7 ms·

Teaching Kids to Love Math

72 points by joeyespo 14y ago

CeRRuTiTo 14y ago
Math skills are essential to everything, so I like this!
Afton 14y ago
I find the most challenging part to putting this into action is the same things that one finds challenging when designing a test for a class: getting the challenge level correct. For a while this summer I'd write out simple math problems for my daughter to do when she got up. I stopped when I realized that I was doing harm by consistently getting the difficulty of my questions wrong. These kinds of articles make it seem easy, but I find it quite difficult. For obvious reasons, I'm continuing to try to help my children with math (and other things!), but like most things, it seems to require patience, careful thought, and practice.
tokenadult 14y ago
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-can-transfer-fear-math-and-undermine-girls-math-performance http://news.uchicago.edu/article/2010/01/25/female-teachers-... http://www.jstor.org/discover/10.2307/41192533?uid=3739736&uid=2&uid=4&uid=3739256&sid=21101177886117 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 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/twodigitmultiplication.htm 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-Mathematics-Understanding/dp/0415873843/ 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.ams.org/notices/199908/rev-howe.pdf http://www.math.wisc.edu/~askey/ask-gian.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/ 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/ 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/ 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 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 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/ 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/ 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/0674057554/ 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.
- lifeisstillgood 14y ago
  As a father of two (both under 3) I want to (obviously) supplement my childrens official education with home encouragement - but resources and kicking off points like that above are - well I am sure they are around, I just don't know where. I am half considering starting the Concerned Parent (tm) github repo Is there any value in a github repo on recommended books and practises to help polyfilla in the cracks in the school system?- different approaches could have their own branches, commits could be discussed (Did you like that book The Teaching Gap? Should it be recommended? etc) No I am not trying to recreate mumsnet. How crazy is this?
  - SiVal 14y ago
    It's not crazy, but it's inefficient. Go hang out in home schooler forums instead. People homeschool for many different reasons, but I think it's great that it's still allowed in the US. The result of a diversity of people with diverse needs meeting a private market that hasn't yet been eradicated by "progressive" teachers' unions and the politicians they employ, is a cornucopia of private resources available to parents and students who exercise some initiative. These resources are under constant discussion on homeschooling forums.
    - lifeisstillgood 14y ago
      Thank you - as a brit I had always assumed it was, well, nutjob territory. Will look into it.
      - SiVal 14y ago
        Yes, the mainstream media and the educational establishments in both the US and UK are dominated by the same leftist statists who, in order to promote their social agenda, need to portray all those who don't surrender themselves to the state programs as obvious nutjobs. Those who leave the herd do so for all sorts of reasons, religious, academic (my case--like many homeschoolers, I'm an atheist), political, special educational need, lifestyle (e.g., frequent travel), or whatever. That's the nature of independent individuals. That level of diversity means that you can find plenty of people you'll consider nutty, which makes it easy for the media to push the message that those who leave the state herd are mentally defective. But look at the effectiveness of the state schools and ask yourself, if the state system were shrunk down to the size of the various small homeschooling alternatives, would the state program be one you would pick for your own kids, or would one of the alternatives seem much more sensible? If you suspect the latter, maybe the state system is the real nutjob territory.
diggan 14y ago
That's great. Can you now teach me how I (~20 year old) can be interested in math? I deeply want to understand it but it gets so boring after a while... Is there any help for me out there?
- GFKjunior 14y ago
  I recommend Calculus Made Easy by Thompson. This book is short but complete. I read it at 22. I gave up on mathematics after my first college course but found that for everything I wanted to get into, Machine Learning & Bayesian stats, it was essential . A year later I know math better than ever before even though I haven't stepped inside a classroom in years. You're probably getting bored because most math books are hundreds of pages long and in addition to the material they include mathematicians bio's, several solution methods, and the explanations of edge cases.
  - debacle 14y ago
