An interesting article, pointing out that mathematics anxiety on the part of adults sometimes limits engagement with mathematics learning opportunities among children. Mathematician W. W. Sawyer wrote about this quite a while ago: "The proper thing for a parent to say is, 'I did badly at mathematics, but I had a very bad teacher. I wish I had had a good one.'" W. W. Sawyer, Vision in Elementary Mathematics (1964), page 5. Elementary school teachers in the United States often fear mathematics themselves,

http://news.uchicago.edu/article/2010/01/25/female-teachers-...

http://www.jstor.org/discover/10.2307/41192533?uid=3739736&#...

and from time to time regret the gaps in their own mathematical education.

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

"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!!!'"

The last time I posted a link to this article on HN, another HN participant kindly posted a link to what is surely the "secret" referred to by the elementary school teacher,

http://www.tech4mathed.com/MAT156/topics%20test%202/twodigit...

pedagogical content knowledge that would be very routine for any elementary mathematics teacher in east Asia.

http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

(book link above, review links below)

http://www.ams.org/notices/199908/rev-howe.pdf

http://www.math.wisc.edu/~askey/ask-gian.pdf

So this advice for parents is good in helping parents provide a supportive environment for their children's mathematics learning.

I have frequent occasion to write about mathematics education here on Hacker News. My occupation is 1) providing supplemental lessons in advanced mathematics to pupils from ten counties in Minnesota through a nonprofit corporation I cofounded and 2) coordinating parent workshops and other aspects of the summer program Epsilon Camp,

http://www.epsiloncamp.org/

perhaps the most advanced mathematics program of its kind for YOUNG learners in North America.

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/

which is a commerical online site (in which I have no economic interest) delivering personalized instruction in mathematics through precalculus mathematics. The ALEKS website includes links to research publicatoins on which ALEKS is based.

I also recommend the Art of Problem Solving (AoPS)

http://www.artofproblemsolving.com/

(where I first took on the screenname that I also use here on HN) for more online mathematics instruction resources, and I also share specific links to specialized sites on particular topics with clients and with my children. I should note for onlookers that the articles on mathematics learning on the AoPS website

http://www.artofproblemsolving.com/Resources/articles.php?

are very good indeed, especially "The Calculus Trap."

My children make quite a bit of voluntary use of Khan Academy (both watching videos and working online exercises) and I am gratified that my previous suggestions to the Khan Academy developers here on HN

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

have been followed up as Khan Academy developers have communicated with me by email about new problem formats available in their online exercises, which are becoming increasingly challenging.

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 rereading The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999) last month. 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 HN participants, have had plenty of thoughtful things to say about ways in which United States mathematical education could improve.

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.

The last time I posted about these issues, a reply asked what I think about essay "Lockhart's Lament." I think it is an interesting read, but less practical for reforming mathematics education than I had hoped. I wonder if Lockhart's forthcoming book Measurement

http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755...

will be a successful attempt to teach mathematical reasoning to students who have already lost confidence in learning mathematics, which would be a great contribution to society.