Pi is not a natural number (Skeptism and Mistakes are a good thing)
Linda Fahlberg-Stojanovska
I am organizing a course for P2PU and our page is http://classroomk12.wikispaces.com/P2PU+CSMI Please feel free to participate in the discussions on the pages (and to join the wiki and create your own pages of course).
This thread comes from 2 thoughts and if / when you have time I would really appreciate any thoughts on this. Even quick reactions are good.
(a) I read this headline - "Teaching with Skepticism and more ..." and
(b) In doing one of the problems for this course, I did not easily see that the diagonals split a rectangle into 4 equal areas and was willing to say so in my screencast.
I enjoy being skeptical and wondering why things work and whether I can prove them and if not - where they go wrong. I don't mind being confused and trying to figure out what confuses me. How do we bring this into the classroom - into the teacher's and student's psyche? The skepticism and the wanting to wonder, to explore, to prove, to make mistakes and by continuing to test, find and correct our mistakes is important.
I have to say that "rich problems" don't do it for me and I don't think they do it for students. Often they take up huge amounts of time trying to understand the text of the problem and not the math.
In contrast, small problems like we are doing in the course and screencasting and/or constructing with GeoGebra work for me. With screencasting, we talk/show our thinking process. And when I think aloud, I become more conscious of what I am saying/doing and skepticism has a chance to enter my work, i.e. "Why am I sure this is right?", "What facts/formulas am I using?" .
===A little more (argh you say) ==
First let me say that the headline was actually for "media messaging" not mathematics, but it seems to me that we teach math so "boldly".
As I said before, we state: the ratio of circumference of a circle to its diameter is pi. Like it is the most natural thing in the world. I think it is EXTREMELY strange. and I will bet that the Greeks (or whoever discovered pi) thought God must be having some fun. Firstly, it is strange that this ratio is always the same number, and secondly, it is strange that pi is not a particularly nice number. But we talk about pi like it is a natural number (pun intended). It is not.
Our students begin to think of math as static and fixed and facts and repetition.
Second it took me a while to see (b). I am not giving it away though - it was a wonderful ah-ha moment. I walked around with my folded rectangle in my pocket. I am relatively certain that in almost any hs geometry class, the teacher would think me slow for not knowing this. (I have found that if I teach something over and over the same way - I begin to think everything is obvious.) My point is that I allowed myself to be skeptical, to be confused and then - using my mathematics and logical thinking skills - I worked it out.
One of my participants (Esther) and I have been talking (see her page) about the skills that make you successful in life and the fact that these skills don't seem to be encouraged in today's classroom. Jumping in head first, being willing to make mistakes, testing our results and if they don't work, figure out why and fix them.
One of our problems was the challenge: Using GeoGebra, draw a line segment. Then construct a square with this line segment as diagonal.
I asked another participant (Steve) if we could give a diagonal to students and ask them to create a rectangle and he said "no" - the non-uniqueness would confuse them.
1. Is confusion bad if we teach them how to logically confront it? (Of course, we must start this process when they are very young.)
And a third participant (Cathy) was doing her "trying to think like my students". Just looking at the "image", I could NOT figure out why her construction wasn't a square. I finally realized that because the construction was done in GeoGebra, I could follow it (view->construction protocol) - like a screencast. Three points here.
2. I could "see" it wasn't right, but I didn't know why - However, I WANTED to spend the time to find the problem (do our students want to solve puzzles?)
3. By following the construction protocol, I could say to myself "Yes, we see the diagonals crossing perpendicularly at the midpoint. That's good. Ahh, C doesn’t look right. What’s up there?" and then to my student "Something about C doesn't look quite right. How did you find it?" The construction protocol is a great bonus from GeoGebra.
4. I see GREAT value in giving this construction as part of the problem and asking "Why isn't this a square?" Why? Because they may go right for the easy construction. To me, an important part of learning is recognizing something is wrong and finding out why.
As teachers, we want to help our students learn. To do this we must allow them to make mistakes, but teach them to test their results. With screencasts and GeoGebra, finding where they are not testing their thinking is easier both for us and for them.
Sorry, I know I go on and on....
Alexander Bogomolny
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Maria Droujkova
I have to say that "rich problems" don't do it for me and I don't think they do it for students. Often they take up huge amounts of time trying to understand the text of the problem and not the math.
Some students, most of them not very strong at math, can't relate to small problems at all. They don't see a point. People who are very divergent thinkers, likewise, need a lot of context to "anchor" them, or they ride away into the sunset of their private associations. The third category of students who crave rich problems are non-sequential thinkers (some call them "visual-spatial" but I dislike the term).
I think small problems vs. rich problems is a very important tag for seeking materials that are best for you personally. I also think everybody needs to learn how to deal with small problems AND rich problems, eventually.
As I said before, we state: the ratio of circumference of a circle to its diameter is pi. Like it is the most natural thing in the world. I think it is EXTREMELY strange.
milo gardner
|
|
Costello, Rob R
if the instructor explained less - or was apparently called away half way through - or demonstrated they were still learning about the object themselves - the child showed measurable tendencies to explore more as well
the last point (being a co-learner) is interesting : i know teaching IT i felt it was easier to position myself as a co-learner and problem solver who might not know everything : harder to model that in maths for some reason (though of course it was certainly true)
Papert also commented that the teacher in IT classes was actually more likely to disclose they hadn't worked things out in advance : and that this tended to encourage more genuine exploration from kids than when they knew the teachers knew all the answers in advance
interesting
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Mike South
Linda, hello:Pi is a pretty natural number.
What needs to be said prior to giving the definition is that all circles are similar. I wrote about that at http://www.cut-the-knot.org/pythagoras/NatureOfPi.shtml
Alexander Bogomolny
--
Hamilton, Eric
I bet some rich problems are satisfying to you, Linda... but just ones that mean something to you, and that is exactly as it should be. One of the great advances in math ed is the understanding that context matters. But we still consider context to be almost person free. Rich versus small might be a useful tag, but I think our challenge is to elicit or find the mathematics that will connect to individuals. All of us like to solve rich, all of us like to solve small – assuming that we “like” or are motivated by underlying context. If you have a problem you want to solve….
As far as a ratio equaling pi… let that forever be known to our students as one of the beautiful and elegant mysteries of mathematics, for such it is. And in so doing, we flexibly give deserved homage both to pure math and, in the paragraph above, to applied math.
kirby urner
On Mon, Jan 31, 2011 at 6:33 AM, Alexander Bogomolny <abo...@gmail.com> wrote:Linda, hello:Pi is a pretty natural number.I couldn't resist, and they just kept coming:
Edward Cherlin
<lfah...@gmail.com> wrote:
> I enjoy being skeptical and wondering why things work and whether I can
> prove them and if not - where they go wrong. I don't mind being confused and
> trying to figure out what confuses me.
