Rotman’s Race to the Top

A few year’s ago, Paul Lockhart wrote an essay entitled “A Mathematician’s Lament”, sometimes known as “Lockhart’s Lament”.  I find this essay to be incredibly inspiring.

If you have not read it, here is a copy: lockhartslament

As far as I know, Lockhart never published this essay. I actually know very little about Paul Lockhart, besides the information that he went to teaching in a K-12 school after being a university professor for a time.  Some brief bio stuff is at http://www.maa.org/devlin/devlin_03_08.html

Okay, back to the essay.  Paul Lockhart is presenting a vision of mathematics as an art, not as a science, where learning is very personal and where inspiration is a basic component.  A fundamental process in Lockhart’s vision is the personal need to understand ideas, often ideas separate from reality.  The ‘lament’ aspect of the title for the article refers to the view that school mathematics has made mathematics into ‘paint by numbers’ instead of an exciting personal adventure.  Where we might focus on insight and creativity, we instead deliver an assembly line where students receive upgrade modules to their skill sets; Lockhart presents the view that these skill sets have nothing to do with mathematics.

I have two reasons for highly appreciating this essay.  First of all, people often assume that I believe that mathematics should be presented in contexts that students can relate to (probably because I advocate for “appropriate mathematics”).  Contexts are fun, and we can develop mathematics from context.  However, contexts tend to hide mathematics because they are so complicated; mathematics is an attempt to simplify measurement of reality, so complications are ‘the wrong direction’.  When I speak about appropriate mathematics, I am describing basic mathematical concepts and relationships which reflect an uncluttered view of mathematics; applicability might be now, might be later … or may never come for a given student.  Education is not about situations that students can understand at the time; education is about building the capacities to understand more situations in the future.

The second reason for appreciating Lockhart’s Lament is the immediate usefulness in teaching.  If we believe that students want to understand, then we can engage them with us in exploring ideas and reaching a deeper understanding.  Here is an example … in my intermediate algebra class this week, we introduced complex numbers starting with the ‘imaginary unit’.  Often, we would start this by pointing out that ‘the square root of a negative number is not a real number’ and then saying “the imaginary unit i resolves this problem.”

Think about this way of developing the idea instead (and it worked beautifully in class). 

1. Squaring any real number results in a positive … double check student’s understanding (because quite a few believe that this is not true).
2. The real numbers constitute the real number line, where squaring a number results in a positive.
3. Think of (‘imagine”) a different number line where squaring a number results in a negative.  Perhaps this line is perpendicular to the real number line.
4. We need to tell which number line a given number is on; how about ‘i‘ to label the number?
5. Let’s square an imaginary number and it’s opposite (like 2i and -2i).
6. Squaring an imaginary number produces i²; since the imaginary numbers are those we square to get a negative, i² must be -1

Any context involving imaginary or complex numbers is well beyond anything my students would understand.  I would always want to explore imaginary numbers; first, it provides a situation to further understand real numbers … second, the reasoning behind imaginary numbers is accessible to pretty much all students.

You might be wondering why I gave this post the title “Rotman’s Race to the Top”.  One reason is just not very important … I wanted a alliteration comparable to “Lockhart’s Lament”; well, I tried :).  The important reason is this:  The “Race to the Top” process is based on  some questionable components that do not connect directly with the learning process (common standards [a redundant phrase], standardized testing, and outcomes.  The world we live in, as teachers of mathematics, is a personal one: We interact with our students, we create situations that encourage learning, and we want our students to value learning for its own sake.  We also want our students to understand some mathematics.  Inspiration and reasoning have a lot more to do with this environment than any law or governmental policy.

Rotman’s Race to the Top is the process of teachers showing our enthusiasm, modeling the excitement of understanding ideas, and helping each student reach a higher level of functioning.  Building capacity and a maintaining a love for learning are our goals.  Rotman’s Race to the Top is about our students and their future.

Will you join this Race to the Top?

 
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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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Implementing MLCS or Mathways or Pathways

If you are drawn towards the new models of developmental mathematics that take a fresh look at the content … and wonder about the “how”, here is some good news.  The Dana Center (University of Texas – Austin) has developed an initial version of an implementation guide.   See http://www.utdanacenter.org/mathways/downloads/new-mathways-project-implementation-feb2012.pdf 

As an example, the implementation guide deals with Examining the culture and capacity of the math department (both developmental and college­‐level math) [page 9], and lists a number of specific questions to help; among them are these:

• How does the math department make a decision about instituting a new program or innovation?

• Are there institutional or departmental policies regarding instruction, assessment, and grading that support or deter implementing Mathways?

• What other innovations are taking place in the math department?

• How does the department view itself in the professional mathematics education landscape?

This implementation guide is VERY thorough; although many items refer to Mathways (the Dana Center project), they all apply to doing a New Life course as well.  The only problem you might have with the guide is that you may think it has too much information … not a bad problem to have.

I encourage you to take a look at this implementation guide.  [And, I thank the hard-working staff at the Dana Center for providing this resource to the profession.]

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Education as Transformation

Much is made these days of ‘value-added’, including the use of student ‘gains’ on standardized tests in the evaluation of teachers.  In colleges, we have defined courses in terms of student learning outcomes … which might reflect a comparable view of higher education (similar to K-12 and emphasis on ‘skills’).

“It must be remembered that the purpose of education is not to fill the minds of students with facts…it is to teach them to think.”  [Robert M. Hutchins]

What is the primary mission of colleges?  We all want our students to get better jobs, and would also like them to have a better quality of life.  Can these goals be achieved by the accumulation of discrete skills and learning outcomes?

Education is what remains after one has forgotten what one has learned in school. [Albert Einstein]  

Community colleges tend to serve the less-empowered segments of society.  People often cite mathematics as a key enabler of upward mobility, with some demographic studies to support this position.  These correlational studies produce a false impression of the processes involved.   The motto is not ‘algebra for all’ … the motto is ‘building capacity to learn and function’.

Education… has produced a vast population able to read but unable to distinguish what is worth reading. [G.M. Trevelyan]

Education should be a transformative experience.  Independent thinking, reasoning with a variety of methodologies (including quantitative), and clear communication should be evidence of this transformation.  In a community college, we can not strive for the same level of transformation as a university or liberal arts college education; however, we stand in the critical first steps for students along this path.

Education is the ability to listen to almost anything without losing your temper or your self-confidence. [Robert Frost]

In developmental mathematics, we have too often been content to provide little snippets of essentially useless knowledge — procedures to deal with a variety of calculations.  Even though it is not easy, and there is always a discomfort involved, our students are capable of much more.  Without reasoning and clear communication, these procedures will not benefit students (beyond a data bit that says they ‘passed math’).

Education is not filling a pail but the lighting of a fire. [William Butler Yeats]

As we work together to build a better model for developmental mathematics, we need to appreciate our place in the education of our students.  A good mathematics course produces a qualitative change in students. We can measure some aspects of this process by examining the reasoning and communication processes that students use.  However, there is no sure-fire and objective measure that says  a student has made progress.  We will develop better tools for this — including some focused on quantitative literacy and reasoning.  The challenges of measurement should prevent us from keeping our proper focus; we need to work to make the important measurable.

Education is the key to unlock the golden door of freedom. [George Washington Carver]

The pre-algebra/introductory algebra/intermediate algebra model of developmental mathematics needs to be re-made into a valid curriculum.  We can include mathematics that is practical, and that is an improvement — however, it  is not sufficient.  A central goal of developmental mathematics needs to be the improvement of quantitative reasoning and communication … a contribution which will enable our students to be educated, free people in a world facing diverse challenges.

 
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