### Beating a Dead Horse: Mathematics Education Reform - Again

I find the opening paragraph in a recent article by Marcus du Sautoy (*The Mathematical Gazette* **97** November 2013 No. 540, 386-397) revealing:

During my year as President of The Mathematical Association the government began a review of the curriculum across all subjects taught in school. Given the constant tinkering with the education system by every government, this is probably a sentence that any MA president could write during their tenure.

The same of course holds for every president of the United States and all relevant organizations (NCTM, NEA, MAA, etc.) The fact is that the educational reform is an unceasing undertaking that in mathematical education started at the end of the nineteenth century and that is still going on strong. In the US, there were some discrete moments of new announcements - not to mention the New Math (1960s), there has been a series of standards from NCTM (Curriculum and Evaluation (1989), Professional (1991), Assessment (1995), Principles (2000), Common Core(2010)). It's a given that the effort will not stop any time soon. The CCSS have already gathered a plethora of critics.

There is one person whose name always comes up when I think of the stream of mathematical reforms that come one on the heels of another - Diane Ravitch. I never met Diane and all I know about her is that at some point in time she changed her mind about reforms and switched the camps so that one camp praised her courage, the other lamented her betrayal. And when I think of changing one's mind, I am reminded of a curious episode involving a well known Russian physicist Yakov Fraenkel (1894-1952). According to legend, he was shown a slide with the graph depicting the outcome of an experiment. He brought it to light and immediately explained why the experiment went the way suggested by the graph. When one of his students pointed out that he was holding the slide upside down, Fraenkel turned it around and produced an explanation of why the graph had to be that way.

The relevance of my perception of the reform process to Diane Ravitch's story is that I feel that most math educators and after them politicians that make decisions come up with reform ideas viscerally or by the sixth sense. The arguments come later: articles written, committees are set, statistics is collected, theories come forth. All this seldom causes anybody to change one's mind. And I keep wondering why, when it comes to educational reforms, it takes making or changing one's mind, and not some kind of deliberate experimentation.

There were successful experiments. Two are well known and are to my liking: an unorthodox geometry course by Harold P. Fawcett at the Ohio State University starting in the 1930s, and somewhat earlier one by Louis P. Benezet. The consensus I believe is that a teacher needs to be a Harold Fawcett to manage a course like Fawcett's or W. Eugene Smith who taught that course in the years 1945-1956. Here I believe lies the main reason for the continuous attempts to standardize mathematical education and keep it under control with standardized testing. Other reasons are usually cited publicly, but I think this is the main one: teachers are not trusted to do the right thing by students on their own.

There is certainly a good reason for mistrust: as in every endeavor, there are excellent, mediocre, and outright bad teachers. For the latter it is easier to follow strict curriculum with testing than to adapt to their class and individual students' progress. Many may not be able to do that.

I strongly believe - what else can I say or how else can I argue - that the only way to find a solution to the real or perceived deficiencies in math education is to admit to that fact. In a country the size of the US, any attempt to teach uniformly, giving every one the same opportunity, so to speak, is simply impossible. Worse, such attempts will always shortchange the better students. As Steven Strogatz wrote in Notices of the AMS,

Though it’s taboo to admit it, I believe there are some kids who have a feel for math.

By extension there are students who do not have that feel, they may be even in majority.

I may be beating a dead horse, but here what Marcus du Sautoy wrote about in the article I mentioned at the beginning:

I often get the feeling that we are still stuck in Napoleonic France, just doing mathematics to serve the state. If only we could find a modem-day Humboldt to take the reins at the Department of Education ....

For the first time in Germany, the study of mathematics formed a major part of the curriculum in the new schools and universities. And mathematicians, freed from the need to model the physical world, began instead to explore mathematical ideas for their own sake. It gave rise to the creation of geometry that lives beyond our three-dimensional universe.

And before that

Just as in English a student isn't meant to grasp the full complexity of a Shakespeare play, we should be prepared to take the risky step of teaching big ideas that a student might not fully comprehend but rather they should be given a way to glimpse something of these great stories. Just as any course in English literature can give just a taste of the great works, a mathematical literacy course would not aim to be complete but to expose students to a sample of what is out there.

I only have one point of disagreement with du Sautoy. It seems to me (I apologize if I am wrong) that he talks of the middle and high school curriculum. I would start earlier by trying to engage little kids with big ideas, and stop ostracizing those who do not get them.

"I feel that most math educators and after them politicians that make decisions come up with reform ideas viscerally or by the sixth sense."

I would like to discuss this part. Because learning and teaching is complex, in the mathematical sense, it's extremely difficult to research. Qualitative methods, including human intuitions as a tool of analysis (maybe a bit more formally than "the sixth sense") have to combine with quantitative methods. But intuitions of an experienced successful teacher are worth a whole lot.

As for politicians, they may follow agendas that are based on entirely other reasons, not how kids learn the best. Consider the case of trig and artillery, the military often being "the high bidder" for what public education should be all about.

March 3rd, 2014 at 11:51 amMy point was that when decisions are made on a political level, or in general by the powers that be, they are global - "one for all". And this, IMHO, is always doomed to failure.

I do believe that there is no global solution, no the best way of teaching, and not every teacher can step into Fawcett's or E. Smith's shoes. But what that experience teachers is that some teachers can do a fantastic job that global regulations may only hinder.

Thus my suggestions is to recognize that there are constraints on what can be done and stop inventing and using battle cries like "No child left behind" and "Race to the top" and let teachers do what they can, some more, some less.

March 3rd, 2014 at 7:50 pm