RE: [Math 2.0] Problem solving - UNIZOR.COM

[z...@unizor.com]

I am working on a project Unizor which aims exactly at the art of problem solving. Basically, it's a relatively complete course of high school level math with problem solving at its core. It contains lectures where theorems are proved, not just stated, where many problems are solved, not presented as properties, and there are some meaningful exams that attempt to test students' abilities to reason.

Some time ago MathFuture had a session about Unizor where Maria Droujkova interviewed me about how it works and what it does, look at http://mathfuture.wikispaces.com/Unizor

Zor Shekhtman
Founder of Unizor Education 
Creative Mind through Art of Mathematics
http://www.unizor.com 

-------- Original Message --------
Subject: [Math 2.0] Problem solving
From: roberto <robe...@gmail.com>
Date: Mon, March 04, 2013 2:47 am
To: "mathf...@googlegroups.com" <mathf...@googlegroups.com>

Dear group,

If you had to suggest some resources about mathematical problem solving skills and processes, based on your experience, what would you choose ?

I'd like to shift problem solving reasoning to a much central place in my courses (age 14 - 18).

Thank you very much.

--
Roberto

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[Joshua Zucker]

I never made it past the front page of http://unizor.com because the message

"Students: You are not looking for fun lifting weights in a gym. You spend time and effort there to achieve meaningful results in physical development. Don't look for fun developing your mind. You have to spend time and effort to achieve meaningful results that last the life time."

is not at all what I want to communicate.  In my view, students should spend time and effort *because* it is fun.

--Joshua

[Melissa Tomlinson]

i would so love to see this type of program for the middle school level. 

[Maria Droujkova]

I think "the f-word" (fun) is too general and generic to be of use, either "pro" or "con." Allow me to push you to particulars! After all, precision is a math value.

Zor, what do you mean math is not supposed to be fun?

Joshua, what do you mean math is fun?

Cheers,
Dr. Maria Droujkova

919-388-1721 

[Alexander Bogomolny]

Joshua has already mentioned the Mathematical Circles Library from MAS/MSRI. 

For the middle school, Anna Burago's "Mathematical Circles Diary" stands out. 

Here is my review http://www.cut-the-knot.org/books/Reviews/MathCircleDiaries1.shtml

Alexander Bogomolny

[Melissa Tomlinson]

Thank you - I will look into this.

[Maria Droujkova]

On Mon, Mar 4, 2013 at 10:50 AM, Alexander Bogomolny <abo...@gmail.com> wrote:

Joshua has already mentioned the Mathematical Circles Library from MAS/MSRI. 

For the middle school, Anna Burago's "Mathematical Circles Diary" stands out. 

Here is my review http://www.cut-the-knot.org/books/Reviews/MathCircleDiaries1.shtml

Alexander Bogomolny

Alexander's site, of course, is a great resource for problem-solving and problem-posing (and more)! Math Future event with Alexander is here: http://mathfuture.wikispaces.com/Cut+the+Knot

I second the recommendation for Anna's book. I just got my copy a couple of week ago. I really like how each problem set has a theme, connecting individual problems into something bigger than the sum of the parts. There are teacher suggestions in each theme. Another strong feature is "discussion of the day" - conversation prompts, clues about student thinking, and other teacher treasures. 

Cheers,

Maria Droujkova

[Melissa Tomlinson]

Thank you Maria - how is it for alignment with common core?  I have a Spec Ed resource room setting that I want to implement something new into.

--

[Maria Droujkova]

On Mon, Mar 4, 2013 at 11:00 AM, Melissa Tomlinson <jmtrh...@gmail.com> wrote:

Thank you Maria - how is it for alignment with common core?  I have a Spec Ed resource room setting that I want to implement something new into.

You will find it aligned exceptionally well with the Eight Practices of the Common Core - and, as you may know, that's the hard part to align! Most of the topics can be linked to particular standards, as well, but the book is not organized by the CC names of curricular areas. This curriculum is appropriate for a resource room or a math circle, rather than as the main, mandatory CC curriculum. Having said that, I would absolutely use it as the main math course for, say, a homeschool semester of math.

