Who is James F Holwell, maths coach?

James F Holwell
e-mail: mathscoach@gmail.com
Blog: www.maths-coach.blogspot.com

PERSONAL INFORMATION
Place of Birth: New York, NY

New York State Certificate Number: 63A9632
(Mathematics, Secondary; Permanent Certificate)

EDUCATION
Brooklyn Preparatory School, Brooklyn, NY -- Science Honors Diploma, 1953

Athenaeum of Ohio (Mount St. Mary's Seminary), Cincinnati, OH
-- AB (Liberal Arts), 1958

Graduate School of Science, Yeshiva University, New York, NY
-- MS (Math Education), 1964

Graduate School, Adelphi University, Garden City, NY
-- MS (Mathematics), 1967

EXPERIENCE
West Hempstead Junior Senior High School, West Hempstead, NY
-– Math Teacher, grades 7-9, 1963 - 66

New York City Schools, Ocean Hill-Brownsville Decentralized School
District -- Math Coordinator, 1967 - 68

Montgomery County, Maryland Public Schools
-- Math Teacher, Senior High School, 1969 - 71

Prince George County, Maryland Public Schools
-- Math Teacher, Junior High School, 1971 - 74

McKinley Tech High School, Washington, DC
-- Math Teacher, 1983 - 85

Riverhead High School, Riverhead Central School Dist., Riverhead, NY
-- Math Teacher, 1985 - 87



EDUCATIONAL PHILOSOPHY

I believe that the applications of mathematics to consumer mathematics, measurement, and evaluation of alternatives are far more important than memorizing rules to get answers.
As far as possible, I use "hands on" concrete experiences.

I am at home in a "math laboratory", having set up
and trained teachers and aides in math labs at elementary schools.

Over the years, I have noticed an alarming tendency in mathematics education. I see this tendency -- concentration on memorized facts and rules -- as responsible for crippling the potential to think and to do mathematics in a large number of children and adults. It is extremely important that we do not dwell overlong on drilling of lengthy processes (calculations).

How often have students struggled with two- and three- digit divisors to the point where they conclude that they are just "no good at math"! It is important that we allow students to enter as soon as possible into the world of "word problems" -- practical applications where math ability goes hand in hand with reading comprehension.

I permit calculators so that understanding and problem solving skills can be developed efficiently. It is essential that these skills be developed regardless of the ability to carry out routine calculations.

It is amazing to me that some educators encourage a child with poor motor skills to type rather than struggle with a pen. Yet some of these same educators are horrified by the idea of allowing a student with poor memorizing skills to use a calculator in order to get past the obstacle of calculation with pencil and paper.

What is happening is that so many teachers regard "skill mastery" as essential "before the child can go on" that we are suffering from a national math anxiety crisis.
PEOPLE CONSIDER THEMSELVES FAILURES AT MATH, JUST BECAUSE THEY WERE ONCE SLOW AT MEMORIZING ‘FACTS’!

...........................................................................


Finally, please let me share with you a quotation from a book that does much to elucidate the way a student's mind learns math:

"MATHEMATICS MADE HARD"

As scientists, we like to make our theories as delicate and fragile as possible. We like to arrange things so that if the slightest thing goes wrong, everything will collapse at once! But that isn't good psychology. It's bad the way we let teachers shape our children's mathematics into slender, shaky tower chains instead of robust, cross-connected webs. A chain can break at any link; a tower can topple at the slightest shove. And that's what happens in a mathematics class to a child's mind whose attention turns just for a moment to watch a pretty cloud.

Teachers try to convince their students that equations and formulas are more expressive than ordinary words. But it takes years to become proficient at using the language of mathematics, and until then, formulas and equations are, in most respects, even less trustworthy than common sense reasoning. Accordingly, the investment principle works against the mathematics teacher, because even though the potential usefulness of formal mathematics is great, it is also so remote that most children will continue to use only their customary methods in ordinary life, outside of school. It is not enough to tell them, "Someday you will find this useful," or even, "Learn this and I will love you." Unless the new ideas become connected to the rest of the child's world, that knowledge can't be put to work.

The ordinary goals of ordinary citizens are not the same as those of professional mathematicians and philosophers who like to put things into forms with as few connections as possible. Children know from everyday experience that the more cross-connected their common sense ideas are, the more useful they're likely to be.

