Proportional Reasoning

What is proportional reasoning?  Are “proportions” truly a different topic than “functions”?  If our students master linear functions, will that enable them to reason with proportional quantities?   Stated another way: Does knowing the mathematical object imply that a person can reason with that object to solve problems?

I think we, as mathematicians, tend to see proportionality as a special case of functions … or as a particular type of equation involving two ratios or rates.  Why does a person need to be able to reason with proportions … or proportionality?

Take a look at this problem, which is paraphrased from my “Math – Applications for Living” course:

A car is driven at an average speed of 44 miles per hour.  At this speed, the car averages 33 miles per gallon.  How much gasoline does the car use per hour?  Follow-up question:  If the tank is full with 14 gallons of gasoline, how long could the car be driven (in hours)?

I’ve seen students struggle with this problem, which I use to highlight proportional reasoning.  Each rate is a statement of proportionality … miles driven is proportional to the time, the gallons of gasoline are proportional to the miles driven, etc.  Further, each statement of proportionality (rate) is equally true in two forms — as stated, and the inverted rate.

By looking at the units we need (like gal per hr in the first question), we set up the rates to provide that answer:

To the extent that our students take basic science classes, this proportional reasoning is very valuable … and has no direct connections to the concepts of functions.  Bringing up concepts of input and output only complicates these problems … because each statement of a rate allows any of the quantities to be the output; identifying an ‘output’ quantity is done by looking at the nature of the output needed for the question at hand.

Instead of addressing proportional reasoning, we often ‘help’ students by teaching them keywords to indicate multiplication or division (if it says “how many pieces can you get” it is division, etc).  You may have noticed how limited this approach is, because problems are phrased differently.  If we look at proportional reasoning with the rates, it becomes much easier.

Proportional reasoning comes up, in a natural way, when we start studying probability.  Take the classic type of problem to introduce probability:

A container has 6 red marbles, 3 blue marbles, and 1 white marble.  What is the probability that a marble, chosen randomly from this container, will be blue?

The concept here is something like “the probability of something happening is proportional to the number of those ‘things’ that are in the entire group”.    Many simple probabilities are based on this, as is the question of ‘drawing two blue marbles’ from this container.

Unfortunately, what many students remember about proportions has limited value (and is often mis-applied) … “cross products”.  Given a proportional situation, the important thing is being able to write two rates or ratios which make the same comparison (they follow the proportionality involved); as described above, it is also important to be able to write products of rates to produce the desired units.

Although a slope is a rate, which might suggest proportionality, the use of linear functions is not proportional reasoning (especially as experienced by novice learners).  The connection between proportionality and linear functions is not an equivalence; it is more of an issue of ‘shared concepts’.  We should not assume that knowledge of linear functions has much to do with proportional reasoning.

 
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Do we need developmental education?

You may have seen the news story about Connecticut considering a law that would eliminate all developmental education in the state … except for imbedded remediation and an ‘intensive college readiness program’. General story: http://communitycollegespotlight.org/content/connecticut-may-end-remedial-requirement_8674/ and more details http://www.insidehighered.com/news/2012/04/04/connecticut-legislature-mulls-elimination-remedial-courses

I see two basic issues raised by this.  An obvious issue is a statement about the perceived value of developmental education.  In the case of mathematics, some developmental programs have 4 courses before the first college-level math class; a logical analysis of this system can easily show that there is a basic design flaw … a two-year ‘getting ready for college’ track is enough credits for a major, but these are courses that do not have value in themselves.  A rejection of this design is basic in our development of the New Life model, where we reduce that pre-college work to 1 or 2 courses, depending on the student’s program.  Does a rejection of the 2-year developmental math program imply that it can be replaced by a ‘just in time’ remediation model, combined with a boot-camp experience?

The other basic issue raised is the change process.  We appear to be in a period when politicians are policy makers in broad areas of education.  It’s not like the state said ‘We are spending way too much money … and not getting enough benefit; we are appointing a task force of experts who are charged with creating a model that meets the needs of our communities in a process that is much more efficient’.  Whatever the process was, the lawmakers believe that they have a solution that can be legislated.  Have we done such a poor job of articulating the power of a good developmental program that lawmakers believe that this is a solution? 

I have no doubt that some students — even many students — would be well served by the ‘imbedded + boot camp’ model; historically, we have underestimated the capabilities of students to cope with challenges … if they have a little more support.  However, I believe that this model will leave many students defeated; these will be the types of students for whom community colleges were created — the ‘first generation college’, the un-empowered and vulnerable, and those for whom the K-12 system did not ‘work’ … as well as the returning adult. 