Cheers,
Dr. Maria Droujkova
919-388-1721

[Christian Baune]

Hi,

I would like to add two more docs I wrote : 

https://docs.google.com/presentation/d/1bc1491b8Ro4PnNpTN01lPSp_E2wK-jhr4IvlX6gf_-U/edit?usp=sharing

https://docs.google.com/presentation/d/11K8gslQ6mGvKpQsE4lTM4IH8AYEDZ-AIJmMtX4tOWhI/edit?usp=sharing

Kind regards,

Christian

[z...@unizor.com]

Maria is right. Fun is too broad a term and saying that math is not supposed to be fun sounds scary for some people.

Basically, I stand by my comparison with a gym. I see people are running, lifting weights, pulling springs and do all kinds of things concentrating on these actions, sweating and clearly not experiencing feelings traditionally associated with a word "fun" (with a drink in a hand and a pleasant person talking to you or with a drink in a hand lying on a beach etc.) Pleasure is not what's on their faces. It's really hard work. They still do it for a sense of accomplishment they achieve in the area of physical development. That accomplishment is the goal and sweat is the price to achieve it. To feel that you have achieved something might be called "fun" if you wish, but it's not the word I would use.

Same with math. Its goal is intellectual development. Its tools are problems and theorems. To achieve the goal you have to prove hundreds of theorems and solve thousands of problems. Then, if you succeed, you'll feel exactly the same - the feeling of an accomplished goal.

"Fun" is traditionally associated with a pleasure you get with almost no efforts. Web search produced the following definition of "fun": "Enjoyment, amusement, or lighthearted pleasure". In this sense math education is not fun. It's not supposed to be easy because easily obtained results would not help to develop intellectual power inasmuch as easy physical exercises that do not require any real load on your muscles will not help in your physical development. You want real results - sweat it. Forget the fun doing it, but enjoy the results.

I am aware about "education is supposed to be fun" slogan, I can support it for small children, but I am very much against it for high school students. We practiced this approach for quite some time and nowadays everybody talks about "crisis of school education". I attribute this desire for fun (actually, desire to avoid any difficult brain exercises) to this crisis.

Zor Shekhtman
Founder of Unizor Education 
Creative Mind through Art of Mathematics
http://www.unizor.com 

Cheers,
Dr. Maria Droujkova

919-388-1721 

[z...@unizor.com]

Predicting a controversy with F(un)-word, I placed on the About page of Unizor some thoughts about it. Just click on a menu item About (at the bottom of a menu) on a front page of unizor.com and enjoy poetical foundation of the approach I consider effective to study math.

Zor Shekhtman
Founder of Unizor Education 
Creative Mind through Art of Mathematics
http://www.unizor.com 

-------- Original Message --------
Subject: Re: [Math 2.0] Problem solving - UNIZOR.COM
From: Joshua Zucker <joshua...@gmail.com>
Date: Mon, March 04, 2013 10:31 am
To: mathf...@googlegroups.com

I never made it past the front page of http://unizor.com because the message

"Students: You are not looking for fun lifting weights in a gym. You spend time and effort there to achieve meaningful results in physical development. Don't look for fun developing your mind. You have to spend time and effort to achieve meaningful results that last the life time."

is not at all what I want to communicate.  In my view, students should spend time and effort *because* it is fun.

--Joshua

[Maria Droujkova]

I love the dramatic way you use in your "about" frame (linking it directly for people's ease of reading) http://unizor.com/Doc/zor_about.html

Loved the poem:

But if, not yielding to their charm,
Your eye can plumb the gulfs of harm
Then learn to love me, read my verses.

I have my doubts about that Inferno quote, though, because of the 20th century history. I think people may associate the quote with "the bad guys" by now.

