Common Core, Common Vision, and Math in the First Two Years

I’ve been thinking about these ideas anyway.  However, a recent comment on a blog post here got me ready to make a post about predicting the future of mathematics in the first two years.  I’d like to be optimistic … past experiences would cause considerable pessimism.   The truth likely lies between.

One of the “45 years of dev math” posts resulted in this comment from Eric:

If Back2Basics is what drifted up to CC Dev Math programs back then, what do you see the impact of CommonCore being on CC Dev Math now?

This post was about the early 1980s, when we had an opportunity to go beyond the grade level approach of the existing dev math courses (one course per grade, replicating content).  Instead of progress, we retrenched … resulting in courses which were subsets of outdated K-12 courses.  Much of the current criticism of dev math is based on these obsolete dev math courses.

We again have an opportunity to advance our curriculum.  This time, the opportunity exists for all mathematics in the first two years.

  • The K-12 math world is changing in response to the Common Core State Standards.  Even if politics takes away the assessments for that content, many states and districts have already implemented a curriculum in response to the Common Core.  (see http://www.corestandards.org/Math/)
  • The college math world is responding to the Common Vision (see http://www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf) which is beginning the process of articulating a set of standards for curriculum and instruction in the first two years.  AMATYC is developing a document providing guidance to faculty & colleges on implementing these standards.  [I’m on the writing team for the AMATYC document.]

The two sets of forces share quite a bit in terms of the nature of the standards.  For example, both K-12 and college standards call for significant increases in numeric methods (statistics and modeling) along with a more advanced framework for what it means to ‘learn mathematics’.

These consistent parallels in the two sets of forces would suggest that the future of college mathematics is bright, that we are on the verge of a new age of outstanding mathematics taught by skilled faculty resulting in the majority of students achieving their dreams.  This is the optimistic prediction mentioned at the start.

On the other hand, we have some prior experiences with basic change.  One example is the ‘lean and lively calculus’ movement (conference and publications in 1986 & 1989).  It is very sad that we had to modify ‘calculus’ with something suggesting ‘good’ (lean & lively) … the very nature of calculus deals with coping with change and determining solutions for problems over time.  As you know, this movement had very little long-term impact on the field (outside of some boutique programs) while the “Thomas Calculus” continues to be taught much like it has been for the past 50 years.

Here are some factors in why we find it so difficult to change college mathematics (the levels beyond developmental mathematics).

  1. Professional isolation:  membership in professional organizations is low among faculty teaching in the first two years.  The vast majority of us lead isolated professional lives with limited opportunities to interact with the professional standards.
  2. Adjunct faculty as worker bees: especially in community colleges, adjunct faculty teach a large portion of our classes … but are separated from the curriculum change processes.  The existing curriculum tends to be limited by these artificial asymptotes  created by our perceptions and the desire to save money by the institution.
  3. Autonomy and pride:  especially full-time faculty tend to place too high an emphasis on autonomy & academic freedom, with the false belief that there is something inherently ‘good’ about opposing all efforts to change the courses the person teaches.  Although most prevalent at universities, this ‘pride’ malady is also a serious infection at community colleges.

I’ve certainly missed some other factors.  These three factors represent strong and difficult to control forces within a complex system of higher education.  Thus, I consider the pessimistic view that ‘nothing will change, really’.

I think there is a force strong enough to overcome these forces restraining progress in our field.  You’d like to know the nature of this strong force?

The attraction of teaching ‘good mathematics’ is fundamental in the make up of mathematicians teaching in college.  If faculty can see a clear path to having more ‘good mathematics’, nothing will stop them from following this path.

If the Common Core, the Common Vision, and the AMATYC new standards can connect with this desire to teach ‘good mathematics’, we will achieve something closer to the optimistic prediction.  The New Life Project has experienced some of this type of inspiration of faculty.  Perhaps AMATYC will create a new project to bring that inspiration to a larger group of faculty teaching in the first two years.

One thing we know for certain about the future:  the future will look very much like the present and the past unless a group of people work together to create something better.  I would like to think that our profession is ready for this challenge.