    You can't justify a $200 price tag for a 100 page text book.
    - lutusp 14y ago
      That might sometimes be true, but the textbook the OP referred to is less than $16.00: http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480 http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/d...
      - debacle 14y ago
        Which is why no one will ever use it, because you can't justify a $200 price tag for a 100 page text books. e.g. no college will ever endorse this book as a text, because they need to sell 60k in calc 101 text books each semester so they can get their cut.
        lutusp 14y ago
        > ... no college will ever endorse this book as a text ... Yes, fair enough, however I think the discussion is not so much about schooling as education. Not all education takes place in college -- and with costs rising as they are, I predict a future with many more autodidacts -- self-educators. "I have never let my schooling interfere with my education." -- Mark Twain
lmkg 14y ago
Children are betting at learning than we are at teaching. You can try teaching 2+2, but if you're uncomfortable with math, then your children will not only learn 2+2 from you, they will also learn "math is hard." And having to un-learn that lesson is a very large, very difficult barrier to have to overcome. When you teach your kids math, don't just teach them 2+2, also make sure that they learn math is fun & easy.
lutusp 14y ago
The article fails to mention the single most effective way to improve math education -- teach mathematics first, then arithmetic later. What do I mean by that? This is mathematics: http://www.hdwallpapers.in/walls/jake_sully_avatar_2009-normal.jpg http://www.hdwallpapers.in/walls/jake_sully_avatar_2009-norm... And this is arithmetic: http://www.scielo.br/img/revistas/bjp/v37n2a/a07frma1.gif http://www.scielo.br/img/revistas/bjp/v37n2a/a07frma1.gif The second part (arithmetic) is required for the first, but it doesn't need to be presented first. The problem is easy to state -- we're teaching mathematics in the wrong order.
- tokenadult 14y ago
  The problem is easy to state -- we're teaching mathematics in the wrong order. Where would we go to find a real-world example of mathematics being taught in the correct order to elementary-age pupils? There are many different schools (and many different settings for learning outside school) in many different countries. What is a good place to look for examples of demonstrably successful practice? AFTER EDIT: Thanks for the reply. The thing I would want to look at in such intervention studies is how the teaching effectiveness is measured. AFTER FURTHER EDIT: Oh, I see the video "Teaching Math Without Words, A Visual Approach to learning Math from MIND Research Institute" is one I have watched before, and have discussed among other mathematics teachers. Similarly, the publication Mathematics Teaching in the Middle School is a publication I subscribe to as a member of NCTM. I should start out in my reaction to both of these links you've kindly shared by pointing out that my own bent in thinking about mathematics is to think visually, and indeed that is why I like Sawyer's book Vision in Elementary Mathematics (which leads off my longer reply in this thread) so well. But although I like visual approaches to teaching mathematics very well, and use them in the mathematics classes I teach, I also like studies of educational interventions to check results. So far, I can't find a publication by independent researchers (after searching Google Scholar and the new Microsoft Academic Search) that verifies the educational effectiveness claims made in the video for the proprietary intervention described there. The company's own website http://www.mindresearch.net/cont/research/re_publications.php http://www.mindresearch.net/cont/research/re_publications.ph... provides only a very sparse set of links to published research on the issue. Plainly, a more visual approach along the lines of the typical school textbooks used in Taiwan, Singapore, and Japan would be a better presentation of mathematics content than is found in typical United States textbooks. That should be possible to deliver without any computers at all, as it was delivered to pupils in Taiwan in my wife's generation, when Taiwan was still a poor, Third World country.
stephengillie 14y ago
The only people who love math are math majors. Most humans don't love math. Teaching kids to love math is pointless. Math is a tool, like a hammer or a DLL. Do you love your screwdrivers and methods? Let's teach kids about how math is useful to them in their lives, and let them take it from there. Ongoing research is shedding new light on the importance of math to children's success. Math skill at kindergarten entry is an even stronger predictor of later school achievement than reading skills or the ability to pay attention, according to a 2007 study in the journal Developmental Psychology. This is like trying to become smarter by listening to classical music. Is it ironic that those suggesting these ideas can't separate correlation from causation?
- tikhonj 14y ago