Good. You might find the draft of my book at
http://booki.flossmanuals.net/discovering-discovery/
interesting.
> As I said before, we state: the ratio of circumference of a circle to its
> diameter is pi. Like it is the most natural thing in the world. I think it
> is EXTREMELY strange. and I will bet that the Greeks (or whoever discovered
> pi) thought God must be having some fun.
Not so much. However, the cube root of 2 is known to be a prank of the
Greek gods. Apollo demanded through his oracle at Delphi that the
Greeks double the size of his cubical altar. First they doubled the
side, which made it eight times larger, then they proved that 2^(1/3)
cannot be constructed with ruler and compass, and sacrificed 100 bulls
instead. Apollo is on record as being pleased with their solution, or
so the Greeks wrote down in their histories.
Euclidean constructions allow addition, subtraction, multiplication,
division, and square roots, just like a five-function calculator. It
is not very difficult to prove that cube roots don't qualify, in
somewhat the same way that one proves that square root of 2 is
irrational.
> Firstly, it is strange that this
> ratio is always the same number,
This follows logically from the existence of similar triangles, and
therefore of similar polygons enclosing and enclosed by similar
circles. The Archimedean construction works for any size circle in
Euclidean geometry.
On the other hand, you are just at the point where we can open up the
discussion of non-Euclidean geometry, where similar triangles are
congruent, and the ratio of circumference to diameter does depend on
size.
> and secondly, it is strange that pi is not
> a particularly nice number. But we talk about pi like it is a natural number
> (pun intended). It is not.
Not like a natural number. A necessary, unavoidable, and beautiful number.
> Our students begin to think of math as static and fixed and facts and
> repetition.
That's because of two centuries of teacher training thought of as
factory automation, intended to give students and teachers just enough
skill to work in a factory, store, or office, and not enough to be
able to understand politics and economics, which were the concerns of
the monarchies and aristocracies at the beginning of the Industrial
Revolution, and of corporate owners and management later on. The basic
idea was that every student could be given the same lesson from the
same book on the same day, and the teacher would not have to
understand the subject like a practitioner. It would be enough to
present the lesson as written, and then to check for right answers to
homework using the teachers' edition.
The actual history of math is much more interesting, particularly at
those points where somebody invented more of it, and others got
furious about such nonsense or even blasphemy as
* Irrationals. Legend tells us that the Pythagoreans were so incensed
at one of their number, who proved square root of 2 irrational, that
they threw him overboard from the ship they were on at the time.
* 0, introduced to Europe during the Crusades and denounced by the
Church as blasphemy, which is why 12-hour clocks still start at XII.
* Arabic numerals more generally (Fibonacci). Roman numerals were
perfectly adapted to use with the abacus, which, however, could not
cope with compound interest. Business people did not accept Arabic
numerals until Pascal and Leibniz invented practical mechanical
calculators in the 17th century.
* negative numbers (still not used in double-entry bookkeeping).
* imaginary numbers (grudgingly admitted in the solution of cubic
equations in the 17th century, but not fully accepted until the 19th).
* non-Euclidean geometry (19th century: Gauss, Riemann, Lobachevsky,
Bolyai vs. "the clamor of the Boeotians").
* higher-dimensional geometry (numerous mathematicians vs. Kant)
* infinite numbers (Cantor vs. Kronecker, and later Brouwer)
* fractal dimension (Hausdorff and Mandelbrot vs. assorted naysayers)
* non-standard arithmetic (Peano vs. Thoralf Skolem)
* infinitesimals (Newton vs. Berkeley, 17th century, until Abraham
Robinson's hyperreal numbers and John Horton Conway's surreal
numbers).
and much more. There is also the astonishing period from about 1890 to
the 1930s when almost every attempt to define mathematics resulted in
contradiction, until most mathematicians were willing to accept that
any consistent, definable mathematics is necessarily incomplete. There
is of course much more to the story than that.
Nearly a thousand years later, Hindu/Arabic numerals, including 0, and
negative numbers have made it into elementary education, but not any
of the others. Irrationals and imaginaries can be discussed in high
school, but the rest is college-level. Children can't possibly
understand whatever most adults can't cope with. ^_^ But see
Calculus by and for young people
> Second it took me a while to see (b). I am not giving it away though - it
> was a wonderful ah-ha moment. I walked around with my folded rectangle in my
> pocket. I am relatively certain that in almost any hs geometry class, the
> teacher would think me slow for not knowing this. (I have found that if I
> teach something over and over the same way - I begin to think everything is
> obvious.)
They say that if the only tool you have is a hammer, everything begins
to look like a nail. I have found that if all you do is pound nails,
everything begins to look like a hammer.
[Other interesting and valuable comments omitted because I have
nothing to say about them.]
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/
Maria Droujkova
.
As far as a ratio equaling pi… let that forever be known to our students as one of the beautiful and elegant mysteries of mathematics, for such it is. And in so doing, we flexibly give deserved homage both to pure math and, in the paragraph above, to applied math.
These are beautiful words, Eric. I find it quite useful to pause before diving into explanations, and acknowledge the high perplexity (Dan Mayer) of mathematics.
Carol (who leads Psychology of Math Learning at our P2PU school) and her son Madison and I just spent a couple of hours drawing fraction division models, as extensions of whole number division models. There was a step in that extension where the analogy breaks and people get stuck conceptually when they learn how fraction division works. The way Madison described the feeling in his story, "So then one quits with logic and goes into pure math." We paused for a good long time drawing and discussing the leap into the mystery. It felt so good.
We will write it up in more detail soon, and share the pictures, but I wanted to thank Eric for reminding to love the mysteries of mathematics
Cheers,
Maria Droujkova
Make math your own, to make your own math.
Mike South
Mike,> Can you elaborate on how you make the r/R argument when you use this approach?Perhaps you can spare another 5 minutes and look at the link
In the framework of similarity, existence of π is self-evident. Since all circles are similar, their relative (or shape) attributes are the same for every circle."
As to the pi being natural, do you question its naturalness? I'd leave the wisdom of using this term with the students to Linda, but is the gist of her quandary is not clear to us the careful readers?Buddy?
Alexander Bogomolny
--
kirby urner
Not so much. However, the cube root of 2 is known to be a prank of the
Greek gods. Apollo demanded through his oracle at Delphi that the
Greeks double the size of his cubical altar. First they doubled the
side, which made it eight times larger, then they proved that 2^(1/3)
cannot be constructed with ruler and compass, and sacrificed 100 bulls
instead. Apollo is on record as being pleased with their solution, or
so the Greeks wrote down in their histories.
