The Way to Teach

[Algot Runeman]

This post seems to apply to many of the current discussion threads on
this list.
http://www.freakonomics.com/2011/11/02/the-way-we-teach-math-and-language-is-wrong/

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-------------------------
Algot Runeman

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[Mike Thayer]

Interesting post, but it seems (as David Wees posted in a comment to the Freakonomics blog) that we've been over a lot of this ground before.  Perhaps its value is in the fact that the ideas presented there are going to a wider audience and thus may bring more people into the conversation?

-Mike

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[Maria Droujkova]

The recommendation for learning to be "idiomatic" caught my eye. People who like math learn it in context: a problem (like "the goat, the cabbage and the wolf") and then a family of problems (river-crossing problems) and then the area of math around the family of problems, for example. Or a task like creating mathematical artpiece. 

What are math idioms? 

Cheers,
Maria Droujkova
919-388-1721

Make math your own, to make your own math

 

On Wed, Nov 2, 2011 at 2:09 PM, Algot Runeman <algot....@verizon.net> wrote:

[Sue Hellman]

I have to admit that most of the discussion on procedural vs. functional programming has gone right over my head, but I decided to take a look at the article referenced in tkosan's response to Linda's question (link-- http://206.21.94.61/misc/permalink/procedural_vs_functional.html). 
 
I think 'math idioms' might be related to 'functional units' in this article. They would be stand-alone skills that allow one to perform real tasks but which can also be nested in/scaffolded into larger tasks in a way that makes them also meaningful and uesful.
 
Using meaning or function, instead of procedure, as the starting point is akin to the way I was taught to teach a second language. The initial phrases learned enable a person to do something real (eg. to say good morning, or introduce yourself, ask for a glass of water). The vocabulary and structures one learns in these 'functional' units in turn become the framework upon which more complex functional units are built. Everthing links to everything else and always to something that was simple, comprehensible, meaningful, and useful in its own right.
 
I'm in the process of creating a unit on fractions for a school in the Caribbean. If I follow this process, I have to start by asking myself what a kid there would want to be able to do that having some 'fraction language' would make possible or easier. That becomes the starting point rather than what I think of as the easiest skill or normal first step. What would they want to be able to do?
 
Suggestions?
 
Thanks,
Sue

What are math idioms? 

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[m...@ms.lt]

Sue, great letter, viewpoint, challenge and question!
Andrius Kulikauskas, m...@ms.lt, http://www.selflearners.net

[Cooper Macbeth]

Sue,

I thought about this in depth during a relief effort to Haiti: what math would help the kids most? My answer is the same for fractions: food.  Everything from water collection to food distribution to storage to farming to recipes to nutrition. What does one do with a gallon of milk that will only last two days in a poorly working ice box? Or what does one do with a pound of flour? The math of food and nutrition, when grasped, makes a very excited student because they understand how to help and contribute!

Cooper

[m...@ms.lt]

Sue,

I'm making a "learning canvas" to illustrate deep ideas in basic math,
such as fractions. So I'm thinking along similar lines.

I'm going to include a large circle of 24 slots for items. Because 24 is
a good number for breaking down into fractions. And 20 or 30 pieces of
data, or stories, or examples, is good for looking for patterns in life,
etc. And if I add one, 25 is good for percentages. Here are my notes for
that:
http://www.selflearners.net/Math/Circle
and for other basic ideas
http://www.selflearners.net/Math/DeepIdeas

I think that fractions (parts of a whole) are important for relative
comparisons, but that's important only if we deal with more than one
problem, if we compare between problems. Otherwise, it would be simpler
to deal with absolutes. The real point of fractions is to be able to
compare, say, 3 out of 8 in one situation with 5 out of 20 in another
situation, which is to say, to compare across two different situations.

That comes up in betting, in probabilities. What's the chance that... ?
And all of the algebra of fractions comes up.