Why do so many schoolchildren learn to fear mathematics? Perhaps in part it is because we try to teach the children those formal definitions which were designed to lead to meaning-networks as sparse and thin as possible. We shouldn't assume that making careful, narrow definitions will always help children get things straight. It can also make it easier for them to get things scrambled up. Instead, we ought to help them build more robust networks in their heads.

Quoted from: THE SOCIETY OF MIND, by Marvin Minsky




Until one is committed, there is hesitancy, the chance to draw back, always ineffectiveness. Concerning all acts of initiative (and creation) there is one elementary truth, the ignorance of which kills countless ideas and endless plans: THAT THE MOMENT ONE DEFINITELY COMMITS ONESELF, THEN PROVIDENCE MOVES TOO. All sorts of things occur to help one that would never otherwise have occurred.

A whole stream of events issues from the decision, raising in one's favor all manner of unforeseen incidents and meetings and material assistance, which no person could have dreamed would come his way. Whatever you can do or dream you can, begin it. Boldness has genius, power and magic in it. Begin it now.
… Goethe


However, I persevere.
Ahh, yes, perseverance.
[these lines are to be read in the manner of W.C.Fields]

Perseverance. Ahh, yes, a good thing, indeed, in - deed. Anyone who perseveres and hates children and dogs can't be all bad.
I do believe in perseverance -- at least until something better comes along.
Ahh, yes, and I can't stand opinionated persons ... they are no good, period.
Persevere to the end, yes... by all means.
This is the end.


James does love to play with words; he can't control himself.
He is always sincere, whether he means it or not.
He always tells a lady the truth; the truth is, of course, whatever she wants to hear.

Children's Books You Will Never See:
"You Were an Accident"
"Strangers Have the Best Candy"
"The Little Sissy Who Snitched"
"Some Kittens Can Fly!"
"How to Dress Sexy for Grownups"
"Getting More Chocolate on Your Face"
"Where Would You Like to Be Buried?"
"Katie Was So Bad Her Mom Stopped Loving Her"
"All Dogs Go to Hell"
"The Kids' Guide to Hitchhiking"
"Garfield Gets Feline Leukemia"
"What Is That Dog Doing to That Other Dog?"
"Why Can't Mr. Fork and Ms. Electrical Outlet Be Friends?"
"Bi-sexually Curious George"
"Daddy Drinks Because You Cry"
"You Are Different and That's Bad"

************************************************

--------- Begin forwarded message ----------
From: TashaHal@aol.com

May your week begin with a chuckle and end with a giggle!
Peace and Love, Tasha

Children's Wisdom: Jimmy went into the pet shop.
"Can I have some birdseed?" he asked.
"Oh, you have a bird?" the storekeeper replied.
"Not yet," Jimmy said, "I want to grow one."

A little boy showed his teacher his drawing entitled
"America the Beautiful."
In the center was an airplane covered with apples, pears, oranges and bananas.
"What is that ?" asked the surprised teacher.
"That's the 'fruited plane'," he replied proudly.

<<>>

Do you realize that the only time in our lives when we like to get old is when we're kids? If you're less than ten years old, you're so excited about aging that you think in fractions. "How old are you?" "I'm four and a half."
You're never 36 and a half-you're four and a half going on 5. That's the key.

You get into your teens; now they can't hold you back.
You jump to the next number. "How old are you?"
"I'm gonna be 16."
You could be 12, but you're gonna be 16.
Eventually.
Then the great day of your life; you become 21.
Even the words sound like a ceremony. You BECOME 21... Yes!!!!!!!

Then you turn 30. What happened there? Makes you sound like bad milk. He TURNED; we had to throw him out. What's wrong? What changed?
You Become 21; you Turn 30.
Then you're PUSHING 40... Stay over there.
You REACH 50.

You BECOME 21; you TURN 30; you're PUSHING 40; you REACH 50; and, then you MAKE IT to 60. "I didn't think I'd make it." You BECOME 21; you TURN 30; you're PUSHING 40; you REACH 50; and, then you MAKE IT to 60.
By then you're built up so much speed, you HIT 70.