On Mon, Mar 4, 2013 at 11:00 PM, <z...@unizor.com> wrote:

Basically, I stand by my comparison with a gym. I see people are running, lifting weights, pulling springs and do all kinds of things concentrating on these actions, sweating and clearly not experiencing feelings traditionally associated with a word "fun" (with a drink in a hand and a pleasant person talking to you or with a drink in a hand lying on a beach etc.) Pleasure is not what's on their faces. It's really hard work. They still do it for a sense of accomplishment they achieve in the area of physical development. That accomplishment is the goal and sweat is the price to achieve it. To feel that you have achieved something might be called "fun" if you wish, but it's not the word I would use.

I think people specify "hard fun" or "hardcore fun" when they talk about such things. Still not the word I would use.

There is a similar conversation going on at LinkedIn - you need to be a member of the group to follow this link, though: http://www.linkedin.com/groupAnswers?viewQuestionAndAnswers=&discussionID=216975274&gid=33207 I think many people on this list are members.

Peter Appelbaum, in that discussion, sent a link to his article on the subject: http://gargoyle.arcadia.edu/appelbaum/scifun.htm

Here is a relevant quote: 

"curriculum materials in the United States construct a contradiction between the instrumental view of science as cultural capital (get a job, increase the US position in a global market, etc.), and the means proposed to reach it (fun)."

[Sue VanHattum]

>

feelings traditionally associated with a word "fun" (with a drink in a hand and a pleasant person talking to you or with a drink in a hand lying on a beach etc.)

Hmm, maybe passionate people use the word differently. I have always wanted my activities to be fun, and having a drink in my hand sounds way less fun than sweating over a math problem, pushing myself hard on a bike, in the water, or in a yoga pose (sometimes painful, but still fun). Putting together a book is a crazy amount of work, but it has mostly been very fun for me.

I have had a great time exploring mathematical territory, even when I didn't solve a problem. There is pleasure in much of it. (Sometimes it makes me feel dumb, and that's not fun.) Even the tedious parts can be fun. When I'm stuck, or typing something into the computer that may help me solve the problem, it's kind of like chewing on a bone. That's fun (for me).

I would recommend that, instead of saying it's 'not about fun', you leave that out entirely, and talk about the hard things people enjoy doing. I don't think the only enjoyment rock climbers get is when they reach the top. It's super hard work, and I think it's 'fun' for those who do it most of the time they're engaged in it.

Another hard thing people enjoy doing (that seems related to math) is making art. It's hard work, and I think there is pleasure all along the way. The painter may not be smiling as she paints, but I imagine the process brings great pleasure, long before there's any certainty that the painting will become what was in her mind's eye. One time, I wrote a story that seemed to come to me from the characters. That was pure joy. Editing it, after it poured out, was fun too. (Sometimes writing is painfully hard for me, and I do need to think about my goal to get anything out of it.)

>Its goal is intellectual development.

I don't think so. I think the goal of math is exploration, and maybe construction, of a world inside our minds. (Or is that what you meant?) There is beauty. There is playfulness. Children pour themselves into their play. In fact, we might find kids' definitions of fun different from adults. Many adults work at alienated labor all day, and feel a need for the 'checking out' kind of fun you describe, in order to recuperate. When I was working 50 hour weeks, I needed something easy for my evenings. I bought tons of young adult novels - my mind candy, and read in the tub. But kids, they (often) work hard when they play.

Yes, math is hard work. And hard work can be fun. If you say what you have to offer is not fun, then I couldn't imagine choosing to do it. I try to only do things I find engaging.

Thank you for giving me a chance to think this through.

Warmly,
Sue

---

From: z...@unizor.com
To: mathf...@googlegroups.com
Subject: RE: [Math 2.0] Problem solving - UNIZOR.COM
Date: Mon, 4 Mar 2013 21:00:53 -0700

[Joshua Zucker]

I'm still having trouble articulating what I mean by "fun" in the context of math -- or of other things for that matter!

Sue has done a pretty good job of saying what I was trying to figure out how to say -- her explanation matches pretty well, though not perfectly, with the ideas I was wrestling to figure out how to communicate.

