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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Towards Success in Developmental Mathematics — A Toolkit

Have you been looking for practices that encourage and support student success, especially in developmental mathematics?  I will share a toolkit of practices that seem to be effective with students in beginning algebra (often the most challenging course for us and for students).

Communication is the key.  To start with, write your syllabus (first day handout) with the reader in mind.  My own syllabus is conversational, and full of textboxes and a few graphic elements.  A traditional syllabus discourages reading; I think they also discourage engagement.  We encourage engagement by the design and tone of our syllabus.

It’s all about the learning.  In the syllabus and your attitude, emphasize that the top priority is to learn to the  best of each student’s abilities.  Homework is not just about practice … homework is a learning endeavor.  In my case, I emphasize the ‘learning cycles’ (see https://www.devmathrevival.net/?p=1229) AND reinforce these ideas by comments and actions every day in class.  repetition and action count; saying ‘it’ once does not matter.

Provide a reward for seeking help.  This is really important as a step to change behavior.  Most students, especially those in developmental mathematics, are reluctant to seek help.  At the same time, help is what makes the difference between passing and failing.  My method for this is to give an assignment (8 to 10 points, out of 1000 for the course) for students who seek help within the first two weeks of the semester.  Not only do students get more help, they feel more connected to the college experience.

Dig deep and build; don’t assume ‘they get it’.  Many of my students could combine like terms … as long as the sum was not zero; they could use exponents … as long as the problems were limited.  This is my most recent change; one of my students this semester wrote me an email (in the first week) that this was the first time she understood algebra.  Even some of the ‘high-performing’ students found some gaps.  Specifically in beginning algebra, I am using language concepts (see https://www.devmathrevival.net/?p=1253) and building processes in great detail: we started from “x + x + x = 3x” and “x·x=x²” … we went through zeros in sums [2x + (-2x) – 8 = -8]. 

Assessment as a routine activity, with instructor feedback.  If a student can go 2 or 3 weeks before getting feedback from me, I am assuming that they are ready for college work (and can make their own judgments about learning).  Everyday we have a quiz or a worksheet; I’ve even run a class where we do both (all classes are 2 hours, twice a week).  Obviously, the more assessment activity the more work we have.  My assessments fit in to the “It’s all about the learning” concept; daily assessments are 5 or 6 points, and I ‘drop’ 3 or 4 over the semester so random absences don’t hurt students.

We are a community of learners in this class.  You might call this ‘group work’, or ‘learning together’.  However, it’s not good enough to have 2 people in the class that help a student … we can all help each other.  If I can create an environment where each student is comfortable asking almost anybody in the class … and where every student is willing to help others, this is a powerful tool for motivation and ‘connecting’.  In my case, I model this behavior during class, and build opportunities for students to work with different people; seldom do I arrange the groups.  (Sometimes I will direct them to work with somebody they have not yet worked with.)

My goals for these practices focus on student engagement and learning capabilities; if the practices ‘work’, I will see better learning this semester and the student will be better prepared for other courses — even if they never use the mathematics we study.   Obviously, these practices are just a part of what I do … I hope you find some ideas within them.

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Math – Applications for Living IV

Percents!  Percent discount!  Percent tax!  Percent increase!  What do our students learn about percents?  Not nearly as much as we would like.

One of my students made a comment about percents and taking his family to a restaurant.  As a result, this problem appeared on a worksheet last week in our Math119 class:

I have $20 to go out to dinner.  All restaurant purchases in my state have a 6% sales tax, and I like to leave a tip that is 15% of the total including sales tax.  How much can I spend on food (menu prices) to stay within my $20?

Every one of my students has shown that they can calculate a 6% tax, and find the total price.  Every one of them can find the 15% tip on a dinner, and the total cost.  Less than 10% of them knew what the ‘base’ for the percent is in the problem.  We had already been talking about the net result of a 10% increase (110%, or 1.10 times the base) as well as a 20% decrease (80%, or 0.80 times the base).  In fact, we started off our work with percents by the classic story:

Boss: Bad times; sorry, everybody gets a 10% pay cut.
Boss (next year): Good times are back; everybody gets a 10% pay raise.