Euclidean constructions allow addition, subtraction, multiplication,
division, and square roots, just like a five-function calculator. It
is not very difficult to prove that cube roots don't qualify, in
somewhat the same way that one proves that square root of 2 is
irrational.
> Firstly, it is strange that thisThis follows logically from the existence of similar triangles, and
> ratio is always the same number,
therefore of similar polygons enclosing and enclosed by similar
circles. The Archimedean construction works for any size circle in
Euclidean geometry.
On the other hand, you are just at the point where we can open up the
discussion of non-Euclidean geometry, where similar triangles are
congruent, and the ratio of circumference to diameter does depend on
size.
> and secondly, it is strange that pi is notNot like a natural number. A necessary, unavoidable, and beautiful number.
> a particularly nice number. But we talk about pi like it is a natural number
> (pun intended). It is not.
> Our students begin to think of math as static and fixed and facts andThat's because of two centuries of teacher training thought of as
> repetition.
factory automation, intended to give students and teachers just enough
skill to work in a factory, store, or office, and not enough to be
able to understand politics and economics, which were the concerns of
the monarchies and aristocracies at the beginning of the Industrial
Revolution, and of corporate owners and management later on. The basic
idea was that every student could be given the same lesson from the
same book on the same day, and the teacher would not have to
understand the subject like a practitioner. It would be enough to
present the lesson as written, and then to check for right answers to
homework using the teachers' edition.
The actual history of math is much more interesting, particularly at
those points where somebody invented more of it, and others got
furious about such nonsense or even blasphemy as
* Irrationals. Legend tells us that the Pythagoreans were so incensed
at one of their number, who proved square root of 2 irrational, that
they threw him overboard from the ship they were on at the time.
* 0, introduced to Europe during the Crusades and denounced by the
Church as blasphemy, which is why 12-hour clocks still start at XII.
* Arabic numerals more generally (Fibonacci). Roman numerals were
perfectly adapted to use with the abacus, which, however, could not
cope with compound interest. Business people did not accept Arabic
numerals until Pascal and Leibniz invented practical mechanical
calculators in the 17th century.
* negative numbers (still not used in double-entry bookkeeping).
* imaginary numbers (grudgingly admitted in the solution of cubic
equations in the 17th century, but not fully accepted until the 19th).
* non-Euclidean geometry (19th century: Gauss, Riemann, Lobachevsky,
Bolyai vs. "the clamor of the Boeotians").
* higher-dimensional geometry (numerous mathematicians vs. Kant)
* infinite numbers (Cantor vs. Kronecker, and later Brouwer)
* fractal dimension (Hausdorff and Mandelbrot vs. assorted naysayers)
* non-standard arithmetic (Peano vs. Thoralf Skolem)
* infinitesimals (Newton vs. Berkeley, 17th century, until Abraham
Robinson's hyperreal numbers and John Horton Conway's surreal
numbers).
and much more. There is also the astonishing period from about 1890 to
the 1930s when almost every attempt to define mathematics resulted in
contradiction, until most mathematicians were willing to accept that
any consistent, definable mathematics is necessarily incomplete. There
is of course much more to the story than that.
Nearly a thousand years later, Hindu/Arabic numerals, including 0, and
negative numbers have made it into elementary education, but not any
of the others. Irrationals and imaginaries can be discussed in high
school, but the rest is college-level. Children can't possibly
understand whatever most adults can't cope with. ^_^ But see
http://www.mathman.biz
Calculus by and for young people
> Second it took me a while to see (b). I am not giving it away though - itThey say that if the only tool you have is a hammer, everything begins
> was a wonderful ah-ha moment. I walked around with my folded rectangle in my
> pocket. I am relatively certain that in almost any hs geometry class, the
> teacher would think me slow for not knowing this. (I have found that if I
> teach something over and over the same way - I begin to think everything is
> obvious.)
to look like a nail. I have found that if all you do is pound nails,
everything begins to look like a hammer.
[Other interesting and valuable comments omitted because I have
nothing to say about them.]
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/
Linda Fahlberg-Stojanovska
With respect to pi not being a natural number, that was indeed just (my attempt on) a pun on both the set N and the fact that we seem to take mathematics for granted. I love that I can get pi from a circle, sqrt(2) from a square of side 1 and sqrt(3) from an equilateral triangle of side 2 or a rhombus with a diagonal and sides 1 (just made a P2PU 45 second challenge http://www.youtube.com/watch?v=5vRd5dKQQvU)
Ed – do I have to register to read?
Rob - too much explanation. I sort of understand this in K-7 (Do I or do I not show the resulting rhombus in my video?). I do not understand “too much explanation” in algebra. That is, I do not understand the logistics, e.g. what might be a specific sample class that gives too much explanation, what is a possible solution?
With respect to “rich problems”. Here is my take – yet again.
I just this moment googled rich problems mathematics and below is the first actual problem in the top link. I could understand the words so I thought “I will give it a try.” to see if it is possible that new and better rich problems have been developed since I last looked.
City Hall has a rectangular lobby with a floor of black and white tiles. The tiles are square, in a checkerboard pattern, lined up with the walls: 93 tiles in one direction and 231 in the other. There are two mouse holes, at diagonally opposite corners of the floor. One night a mouse comes out of one mouse hole and runs straight across the floor, and into the other mouse hole. How many tiles does the mouse run across? A complete solution and handouts are provided.
Question: What is this teaching? How can I possible get started thinking about it? (I tried drawing it and gave up immediately – the grid was incredibly small. I tried finding the gcd to reduce it – it is 3 so not much of a reduction.) Do I count 2 tiles or 4 tiles if the mouse crosses a diagonal point? How long will I spend setting it up before any math comes up? Argh!
And speaking of teaching and learning math …. Here is the link for the article on math sophistication in Maria’s note about Ko’s class: http://ed526b.wikispaces.com/file/view/Mathematical_Sophistication.PDF
Please scroll down to page 9 and then page 10.
Am I nuts?
MMSI Sample problem 2 so offends me mathematically I can almost scream. Either set HF1=0 or start at HF1=6. I don’t understand how HF1=1 and even if I did, I wouldn’t write that in this problem. It is deliberately confusing.
Are people altruistic? (ok - I have been reading super freakonomics)
MMSI Sample problem 3 sounds like the “Are people altruistic?” survey. Hmm –I wonder what the “correct” answer is? Do I actually have to spend time? No? Then, let me choose the one where I look like I am interested.