Are you ready to become engaged with the process of creating a better future for college mathematics?

 
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The Big Missed Opportunity: Forty Five Years of Dev Math, Part III

This is part of a series of posts reflecting on our history in developmental mathematics … especially at community colleges in the USA.  We’ve talked about the ‘origins’, about a ‘golden age’ (or not), and now we move to the first half of the 1980s.

Two major movements were active at about the same time in the early 1980s … one dealt with placement policies, and the other dealt with the content of mathematics courses at this level.  When more than one movement is impacting a profession at the same time, there is always an opportunity for fundamental change.  That is not what happened in this case, and we continue to deal with the ‘incorrect’ responses to that opportunity.

The use of standardized assessments for placement was widespread (though with varied instruments) at the start of this period, as we moved from home-grown placement measures to assessments used at a larger scale (state, region, or nation).  Those tracking data quickly noticed that these measures, often used with mandatory placement, were impacting certain groups at a disproportionate rate.  In some cases, the items on the assessments had been tested for bias; even with tests using only these tested items, the results showed an uncomfortable level of differential impact.

Clearly, “something” had to be done.  A professional response might have been to develop an effective short term intervention that would equalize the results.  Another professional response might have been to establish collaborations between community college math faculty and the local K-12 school’s math program.  In general, neither of those responses occurred.  Instead, there was a decline in the rate of mandatory placement:

Students have the right to fail.  If they disagree with the placement measures, they can take the higher course.

I still hear this “right to fail” statement, which I see as a abrogation of our responsibilities:  We let students make a decision known to put them at unnecessary risk (we knew they were likely to fail).  Most colleges did not continue this ‘worst practice’ (as opposed to best practice), with the result that the placement system continued to have a differential impact on known groups of students.  That problem continues to the present day, as a general condition.  [Some colleges, systems, and states use either placement systems that moderates the impact (true multiple measures) OR have implemented new curricula which make the results more tolerable (pathways).]

For some history of placement policies, see https://ccrc.tc.columbia.edu/media/k2/attachments/college-placement-strategies-evolving-considerations-practices.pdf  .

The content movement impacting developmental mathematics in the early 1980s was a ‘trickle-up’ reaction to K-12 math reform in the prior decade or two.  The K-12 math reform is usually called “new math”, which failed because the curriculum was designed by university math education professors with little attention to the teachers who would try to deliver it.  Even though we can see the “DNA” from this New Math within the modern curricular standards of NCTM, AMATYC, and MAA, there was a back-lash in K-12 that drifted up to college … “BACK TO BASICS”.

There were very few college level books that implemented New Math designs; most were (and still are) very similar to the K-12 math predated New Math.  However, here was an opportunity for college math faculty to create developmental mathematics courses with balanced and effective approaches to multiple levels of learning — including reasoning and communication.  Our collective response was to regress even further on the levels we sought to deliver in our curriculum.  We reduced the amount of reading in our books, added examples, grouped the student practice by type, and generally made choices guaranteed to limit the student benefit for their efforts.

The two movements (right to fail, back to basics) involved forces that could have had that synergy necessary for significant long-term change.  We should have had one response to resolve both issues … change our curriculum in a basic way so that entering memory levels of particular skills do not determine success; rather, the entering level of understanding would determine success.

In my view,  the “New Life Project” represents this type of approach with developmental courses that are far less sensitive to remembered skills (Math Literacy, Algebraic Literacy), which means that they are far more accessible to all parts of our student population.  The fact that this solution appeared and gained support 30 years after the first opportunity indicates to me that our profession has been resistant to progress.  It’s not that dev math did not change between 1985 and 2010; it’s that all of the other changes did not address the core problems we face.  We needed other external forces acting upon our work before we were willing to try something different enough to possibly make real progress towards helping all students succeed.

We currently are in the ‘next big opportunity’ to make progress.  Let’s be sure to do things this time that will get us significantly closer to our goals.

