"I now view math more as an art..."

At the beginning of the term of a course for future elementary school teachers, I ask my students to write a math autobiography.  Normally this is done in the first course in the sequence for future elementary teachers.  This time I have done it in the third course.  One of my colleagues, who taught the first two courses via guided inquiry is on leave this term.

As I read through the math autobiographies, I can see those who have had negative experiences noticing a change of heart.  The seeds of hope have been planted, and they see themselves liking math more than they used to.

Most of the negative attitudes about math appear to develop or at least surface somewhere between upper elementary school and HS Geometry.  This is a pattern in the biographies, so I won't try to explain it. I'll just report it.  Why this is the case is not the point of this post.  What I'd like highlight is one student's poignant statement:

I now view math more as an a art because of the diversity in how to get one answer.  Growing up I was only taught one way to solve a problem, so I did not even think about the other methods that may have been easier for me personally.  It is so important to teach children today that there is not always one correct answer and not one method to get to that answer.

Beautiful!

Let us focus on what we can do for students in our classes, and let us be reminded that students have the capacity to change, if given the opportunity to do better.  There is a way to make it happen!

AIBL Small Grants Program: (For Faculty at U.S. Institution Only)

The Academy of Inquiry Based Learning has small grants for mathematics faculty working at U.S. institutions.  Click on this Link for descriptions of the categories, eligibility, and application process.  The deadline for applications is November 8th.

Video: Ken Robinson "Do Schools Kill Creativity?" (2006)

When I start a new academic year, I like to revisit sources that help me re-center as I plan.  It's like being an athlete and doing drills and base training to get the body ready for an upcoming season.  I revisit why we teach as a check to keep me from straying from the real reasons why we love our jobs as teachers.  We are also privileged as math experts to have the capacity to see the wonder and beauty that lies far beneath what nearly everyone else considers mathematics.

What we are presented with every academic year are opportunities.  Opportunities to use Mathematics (or whatever subject you teach) as a vehicle to provide transformative experiences for students.  We very easily can get bogged down in the details of covering chapters 1-8, making sure all the t's are crossed and techniques are sharply executed.  But what we are really after is, if you think for a moment, is helping students become powerful, creative learners.   

Ten or fifteen years later, what do you want your students to have retained from your class?  My guess is that knowing how to compute $\frac{d}{dx}\arctan(x)$ isn't highest on the list.

Nuts and Bolts: Homework Templates

One big idea I learned in college English and Literature courses was the notion of writing and rewriting drafts.  Constructing an essay is a process -- you read, you think of some idea you want to argue for, you construct an outline, you write a draft, and then you rewrite and reprocess until a finished product (or the deadline) arrives.

Of course you learn this in any subject.  To master any discipline require a long, long process of thinking, working, reworking,... Discipline and focus lead over time to an increase in skill and understanding.

All too often in math courses, the content is broken down into bite size pieces.  That is, the material is overly preprocessed for "easy" consumption.  Students follow.  Teachers get high marks.  Life is good.  But we know that's not good enough.  Certainly this doesn't develop the kinds of work ethic, habits of mind, and problem-solving ability we value.

One issue that needs to be dealt with is the specific process students use in their daily practice.  Do they just try something and either (a) get it or (b) give up after getting stuck?  This is probably the case for far too many students.  Poor process and practice leads to poor results, and diminished intellectual develop in the long run.   To shift the nature of the practice is not an easy task, and what I propose is not *the* solution -- it's just a start to one.

How does it work?
In my classes, I require students to use a homework template.  It's just a word document with the name of the course at the top and two little prompts.  The first is the statement of the problem or task.  The second is the solution/work/whatever the student has.   Students are instructed to do scratch work on separate paper, and most importantly analyze their work and transfer it to a final version onto the template.  One problem per page, unless the problems are really short.  Students are given the option of including their scratch work to the problem.

This system forces the drafting process that is much more explicit in the humanities and is easy to pull off in math classes.

A sample Template

Does it work? Yes.  The quality of student work improves considerably.  Instead of getting scratch-work quality written work, students are now required to take their homework more seriously.  Students also comment that they feel proud of their work, and why not?  If you think of an original idea, and present it well, that's something you should be proud of.  The homework has good content, it looks good, feel professional,... a job well done!

Does it solve all problems in the math process "pipeline"?  No. Templates and requirements of drafting do not address all issues.  Teaching is a complex system, and addressing students' process of doing homework is but one part of the picture (though a very big one).  It is, however, a very welcome step in the right direction.

Farewell, Lecture?

Eric Mazur, Harvard University, is a well known physicist, who has championed active learning teaching methods in Physic.

Here's the abstract.

This article presents a method for teaching in large introductory required courses that is different from the traditional lecturing. The responsibility for gathering information now rests on the shoulders of the students. Class time is devoted to discussions, peer interactions, and time to assimilate and think. Instead of teaching by telling, we use teaching by questioning students. Research shows that learning gains nearly triple with this approach. Students not only perform better on a variety of conceptual assessments, but also improve their traditional problem- solving skills.

 http://www.sciencemag.org/content/323/5910/50.full

Eric Mazur's talk "Confessions of a Converted Lecturer" is highly compelling.