  That's like claiming that the only people who love literature are English majors. It's simply not true. For one, there are plenty of people in all sorts of disciplines who had a good exposure to and still like math. More importantly, there are many people who enjoy logic puzzles and games but hate "math" because they have a warped perception of what "real math" actually is. A good introduction to math would turn a person's innate interest in thinking and logic into an interest in math. After all, at its base, math is really just a way to approach logic in an organized and systematic fashion. I think at least a basic understanding of some more abstract math--especially formal logic--is as valuable to a well-rounded person as an appreciation of literature or knowledge of basic history. Sure, the basic person on the street doesn't need to be an expert on algebraic topology, but they don't need to be experts on romantic literature or 17th century Belgian history either. This doesn't mean that they shouldn't be well versed in some--and probably a fair bit--of literature and history, and, in the same way, they should be well versed in at least basic mathematics. Now, one of the problems with math education is that what they teach is not really basic math. Rather, they teach subjects that are readily applicable at a fairly superficial level. I think a good grounding in formal logic and set theory, for example, is far more valuable in general than a thorough understanding of differential calculus. And yet it's the latter that is considered basic and widely taught, probably because it is immediately useful to engineers and physicists.
- lutusp 14y ago
  > The only people who love math are math majors. Most humans don't love math. Oh, this is so wrong. Did you like Avatar? Then you like math. Avatar is one long, beautiful mathematical expression, from beginning to end. > This is like trying to become smarter by listening to classical music. Granted the problem with the basic idea, this has it all over trying to become smarter by listening to Country & Western music. > Is it ironic that those suggesting these ideas can't separate correlation from causation? Math is neither a cause nor an effect -- it is both. I could argue this point in detail, but instead I will get Bertrand Russell argue it for me: "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry." Richard Feynman said, "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature." I can't think of a place that equals America for complete and willful misunderstanding of mathematics.
001sky 14y ago
Numeracy is the foundation for math. That is what we should be teaching kids. It makes it much easier to transition to more logic-intensive modes of analyisis later on.
VSerge 14y ago
How is learning maths playfully not insisted upon in this article or the comments? 2 examples: - many card games require players to count points depending on the cards they won. lots of adding and mental calculation training in a fun setting. - more advanced games like DragonBox, which blew me away from a purely game design point of view. FYI, I'm a game designer and I had never seen a game teach players, 5yo and adults alike, how to solve an equation for x in a matter of hours, while actually being fun, and WITHOUT triggering math-phobic reflexes.. (http://dragonboxapp.com/ http://dragonboxapp.com/)
orangethirty 14y ago
They dont need to learn how to love math, they already do. What you cant teach them is to love a system designed two hundred years ago to teach very ignorant people about math. Most of the people who "hate" math do so because they were forced to "learn" it through an antiquated system. I hated my math classes, yet was always the top score on the annual standarized tests. Its not about math, but about how they teach it. Best way to teach math? Same way as with other science. Use it. Dont teach kids fractions because of fractions. Show them how fractions work in the real world. My 10 year old niece was having trouble with the multiplication. Why? Because her teacher made them say it in front of the whole class. I went ahead and sat her down. Booted up ubuntu and taught her how multiplication works with javascript. She sat for around one hour toying with the code I wrote for her. What do you think is more fun? - Standing in front of other kids who are making fun of you and saying the multiplication tables. - Writing done the following code and tinkering with it and calculating what the answer will be. var base = 5; var num = 4; var total = num * base; document.write(total); //She spent one hour with those four lines of code, and learned more about multiplication. She is now learning python. :)
- 001sky 14y ago
  Best way to teach math? Same way as with other science. Use it. - Agree with this. "By doing, we Become"
oob205 14y ago
Worth noting that a way NOT to teach kids to love math is to inundate them with arithmetic. My math education, like many in the US, was an endless barrage of memorizing facts, solving equations, then memorizing some more facts. It was not until college that I began to see the essence of math is creativity. Kids are naturally creative and inquisitive and we should use this to our advantage in early education. We should explore topics like topology and infinity, topics that still blow my adult, math-major mind. We also need to encourage students to be creators in math. One of the ways we learn to love reading is by writing and making up our own stories. We can have students make puzzles, write computer programs, and see that they can invent things with math. For a more informed perspective on how we teach math all wrong, I recommend Lockhart's Lament: http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf
- joezydeco 14y ago
  I spent an hour last night working on a math homework assignment with my second-grader. The practice was adding up single digits to sums greater than 10. The idea behind the problem (say 9+5) is to "make a 10" and then figure out the sum from the remainder. So 9+5 -> 10+4 = 14. But he didn't know which numbers routinely added up to 10. That was never memorized. The assignment was so focused on strategy and technique that it neglected the fact that the basic tools aren't there. There's a point where you need to memorize something to make the strategy and discovery easier.
- aswanson 14y ago
  We should explore topics like topology and infinity, topics that still blow my adult, math-major mind... One of my happiest conversations was with my 7 year old inquiring into infinity...days after our conversation I overheard him saying to a classmate, 'Inifinity is not a number, it's a statement saying numbers never end...'
marcpaul22 14y ago
I loved "Prof. E. McSquared's Calculus Primer" when I was learning Calculus (look for used prices): http://www.amazon.com/gp/product/0913232475/ref=olp_product_details?ie=UTF8&me=&seller= http://www.amazon.com/gp/product/0913232475/ref=olp_product_... Home page here: http://www.math.sjsu.edu/~swann/mcsqrd.html http://www.math.sjsu.edu/~swann/mcsqrd.html Fun and funny, written in a comic book format. It illustrates the concepts in such an entertaining style, but makes the point better than many textbooks.
- calibraxis 14y ago
  Along those lines, you might enjoy _Thinking Physics_. (http://www.amazon.com/Thinking-Physics-Understandable-Practical-Reality/dp/0935218084 http://www.amazon.com/Thinking-Physics-Understandable-Practi...)
5xz41s0P8T5N 14y ago
As a math undergrad dealing with mathematical anxiety (how silly, I know), I wish math education started with logic. Is there evidence that learning discrete math first leads to better skill, or that it is easier/more fun for kids? or even teachers? I suspect I only feel this way because discrete math seems so foundational now, but perhaps what is taught matters less than how it's taught. I wish education was more empirical.
csense 14y ago
Most elementary schools have all subjects taught by one teacher. (At least mine did; my K-12 education was entirely in the US public school system.) Kids taking different subjects from different teachers doesn't happen until middle or high school. So I'd guess that most teachers of young kids probably chose to train as elementary-education generalists, and math was hard and scary for them. There are several mechanisms by which this might rub off on their pupils: 1. If the teacher doesn't have an innate joy and passion for their subject, the lack of enthusiasm may be contagious. 2. The teacher may have no idea why things like the addition algorithm work, and no idea how to explain it to their students. 3. Most of the math classes in elementary school were largely spent reviewing material covered in previous years. Having an expectation that students should know material that's been covered in previous lectures is simple common sense. I'm not against an occasional review class, mind you, but spending an entire quarter or semester on nothing but previously covered material strikes me as a colossal waste of everyone's time. 4. High-stakes standardized testing is a recipe for disaster. The tests don't accurately measure anything but the amount of time and effort teachers spend teaching to the test. I was pretty far out of elementary school by the time No Child Left Behind hit the fan, but I'd imagine even knowledgeable, passionate teachers who want to motivate their students with the joy of learning to use their imagination to manipulate abstract ideas get immense pressure from above to turn their class into nothing but drill, drill, drill. 5. The very name, "No Child Left Behind," implies is that the goal is to have the entire class proceed at the pace of the least capable student. On the face of it, this is deliberately aiming for poor results. For those of us who have never observed the human race in action before, let me explain: People are different; they have different levels of mental strength, motivation, talent, and parental support; they have different areas of interest, different areas of talent, put in different amounts of effort, and think in different ways. It is inevitable, then, that some people will be better at math than others. But we seem to be trying extremely hard to deny this simple and obvious truth, to bring the least capable students to a level of competence that they may never achieve, at great cost to taxpayers and immense frustration for everyone. Meanwhile, the middle- and high-capability tiers of the class are bored, frustrated and alienated by the endless repetition of very dull tasks -- which is most assuredly not what mathematics is about, especially since the invention of cheap, ubiquitous computers.