BTW: How on earth can you “make a guess about how other functions might be used to model data”? I can understand making a guess that other functions might model other kinds of data, e.g. “Hey, maybe some data is better modeled by an exponential function?” and I can understand asking whether or how other ….. But I would be quite interested in how one makes a guess about how …
I did like the first sample problem and I think the hypothesis and conclusions of the article very important, but I would certainly want to see all the questions before drawing any firm conclusions.
Alexander Bogomolny
--
Linda Fahlberg-Stojanovska
Yes – so why can the problem say 6x11 and not 231x93? I totally agree.
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Alexander Bogomolny
Sent: Wednesday, February 02, 2011 7:47 PM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Pi is not a natural number (Skeptism and Mistakes are a good thing)
Linda,
Alexander Bogomolny
Mike South
Yes – so why can the problem say 6x11 and not 231x93? I totally agree.
Mike South
The reason may be in that I never red that "rich" problem in the first place. I also think it's unrealistic for a City Hall to be so void of furniture as to allow straight diagonal runs.
kirby urner
Costello, Rob R
HI Linda,
i really was just making general comment on your approach
"I don't mind being confused and trying to figure out what confuses me. How do we bring this into the classroom - into the teacher's and student's psyche? The skepticism and the wanting to wonder, to explore, to prove, to make mistakes and by continuing to test, find and correct our mistakes is important."
i like that sort of mindset : like you i spend a lot of time with GeoGebra (and formerly with Scratch and various introductory programming environments - trying to seed something of these possibilities for kids) - something about constructing with these tools opens that approach up for me
recently read Donald Knuths paper : "computer science and its relation to mathematics"(1974) which is an interesting take on it all - (before there were many computers in schools) but sometimes feels like school curricula hasn't really got much further down the track in thrashing out its idea of the relationship
how do you do that with particular classes? ... i'm not going to pretend i have the answers there; your experiments and approaches sound good; i was just meaning to agree with the experimental mindset that these tools can help open up
instruction does make for efficient learning : but sometimes overly sculpts the thinking : the tightrope a teacher walks on i guess :)
there's a book called All the Green Year, fictional account of growing up in the 30s, and the narrator describes a turn for the better in his maths class, when a new teacher arrived
"she taught simultaneous so well i believed i had discovered the art for myself"
wouldn't mind watching that :)
cheers
rob
-----Original Message-----
From: mathf...@googlegroups.com on behalf of Linda Fahlberg-Stojanovska
Sent: Thu 2/3/2011 5:30 AM
To: mathf...@googlegroups.com
Subject: RE: [Math 2.0] Pi is not a natural number (Skeptism and Mistakes are a good thing)
With respect to pi not being a natural number, that was indeed just (my attempt on) a pun on both the set N and the fact that we seem to take mathematics for granted. I love that I can get pi from a circle, sqrt(2) from a square of side 1 and sqrt(3) from an equilateral triangle of side 2 or a rhombus with a diagonal and sides 1 (just made a P2PU 45 second challenge http://www.youtube.com/watch?v=5vRd5dKQQvU)
Ed - do I have to register to read?
Rob - too much explanation. I sort of understand this in K-7 (Do I or do I not show the resulting rhombus in my video?). I do not understand "too much explanation" in algebra. That is, I do not understand the logistics, e.g. what might be a specific sample class that gives too much explanation, what is a possible solution?
With respect to "rich problems". Here is my take - yet again.
I just this moment googled rich problems mathematics and below is the first actual problem in the top link. I could understand the words so I thought "I will give it a try." to see if it is possible that new and better rich problems have been developed since I last looked.
City Hall has a rectangular lobby with a floor of black and white tiles. The tiles are square, in a checkerboard pattern, lined up with the walls: 93 tiles in one direction and 231 in the other. There are two mouse holes, at diagonally opposite corners of the floor. One night a mouse comes out of one mouse hole and runs straight across the floor, and into the other mouse hole. How many tiles does the mouse run across? A complete solution and handouts are provided.
Question: What is this teaching? How can I possible get started thinking about it? (I tried drawing it and gave up immediately - the grid was incredibly small. I tried finding the gcd to reduce it - it is 3 so not much of a reduction.) Do I count 2 tiles or 4 tiles if the mouse crosses a diagonal point? How long will I spend setting it up before any math comes up? Argh!
And speaking of teaching and learning math .. Here is the link for the article on math sophistication in Maria's note about Ko's class: http://ed526b.wikispaces.com/file/view/Mathematical_Sophistication.PDF
Please scroll down to page 9 and then page 10.
Am I nuts?
MMSI Sample problem 2 so offends me mathematically I can almost scream. Either set HF1=0 or start at HF1=6. I don't understand how HF1=1 and even if I did, I wouldn't write that in this problem. It is deliberately confusing.
Are people altruistic? (ok - I have been reading super freakonomics)
MMSI Sample problem 3 sounds like the "Are people altruistic?" survey. Hmm -I wonder what the "correct" answer is? Do I actually have to spend time? No? Then, let me choose the one where I look like I am interested.
BTW: How on earth can you "make a guess about how other functions might be used to model data"? I can understand making a guess that other functions might model other kinds of data, e.g. "Hey, maybe some data is better modeled by an exponential function?" and I can understand asking whether or how other ... But I would be quite interested in how one makes a guess about how .
I did like the first sample problem and I think the hypothesis and conclusions of the article very important, but I would certainly want to see all the questions before drawing any firm conclusions.
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Maria Droujkova
With respect to “rich problems”. Here is my take – yet again.
I just this moment googled rich problems mathematics and below is the first actual problem in the top link. I could understand the words so I thought “I will give it a try.” to see if it is possible that new and better rich problems have been developed since I last looked.
City Hall has a rectangular lobby with a floor of black and white tiles. The tiles are square, in a checkerboard pattern, lined up with the walls: 93 tiles in one direction and 231 in the other. There are two mouse holes, at diagonally opposite corners of the floor. One night a mouse comes out of one mouse hole and runs straight across the floor, and into the other mouse hole. How many tiles does the mouse run across? A complete solution and handouts are provided.
Question: What is this teaching? How can I possible get started thinking about it? (I tried drawing it and gave up immediately – the grid was incredibly small. I tried finding the gcd to reduce it – it is 3 so not much of a reduction.) Do I count 2 tiles or 4 tiles if the mouse crosses a diagonal point? How long will I spend setting it up before any math comes up? Argh!
These are exactly the sort of questions that make the problem rich.
- How can I represent something that is larger than my paper? How DO people visualize billions or parsecs, anyway?!!!
