Pathways and New Life presentation

If you would like a ‘quick’ summary and comparison of the Pathways (Statway™ and Quantway™) and New Life model, take a look at this presentation.

  Pathways and New Life session MDEC 2012 final

There is also a ‘handout’ — references for the models, and the current visual for the New Life model.   Here is that handout: References_EmergingModels_March2012

 
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A New Vision of Mathematics Pathways

The New Life project has been active now for 3 years; faculty have been supportive of the model, and even inspired.  With the help of many people, the ‘word is out’.

The basic model has remained as that developed in 2009 by a variety of experts and practitioners.  However, based on questions people have asked, I have updated the visual image that describes the  model.  Here it the updated chart:

I realize that this chart may be ‘hard to read’ in a browser, so here is a file to download (pdf format):  New Math Pathways General Vision 4 2 12

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Reducing Costs of Developmental Math

The ‘cost’ of developmental mathematics is one of the major issues being faced by states and institutions.  Although this is commonly stated as a financial cost, an equally important cost is present — the cost to our students (time, credits).  There is also a risk involved, given that most studies of developmental education seem to report that students placed into developmental courses have a lower chance of completing programs.

Is there a solution?  Is there a simple solution?

In a recent post (https://www.devmathrevival.net/?p=756) I talked about what a reasonable prerequisite to beginning algebra could be.  That post hinted at some solutions which could be implemented to reduce costs.

Here is a simpler solution that can be done right away, and may not have the kinds of problems you might predict:  Place students into beginning algebra, even if their placement test suggests something before that.

I admit that this is a strange suggestion.  However, think about how ‘strange’ our current system can be … at many institutions, students who start in pre-algebra have about a 20% ‘chance’ of completing their college level math requirement.  Are we helping that 20% so much that this process is worth the risk to the other 80%?

Before you jump up and down, screaming “THIS IS NOT GOING TO WORK” … look at some potential numbers.  If we assume that 70% of the students placed into pre-algebra pass that course, and that 50% of those who proceed to beginning algebra pass that second course, we have a net 35% who complete beginning algebra in the second semester.  This 35% assumes that ALL students will pass pre-algebra continue to beginning algebra; this is not reasonable.  Based on estimates from my data work at my college, from 70% to 80% actually go on to the second course.  Applying the highest rate (80%) to the 35% value gives us a realistic net of 28% … about 28% of students who start in pre-algebra complete the beginning algebra course the second semester.

What would we expect to happen to students who go directly to the beginning algebra course?  Would they be half as likely to pass that course, compared to having taken pre-algebra?  This “half” seems like a reasonable estimate (and may be too low).  Half of 50% … is 25%.  Since 25% is generally not statistically different from 28%, there is a good chance that placing all students in to beginning algebra would not create any additional risk to the student — and would save a semester of credits.

There is actually evidence that suggests this 25% ‘direct’ rate is too low.  A study (http://ccrc.tc.columbia.edu/Publication.asp?UID=1030) shows the predicted pass rates for students above and below the cutoff on a placement test (Accuplacer in this case); the predicted values for rates of C or better are above 30% for all placement test scores.  If this is accurate, then it would actually help students to never place them in to pre-algebra.

Based on years of talking with students struggling in beginning algebra, there is another reason why ‘skipping’ pre-algebra might help quite a few students: of the students who pass pre-algebra, quite a few of them were not challenged by the material … in fact, many do not study … and still pass.  This “no study, and pass” experience is exactly the opposite of what most students need; students need to know that working hard and continuing are critical for academic success.  As long as a pre-algebra course is primarily procedural, with a focus on correct answers, it will not contribute to habits that help students in later courses.

Think of that … a simple solution that saves a lot (money & credits for students, costs and resources for colleges), with either no risk or even some significant benefits.  Let’s agree to not place any student into pre-algebra (or whatever your course is called); if their placement test suggests that they don’t have enough ‘basic skills’, we would be better off placing them into beginning algebra anyway, perhaps with a sheet of references for refreshing those skills.

 
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Pre-Algebra … an Oxymoron?

One of the issues our department is looking at is the course prior to beginning algebra … we call it Pre-Algebra.   The current incarnation of this course includes early work with variables, expressions and equations — based on the thought that this will help prepare students for algebra.  The results are not what we would hope for.

Is “Pre-Algebra” an oxymoron?  The word oxymoron (apparently) literally means “sharp dull”, and the implication of the word is that the two concepts are contradictory.  What really comes before algebra (‘pre’)??

