A Young Aspie Figures Out Figures


Derp-Alert:  Run away! Head for the next post. This post is of possible interest only to:  Math nerds, some teachers, bored Aspies, parents and friends of Aspies, and regular folk who have finished the back of the toothpaste tube for the n-teenth time and are looking for something with a little more bite (heh, heh–oh, Outlier Babe, you are a laugh riot…).

If This is What Looks Back At You Each Morning, This Post Is For YOU

At age 6, I starting thinking:  “Numbers don’t make any sense.” Why did we decide to represent four objects by the thing that looks like this ?:

The Number 4 Asks Nu

Nu? You’re Asking ME?

Roman numerals were very logical.

...Four Dead Christians, Five Dead Christians...

Arabic ones seemed arbitrary, but I reasoned that there must be a logical basis for them, too. So I sat in Mrs. Thompson’s 2nd grade class writing the numbers 1 through 9 over and over on a piece of paper.

I kept experimenting until I finally figured it out:   It made so much sense!  It all had to do with angles:

Click the Pic for a Lovely Animation of Types of Angles

Just like the Arabs (as I surmised) I drew straight-lined angular digits: A 1 with a single top hook forming 1 angle, a zig-zag 2 having 2 angles, and so on. I was so proud I’d solved the puzzle!

Interfering somewhat with my joy, I’d had to add a distinctly maze-like curl to the tail of the 9 in order to provide sufficient angles, and a base and a European crossbar to the number 7. 

But I was far too pleased with myself my hypothesis to abandon it over these huge, gaping minor failings. 

Here are the numbers exactly as I drew them:

From Jeff's Lunchbreak:  Origin of Arabic Numerals (http://www.jefflewis.net/blog/2009/10/origin_of_arabic_numerals_was_1.html)

1 Through 4 Were No-Brainers

Roman Numeral Trivia: Even after Arabic numbers were adopted, Roman numerals were still sometimes preferred to Arabic ones during medieval times for their reasonable security against fiscal fraud.

For example, any digit of the number 2381 can be altered to cause a substantial difference (as much as 7000 by changing the 2 to a 9), but the equivalent MMCCCLXXXI cannot be easily altered to produce a new number.

– From Math Lair, Roman Numerals

http://www.jefflewis.net/blog/2009/10/origin_of_arabic_numerals_was_1.html

5, 6, and 8 Were Easy, Too. But 7 Is Quite the Cheat...

(I can’t tell you how stunned I was to find my exact original numerals online–as well as other angle-based versions–and to discover for the first time when writing this post that others had also theorized that angles were behind the shapes of Arabic numbers.

Shout-out to my fellow geeks!

http://www.jefflewis.net/blog/2009/10/origin_of_arabic_numerals_was_1.html

...And What a Curly-Tailed Workaround For Number 9!

I liked that my hypothesis explained why Mrs. Thompson insisted we write the number 4 the closed-top triangle way, like it appeared in our math book, instead of the way most people wrote it:

Numberjacks Howdy-Do Four

The “Howdy!” Way of Writing the Number 4 (From “Numberjacks”

Written the “Howdy Do!” way, the number 4 had five angles instead of four, so of course that way was wrong!

Another Reason To Use the Triangle-y Four

I further theorized that over time, people had gotten tired of drawing the 9’s complicated loop-de-loop tail, so they just kind of unwound it until the digit’s whole spine developed scoliosis.

The 7 lost its extra horizontal lines due to laziness, too (except in England, where they aren’t as lazy as we  Americans). 

Yes, I was quite the scientific thinker at my grand old age of 6.

An Even Dumber Classmate

A bonus benefit to my hypothesis was that I was able to use touch math from then on whenever I did addition:

I had already been using it for the digits 2 and 3, touching each pointy digit tip with my pointy pencil tip. (There are two points on the left side of the 2, three on the left of the 3.)  When I added 2 plus 3, I just counted up as I touched each point:

1-2-3-4-5” .

Hear the Strauss? 1-2-3, 1-2-3...

Now I could do the same thing for the bigger digits, by touching each angle (or its imagined historical position) and counting up on those.  Nice.

Many children devise digit-touch systems. A highly-talented Math instructor once told me that these are the children who become stronger in math concepts.

One of My Old Math Teachers Just Heard That...

He said research suggests that children who don’t develop their own touch systems can be taught one to gain the same advantages.

There are commercially-available and cost-free touch math systems. The touch positions on digits differ among systems.

The most prevalent commercial system is Touch Math, described in the References. I take issue with that program’s name, and some of its features, also explained in the References.

