Algebraic Literacy: Finding a Textbook

One of the new courses coming to a college near you is “Algebraic Literacy”, a modern course that prepares students for a STEM path (and related work).  This position in the traditional curriculum is held by ‘intermediate algebra’.

For a brief comparison of these courses, see the chart below:

In this chart, “MLCS” refers to the Mathematical Literacy for College Students course (also known as Math Lit, and similar to Quantway I).

One of the issues with the Algebraic Literacy course is finding textbook materials.  Books being written for this course are not available yet.  However, there are materials available which have enough similarity to be used.

One book I have learned about recently is “Algebra: Form and Function” (Wiley publishing, 2010).  This book was written by a team connected with the calculus reform efforts, and is designated as a ‘college algebra’ textbook.  However, the book does not assume that students have the higher background; it’s quite accessible by students in an Algebraic Literacy course.  For a quick look, see this link to the Course Smart page:  http://instructors.coursesmart.com/9780471707080

You can also find more information on this text at the Wiley page http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000346.html

 
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Mathematical Literacy: Factoring in the Mathematics Curriculum

Some people will be surprised that our Mathematical Literacy course includes some factoring.  Over the years, the topic of factoring has been a focal point of conversations — almost with the assumption that a reform math course would not cover any factoring.  Sometimes, we go to the extreme view of “anything not practical right now … will be omitted”, and factoring is usually not very practical.

In our Mathematical Literacy course we covered factoring  last week — true, this is just the GCF (no trinomial methods nor special formulas).  Since we only include GCF as a method students have an easier time.  However, if we had time, I would not mind if we covered a little more factoring.

For language skills, it is important that people be able to express thoughts concisely (simplify); in some important situations, it is even more important to be able to express thoughts in a more complex way that maintains the equivalent message — persuasive writing and speaking are particular modes in this style.  In a general way, learning (or a process) that can only be used one direction is usually learned only partially.  Deeper learning depends upon a variety of experiences with objects or ideas.

Factoring plays a comparable role in any course emphasizing algebraic reasoning.  A basic issue in algebraic reasoning is “Adding or multiplying?”  Many of our students believe that parentheses always show two things — what to do first (under the curse of PEMDAS) and “this is a product”.  Our work with the GCF puts students right in the middle of this confusion; in other words, the GCF is a great opportunity for students to better understand basic algebraic notation.

Of course, one risk of this work with the GCF is that students get even more confused.  We need to be careful that assessments help students understand better; within the Math Lit class, I need more experience designing the class work so better assessments can be delivered to students.

Of the traditional developmental algebra content, factoring is not my lowest priority — it connects with basic issues of algebra.  I can’t say the same thing for radical expressions, where we deal with procedures only vaguely connected with exponents.  I also place ‘rational expressions’ lower in priority than factoring; outside of the very basic ideas of reducing simple rational expressions, our time on operations and equations with rational expressions list mostly wasted … the emphasis ends up on procedures, not concepts and understanding. Such topics have been included in developmental courses because they are seen as needed in pre-calculus courses … because they are seen as needed in calculus courses.  We should strengthen this flimsy curriculum design based on student needs AND content needs in deliberate ways.

All of us have a role in this process so that mathematics becomes an enabling process rather than a inhibiting process.  Factoring polynomials is not necessarily an evil to be avoided.

 
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New Life and Reform at AMATYC 2013

The AMATYC conference (Anaheim, October 31 to November 3) will include several sessions on New Life-inspired efforts to reform developmental mathematics.  I will be doing a general session on the New Life model at the conference, and other sessions will focus on particular implementations.  Over the next month or so, I will be posting a detailed schedule.

This conference will not include a workshop on the New Life courses; this workshop was done at last year’s conference and the materials are still available at http://dm-live.wikispaces.com/workshop2012  If you want to know more about the details of MLCS (Mathematical Literacy for College Students) and AL (Algebraic Literacy), I plan to create some additional 5-minute presentations about each — they will be posted on the “Instant Presentations” page (https://www.devmathrevival.net/?page_id=116)

Some related work will also be available at the conference — the Dana Center “New Mathways Project” will have sessions.  When the mini-program is available, I will post a summary of all reform-related sessions.

 
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Mathematical Literacy: Making Sense of Negative Exponents

One of the unique characteristics of our math literacy class is the organization of content … in a given week, we usually have a blend of numeric and symbolic work mixing in some algebra within a set of concepts that have been building during the semester.

The last class had two primary tasks: Integer exponents in problems, and equations of lines.  In a typical algebra course, integer exponents are ‘covered’ as part of a concentrated sequence; in the Math Lit course, we have already been using basic exponent properties for several weeks (products, quotients, and simple powers).  The idea is to have the work make sense to students, as much as possible.  We had, in fact, done work with scientific notation including small numbers — before we dealt with negative exponents in general.

Since this was our first time doing negative exponents as a general idea, we started with a very basic problem:

Simplify x^2/x^4

We wrote out the powers of x and reduced; then we subtracted exponents.  Very typical stuff.  The difference was the next step:

The two answers are both correct (1/x^2 and x^-2).  What do you think negative exponents mean?

The first student response was ‘do the opposite’, meaning divide.  The second response was ‘turn it over’ meaning reciprocal.  Nonverbal clues indicated that most students understood one of these meanings, and several got both of them.  Only after this interaction did I say anything about what it meant — as we dealt with the idea of writing an expression with positive exponents.

It’s not that this was a magical moment.  Several students in class could not apply what they had just said, and some had a very incomplete understanding of exponents and coefficients.  I’m still looking for the magic path for this ‘sum versus product’ understanding.  However, less of the struggle was about negative exponents than I usually see.

Some progress is encouraging, and evidence of an idea making sense looks like progress.

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