math pedagogy

Can the Lesson Study Model work in the US? Look to US manufacturing for the answer.

Can the Lesson Study Model work in the US?   Look to US manufacturing for the answer.

There is a lot of interest in the US Education establishment about implementing Japanese style “lesson study models”. Consultants try to make it sound like it is a new idea. It may be new to US education, but it is not a new idea to the US. It is one manifestation of the continuous improvement quality movement in the Education arena. This quality movement requires three things: the desire to strive for continuous improvement, the willingness and ability of all members to work together to improve and a system for improvement.    In the 1950’s and 1960’s the Japanese were trying to rebuild their manufacturing infrastructure after WWII and the had a desire to get better and were willing to work together to do it, but they needed a system. They looked to the US for their inspiration. They found it in W.E. Deming, a statistician. Ironically, the US manufacturing establishment ignored his system of continuous improvement, but the Japanese embraced it and his system of 14 points permeated their society. The Japanese improved their quality so much that in the 1970’s Japanese automobile quality (which was a joke in the 1960’s) became the standard by which other goods were judged. At the same time, US manufacturing quality became a joke. The US manufacturing industry, after losing significant market share (e.g. in automobile manufacturing) to the Japanese, embraced Deming’s concepts in the 1980’s and 90’s. The quality of US goods is now equal to the rest of the world. There are many things that the education community can learn from Deming and US manufacturing as we help our students compete with other students from around the world.

For example, how did the US manufacturing industry make the difficult change?

  • They realized they did not have a choice. They were losing significant market share every year.
  • They realized that it was a systemic change that required all members of the organization to embrace Deming’s 14 points. This was not a “flavor of the month” program some consultant thought of. It was a way of being.
  • They realized it was a long term process that would take years if not decades.

Does Deming’s 14 points actually apply to Education? Yes. In fact many of the “new” ideas that are being rolled out in public education are just Deming concepts put into education lingo.

For example, it is becoming common thinking that high stakes end of the year testing is not useful and may be harmful. Deming’s points out that we should “Cease dependence on inspection to achieve quality. Eliminate the need for inspection on a mass basis by building quality into the product in the first place. Eliminate fear.” In education lingo Deming would encourage formative assessment and student feedback as part of the learning process and there would be no need for end of the year tests, when it is too late to affect student learning.

I have attempted to put Deming’s 14 points into Education lingo. Below in bold are my version of Deming’s 14 points. The non-bold words are Deming’s 14 points from the website: . I have grouped some of them together so that there are 10 points, not 14. As I wrote these, I thought of the principals and assistant principals as the supervisors and the teachers as the workers, but these points apply just as well to the teachers as supervisors and the kids as the workers.

1) It’s about the kids and getting them to learn and perform on par with the best students in the world. Everyone from the janitor to the principal should be focused on that.

  • Create constancy of purpose toward improvement of product and service, with the aim to become competitive and to stay in business, and to provide jobs.
  • Put everybody in the company to work to accomplish the transformation. The transformation is everybody’s job.
  1. Teaching (and leading) the way we have always taught is not working.
  • Adopt the new philosophy. We are in a new economic age. Western management must awaken to the challenge, must learn their responsibilities, and take on leadership for change.
  1. Standardized tests at the end of the year are ineffective because it is too late at that point. Frequent formative assessments are critical. Build the quality into each day’s teaching with many feedback loops.
  • Cease dependence on inspection to achieve quality. Eliminate the need for inspection on a mass basis by building quality into the product in the first place.
  1. There is no end to improvement. No more “this is the way I have always taught and it has worked for me.”
  • Improve constantly and forever the system of production and service, to improve quality and productivity, and thus constantly decrease costs.
  1. Teachers need to be learning and improving constantly and this is a group as well as individual process. Teachers need to collaborate and share best practices. Lesson study models and microteaching are effective ways to improve teaching.
  • Institute training on the job.
  • Institute a vigorous program of education and self-improvement.
  1. Administrators and department chairs need to move from evaluation once or twice a year for a ranking/grade of the teacher to many smaller less formal evaluations with the goal to help the teacher get better.
  • Remove barriers that rob people in management and in engineering of their right to pride of workmanship. This means, inter alia, abolishment of the annual or merit rating and of management by objective.
  • Institute leadership. The aim of supervision should be to help people and machines and gadgets to do a better job. Supervision of management is in need of overhaul, as well as supervision of production workers.
  • Drive out fear, so that everyone may work effectively for the company.
  1. Teachers need to talk to and work with each other. Physics needs to talk to math (e.g. math needs to teach vectors before Physics uses them). English needs to talk to History. Precalc teachers need to talk to Calc teachers.
  • Break down barriers between departments. People in research, design, sales, and production must work as a team, to foresee problems of production and in use that may be encountered with the product or service.
  1. Focus on the learning process and the kids, not standardized test scores. Do that and the test scores will take care of themselves. Take down those posters that say “your altitude is determined by your attitude”.   Teachers and administrators need to MODEL grit, continuous improvement, curiosity, collaboration, positive attitude etc…
  • Eliminate slogans, exhortations, and targets for the work force asking for zero defects and new levels of productivity. Such exhortations only create adversarial relationships, as the bulk of the causes of low quality and low productivity belong to the system and thus lie beyond the power of the work force.
  • Eliminate work standards (quotas) on the factory floor. Substitute leadership.
  • Eliminate management by objective. Eliminate management by numbers, numerical goals. Substitute leadership.
  • Remove barriers that rob the hourly worker of his right to pride of workmanship. The responsibility of supervisors must be changed from sheer numbers to quality.



