Teaching Kids to Love Math
online.wsj.com
online.wsj.com
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. :)- Agree with this. "By doing, we Become"
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?
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.
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.
Can you elaborate? To me, the interesting part of mathematics is the process of manipulating expressions, not the expressions themselves. For example, the number pi would be pretty boring if it weren't for the theorems related to it.
I think most mathematicians and physicists would disagree. I think such people wouldn't want distinguish between generating equations and studying them and their effects, especially with physical equations that must reflect reality to have any value.
For example, I can make elaborate predictions about a lot of physical reality with:
f = G M_1 M_2 / r^2 (gravitational force)
And if I want to study tidal forces, all I need to do is take the first derivative of the above:
f = 2 G M_1 M_2 / r^3 (tidal force)
I have just "manipulated" an equation, but with an awareness of the equation's physical consequences and effects, and I can now use the second equation to make predictions about ocean tides, the future position of the moon, and the temperature of the volcanoes on Io (which arise from the energy of tidal forces). So for me, the equation itself is immensely valuable for modeling reality -- it's more than a case of manipulating terms.
> For example, the number pi would be pretty boring if it weren't for the theorems related to it.
Pi, as it turns out, may serve as a source for random sequences of digits. So apart from its geometric meaning, Pi stays interesting.
Right, but the fun part (for me) is finding out how and why pi can be used as a source of random sequences of digits, not the fact itself.
> I think most mathematicians [...] would disagree.
I would be surprised to meet a mathematician who enjoyed learning theorems/proofs more than finding theorems and proving things on their own. That's why I said the interesting part is the process of manipulating expressions rather than the expressions themselves.
Based on your gravity/tides example, I think you would take the same stance (Do you?). I just wasn't clear enough in my previous comment.
One must crawl before one can walk. Reading the work of others is quite enjoyable. And not everyone yearns to reinvent the wheel -- for many problems, understanding the prior work gives one more than enough satisfaction.
> That's why I said the interesting part is the process of manipulating expressions rather than the expressions themselves.
A finished equation can stand alongside the finest art, and garner the same kind of appreciation. People still read Einstein's relativity equations, and Maxwell's electromagnetic equations, with a deepening appreciation of their beauty, quite apart from the degree to which they describe reality.
Also, there is the interesting task of applying equations to real-world problems. I don't need to rewrite the gravitational and tidal equations to discover new things while applying them.
> Based on your gravity/tides example, I think you would take the same stance (Do you?).
Not really. It would be like asking someone whether they prefer reading, or writing. Obviously the full experience of literacy involves both.
Here's a comparison -- a common problem for student writers, usually pointed out by someone with more experience, is that they haven't read enough to be able to write effectively. There's a parallel in mathematics -- those who take the trouble to read enough mathematics, by so doing learn how to express themselves more efficiently.
For me, the two equations I posted earlier, one that describes the gravitational force, and the other the tidal force, the mathematically interesting thing is the relationship between them, not so much the equations themselves -- the fact that a simple derivative operation produces the second equation (in physical terms it's because the tidal force is felt by any two adjacent masses placed arbitrarily close together).
For physical equations, working with them means either imagining their consequences, or modeling them, usually with a computer. In that case, you don't manipulate the equations, you use them to model reality. So having an equation that's known to represent some aspect of reality is just the beginning of a research program that models the consequences of the equation and compares the model to reality.
Here's an example. Observations of Jupiter's moon Io revealed the possibility of a large, static tidal force on its mass. You may be aware that a static force doesn't require any energy expenditure (imagine a book lying on a table). Then someone pointed out that Io has an elliptical orbit, which means Io is constantly moving toward, and away from, Jupiter. This would have the effect of changing the tidal force, and a changing tidal force would perpetually change the moon's shape -- and changing the moon's shape would require energy. This would generate a lot of heat. Shortly thereafter, volcanoes were observed on Io, a moon too small to have the kinds of volcanoes we have here, and the explanation was the elliptical orbit and tidal force.
That's mathematics.
Your parent comment missed the point of his parent.
That might be a fair objection if they could be separated. If I say y = cos(x) * e^-x^2, when expressed in two dimensions I get this:
And in three dimensions I get this:
But I'm still looking at y = cos(x) * e^-x^2 -- the picture only confirms the mathematical identity. So it is with Avatar -- when we watch Avatar, we're looking at math.
And much of nature is defined using math -- note that I said, not described, but defined. Here's an example -- there are species of locusts (cicadas, actually) that reproduce at 13-year and 17-year intervals. Until recently, no on knew why. It turns out that both 13 and 17 are prime numbers, and this ties into a survival strategy hatched by natural selection (quite by chance). All explained here:
http://arachnoid.com/prime_numbers/index.html#Mathematical_L...
