Colin once again submits an article from the United States popular press, here a co-authored op-ed piece, about mathematics education reform in the United States, one of my main topics of personal research for more than a decade. The regular opinion column to which this guest piece was submitted, the "Answer Sheet" edited by Valerie Strauss, is basically propaganda for the current school system, and has been caught here on HN before stretching facts beyond all recognition to make political points.

https://news.ycombinator.com/item?id=3327847

Michael Paul Goldenberg has such a lengthy online trail of writings about mathematics education that as I entered his name in Google, autocomplete finished the search query

https://www.google.com/search?q=michael+paul+goldenberg+math...

and I was led to some of his more recent writings. (I used to interact with him quite regularly online in specialized email lists about mathematics education reform, about a decade ago.) He, um, definitely has a point of view in his approach to education reform. It's all about the providers for altogether too many people who look at education results and school practices in the United States.

The response to the usual excuses for United States school performance by another observer,

http://www.nationalreview.com/agenda/255997/are-tino-sananda...

who points out how often critiques of schools in the United States are responded to by excuses that shift blame from providers of "education" to the learners in their care:

"Consider that Americans tend to have more disposable income than citizens of other advanced market democracies, at least some of which can be devoted to supplemental instruction. After all, parents have fairly strong incentives to secure educational advantages for their children. This suggests that our schools are performing very poorly indeed.

"Don’t believe the hype."

is closer to reality than many of the critiques of outside-the-box approaches to mathematics education in the United States.

That said, I have been up-front here on HN in suggesting ways that Khan Academy can improve, for example by building more online practice that is truly problems rather than exercises (379 days ago),

http://news.ycombinator.com/item?id=2760663

"Just for friendly advice to the Khan Academy exercise developers, I'll repost my FAQ about the distinction between "exercises" and "problems" in mathematics education. It would be great to see more problems on the Khan Academy site."

and the Khan Academy developers have been listening, and I have had interesting off-forum email interaction with them as they attempt to improve the instructional model at Khan Academy.

To date, I recommend to my own children and to my clients in my own supplemental mathematics education program that they also turn to ALEKS

http://www.aleks.com/

(Yet another edit. About the time I posted this, someone else asked below another comment,

So who is making the site that will deliver more personalized instruction? Where is the research that site will use, telling all about which kinds of personalization are proven and how much effect they will have?

and ALEKS is an answer to those questions in large part. Browse around the ALEKS site to see its links to its research base.)

and to Art of Problem Solving

http://www.artofproblemsolving.com/

for more online mathematics instruction resources, and I also share specific links to specialized sites on particular topics with clients and with my children. Besides that, I fill my house with books about mathematics, and circulate other books about mathematics frequently from various local libraries.

I also recommend that all my students use the American Mathematics Competition

http://amc.maa.org/

materials and other mathematical contest materials as a reality check on how well they are learning mathematics.

In general, I think mathematics is much too important a subject to be single-sourced from any source. Especially, mathematics is much too important to be left to the United States public school system in its current condition.

I was just rereading The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999) the other day. It reminded me of facts I had already learned from other sources, including living overseas for two three-year stays in east Asia.

"Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x

"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77

Plenty of authors, including some who should be better known and mentioned more often by the co-authors of the article Colin kindly submitted here, have had plenty of thoughtful things to say about ways in which United States mathematical education could improve.

In February 2012, Annie Keeghan wrote a blog post, "Afraid of Your Child's Math Textbook? You Should Be,"

http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_o...

in which she described the current process publishers follow in the United States to produce new mathematics textbook. Low bids for writing, rushed deadlines, and no one with a strong mathematical background reviewing the books results in school textbooks that are not useful for learning mathematics. Moreover, although all new textbook series in the United States are likely to claim that they "expose" students to the Common Core standards, they are not usually designed carefully to develop mathematical understanding according to any set of standards.

In a January 2012 lecture,

http://math.berkeley.edu/~wu/Lisbon2010_4.pdf

Professor Hung-hsi Wu of UC Berkeley points out a problem of fraction addition from the federal National Assessment of Educational Progress (NAEP) survey project. On page 39 of his presentation handout (numbered in the .PDF of his lecture notes as page 38), he shows the fraction addition problem

12/13 + 7/8

for which eighth grade students were not even required to give a numerically exact answer, but only an estimate of the correct answer to the nearest natural number from five answer choices. Even at that, very few students chose the correct answer.

Patricia Clark Kenschaft, professor of mathematics at Montclair State University in New Jersey, reported in her article "Racial Equity Requires Teaching Elementary School Teachers More Mathematics" in the Notices of the American Mathematical Society

http://www.ams.org/notices/200502/fea-kenschaft.pdf

about elementary teachers' knowledge of mathematics in New Jersey:

"The teachers are eager and able to learn. I vividly remember one summer class when I taught why the multiplication algorithm works for two-digit numbers using base ten blocks. I have no difficulty doing this with third graders, but this particular class was all elementary school teachers. At the end of the half hour, one third-grade teacher raised her hand. 'Why wasn’t I told this secret before?' she demanded. It was one of those rare speechless moments for Pat Kenschaft. In the quiet that ensued, the teacher stood up.

"'Did you know this secret before?' she asked the person nearest her. She shook her head. 'Did you know this secret before?' the inquirer persisted, walking around the class. 'Did you know this secret before?' she kept asking. Everyone shook her or his head. She whirled around and looked at me with fury in her eyes. 'Why wasn’t I taught this before? I’ve been teaching third grade for thirty years. If I had been taught this thirty years ago, I could have been such a better teacher!!!'"

A discussion of the Common Core Standards in Mathematics, "The Common Core Math StandardsAre they a step forward or backward?"

http://educationnext.org/the-common-core-math-standards/

gets into further details of how mathematicians look at the general school curriculum in the United States. It is not the worst curriculum possible, and survivors of the system often have access to outside resources to supplement school lessons, but the public school instruction in mathematics in the United States still shows plenty of room for improvement.

After edit: I was asked in a reply what I think about the essay "Lockhart's Lament." I think it is an interesting read, but less practical for reforming mathematics education than I had hoped. (I say the same in general about articles by Keith Devlin, the mathematician who popularized Lockhart's Lament.) To reform education, it is important to be relentlessly empirical, and look again and again and again at the best practices of the highest-achieving countries. That's why I prefer several of the links I submitted to Lockhart's interesting essay as policy guidance for United States parents, taxpayers, and learners.

Another edit: HN user danso just kindly posted

http://news.ycombinator.com/item?id=4301758

a link to a response by Sal Khan in the same Washington Post op-ed column about education. Direct link is

http://www.washingtonpost.com/blogs/answer-sheet/post/sal-kh...