72 points by joeyespo 14y ago

Math skills are essential to everything, so I like this!

Afton 14y ago

I find the most challenging part to putting this into action is the same things that one finds challenging when designing a test for a class: getting the challenge level correct. For a while this summer I'd write out simple math problems for my daughter to do when she got up. I stopped when I realized that I was doing harm by consistently getting the difficulty of my questions wrong. These kinds of articles make it seem easy, but I find it quite difficult. For obvious reasons, I'm continuing to try to help my children with math (and other things!), but like most things, it seems to require patience, careful thought, and practice.

tokenadult 14y ago

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-can-transfer-fear-math-and-undermine-girls-math-performance http://news.uchicago.edu/article/2010/01/25/female-teachers-... http://www.jstor.org/discover/10.2307/41192533?uid=3739736&uid=2&uid=4&uid=3739256&sid=21101177886117 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 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/twodigitmultiplication.htm 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-Mathematics-Understanding/dp/0415873843/ 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.ams.org/notices/199908/rev-howe.pdf http://www.math.wisc.edu/~askey/ask-gian.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/ 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/ 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/ 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 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 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/ 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/ 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/0674057554/ 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.

lifeisstillgood 14y ago
As a father of two (both under 3) I want to (obviously) supplement my childrens official education with home encouragement - but resources and kicking off points like that above are - well I am sure they are around, I just don't know where. I am half considering starting the Concerned Parent (tm) github repo Is there any value in a github repo on recommended books and practises to help polyfilla in the cracks in the school system?- different approaches could have their own branches, commits could be discussed (Did you like that book The Teaching Gap? Should it be recommended? etc) No I am not trying to recreate mumsnet. How crazy is this?
- SiVal 14y ago
  It's not crazy, but it's inefficient. Go hang out in home schooler forums instead. People homeschool for many different reasons, but I think it's great that it's still allowed in the US. The result of a diversity of people with diverse needs meeting a private market that hasn't yet been eradicated by "progressive" teachers' unions and the politicians they employ, is a cornucopia of private resources available to parents and students who exercise some initiative. These resources are under constant discussion on homeschooling forums.
  - lifeisstillgood 14y ago
    Thank you - as a brit I had always assumed it was, well, nutjob territory. Will look into it.
    - SiVal 14y ago
      Yes, the mainstream media and the educational establishments in both the US and UK are dominated by the same leftist statists who, in order to promote their social agenda, need to portray all those who don't surrender themselves to the state programs as obvious nutjobs. Those who leave the herd do so for all sorts of reasons, religious, academic (my case--like many homeschoolers, I'm an atheist), political, special educational need, lifestyle (e.g., frequent travel), or whatever. That's the nature of independent individuals. That level of diversity means that you can find plenty of people you'll consider nutty, which makes it easy for the media to push the message that those who leave the state herd are mentally defective. But look at the effectiveness of the state schools and ask yourself, if the state system were shrunk down to the size of the various small homeschooling alternatives, would the state program be one you would pick for your own kids, or would one of the alternatives seem much more sensible? If you suspect the latter, maybe the state system is the real nutjob territory.

diggan 14y ago

That's great. Can you now teach me how I (~20 year old) can be interested in math? I deeply want to understand it but it gets so boring after a while... Is there any help for me out there?

GFKjunior 14y ago
I recommend Calculus Made Easy by Thompson. This book is short but complete. I read it at 22. I gave up on mathematics after my first college course but found that for everything I wanted to get into, Machine Learning & Bayesian stats, it was essential . A year later I know math better than ever before even though I haven't stepped inside a classroom in years. You're probably getting bored because most math books are hundreds of pages long and in addition to the material they include mathematicians bio's, several solution methods, and the explanations of edge cases.
- debacle 14y ago
  You can't justify a $200 price tag for a 100 page text book.
  - lutusp 14y ago
    That might sometimes be true, but the textbook the OP referred to is less than $16.00: http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480 http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/d...
    - debacle 14y ago
      Which is why no one will ever use it, because you can't justify a $200 price tag for a 100 page text books. e.g. no college will ever endorse this book as a text, because they need to sell 60k in calc 101 text books each semester so they can get their cut.
      - lutusp 14y ago
        > ... no college will ever endorse this book as a text ... Yes, fair enough, however I think the discussion is not so much about schooling as education. Not all education takes place in college -- and with costs rising as they are, I predict a future with many more autodidacts -- self-educators. "I have never let my schooling interfere with my education." -- Mark Twain