In other words, the locust survival strategy is math speaking out loud.
If you read a book in which a seashore is described, do you argue that the description isn't germane to the thing being described? If the writer isn't skilled, or the reader is lacking in the capacity for visualization, then that is perhaps a legitimate objection, but for most people, words convey meaning. So does math. Math is a language in much the same way that words are a language.
The distinction between "loving math" and "loving things made of math" is dubious at best. When P.A.M. Dirac wrote his now-famous equation that describes how relativistic electrons behave, he noticed that it had two solutions -- sort of like a quadratic equation.
At first he doubted that there could really be two kinds of matter, as his equation suggested. But within a few years antimatter was observed, and Dirac forever afterward wished he had been willing to take his own equation at face value and predict antimatter himself.
My point? The math told him something about reality that no one knew, even him. To put it another way, the math was reality.
Do you mean the computer-graphics movie, or The Last Airbender? None of those really has any mathematical interest for the viewer -- though The Last Airbender rocks.
I meant Avatar, the computer-graphics movie, although I really like The Last Airbender, and I wish it had gotten more attention.
Avatar is nothing more or less than a very long mathematical expression, sort of like a computer plot conducted very carefully for aesthetic reasons. When they watch Avatar, viewers are looking at mathematics.
> None of those really has any mathematical interest for the viewer
People who watch Avatar, who find it interesting, are appreciating mathematics even though they may not realize it. All the lighting, the colors, the motions, are expressed by the mathematics of physical reality.
When a real tree falls, it's very mathematical. When the big tree fell in Avatar, that was also mathematical. As to the first, mathematics could be used to predict exactly how the real tree falls. As to the second, the falling tree was the result of a computer solving a differential equation that described the tree and gravity.
They're really the same -- mathematics predicting reality, and mathematics generating an imaginary reality.
What do I mean by that? This is mathematics: http://www.hdwallpapers.in/walls/jake_sully_avatar_2009-norm...
And this is arithmetic: 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.
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.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.
Good question -- and it seems this idea is being put into practice. Here are some examples:
"Using Art to Teach Fraction, Decimal, and Percent Equivalents" : http://mason.gmu.edu/~jsuh4/math%20masterpiece.pdf (PDF)
"Teaching Math Without Words, A Visual Approach to learning Math" (TED talk) : http://www.youtube.com/watch?v=7odhYT8yzUM
"Teaching with Visuals: Students Respond to Images" : http://www.edutopia.org/visuals-math-curriculum
There are many other similar examples. Here's one from an article I wrote ("each member of the running sum of odd numbers is a perfect square"):
http://arachnoid.com/example/index.html#Math_Example
The graphic says it all.
> The thing I would want to look at in such intervention studies is how the teaching effectiveness is measured.
That will take much longer and is fraught with measurement difficulties (as with all social science questions). We might end up with people possessed of self-confidence about their math skills, but who can't balance a checkbook. (I say this as the devil's advocate, not because I actually believe that will be the outcome.)
> 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.
Yes, a perfectly legitimate concern, and one especially important if one is to openly advocate a change in public school curricula. It is equally clear that there are no reliable data on outcomes at this time.
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.
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.
For a more informed perspective on how we teach math all wrong, I recommend Lockhart's Lament: http://www.maa.org/devlin/LockhartsLament.pdf
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.
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...'
http://news.uchicago.edu/article/2010/01/25/female-teachers-...
http://www.jstor.org/discover/10.2307/41192533?uid=3739736&#...
and from time to time regret the gaps in their own mathematical education.
http://www.ams.org/notices/200502/fea-kenschaft.pdf
"The teachers are eager and able to learn. I vividly remember one summer class when I taught why the multiplication algorithm works for two-digit numbers using base ten blocks. I have no difficulty doing this with third graders, but this particular class was all elementary school teachers. At the end of the half hour, one third-grade teacher raised her hand. 'Why wasn’t I told this secret before?' she demanded. It was one of those rare speechless moments for Pat Kenschaft. In the quiet that ensued, the teacher stood up.
"'Did you know this secret before?' she asked the person nearest her. She shook her head. 'Did you know this secret before?' the inquirer persisted, walking around the class. 'Did you know this secret before?' she kept asking. Everyone shook her or his head. She whirled around and looked at me with fury in her eyes. 'Why wasn’t I taught this before? I’ve been teaching third grade for thirty years. If I had been taught this thirty years ago, I could have been such a better teacher!!!'"
The last time I posted a link to this article on HN, another HN participant kindly posted a link to what is surely the "secret" referred to by the elementary school teacher,
http://www.tech4mathed.com/MAT156/topics%20test%202/twodigit...
pedagogical content knowledge that would be very routine for any elementary mathematics teacher in east Asia.
http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...
