about a process that creates a qualitative and quantitative change in the capacities of the student.

If a student leaves a class, or a college, with the same capacities with some added skills, we have not educated the student — we have provided some training.  Training is all about skills; education is about capacities.  This is the reason why college graduates do better in jobs and quality of life measures. 

Mathematics education is about a process that creates qualitative and quantitative change in the mathematical capacities of the student.

Knowing the answer to a problem like 8×4 is not an issue of capacity.   However, needing to use a calculator to find the answer to simple problems often means a lack of mathematical capacity.   Capacities are based on understandings and connections; a specific missing fact is not a matter of capacity.  Having a grasp (call it an intuitive grasp) of number relationships begins the network of quantitative structures that make up mathematical capacities.

At some point in reading this, it is likely that you thought of the word ‘memorization’.  When calculators are not allowed in classes such as developmental mathematics, we often justify it by saying that students need to memorize basic facts.  My guess is that students in such classes store number facts in special locations in their brain with an index like “stuff I have to remember verbatim in order to pass”; I would like to see good research on this learning issue.  I want the number facts stored in a more complex way related to indices such as “factors”, “multiples”, “sums”, “differences”, “divisors”, and “properties of numbers”.

In my own classes, I require a calculator for all students.  This happens to be a department policy, though I would do the same thing if it were my choice only.  The issue is not ‘memorization’ — the issue is ‘understandings’ (as part of capacities).  Allowing the calculator implies that I need to observe students and provide feedback about the goals of a math course (understanding).  This is admittedly tricky, and I know that I do not provide enough feedback to enough students. 

A professional use of calculators is to focus on the contributions to learning.  The presence of the calculator provides learning opportunities that I value — such as understanding the difference between (-5)² and -5².  As you probably know, the confusion between these forms is common and problematic; I have students (this week, in fact) who have learned to state the correct words (memorized) but enter it incorrectly on the calculator.

Another example:  One of the most common relationships in the world (natural and societal) is repeated multiplying.  These exponential relationships require sophisticated methods to solve symbolically.  However, a numeric and graphical exploration is within reach — IF we use a good calculator.  Exponential relationships, in fact, are behind many of the general education goals in colleges (science, economics, and politics as examples).  Without a calculator, we are saying that a student needs to complete the advanced symbolic work of a strong pre-calculus course in order to be generally educated.  This is exactly the approach of many universities, including a large institution located a few miles from my college.  Pre-calculus is not general education; it is STEM education, and using that course for general education is part of the larger problem in college mathematics.

One final thought on learning opportunities with calculators — with calculators, we can present reasonable approximations for ‘real world problems’.  The world is messy; few calculations out there deal with integers only, and many involve very large numbers … or very small numbers.  It might not actually help students transfer what they are learning, but it feels better in class.

Can calculators be a problem in a math class?  Obviously yes — depending on many factors.  NOT using calculators is also a problem; knowing how to use technology is an employment skill, and also can support learning mathematics.  Not using calculators puts mathematics in a make-believe world that has no connection to a student’s life; after all, almost all students have cell phones that they use as a calculator … some have a smart phone with a ‘math app’.  We might argue that a spreadsheet is a better mathematical tool than a calculator; as a learning tool, a spreadsheet has a learning curve and some limitations that make it more difficult.

Calculators, then, are both a good thing (resource) and a bad thing (problem).  The important decision is not ‘calculator’; rather, the important decision is ‘learning as building capacity’.

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