Meanwhile here's a quote from astronomer Phil Plait about a recent image of Saturn, from http://www.slate.com/blogs/bad_astronomy/2013/03/05/cassini_sees_venus_spacecraft_sees_venus_while_orbiting_saturn.html:

"I love Cassini pictures from Saturn. Not just because they’re so lovely, and show such a magnificent view of our Universe—though there is that. But it’s also because they so commonly twist my brain up, giving me just enough information to figure out what I’m seeing, but not making it so screamingly obvious. It sometimes takes a bit of thought to unravel what the pictures are saying, and what the eye is telling the brain.

It’s a bit of mental gymnastics that always makes me smile. Science is fun."

It's a pretty picture, too.

--Joshua

[Alexander Bogomolny]

In the preface to "Winning Ways" Berlekamp, Conway, Guy wrote, in particular,

"It's not a book on recreational mathematics because there's too much serious mathematics in it. On the other hand, for us, as for our predecessors Rouse Ball, Dudeney, Martin Gardner, Kraitchik, Sam Loyd, Lucas, Tom O'Beirne and Fred. Schuh, mathematics itself is a recreation."

I never saw a discussion on whether mathematics is a tool for developing intellectual abilities - among mathematically educated fellows, of course - which uniformly led to a certain conclusion. To me it says that the thesis is wrong for one of two reasons:

1. Either intellectual development does not imply logical thinking, or

2. Logical thinking may not be a secure way to arrive at conclusions since the premises for the derivation have no logical foundation.

Exactly as in sports, the achievements of anybody who was not born to be a (math)athlete will forever remain mediocre at best, regardless of the amount of effort invested into the preparation and exercise. On the other hand, a focus on development of motivation may lead to extra intellectual development. Fun or no fun, without motivation there is little hope for any progress in education.

Alex B

[David Wees]

"Exactly as in sports, the achievements of anybody who was not born to be a (math)athlete will forever remain mediocre at best, regardless of the amount of effort invested into the preparation and exercise."

But this is not true!

You've implied that our "talents" are based entirely on our genetic make-up while we keep finding evidence that the contrary is fact true - that our environment and how we are raised has an enormous impact. Given the plasticity of the human brain, enormous changes and gains can be made in one's intellectual ability through sufficient effort. 

Now, as we age, it is true that our capacity for change diminishes, if only because more change needs to happen, but I do not believe for one instant that one's intellectual (or athletic) achievements are predetermined by your parents and your early upbringing. If I did, I quickly leave education as a field for I would see it as a hopeless endeavor.

David

[Alexander Bogomolny]

You see, David, I was right in part, or have you missed that? I assume we both have had reasonably good math education. So why do we disagree?

May it be that somehow we managed to talk at cross-purposes? Are you sure that you understood what I meant? Did I say that education has no role in intellectual development? Are you sure that we agree on the meaning of the term "mediocre"?

To settle on the latter, could you give me an example of anybody who - in your view - has uniquely excelled in a field to which he/she had no innate talent?

Alex

[Sue VanHattum]

Alex,

I am a slow learner when it comes to singing. I don't know how much 'innate talent' I have, but it's not impressive. Will I ever be a great singer? Probably not. But I do think it's possible. (I love singing, but don't do it much. I am not good enough to perform. I am good enough to sing together with friends.)

We have a wide range of what we can achieve with our 'innate talents', and how much we achieve depends on how much work we put into it, which partly depends on how much joy we get from the work.

I am much better at math and at writing than at singing, though even with those, you might use the word 'mediocre'. I wouldn't. I hope to someday excel as a writer, and I didn't start out looking like much of a writer when I was younger. (Ahh, the run-on sentences...)

One way people excel is to do something new. Is my combination of talents and passions different enough that I can come up with something excellent? I think maybe.  ;^)

Warmly,
Sue

---

Date: Tue, 5 Mar 2013 14:35:55 -0500

Subject: Re: [Math 2.0] Problem solving - UNIZOR.COM

From: abo...@gmail.com
To: mathf...@googlegroups.com

[David Wees]

How can I? Anyone who is highly successful in a field does an enormous amount of work in order to get that stage. One might say that they were started down that path because of their talents, another might say that the affordances of their life enabled them to put the effort into an area in order to become highly successful.

I can offer some anecdotal evidence that there is more to exceptional ability than just talent (even if the question of whether talent itself exists at all must be left open).

Wayne Gretzky (arguably one of the best hockey players ever, if not the best) is a perfect example. It seems obvious that he must have tremendous talent in order to achieve at the level that he did, but he has some advantages that make this analysis a bit thin.

• He started practicing shooting a hockey puck at the age of 2! His parents let him hit the puck around soon after he learned how to walk, and he ended up damaging the washing machine in his basement as a result. This suggests that he had a strong parental influence that encouraged him to get started early, because no two year old knows about hockey or will play with a hockey stick in this way without someone mentoring them.
• His father built a hockey rink in their backyard so that Wayne could practice whenever he wanted. How many six year old children do you know that have a hockey rink in their backyard? It is well known that an early advantage can lead to widening gaps in learning over time (look at the difference in the "achievement gap" in the U.S. at a young age and at an older age for an example of this).
• He was born in January. The cut-off age for the leagues in Canada is the end of December, which means that he was almost a year older than the youngest kids in his group. Usually he ended up playing the league above him as a result, and faced much stiffer competition early on.
• He moved to Toronto to play in a different team when he was 14. This would have involved significant financial and emotional support from his parents, which not many kids have (unfortunately).
• His early scoring records were set with the Edmonton Oilers of the 1980s, who included some of the best players in hockey at the time. How much of his early scoring records were the result of the hard-working and dedicated exceptional players with whom he played?
• Through out his long career, he nearly always showed up early to practices and stayed until the end, working extremely hard all the time.

So was Wayne Gretzky just a talented player? Or did he have a life-time of advantages that gave him a competitive edge over everyone else in his field?

David

[Alexander Bogomolny]

Sue, first off, I like singing, even though, every time I do it annoys everyone about. (That's a joke of sorts.)

Here's a definition from wikipedia (that is not necessarily the greatest authority):

// start wikipedia definition

Talent can refer to:

• Talent (measurement)
• Aptitude, a talent is a group of aptitudes useful for some activity, talents may refer to aptitudes themselves

An aptitude is a component of a competency to do a certain kind of work at a certain level, which can also be considered "talent". Aptitudes may be physical or mental. Aptitude is not knowledge, understanding, learned or acquired abilities (skills) or attitude. The innate nature of aptitude is in contrast to achievement, which represents knowledge or ability that is gained.

// end wikipedia definition

Let's agree on that, or suggest anything else. But we need to know that our premises are the same.

I would combine motivation and passion into one component of success and enjoyment. What I wrote, i.e., the part of my thinking that I wanted emphasized is that it should be an important part of education to develop this component. For, without it, even a talent will not lead to success, much less knowledge expanded.

Alex

[Alexander Bogomolny]

Please, David, what did you try to prove? That work, environment, education help develop a talent. But, of course. 

What I said could be translated in terms of Wayne Gretzky's example as, 

If Wayne Gretzky did not have a talent to start with, all the parent's effort would not lead to him becoming the Wayne Gretzky we know. I doubt that if his talent had not shown at the age of 2, his father would have built a hockey rink.

I've been dragging my kid from age 3 to 5 to a gym, because I like gymnastics. Eventually I gave up and tried swimming. I was lucky on the second attempt. The boy has an affinity to competitive swimming. That the environment is entirely different (unlike in gymnastics where they took turns on a single device, they swim in a loop all together, with no time to cool down and lose interest) might be a factor - I do not know.

Talent needs work, too. I recommend Ray Bradbury's "Zen in the art of writing" as a testimony. 

I do not disagree with you. Nor your examples disprove anything I said. So what is it all about?

