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 (https://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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Comparing Dana Center Mathways and AMATYC New Life

I have been at the MichMATYC conference where we spent quite a bit of time talking about how similar the three models are — AMATYC New Life, Dana Center New Mathways, and Carnegie Pathways.  Tomorrow, I go to the CBMS Forum on math in the first two years … where I will be doing a presentation involving all three models.

As part of that work, we have an updated chart showing how similar the 3 models are (and highlighting the differences).  Take a look:

Summary of Three Emerging Models for Developmental Mathematics Updated 2014

Another thing I am currently doing is updating the summary of New Life course implementations (Mathematical Literacy or Algebraic Literacy courses).  I’ve got some more to include; a preliminary estimate is that there are currently about 500 sections of these courses offered.  Some are ‘pilots’, but most are regular sections as part of the curriculum.

 
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Basic Math or Pre Algebra or Nothing

What do students need before a ‘beginning algebra’ course?  Several of us (math faculty at my college) are working on this problem, with a goal of helping more students make a good transition to algebra while being aware of other expectations or demands.

My college does not have a basic math class, having eliminated that quite a few years ago.  There is still a prerequisite for the pre-algebra course (a placement test) though the cutoff is not very high, which means that one of the issues is students with extensive gaps in numeracy.  Our pre-algebra course has these components:

• variables and expressions used from the first chapter
• signed numbers start next
• solving first degree equations (some with simplifying first)
• fractions
• geometry (formulas primarily)
• units and conversions (the only math course doing this, for most students)
• percents and applications (tends to be uncomplicated)

One of the issues I see us dealing with is our own views on “what students should know”.  In our course, we designate the first part ‘calculator free’ because students “should know” their basic facts about numbers; the remainder of the course allows a calculator.  We also expect students to use arithmetic procedures for fractions, though we do not check to see if they understand ‘why’.  We cover classic percent problems, because students “should know” these.

So, what essentials are needed to help students succeed in basic algebra?  In some ways, the answer has been “do some basic algebra”; the last course revision integrated algebra throughout.  We’ve looked at the data for the progression, and it is my opinion that the alumna of the newer course have similar struggles in basic algebra compared to the older course (with less algebra).  One observation is that the students struggle with the expressions and first degree equations that they ‘had’ in the pre-algebra course, whether the algebra was integrated or covered separately.

Here is the basic need I would identify for success in basic algebra:

Students need a core of understanding about numbers and properties, and need a sound beginning on procedural flexibility.

The traditional percent material focuses on correct answers, often using memorized procedures.  I would shift to questions about equivalence and multiple solution methods … because these are core issues in algebra.  My class work and assessments would focus on creating as well as identifying alternate correct methods.  The traditional geometry work in this course also tends to have a focus on correct answers (though we do not memorize formulas).  I would instead deal with how parts of shapes relate to the whole, and concepts of perimeter/area/volume; the same focus on multiple solutions would be appropriate.

The numerical demands of a basic algebra course are quite limited; we are not going to solve a lifetime of numeracy problems in 15 weeks of a basic math course.  A pre-algebra course gains little by making the attempt.   A reasonable goal is to develop a significant set of understandings about numbers and objects, along with the flexibility that this understanding supports.  Deliberate design, sophisticated pedagogy, and faculty expertise are required for this … just as is the case for most math courses that we should place in front of our students.

One of my colleagues used to say:

The student’s fragile understanding of mathematics begins in the pre-algebra course.

We need to shift our focus.  Without understanding, any math course becomes just a barrier to student success.  Without understanding, math is that subject that everybody says they are bad at.  With a focus on understanding, we offer an honest math course that can provide real benefits for students. With a focus on understanding, we demonstrate our commitment and respect towards all students … starting from the first day of our first math course.

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