How Do You Spark a Love of Math in Children?
blogs.kqed.org
blogs.kqed.org
He brought home his report card the other day and got a failing grade in math, apparently, he won't draw lines between numbers (eg. connect two apples to the number two.) His mom went to talk to the teacher with my son, the teacher explained all the tasks, in front of the teacher she asks my son what's 32+21, 53 he says, the teacher gets wide eyed in amazement and then informs his mom that they can't put him into a more advanced math class because the first years are concentrated on getting the english kids conversational in french. According to the school French > Math, C'est la vie!
Oddly enough he just got an award for embracing the french culture, I personally think it's very french of him to avoid doing pointless work.
I had the same problem in school, I wouldn't do the homework, would ace the tests and get questions about cheating because I never did the homework. I avoided higher education for much the same reasons, I'd rather just open a book, teach myself and move on rather than engage in random excersizes. You only need to do the a^2+b^2=c^2 a few times to understand the concept.
The school system has no interest in mathematics but has an intense interest in making kids perform strange rote tasks that have something to do with numbers.
I see here and there (khan academy, psychology publication in Europe) a tendency to kick painful myth out of learning, but I'd like to see a cohesive effort to rethink the learning systems we call schools. Especially the early years. If any of you here knows people working on this I'd be happy to read about their thoughts.
My 3-year old daughter has been at a Montessori school since August last year and the progress she's made --in manual work, counting, drawing, and reading, is incredible. Her vocabulary is now equivalent to a 5-year old.
See http://www.youtube.com/watch?v=GcgN0lEh5IA for a fun, decent exposition of what a Montessori education does by contrast with traditional schooling.
"math" is the game of rote memorization that kids play in school where they try to memorize arcane rules that seemed to have been pulled from some greek dudes ass with no explanation.
Math is a legendary construct whispered about as a hypothetical possibility in high school classrooms. A problem solving exercise that has you think, with a logical progression from base rules and assumptions.
"Math" and Math both use the same language and the same symbols, with a lot of the same concepts. I've never seen Math, and only know of it's existence through hear say from mathematicians, but been through a lot of "Math". Imagine if we tried to teach English this way. It would look a lot like memorizing words out of the dictionary. And then playing ad-libs with out of context sentences.
What's an eigenvector?
It's the solution to an eigenfuction!
What's an eigenfunction?
It's an equation that has an eigenvector as the solution - now goto the next chapter...
Now kids, what did you learn?[0]
Don't guess the teachers password?! Excellent! Next chapter...
[0] As I was writing that I thought that there could be a paradox somewhere in there if you phrased it right.
Then again, functional analysis as an admission requirement for a graduate science program seems like it'd be a bit of a stretch in most places.
An eigenvector (for a linear map `T`) is a non-`0` solution `v` to an equation of the form `Tv = λv`. (I suppose one could call the equation an eigenequation, but I've never seen anyone do that.)
An eigenfunction is exactly the same thing; we just tend to use the terminology when `T` is operating on a space of functions. (For example, one may refer to an exponential map as an eigenfunction of the differentiation operator on the space of smooth functions; it would be equally correct, but probably confusing, to call it an eigenvector for that operator.)
That was a big part of me learning English (my fourth language). You need to memorize something when learning something — just to get a foundation to operate on.
But they can tell you all sorts of arbitrary rules about English, and probably something about iambic pentameter.
Arithmetic can help show some deep mathematical concepts that might not be obvious if you study theory alone. For example, the advantages of arabic numerals over roman numerals. There are important mathematical distinctions between the two representations, but without actually using them both, it's hard to understand why arabic numerals are so much more useful for so many purposes.
I think it's best to switch between concepts and practical applications. I also don't think there is a problem with memorization. In any field, to exercise good judgement or gain deep insight, logic and reason are not enough. You need a collection of facts to guide you as well as other people's ideas to draw from.
This article and research is a non-starter. It doesn't address the core problem. Mathematics is a beautiful way of thinking about the abstract, yet it's taught as nothing more than a procedural tool.
Teach it like music. Instill a love of mathematics as the art that it is. Reconstruct how we present mathematics to children and they will have a chance to love it, like they might love music or art or science.
These micro-problems are just hairline cracks in the broken pieces of a shattered cup. Why patch the cracks if the cup will never hold water? We need to forge a new one.