Why True but Not Provable?

Kurt Godel is one of my heroes (along with Norman Borlaug – but that is for another blog entry).  In the early 1900’s Bertrand Russell and Hilbert were trying to prove that for certain number systems that all true statements could be proven by axioms WITHIN the system (completeness).  Tying up number systems with a nice neat, consistent, complete bow was very attractive to these mathematicians.  Unfortunately, Godel came along and showed that for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers that:

1.  If the system is consistent, it cannot be complete (If the system has only statements that can be proven, then there are others outside of the system that are true, but cannot be proven).

2.  The consistency of the axioms cannot be proven with the system.  (There may be external proofs that exist, but not ones within the system)

Some great explanations of the theorem and it’s philosophical ramifications are here:

There are many ramifications of Godel’s Incompleteness theory, but the one’s I particularly like are:

1)  As Douglas Hofstadter says “Provability is a weaker notion than truth”.  True but not provable!

2)  Led to the birth of computer science and showed Alan Turing the way.  The

“Godel-numbering” system took syntax and represented it by numbers, effectively showing the how to take 0’s and 1’s and represent logic (arithmetization of syntax).

3)  Established bounds on what is and what is not computable.  Does the “human mind infinitely surpass the powers of any finite machine?”  Godel believed that this was a highly likely conclusion based on his theorem.

Godel uses logic to play the beautiful music of mathematics and computer science by showing that Truth may not be Provable.



Do not teach your students to imitate math

Mathematician’s Delight by English Mathematician W.W. Sawyer (1911-2008)

“It would, I suppose, be quite possible to teach a deaf and dumb child to play the piano. When he played a wrong note, he would see the frown of its teacher, and try again. But he would obviously have no idea of what he was doing, or why anyone should devote hours to such an extraordinary exercise. He would have learnt an imitation of music. And he would fear the piano exactly as most students fear what is suppose to be mathematics.”

Teaching Math is Like Teaching Music

I have always wanted to start an Algebra II class in a very different way to show the students that mathematical notation is not Math.  It just describes and helps us communicate Math.

I want to print off one page of sheet music and set it on the Algebra II students’ desks and immediately start talking about Major keys and Minor keys and how to do inverted note transformations.  The whole time I will be firing off definitions of deceptive cadence, Gavotte and Klangfarbenmelodie. Then I would assign them homework that is designed to have them practice the transformations over and over again.  I would do all of this without ever letting them hear the music.

Unfortunately, Math class can become this type of experience for many students.  How often do we as teachers throw a worksheet on the projector to introduce a new concept and then walk through the worksheet showing the student how to manipulate the letters and numbers on the page while firing off definitions of standard deviation and variance and think that they have just taught Math?

Math textbooks are perfect examples of presenting the notation first.  I was reading my son’s Algebra II textbook and the very beginning of the chapter begins with the words

 “An exponential function has the form y=abx where a<>0…”.

Most people look at that sentence and think that it is gibberish.  As a Mathematician, I look at that sentence and I actually see the graph and think of a real world business situation that it can represent.  I feel the flow of the numbers and the elegance of it getting infinitely close to “0” as it goes to the left.  I hear the music.  Just as a conductor can hear the music in his head when he reads a score, most math teachers hear the music just by reading the “notes”.  Unfortunately, most students cannot do this.

I have not taken a lot of music classes, but one of the best ones I have ever taken is one given by Professor Robert Greenberg on Beethoven’s symphonies.  In each lesson, he starts out by playing a whole movement of the symphony and then leading a discussion about how it makes us feel and what do we hear in the music.  He then plays small chunks of the music and shows, using musical notation and definitions, how Beethoven is able to accomplish such amazing work.

Math class should be a similar experience, but teaching students to hear the music of math is much harder than Professor Greenberg has getting his students to hear the music.  All he has to do is turn the stereo on and the sound flows around the room.

Even though it is hard, Math teachers must find ways to let the Math music flow around the room.  What student wants to just manipulate letters and numbers and memorize steps?  You want to know why many students say “I hate Math”?  It is because they do not hear the music, all they are doing is writing notes.

Pilot / Surgeon Training is a Good Template for Math Pedagogy — WHAT?!

I stumbled on this website and thought it held promise.  But, after following the link to

and to

I have to say that I was dismayed to say the least.  All of our math students should be like pilots and surgeons?  Wow.

I have spent the last 25 years doing a variety of jobs that use math (financial analyst, computer programmer, real estate investment and management, even elevator electrician) and I can confidently say that I have used lots of math and there was no way that a “pilots checklist” would have been helpful for any of the jobs.

Also, for the vast majority of jobs, you don’t have to get the math perfect the first time.  You are usually using imperfect data to make imperfect (but hopefully close) decisions.

I am all for differentiating math for individual learning styles and multiple intelligences, but I definitely do not like the term “personal learning” being a euphemism for rote memorization of steps to solve math problems with the emphasis on accuracy and not understanding.