After that, it's a day by day thing.
After that, you HIT Wednesday...
You get into your 80's; you HIT lunch.
You HIT 4:30.
My grandmother won't even buy green bananas.
"Well, it's an investment, you know, and maybe a bad one." And it doesn't end there.
Into the 90's, you start going backwards.
"I was JUST 92."
Then a strange thing happens; if you make it over 100, you become a little kid again. "I'm 100 and a half."



Truth Can Be Simple: Nobody will ever win the battle of the sexes... There's too much fraternizing with the enemy. ~
A baby first laughs at the age of four weeks. By that time his eyes focus well enough to see you clearly. ~
Drive carefully. It's not only cars that can be recalled by their maker. ~
Don't worry about the world ending today...It's already tomorrow in Australia. ~
Character is what you are. Reputation is what people think you are. ~
Thousands of years ago, cats were worshipped as gods.
Cats have never forgotten this. ~
Friends may come and go, but enemies accumulate. ~
A man who says marriage is a 50-50 proposition doesn't understand two things:
1. Women.
2. Fractions. ~
The facts, although interesting, are irrelevant. ~
There is always one more imbecile than you counted on. ~
The only gracious way to accept an insult is to ignore it.
If you can't ignore it, top it.
If you can't top it, laugh at it.
If you can't laugh at it, it's probably deserved. ~
He who hesitates is sometimes saved. ~
The only difference between a rut and a grave is the depth ~
--------- End forwarded message ----------



If you like this kind of thing, then send a message to Tasha Halpert and ask her to include you on her list.

TashaHal@aol.com wrote:
You will all enjoy this, I think.
Peace to your hearts and Blessings, Tasha
-----------------
Keep On Singing

Like any good mother, when Karen found out that another baby was on the way, she did what she could to help her 3-year-old son, Michael, prepare for a new sibling.
They find out that the new baby is going to be a girl, and day after day, night after night, Michael sings to his sister in Mommy's tummy.

The pregnancy progresses normally for Karen, an active member of the Panther Creek United Methodist Church in Morristown, Tennessee.
Then the labor pains come. Every five minutes ... every minute.
But complications arise during delivery. Hours of labor.
Would a C-section be required?
Finally, Michael's little sister is born.
But she is in serious condition.
With siren howling in the night, the ambulance rushes the infant to the neonatal intensive care unit at St. Mary's Hospital, Knoxville, Tennessee.

The days inch by. The little girl gets worse. The pediatric specialist tells the parents, "There is very little hope. Be prepared for the worst."
Karen and her husband contact a local cemetery about a burial plot.
They have fixed up a special room in their home for the new baby -- now they plan a funeral.

Michael keeps begging his parents to let him see his sister;
"I want to sing to her," he says.

Week two in intensive care.
It looks as if a funeral will come before the week is over.
Michael keeps nagging about singing to his sister, but kids are never allowed in Intensive Care.
But Karen makes up her mind.
She will take Michael whether they like it or not.
If he doesn't see his sister now, he may never see her alive.
She dresses him in an oversized scrub suit and marches him into ICU.
He looks like a walking laundry basket, but the head nurse recognizes him as a child and bellows, "Get that kid out of here now! No children are allowed in ICU."

The mother rises up strong in Karen, and the usually mild-mannered lady glares steel-eyed into the head nurse's face, her lips a firm line.
"He is not leaving until he sings to his sister!"
Karen tows Michael to his sister's bedside.

He gazes at the tiny infant losing the battle to live.
And he begins to sing.
In the pure hearted voice of a 3-year-old, Michael sings:
"You are my sunshine, my only sunshine, you make me happy when skies are gray ---"
Instantly the baby girl responds.
The pulse rate becomes calm and steady.
Keep on singing, Michael.

"You never know, dear, how much I love you,
Please don't take my sunshine away---"
The ragged, strained breathing becomes as smooth as a kitten's purr.
Keep on singing, Michael.

"The other night, dear, as I lay sleeping, I dreamt I held you in my arms..."
Michael's little sister relaxes as rest, healing rest, seems to sweep over her.
Keep on singing, Michael.
Tears conquer the face of the bossy head nurse. Karen glows.

"You are my sunshine, my only sunshine. Please don't, take my sunshine away."

Funeral plans are scrapped. The next, day -- the very next day -- the little girl is well enough to go home!