"I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers—the kind of thing a real mathematical education might provide." -- Paul Lockhart
Any competent music teacher (I write as the husband of a piano teacher) can perform music with musical expression, and thus show students examples of the beauty of music. But the very best music teachers are also intimately familiar with all the isolated subskills that build into understanding a piece of music, and controlling the performer's muscle movements, and responding to the audience in a live setting to build a coherent, musically expressive performance. My wife teaches skills such as "music mapping,"
http://www.amazon.com/Mapping-Music-Learning-Teachers-Studen...
proper hand position,
http://www.amazon.com/Playing-Less-Hurt-Prevention-Musicians...
and how to tie those and many other skills together
http://www.amazon.com/Art-Piano-Playing-Heinrich-Neuhaus/dp/...
as part of a comprehensive process of teaching making music.
The basic problem with mathematics education at the elementary school level in the United States (see my previous reply to this thread
http://news.ycombinator.com/item?id=4034248
generally commenting on the submitted article) is that elementary school mathematics teachers can do NONE of the comparable things with mathematics that a good music teacher can do with music. They cannot isolate and focus on useful techniques, they cannot put on an example performance of solving an interesting, challenging problem, and they cannot make connections between their (poor) understanding of the problems found in elementary mathematics and their students' (often better, but different) understanding of the same problems. In the United States, people mostly seek music instruction for young learners from the private enterprise system. My wife gains most of her new clients, who are crammed into her busy teaching schedule as previous clients graduate from secondary school and go off to university, from friends of current clients who are happy with her work. By contrast, an elementary school teacher in a typical government-run school in the United States teaches on a take-it-or-leave-it basis, with attendance being compulsory in default of a government-approved alternative, and funding guaranteed to the school, and thus employment guaranteed to the teacher, whether the learners learn mathematics or not. Systemic change is necessary to get mathematics taught as music is taught to elementary-age pupils in the United States.
For an eye-opening look at how elementary mathematics teachers could be prepared, and how that would help learners, see Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma.
http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...
That book is a very enjoyable--but rather shocking--read, full of information about how to teach mathematics for "Profound Understanding of Fundamental Mathematics."
I remember my elementary school music classes with deep chagrin. The teacher had a deck of cards with photographs of musical instruments.
"This is an oboe. It has a double reed."
And so on, for fifteen minutes or so.
Most people do not have the intelligence for much abstract thought. Look at how much trouble CS students have with recursion, and they are in the top few percent of intelligence.
> Teach it like music.
The majority of people do have enough tone perception, coordination, and muscle memory to make music. It does not compare to math.
they are in the top few percent of intelligence
"What about geometry? Don’t students prove things there? Isn’t High School Geometry a perfect example of what you want math classes to be?"
His reply
"There is nothing quite so vexing to the author of a scathing indictment as having the primary target of his venom offered up in his support. And never was a wolf in sheep’s clothing as insidious, nor a false friend as treacherous, as High School Geometry. It is precisely because it is school’s attempt to introduce students to the art of argument that makes it so very dangerous.
Posing as the arena in which students will finally get to engage in true mathematical reasoning, this virus attacks mathematics at its heart, destroying the very essence of creative rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject, and permanently disabling them from thinking about math in a natural and intuitive way.
The mechanism behind this is subtle and devious. The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.”
As Lockhart notes "Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions."
a) My teacher sat down with me and explained the WHY. It wasn't just formulas and rules but actual explanations that allowed me to approach problem questions with logic rather than a formula book.
b) Khan Academy. The ability for me to go back to the very start and become proficient in the very basics of maths without looking like and idiot and at my own pace was amazing. Some people laughed when I started at the very bottom in khan academy and worked through most of the program but the things that I have learnt and were never taught to me (or I ignored them when they were taught) was just astounding.
c) Choosing physics as my main science where I had to use maths to solve real world problems. When I discovered that maths could actually be used in real life to solve real problems I extremely motivated to read more about the formulas and their deeper roots.
Now I have graduated high school and when ever younger kids ask me what they should do and focus on I tell them that they should focus on maths and make sure they understand everything when its taught because learning it later when you are on more advanced topics is extremely painful.
I think the biggest problem school's today in relation to maths is that we are allowing people to move on in the curriculum with a maths score of 50%. In other subjects it doesn't matter because the content next year isnt based completely on the content of this year but in maths it does. If only I had been told in year 6,7,8,9,10 "You are going to have to repeat this year unless you improve you maths score".
All it takes with maths is practice and giving kids a reason to learn it other than "you have to". Maths is one of the most amazing things I have ever come across in my (short) life and once I truly understood its power, my view of the world was never the same. Maybe my experience is unique but it truly think that many kids are suffering the problems as i had.
I mean that's the whole point of math - to provide an explanation in which every step is necessary, and understood to be necessary.