Alex

[John Sharp]

You might look at the website

http://www.mathsisfun.com/index.htm

John S

[Maria Droujkova]

On Tue, Mar 5, 2013 at 2:35 PM, Alexander Bogomolny <abo...@gmail.com> wrote:

To settle on the latter, could you give me an example of anybody who - in your view - has uniquely excelled in a field to which he/she had no innate talent?

Alex

Alex, I am concerned, because the above description does not seem falsifiable (refutable). http://en.wikipedia.org/wiki/Falsifiability

So I am not sure, even with definitions, that it can be discussed as is. Proving lack of innate talent is about as easy as proving the lack of the Flying Spaghetti Monster (may you be touched by Its Noodly Appendage).

I suggest we tackle this same general area somewhat differently, using whole-culture approach. For example, deductive reasoning and abstract math are non-universals among cultures. Speaking in terms of cultures, not individuals, everybody laughs, and everybody plays games, but not everybody makes axiomatic proofs. I recently read about studies that seem relevant: http://www.psmag.com/magazines/pacific-standard-cover-story/joe-henrich-weird-ultimatum-game-shaking-up-psychology-economics-53135/

It was in the 1960s, for instance, that researchers discovered that aspects of visual perception were different from place to place. One of the classics of the literature, theMüller-Lyer illusion, showed that where you grew up would determine to what degree you would fall prey to the illusion that these two lines are different in length:

Researchers found that Americans perceive the line with the ends feathered outward (B) as being longer than the line with the arrow tips (A). San foragers of the Kalahari, on the other hand, were more likely to see the lines as they are: equal in length. Subjects from more than a dozen cultures were tested, and Americans were at the far end of the distribution—seeing the illusion more dramatically than all others.

The growing body of cross-cultural research that the three researchers were compiling suggested that the mind’s capacity to mold itself to cultural and environmental settings was far greater than had been assumed. The most interesting thing about cultures may not be in the observable things they do—the rituals, eating preferences, codes of behavior, and the like—but in the way they mold our most fundamental conscious and unconscious thinking and perception.

For instance, the different ways people perceive the Müller-Lyer illusion likely reflects lifetimes spent in different physical environments. American children, for the most part, grow up in box-shaped rooms of varying dimensions. Surrounded by carpentered corners, visual perception adapts to this strange new environment (strange and new in terms of human history, that is) by learning to perceive converging lines in three dimensions.

Cheers,

MariaD

[Sue VanHattum]

How does this website figure into our discussion?

• I see at least one cute, fun for me, hard enough puzzle.
• Mostly the stuff is basic, and I can't judge how engaging it would be for young students.
• My first take is that they just used the word fun for marketing purposes. That goes with what I think Maria would say - that the word 'fun' is somewhat empty. (Is that why fun is the f-word ,Maria?)

Warmly,
Sue

---

Date: Tue, 5 Mar 2013 21:58:31 +0000

Subject: Re: [Math 2.0] Problem solving - UNIZOR.COM

From: js.sli...@gmail.com
To: mathf...@googlegroups.com

You might look at the website

http://www.mathsisfun.com/index.htm

John S

[Alexander Bogomolny]

Maria,

I do not particularly care about specific definitions; I do care about choosing and agreeing on one. It does make sense to agree on the topic of discussion. Environment, culture, neighborhood, financial circumstances, what not, do effect children development - whatever definition we may choose. I never questioned that.

I heard it said that when kids were polled, most preferred a 4x3 rectangle to the golden one. This will probably change with wider adoption of wide screen.

I hope we agree on that. What remains, in your view?

Alex

--

[Maria Droujkova]

On Tue, Mar 5, 2013 at 5:33 PM, Sue VanHattum <suevan...@hotmail.com> wrote:

• My first take is that they just used the word fun for marketing purposes. That goes with what I think Maria would say - that the word 'fun' is somewhat empty. (Is that why fun is the f-word ,Maria?)