We need to do a better job of articulating the power of what we do everyday.  Our courses are not just about some collection of basic skills, that our goals include developing learning and thinking in our students; we need to tell people in authority that we have expertise and methods that produce results.

We also need to be willing to ‘take the criticism’ … that our developmental programs have become entrenched and stagnant systems that do not serve enough students nor well enough for all students, that we can develop models that better serve our students with dramatically reduced credits and costs.  If we continue to insist on the same-old programs, or even if we fail to recognize this problem, then we deserve to have others (like politicians) determine a better system.  I believe that we are wise enough to do the right thing.

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MLCS Webinar – April 24 !!

MLCS is “Mathematical Literacy for College Students”, the first course in the New Life model.  Kathleen Almy, outstanding faculty and pioneer in this work, is offering an AMATYC webinar on April 24 (3pm Eastern, 2pm Central, 1pm Mountain, noon Pacific).  

The webinar is entitled “New Pathways for Developmental Math: A Look into Mathematical Literacy for College Students”, and is open for any AMATYC member.  Registration for the webinar is available at https://netforum.avectra.com/eWeb/DynamicPage.aspx?Site=AMATYC&WebCode=HomePage 

Here is the description for this 1-hour event:

Mathematical Literacy for College Students (MLCS) is a new course that is part of an AMATYC initiative called New Life for Developmental Math as well as the Carnegie Quantway project.  It is an innovative way to redesign the developmental curriculum, providing pathways for the non-STEM student.  The course uses integrated, contextual lessons to develop conceptual understanding and technology to improve mastery of skills.  In one semester, a student placing into beginning algebra will gain the mathematical maturity to be successful in statistics, liberal arts math, or intermediate algebra.  Reading, writing, critical thinking, and problem solving are key components to reaching that goal. Webinar participants will learn much more about the course as well as receive ideas for course development including a sample course outline and a sample lesson.

The live webinar is open to AMATYC members; recordings of AMATYC webinars are available after the event from the AMATYC webinar page http://www.amatyc.org/publications/webinars/index.html.

To register for the webinar on April 24, go to at https://netforum.avectra.com/eWeb/DynamicPage.aspx?Site=AMATYC&WebCode=HomePage 

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Developmental: Skills or Capabilities?

At a recent conference (MDEC, the Michigan affiliate of NADE), we were having a conversation with Hunter Boylan about developmental education. One of the participants commented that a major concern was that students sometimes leave developmental courses as developmental students.

What did they mean by ‘developmental students’?  I think the basic concern is that students were leaving our courses without the capabilities (not abilities) to handle college academic work.  One of my colleagues who is a ‘reading’ faculty commented that it seems like the developmental course was a collection of discrete skills which did not add up to any additional capabilities.

There is a somewhat different point of view for professionals engaged with NADE or the National Center for Developmental Education (which is directed by Hunter Boylan).  Their framework specifically includes ‘personal growth’, referring to a collection of cognitive and affective factors … which I categorize as ‘capabilities’.    [The “NADE-type” definition of developmental implies that it is not a nicer name for remedial; most of us in the mathematics community equate the two phrases.   As implemented, most developmental math programs are ‘remedial’; I wish they were not.]

In reading, for example, parsing a phrase … vocabulary … decoding … these are groups of skills; however, without additional capabilities, students remain developmental in their functioning — resulting in a higher risk of failure in college courses.  That is, basic literacy skills are not sufficient in a good developmental reading program.

How does a typical developmental math program compare?  Sadly, I think we are the epitome of skill courses that do not impact the capabilities of our students.  A beginning algebra course usually has 8 to 10 chapters of material, with a preponderance of … parsing phrases … vocabulary … procedures; our ‘applications’ are mostly stylized puzzle problems which avoid the need to think deeply about relationships.  In fact, we sometimes take pride in providing rules or tools to cope with word problems so students do not have to analyze them. 

A basic reason behind the New Life project is this:  serving up skills with symbols does not change the capabilities of our students.  Dealing with basic concepts, connections, transfer, analysis … this process changes the capabilities of our students.  It is our belief that good preparation for college work is based on an emphasis on deeper academic work in ‘developmental’ courses.

As you look at the learning outcomes for New Life (or the New Mathways), keep in mind that the model is making a serious attempt to build student capabilities.  Since there is not a linear sequence of basic skills, you will have to work harder to understand what the curriculum is trying to accomplish for our students.

Any course — ‘developmental’ or not — that only seeks to add skills to a student, without a larger focus on capabilities, is a missed opportunity.  When that course is used in a gate-keeper fashion (like mathematics is), we need to move towards a design that truly helps our students.

 
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