Woman's Day magazine called it "the miracle of a brother's song." The medical staff just called it a miracle.
Karen called it a miracle of God's love.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

The Athenaeum Philosophy -- Guiding Principles

What do we really mean when we advocate 'freedom with responsibility'?

To many young persons this slogan means that they can be free . . .
IF they submit to authoritarian rules whether reasonable or not;
IF they do what they are told whether they see the point or not:
IF they learn to value material wealth above everything else;
the list of IFs goes on and on.

Of course everyone will agree to the slogan "freedom with responsibility".
In many cases, however, the list of "responsibilities" which are the price of freedom is virtually endless.

On the other hand, the adolescent may choose to live in a way that ignores the talents waiting to be developed. He or she may resist all requests by a parent or teacher to learn in a way that these talents can be brought forth [e-duc-ated]. This desire for 'freedom', which is really the 'license' to do whatever feels good, regardless of the consequences, needs to be corrected.
It is an example of an extreme overreaction to a problem, resulting in another problem. Parents and educational resource people need to act as mentors or coaches. They are to provide guidance to have our young people pay attention to the long view, the big picture.

Freedom does not include the liberty to disregard our talents and to value everything based on the momentary pleasure it gives us.
Often the pleasure is nothing more than the avoidance of the inner pressure to learn and to excel at something of value. We need to decide whether, in the name of freedom, we are willing to allow our children to spend eight hours a day with video games, if that is their "free choice". It seems clear that many "choices" of young people are not valid, because they are based on insufficient information.

They often resist doing or learning something because it takes an effort. They do not like the way something sounds. All of us have struggled for the right to make our own decisions (autonomy). This is good and natural, of course.

At the same time, those of us who happen to be older have a wealth of experience as to what works, and what does not work, in the game of interacting with the so-called "real world".

There is a way to impart this information to young people so that it makes a difference in their lives NOW. We do not do this by "giving up" or by allowing them to do whatever they want. We have the responsibility to lead them in acquiring knowledge on a firm base of experience. "Freedom" needs to be understood not as "freedom from" but as "freedom to".

We need to find ways of freeing young people "to" play a substantial, active role in determining their participation in the life of the community. When we do this, they are not so concerned any more with "freedom from". We ought to allow and encourage more autonomy at the earliest age possible.

"See you later" is an expression our young people are fond of using.
Yet we ought to teach them to beware of the "later" trap.
None of us has a guarantee that we'll be here "later".
The younger we are, the harder it is for us to grasp this concept.


"Principle-Based Instruction"
Knowledge -- the real knowledge that makes a difference in our lives -- comes from application of "Principles" far more frequently than the recall of memorized information.
The human mind craves understanding. Students need to have time to absorb the Basic Principles in every subject, so that the "facts" fit in or make sense in a larger picture of things.
"Principle-Based" instruction makes just as much sense in mathematics and science as it does in history, English grammar, and philosophy. Each day of our life is a precious gift.

All of us, even children, have one day fewer to live than we had yesterday. "Carpe Diem" does not mean "Eat, Drink, and Be Merry for Tomorrow We Die." This ancient wisdom is understood to mean, instead: "The time is ripe; 'pluck' the present day."

To all you ten-year olds who are ready for algebra --
Why wait till you are high school age?

To you twelve year olds who know how to write a paragraph --
When will you start to write your first book?

To you fourteen year olds who have mastered elementary algebra and some geometry -- why not study the calculus today?

To young people of all ages -- have you ever memorized a poem?

Why are you waiting to learn in depth what you have an interest in?
Are you waiting for it to come up in "the school curriculum"?

If so -- what a waste of time!
Never in the future will you be more able to learn than you are right now, whatever your present age.

What book are you reading today, or are you waiting for a book to be "assigned" by a teacher?

What science experiment are you conducting TODAY that will increase your understanding of the universe?

Why is there a need for "The Athenaeum"?
Our public schools are hard-pressed to educate our children.
At best, they may "teach" successfully; that is, present a curriculum and have students pass tests on it.

Attention to grades takes the place of the satisfaction that comes from the intrinsic awareness of having mastered an objective. Furthermore, what real "attitudes of teamwork and cooperation" can be developed in a system of competition for grades?

