Meaningful Mathematics … Learning Mathematics

Among the pushes from policy makers is ‘meaningful math’ … make it applicable to student interests.  Of course, this is not a new idea for us in mathematics; we’ve been using ‘real world applications’ to guide some reform efforts for decades. #realmath #collegemath #learningmath

Some current work in reforming mathematics education in college is based on a heavy use of context, where every new idea is introduced with a situation that students can understand.  We know that appropriate contexts with meaning to students helps their motivation; does it help their learning?

Before I share what I know of the theory and research on these approaches, I would like you to envision two types of textbooks or instructional materials (print, online, or whatever).

  1. Opening up the start of a lesson or section, repeatedly and randomly, generates a short verbal introduction followed by formal mathematical symbolism related to the new idea(s) in almost all cases.
  2. Opening up the start of a lesson or section, repeatedly and randomly, generates a variety of contextual situations and mention of a mathematical idea or tool that might be used.

Many traditional textbooks are type (1), while a lot of current reform textbooks are type (2).  Much of the change to (2) is based on instructor preferences, and I am guessing that much of the resistance to change is based on instructor preferences for (1).

Let’s take as a given that contexts that are accessible to students improve their motivation, and that we have a goal to improve student motivation; further, let us assume that we share a goal of having students learn mathematics (though that phrase means different things to different people).  There seem to be a number of questions to answer dealing with how a context-intensive course impacts student learning of mathematics.

  • How uniform is the impact of context on different learners?  (Is there an “ADA-type” issue?)
  • Do students learn both the context and the mathematics?
  • Is learning from a context more or less likely to be used and transferred to new situations?

I know the answer to the first question, based on research and experience: The impact is not uniform.  You probably understand that there are language issues for quite a few students, perhaps based on a class taught in English when the primary language is something else.  However, most of the contexts have a strong cultural factor.  For example, a common context for mathematics work is “the car”; there are local cultures where cars are not a personal possession, as well as cultures outside the USA where cars are either generally absent or relatively new (and, therefore, people know little).  The cultural problems can be overcome with sufficient scaffolding; is that how we want to spend time … does it limit the mathematical learning?  There are also ADA concerns with context: a sizable group of students have difficulties processing elements of ‘a story’ … leading to problems unpacking the context into the quantitative components we think are ‘obvious’.

The second question deals with how the human brain processes different types of information.  A context is a type of narrative, a story; stories activate isolated memories and create isolated memories.  To understand that, think about this context:

You are standing on a corner, and notice a car approaching the intersection.  When the light turns red, the car applies the brakes so that it stops in about 2 seconds.  You estimate that the distance during the stopping process is about 100 feet, and the speed limit is 30 miles per hour.  Let’s look at the rate of change in speed, assuming that this rate is constant through the 2 second interval.

The technical name for a story in memory is “episodic memory”.  This particular story might not activate any episodic memories for a given student; that depends on the episodes they have stored and the sensory activators that trigger recall.  Some students will respond strongly and negatively to a particular story, and this does not have to depend upon a prior trauma.  More of a concern are students who have some level of survival struggle (food, shelter, etc); many contexts will activate a survival mode, thereby severely limiting the learning.  Take a look at a report I wrote on ‘stories’,%20Ignore%20the%20Story.pdf

What happens to the mathematics accompanying the story?  If we never go past the episodic memory stage, the mathematics learned is not connected to other mathematics; it’s still a story.  In the ‘car stopping at an intersection’ story, the human brain might store the rate of change concepts with the rest of this specific story, instead of disassociating the knowledge so it can be used either in general or in new ‘stories’ (context).  Disassociating knowledge is another learning step; many context-based materials ignore this process, and that results in my biggest concern about ‘problem based learning’.

