The Mathematics conundrum...

As I write the Seattle School Board is debating on how they will address mathematics in their schools for the coming year.  The television news have reported on it and I suspect if I check we would find that the newspaper and the blogs have also made comments as to what is happening.  But before you think you are on top of all this, let me explain that this problem of how to present mathematics to our students has been around since slightly after world war II.  And we still haven't solved the issue.

I hope you remembered my simplistic explanation of the three philosophies of education some blogs back, the one involving circles.  Let me explain the math problem from that point of view.  There are two basic methods in teaching mathematic.  One is from top down--the idealistic method.  You start by explaining what numbers are and how to manipulate those numbers.  You work from the simple to the complex, from adding of small numbers to the big bang theory and quantum physics.  It is primarily a rote method of instruction. 

The other method (note that I did not say the second method as that might imply the 1st being better then the 2nd--see how numbers can be viewed?) is the realistic method of teaching where the students have a problem to solve and use numbers in a way to "discover" or "own" the solution to the problem.  They may actually use something else besides the numbering system we now use but in essence they then understanding "numbers" as well as the solution to their problem.  

So what we have then is the "rote" method vs. the "ownership" method.  Which is best?  In spite of studying the mathematics curriculum problem for most of forty years, I still haven't the foggiest idea which method is right.  And if you think "politics" or "religion" are super sensitive subjects, try asking the Mathematics Department at your university as to which is best and watch the fur fly.  A hot question to ask any math prof is...."How much mathematics does a K-12 teacher need to know to be able to teach?"  And....."how should they teach mathematics?"  This is one of the more perplexing problems of the educational system.

I have a sense after teaching all these years that the answer to this question of learning mathematics is in our genes.  Some children seem to understand the abstraction of numbers and enjoy manipulating them.  Other children see no fascination in arithmetic and are bored with the subject.  It seems to last throughout one's lifetime.  But I have no proof (pun intended).  

I wonder what the Seattle School Board decided?  [blogger's update]  Seattle decided last night to forego the discovery approach to mathematics and by a 4 to 3 vote picked a more top down approach.  Many reasons for the change, many people unhappy.  You can read Seattle P.I. article about it at:  http://www.seattlepi.com/local/405961_math07.html  [May 7, 2009]

Do you know how to count and make change?  Remember to thank a teacher for that skill.

How To Look at What Teachers Do

Before we talk more about teachers and teaching we need to have a quick overview of teaching philosophy.  No, this won't be a long dissertation debating what words mean or how the world is perceived.  I love educational philosophy but I'm not good at it--sort of like being a poor amateur golfer.  I can barely stay in the fairways of thought.  We're going to look at three major philosophies of education--none being better then the others.  Consider them more like driving instructions on how to get from here to there.  Each of the three philosophies will get you there.

If you have a piece of paper nearby and a quarter and a penny I want you to circle around the quarter making a circle and then place the penny inside that circle and draw another circle in the center.  Got a doughnut?  Okay then, in the center circle write the capital "I".  Now from the outer circle draw arrows toward the inner circle.  Four or five will do for this illustration. Remember neatness counts--put your pencils down.  [I'm teasing]

Once you have this illustration completed you have a good representation of Idealism in Education.  The outside forces, i.e., adults are at work on the Individual, in this case the student.  The teacher tells the student what to learn and how to learn it.  A good example of an Idealistic school system is the Catholic schools in which the Pope tells the Cardinals who tell the Bishops who...... and finally it gets to the teachers and nuns who tell the student what to learn.  A good example of the curriculum are the Great Books.  Someone in charge has figured out what is important to learn.  The objectives or Truth are chosen by someone higher up.

In the early days of learning it was a simple curriculum--how to read, write and do simple math.  Health, science, geography had yet to be invented.  In today's society the state (society) mandates a minimum curriculum to the school districts by way of the WASL or NCLB tests.  Knowledge that we can count on, so to speak.  

So now, draw the same circles once again with the penny circle inside the quarter circle.  Good. This time draw arrows going from the inner circle to the outside circle.  Then place an "I", representing the individual, in the center circle.  You now have a good illustration of Realism in Education.  Here the individual decides what they want to learn and the objectives are chosen by the individual. Of course skilled teachers can guide the learning.  Do you remember Mrs. Donnelly who asked her first grade class what did they want to tell the person that was walking on their blackboards?  She was letting them decide and they were deciding they wanted to communicate, hence, they had to learn how to read and write.

There are many fine secondary teachers who use this philosophy.  There is a private school, Lakeside School, in Seattle that uses this approach to learning.  By buying a used, refurbished IBM 360 computer years ago, two of their students learned how to program it, play with it, and then they started Microsoft.  Well, maybe a few years later....but you get my point.

The world is their curriculum.  What interests the student?  How should they proceed?  Truth is the world

Once again into the fray--draw those two circles once again.  And again but an "I" in the inner circle.  Good.  This time have three arrows going from the outer circle to the inner circle and three going from the inner circle to the outer circle.  In essence, we have arrows of intention going both ways.  In the Pragmatism of Education philosophy we have the student deciding upon objectives and the outside world acts upon those objectives.  Change is the curriculum for it is always different.  One of the more difficult parts of Pragmatism is the question of "which knowledge is of most worth?" What should we learn?  A good example of a pragmatic learning system may well be the Boeings Learning Center which some employees learn today how to do something on a plane and then in a few months learn something different for a different plane.  Truth is validated by if what you learn works.

Again I reiterate that all three philosophies are viable, useable in the classroom.  Neither one nor the other is preferred although some school districts will favor one of the philosophies.  In recent years given the "No Child Left Behind" tests, districts have favored the idealistic philosophy to some degree as it was convenient to set the curriculum to fit the tests--teach to the tests.  

In the coming weeks I will be writing about more teachers I have known and will try to point out their philosophy as it relates to their teaching style.  However, not all that goes on in the classroom can be attributed to philosophy--sometime it is just chaotic pure and simple.