- GCD does not work. Now what?
- There is ambiguity about corners. What is the appropriate math way to deal with such ambiguities?
- How long will this problem take me? How can I manage my time?
There is a relatively old and developed tradition of problem solving in Olympiad and Math Circle communities. If you want to learn more, there is a free book called "Circle in a Box" that has good examples:
http://www.mathcircles.org/GettingStartedForNewOrganizers_WhatIsAMathCircle_CircleInABox
A more classic list for dealing with these issues comes from Polya's "How to solve it" (1948): http://en.wikipedia.org/wiki/How_to_Solve_It
And I can tell you my rule of thumb for "how long to spend before math comes up" - for a good problem, it's quarter to half an hour of context investigation and "image making" before formal math starts to emerge.
And speaking of teaching and learning math …. Here is the link for the article on math sophistication in Maria’s note about Ko’s class: http://ed526b.wikispaces.com/file/view/Mathematical_Sophistication.PDF
Please scroll down to page 9 and then page 10.
Am I nuts?
Cheers,
MariaD
Linda Fahlberg-Stojanovska
Rob, I like your line: == the tightrope a teacher walks on i guess == No kidding and usually without any training.
Please can you give me a link to any of your stuff from Scratch or GeoGebra? These are my most absolutely favorite programs for getting kids engaged. I use these programs to test everything and encourage my kids (college students in computer science who do not even know how to enter math on a computer!!!) to use these to understand/check/test their work. I use scratch to test most anything with probability. Do your kids try to make their own stuff? I have pretty good luck getting my kids to make small simulators like this: http://geogebrawiki.wikispaces.com/DIY+Simulator+of+Boats+Colliding. I wanted my kids to make this: http://tinyurl.com/67bjgrz with scratch but that did not go so well.
Rich problems:
Let me see if I get this (and yes I realize I am being particularly simplistic and obtuse).
1. A rich problem is a normal problem that we (a) add an unlikely story, (b) make completely unwieldy and (c) make deliberately ambiguous just so we can spend valuable class time to (a) find a simpler problem that we can handle, (b) remove the ambiguities for the moment and (c) draw a mathematical simulation (thus taking away the story.) - i.e. in this case, get them to reduce it to Alex’s problem. Meanwhile, this takes great skill on the part of the teacher, keeping them on task - particularly in a classroom of 25 children - without just resorting to just giving them Alex’s problem.
2. So why not just start with a normal problem like Alex’s and work up to a rich problem? If you want, make it into a board game with movable tiles and use either a stick or a laser pointer for the line. Work on that and get some results. Then have the kids create their rich problems with their own unlikely stories (which automatically creates ambiguities and unwieldy expansions), tell them they must actually solve their problem and then do an “exchange” with other kids in classroom.
Anyway – for the sake of argument – l will accept that rich problems have value. Frontal instruction (teacher teaching) has value and there is plenty – if not more- valid research to back up that assertion. Testing has value and there is lots of research that backs up that assertion.
Also – for the sake of argument – I am going to assume rich problem advocates know that there are children like me. I like small problems and breaking big problems into small problems. This is a totally different concept than rich problems! Most problems are small in life. We don’t really do too much with gazillions and parsecs. And I don’t see how unwieldy problems help me understand functions, limits, asymptotes, infinity ... (I have yet to see a rich problem that helps me there.)
Relating all of this – I think there must be variety of teaching methods used. Off the top of my head - Let Monday be “frontal instruction day”. Let Tuesday be “kill and drill day”. The Wednesday be “small problems day. Let Thursday be “rich problems day”. Let Friday be “test day” where everybody gives problems to everybody day (including students get to make up (and first solve themselves) and then give problems to their teachers).
--- A Wednesday problem - realistic solvable interesting problem. I love this problem (and it is a great DIY simulator in GeoGebra) I learned a ton of stuff from it, David Cox’s kids had fun figuring out how to crash the boats….
Scenario: Two ships are sailing in the fog and are being monitored by tracing equipment. As they come into the observer's rectangular radar screen, one ship, the Rusty Tube, is at a point 900 mm to the right of the bottom left corner of the radar screen along the lower edge. The other ship, the Bucket of Bolts, is located at a point 100 mm above the lower left corner of that screen. One minute later, both ships' positions have changed. The Rusty Tube has moved to a position on the screen 3 mm left and 2 mm above its previous position on the radar screen. Meanwhile, the Bucket of Bolts has moved to a position 4 mm right and 1 mm above its previous location on that screen.
Kirby – I have to admit I didn’t read all of your words, but a couple of things stuck.
The teacher tends to exude a kind of moral judgment, as the impatience level rises and it’s time to move on. I told my son “I will pay for your college education as long as you don’t go to the college of natural math and science.” The reason is that I was so worried - not that he wouldn’t graduate (he is stubborn like his dad; luckily not argumentative like his mother), but rather that they would kill his ability to think. Not just moral judgment and superiority (engineers have these too), but also an unwillingness to tolerate skepticism and teach mistakes testing/checking – my original thought for this thread.
My experience is that they do not teach problem “solving” so much as they teach problem “realizing”. They have no consistent approach to looking at a problem. Rather they have solved so many problems that they can “see” the way to go. This is another complaint I have with “rich problems”. If I am a teacher and I have to check out the handout in order to solve the problem, then it is too rich.
Kirby I wonder about problem solving. Children love to solve problems and overcome difficulties – until we take that love from them. I don’t think the problem is “problems”. But I don’t know exactly what happens.
---------------
Maria – My experience here with Olympiads is totally negative (see brown text above). So I definitely don’t want to go there.
My reaction was to 2 of the 3 sample questions presented and I gave my reasons for thinking both #2 and #3 inappropriate under “Am I nuts?”. I also did not like #1 (it seems typical of the kind of problem/answers that students abhor about mathematics), but its content was in line with the goals of the survey (in my opinion). However #2 and #3 are a different story and my reasoning is in the thread. (Further they use HF like it is accepted terminology for “hexagonal factor”. So I went and looked everywhere online for it. It is not.) I first heard of “mathematical sophistication” about 40 years ago when I wanted to take General Topology when I was 18. I fulfilled the prerequisites, but my guidance counselor did not think that I had the “mathematical sophistication”. (Of course I worked my ass off to prove him wrong and learned so much that to this day I am grateful to him.) Anyway the term stuck in my mind as a useful way of describing an important quality and hence my interest in the article.
