Geometry is taught correctly when proofs are used as a tool to convey a deep intuition about a mathematical pattern. Check out Lockhart's book  if you want to see what it's like when done right, though there are many more examples. Two-column proofs are simply a tool for the lazy/unknowledgeable teachers to fill a geometry class.
I am aware of Lamport's work, and it's a specific tool for a specific subfield in which there is a plethora of false results. Ignoring the fact that most of theoretical computer science research and most of math research more does not fall into the category that Lamport is critical of (distributed computing), these temporary issues about rigor in academic publishing should have no effect on high school pedagogy. Instead, we should listen to the world's finest math teachers, who pretty much all agree that two-column proofs are awful. A quote from Lockhart :
> Geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming. There is nothing charming about what passes for proof in geometry class. Students are presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a format as unnecessary and inappropriate as insisting that children who wish to plant a garden refer to their flowers by genus and species.
It's hard to keep at it, learning on your own. I sometimes find problems where the usual solutions or explanations feel kind of ugly, and try to make them cleaner, like rewriting someone's code. (A couple days ago it was Snell's law: this optics tutorial http://www.bigshotcamera.com/learn/imaging-lens/refraction linked from HN just dropped this formula down, and to most kids it's going to be magic. Can you formulate the law of refraction in a more elementary way and derive it from some simple assumptions? I did come up with a version that never mentions sines, but I'm not really satisfied and it's gone back on the to-do list to try to take it further. See: hard to keep at it.)
More generally, this kind of work can come up all the time when programming if you say, "No, I'm not going to look up the algorithm, I'll work one out for myself and then see what's been done." Occasionally you find something kind of new that way, besides often deepening your appreciation of the usual solutions. For example, last week I found a new way to avoid the epsilon-loops in Thompson's regular-expression search algorithm -- new to me, at least. This has minor significance and came out of a ridiculous amount of work rediscovering things taught in automata-theory classes, but Lockhart wasn't kidding: it's a rush when you figure it out.
and from time to time regret the gaps in their own mathematical education.
"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,
pedagogical content knowledge that would be very routine for any elementary mathematics teacher in east Asia.
(book link above, review links below)
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,
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,
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)
(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
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
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
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?"
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
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.
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