When Are we Going to USE this?

I think all students receive the same email when they register for a math class; this email says “The BEST way to irritate your instructor is to ask ‘When are WE going to USE this?’ at the first sign of confusion.  Not only does this irritate your instructor, but this will distract the instructor from your lack of understanding.”

Of course, my introduction above is a weak attempt at humor (a dangerous choice on my part!!).

Seriously, the issue is this:  Do students NEED to see the application of everything that we tell them to learn?  Are some concepts okay to learn, even if there are no applications that the student values?

Underneath this issue is the curricular issue:  What is the purpose of this course?  For developmental mathematics, we are in a “preparing for” business.  Our students need to take further mathematics (of various kinds), science classes, and technology classes; we also prepare students for college in general.  These future situations are a justification for the mathematics they learn … is it reasonable for us to expect students to appreciate the applications while they are learning the mathematics?

 In a recent post (https://www.devmathrevival.net/?p=282 ) I emphasized mathematics as a practical science.  For a recent conference in Michigan, I gave a talk on general education (http://jackrotman.devmathrevival.net/General%20Education%20Mathematics%20in%20Michigan%20May%202011.pdf) in which I highlighted the ‘usefulness’ of mathematics we require students to take.  It’s pretty common for people to conclude that I think developmental mathematics should be applied and in-context, in almost all topics we teach.

However, that is not my judgment about what is appropriate.  The traditional developmental mathematics courses are predominantly procedures; we do ‘applications’, but 90% of these applications are puzzles (you have to know the answer in order to write the problem).  This is not enough practicality to show students that mathematics is powerful and practical.  We need more applications meaningful to students, and we need content which will benefit our students.

At the same time, it is obvious to me that we would make a mistake to limit the mathematics we teach to those ideas for which we have applications that students would understand at that time.  A math course is not employment preparation (only), nor is it solving problems (only) … just like it is not beautiful mathematics (only).  We need a balance.  Students will need to learn material with no clear applicability to them in other classes, and learning in the face of ‘not useful right NOW’ is a critical survival skill.

We can also use learning research to provide guidance.  My own reading of theory and research indicates that context, and applications that students value, plays a positive role for motivation when done in moderation; an emphasis on context complicates the learning process, and may make the important seem invisible.  [For some references, see http://jackrotman.devmathrevival.net/sabbatical2006/9%20Situated%20Learning.pdf.]

I hope you will think about the purposes of our math courses, and reach your own conclusion about the appropriate role of applications that students will understand to support the learning of mathematics.  I am sure that we can design our courses so that students can learn powerful mathematics that WILL be useful to them, and that we can incorporate applications along the way.

 
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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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All of The Above

Somehow, I am on my congressperson’s email list; fortunately, he does not produce too much spam.  However, sometimes the emails sent from his office provide an interesting thought.

A recent email focused on energy policy, and touted an “all of the above” approach (exploration, federal lands, nuclear, coal, green, etc).  I started thinking … policy is based on a goal (or problem) and reflects the understanding of how we can achieve a desired outcome (also known as progress).  In the context of energy policy, ‘all of the above’ is a non-policy.  It strikes me more as a desperate person thrashing wildly in the water to avoid drowning.

In developmental mathematics, though, we have been practicing “all of the above” for many years.  We work from where we are, and we add ideas that sound productive.  I think we appear to many others to be ‘thrashing wildly in the water’, and I know that some of us actually think this is accurate. 

Is there an alternative?  Yes, of course … though it requires going back to an empty page in many ways.  We have a small set of basic questions that should guide (and somewhat determine) all of our work: What is important mathematics for all students?  What does it mean to ‘learn’ this mathematics?  How can we determine if a student has achieved this learning?

Our understanding of these questions is critical, for we know that various methodologies have different strengths … their impact on learning is different, and each is better suited to particular learning goals.  For example, we generally give broad support to ‘problem solving using mathematics’, and this means capacity to transfer learning in our domain; research has determined how this outcome can be enhanced, and which methodologies are likely to be more effective.  One specific point: Contrary to popular mythology, ‘drill’ is not ‘kill’ — repetition of skills forms a critical basis for development of problem solving; the problem is not ‘drill’ … the problem is ‘only drill’.

Think about this ‘all of the above’ idea.  Take a look at totally new models (like New Life), and consider your own ‘answers’ to the basis questions.  Our professional standards (Beyond Crossroads, http://beyondcrossroads.amatyc.org/) suggest that our work be focused on achieving our shared goals.  “Thrashing wildly” in an “All of the Above” mode does not appear in the improvement cycle.

As long as we continue an “All of the Above” approach to our work, we actually achieve “None of These”.

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Pathways Blog Available

Our friends at the Carnegie Foundation for the Advancement of Teaching have launched a blog for their ‘pathways’ work.

The blog is at Math Pathways, and I encourage you to take a look; you can also join the community on that page to receive updates.

I like the fact that the current post at the Pathways blog deals with quantitative literacy; we would be better off if we focused on quantitative literacy instead of ‘developmental mathematics’.  Of course, I would rather it be called ‘mathematical literacy’ (to focus on the scientific aspects of mathematics, not just the tools) — but that is a relatively minor point.

I hope you take a look!

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