lmkg 14y ago

Children are betting at learning than we are at teaching. You can try teaching 2+2, but if you're uncomfortable with math, then your children will not only learn 2+2 from you, they will also learn "math is hard." And having to un-learn that lesson is a very large, very difficult barrier to have to overcome. When you teach your kids math, don't just teach them 2+2, also make sure that they learn math is fun & easy.

lutusp 14y ago

The article fails to mention the single most effective way to improve math education -- teach mathematics first, then arithmetic later. What do I mean by that? This is mathematics: http://www.hdwallpapers.in/walls/jake_sully_avatar_2009-normal.jpg http://www.hdwallpapers.in/walls/jake_sully_avatar_2009-norm... And this is arithmetic: http://www.scielo.br/img/revistas/bjp/v37n2a/a07frma1.gif http://www.scielo.br/img/revistas/bjp/v37n2a/a07frma1.gif The second part (arithmetic) is required for the first, but it doesn't need to be presented first. The problem is easy to state -- we're teaching mathematics in the wrong order.

tokenadult 14y ago
The problem is easy to state -- we're teaching mathematics in the wrong order. Where would we go to find a real-world example of mathematics being taught in the correct order to elementary-age pupils? There are many different schools (and many different settings for learning outside school) in many different countries. What is a good place to look for examples of demonstrably successful practice? AFTER EDIT: Thanks for the reply. The thing I would want to look at in such intervention studies is how the teaching effectiveness is measured. AFTER FURTHER EDIT: Oh, I see the video "Teaching Math Without Words, A Visual Approach to learning Math from MIND Research Institute" is one I have watched before, and have discussed among other mathematics teachers. Similarly, the publication Mathematics Teaching in the Middle School is a publication I subscribe to as a member of NCTM. I should start out in my reaction to both of these links you've kindly shared by pointing out that my own bent in thinking about mathematics is to think visually, and indeed that is why I like Sawyer's book Vision in Elementary Mathematics (which leads off my longer reply in this thread) so well. But although I like visual approaches to teaching mathematics very well, and use them in the mathematics classes I teach, I also like studies of educational interventions to check results. So far, I can't find a publication by independent researchers (after searching Google Scholar and the new Microsoft Academic Search) that verifies the educational effectiveness claims made in the video for the proprietary intervention described there. The company's own website http://www.mindresearch.net/cont/research/re_publications.php http://www.mindresearch.net/cont/research/re_publications.ph... provides only a very sparse set of links to published research on the issue. Plainly, a more visual approach along the lines of the typical school textbooks used in Taiwan, Singapore, and Japan would be a better presentation of mathematics content than is found in typical United States textbooks. That should be possible to deliver without any computers at all, as it was delivered to pupils in Taiwan in my wife's generation, when Taiwan was still a poor, Third World country.

stephengillie 14y ago

The only people who love math are math majors. Most humans don't love math. Teaching kids to love math is pointless. Math is a tool, like a hammer or a DLL. Do you love your screwdrivers and methods? Let's teach kids about how math is useful to them in their lives, and let them take it from there. Ongoing research is shedding new light on the importance of math to children's success. Math skill at kindergarten entry is an even stronger predictor of later school achievement than reading skills or the ability to pay attention, according to a 2007 study in the journal Developmental Psychology. This is like trying to become smarter by listening to classical music. Is it ironic that those suggesting these ideas can't separate correlation from causation?