If there's one class that should be Pass / Fail, this is it, and the bar for passing should be set VERY high. I agree that this would create problems, but those would fall squarely on the schools, who - as many others have mentioned - teach "math", not math.
One of the best books I´ve read in scientific divulgation is Fermat´s last theorem by Simon Singh . It tells the story of the assault to the theorem beautifully. You follow different mathematicians through history and see how they advance and how they fail, how it affects their lives, their careers and also human history.
Although the mathematic concepts involved at the end of the book are pretty advanced (the book is not heavy in mathematic formulas) you feel able to understand them and I feel that one of the reasons is due to the beautiful context every thing is set on.(the other of course is the great writing job Simon Singh does). I still remember some of the developments involved, and more than 7 years have passed since I read the book.
It sparked in me a deeper interest in mathematics, I used to like them, but it was not the kind of curiosity I have now.
Of course you can not "learn" with this book, but it is a great tool to put everything in a context, and learn how it advances. I really think that teaching some kind of history of mathematics (greeks, Egyptians, arabs, etc..what they discovered, how, where they got stuck and why) + mathematic concepts (no exercises needed in this class) could be a great way to spark the interest of students, and see mathematics in a human and achievable way.
edit: some typos, and some excess of "mathematics" to be cleared.
Article suggests a few strategies for addressing those problems, and references the source research paper [2].
1. http://blogs.kqed.org/mindshift/2012/03/how-to-deal-with-kid...
2. http://psycnet.apa.org/index.cfm?fa=search.displayRecord&...
A singular teacher will be able to integrate all subjects together during the school day.
When you switch to department teaching there is less emphasis on the importance of a particular subject. For example a students English teacher will not show the importance of Math.
Also teachers seem to hesitate to tell a student that they are incorrect. Teacher cushion their responses by saying this like "hmmm, ok... lets see if Johnny agrees" or "I'm not sure about that, lets try it this way". This works alright for most subjects, but Math is pretty black and white. You either get the answer or you don't.
Students also don't get grades in elementary school. Getting a poor grade in a difficult subject like math could turn them off.
It's like playing a reasonably enjoyable, if a little tedious, puzzle game. As a side-effect of playing it they will just happen to learn all K through 12 math. I went through the whole thing when they had 280 exercises over the course of a month. It was a great review.
Here's a link: http://www.khanacademy.org/exercisedashboard
I'm not an educator, but an observer with a particular interest in teaching hard science/math/cs subjects to people who are not naturally talented at them.
Based on what I've read of neuroscience and our deepening understanding of the way the brain works, I suspect it's possible, but requires an entirely different way of going about it than currently. Anyone know of any research in this area?
First, you can advance grades without a full understanding of what you were taught. I'm not saying students need 100% comprehension, but letting someone go up in math with only ~50-60% comprehension is a terrible idea. Math builds, and like Jenga, you can't build without a firm foundation.
Math is taught as a set of rote "exercises" with loosely- (and vaguely-) connected theorems. Students are never really given an explanation of why something is useful or why they should even care. You're just expected to memorize it, recite it for the test, and forget it until the final rolls around. This is the mindset school has got us in: so many students don't realize the building nature of math before it is far too late.
Perhaps the most damning thing about math education is its focus on the what, not the why. For example, the quadratic formula: it is introduced and students are told that if you have an equation "of the form" ax^2+bx+c=0, you can "find x" by plugging in the numbers. Never are you given an explanation of where the quadratic formula comes from, never are you given an explanation of why you would want to solve a quadratic.
In my math classes, I was always trying to figure out the "why" behind something. Why were sine, cosine, and tangent all positive in the first quadrant, while only sine was in the second quadrant? We were never told, and most students never even bothered to question why this was so. In fact, when they were told "all students take calculus" and shown the pattern, they just happily nodded and thought "I'll remember that for the test." But if you understand the real reason behind "ASTC," you don't need a silly phrase to remember it.
And, finally, you have the "exercises." You're given a generic 'class' of problem, told how to solve it, and move on. Really good teachers will make some cursory attempt to explain why you can solve a problem how you can, but if a student already has been pushed up through a few math classes he should not have passed, it is likely to be well over that student's head. That feeling of mathematical incompetency just makes them tune the teacher out. And all the while, since grades are the de facto measure of your "worth" in school, the students are thinking "this explanation is too difficult: it won't be on the test." That's a dangerous mindset to cultivate... but the way the system is designed, it doesn't matter one whit.