If a teacher gave everyone in the class the same grade, you know where that teacher would wind up.
We are, however, strongly in favor of high standards, VERY high standards.

We understand, however, that what is a high standard for one student may be an unacceptably low standard for another! YOUR standard of excellence is determined by your gifts, not based on the abilities or commitment level of the others in your class!

The amount of individual attention that can be given to each child is very limited at the elementary level, and at the secondary level, where teachers see each class for only one period per day, there is almost no time for one-on-one instruction.

Of course, students who are 'discipline problems' manage to get attention in their own way -- and students who are willing and able to learn something often wind up with very little attention paid to them.
Teachers are faced with a wide distribution of ability levels ranging from the very slow to the very quick. Attention spans range from jittery "ADHD" children to the studious and thoughtful.

The teacher does his or her best to coax everyone along a prescribed path.
The brightest children become the "Bored of Education", while the least able ones can't keep up, and tragically come to accept "the truth" that they are "failures".

Self esteem doesn't result from pats on the back and other 'strokes';
self esteem occurs quite naturally when we know inside that we have made the "progressive realization of a worthwhile goal"
-- the classic definition of Success offered by the late Earl Nightingale.

The goal may be vastly different from one child to another; the practice of competitive grading will be seen one day as one of the great crimes against children. Not one group benefits from this travesty; the bright tend to have a false sense of their excellence, even when they are not really using their full potential. The slower are left with an anxiety over adequacy that they carry with them all their lives.

What courses would be available through the Athenaeum?

I have truncated the file here -- this ends the philosophy section, the ground of being from which I am coming.


&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

There once was a farmer who took a young miss
in back of the barn where he gave her a

lecture, on horses and chickens and eggs,
and told her that she had such beautiful

manners, which suited a girl of her charms,
a girl whom he'd like to take in his

washing and ironing, and then if she did,
they could be married and raise lots of

sweet violets, sweeter than the all the roses
covered all over from head to toe,
covered all over with snow ...
[to be continued]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

How does one keep a turkey in suspense?
_________________________________________________





^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Today while the blossoms still cling to the vine,
I'll taste your strawberries, I'll drink your sweet wine.
A million tomorrows shall all pass away
Ere I forget all the joy that is mine ... today.

James F Holwell
mathscoach@gmail.com for private messages
Everyone is invited to post a comment in this blog.

What is a Mathscoach?

To: Students of All Ages
From: James F Holwell, Learning Coach

What, then, is a "learning coach" -- ?
First of all, it is a title given to me by John Seal in 1986 when he was in 9th grade, trying to learn enough math to get a good grade.

John was a member of the wrestling team and he admired his coach.
One day he stayed for extra help in math and I spent the time with him, asking him questions that brought forth his abilities.
I was not trying to pound 'material' into his head.

My reward for that was the title "Learning Coach" which John bestowed upon me.
A coach is someone who knows that the ability to achieve is present, in you, now.
A coach works with you on the basis of declarations, requests, promises, and assertions.

Declaration -- We align on a vision for your life.

Requests -- Coaching includes asking you to do things.

Promises -- The coach lets you know what benefit
you may expect to gain from everything you do.

Assertions -- Statements for which evidence is available.


Your Declaration is what You say is the purpose for your life / why you are here.
Your Promises to the coach let him know what you can be counted on for. (They are NOT what you think he wants to hear!)
The work you will do is not to be confused with 'homework for the teacher'; the assignments are requests made of you by your coach,
which you do because you said you would; you do not do them for him, but for yourself, because you remember your vision.

Your coach's Requests of you are based on what you need to know,
and what experiences are useful for you, in order for you to be, to do, and to have all that supports the purpose of your life.

If he makes an Assertion that something is so, he can back it up with evidence: proof that it is so. A coach will speak to you in a way which supports you in achieving what you said you intend to achieve!

A coach can't make you do anything -- you are always free to fire the coach.

In sending you this activity (Searching for the Golden Rectangle)
I am assuming that you are aligned with me in being willing to do things that will result in your increased understanding and ability in mathematics.


INTRODUCTION

Recently I was searching the key words [dogs friendly] for a young friend who loves dogs, and found this little wisdom from a website about dogs:

There are those who teach basic ways to survive in the human world,
and there are those who teach us the ways of love, faithfulness,
reciprocity, forgiveness, spontaneity, and a natural predisposition for joy and companionship.

