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.
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
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,
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),
"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
(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.)
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
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,"
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.
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
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?"
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.
""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."
The "Secret":
http://www.tech4mathed.com/MAT156/topics%20test%202/twodigit...
Since there is a strong, strong chance that many of the readers who were taught in a U.S. classroom never saw this, I have the above link included.
The multiplication algorithm I was taught in school was something like this:
for each digit in the bottom number, starting with the ones digit:
begin a new line beneath the current state.
write a number of trailing zeroes equal to the number of digits of the bottom number you've already processed.
for each digit in the top number, starting with the ones digit:
write (carry + top digit * bottom digit) % 10 to the left of the leftmost digit on the bottom line of the state.
the carry for the next digit of the top number will be (carry + top digit * bottom digit) / 10
use the addition algorithm to compute the sum of all the numbers you wrote this way.
If you hard code it to only handle two digit numbers, it's a bit simpler. It computes the product of two numbers 10A+B, 10C+D for one digit A,B,C,D as (D * B + 10 * D * A) + 10 * (C * B + 10 * C * A). That sounds really confusing, because it is. The visual explanation relates it to something much less confusing in an obvious way.
and the Khan Academy developers have been listening, and I have had interesting off-forum email interaction
And THERE is the difference between genuinely-given constructive criticism and the questionably-motivated smearing that those folks working with the Washington Post are attempting to deliver.
Thank you very much for the links. I've already sent some of them on to my wife so that she and I can research them a bit more as instructional aids for our children.
As one of those who worked on one of the Washington Post pieces, I would invite you to come by my blog, <a href="http://christopherdanielson.wordpress.com>Overthinking My Teaching</a>, where I think you will find my motives to quite pure-better understanding of student learning of mathematics.
In fact it was very important to me that the piece I co-wrote <strong>not</strong> be a smear piece. I can't speak for the comments I did not write. But a careful read of the post itself should reveal a claim backed up by evidence. The claim is that Sal Khan is lacking some important knowledge for teaching. That's not a smear. I would gladly engage in a discussion that pointed to evidence that he does know some important things about how people learn.
I looked at your blog. From my perspective, you folks are down in the weeds. Khan isn't using the optimal technique for teaching decimals? Who cares? The problem isn't that kids aren't learning optimal methods.
The problem with education in this country is that we funnel most all of our public education dollars into a system and related organizations that are only tangentially motivated to educate our children. Watch how the public school systems and teachers unions react to vouchers and charter schools and you'll see how much they care about educating kids vs protecting their turf and power base.
I've noted that those within the education system or closely aligned with it absolutely hated "Waiting for Superman". They nitpick it and try to blow up minor faults with it in a similar way to how you're going after Khan. To those of us outside of that power structure, that documentary showed how the establishment is utterly opposed to actually solving problems in education in this country if those solutions occur outside of that establishment.
I view your articles vs Khan to be in a similar vein. What Khan has done is he's added a solution to our society for our education problems that operates outside of the current public education power structure. Rather than build on what he's done and improve on it through working with Khan or doing something better -- you folks have taken the route of trying to publicly take him down in the media.
There is a difference between being interested and doing something. All I can say is, trusting a site run by a former middle school teacher/math coach as the "savior of education" is completely stupid.
Yet we have all these influential people saying such things. You would think Khan Academy would be more vocal about being a supplement and not a substitute.
His seems to advocate that children should be taught to explore math problems and make discoveries rather than memorize and practice given truths. My layperson's read of your comment finds some commonality between your positions.
I used to teach math and reading Lockharts Lament was deeply satisfying. He is much closer to the light than our current system, but in retrospect I think he's got his head in the clouds a little.
Thank you for such a detailed response. It's particularly timely as I'm trying to figure out how to introduce my kids to mathematics.
Granted that there is a lot wrong with the current system, I'm curious what you think about the specific criticisms levelled against Khan in the OP's article. Is there any merit in the charge that he introduces the equality operator without enough explanation? Or that there ought to be more discussion in his lessons about the meaning of the decimal point?
Karl, I'm currently working fulltime on a summer project in the online education space. The differentiator is that for a given topic, there can be multiple teachers and multiple videos each trying to teach it in the best way. http://neoteach.com. I imagine you're very busy, but I'd love to hear feedback from someone who has so much education experience. And from anyone else in this thread, for that matter.
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...