Even Our Puzzles Are Outdated … Mathematics for 2025 (and today)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
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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Acceleration, Pathways, and the Forum on the First 2 Years of College Mathematics

I will be posting more about the actual “CBMS Forum” (Conference Board of Mathematical Sciences) held earlier this month in Reston, Virginia; several of the talks made relevant points about our work.

One of the breakout sessions was a one-of-a-kind: A single session covering all three models for pathways and acceleration (Carnegie Pathways, Dana Center New Mathways, and AMATYC New Life).  You can view the slides for that session here CBMS Pathways to Success Oct 2014 or at the “Instant Presentations” page (http://www.devmathrevival.net/?page_id=116).   The three of us (Bernadine Fong of Carnegie, Uri Treisman of the Dana Center, and myself) were impressed by the standing-room only crowds at both of our sessions.

Much of the motivation for faculty and colleges falls under the heading of ‘acceleration’, which is fine.  However, my own view … and much of what I heard at the Forum … dealt with the nature of the mathematics courses we offer (developmental and ‘college’ mathematics).  Issues surrounding the curriculum will be the focus of my comments-to-come in response to “Forum 5″.

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