The student can memorize some facts, never understand why they are as they are, and then study the generic classes of problems that will be present on the test. Then the student can ace the test, feel good about his or her self, and move on, all without any real comprehension. But that's the way the system is designed: you can't apply a teaching process that would really give a full education of math to everyone. For one thing, we don't have teachers skilled enough in math to do so. (A chicken-and-egg problem.) So we are stuck with these half-baked attempts at a curriculum, with kids gaming the system to get their A (or whatever standard they set for themselves) and nothing more.
It's sad. Math is such a beautiful field, full of mysteries and interesting connections. But thanks to the terrible math education system, the vast majority of high school graduates think math is a worthless field full of arcane formulae jumbled about in a seemingly random way all with no real logical structure.
This is not entirely the education's fault.
When I was taking algebra in 7th grade, the teacher did derive the quadratic formula for us. We studied completing the square first, and then he did it with generic symbols for A, B, and C. I thought it was WAY cool. I never memorized it; instead, for the next two years, any time I needed it, I simply re-derived it. You could find the derivation on the back of just about every math test I took in high school.
None of my classmates took that route, though. Every last one of them memorized the thing.
I was the sort of person who automatically thought the why of something was much cooler than the what, but I don't think most people are that way. I don't doubt math education could inspire an interest in the why, but I'm not optimistic that it will inspire everyone. At the very least, I think that would take more than merely presenting it.
I don't know. Maybe tests have pounded the natural curiosity out of kids, maybe it's just cynicism. But it's certainly my perception that the vast majority of them care more about the what.
That's not [exclusively] how I was taught the quadratic equation and it's solutions, in high school in the UK, either.
I wish people would remember to provide geographical context on HN. The article is about Australian middle school so should I assume that's what everyone is referring to with their generalised statements of "this is what 'math' is like"?
>I simply re-derived it //
This is why I did well at maths. Practically no memorisation required; you can start with something you know and derive what you need to answer the question.
That, IMO, is the big thing: students that are relatively poor at abstraction cannot see commonalities between problems that those with more talent find trivial.
Certainly later on, like with QFT, I was left to grope in the dark recesses of memory for the next step in a proof of some corollary or other but I found that understanding how a proof works means that the steps make sense in the same way as having to pull down your trousers before pulling down your underwear. Yes there is memorisation involved but nothing like that required to establish who was the King of France in 1492.
I did say "practically", perhaps "comparatively" would have been more to the point.
You were not endlessly drilled with diagrams of circles that made the signs obvious? Trigonometry without diagrams is like pie without fruit!
http://www.educ.msu.edu/content/sites/usteds/documents/USTED...
a few days ago, as I discovered the book Teacher Education Matters: A Study of Middle School Mathematics Teacher Preparation in Six Countries
http://www.amazon.com/Teacher-Education-Matters-Mathematics-...
at my alma mater university library as I searched for books about mathematics education, my occupation. (The book, in turn, appears to be based on a publication from the study
http://www.educ.msu.edu/content/sites/usteds/documents/MT21R...
that I was able to view in one Web browser but not another. Perhaps most of you HN participants can read the study publication directly online.)
The study found and the book reports that "Putting more resources into U.S. middle school mathematics teachers' education could significantly raise future teachers' mathematics skills but may not be sufficient to equal those in countries where mathematics skills are substantially higher or produce sufficient numbers of more highly skilled middle school mathematics teachers, for two reasons. Average mathematics knowledge among U.S. college students is much lower than in Taiwan, South Korea, or Germany, and because of the relatively low salaries and prestige of teaching in the United States, the college students enrolled in teacher education are likely to average much lower mathematics skills than the large number of students in science, engineering, and economics/business." (Pages 278-279) The book also reports, especially relevant as a commment on the submitted article here, "South Korean and Taiwanese future teachers included both simple and complex examples in their lessons, usually including these in the beginning and middle of the lesson. By contrast, sampled U.S. future teachers tended mostly to use simple examples and to include them at the very end of the lesson." (Page 289)
Teachers in the early grades having adequate mathematics preparation to help young learners advance in their understanding is a very severe problem in the United States, where it has been reported that most elementary school teachers in a sample of teachers in New Jersey did not know a general rule for finding the area of a rectangle if the side lengths of the rectangle are known.
http://www.ams.org/notices/200502/fea-kenschaft.pdf
The dramatic differences in teacher preparation result in dramatic differences in mathematics achivement between countries.
http://pirls.bc.edu/timss2007/PDF/T07_M_IR_Chapter1.pdf
(See Exhibit 1.1 on pages 34 and 35 of the .PDF document for an example of an excellent use of parallel boxplots to compare the centers of various groups.) In general, United States "average" students are at the bottom level of top-performing countries, while even "average" students in those countries are at a "gifted" level for the United States.