Andrius

Andrius Kulikauskas
m...@ms.lt
(773) 306-3807

[Sue Hellman]

Thanks for the responses
 
@ Andrius -- Before I read your idea, I had thought of using a measuring tape (up to 16ths) because it leads nicely into the carpentry trade analogy. But there's another part of the same course that has to do with circle graphs, so now I'm thinking about a clock and starting with 60 minutes/slots. Young teens already have an intuitive sense of  the numbers of minutes in fractions of an hour and so when asked to add 1/2 + 1/3 + 1/4 of an hour, some will naturally think of changing to a common unit (i.e. minutes). From there it's not too big a leap to the idea of common denominators. (This grade covers addition and subtraction but not multiplication or division of fractions).
 
@Cooper -- I'm with you about the 2 themes of social responsibility and food. Trinidad and Tobago are in an active zone for earthquakes and tsunamis and also have a big oil and gas industry so oil spills and environmental damage are also an issue as are (I imagine) the possibility of water shortages and landfills. I like the idea of having the students work on infographics but the students cannot access the internet from school so I would have to provide all the data sources from which to work. I also thought I might have them do a mock up of a Carnival (or other event) food concession.
 
-Sue
 
 
 
 
 
 

[Bradford Hansen-Smith]

Since you are drawing pictures of circles and making divisions to look at fractions, why not just fold them. The 9 creases that makes the tetrahedron net yields 24 proportional divisions on one side of the circle, another 24 on the other side, add 12 more for the circle ring that holds the two sides together. All considered you have 60 divisions in the circle;  just like minutes in an hour. You also have a tetrahedron. The areas in the circle are more interesting as fractions being proportionally related and not everything the same.  If 60 is too big a number then start with just 3 diameters;  6 areas on each side and the ring making 18 individual areas. By using only one side of the circle there are 6 equal areas and 720 different combinations with which to explore fractions. Brad --- On Sun, 11/20/11, m...@ms.lt <m...@ms.lt> wrote:

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[Maria Droujkova]

Sue,

I would like to invite you to join the fraction seminar we will start on December 1, at P2PU School of Math Future: http://www.p2pu.org/en/groups/fraction-interactives-seminar/

The goal is to make a Moodle-based OER about fractions, using open interactives.

Cheers,
Maria Droujkova
919-388-1721

Make math your own, to make your own math

 

 
 
 
 
 
 

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[michel paul]

A long time ago I started contemplating the 'Rational Number Plane'.

Consider a Cartesian lattice where the x axis is the 'denominator' axis and the y is the 'numerator' axis. This will be a discrete world - the only points we will be interested in will have integer coordinates.

In this world, the faction 2/3 corresponds to the ordered pair (3,2), and the fraction 3/3 corresponds to the ordered pair (3,3). Notice that the rectangle whose diagonal is [(0,0),(3,2)] has an area 2/3 that of the square whose diagonal is [(0,0),(3,3)].

So each ordered pair (x,y) in this lattice has an associated rectangle with diagonals of [(0,0), (x,y)] and [(x,0), (0,y)]. All ordered pairs (a,a) represent various zoom levels of the primary unit (1,1). Equivalent ratios for any y/x will produce a set of rectangles sharing the diagonal [(0,0), (x,y)]. And again, we can regard similar rectangles as being zoomed in versions of the initial rectangle where y and x are relatively prime.  

A nice exercise - find all ordered pairs (x,y) where x and y are relatively prime. The pattern is kind of cool to watch as it develops, but doing it by hand can be frustrating - it always leaves you wanting to see more. So that's where programming helps. It wasn't until I could express this in Visual Python that I got to have a really good look at this pattern.

This gives us a nice way to visualize the countability of the rationals as well as the Dedekind cut -

Here's a fun way to generate relatively prime ordered pairs - start with (0,1) and (1,0) and find mediants. (1,0) corresponds to 0/1, and (0,1) corresponds to 1/0, which for this purpose we'll call 'inifinity'. The mediant of two fractions (a,b) and (c,d) is simply (a+c, b+d). It is a kind of average. Batting averages are mediants. So if you start with 0/1 and 1/0 and successively find mediants, you will generate a list that will eventually include ALL the positive rationals, and this can essentially turn into Cantor's demonstration of the countability of the rationals.

Now consider the line y = sqrt(2)*x. This line will never encounter another lattice point other than (0,0) and will cut the Rational Number Plane into two distinct sets. This is essentially the Dedekind cut.

