Intermediate Algebra … the Bridge to Nowhere

Yes, I am using an emotional label being used about developmental education … yes, I am saying intermediate algebra might be such a thing.  A bit of a cheap trick, but I hope that you will continue reading anyway!

The content of our intermediate algebra courses is usually based on topics that were once covered in a second year high school algebra course.  That course, in turn, was created by companies and teams of authors (often a combination of university mathematics educators and high school math teachers).  I have not seen any documents relating to how the companies and authors determined the content; I suspect that much was based on a view “well, this topic would be good for them”.

All of this work occurred long before a general emphasis was placed on understanding, application, and cognitive science.  Procedural accuracy is the hallmark of our intermediate algebra courses — even more so than the high school algebra II course; it’s like we copied the content but limited our work to the lowest levels of learning.

We actually have some helpful stuff in there, if students can remember it later when (and if) they take more advanced courses (whether a pre-calculus/analysis course or in calculus).  The better students may do this; most do not, because the material is not usually taught in a way to create long term use.

So, here is an initial list of reasons why intermediate algebra is the biggest ‘bridge to nowhere’:

• content created over 50 years ago outside of our curricular process
• textbooks focus on procedural accuracy
• learning heavily weighted towards lowest levels of learning

Students who pass an intermediate algebra course meet the prerequisite for some college math courses; however, the intermediate algebra course did not prepare students for that course.  Nor does the intermediate algebra course contribute to mathematical understanding, nor to positive attitudes about mathematics.

Fortunately, we have a model for replacing intermediate algebra — the Algebraic Literacy course from the New Life model.  The outcomes for this course were extracted from what students need in subsequent courses, and these outcomes include both procedural and understanding emphases.   In addition, the Algebraic Literacy course includes the use of mathematics to understand the quantitative components of issues in the world — such as the spread of infectious disease.

The Dana Center work on a Stem Path is also involved in creating a replacement for intermediate algebra.  Those teams are approaching the problem from a similar viewpoint, so I expect their results to be compatible with the New Life Algebraic Literacy course even if their content has some significant differences.

To learn more about the Algebraic Literacy course, I encourage you to come to my session next week at the AMATYC Conference (Nashville); this session is at 8am on Friday (November 14).  [I am also doing a general session on the New Life model that Saturday (November 15) at 2:15pm; this session will include basic information about Algebraic Literacy.]

If you are not able to be at the AMATYC conference, take a look at the Instant Presentations page on this blog https://www.devmathrevival.net/?page_id=116 .  After the conference, I will be posted the materials from the session on that page.

Of course, if you have any questions about the Algebraic Literacy course, just contact me!

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Even Our Puzzles Are Outdated … Mathematics for 2025 (and today)

Earlier this month, the Conference Board of Mathematical Sciences (CBMS) held a forum on mathematics in the first two years; many of the presentations are available on the web site (http://cbmsweb.org/Forum5/)

As part of one of the first plenary sessions, Eric Friedlander commented …  Students in the Biological Sciences now outnumber those in the Physical Sciences in the standard calculus 1 course.  (David Bressoud shared some specific data on those enrollment patterns.)

Historically, the developmental mathematics curriculum was all about getting students ready for pre-calculus.  Our “applications” tended to be puzzles created with physical sciences in mind — bridges, satellites, pendulums, and the like.   Few problems in our developmental courses draw the attention of those in biologically-oriented fields (including nursing).

We could include:

• Surge functions to model drug levels
• Functions to estimate the proportion of a population needed to be immunized to prevent epidemics (P_sub_c = 1 – R_sub_0)
• Models for spread of cancer … and for treatments
• Pollution prediction (simplified for closed systems)

This list is a ‘bad list’ because there is no common property (except being related to biology) … and because I do not know enough to provide a better list.  Take a look at books in applied calculus for the biological sciences; you will see applications that are perhaps better than those above.

There is a trend in the new models for developmental mathematics (AMATYC New Life, Dana Center New Mathways, and Carnegie Foundation Pathways) to include a balance of applications — including more from biology.  We need to bring in more of these applications throughout our curriculum (from the first developmental course up to calculus).

Most of us realize that the ‘applications’ in our courses and textbooks are puzzles created by somebody who knew the answer; generally, these problems do not represent the use of mathematics to solve problems and answer questions in the world around us.  Sometimes, we are not able to provide enough non-mathematical information to provide representative problems … in those cases, some reduction to the ‘puzzle state’ is acceptable.

Our puzzles should represent the diversity in the uses of mathematics, with a significant portion of applications being realistic in nature.

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Status of New Life Math Courses; AMATYC 2014 sessions

Here is a summary of “where” New Life courses are being taught currently:

Arizona California Colorado Florida Iowa Illinois
Kentucky Massachusetts Michigan North Carolina New York Texas
Wisconsin
Alaska Minnesota Ohio Oregon Utah

These 18 states involve over 50 colleges. Over 500 sections with enrollment over 10000 students are represented by those colleges.

Mathematical Literacy is the most common course being implemented; Algebraic Literacy is being taught at the same level that Math Lit was two years ago. I expect the Algebraic Literacy course implementations to follow the same trend as Math Lit; Algebraic Lit is about two years behind.

At the AMATYC 2014 conference next month (https://amatyc.site-ym.com/?page=2014ConfHome) I will be doing two sessions on the New Life courses.
On Friday (November 14, 8am) the session is The Missing Link: Algebraic Literacy to Replace Intermediate Algebra  .  I will describe the purposes for the Algebraic Literacy Course and provide details on the learning outcomes.  Included in the handouts will be a sample lesson representing what might be done in an Algebraic Literacy course.

On Saturday (November 15, 2:15pm) the session is Accelerate and Improve Developmental Mathematics: The New Life Model  .   I will provide an overview of the New Life Model and how it fits in to a curriculum to provide acceleration along with improved content.  Each course (Math Lit, Algebraic Lit) will be reviewed, and handouts will include the learning outcomes for each course.

I hope to see you there!

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The Common Core State Standards and College Readiness

At the recent Forum on mathematics in the first two years (college), we had several very good presentations — some of these very short.  Among that group was one by Bill McCallum, a primary author of the mathematics portion of the Common Core State Standards.  Bill focused his comments on 9 expectations for the high school standards intended to represent college and career ready.

The expectations listed are:

• Modeling with mathematics
• Statistics and probability
• Seeing algebra as based on a few coherent principles, not a
  multitude of unrelated techniques
• Building and interpreting functions to represent relationships between quantities
• Fluency
• Understanding
• Making sense of problems and persevering in solving them
• Attending to precision
• Constructing and critiquing arguments

Of these, Dr. McCallum suggested that fluency is the only one commonly represented in mathematics courses in the first two years.  The reaction of the audience suggested some agreement with this point of view.

So, here is our problem:  We included all 9 expectations when the Common Core standards were developed.  We generally support these expectations individually.  Yet, students can … in practice … do quite well if they arrive with a much smaller set of these capabilities.  Clearly, the Common Core math standards expect more than is needed.

What subset of the Common Core math expectations are ‘necessary and sufficient’ for college readiness?

For example, even though it is critical in the world around us, modeling does not qualify for my short list; neither does statistics and probability.

We are basically talking about the kinds of capabilities that placement tests should address  Measuring 9 expectations (all fairly vague constructs for measurement) is not reasonable; measuring 4, perhaps 5, might be.

I think we should develop a professional consensus around this question.  The answer will clearly help the K-12 schools focus on a critical core, and can guide the work of companies who develop our placement tests.

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