Using knowledge (Transfer … question 3) depends upon the brain receiving sensory input that activates the knowledge.  This is the fundamental problem with learning mathematics … our students do not see the same signals we see, ones that activate the appropriate information.  For this purpose, traditional symbolic forms and contextual forms have the same magnitude of difficulty: the building up of appropriate triggers to use information, as well as creating chunks of information that work together.  We need to be willing to “teach less mathematics” so that we can focus more on “becoming more like an expert with what we know”.  For more information on learning mathematics based on theoretical (and research-based) points of view, see and 

I can’t leave this post without mentioning a companion issue:  Contextual learning is often done by ‘discovery’.  Some reform materials have an extreme aversion to ‘telling’, while traditional materials have an extreme aversion to ‘playing around’.  From what I know of learning (theory and research), I think it is safer to take the traditional approach … telling does not provide the best learning, but relying on discovery often results in even more incomplete and/or erroneous learning.  Just for fun, take  a look at,%20Explaining,%20and%20Learning.pdf

Looking for a brief summary of all of this stuff?

Contextualized learning comes with significant risks; use it with caution and a plan to overcome those risks.
Mathematics is, by its nature, a practical field; all math courses need to have significant context used in the process of learning mathematics.

These ideas about context are related to the efforts of foundations and policy influencing agents (like CCA).  We need to keep the responsibility for appropriate instruction … including context.

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GPS Part IV: CCA as a Dot Com Bubble

Many states and colleges are engaged with the Guided Pathways to Success (GPS) program and other methodologies supported by Complete College America (CCA).  In this post in the series, I will suggest that this observation if essentially true:

The influence of CCA will be similar to the dot com bubble of the 1990s.

In other words, the CCA is advocating dramatic action using unproven methods for a large group of investors (states and colleges).  Some methods involve components which have sufficient evidence for scaling, but the magnitude of change being created exceeds any reasonable prediction for a positive return on investment.  Even if the labels (like GPS) stick, the market will collapse within a few years as states and colleges get data indicating the large amounts of money are being lost with little gain for students.

To understand why this observation is made, take a look at a quote from the CCA materials:

But game changers don’t spontaneously happen: They are caused by people who act boldly and decisively in response to challenges

The ‘game changer’ reference is designed to pull in the big investors; investors are drawn to promises of large returns, especially when there is an apparently simple plan for the large returns promised.  The declaration of ‘boldly and decisively’ is a propaganda tool meant to turn off any inclination to be skeptical of the rationale for the components of the plan.

The question is this:  Why do we need ‘game changers’ in the first place?  Few of us would like the process of education being equated with any game or set of games; let’s set that valid concern aside.  “Game changer” is defined (Merrian-Webster) as “a newly introduced element or factor that changes an existing situation or activity in a significant way”.  Some components of the methods suggested by the CCA would meet this definition (such as ‘full time is 15′); however, the methods are more accurately summarized as “changing the game” rather than “game changers”.

The push toward GPS and other ‘game changers’ is accompanied by a rationale that sounds reasonable to those with smaller amounts of understanding of culture of our institutions … community colleges in particular.  I am reminded of the many novice arguments presented by my students for why their incorrect mathematics was actually ‘correct': such arguments convince other novices, and perhaps some professionals who turned of their skeptical (critical) functions.

In spite of the obvious and reasonable doubts about the “CCA Game”, their marketing has worked very well.  Several states are deep in to the “CCA com” (like dot com) bubble.  The press for CCA has been extremely one-sided … partially because they create much of the press themselves.  No organization has stood up to question the CCA messages, even though the messages lack significant professional history.

I commend the CCA for a hustle well played.  It’s disappointing that so many leaders and policy makers have been hustled like this.  The prediction for the collapse of this CCA bubble is supported by the track record of prior changes … prolonged change tends to be consistent with, and supported by, the work of professional organizations.  The CCA bubble is supported by a network of change agents, much like the ‘dot com’ bubble.

The unanswered question: How long will the CCA bubble last?

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GPS Part III: Guided Pathways to Success … Informed Choices and Equity

In the structures used for Guided Pathways to Success (GPS), colleges are encouraged to provide information to students about selecting a major.  That is great, obviously, until one reads the next detail:

Colleges use a range of information such as past performance in high school to provide recommendations to students about programs of study that match their skills and interests.

In other words, we would limit information to each student based on our interpretation of their background.  I want every student coming to my institution to consider (even dream) about goals that exceed their history and the accidents of their background.

The push to have students select a program of study is well-intended … we all want students to ‘succeed’.  However, we can not be so short-sighted that we encourage students to only consider goals that seem reasonable based on the data we might happen to have available.  Such methodologies will tend to maintain social class and economic standing; therefore, I see a fundamental conflict between this GPS method and the basic purposes of community colleges … upward mobility.