I like to say that 'Fractions are objects, not unfinished division problems.' A fraction is a two-part data structure. It is a two-dimensional object, and so visualizing fractions in two dimensions seems better than trying to map them onto a one-dimensional number line before we've explored their two-dimensional structure.

After we've explored the rational number plane a bit, we can see how each (x,y) can be mapped to a (1, y/x) if we want a linear graph of their magnitudes. Later on we can also map each (x,y) to a point (cos(theta),sin(theta)) on the unit circle where theta = arctan2(y,x).

- Michel Paul

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===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[mok...@earthtreasury.org]

You are planning a party for n guests, and you want to make sure that you
have enough pizza for them, allowing 3 slices each (big pizza eaters, your
guests, like my son's gamer friends who convene twice weekly). How many
pizzas do you need to order? Divide n by 3/8 and round up.

Ask the identical question, but dress it up as a problem of paying pirates
with pieces of eight broken into eighths.

I want to make two and a half times a recipe in English/American units,
not in metric. So I have to double all of the fractional measurements.
What is 2 1/2 times 3/8 cup?

However, your real question is, what do your students measure in fractions
rather than decimals, in addition to the possibilities of pie, cake, and
cup measures, and odd sizes of historical money? Ask your students, not
us. And please let us know what you find out.

>> Suggestions?
>>
>> Thanks,
>> Sue

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Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://wiki.sugarlabs.org/go/Replacing_Textbooks

[mok...@earthtreasury.org]

Any language with good arithmetic and good graphics will do, of course. I
do stuff like that in J. Let us define the mnemonics

gcd=.+.
rp=.(1:=gcd)"0

so that

48 gcd 66
6
48 rp 66 NB. 48 (1:=gcd) 66; 1=48 gcd 66 NB. 0 is false
0
49 gcd 66 NB. 1 is true
1
49 rp 66
1
and
48 gcd
and consider

]l=.1+i.8
1 2 3 4 5 6 7 8
l rp/ l
1 1 1 1 1 1 1 1
1 0 1 0 1 0 1 0
1 1 0 1 1 0 1 1
1 0 1 0 1 0 1 0
1 1 1 1 0 1 1 1
1 0 0 0 1 0 1 0
1 1 1 1 1 1 0 1
1 0 1 0 1 0 1 0

> This gives us a nice way to visualize the countability of the rationals as
> well as the Dedekind cut -
>
> Here's a fun way to generate relatively prime ordered pairs - start with
> (0,1) and (1,0) and find mediants. (1,0) corresponds to 0/1, and (0,1)
> corresponds to 1/0, which for this purpose we'll call 'inifinity'. The
> mediant of two fractions (a,b) and (c,d) is simply (a+c, b+d). It is a
> kind
> of average. Batting averages are mediants. So if you start with 0/1 and
> 1/0
> and successively find mediants, you will generate a list that will
> eventually include ALL the positive rationals, and this can essentially
> turn into Cantor's demonstration of the countability of the rationals.

0/1 1/0
0/2 1/1 2/1
0/3 1/2 2/1 2/2 (=1/1) 3/2 0/4 1/3 2/3 4/2 (=2/1)
...
At each stage, we add the numbers in the last line generated to all of the
numbers encountered so far, including each other.

> Now consider the line y = sqrt(2)*x. This line will never encounter
> another
> lattice point other than (0,0) and will cut the Rational Number Plane into
> two distinct sets. This is essentially the Dedekind cut.

Actually, we need to plot y^2=x, so that we are not using the result of a
cut to define the cut.

> I like to say that 'Fractions are objects, not unfinished division
> problems.' A fraction is a two-part data structure. It is a
> two-dimensional
> object, and so visualizing fractions in two dimensions seems better than
> trying to map them onto a one-dimensional number line before we've
> explored their two-dimensional structure.

J includes a rational data type suitable for reinforcing this point. The
expression 1/2 is a command in J. The expression 1r2 is a number just as
it is. Some other languages do likewise.

Michel's approach is also good for introducing the idea, fundamental to so
much of mathematics, of equivalence classes.

--

[michel paul]

Well, here's a surprise. Turns out that Minkowski dealt with this Rational Number Plane! I had no idea. Yeah, that first picture you see in the top right, that is definitely what the Rational Number Plane looks like as it unfolds.  