Because I was weak in math, it’s too bad no teacher back then introduced me to dice math: 

To instantly recognize the numbers up to 12 visualized like they are on die faces, and to automatically add and subtract by these groups.  Now THAT would have been at least as helpful as touch math.

Dem Bones, Dem Bones Is...Great Fo' Math!

A gentleman named Owen Prince realized this many years back, and developed a dice-based touch math system he copyrighted as Dot Math:

Dot Math Keypad, copyright Owen Prince, from DotMath for kids, http://dotmath.tripod.com/index.html

Dot Math Keypad, copyrighted by Owen Prince, from DotMath for Kids

Some of the opinion portions of Mr. Prince’s site are worded in a way that sounds a little…well…hmmm…perhaps it’s better not to say.

However, I recommend a visit there, anyhow, because his concept is worthwhile and because, after all the “copyright dispute” text (you’ll see what I mean), he gives a lovely, if oddly-worded, helpful review of math resources.

Please see also his Comment on this post, because he gives a detailed, impassioned and, to me, convincing argument in favor of using his system over the commercial Touch Math’s approach.

Okay, this post should end right here, but it’s just gonna keep going and going, so if you’re wise, you’ll stop reading now and go have a nice cup of tea. Enjoy!


Oh–I guess you may as well read the rest while it cools down, then, yeah?

Entirely off-topic, but another helpful idea: 

That same talented math teacher, of 1st graders, did not teach or use the English words for 11 through 19 in his classroom until the end of the year. 

He had his children use the Asian nomenclature “ten-one (versus “eleven”), ten-two (for “twelve”), ten-three…”.  He continued this pattern when using his number poster that displayed numbers up to 100 (so, for “55” he said “five-ten-five”).

This teacher considered this a key feature in enabling his students to retain the place value concepts that he felt they developed naturally but which were otherwise interfered with by English number terminology.

Happy Grandma Ratty: “Let’s See: That’s 5-ten Baby Ratties In Those Last Litters, Each Baby Will Have 10 More Babies…

Each year, all of that talented teacher’s 1st grade students ended their first year in school testing at a 3rd grade Math level.

(BTW, I can say from budget-driven experience that Cheerios threaded onto coffee stirrers stuck vertically into clay work just fine for abacuses, if you can prevent snacking to hide evidence of miscalculations.)

Relevant:  From a June, 2010 review of  literature on cross-cultural mathematics instructional and learning, Chinese Number Words, Culture, and Mathematics:

“Although it is not possible to disentangle the influences of linguistic, cultural, and contextual factors on mathematics performance, language is still seen as contributing to early cross-national differences in mathematics attainment.”

Imagine if every classroom across America instituted that teacher’s simple terminology change today. 

Easily implemented, easily taught to teachers with five minutes of instruction (“Start doing it, it will feel odd, but it works, it’s easy to do, you’ll get used to it”). 

Or, pick the 10 lowest-performing schools in each state and implement this in 5 of them.  Compare the 10 schools in two years.  Bet you’ll net impressive results in those 5 out of 10!

More on Asian Number Names

Another interesting reference on English vs. Asian Place Value Concepts and Number Words

Touch Math vs. touch math

Note the capitalization: In this post, lowercase “touch math” means the generic touching of digits. It is irksome that TouchMath was permitted to copyright a phrase that can apply equally to non-commercial uses, and to uses other than touching pencil to paper.

The two words “touch math” could as easily mean the same as “hands-on Math”, such as the touching of beads or any items used to help in counting.

More on Touch Math

The commercial “Touch Math” program’s addition demonstration in 50 seconds (the program also supports other operations):

TouchMath can be effective, but it would have caused problems for me and my particular flavor of Aspie-ness, and I bet the same factors could bother other kids, too:

A) “Double touch points“.

As shown in the video, some touch points/dots count as 1, but others as 2. Remember that each spot will be an imagined tiny speck on the child’s own hand-drawn digits when actually used.)

I was a Math-phobic Aspie child. Different-valued dots would have given me the mental screaming-meemies.

B) Random positions of touch points.

The digits 7 and 9 in particular look like a confused scattering of dotted nonsense.

My particular flavor of Aspergers still freaks when confronted with this type of disorganization.

We're Just Going to Put a Touch Point Here...and Here...and HERE!!

C) Touch points on top of digits, rather than adjacent.

This is off-putting for me as an adult; when a child, I believe it would have made it difficult for me to use the touch points with my handwritten digits.