 
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Math Literacy: Placement, Prereqs, and Access

In response to a recent post on placement tests, a colleague made this comment:

In my experience, many of the people who struggle with arithmetic really aren’t ready for Math Literacy. [S. Jones]

This colleague teaches at one of the premier colleges in the “Math Literacy Movement”, with experience and wisdom.  I think this is one of the most important issues we face in community colleges.

The intended curriculum in a Math Literacy course has very limited prerequisites.  Among these are basic number sense (place value and order), and some understanding of basic operations within contextual situations.

If a student struggles with these items, yes … they are likely to be ‘not ready’ for Math Literacy:

  • Add 24.1 + 1.3     [know place value ‘across’ the decimal]
  • Which of these is smallest?   0.23, 0.201, or 0.1065?  [order of numbers]
  • A set of bleachers has 6 rows, and 10 people can sit in each row.  How many people can sit on the bleachers?  [operations in context]
  • A recipe calls for 24 ounces of diced dried fruit.  The packages I’m buying contains 3.5 ounces; how many packages will I need for that recipe?  [operations in context]

On the other hand, struggles with these items are far less related to readiness for Math Literacy:

  • Add 3/4 + 5/6 and write as a mixed number if necessary
  • Which of these is smallest:   3/13, 5/14, or 2/7?
  • Divide (without a calculator):  19.3 รท 2.56
  • If the area of a rectangle is 56 square feet, find the width if the length is 6 feet.

To understand what the prerequisites are for a course like Math Literacy, we need to think about the end point of that course.  The goal of Math Literacy is to build readiness for the next math course (quantitative reasoning, statistics, or Algebraic Literacy).  This goal drives the content of Math Literacy, which is outlined in four areas in the document mlcs-goals-and-outcomes-oct2013-cross-referenced

This goal, as operationalized in the content, seeks to have students meet the necessary and sufficient conditions for readiness in the next math course … any of those next math courses.  None of these courses are arithmetic in nature, though all of them depend upon numeracy skills to some extent.

The problem with our conceptualization of arithmetic in a college setting is that we attribute “here is what we would like students to know” to that content.  Of course, we’d like people to be able to perform fraction operations and decimal operations without depending upon a calculator.  Of course we would like students to know some basic geometric relationships.  In fact, most implementations of Math Literacy will assist students in those areas, but not as a core goal of the course nor as a prerequisite.

The truth is … that arithmetic was never a prerequisite for algebra, based on content structure.  Sure, some parts of arithmetic had a role.  In fact, we might call those parts ‘numeracy’ just like we do in the conversations about Math Literacy.  However, competency in arithmetic procedures is (and has been) unrelated to readiness for a subsequent math course.

Too often, we create artificial barriers to students reaching their goals.  One of the largest barriers in a college environment is the “arithmetic placement test”.  We have a situation where:

  1. A content analysis does not support the treatment of arithmetic as a prerequisite to a math course.   AND
  2. No data exists to suggest that there is any practical connection between competence in general arithmetic and readiness for a math course.

My college is currently using an arithmetic placement test merely for the purpose of sorting students relative to our two Math Literacy courses … the Math Literacy with Review course has a lower cutoff than the Math Literacy without review.  We no longer offer any math course ‘before’ Math Literacy.  Eventually, we might be able to make the determination about which Math Literacy course from other measures.

Think about this aspect of the situation: Most of the students who might take an arithmetic test have experienced 12 years of mathematics with over half of this time focused on arithmetic.  Do we expect to ‘fix’ most of their problems in arithmetic within a few months?  Also, what do the students look like who get lower scores on placement tests (especially arithmetic)?  The polite phrase is “this group is very diverse”.  The fact is that tests on arithmetic impact certain minority groups (race, poverty) more than others.  Unless we can show a very strong connection between ‘arithmetic’ and success (in a specific math course, or in general), we have a moral obligation to NOT impose an arithmetic barrier.

Using an arithmetic placement test to identify students required to take an additional math course is a fundamental access issue.  Such courses are obsolete relics of a different era, and lack connections to both school mathematics in this century and to other math courses in colleges.  We can help thousands of students by following one simple plan.

