Emporium Model … A Good 3-Year Solution to the Wrong Problem

Many of us are involved with emporium-type projects; these changes are often in the popular media … such as the article about college algebra at the University of Texas – Arlington (http://www.nytimes.com/2012/09/07/us/ut-arlington-adopts-new-way-to-tackle-algebra.html).  I refer to emporium-type projects as ‘3-year solutions’ because they do not address basic problems … the main focus is a symptom of a problem (low pass rates) and a generalized threat to higher education (costs). 

A particular project might in the emporium style might last longer than 3 years, but mostly due to the investment the institution has made in space and infrastructure.  And, most emporium-type projects will produce better data (the symptom) especially in the first two years … the motivational effects of ‘something new’ can help faculty and students, at least for a while.  Over time, the data will tend to degrade from the early improvements.  The cost savings will be difficult to maintain in the long term; since the primary savings comes from limiting of faculty time, administrators will face pressures to increase faculty assistance for students.   One of the forces that supports ‘survival’ of the project (investment in space, especially) will eventually turn out to be a challenge for survival, as other demands develop in the institution.

Like ‘module solutions’, emporium-type solutions tend to avoid curricular problems.  Procedural techniques in the absence of understanding mathematical concepts, with a pronounced lack of applications to useful situations (in academia and in life), within a context of “you have to pass this math course in order to do what you want” … these are some of the basic curricular problems we face.  In theory, an emporium-type model COULD address these issues; however, doing so is likely to be more difficult (perhaps much more) than in other designs which place faculty in a more active role as facilitator of learning.

Part of these difficulties are a result of using technology as a foundational component in the learning process.  Most of the technology used is mass-market software focusing on the ‘greatest common factor’ (often mis-spoken as ‘least common denominator’) of math faculty — these technology solutions deal primarily with problem types that most faculty can agree to … in other words, procedural techniques and routine applications that can be done repeatedly without understanding.  Addressing the curricular problems with technology would involve large investments of resources in development, which no single institution can afford … and publishers are reluctant to provide it without the ‘market’.

Let’s put it this way:  The curricular problems must be addressed first, before we can identify or build appropriate instructional systems.  The problem solving of solving the curricular issues is the strength of faculty, which means that the solutions will tend to be very faculty and classroom-based for quite a while.   Eventually, we might be able to use a model like the emporium in a long-term solution; right now, this is just not possible.  Technology follows curricular change … curricular change CAN be inspired by technology, but this is unusual.  (As an example, the curricular changes due to graphing calculators turned out to be less substantial than many thought or some feared — even though there have been changes, the long-term effect was mediated by the curricular problems.)

If you are involved with an emporium-type project, I would say that you should enjoy it as much as possible … and do not count on the project to last past a few years.  In the meantime, become familiar with the emerging models for developmental mathematics (AMATYC New Life, Dana Center Mathways, Carnegie Pathways) and the reform work in college mathematics (MAA CRAFTY, AMATYC Right Stuff).  You will need to be prepared for the day when your institution decides that the emporium-type project is not good enough.  Perhaps you will even be involved in convincing your institution that there is a better path forward, a path focusing on solutions that address basic problems. 

Emporium does not mean forever.

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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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The Calculator Issue

I was talking with an editor from one of the larger publishing companies earlier this week, and one of the issues the editor saw as critical was “the calculator issue”, which this editor saw as both basic and divisive in the profession.

I would like to start by asking “When do YOU reach for your calculator?”

For my own work, and perhaps yours, I use calculating devices for several categories of work.  First, if the quantities are ‘complex’ and a precise answer is needed.  Second, if a procedure will need to be repeated more than twice (like finding a table of values for a function).  Third, if the calculation deals with a high-stakes question (like grades for my students).  Fourth, if the situation involves the exploration of ideas which are still in the ‘learning process’ (like a new mathematical concept or a review of long-lost treasures).  There might be a few other situations.

You might wonder why I start with our use of calculators.  So often, the comments we make are about our students’ “over-use” or “dependency” on calculators; we see calculator use as creating a risk for learning mathematics.  Many of us do allow calculators, and even embed their use in the learning process.  Some of us forbid their use, and some of us have a blended approach.  Most of us, however, believe that there is an issue with calculator dependency.

My conclusion is that the problem is not with calculators being used.  The problems occur when student attitudes about mathematics and their own efficacy create motivation to use calculators when the human brain is a better device.  If a student is doing a problem in the homework, or an example in class, and reaches for their calculator to add two one-digit numbers, this is part of the problem — the human brain is a better device, and the use of the calculator in this situation provides a clear ‘bad at math’ message about the student (and the student is the one sending the message).

Of course, we can not ignore the impact of culture on the use of tools, even calculators.  Some students temporarily feel ‘smarter’ when they use a technological tool frequently; I suppose this is not so much ‘smarter’ as ‘good’.  In the case of mathematics, the cultural bias towards technology combines with a norm that “it is okay to be bad at math” to encourage over-use of calculators.  These comments about culture are not universal for our students, some of whom come from cultures where very little technology is available … some even come from cultures with a positive attitude towards mathematics.

If this analysis is correct, then the issue is not whether we allow calculators or not.  The first basic issue is our outlook on learning — maximizing understanding, connections, and abilities to work with quantities … these are traits of the emerging models of developmental mathematics.  Students should develop their strategies for good uses of calculating technology, and they can do this as long as we do not focus so obsessively on ‘correct answers’; if we assess representation and communication, the use of calculators will not create problems.  As you probably know, technology does not really “solve” problems either; problems are solved by people and what we do.

I encourage you to ponder your approach to calculators in your classes.  Are you creating a calculator policy to make a personal statement, or are you creating a calculator policy which reflects your understanding of how technology affects learning of important mathematics?  Does your calculator policy encourage, or does it discourage, the development of mathematical proficiency in your students?

These issues are complex, and will not be solved by a simple yes/no policy on calculators.

 
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