The Dot Math’s Mr. Prince claims that he has research demonstrating that for some students this interferes with transference of skills to regular non-dotted digits. That is why his system, which originally had dots atop digits, was revised to have them adjacent. I would love to see research on this issue.

An ideological objection:

No generic phrase should be cornered by a commercial product. One can no longer perform a google search for anything to do with the generic concept–I had to plod through 29 pages of results before landing on one reference to non-copyrighted material: a lone research paper I found on Eric (and even that one may refer to the Big Touch).

Further, the two words “touch math” can easily be applied to any mathematical manipulation of real or virtual three-dimensional objects (e.g. counters, graph paper, a balance…).  How about if I copyright the phrase “Edit Post”, or “Code HTML”? How about this one?:

Bite. Me.

Okay, the post is finally finished. You’d better go stick that tea in the microwave. Careful this time…

 

Advertisements

Geometer


Polygons

Perimeter, perimeter,
Around the edge I roam;
I look into the middle,
Where the area has its home.

Circular Circulation Equals Perimeter Rotation


We measure perimeter by adding sides:
How long is every one?
That tells you how much fence you’d need,
Or far you’d have to run.

Perimeter

We measure area with squares:
How many can we glue?
Just multiply the base times height
And area is through.

A-ha! That is One Square Kilometer of Area!

(But if the shape’s a triangle,
You then divide by 2.)

Base Times Height (Do You See Why We Divide By 2 For Triangles?)

For complicated shapes,
Cut into smaller shapes you know,
If lengths aren’t labeled,
Look across at opposite lengths that show.

Not So Scary When You Cut It Up

We measure volume space with cubes:
How much space will there be?
From any corner, look three ways
And multiply what you see.

(That’s base times height times depth
to give dimensionality.)

Volume

To see how triangles really behave when they’re letting all their points hang out, check out the polygon partiers in this video:


Circles

Outside of every circle,
Let’s build a circle fence;
And give that circle fence a name:
It’s called “Circumference”.

http://img.locationpartnership.com.s3-website-eu-west-1.amazonaws.com/dbims/PL240_23.JPG

Let’s stretch a cord across that fence,
Straight to the other side;
“Diameter” is the fancy name;
Our circle it divides.

 

Clever Stretchy Triangle Diameter

This Diameter Is Stretchy and Clever, No?”


 

With two of our diameter cords,
We try to wrap around,
But two won’t reach;
With D times 3 circumference is found.

One C Unrolls, We Get Three Ds–A Bit More Pi/e Is Left For MEs!(Refresh Your Screen If It Won’t Go: The “One C Equals Three Ds” Show!)


 

(Of course, the opposite is true:
Divide circumference by 3 to find diameter, too.)

A more exact result you’ll see,
If you use pi (Π) instead of 3.
(That’s 3-point-1-4 and some more,
But it’s close enough with 3-1-4.)

They’ll try to trick you:
They’ll give radius instead of D;
But r times 2 makes D,
And D times 3 will give you C.

(Or 3-point-1-4, pi, to get
A more exact circumference yet.)

r Times 3 Gives Half a C. Yikes! Stick With D Times 3, Like Me!   And Just Keep Quiet About Tau, For Now… (Refresh for Action)

But if you can’t remember “r”
or “radius”, think “ray”:
Two rays of sun join up as one;
And start a brand-new Day.

Good Day!

I’d be remiss to end without including two Pi items well-known in nerd circles (she said, sneeringly, while secretly enjoying the videos as well):




edited 11/13 to shrink some images for phones, to note that ya’ need to click “Redisplay” to reanimate some .gifs by the time you get to ’em, to add a coupla’ more nice .gifs while I was in here, and, last, to toss off another verse which ain’t so hot, but I felt like it anyhow…and lastly last, to add a tau reference (but I draw the [arc?] at eta!).
 

Credit Cards for Dummies?


Use With Caution

Credit Card Disclosure

Over half of Americans don’t understand what percents are or how they work. They might think they do, because they can tell you 50% means “half”, but they can’t answer the very easy question of how much 25% of 8 pennies is.

Half of Americans can’t read well enough to really understand what they’re reading.

People who can’t read well or can’t understand what a percent is cannot understand today’s “disclosure” documents when they take out a loan. A disclosure document is the piece of paper that says you understand the rules for borrowing money, and you understand what might happen to you if you don’t pay it back, or don’t pay it back quickly enough.

But if you can’t read well, or understand math well, you do NOT understand what might happen when you borrow money with interest due.

(That includes when you get a credit card. Every time you use a credit card, you are borrowing money.)

