Tuesday, August 28, 2012

Response to "Is Algebra Necessary?"

A month ago, Andrew Hacker published an Op-Ed piece in the NY Times, called "Is Algebra Necessary?"

Here’s the short version of my response:  Yes, Algebra and algebraic reasoning is necessary.  But, the existence of people who write pieces like “Is Algebra Necessary?” is in actuality a symptom of the larger failing of our educational system. 

Many of my colleagues scoffed at Hacker's piece.  It is sad indeed to see this printed in the NY Times.  But that made me think of the question, "How did we get here?"  One thing to note is that lots of different issues are mingled in Hacker's piece and the subsequent responses.

Hacker’s piece is as if it was like something out of the past held in a time capsule.  If he has been reading Math Ed articles during the last couple of decades, it would have been clear that changes were needed in instruction, curriculum, and learning cultures, and that there were innovations and implementation efforts underway.

We should view Hacker as [Hacker].  That is, [Hacker] is an equivalence class of people with his somewhat naive views and opinions about Math Education.  The existence of “Is Algebra Necessary?” could be a symptom of larger failings or shortcomings in the way we teach and learn Mathematics in the U.S.  Further, mathematicians have not done a good job of communicating what math really is.

Let’s take a step back and look at the situation broadly.  The feelings of is dissatisfaction with math education in the United States (in this case Algebra) is unfortunately widespread and has been prevalent for a long time.  There exists deep seeded nonavailing attitudes and beliefs about math.  A nonavailing belief or attitude is one that does not support or inhibits the learning of mathematics.

One of the unintended consequences of traditional math educational is the cultivation of nonavailing beliefs about math.  Krista Muis, Simon Frasier University (Link) conducted a meta-study, which resulted in collecting a list of nonavailing beliefs, including the 15 selected below.  This data is unwelcome indeed.

  1. Memorizing facts and formulas and practicing procedures are sufficient to learn mathematics.
  2. Mathematics textbook problems can only be solved using the methods described in the textbook.
  3. Teachers and textbooks are the mathematical authorities.
  4. School mathematics is driven by rules and memorization, and is driven by procedures rather than concepts.
  5. If a problem takes longer than 5-10 minutes, then there is something wrong with the student or the problem.
  6. The goal of mathematics is to obtain one correct answer and do it quickly.
  7. The teacher is the only source of determining whether an answer is correct or incorrect.
  8. Students’ role in the classroom is to receive knowledge by paying attention in class and to demonstrate it has been received by producing right answers.
  9. The teacher’s role is to transmit knowledge and verify that it has been transmitted.
  10. Only geniuses have what it takes to be good at mathematics.
  11. Students prefer to have only one way of solving a problem, because it is less to memorize.
  12. The processes of formal mathematics have little or nothing to do with discovery or invention.
  13. Students who understand mathematics can solve assigned problems in 5 minutes or less.
  14. One succeeds in school mathematics by performing the tasks, to the letter, as described by the teacher.
  15. The various components of mathematics are unrelated.

Nonavailing beliefs affect perceptions of what mathematics is, contributes to “I hated Math in school!” comments, and could lead policy makers to inappropriate conclusions.  Students who believe that math should not be understood, but merely memorized engage in the study of mathematics in a completely different way than a mathematician does.  This massive gap explains to some degree the differences between those who can and those who can't.

Curiously, some of the major problems we face as a society have the commonality that they are complex, long-term, and gradual.  Understanding something as complex as the educational system in the U.S., with all its intricacies, failures, successes, and fluidity is a big task.  It is difficult, even for well educated people.  A fundamental difficulty when evaluating education is that it is extremely hard to know where one’s knowledge ends and one’s ignorance begins.  We’ve all been to school.  We know what it’s like, and so of course we know how it should be done.  I mean, I've been to elementary school, right, so I should know how it ought to work.  Of course, we wouldn't say that about going to the hospital or dentist.  Ultimately this leads to implicit, perhaps unconscious oversimplification of the problem.

Technical knowledge in education is far beyond the general public.  Most people are blind to many key issues in K-12 math, which are very technical in nature.  How technical?  Here’s an example from elementary school:  What is the difference between quotative and partitive division, and which model is more appropriate for developing and understanding of division of fractions?  This is an important issue to deal with in upper elementary math. Most people I know, however,  do not know what I am talking about if I ask them this question.  Teaching math is hard, technical stuff.  Designing student-centered, meaningful math curriculum and lesson plans is a big, complex task.  Yet, everyone has an opinion about how math should be taught and what policies should be put in place, without knowing major bodies of knowledge that are critical in the development of children’s mathematical thinking.

A main point here is that people who should not be making policy recommendations are doing just that.  As countries have technocrats working in central banks to guide monetary policy, countries should have "education technocrats," who can do the hard, technical work of guiding education policy, curriculum develop, etc.  Education is hard stuff, and it’s well beyond the knowledge and skill set of the general public, even the well-educated sector of the public.

Another issue is that teaching and learning, broadly speaking (and excluding the star teachers out there), is failing on some significant levels.  U.S. students perform poorly on international comparisons.  Our students generally don’t like math.  High school dropout rates are too high.  Many students are developing nonavailing beliefs, and they do poorly on problem solving, proof writing, reading or using the mathematical language, etc.  This makes [Hacker] revolt against mathematics education as constructed today.

“Is Algebra Necessary?” should then be viewed in part as an alert to mathematicians. The notion that the captain goes down with the ship seems appropriate here.  Mathematicians are the leaders, naturally, of mathematics, and we have a tremendously influential and important role in mathematics education.  Thus, whether you disagree with [Hacker] isn't really the point.  There exists a large, unhappy group of people who do not like math, primarily due to traditional paradigms of education.  Sure some members of this class are successful, but they missed the point of mathematics primarily due to how classes are/were taught.  And it is duly noted that the cardinality of [Hacker] is much larger than the cardinality of [Mathematicians].  To provide more emphasis to this point the theme that mathematics education in the U.S. is not up to par is very critically evident in the PCAST document that we released last month, where one of the messages is the frustration and disappointment of the math community in the slow uptake of modern, student-centered pedagogies.  We have been put on notice in more than one way recently by those outside of mathematics.  (See David Bressoud's notes on  PCAST HERE and the MAA response HERE.)

What can you (mathematician or math teacher) do specifically? The easiest way is to start using more empirically validated, student-centered teaching methods in your classroom right now, attend MAA and NCTM conferences that have sessions on innovations in teaching, and sign up for workshops that provide rich experiences for transforming your teaching.   Additionally you could work with your School of Ed in outreach programs to local K-12 schools or with a regional Math Teacher Circle group.

What else?  These types of blog posts are dancing around the issue of the point of school.  If you continue the line of reasoning, "Is Algebra Necessary?" to "What's the point of Algebra?" to eventually "What's the Point of School?", then you start to get somewhere.   (See Guy Claxton's book "What's the Point of School?" and Mike Starbird's talk from June 2012.)

Where do we stand?  We are in the Era of Implementation.  It is clear that we have plenty of innovations in the U.S. about effective teaching.  Some high performing school systems, such as the Finnish system (See Pasi Sahlberg), actually obtain much of their innovative techniques from American researchers.   Simultaneously we have is an educational system that doesn’t provide enough access and support to implement these teaching innovations.  Thus, the challenge we face as a profession is in widespread adoption of empirically validated teaching methods.  The question is whether we will rise to the challenge and do what is necessary to transform our system.

Game on.