ESCAPING THE TRACTOR BEAM

Encasing Math Education for the Last Century

Michael Round

May 11, 2009

The Star Ship Enterprise is caught in an alien tractor beam - a space "grappling hook", if you will - preventing it's escape.  Captain Kirk orders all engines full reverse, and applies a shearing maneuver to break free of the force.

The effect not only frees the Enterprise, but leaves the alien ship powerless.

The metaphor comes to mind when I think of math education in the 20th century in the United States - now nearly a decade into the 21st century.

We're caught in a similar tractor beam.

As a simple test, ask 100 high-school students, randomly, about the following two problems:

To the half getting the problems right, ask "Are you sure?" and the percentage responding in the affirmative will drop like a thermometer in December. 

For that matter, ask adults the same questions!  Likely, you'll hear 3/4 of them proudly announce their disdain for math, instead of providing right answers!

Santayana warned us in “Life of Reason”, “Those who cannot remember the past are condemned to repeat it.” The use of this phrase is only exceeded by the number of times it sadly rings true. How do we explain this contradiction? What does it mean to “remember the past”, and how can we do so to ensure we learn from our mistakes – to not repeat the past?”

Mathematical education falls into this category.

The New Math, implemented in the 1960s, had noble origins and good intentions. The 19th and early-20th centuries had seen the erosion of “certainty” from the once-solid foundations of math. Euclidean geometry, the calculus, and set theory were under assault at the ground level for inconsistencies, contradictions, and non-intuitive results. As a result, a paradigm shift in mathematics took place, the goal: shore up the foundations of math.

When the Soviet Sputnik created the USA mathematical / scientific panic igniting the New Math movement, the above curriculum paradigm shift was put into action. The strategy and tactic? In order to achieve good mathematical results, we must start our kids on the road to certainty at an early age.

Of course, it did not take long to realize we were not achieving that noble goal with that tactic, and it also did not take long to realize why not! There was a crucial assumption that had not been considered: the pedagogy is sound.

We quickly realized the flaw in the “new-math” implementation plan: our kids were not ready for the material. It was no surprise. It had taken thousands of years to reveal the inconsistencies, contradictions, and non-intuitive results in math, we sought to restructure the math, and we did not give a moment’s thought to whether this was good – for kids? We knew, having practiced it in academia for ½ century it was good for adults, and assumed what’s good for adults is also good for children. This was the underlying assumption in judging the pedagogy of the New Math.

As I said, it did not take long to realize the error of the plan: as adults laboring through years of various mathematical disciplines, we saw the need for new fundamentals. Coupled with the assumption “what’s good for adults is good for children”, we inferred a solid pedagogy, and marched ahead. The results should have been anticipated. Confusion. Lack of joy. No increase in performance.

For Goodness Sakes

But we fixed all this, didn't we?  The New Math?  Are you kidding me!  This was when I was a kid, for goodness sakes.  We've had revision - after revision - after revision - of math curriculums.  We were "A Nation at Risk" under the Reagan Administration, moving through the decades to "No Child Left Behind" in the 21st century.

This should give us a hint something's not right in the mathematical world.

And right now are in the process of repeating the same mistakes again.  Consider the teaching of statistics, data, and probability. The parallels are ominous, and the conceptual framework identical.

The Fractal World of Repetition

We live in a data-rich world, bombarded with numbers, statistics, data, information, etc., on a continual basis. The curriculum of the past is not sufficient for the new-millennium worker, we are told. As adults, we see the crucial role statistical inference, deduction, data understanding, etc. And “what’s good for adults is good for children”, right? Again, we jump straight to the assumed conclusion “the pedagogy is sound”. Do you see the ominous parallels? Differing only in specifics but conceptually equal, the fractal nature of the argument is readily apparent!

And the urgent plea from the professional community?

“Our K-2 students need to: collect data and discuss meaning; picture data to summarize meaning; design data to solve problems; and learn that inference from data to conclusions is as important as deduction from theory to conclusions” OUR K-2 STUDENTS? IS THIS SOUND PEDAGOGY?

“To be better educated, our 3-5 students need to: describe location and spread of data to begin, nut not end, statistics as it does now; use graphs and tables to serve as simple models; analyze data relationships to serve as more valuable models, employ some prealgebraic thinking to understand the variables in data; and apply both deduction and inference, as well as compare and contrast them.” OUR 3-5 STUDENTS? IS THIS SOUND PEDAGOGY?

“To be better educated, our 6-8 students need to: know that viewing data relationships as a function plus an uncertain error adds both meaning and usefulness; understand that data that appear to almost follow a line help give meaning to mathematics of the line; describe the line’s analytic geometry from Descartes (historical context adds life); find the best linear data relationship by Gauss and Legendre least squares, and discuss correlations of Galton and Pearson.” OUR 6-8 STUDENTS? IS THIS SOUND PEDAGOGY?

“To be better educated, our 9-12 students need to: recast least squares as estimating coefficients for best line; know that the frequency distribution of its residuals leads to the introduction of probability distributions; and understand that probability distributions form the basis for constructing confidence intervals for estimates, and for accepting or rejecting hypotheses.” OUR 9-12 STUDENTS? IS THIS SOUND PEDAGOGY?

All of the above was initially printed in the California ComMuniCator, Volume 28. No 4, and reprinted in Missouri Council of Teachers of Mathematics, February 2006.

And what do we see in reality: how do we implement such material in the K-12 environment? For example: we see median is a necessary measure – in certain contexts – as adults, and teach median to young children. Calculate the median of the following set of numbers:

44, 48, 52, 30, 60, 12, 14

Of course, since there is no use for median at this age, we need to remind the young child to put the numbers in ascending order.

12, 14, 30, 44, 48, 52, 60

And supposing the child answers correctly, what have they gained?

We see “box-and-whiskers” plots as helpful in understanding complex tables of data – as adults – and ask the young student to similarly create the plot. What’s the nature of the data? Why is a box-and-whiskers plot effective in this context? No answers are given. We only know “as adults it’s helpful”, and “what’s good for adults is good for children”. More box-and-whiskers plots! More stem-and-leaf plots! More linear regression! Why? Because as adults we see they’re important, and “what’s good for adults is good for children!”

As adults, we see graphing and pictorial representation as helpful in understand data, and assuming “what’s good for adults is good for children”, assign graphing lessons to students. Of course, graphing data assumes one has data, and if the child has no meaningful data to graph, is provided “meaningful” tables. What does this mean? M&M’s, hair color, and teeth lost? And to ensure interest is not lost, we make these graphs even less meaningful by making them extremely colorful! Where is the pedagogy in any of this?

Observe the noble intentions: Children should learn “statistical thinking”. Who would argue with the statement’s alternative? Should we not teach statistical thinking? The question then becomes: what constitutes “statistical thinking”, and herein we see the ominous parallels with the New Math. Professional educators, academics, etc., see “what’s good for adults”, and conclude, once again, “what’s good for adults is good for children”.

And the results, like the New Math movement, lead to a curriculum without a solid link to reality. To make the curriculum relevant, the applications are either too difficult – and therefore meaningless, or very trivial – and again, meaningless. The huge learning gap in the middle where real learning can take place is left unattended.

The Fractal Nature of Mathematical History

“The repetition of a similar structure in a different context on a different scale” - With noble origins and good intentions …

HOW CAN THIS BE?  A LOOK AT PARADIGMS AND PARADIGM SHIFTS

Consider a math classroom.  20 kids, all different.  Different abilities, different interests.  You're the teacher.  A daunting task!  What do you do?  What do you teach?  You teach - what you've been taught to teach - what you're told to teach.  This appeals to some kids.  Some kids love it.  Others don't.  OK - let's be honest.  MANY don't!  But some advance.

Of all of the original 20 students, who's likely to become a teacher?  Any who don't like math?  Of course not.  They've long ago dropped from the math scene, instead joining the growing ranks of people proudly proclaiming their dislike for the subject.  What about all of those who do like the subject?  How many of them will go on to become a teacher?  A fraction?  Fine.

What does this "fraction" do?  They go back into the classroom.  What do they teach?  They teach - what they've been taught to teach - what you're told to teach!

Are you recognizing a pattern here ...

But you, as the teacher, know something's not right.  You see many students "not getting it".  You probably see, by now, your OWN children struggling!  You know something is wrong.  What can you do?  You can change things!

After all, you're a member of the National Council of Teachers of Mathematics.  They SET the standards!  They help write the tests!  You can make things better!

What are you going to do to revise the curriculum?  You're with other similar-minded people, and you see something's not right.

And the circular argument only intensifies!  Teachers taught a certain way help redesign the new system!  What can they do differently?  How will things be different?  How do they "break out of the box"?  Evidence suggests they've been unable to do it, given the history of math education in the United States!

The future seems ominous!  In this system, it IS ominous!  Can we break free of the educational tractor beam?  Is there a shearing force we can apply?

Stay tuned!