The only way to be fair to all Americans–the ones who read and do math well, and the ones who don’t–is to write the clearest disclosure papers ever (and to start educating our children in math and reading when they are preschool age instead of waiting until it’s harder for them to learn).

YOUR LOAN

We are loaning you our money so that you can buy something you want now, even though you don’t have enough money now.  You’ll have to pay us back not only the money we loaned you, but also extra money.  Interest is name for the extra money you will have to pay us back.

Interest uses a percent, or interest rate.  Different percents work different ways, but here is one example: If you borrow ten beans, and the interest rate, or percent, was 10%, you would have to pay back ten beans plus one more. If the interest rate was 20%, you’d have to pay back ten beans plus two more.

What if the interest rate was still 20% but you borrowed MORE beans?

BEANS   Extra Due (Interest Rate 20%)
10                       2
20                       4
30                       6
40                       (How many extra beans do YOU think?)

The more you charge on your credit card (the more money you borrow from us), the more interest you’ll have to pay us.

Let’s say your interest rate is 9.99%.  That looks like a big, scary 999% but the . after that first 9 means it is really very close to 10%, so let’s pretend it is.  At 10% interest, if you charged, or borrowed, $100, you’d have to pay us back $100 plus ten dollars extra in interest.  But if you charged $1,000, you’d have to pay us back $1,000 plus ONE-HUNDRED dollars in interest.  You can see that interest really adds up when you charge a lot. 

If you don’t pay your whole bill every month, you’ll have to pay even more interest: You’ll pay interest on any new charges, just like always, but you’ll also have to pay more interest on the leftover billed amounts you haven’t paid for yet–the ones from last month that ALREADY had interest amounts added in to them last month!

That’s right: The credit card company charges you interest on interest. Yes. We calculate interest all over again on the same money we calculated it on last month, and add that NEW interest in on top of the OLD interest that was charged for the same borrowed amount!

There are different kinds of interest and different ways of computing it, but this gives the main idea.

We can change your interest rate, or percent, each month if we want.  Sometimes, the more money you owe us, the more we will raise your interest rate, so that your Total Owed, or debt, grows bigger even faster. This can feel scary if it happens to you when you owe a lot on your credit card.

Besides interest, if you pay a credit card bill late, or even if you pay on time but pay less than the “Minimum Payment” amount shown on the bill, you’ll have to pay us late fees as a punishment for paying late. (This is because when you pay your bill late, it costs our company money, so we charge you extra money in late fees–that seems fair to us.)

Interest and late fees can make your credit card bill get high very quickly.  Try to pay your bill on time and pay everything you owe all at once.  If you can’t, try to stop buying things using your card until your your whole bill is paid off and your Total Owed is $0.   Sending extra payment amounts—even small ones—helps pay your debt faster.  Some people send their tax refunds.

To Think About:

If you do have any extra money sitting in a savings account (money not being used for everyday needs, or reserved for emergencies), that money is not being saved.  It is being lost.

Until all credit card debt is paid off, saving money is losing you money. That is because what a bank pays you in savings interest is never as much as what a credit card costs you in interest and fees for the money you’ve charged (borrowed) from the credit card company.

(Figures below totally bogus–just used for an example.)

.

The Reveal Card
Total Owed (Debt):
$10,724.23                                                                           (CALCULATIONS AT BOTTOM)
Payment Due Date:
April 23, 2011
Payment Choices:                                                               (ESTIMATES)
$10,724.23 will pay your entire debt now
With No New Charges:
$     1,281.90      a month will pay off your debt in 1 year. 
$        742.56      a month will pay off your debt in 2 years.   
$        299.99      a month will pay off your debt in 3 years     (“MINIMUM PAYMENT”)
$      199.99     a month will only keep your debt the same size.
Any less than that will make your debt grow bigger, and,
If you keep using your card, even BIGGER payments will be needed.
____________________________________________________________
CALCULATIONS
A.  $aa,aaa.aa   Still unpaid from your last bill
B.  +  b,bbb.bb  New charges for this month (the “posted” ones we know about) 

C.  $cc,ccc.cc    Unpaid plus new charges 
D.  +    _  ii.ii  The interest amount we’re charging you (cc,ccc.cc  X  pp.pp%)

E.  $10,724.23   Your new total debt, or Outstanding Balance

Your card’s interest rate, or percent, is now pp.pp%.  Your card uses the average daily balance method to calculate the amount on line “B.”. 
(All your charges for the month divided by the number of days in the month).

(Orig. posted 4/22 on Blogger)

edited 2013/11/22 to add an excerpt, try to improve readability

%d bloggers like this: