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.

Monday, August 20, 2012

Learn by Making Mistakes: Diana Laufenberg

Diana Laufenberg is a high school teacher.  Her insights into learning are straight to the point. Let students explore and make mistakes.  Great stuff in 10 minutes!

Here's Diana Laufenberg's TED talk.


Friday, August 10, 2012

AIBL's Mission

The purpose of this post is to outline the AIBL mission.

The overarching AIBL mission is to support and sustain a growing community of IBL instructors at a national level.  AIBL supports more than just particular IBL courses or a narrowly defined mode of instruction.  AIBL supports individual instructors at all levels of experience.  Indeed, transforming math education will not be achieved without a strong, community working together towards common goals. 

Community: Reform based primarily on new tests or books, without considering larger goals (e.g. effective thinking and advanced problem solving) and environment (e.g. learning culture) are good, necessary steps, but not sufficient.  AIBL's mission, to develop an IBL community, addresses issues of sustainability and continuous professional growth.  Knowing about something is a far cry from being able to do it.  In professions as complex and perhaps daunting as education, a community of support is essential.

Big Tent Philosophy:  AIBL is founded upon the principle of inclusiveness.  There exists a lingering belief that instruction falls into two, distinct categories.  Lecture is category 1, and the Moore Method is category 2.  This is incomplete, and AIBL supports instructors interested in a wide range of inquiry-based methods.  While it is our belief that full IBL courses have the most potential for transformative experiences for students, it is understood that environment, instructor experiences, student body, and other factors affect teaching decisions.  We understand from experience that instructor change is a process that takes time, and encourage instructors to take a responsible, long-term approach to changes in practices.  Go your pace, while at the same time don't wait too long (for your students' sake).  Whatever the situation, instructors who aspire to involve their students in rich mathematical tasks and allow ample opportunities for student collaboration (broadly defined), are welcome and encouraged to participate in AIBL activities.

AIBL is here for the long haul. AIBL is here to help!  Get involved, and get your students doing Mathematics!

Tuesday, August 7, 2012

Marketing IBL to Your Students

One of the common issues new IBL instructors face is student buy-in.  This is a real concern for all instructors.  In this post, I outline the issue broadly, and then provide some tips for how to ensure that your students get with the program in their minds and in their hearts.

First, let's talk about the issue broadly.  One important thing to remember is that Math is culture.  There exists default expectations about what a math class is and the roles for students and instructors.  These default expectations are often unconscious -- we don't think about them.  When you meet someone new or talk with your boss, you probably do not realize all of unconscious things you do (or don't do) in these interactions.  Likewise, in a math class students have certain expectations that are almost always aligned to traditional instruction.  Students expect instructors to show, and their job is to follow dutifully and write down notes and perform these tasks on exams.

IBL classes are aligned differently, of course.  Students are asked to solve problems they do not know the answers to, to take risks, to make mistakes, and to engage in "fruitful struggle."  These are all very different from normal expectations (as of today -- hopefully that will change).

Tools for making sure your students are on board are
  1. clearly defining students' role in the class
  2. providing a clear rationale for IBL (regularly)
  3. creating a safe and successful classroom environment.
Students need to know what their job is in an IBL class, and it is the instructor's job to make this clear.  Students must know what they are supposed to do (solve problems, write math proofs/solutions, communicate effectively,...).  Students need to know the instructor's role (provide appropriate tasks, coaching, mentoring, adjusting the challenges as needed, moderating discussions,...)

Why IBL?  Well there are lots of reasons.  Research shows it's better for students.  We are now in the era where information about anything is available on your cell phone, and one can run Wolfram Alpha on a cell phone, too!  In other words, all lower-order thinking levels (as per Bloom's Taxonomy) are now nearly worthless due to advances in technology.  Effective thinking is now where it's at.  Thus, IBL is the way forward for students.  This line of reasoning addresses items 1 and 2 above.  What about 3?

The heart is the heart of the matter.

Telling people, "Medicine is good for you!" isn't sufficient.  Students need to know that the instructor is their advocate for learning.  Students need to see themselves as successful mathematicians (where they may never have thought this before in their lives).  Thus it is important for students to struggle, but struggle within reason.  It is suggested that IBL units start off at a basic level, where all students in the class can achieve some success.  Then the problems should ramp up in difficulty as appropriate for your students.  When in doubt, include more easy problems than less, especially at the beginning of the course and at the beginning of new material.  The worst case scenario is that you spend a few extra minutes on them or just skip them entirely in class, and assign them as homework.  There is no cost to including more problems.

In this sense, establishing a safe and successful classroom environment is asymmetrical.  Erring on the side of being "tougher" is fraught with perils.  First, you are going again previous experiences and the traditional classroom culture.  Second, many students have negative attitudes about math.  Third, telling students that are stuck to "just keep going" can lead to the perception that the instructor is not helpful, and thus not teaching.  Struggle is good if the students feel that the struggle leads somewhere.  This type of scenario is often the case for students writing negative comments on course evaluations.  Students may in their minds understand that IBL is good for them, but they experienced too much frustration to truly enjoy the experience in their hearts.  In other words, the experience was not an aesthetic experience.

I also note that saying that you told them is not enough.  You need to know if the students feel it in their hearts.  Look at them and see if they are enjoying the math and interacting positively.

Of course, there is good reason for students to struggle and perhaps not solve a problem.  Such experiences are fruitful on many, many levels of learning.  BUT this is something that should happen down the road, once students are off and running, enjoying math and doing math successfully in a positive and supportive learning environment.  Training for a marathon has similarities to teaching an IBL course.  You don't coach a new runner with hard interval training on day one followed by long 20-mile tempo runs.   Athletes train by exerting an appropriate training load and recovering.  Then they repeat and then move on to new things gradually.  Math is no different.  Tasks should match students' experiences and abilities and grow with them.

If you are not positive, how can your students be positive?  If you never smile, why would your students smile back at you? Be positive!  It's important to let your students know that they are working hard and progressing.  I thank my students for their participation, and I try as best I can to make classes a supportive environment.  Pointing out the good parts of solutions, ideas, and efforts should be a part of daily practice.

Start easy. Establish the learning culture from day 1.  Build on positive class experiences to challenge students to do more and more.

General IBL Points
Some points you can use as a base for discussing IBL classes with your students.  This is a list of talking points to help you find your own way of conveying the message that IBL is good for the mind.

  1. IBL is a student-centered method of teaching similar to the Socratic method. It requires more work for me (the instructor), but it's better for you.  Research shows that students who are actively engaged learn better.  While you may not be used to it, I'll do my best to make sure you are comfortable with it and will be successful in this class.
  2. One goal of IBL is to help students learn to think independently, and become a successful problem solver.  In other words, a goal of IBL is to help you get better at thinking effectively.  That's really what we will work on.  And you can't learn to think effectively, if someone does all the thinking for you...
  3. IBL emphasizes the process of problem solving and theorem proving rather than the memorization of facts.
  4. IBL is not experimental.  It has been employed successfully since the days of Socrates.
  5. The reason why books and other outside resources are not allowed is because we will discover the ideas ourselves.  We will collectively work on the tasks and come up with our own ideas.
  6. It’s OK to be stuck.  Being stuck is a noble state of mind.  It means you are just about to learn something new!
  7. It’s OK to be frustrated.  You’re doing fine -- try to slow down and enjoy the process.  We'll get it eventually.
  8. Being stuck is natural.  Whatever you do, don’t give up.  If you're stuck, there has to be question in there that you can ask.  
  9. What's the best way to learn to play the piano?  Should you just watch videos of pianists?  Do you need to do something else besides watch someone else play?
Teaching is more than content delivery.  Addressing the learning challenges as part of the course is a good thing to do, and acknowledging where students are coming from and building a bridge for them to cross is a core component of effective IBL teaching.