I go on and on…. My apologies. Linda
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Maria Droujkova
Sent: Thursday, February 03, 2011 2:18 PM
To: mathf...@googlegroups.com
--
Edward Cherlin
>
>
> On Mon, Jan 31, 2011 at 10:15 AM, Edward Cherlin <eche...@gmail.com> wrote:
>> The actual history of math is much more interesting, particularly at
>> those points where somebody invented more of it, and others got
>> furious about such nonsense or even blasphemy as
...
>> * 0, introduced to Europe during the Crusades and denounced by the
>> Church as blasphemy, which is why 12-hour clocks still start at XII.
>
> The way I learned it, Arabic algorithms made reckoning with
> ciphers just too dang easy and the Church was making a tidy
> sum doing everyone's book figuring, for which they collected
> service fees and other extraneous charges.
> This was called "the dark ages" for a reason (the Church tends
> to be dreary, especially when complacent).
You raise complicated questions covering the period from Charlemagne's
ninth-century schools and proto-Renaissance through the second
proto-Renaissance in Spain in the 11th century, to the definitive one
in 1453, when Greeks fleeing from Ottoman Turkish conquest met with
the new-fangled printing press in Italy, where Aldus Manutius created
the first Greek printer's font. Questions of great interest to me,
which I will not go into today, except to quote:
http://en.wikipedia.org/wiki/Hindu-Arabic_numeral_system#Adoption_in_Europe
"Fibonacci's introduction of the system to Europe was restricted to
learned circles. The credit of first establishing widespread
understanding and usage of the decimal positional notation among the
general population goes to Adam Ries, an author of the German
Renaissance, whose 1522 Rechenung auff der linihen und federn was
targeted at the apprentices of businessmen and craftsmen."
It took another generation to get businesses in general to start
switching over in significant numbers, and the process was not
complete until the invention of practical decimal calculators.
> The idea of Jeffersonian farmers, each landed and independent,
> able to keep their own books on PCs, was threatening to
> established power, so the use of Algorithmics (aka Iraqi Math
> from the Wisdom School in Baghdad) was strictly verboten.
> Breaking with Church law (Xtian Sharia) led to the Italian
> Renaissance and renewed hope in humanity (until the
> Black Plague, which some may have regarded as the
> gods' punishment of humans for enjoying enjoying so
> Promethean fire).
>
> Please pardon the creative anachronisms in the above
> telling (use of contemporary analogies).
OK.
>> * Arabic numerals more generally (Fibonacci). Roman numerals were
>> perfectly adapted to use with the abacus, which, however, could not
>> cope with compound interest. Business people did not accept Arabic
>> numerals until Pascal and Leibniz invented practical mechanical
>> calculators in the 17th century.
>
> I thought Venice came to power on the backs on the nouveau rich
> merchant class who used double-entry bookkeeping with Arabic
> algorithms to foster world trade.
True, and it is also true that Luca Pacioli used Arabic numerals in
his treatise.
But the merchants translated Luca's methods back into Roman numerals.
> From Wikipedia:
> Luca Pacioli's "Summa de Arithmetica, Geometria, Proportioni et
> Proportionalità"
> (Italian: "Review of Arithmetic, Geometry, Ratio and Proportion") was first
> printed and published in Venice in 1494. It included a
> 27-page treatise on bookkeeping,
> "Particularis de Computis et Scripturis" (Italian: "Details of Calculation
> and Recording").
> It was written primarily for, and sold mainly to, merchants who used the
> book
> as a reference text, as a source of pleasure from the mathematical
> puzzles it
> contained, and to aid the education of their sons. It represents the first
> known
> printed treatise on bookkeeping; and it is widely believed to be the
> forerunner
> of modern bookkeeping practice. In Summa Arithmetica, Pacioli introduced
> symbols for plus and minus for the first time in a printed book, symbols
> that
> became standard notation in Italian Renaissance mathematics. Summa
> Arithmetica
> was also the first known book printed in Italy to contain algebra.[32]
>
> http://en.wikipedia.org/wiki/Accountancy#Luca_Pacioli_and_double-entry_bookkeeping
>>
>> * negative numbers (still not used in double-entry bookkeeping).
>> * higher-dimensional geometry (numerous mathematicians vs. Kant)
>
> You say vs. Kant because of Critique of Pure Reason and his notion
> that we're subjectively confined to a specific dimensionhood?
> Was that three or four dimensions?
Three.
> The early 20th century was a time of ferment as "the fourth dimension"
> struggled for a place in the sun. Linda Dalrymple Henderson wrote
> a definitive book on this subject.
> The 4D meme was adopted by more than one school of thought,
> Coxeter's among them (Coxeter was at one time a student of
> Wittgenstein's in Cambridge, the latter being the great
> philosopher of mathematics who helped contextualize
> Russell's approach).
Coxeter wrote an excellent introduction to
kirby urner
Kirby – I have to admit I didn’t read all of your words, but a couple of things stuck.
The teacher tends to exude a kind of moral judgment, as the impatience level rises and it’s time to move on. I told my son “I will pay for your college education as long as you don’t go to the college of natural math and science.” The reason is that I was so worried - not that he wouldn’t graduate (he is stubborn like his dad; luckily not argumentative like his mother), but rather that they would kill his ability to think. Not just moral judgment and superiority (engineers have these too), but also an unwillingness to tolerate skepticism and teach mistakes testing/checking – my original thought for this thread.
My experience is that they do not teach problem “solving” so much as they teach problem “realizing”. They have no consistent approach to looking at a problem. Rather they have solved so many problems that they can “see” the way to go. This is another complaint I have with “rich problems”. If I am a teacher and I have to check out the handout in order to solve the problem, then it is too rich.
Kirby I wonder about problem solving. Children love to solve problems and overcome difficulties – until we take that love from them. I don’t think the problem is “problems”. But I don’t know exactly what happens.
Maria Droujkova
Rich problems:
Let me see if I get this (and yes I realize I am being particularly simplistic and obtuse).
What follows is an EXCELLENT explanation of how rich/Olympiad problems feel to the person of the type we shall name after Linda, when we build the mother of all learning type taxonomies :-)
1. A rich problem is a normal problem that we (a) add an unlikely story, (b) make completely unwieldy and (c) make deliberately ambiguous just so we can spend valuable class time to (a) find a simpler problem that we can handle, (b) remove the ambiguities for the moment and (c) draw a mathematical simulation (thus taking away the story.) - i.e. in this case, get them to reduce it to Alex’s problem. Meanwhile, this takes great skill on the part of the teacher, keeping them on task - particularly in a classroom of 25 children - without just resorting to just giving them Alex’s problem.
I could share how someone who loves such problems views them, in parallel to the above list, but I don't see a point. It is sufficient to say that there is a minority of people in each generation, rapidly growing in some places in the world, who find that math, possibly all learning, is all about solving rich problem.
