Pathways Summer Institute

This week, over 100 faculty (along with over 20 administrators and 20 institutional researchers) met in Palo Alto (California) for the Quantway™ and Statway™ “Summer Institute”.

These “pathways” are being developed under the coordination of the Carnegie Foundation for the Advancement of Teaching (see http://www.carnegiefoundation.org/developmental-math for more details).  AMATYC is a partner in this work.

I see two exciting parts of this work.  First of all, the Carnegie Foundation thinks it is important to have official AMATYC involvement in this work; this allows two of us (Julie Phelps of Florida, and me … Michigan) to be engaged at a deep level in the work.  Both of us were part of the Summer Institute, and have been involved with the planning for it as well, along with work on the actual instructional material.

The instructional materials are the other exciting part of this work.  This is due to multiple factors … the materials are being developed by the Dana Center (see http://www.utdanacenter.org/), known for its quality work in this area.  Another factor is the fantastic fact that the materials will be open resources in 2012, under a “Creative Commons License” which allows general use with source credits.  The materials will include an online homework system of high sophistication.

We also have considerable synergy between the work of the Pathways and the New Life project (AMATYC Developmental Mathematics Committee).  This synergy can be seen in the high degree of agreement between the learning outcomes used in both projects, as well as the multiple people engaged with both projects.

The Pathways are a very sophisticated solution which addresses several needs and problems in developmental mathematics.  I encourage you to become familiar with the work of the Pathways … we live in the best time for developmental mathematics in recent memory.

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Roots and the Mythology of Mathematics

We teach ‘mathematics’.  We believe ‘mathematics’ to be valuable.  What is this ‘mathematics’?

Our students firmly believe that ‘mathematics’ is a difficult mystery hidden from normal people.  Why do they have to ‘learn’ this ‘mathematics’?  When will they use it?  Does anybody really care (outside of a math class)?

Whether we are in a developmental classroom, or pre-calculus, or some other ‘math class’, we do not generally deliver an honest presentation of our subject.  How can I possibly make that statement?  Well, I’ve been thinking for years … and reading other peoples’ informed judgments … and conclude that the core property of mathematics is “the science of quantitative relationships”.  Mathematics is a science, not an abstract play ground; neither is mathematics a complex set of occasionally connected manipulations on various symbols and statements.

Mathematics enjoys a privileged position in American society, a position based more on the mythology of of mathematics than any reality.  Decision makers think ‘more mathematics’ is a good thing, and they can find statistical data that supports that position.  Our skeptics (and there are a few) can present better statistical studies that show that it is actually not the mathematics that makes the difference — there is a common underlying cause.

One of my students said this week (as she asked another question) “How can you stand to teach something that everybody hates so much?”  This was a spontaneous comment, and shows the type of mythology that I speak of.  If ‘mathematics’ was valuable, as we teach it, students would (to varying degrees) understand the benefits and gain motivation for working hard.

Instead, ‘mathematics’ is normally experienced as that complex set of occasionally connected manipulations on various symbols and statements.  We have students ‘simplify variable expressions’, but we have no clue that they realize we are talking about representations of quantities in their lives.  They ‘solve equations’, with no clue of how equations state conditions that people, objects, and properties must meet in specific ways.  We make students ‘graph functions’, without either making sure that they know how functions express the central relationships of quantities important to them or letting them in to the powerful tools of ‘rate of change’.

The roots of mathematics are in the rich intersection of practicality and science.  We have lost our roots, and cover neither side of this intersection.  We survive only because of the mythology surrounding ‘mathematics’; this mythology is not correct, and is offensive to a mathematician (in my view).  We teach mythology instead of mathematics.

Get up!  Look back at our roots as a practical science.  Do all you can to dispell the myths held by people concerning mathematics.  A central part of this work is to build a curricular structure that emphasizes actual mathematics.  You can begin this process by looking at the New Life model for developmental mathematics, as one model based on mathematics not mythology.

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The Capabilities of Developmental Students

What are our students capable of?

I think we end up taking a ‘bipolar’ position on this.  On the one hand, we believe that our students can achieve their goals; we encourage them, nudge them, motivate them, and suggest that they might be capable of higher goals.  Our greatest satisfactions come from watching our students — who needed developmental course work — graduate with a completed degree.  Gowns, in college colors, form a visible symbol of this hope for all of our students.