The FAQ page for Epsilon Camp collects some other writings about producing challenging (and thus engaging) lessons for mathematics learners, preparing them to go far in mathematics with a love for the subject.
http://tux4kids.alioth.debian.org/tuxmath/
TuxMath is part of the "Tix4Kids" project which provides similar software for other topics:
Have a read:
http://educationnext.org/miracle-math/
----- It was another body blow to education. In December of 2004, media outlets across the country were abuzz with news of the just-released results of the latest Trends in International Mathematics and Science Study (TIMSS) tests. Once again despite highly publicized efforts to reform American math education (some might say because of the reform efforts) over the past two decades, the United States did little better than average (see Figure 1). ... And in three consecutive TIMSS test rounds (in 1995, 1999, and 2003), 4th- and 8th-grade students in the former British trading colony of Singapore beat all contenders, including math powerhouses Japan and Taiwan. United States 8th graders did not even make the top ten in the 2003 round; they ranked 16th. Worse, scores for American students were, as one Department of Education study put it, “among the lowest of all industrialized countries.” -----
Further reading: http://personal.anderson.ucla.edu/jason.frand/math_enrichmen...
It sounds like the way math is taught in many public US classrooms is killing students' confidence and interest in math. There were a few news articles talking about how many students became interested in math again after the Singapore program was introduced in their school (see here: http://singaporemathsource.com/curriculum/schools-in-the-new... ). A common refrain was that the kids simply weren't understanding what was going on in math - as if they were stumbling around in the dark with no idea what the computations they were doing meant.
I've also read good things about other foreign curricula (e.g. "Russian math") being used in American schools and by homeschoolers, but none are as common as Singapore Math. Here's some praise about SM from American mothers, including sample word problems: http://www.redshift.com/~bonajo/singapore.htm - where you'll see simple algebra problems that are not solved by traditional algebra at all.
I can tell you as a Singaporean that I think the curriculum gave me a solid base and understanding of school math, and really stretched me at its harder portions. I hope more people will give it a try, especially if you have kids who aren't "getting" math or are bored of their current math curricula. Get the additional 'tougher' practice books (IP and CWP) for your bright kids to experience the full brain training of the Singaporean system.
Apparently planes are grounded, late children are driven to school by police officers and people actually gather at the schools to cheer on their relatives/peers on their way into school. What these events convey to me is a deep, cultural understanding of the importance of education.
In my high school, it seemed as though respect was garnered among students for (Excuse my language) "Not giving a fuck". The ability not to care seemed to be something to aspire to. Of course, this is entirely anecdotal, but a point of interest for me after seeing the incredible dedication students in other countries have.
Can you tell me a little about the culture surrounding education in Singapore? Is it at all similar to that of South Korea?
Test results matter when a majority of successful people got where they were because of their results. They did well in school and got good jobs (often in foreign MNCs); not many started their own businesses. Add to this the cultural motivation of wanting to keep up with the Joneses (being 'kiasu', in our lingo) and you have a strong incentive for parents to be all "responsible" for their kid by making sure she gets tutoring in whatever subjects she's weak in and she does additional practice outside of homework, often at a harder level than in school.
And of course, in the good schools, the ones that take in a higher-scoring crop of students, the teachers create harder test papers for their students to keep them motivated (we can't have everyone getting As now, can we?), and so parents go to tutoring companies with "challenging" or "enriching" programmes to make sure their kids keep up. The problems at this end of the pool get a bit ridiculous, and people end up complaining: http://everythingalsocomplain.com/2011/09/10/primary-6-maths...
The tide is turning slightly as new parents, freshly emerged from the competitive sea of schooling, take a less intense view of the importance of academic performance, and recognise that there are other paths that their kids can take in life that don't rely on test scores. We've opened specialist schools in arts, music, sports and design in recent years. But there's still a heavy economic pressure to do well and get good jobs in finance, medicine, law and so on, so I don't expect the culture of focus on education will weaken significantly in the near future.
I think the fundamentally different culture of American schools means that many Asian approaches will never transfer well; but there are still improvements that can be made, e.g. in the area of curriculum, in teacher training and ongoing development, etc. American schools are incredibly diverse in populations and practices; they have big weaknesses and big strengths. I think it would be good to have a unified tough-but-capable program for the struggling schools in poorer neighbourhoods, while letting strong/specialised schools do their thing. That would be something like the best of both worlds.