An excellent online community for these people is The Art of Problem Solving. I interviewed its founder Richard Rusczik last year: http://mathfuture.wikispaces.com/Art+of+Problem+Solving He said it's NOT a good idea to do this sort of work with everybody in general classes, and he does not attempt it. It's also a bad idea not to teach "his" kids this way, and in his opinion, mainstream ed groups like NCTM don't understand their needs and don't aim to help them.
---------------
Maria – My experience here with Olympiads is totally negative (see brown text above). So I definitely don’t want to go there.
Naturally.
My reaction was to 2 of the 3 sample questions presented and I gave my reasons for thinking both #2 and #3 inappropriate under “Am I nuts?”. I also did not like #1 (it seems typical of the kind of problem/answers that students abhor about mathematics), but its content was in line with the goals of the survey (in my opinion). However #2 and #3 are a different story and my reasoning is in the thread. (Further they use HF like it is accepted terminology for “hexagonal factor”. So I went and looked everywhere online for it. It is not.)
Problem number 2 is the following picture:
http://screencast.com/t/Uyb1c3gIB3t
Linda, you write: "MMSI Sample problem 2 so offends me mathematically I can almost scream. Either set HF1=0 or start at HF1=6. I don’t understand how HF1=1 and even if I did, I wouldn’t write that in this problem. It is deliberately confusing."
This problem has a target audience, for whom it's not confusing - or maybe, I should say, it's a pleasant tease and challenge, rather than "get me out of here" confusion. What these people accept that you don't?
- Made-up words and definitions
- Numeration of sequences starting from a random point; you say it HAS to start with HF1=0; these people would say "or we could claim HF225=1 and go from there, because we defined it so"
- "Patterning" (for the lack of better word) - that is, definitions from examples rather than top-down deductive (Greek) style
http://www.flickr.com/photos/26208371@N06/tags/naturalmathclub01272011/show/
Cheers,
MariaD
kirby urner
ck of better word) - that is, definitions from examples rather than top-down deductive (Greek) style
My seven-year-olds today concluded that they can clearly see infinity in every eye. To get there, they spent about 2 hours, over two meetings, drawing and discussing infinity in different ways. That involved a lot of acceptance of other people's made-up definitions and a lot of patterning. We are heading for sequence numeration next week :-) But we aren't doing what Richard does with his problem-solving kiddies.
http://www.flickr.com/photos/26208371@N06/tags/naturalmathclub01272011/show/
Cheers,
MariaD
I'd be interested if you have seven-year-olds in other groups, or groups of other ages, who come to a different conclusion re infinity and eyes.
How much diversity have you experienced on this issue?
Kirby
Maria Droujkova
There is a conversation at the Natural Math list, which came from Carol Cross's P2PU psychology of learning mathematics course, about divergent and convergent thinking: https://groups.google.com/forum/#!topic/naturalmath/yW5Pdr8_WhI
I start from the premise that we need a flow channel growing and expanding BOTH divergent and convergent thinking: http://naturalmath.wikispaces.com/Flow+Channels
In particular, all activities I design have strong divergent components that produce a lot of diversity in student outputs. But we also go for people understanding one another. I think this is the area where "problem-solving people" could do much better than they currently do, judging by Linda's reaction.
Cheers,
MariaD
Alexander Bogomolny
--
Maria Droujkova
The formula I see is:
6(n-1) for n>1
1 for n=1
I am CCing Carol, one of the authors of the paper, who can probably shed some light on the issue for us. The pattern makes geometric sense to me as an extension of "dot hexagons" to the left.
Cheers,
Maria Droujkova
Make math your own, to make your own math.
Alexander Bogomolny
Maria Droujkova
Is it a proper paradox?
Incidentally, the issue of the first number in the sequence seems to have no bearing on the question there: "What is HF4?"
Alexander Bogomolny
Incidentally, the issue of the first number in the sequence seems to have no bearing on the question there: "What is HF4?"
Alex
Maria Droujkova
It took me, what, five emails to even become aware of the ambiguity there. I think I am that "anybody."
Suppose we replace the first term with an empty spot. What will the fourth term be then? I say it will be the same.
Cheers,
MariaD
Alexander Bogomolny
--
Linda Fahlberg-Stojanovska
Sorry – Linda went to bed :) But your discussion Alex and Maria was great and clear (and without my droning on and on.)
I don’t object specifically to setting HF1=1. Certainly we CAN define a pathological (degenerative) element anyway we want. And this is indeed an interesting discussion point in a “non-testing” situation just as we discuss here whether a paper plate is 2D or 3D. And of course we must learn to “deal with” outliers. This too is important.
=
However, in the context of (a) survey question with limited explanation and pictures, (b) a question aimed toward JUDGING mathematical sophistication among “non-mathematicians” and (c) a question whose conclusion clearly requires seeing “6-sidedness” and concluding “adding 6”, I think it is inappropriate to introduce a “discussable factor”, particularly as it gives a non-standard arithmetic series (unless of course this was their goal – in which case they should explicitly state that and all of the consequences of such a decision).
(Not to mention all the problems in the future when we give these students the general term of an arithmetic sequence (an=6n) and tell them to write out the first 4 terms and give an example of where such a sequence might arise…)
==
Also, creating the impression that “hexagonal factor” is standard formal terminology is not appropriate anywhere (in my opinion).
==
So much fun to discuss, Linda
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Alexander Bogomolny
Sent: Friday, February 04, 2011 1:58 AM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Pi is not a natural number (Skeptism and Mistakes are a good thing)
Then, I am sure, Linda would have had no problem guessing the missing term.
Linda Fahlberg-Stojanovska
Oh Maria, you can’t name it after me – Linda means “pretty” in Spanish and that would create bias :)
My REAL problem is the “either/or”. Why can’t we do a bit of everything? Use varied teaching methods. I understand that kill and drill is boring. Learning to parallel park is really boring. But it must be done. However good driving skills and enjoying driving includes much other learning.
Also, I would like to point out that Linda the person has not formed an opinion on rich problems. This is Linda the math teacher speaking. I find teaching rich problems a very slow go (see below). And I think most students have a hard go at it, but I would be willing to consider that this might be a training/expectations problem.
There are many types and approaches to learning problem solving. Being able to “realize” problems after killing and drilling a whole bunch is probably the worst. But being able to work with and solve rich problems is not a sign of mathematical genius. I get the impression that some people think so and discard others as mathematically unworthy. Neither is being able to work with and solve standard problems is not a sign of future success in life. But “standards” people think so and swat away the thinking of others as useless in a market economy.