Mistakes Are Good!

One of the messages in an IBL that goes along with "Being stuck is okay!" is "Mistakes are Good!"

Mistakes are generally stigmatized in U.S. Math Education.  When a students does something wrong, it is unusual if the student thinks of the mistake or error as an opportunity for greater, perhaps even profound insight.

There exists many reasons why we build prototypes or practice in a batting cage or simply use scratch paper.  We need to see how things work.  We need to practice and fail, so that we can learn to do what is right.  In short, practice and experimentation.

One cannot grow without experimenting or trying things.  It would be nice if our students could all have the disposition to say things like "Let's see if the idea works for a special case..." or "Let's see if we can check our thinking..."

If students fundamentally believe that mistakes are bad, then the very nature of their interaction with mathematics is limited.  Over time this leads to poor self image and then ultimately poor habits of mind and work ethic.  That would be the nail in the coffin.

One of the ways to get students over the negative image of making mistakes is to provide opportunities for students to experiment, and to allow for mistakes to play a central role in the learning process.  In fact, in an IBL class students make *great* mistakes.  They say or do things in ways that maximize their learning.  As an instructor I no longer make these mistakes, because I already know the material.  First time learners of a subject reveal, through their mistakes, what they know and what they don't know.  This is where the learning zone is, and this is where one can create magical learning experience!

Student: <Writes or says something that is incorrect>
Teacher: "Oh, did you just say/write... Well I'm really glad you brought this up.  How many of your were thinking about this the same way?  Good!  Let's rewrite this as a question, and then investigate it further to get to the bottom of this."

A Good, But Not Good Enough Idea

In the NYTimes, Sol Garfunkel and David Mumford published a piece titled, How to Fix Our Math Education.  In this column, Garfunkel and Mumford claim that the curriculum should be changed to include more realistic and practical situations that are meaningful to students.  This I agree with 100%.

The problem with their piece is that the curriculum (AKA the specific content in courses) is only one of many issues.  Their analysis is limited, and here's why.

There are numerous examples from the K-12 system, where schools have adopted a research-based, NSF funded curriculum like IMP or CPM.  Many times these programs are shut down by some exogenous force.  For example parents, who have good intentions in their hearts have organized and shut programs down. They say things like, "Well that's not how I learned it...," never mind that the world has changed and students are being prepared for new challenges, not challenges from the 19th century.   Or a new superintendent wipes out the programs, or a new principal comes in and moves teachers to new classes or grade levels, or budget cuts,..., the list could go on and on.  I note that none of these people do this intentionally.  People in education and parents are well-intentioned.  They mean well.  It's just that we don't know better as a society, and look for simple fixes for long-term, multi-layered problems.  Insufficient.

The problems we face now are complex, and my sense is that they are on the order of magnitude to problems related to ecosystem sustainability.  Our educational system is vast and complex, with outdated doctrines interfacing with modern challenges, and a currently fashionable and misguided desire to apply business models to non-business systems.  So when someone says, "Just fix X, and all will be good," I know that it just won't.

Content (i.e. textbooks or materials) is just one small slice of the education pie.   In addition to content is

• Instruction (and all this huge category entails)
• Student attitudes, beliefs, habits of mind, experience with math, and their natural cognitive development mathematically.  (Example: the developmentally appropriate order of topics is sometimes different than the logical order in a subject.)
• External issues to the classroom like tenure and promotion for research (but not teaching), pacing plans, poverty, etc.
• Systemic issues like standardized testing (K-12) or standardized courses (calculus)
• The factory model mindset held by most people regarding education.  We value certain kinds of intelligence.  Math and reading are at the top....Art, Dance, Music, craft, and manual skills are at the bottom, which leaves some students feeling marginalized for being born with the "wrong" abilities.    
• We educate students in disciplines in ways such that they are not connected to each other, even though they often are connected and of more value when studied as a whole.
• Community and culture.  Most people think math is "2+2 = 4."  That there is only one answer.  That there is only one way to get to that singleton answer.  That math is immutable and the same across all time.  Learning math means you know how to calculate.  Additionally, Parents, even well educated ones, can destroy a positive change, thinking that they are doing a good thing.  I do not doubt parents' love of their children or desire for good education. It's just that we as a society are generally naive about what math is.  What we assume to be true about math is actually false.  So based on false assumptions we act unwittingly against our own interests. Mathematicians bear some of this responsibility, as we are not good ambassadors of our wonderful subject. 

Viewed even from this slightly broader perspective, changing the content without thinking about the education ecosystem is limited.

So how does IBL come into play?  At the college-level, a math instructor, who uses IBL, can control enough of the ecosystem within the classroom to construct a little, temporary greenhouse.  This little greenhouse has enough of the conditions necessary to foster intellectual growth and provide opportunities for students to undertake the effort required for transformative changes.  Considering how to scale these little greenhouses up is one way to realize the enormity of the challenge in education reform, and emphasizes that changing a textbook or the curriculum is like upgrading the rake or the shovel to a new, more effective one.  While it's a necessary step in the right direction, it is insufficient.