tikhonj 14y ago
That's like claiming that the only people who love literature are English majors. It's simply not true. For one, there are plenty of people in all sorts of disciplines who had a good exposure to and still like math. More importantly, there are many people who enjoy logic puzzles and games but hate "math" because they have a warped perception of what "real math" actually is. A good introduction to math would turn a person's innate interest in thinking and logic into an interest in math. After all, at its base, math is really just a way to approach logic in an organized and systematic fashion. I think at least a basic understanding of some more abstract math--especially formal logic--is as valuable to a well-rounded person as an appreciation of literature or knowledge of basic history. Sure, the basic person on the street doesn't need to be an expert on algebraic topology, but they don't need to be experts on romantic literature or 17th century Belgian history either. This doesn't mean that they shouldn't be well versed in some--and probably a fair bit--of literature and history, and, in the same way, they should be well versed in at least basic mathematics. Now, one of the problems with math education is that what they teach is not really basic math. Rather, they teach subjects that are readily applicable at a fairly superficial level. I think a good grounding in formal logic and set theory, for example, is far more valuable in general than a thorough understanding of differential calculus. And yet it's the latter that is considered basic and widely taught, probably because it is immediately useful to engineers and physicists.
lutusp 14y ago
> The only people who love math are math majors. Most humans don't love math. Oh, this is so wrong. Did you like Avatar? Then you like math. Avatar is one long, beautiful mathematical expression, from beginning to end. > This is like trying to become smarter by listening to classical music. Granted the problem with the basic idea, this has it all over trying to become smarter by listening to Country & Western music. > Is it ironic that those suggesting these ideas can't separate correlation from causation? Math is neither a cause nor an effect -- it is both. I could argue this point in detail, but instead I will get Bertrand Russell argue it for me: "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry." Richard Feynman said, "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature." I can't think of a place that equals America for complete and willful misunderstanding of mathematics.

001sky 14y ago

Numeracy is the foundation for math. That is what we should be teaching kids. It makes it much easier to transition to more logic-intensive modes of analyisis later on.

VSerge 14y ago

How is learning maths playfully not insisted upon in this article or the comments? 2 examples: - many card games require players to count points depending on the cards they won. lots of adding and mental calculation training in a fun setting. - more advanced games like DragonBox, which blew me away from a purely game design point of view. FYI, I'm a game designer and I had never seen a game teach players, 5yo and adults alike, how to solve an equation for x in a matter of hours, while actually being fun, and WITHOUT triggering math-phobic reflexes.. (http://dragonboxapp.com/ http://dragonboxapp.com/)

orangethirty 14y ago

They dont need to learn how to love math, they already do. What you cant teach them is to love a system designed two hundred years ago to teach very ignorant people about math. Most of the people who "hate" math do so because they were forced to "learn" it through an antiquated system. I hated my math classes, yet was always the top score on the annual standarized tests. Its not about math, but about how they teach it. Best way to teach math? Same way as with other science. Use it. Dont teach kids fractions because of fractions. Show them how fractions work in the real world. My 10 year old niece was having trouble with the multiplication. Why? Because her teacher made them say it in front of the whole class. I went ahead and sat her down. Booted up ubuntu and taught her how multiplication works with javascript. She sat for around one hour toying with the code I wrote for her. What do you think is more fun? - Standing in front of other kids who are making fun of you and saying the multiplication tables. - Writing done the following code and tinkering with it and calculating what the answer will be. var base = 5; var num = 4; var total = num * base; document.write(total); //She spent one hour with those four lines of code, and learned more about multiplication. She is now learning python. :)

001sky 14y ago
Best way to teach math? Same way as with other science. Use it. - Agree with this. "By doing, we Become"

oob205 14y ago

Worth noting that a way NOT to teach kids to love math is to inundate them with arithmetic. My math education, like many in the US, was an endless barrage of memorizing facts, solving equations, then memorizing some more facts. It was not until college that I began to see the essence of math is creativity. Kids are naturally creative and inquisitive and we should use this to our advantage in early education. We should explore topics like topology and infinity, topics that still blow my adult, math-major mind. We also need to encourage students to be creators in math. One of the ways we learn to love reading is by writing and making up our own stories. We can have students make puzzles, write computer programs, and see that they can invent things with math. For a more informed perspective on how we teach math all wrong, I recommend Lockhart's Lament: http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf

joezydeco 14y ago
I spent an hour last night working on a math homework assignment with my second-grader. The practice was adding up single digits to sums greater than 10. The idea behind the problem (say 9+5) is to "make a 10" and then figure out the sum from the remainder. So 9+5 -> 10+4 = 14. But he didn't know which numbers routinely added up to 10. That was never memorized. The assignment was so focused on strategy and technique that it neglected the fact that the basic tools aren't there. There's a point where you need to memorize something to make the strategy and discovery easier.
aswanson 14y ago
We should explore topics like topology and infinity, topics that still blow my adult, math-major mind... One of my happiest conversations was with my 7 year old inquiring into infinity...days after our conversation I overheard him saying to a classmate, 'Inifinity is not a number, it's a statement saying numbers never end...'

marcpaul22 14y ago

I loved "Prof. E. McSquared's Calculus Primer" when I was learning Calculus (look for used prices): http://www.amazon.com/gp/product/0913232475/ref=olp_product_details?ie=UTF8&me=&seller= http://www.amazon.com/gp/product/0913232475/ref=olp_product_... Home page here: http://www.math.sjsu.edu/~swann/mcsqrd.html http://www.math.sjsu.edu/~swann/mcsqrd.html Fun and funny, written in a comic book format. It illustrates the concepts in such an entertaining style, but makes the point better than many textbooks.

calibraxis 14y ago
Along those lines, you might enjoy _Thinking Physics_. (http://www.amazon.com/Thinking-Physics-Understandable-Practical-Reality/dp/0935218084 http://www.amazon.com/Thinking-Physics-Understandable-Practi...)

5xz41s0P8T5N 14y ago

As a math undergrad dealing with mathematical anxiety (how silly, I know), I wish math education started with logic. Is there evidence that learning discrete math first leads to better skill, or that it is easier/more fun for kids? or even teachers? I suspect I only feel this way because discrete math seems so foundational now, but perhaps what is taught matters less than how it's taught. I wish education was more empirical.

csense 14y ago

Most elementary schools have all subjects taught by one teacher. (At least mine did; my K-12 education was entirely in the US public school system.) Kids taking different subjects from different teachers doesn't happen until middle or high school. So I'd guess that most teachers of young kids probably chose to train as elementary-education generalists, and math was hard and scary for them. There are several mechanisms by which this might rub off on their pupils: 1. If the teacher doesn't have an innate joy and passion for their subject, the lack of enthusiasm may be contagious. 2. The teacher may have no idea why things like the addition algorithm work, and no idea how to explain it to their students. 3. Most of the math classes in elementary school were largely spent reviewing material covered in previous years. Having an expectation that students should know material that's been covered in previous lectures is simple common sense. I'm not against an occasional review class, mind you, but spending an entire quarter or semester on nothing but previously covered material strikes me as a colossal waste of everyone's time. 4. High-stakes standardized testing is a recipe for disaster. The tests don't accurately measure anything but the amount of time and effort teachers spend teaching to the test. I was pretty far out of elementary school by the time No Child Left Behind hit the fan, but I'd imagine even knowledgeable, passionate teachers who want to motivate their students with the joy of learning to use their imagination to manipulate abstract ideas get immense pressure from above to turn their class into nothing but drill, drill, drill. 5. The very name, "No Child Left Behind," implies is that the goal is to have the entire class proceed at the pace of the least capable student. On the face of it, this is deliberately aiming for poor results. For those of us who have never observed the human race in action before, let me explain: People are different; they have different levels of mental strength, motivation, talent, and parental support; they have different areas of interest, different areas of talent, put in different amounts of effort, and think in different ways. It is inevitable, then, that some people will be better at math than others. But we seem to be trying extremely hard to deny this simple and obvious truth, to bring the least capable students to a level of competence that they may never achieve, at great cost to taxpayers and immense frustration for everyone. Meanwhile, the middle- and high-capability tiers of the class are bored, frustrated and alienated by the endless repetition of very dull tasks -- which is most assuredly not what mathematics is about, especially since the invention of cheap, ubiquitous computers.