Also, the article itself goes on to develop how this is actually useful - related to 'resonance phenomena' and 'oscillating systems'. So this is turns into an interesting answer to the original question!

- Michel

[milo gardner]

This system of unit measurement was continuously used for 3,700 years. Only with the adoption of decimals in 1600 AD were the two forms of ancient unit fraction systems disappeared from our medieval and modern school systems.

The oldest system is the hardest for modern students to understand.. Rational numbers were ciphered until 800 AD, hiding the beauty and utility of the ancient unit fraction system that managed commodity based economic systems. Weights and measure systems were scaled to (64/64) and (320/320) unity statements within quotient and remainder statements that retained algebraic identities of each level of the system.

After 800 AD Hindu-Arabic numerals wrote out rational numbers in an algorithmic context. By 999 AD Pope Sylvester required the updated unit fraction system to be taught in European Latin schools. A formal text book did not arrive until 1202 AD. Leonardo from Pisa, Italy (Fibonacci) revised Europe's first unit fraction arithmetic book until his 1254 death. The book continued in use for another 200 years, until the Ottoman Empire conquered the Byzantine Empire in 1454 AD. After 1454 AD, Europeans searched for new trade routes to India and China, and new trading units to conduct business.

Today, five copies of the Liber Abaci, the name of Fibonacci's book exist In 2002 AD the 500 page book was fully translated into English by Sigler. I had failed to grasp many details of the older unit fraction system until the English translation of the  Liber Abaci was read in 2005. My view of the ancient arithmetic changed for the better after reading the first 124 pages.

The everyday problems that requested to be discussing in this thread were explained on pages 125 to 500 of the Liber Abaci The most difficult problems solved indeterminate equations, topics that go well beyond this thread. I'd be happy to cite a few medieval examples of the system, considering this email as background considerations for teachers.

Best Regards,

Milo Gardner

[Sue Hellman]

As always, what I thought was a simple question is spinning off into many rich directions. Thanks to everyone for their help. -Sue

[m...@ms.lt]

Brad,
Thank you. Do you have a picture? Or more detailed instructions? How do I
make the folds?
Andrius

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[Bradford Hansen-Smith]

Andrius, go to my web site http://www.wholemovement.com/  then to the How to Fold page and that will take you though folding the tetrahedron. The rest is observation, counting, applying what you already know to what has been generated in the circle. This is not complicated, it is just that we do not look for anything in the image of something that we have decided represents nothing. Brad

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[Linda Fahlberg-Stojanovska]

This reminds me of a “rich problem” I was complaining about (yes, yes I know – it is hard to remember this particular complaint – there are so many J).

Anyway, it was the: A mouse crosses diagonally across a rectangular floor with a zillion square tiles. How many tiles does he touch? (Problem: http://www.ohiorc.org/pm/math/richProblemMath.aspx?pmrid=56  )

 

A friend of mine from GeoGebra asked about how to highlight the squares crossed by a diagonal. Luckily I did NOT have time to think about this and he posted it on the GeoGebra forum. It is incredible – people solved and made beautiful applets of this – one has only 7 objects (including the text object). Another added sliders to allow for rectangular tiles. I posted a couple of them at: http://geogebrawiki.wikispaces.com/Mouse+Diagonal+Problem  and there is a link there to the forum page.  (You can click on View file in your browser to see the uploads without downloading them.)

 

I then realized that this question was in fact the mouse problem that I found so tedious.

 

I am amazed that (a) so many people find this problem interesting, (b) how elegant the solution is in GeoGebra (and hence must be mathematically) and (c) how incredibly active are these math forums (the GeoGebra forum, the LinkedIn forum, this forum, et.al.).

 

All of these people fascinated by mathematics – if only our students could catch our enthusiasm. This is the way to teach. Show them what we do and how we use our spare time to think about these things. Okay, done blabbing.

Linda

What are math idioms? 

 

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[Maria Droujkova]

This problem is a magnet for trouble. I call this property "rich," lol.

Check out the recent strife around it at dy/dan's blog, with 60+ total comments:

http://blog.mrmeyer.com/?p=12256

http://blog.mrmeyer.com/?p=12004