Using a variety of teaching methods would ameliorate lots of teaching problems too. I actually am good at frontal instruction and can interact in mathematics with a large group of students and see what is working and what is not. But I stink at letting people explore and discover (even though I myself explore ALL the time). So my classes are over-organized. That is why I suggested the M-F idea. Every so often, force me to give a rich problem and make me sit at my desk for 15 minutes while the kiddies work.
Also my experience with Olympiads is not a personal experience – it is a teacher experience. The kids get so discouraged by the Olympiads that I know longer recommend that they go. Grades 1-4 here are taught by persons with a degree in education (2 courses in mathematics methodology and one course in mathematics). We have gotten aid from the US and our 1-4 curriculum has been heavily influenced by the “rich problems and let’s get them started with variables before they know how to count and no need to memorize the multiplication tables” theories. Then in 5-12 they get teachers with degrees in theoretical mathematics (and although now there is a mathematics-education degrees – the professors are still theoretical mathematicians). This attitude and training difference is astronomical. In 9-12, the math curriculum remains extremely old world (content and memorization heavy) PLUS statistics/probability has been added. This dichotomy has proven unbelievably difficult and the quality of students I get (college mathematics) is now unbelievably poor.
Re: Varied teaching. My assistant was showing 10th graders how you get the sine curve from the unit circle on a video projector (using GeoGebra) and a friend was filming him from the back. They were in a computer lab because the equipment was there. When we watched the video we saw that many of the kids were working out the steps on their own computers and creating the exercise without any specific instructions. This would never occur to me.
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Maria Droujkova
Sent: Friday, February 04, 2011 12:42 AM
To: mathf...@googlegroups.com
Cc: Carol Seaman CESEAMAN
Subject: Re: [Math 2.0] Pi is not a natural number (Skeptism and Mistakes are a good thing)
--
Maria Droujkova
Sorry – Linda went to bed :) But your discussion Alex and Maria was great and clear (and without my droning on and on.)
I don’t object specifically to setting HF1=1.
Linda, if the problem were stated identically, except the first member of the sequence was empty space and HF1=0, would it help you to solve for HF4?
Cheers,
MariaD
Linda Fahlberg-Stojanovska
It is/was obvious to me from the answers given that they “want” the answer with 18.
The point is not whether I can answer it, but whether the question is appropriate for this survey.
----
If you are asking me as a person off the street I would say “What??”
If you are asking me as a teacher of K-6, I probably would say HF1 should be 1.
If you are asking me as a teacher of 7-13, I would immediately say HF1=0.
If you are asking me as a mathematician, again I would say “neither” definition is totally correct.
This is my complaint against the problem.
It is mathematically and statistically that I am offended.
1. The goal of the survey is to determine mathematical sophistication of non-mathematics people.
2. From the list of answers, I believe the focus of this problem is whether you can logically determine the pattern 6,12,18.
Statistically:
(a) Survey questions should be carefully formulated towards their goal. Having HF1=1 (OR HF1=0) is a distraction and this distraction makes the problem inappropriate. (And using the language “hexagonal factor” is inappropriate – it add pedagogical “weight” to the definition.)
Mathematically,
(b) There is no way to logically determine the next member of the sequence: 1,6,12,?. One can only guess – so how does this indicate mathematical sophistication?
(c) If asked for a written explanation of your choice (as opposed to being required to circle an answer), what would you say? “I discarded the first hexagonal factor.”? Oh my gosh – a non-mathematical person Is never going to take the risk of saying that to a math person.
Sorry – not a simple answer to your question :)
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Maria Droujkova
Sent: Friday, February 04, 2011 12:40 PM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Pi is not a natural number (Skeptism and Mistakes are a good thing)
On Fri, Feb 4, 2011 at 3:33 AM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:
--
Maria Droujkova
It is/was obvious to me from the answers given that they “want” the answer with 18.
The point is not whether I can answer it, but whether the question is appropriate for this survey.
Has anyone here argued for any possible answer beside 18 OR was unable to reach 18? I am just curious. If not, it means we share some sort of consensual understanding about the nature of the question and the correct answer.
Cheers,
MariaD
kirby urner
It is/was obvious to me from the answers given that they “want” the answer with 18.
The point is not whether I can answer it, but whether the question is appropriate for this survey.
----
If you are asking me as a person off the street I would say “What??”
If you are asking me as a teacher of K-6, I probably would say HF1 should be 1.
If you are asking me as a teacher of 7-13, I would immediately say HF1=0.
If you are asking me as a mathematician, again I would say “neither” definition is totally correct.
This is my complaint against the problem.
It is mathematically and statistically that I am offended.
Alexander Bogomolny
Edward Cherlin
Did you ever hear about the physicist who (in legend) claimed in court that the red light looked green to him because of blue shift, so they shouldn't fine him for running the red light? The judge gave him a severe sentence for speeding, considering that blue-shifting red to green means he was going at something like .9 c. (Take ratio of frequencies, then solve r= (1-(v^2)/(c^2))^(1/2) for v, getting c(1-r^2)^(1/2).)
I have maintained for some time that we should use sports statistics and physics for our "real-world" rich examples, allowing each student to choose a sport of particular interest: men's or women's; school, Olympic, or pro; baseball, basketball, football, cricket, soccer...and then move on to economics and politics that affect the student or others in the students' families directly. Those who are resolutely uninterested in sports will have some other subject of intense interest.
--
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Edward Cherlin
be a natural number, considering that it is the base of the natural
logarithms.
Once there was a mathematical tree with square roots. It woke up one
day and said, "Gee, I'm a tree!" But they cut it down to make log
tables.
Told to me by my father, a Ph. D. mathematician.
kirby urner
Kirby,> I do like number sequences with a geometric basis howeveryou may like "Number Mosaics" too
On Fri, Feb 4, 2011 at 12:06 PM, kirby urner <kirby...@gmail.com> wrote:
Linda Fahlberg-Stojanovska
Edward. We (my bro Tim Fahlberg and I) loved the jokes. Thanks so much.
Kirby - I agree totally about "double the size". Size of what - the perimeter? the area? the volume?
(I thought a long time before deciding to call my exercise "largest triangle in circle" and decided that although I was after area and state so in the explanation, the same answer gives the largest perimeter so I was “covered”).
-----Original Message-----
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Edward Cherlin
Sent: Friday, February 04, 2011 7:21 PM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Pi is not a natural number (Skeptism and Mistakes are a good thing)
It occurred to me yesterday to ask whether people here consider e to be a natural number, considering that it is the base of the natural logarithms.
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