(book link above, review links below)
http://www.ams.org/notices/199908/rev-howe.pdf
http://www.math.wisc.edu/~askey/ask-gian.pdf
So this advice for parents is good in helping parents provide a supportive environment for their children's mathematics learning.
I have frequent occasion to write about mathematics education here on Hacker News. My occupation is 1) providing supplemental lessons in advanced mathematics to pupils from ten counties in Minnesota through a nonprofit corporation I cofounded and 2) coordinating parent workshops and other aspects of the summer program Epsilon Camp,
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)
http://www.artofproblemsolving.com/
(where I first took on the screenname that I also use here on HN) for more online mathematics instruction resources, and I also share specific links to specialized sites on particular topics with clients and with my children. I should note for onlookers that the articles on mathematics learning on the AoPS website
http://www.artofproblemsolving.com/Resources/articles.php?
are very good indeed, especially "The Calculus Trap."
My children make quite a bit of voluntary use of Khan Academy (both watching videos and working online exercises) and I am gratified that my previous suggestions to the Khan Academy developers here on HN
http://news.ycombinator.com/item?id=2760663
have been followed up as Khan Academy developers have communicated with me by email about new problem formats available in their online exercises, which are becoming increasingly challenging.
Besides that, I fill my house with books about mathematics, and circulate other books about mathematics frequently from various local libraries.
I also recommend that all my students use the American Mathematics Competition
materials and other mathematical contest materials as a reality check on how well they are learning mathematics.
In general, I think mathematics is much too important a subject to be single-sourced from any source. Especially, mathematics is much too important to be left to the United States public school system in its current condition. I was rereading The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999) last month. It reminded me of facts I had already learned from other sources, including living overseas for two three-year stays in east Asia.
"Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x
"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77
Plenty of authors, including some who should be better known and mentioned more often by HN participants, have had plenty of thoughtful things to say about ways in which United States mathematical education could improve.
A discussion of the Common Core Standards in Mathematics, "The Common Core Math StandardsAre they a step forward or backward?"
http://educationnext.org/the-common-core-math-standards/
gets into further details of how mathematicians look at the general school curriculum in the United States. It is not the worst curriculum possible, and survivors of the system often have access to outside resources to supplement school lessons, but the public school instruction in mathematics in the United States still shows plenty of room for improvement.
The last time I posted about these issues, a reply asked what I think about essay "Lockhart's Lament." I think it is an interesting read, but less practical for reforming mathematics education than I had hoped. I wonder if Lockhart's forthcoming book Measurement
http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755...
will be a successful attempt to teach mathematical reasoning to students who have already lost confidence in learning mathematics, which would be a great contribution to society.
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?
These resources are under constant discussion on homeschooling forums.
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.
Do you have an opinion on it?
http://www.amazon.com/gp/product/0913232475/ref=olp_product_...
Home page here: 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.
The popular books by mathematician Ian Stewart
http://www.amazon.com/s/ref=ntt_athr_dp_sr_1?_encoding=UTF8&...
are very interesting and mathematically accurate. Some readers also like the books by Keith Devlin,
http://www.amazon.com/Keith-Devlin/e/B000APRPC6/ref=ntt_athr...
one of which I am reading right now.
I like almost every book by John Stillwell
http://www.amazon.com/John-Stillwell/e/B001IQWNS2/ref=ntt_at...
and especially recommend the latest edition of Mathematics and Its History
http://www.amazon.com/Mathematics-Its-History-Undergraduate-...
as a book you should try to obtain from a library to see what a book with challenging, interesting, but accessible problems looks like.
Many people like the videos that feature Edward Burger
http://www.thegreatcourses.com/tgc/professors/professor_deta...
or Arthur Benjamin lecturing about math in the Great Courses (Teaching Company) video lecture series, which you may be able to find at a library.
AFTER EDIT: Here is a link for Calculus Made Easy, a book recommended by another participant here.
http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/d...
First step: choose a problem that you need to solve -- a goal. Think about the problem, how valuable it would be for you to get a grip on it. And it doesn't matter what the problem is, as long as it's important and tangible.
Second step: learn how to solve that problem. Don't necessarily understand all the details at this stage, but know how to solve it in practice, for any statement of the problem.
Third step: learn the details of why the above solution works.
And guess what? The above is a very effective way to acquire an understanding of mathematics, and acquire self-confidence as well. And it is exactly the reverse of our normal math education curriculum.
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.
http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/d...
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.
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
The authors have a way of digging deeper into problems than you might expect. It's not a book where you learn some theorems, then apply them in the problems.
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.