I’ve commented previously that arithmetic, as an academic topic, is more difficult than algebra and therefore it is illogical to use arithmetic as a prerequisite to algebra.  If we look at ‘tests’ of readiness for algebra (a typical one is http://www.algebra-class.com/algebra-readiness-test.html) the content is a mixture of arithmetic of whole numbers and fractions, order of operations, ‘simple’ algebraic expressions and equations, and perhaps some basic geometry knowledge.  Our own pre-algebra course looks a lot like this. 

What does a real student take away from (gain from) this pre-algebra experience?  Here is a (somewhat cynical) summary:

• Whole numbers — “Okay, got that; thanks for making me go through this for the 20th time”
• Fractions — “Are we done with those yet?  I hope you don’t expect me to remember this; I certainly do not understand fractions.”
• Order of operations — “Look for parentheses; there is also something about multiplying before adding, I just have trouble remembering what it is.”
• Algebraic expressions — “I don’t get why x and y are different … they are both unknown; I remembered the rule for simplifying until I took the test … not a minute longer.”
• Basic equations — “I like finding x; just don’t give me a word problem.  Oh, and I prefer to not show steps for solving … is that okay?”
• Basic geometry — “Don’t give me too many formulas to memorize; I can do the geometry.  What’s the deal with area having a different unit than perimeter?”

Like many of us and our courses, our course is mostly about using procedures to get correct answers … and this shows in what students get out of the class, and painfully shows when the students take an algebra course.  Some parts of a typical beginning algebra course can be done by simple procedures; some topics are ‘procedure challenged’ (like graphing, and systems).  If the list above is an accurate summary of what a student gains from pre-algebra, then it has nothing to do with being ready for algebra … pre-algebra has become an oxymoron.

Once upon a time, pre-algebra did not exist.  We offered some arithmetic courses, and sometimes used arithmetic as a prerequisite to algebra (reinforcing the myth that all mathematics is a linear and dependent sequence of steps).  Some pre-algebra courses are really arithmetic courses with a less remedial name; some are an honest attempt at improving readiness for algebra.  Is there ANY evidence that a prerequisite to algebra exists in a form that could be the basis for a course?

Some research has been done, mostly at the middle school level.  For example:  http://www.mheresearch.com/assets/products/ea5d2f1c4608232e/CA_Algebra_Readiness_Research_Base.pdf  and http://www.rachaelwelder.com/files/vitae/Welder_Prereq_Know_Algebra.pdf  .   The research cited by such articles is often descriptive in nature, which can show (at best) some correlation.  [The latter reference even includes a 1991 presentation I made at an AMATYC conference!   That report might be helpful, in spite of that.]

Some middle school programs might do a better job of algebra readiness than colleges; we, in the college setting, generally do a procedural course with little designed to help students mature their thinking and reasoning.  I do not have a definite answer … I can not say “THIS stuff is the real prerequisite to algebra!”  However, I can tell you that having a student generate 2000 correct answers to a variety of problems has nothing to do with being ready for algebra.  We might be better off using a general test of reasoning as the prerequisite (ie, active writing skills … as opposed to reading).

In the New Life model, we do not have a pre-algebra course or level.  The first course, Mathematical Literacy for College Students (MLCS), stands mostly on its own.  The people involved in designing MLCS have identified some prerequisites to MLCS — which fall in the category of ‘basic numeracy’.  Here is that list:

• Use arithmetic operations to represent real-world operations, such as putting together, comparing, distributing equally, etc.
• use real-number arithmetic to solve stated problems.
• Use graphical representations on a number line to demonstrate fluency in
• interpreting interval notation,
• ordering numbers,
• representing operations (i.e., addition, subtraction, doubling, halving, and averaging)
• representing decimal numbers, including negative numbers.

This list was created by ‘backwards analysis’ — looking at the learning in MLCS, what specifically does a student need to know before then?  This list is quite short, and our dream is that students who need this content can be served WITHOUT taking another course — whether this is done via a boot-camp review, or ‘just in time remediation’ within MLCS.  [The list is from https://dm-live.wikispaces.com/file/view/MLCS_Numeracy_March2010.pdf]

It is possible … really possible … that we could start students directly in MLCS (or the beginning algebra) without a prerequisite; a little bit of support on prerequisite knowledge, and some scaffolding within MLCS or algebra, would be enough for almost all students.

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