Worker: Am I back to where I was?

Many percent situations in the world involve a chain of percent increase or decrease factors operating on a moving base.  In my dinner example, the goal is to see the situation as

1.15(1.06n)=20

The solution here ($16.41) is pretty good, as the rounding happens to work out well; in general, this method is a ‘good approximation’ — an idea that is not brought up in this class.  We are still going through a lot of struggle to identify the base in percent problems.  Later in the semester, we will connect this repeated percent concept to exponential functions; identifying the base correctly will continue to be an issue then.

Whatever you do with percents in your courses … please focus on identifying the base. Being able to calculate a tax or a decrease is nice, but of limited usage.  Percent change is all around us, and we often deal with an unknown base (frequently hidden within the context of the problem).  We don’t need to disguise problems to the point that finding the base is a horrendous exercise for students.  On the other hand, creating cookie-cutter exercises where no thought is needed is a self-defeating practice.  “Percents” and “thinking” go together in the world around us, as they should in our math classes.

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It’s Time for Algebra Class … Do You Know Where Your Linguist Is?

We’ve heard … and many of us say … “math is a language” or “algebra is like a foreign language”.  In our classrooms, these statements are often intended to motivate students to pay attention to vocabulary and syntax.  In general, I think the net result is neutral or even negative.  [Students are told to attend to something that they do not understand, and also lack a structure for learning.]

Twenty-five years ago, the Center for Applied Linguistics (http://www.cal.org/) published a pair of books on “English Skills for Algebra”, authored by Joann Crandall et al (Crandall, Dale, Rhodes, and Spanos).  One book was a student workbook … the other a tutor guide; the goals were:

“provide practice in manipulating the specialized language of mathematics and algebra through listening, speaking, reading, and writing activities in English; and

“provide practice in using language as a vehicle through which they can think about and discuss the processes used to perform basic operations in beginning algebra.”

I notice that the authors (linguists) include four modes of langauge usage (active — speaking & writing; passive — listening & reading).  I suspect that this is obvious to linguists … but not to mathematicians … that fluency depends on prolonged and deliberate efforts in all four modes.  Our math classes tend to focus on the passive modes; we consider ourselves progressive if we include talking in small groups. 

You probably will have difficulty finding the books mentioned.  I am adopting some of the content for my beginning algebra classes, and can provide a sample of one activity.  This is a worksheet, delivered through our course management system, with the purpose being to understand both one correct meaning for an algebraic statement AND to identify a correct paraphrasing.  Here is an image (you may need to right click on it, and open separately so you can enlarge it):

This activity has 7 questions, and I have a series of 3 for students to use. 

As you can see, this still works on the passive modes.  For active modes, here are two things I do in class:

Speaking — I ‘cold-call’ on students to have them explain how to do problems (they have had a couple of minutes to work on the problem, which is related to an example I have worked with verbal explanations).  I am able to encourage correct spoken language, as well as identify gaps in language or understanding.

Writing — I use “no-talk quizzes’, where students review other student’s work and provide feedback in writing phrases or sentences.  The focus is on explanations; feedback must be verbal (can not be symbolic).” (pg iv)

I encourage you to think more deeply about ‘algebra is a language’.  If you are fortunate enough to have a linguist nearby (which I was for a few years), talk to them; you might need to draw an analogy to learning a foreign language.  [Most linguists actually have some background in applied mathematics, but not so much in learning issues in mathematics.]  My own work in this regard is unfinished … I am most concerned about getting a process for spoken algebra with feedback, and I want to add more writing with feedback.

Writing across the curriculum is wonderful; however, the language within mathematics is more fundamental to our work.  If we conceptualize algebra as a language, we should have a deliberate plan for developing the fluency of our students in all modes of usage.  Just saying “it is a language” is a bit like saying “don’t you understand this yet?”.  The langauge learning process is not just a matter of a label like that, or motivation; language learning has its own processes.

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