Statistics does not work for limiting choices at the individual level.  In the medical uses of data, providers can get very close to a valid ‘limiting’ of choice … when the statistical analysis has a small margin of error, due to understanding a physical process well.  Education does not deal with small margins of error (not in this decade, anyway).

The CCA website repeatedly shows ‘data’ with the implication that the results are statistically determined.  For example … community college students average 81 credits for a 60 credit degree, proving that students accumulate ‘too many’ credits.  That only makes sense if you look at the 130% credit count as measuring wasted effort; this has never been determined for a group of students, even though the CCA would like us to believe that it has been.

What’s your thought on what the ‘extra’ 30% represents?  Personally, I look at that 30% as being composed of several parts:

  • Excess remediation (should be 10% or less); is likely to account for 15%.
  • Intentional program credits (programs requiring 61 to 64 credits are common).
  • Intentional student choices (deliberately taking a course at CC … often because it’s cheaper).  This one probably accounts for 10% in that 30%.
  • Uncertainty causing choice of courses inappropriate for the student

I do not see the rationale that says this 30% ‘excess’ means that students must make a choice of program of study EARLY and that we should direct them to fields appropriate to their background.  I believe that the CCA does not understand the community college environment, with the factors influencing student choices about courses; this, combined with a bad use of a piece of data, results in a socially unacceptable suggestion (that we track students based on their background).  Following the CCA advice seems to amount to “keeping the wrong people out of the important programs”.

Equity is a fundamental part of our work in community colleges.  Equity and upward mobility are more important than arbitrary metrics of credits earned in community colleges.  State and local policy makers should be very concerned about following the CCA advice to implement GPS with a heavy emphasis on selecting the best program of study right away.

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GPS Part II: Guided Pathways for Success, a Mathematician’s View (Part II)

Guided Pathways (GPS) is one of the current ‘movements’ in higher education, both at the associate degree and bachelors degree level.  A description of GPS is available at (although quite a bit of this document is rhetoric designed to convince the reader of a point of view).

At the heart of GPS is the concept of one set of courses for the student to take for their program, starting (hopefully) in their first semester.  Mathematics is specifically addressed in the GPS model … “Math Alignment to Majors”, and this echos movements within the mathematics profession to create pathways leading to multiple end-points (college algebra/calculus, statistics, quantitative reasoning or QR).

This apparent congruence is a concern for me.  Here is the issue … math alignment is intended to divert students out of the college algebra path as early as possible.  This is somewhat true of pathways in general, but the GPS work tends to create rigid walls around the paths.  A student declares a major like nursing (which the CCA considers “STEM”, by the way) … and is likely to take statistics as their math course (possibly QR).  What happens when this student gets inspired to pursue a truly STEM field, such as biology or pre-med?  Actually, the student will not have much chance to be inspired in their math courses; the GPS work has a goal “as little as possible” when it comes to mathematics.

One of the reasons I believe so strongly in the QR course we offer is that it builds algebraic reasoning (as well as statistical reasoning and proportional reasoning).  If all QR courses did this, I would have fewer concerns about GPS paths … if QR was the default math path.  In many parts of the country, statistics has become the default math path (outside of STEM); I am concerned about a student’s only college math course being in one field of mathematics when the student’s program does not call for specialized or focused mathematics.

GPS also presents the idea of milestone courses; mathematics is likely to be on an institution’s short list of milestones, especially in the first year.  I do not want students to see that the world shares their desire to get math out of the way, nor do I want to see mathematics used as gatekeepers for programs.  Certainly, if the student’s program involves courses which will actually use the mathematics in their math course, by all means … require the math in semester #1.

Too often, however, our colleagues in other disciplines have de-quantified their subject … even STEM disciplines.  Intro science courses are often presented (at the associate degree level in particular) in a conceptual way, without the mathematical methods (or ideas, even) used in current work in those fields.

GPS holds promise, and our students can benefit if we do a good job.  We need to avoid the pressure to swallow the GPS pill whole; each component needs critical thinking and the professional expertise we can bring.

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