Just stop it!!

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The Placement Test Disaster ?

For an internal project at my institution, I’ve been looking at the relationships between Accuplacer test scores, ACT Math scores, and performance in both developmental and college-level courses.  Most of the results are intended for my colleagues here at LCC.  However, some patterns in those relationships are important for us to explore together.

So, the first pattern that is troubling is this:

Students who place into a pre-calculus course based on their ACT Math score have lower outcomes than those who place based on the Accuplacer “College Level Math” test … and lower than those who needed to take intermediate algebra before pre-calculus.

We use the ‘college readiness’ standard on the ACT Math test of 22 (see https://www.act.org/content/act/en/education-and-career-planning/college-and-career-readiness-standards/benchmarks.html ).  The pattern in our data for the ACT Math is similar to some references found at other institutions … though we tend not to talk about this.

Of course, the use of an admissions test (ACT or SAT) for course placement is “off label” — the admissions tests were not designed for this purpose.  We tend to use the ACT option for placement in response to political pressure from administrators (internally) and from stakeholders (externally), and sometimes under the guise of “multiple measures”.  The patterns in our data suggest that the ACT Math score is only valid for placement when used in a true multiple measures system … where two or more data sources are combined to create a placement.  However, most of us operate under ‘alternative measures’, where there are different options … and a student can select the highest outcome if they wish; alternative measures is guaranteed to maximize the error rate in placement, with a single measure placement test almost always providing better results.

The second pattern reflecting areas of concern:

The correlations are low between (A) the ACT Math and Accuplacer College Level Math test, and (B) the Accuplacer Algebra and Accuplacer Arithmetic tests.

The second combination is understandable, in itself; the content of the Algebra and Arithmetic tests have low levels of overlap.  The problem deals with our mythology around a sequence of math courses … that the prerequisite to algebra is ‘passing’ basic math.  Decades of our research on algebra success provide strong evidence that there is little connection between measures of arithmetic mastery and passing a first algebra course.  In spite of this, we continue to test students on arithmetic when there curricular needs are algebraic:  that is a disaster, and a tragedy.

The first ‘low correlation’ (ACT Math, College Level Math) is not what we would expect.  The content domains for the two tests have considerable overlap, and both tests measure aspects of ‘college readiness’.  As an interesting ‘tidbit’, we find that a higher proportion of minorities (African American in particular) place into pre-calculus based on the more reliable College Level Math test compared to majority (white, who have a higher proportion placed based on the ACT Math) — creating a bit of a role reversal (whites placed at a disadvantage).

Placement testing can add considerable value … and placement testing can create extreme problems.  For example, students with an average high school background will frequently earn a ‘college ready’ ACT Math score when they have too many gaps in their preparation for pre-calculus.  A larger problem (in terms of number of students) comes from the group of students a bit ‘below average’ … who tend to do okay on a basic algebra test but not-so-good on arithmetic, which results in thousands of students taking an arithmetic-based course when they could have succeeded in a first algebra course (or Math Literacy).

Those two problems are symptoms of a non-multiple-measures use of multiple measures, where alternative measures allow students to select the ‘maximum placement’ while other measures (with higher reliability) suggest a placement better matched for a success situation.

As a profession, we are under considerable pressure to avoid the use of placement tests.  Policy makers have been attacking remediation for several years now, and more reasonable advocates suggest using other measures.  The professional response is to insist on the best outcomes for students — which is true multiple measures; if that is not viable, a single-measure placement test is better than either a college-admission test or a global measure of high school (like HS GPA).

And, all of us should deal with this challenge:

Why would we require any college student to take a placement test on Arithmetic, when their college program does not specifically require proficiency in the content of that type of test?

At my institution, I don’t think that there are any programs (degrees or certificates) that require basic arithmetic.  We used to have several … back in 1980!  Technology in the workplace has shifted the quantitative needs, while our curriculum and placement have tended to remain fixated on an obsolete view.

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