The Math Dance

A step forward, step to side, bring feet together … a step backward, step to side, bring feet together.  Dance the waltz enough, and a person can do this sequence without any difficulty.  Many can become experts at the dance, and all can be included. 

Lost is why these are used as the dance steps. 

Of course, the ‘why’ does not really matter — it’s just a dance!!

I have taught a lot of students to dance.  The majority have been able to do dances like the waltz and two-step.  Sadly, the math dance has no particular value if a person does not know why the steps are done like this.  To understand means that a person can improvise; a little understanding allows helpful flexibility, and much understanding allows an artist’s rainbow of insight, logic, and problem solving.

Mathematics has become a dance, one that can be taught as remembered moves to particular musical themes.  There are some experts who assert that this the only possible outcome when society decrees that ALL persons must complete a subject, that mandatory always translates into a lowering of the value of this learning.  The evidence for this view seems abundant, and it is easy to accept this result (especially with the bright and blinding light of accountability shining on education).

We must not give up on mathematics so easily.

Mathematics has much to offer every student, our society, and the future.  Not the math dance — the real mathematics, science of relationships between quantities. 

We can create sound mathematics appropriate for all learners.  All students can learn, given the proper resources and conditions.  I might grant that the more extreme learning disabilities might present obstacles too large for a very small minority; this group is at least 2 standard deviations below the mean.

I encourage you to avoid the current rush of methods that might be more efficient at teaching the math dance.  We have seen these types of improvements before, which provide change but not progress.

I invite you to work with me to imagine a better mathematics program for all of our students, a program that shows the practicality and beauty of mathematics.  We do not need to make mathematicians of all students, just like we do not need to create math dancers … however, I believe that we can create a program that inspires more students to seek out more mathematics.

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Contextualized Mathematics

Should mathematics be learned within the context of a situation that creates (or at least, shows) the need for the mathematics?  Do students learn better when we do?  What do we want our students to gain?

Some of my recent posts might leave a person with a belief that I think mathematics should be highly contextualized.  However, I definitely do not think this is true.  My reasons deal with two discrete issues that connect within our classrooms — the impact of high context on learning, and identifying the goals (what we want students to gain).

The research on contextualized learning is not particularly strong at this point.  Certainly, the general researchers in learning & cognition conclude that context can actually interfere with learning; this is simply a corollary to the principle that learning is improved by making the target (the thing to be learned) as visible as possible.  Context can hide the big ideas.  For a good summary, see this article about cognitive psychology in mathematics http://act-r.psy.cmu.edu/papers/misapplied.html  — one of the best single sources I’ve ever seen.

A second component of the learning dimension is language and culture.  As soon as we present a context, we make demands of our students about other knowledge … sometimes fairly unrealistic.  One example I saw recently involves ‘shooting free throws’, rebounds, and ‘dunks’; another dealt with a baseball infield.  Sports are not uniformly followed by our students.  The same difficulty arises when we talk about projects around the house (whether it is sewing or woodworking).  These language and cultural factors affect both native speakers and those who learn English as a second language, and the problems cut across economic standing as well.  

The other dimension of my concern deals with our goals … what do we want our students to gain?  Some people bring in contextualized learning so that students can experience ‘doing mathematics’ like a mathematician does.  Other people use high-context because it motivates students.  To me, both of these goals are important … however, they are not the whole story.  A major goal of any math class should be to provide general tools that can be used, especially in future classes that the student needs to take, and this suggests a need to be able to transfer learning to other situations.  This transfer is inhibited when a learner has not practiced a skill or process repeatedly; meeting this threshold is very difficult if we contextualize most problems.  See http://jackrotman.devmathrevival.net/sabbatical2006/3%20Life%20in%20The%20Grey%20Zone.pdf, which deals with ‘how much practice is enough’.

I am especially concerned about preparing students to cope with the quantitative needs in their science and technology classes.  These needs vary from the very specific context to quite broad conceptualizations, and we seldom know which mixture of needs a particular student will have.  Developmental mathematics needs to deal primarily with broad sets of needs.  I do not think we can limit the mathematics to that which there are contexts that the student will understand.

It would be simple to say that we need ‘balance’ in our curriculum, and this would be true.  However, we should talk about what students should gain.  Some of their future classes are actually after some very specific skills (such as equivalence of different forms, or dimensional analysis); others are general … almost theoretical (such as behavior of types of functions for biology).  For a particular college, the needs might shift the ‘balance’ more strongly in one direction or the other.

The New Life reference material (at http://dm-live.wikispaces.com/) was developed to summarize many of these needs.  I encourage you to look at the sound mathematics described, most of which can be well served by combining learning about the process along with dealing with various contexts.  Needs exist for which contexts may not exist; some needs deal with theory, where context is a temporary step off the trail.

Yes, there are “what works” studies that conclude a high-context approach improves math classes.  These are not proofs of a result.  Like other ‘what works’ studies, there are many factors involved and only a few measured for inclusion in analysis.  From a learning standpoint, the best to be expected is “no significant difference” with high context … and this would reflect a great deal of effort to avoid the known difficulties with high context.

We are preparing students for success, and their success involves a multitude of needs.  Our math classes should focus primarily on general concepts, with a limited role for contextualized learning.

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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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