The site may be an ostensive definition of "math fun" (pointing at a list of examples): http://en.wikipedia.org/wiki/Ostensive_definition

For me, the word fun is not well-defined or precise. It does not convey the same meaning to different people, as our discussion here demonstrates so well. Therefore, the word is often useless - but much can be gained if you ask people what it is they meant by "fun"! This is similar to the effect of the words "interesting" or "important." Writers and presenters should probably stay away from those words, just like curriculum and game developers should stay away from the word "fun."

Cheers,

MariaD

[Maria Droujkova]

On Tue, Mar 5, 2013 at 5:54 PM, Alexander Bogomolny <abo...@gmail.com> wrote:

Maria,

I do not particularly care about specific definitions; I do care about choosing and agreeing on one.

I don't know of a definition of talent that could tease it apart from the environment. That's why I said claims about "achievement without innate talent" can't be proved or refuted at all.

In math in particular, the best predictor (not cause, but statistics predictor) of success is the number of hours spent with the subject. But how does it relate to the innate abilities, and can we measure that relationship? 

As a kid, I played Olympiads competitively. Last I tried to solve an Olympiad-type puzzle fast, I beat a roomful of a (math-related) conference attendees on time. Yet I always feel slow and not-innately-good at math, because it takes me a long time to develop these patterns of solutions. I have developed a lot of the problem-solving patterns by now, so I am a powerful learner at that puzzle level. But I don't feel particularly talented at it. And I realize that nothing in this story would work as a proof, or a refutation, of any claims about work vs. talent. My mom did work on math puzzles with me starting at the age of three or younger.

To use David Wee's example, some sort of rudimentary "hockey rinks" (ice fields) are routinely, naturally available to kids living in appropriate climates. Parents in the north of Russia would routinely make rinks around apartment buildings, for example. You just have to use hot water, so it freezes smoothly. I remember that trick from when I was little - everybody knew that. This got to influence hockey abilities, if only the culture also includes that game!

It does make sense to agree on the topic of discussion. Environment, culture, neighborhood, financial circumstances, what not, do effect children development - whatever definition we may choose. I never questioned that.

I heard it said that when kids were polled, most preferred a 4x3 rectangle to the golden one. This will probably change with wider adoption of wide screen.

I hope we agree on that. What remains, in your view?

Should we teach everybody? - that is the question that endures for me.

Should we teach different people differently? - another one. 

These questions connect to the issue of "fun" and to the issue of "nature vs. nurture" (talent/ability). 

Cheers,

MariaD

[Ted Kosan]

Maria wrote:

>In math in particular, the best predictor (not cause, but statistics predictor) of
>success is the number of hours spent with the subject. But how does it relate
>to the innate abilities, and can we measure that relationship?

I think the first step in the process of measuring innate abilities is
to precisely define what abilities are being measured. The research
that was done in the 1970s that I described in another post indicates
that numerous rules that are needed to perform mathematics properly
are not taught explicitly. One conclusion I draw from this research is
that most students are not actually being taught mathematics.

What are they taught that is being called mathematics? I don’t know.
What would the game be called that operated exactly like chess, except
the rules for how the rook, the knight, and the queen moved were never
written down and never taught explicitly? Lets call it c_es_. One
critical ability that is needed to be good at c_es_ that is not needed
for being good at chess is that of figuring out how these pieces are
moved by only observing how other people move them.

Can one conclude that students who don’t have the innate ability to
play c_es_ also don’t have the innate ability to play chess?

Also, I think the reason that the number of hours spent working with
“math” is the best predictor of success is because it takes a very
long time for people to learn a game when a significant number of its
rules are hidden from them. One reason I think this is because we
homeschooled our two sons, and the oldest spent less than ⅓ the time
studying mathematics than a typical student usually does in grades
K-12. Before he went to college, he only worked through 2 ½
mathematics books, and he never took even one test. During the first
2/3 of the time he spent studying mathematics, he was so mediocre that
one would think that he did not have any ability for mathematics at
all. During the last ⅓ of this time, however, he ended up
understanding it at a deep level.