On the other hand, we seem to design courses which say “I get it … you can’t understand mathematics, really; so I will just expect you to recognize some patterns for which you have a solution in memory.”  We build instruction around the goal of maximizing correct answers for students.  We select textbooks which simplify the presentation and provide clear examples of the procedures, and avoid textbooks which discuss the ideas outside of examples.  We observe that our students do not remember much of what they had last semester, and conclude that this reinforces our design of ‘simplify’.

In fact, our ‘simplify design’ paradigm is part of the problem.  As long as learning focuses on remembering procedures, the powerful brain work that enables long-term changes and transfer of learning do not have a chance to occur (except by accident).  In some ways, most of our students leave our classrooms with the same condition that they arrived … summarized by the one word “unable”.

I can not accept the ‘simplify design’ of curriculum due to its message about the capabilities of our students.  Our students are capable of achieving much, and our society actually depends upon them achieving much.  We can not avoid building this capacity within our ‘developmental’ classrooms.  (It’s ironic that we call our courses ‘developmental’ but tend not to develop capacity.) 

Now, I am not under the influence of some ‘just be happy’ medication.  Obviously, students in developmental mathematics classes have some current limitations.  Our response must be to overcome limitations and build capabilities.  This response is not easy, certainly not just ‘pick the best homework system’.  Just like our students, we will achieve more than we thought possible when we face challenges directly.

And, just like our students, we will find the work is easier … and we understand more … when we work with each other.  You are not alone, and we are capable of designing courses which build capacity within our students.

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Lessons from a Struggling Student

Teresa came to my office this week, in something of a desperation move.  She has been struggling all semester in my beginning algebra class, and did not know what to do.  Teresa  is very thorough about doing homework; as a person returning to being a student after 15 years in the workforce, she knew how to follow through on her responsibility.  

Teresa knew the course was going to be hard work, but it is part of her plan to get an associates degree in 18 months.  When her performance was not at the passing level on the first test, she took immediate action; she hired a tutor, and bought an extra book to study.  After the second test and still lower scores, she changed tutors and bought more books.  Her motivation is outstanding; her commitment to the course is unequaled, and she seeks help at every opportunity.  Routinely, Teresa would understand a topic but be unable to retain her knowledge through time.  Teresa’s frustration was evident; she did not need to tell me that discouragement had caused her to shed quite a few tears.

How could I help such a student?  What was wrong with her strategies? 

Essentially, the problem with Teresa’s strategies is that they were oriented towards external sources, especially experts.  People were answering her questions, and telling her what she needed to know.  Teresa needed to move from a helpless perspective to an active role, one based on confidence in her own learning skills in understanding mathematics.  In addition, Teresa needed to stop expecting frustration and failure … even though failure has been her constant companion for ten weeks in this course.  Learning when expecting failure is nearly impossible.

Do we, as a profession, approach our work like Teresa?  Are we looking for somebody else (‘experts’) to tell us the answer?  Maybe if we just offer modules, mastery learning and extra help we will be successful.  Do we expect to continue to get results like we have had before?  Are we applying our impressive learning abilities to understand the root problems?  Perhaps we can find the ideal text format and content organization … and solve the problem by using contexts that students see as relevant to their lives.   Who owns the problem that is developmental mathematics?

I am completing my 38th year as a professional in developmental mathematics, and there have been ‘movements’ and ‘trends’ before.  For the first time, there is actually an opportunity for us to build something new in developmental mathematics.   Resist the temptation to accept solutions from the outside.  We are mathematicians with keen insights into helping students understand mathematics, and we have access to information on designing a program to respond to the diverse mathematical needs of our students. 

Come to the revival of developmental mathematics … and learn what that means for you and your students.  Come to the revival of developmental mathematics … and imagine our courses dealing with some essential mathematical ideas in a manner that enables students to apply those ideas across other disciplines.  Come to the revival of developmental mathematics … and reach for the goal of a program that enables students to reach their goals, and perhaps inspires them for higher goals.  Come to the revival of our profession!!

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