He is currently a junior in college majoring in mathematics and
physics, and he has a 4.0 GPA. How was he able to spend so little time
studying mathematics and still accomplish this? I think its because he
spent his time learning mathematics and not _a_he_at_cs :-)

Ted

[Alexander Bogomolny]

>I don't know of a definition of talent that could tease it apart from the environment. That's why I said claims about "achievement without innate talent" can't be proved or refuted at all.

And this is why I said I do not care about the definition. But I am sure we know what is the purpose of the discussion. It is most certainly not the sources of, or reasons for, talent. My goal at picking up the discussion was to clarify how much and to what extent studying mathematics at school, online, or wherever else, may affect intellectual development in the absence of innate aptitude and motivation. I claim and believe that everyone's experience bears witness to that in the absence of motivation studying mathematics in the present format makes a minuscule contribution, if any. This also holds for those who have innate aptitude: they - if lucky - learn to solve math problems, but rarely become "smarter overall" just for that reason. 

My main thesis is this: there are umpteen ways to enhance intellectual development of children, and studying mathematics is not one of them, or at least far from the best, except in some rare cases. In any event, mathematics should not be forcibly studied under this pretext, but rather for its intrinsic value - and this is impossible in the absence of motivation and innate abilities.

In the hope of preventing another digression from the topic that interests me, I do not believe that in what I said there is anything against teaching mathematics. Mathematics (its rudimentary aspects mostly) is an extremely useful tool, mastering which may benefit many a citizen. It's an educational calamity that for the sake of that modicum of knowledge children are made to undergo years of frustration.

>Should we teach everybody? - that is the question that endures for me.

>Should we teach different people differently? - another one. 

These are very relevant questions.

Alex B

--

[Maria Droujkova]

On Wed, Mar 6, 2013 at 9:37 AM, Alexander Bogomolny <abo...@gmail.com> wrote:

I claim and believe that everyone's experience bears witness to that in the absence of motivation studying mathematics in the present format makes a minuscule contribution, if any. This also holds for those who have innate aptitude: they - if lucky - learn to solve math problems, but rarely become "smarter overall" just for that reason. 

Ooh, so you are talking about coercion?

The question of coercion of children is harsh and thorny. First, because minors can't freely give consent (neither can older students, and that's why professors aren't allowed to date them, for example). Second, because the states mandate legal guardians to coerce children along many dimensions, including mandatory mathematics education.

My answer to this question, "Should we force people to study math?" lies outside of the realm of mathematics and mathematics education. I have moral objections to coercing people. Whether math works for intellectual development or not has no bearing on my decisions around these issues, because moral arguments override utility arguments, in my reasoning. 

For a field with related moral reasoning dilemmas, consider research on humans and issues of human subject protection, especially in the face of horror stories of the twentieth century. No amount of utility in research overrides the need for consent of the participants. 

http://en.wikipedia.org/wiki/Human_subject_research

My main thesis is this: there are umpteen ways to enhance intellectual development of children, and studying mathematics is not one of them, or at least far from the best, except in some rare cases. In any event, mathematics should not be forcibly studied under this pretext, but rather for its intrinsic value - and this is impossible in the absence of motivation and innate abilities.

I think math is a good way to grow - emphasis on "a"! We can work on making it a better way, too. For example, what you do in your Eye Opener series enhances the abilities of people to grow using math objects - by transforming the objects into eye openers: http://www.cut-the-knot.org/pythagoras/tricky.shtml

And I absolutely agree that X should not be forcibly studied, for any X. I consider forced study immoral.

[roberto]

Do you know if the book from A. Burago's has a version for Grade 5 to 7 only ?

I wondered if there is also something for high-schoolers.

Thanks so much for this intriguing discussion.

Cheers,

Roberto. 

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--
Roberto

[Alexander Bogomolny]

The book is subtitled "Year 1". For the preface, the circle ran for about a decade. So I expect a "Year 2" sequel in the near future.

Best wishes,

Alex