One is called human wisdom, the other instinctive wisdom.
One teaches obedience (and silly tricks), while the other offers us
a window into a world that we either lost or never knew.

Excerpt from: http://www.lighthousedogs.com/index.html
.................................................................................................

Note to parents and mentors:
We are used to learning in a Sequential Curriculum modality,
with one bit of information following logically from the previous one.
This is convenient for a systems approach to education, perhaps,
but there is one problem with it:
It does not necessarily represent the way many of us learn.

The approach in my coaching, whether in Mathematics, Chess, English Grammar or whatever -- is what I have called the Chaotic Curriculum.
You get dropped into the middle of a lake and you have to learn to swim!
It is scary, but remember, a coach is not a teacher; a coach is not about to test you to find out if you have been 'working up to your ability' -- unless you ask the coach to do that for you.

So, in this method of learning, you announce happily to the coach:
"Guess what, I don't know how to do that!"

To that your coach replies:
"Guess what else, you don't know how to do that YET!" --
and then your coach takes you to that part of your mind which easily comprehends all you need to understand and to do what you intend.

If you are fearful of letting the coach know when you need more explanation or examples, that may be because you have learned in the past that you suffered in some way when someone found out you didn't know something you were supposed to know.
Maybr you got a 'Zero' or an 'F'!

YOU NEED TO UNDERSTAND THAT IT WAS NOT YOUR FAULT YOU DIDN'T KNOW IT! You need to forgive yourself totally for what you believe to be your failures. You begin again, but with a different direction, in a different context, with a coach to guide you.

Here is a simple way of looking at forgiveness:
FORGIVENESS IS GIVING UP ALL HOPE FOR A BETTER PAST.
If you feel that you have failed in the past, then for you the first activity is to forgive yourself.
You need to recognize that these 'failures' were there to help you learn what kind of teaching does not work for you.

Only then will you be able to start over and to expect success.
In this way you achieve joy and satisfaction in your life, starting today.
Like the dog teaching his master, I offer you a window into a world you "either lost or never knew".

Activity Cards are lists of things for you to do.
The purpose of activity cards is to assist you to understand, apply, analyze, synthesize and evaluate what you learn.

What you will learn while searching for a golden rectangle
will be many of the most important principles of mathematics
and how to use them to solve many kinds of problems.

That means you'll be learning why things are the way they are.
You will understand what you are doing.
You will be discovering many things for yourself.

So let's get started...


To Students:
"Searching for the Golden Rectangle"

This first unit is divided into three groups of activities:
Activity One -- Exploring the Size [area] of a rectangle
Activity Two -- Exploring the Shape of a rectangle
Activity Three -- Searching for a rectangle with the Golden Shape
____________________________________________________

Activity Card One

You need:
notebook for recording principles, activities and your discoveries
graph paper (four squares to the inch)
straightedge (may be a ruler or anything with a straight edge)
pencil, eraser
your willingness to understand by DOing!

1. Make six rectangles on the graph paper.
Their names are: A, B, C, D, E, F.
Here are the height and width of each one:
A) 1 by 1
B) 1 by 2
C) 2 by 3
D) 3 by 5
E) 5 by 8
F) 6 by 10.

2. One way to determine the 'size' of a rectangle is to count the number of 'unit squares' inside of it. [This is called the 'area'.]
The unit square is a '1 by 1', or just one box on your graph paper.
Determine the size [area] of each one of the six rectangles.

3. See if you can finish this chart by filling in the missing numbers:
Height Width Size [Area]
A) 1 1 _1_
B) 1 2 ?___
C) 2 3 _6_
D) 3 5 ?___
E) 5 8 ?___
F) 6 10 _60_

4) (Optional activity) -- not part of the Search for the Golden Rectangle -- but If you would like an interesting way to learn your multiplication facts, and to prepare yourself to be able to divide, do this:
Take a new sheet of graph paper and write the number 24 at the top.
Then, see how many different rectangles you can make --
ALL OF THEM HAVE HAVING AN AREA OF 24!

When you are finished, write down the height and width of each one.

Now you know all the numbers that make 24 when you multiply them!

You may be interested in exploring other sizes of rectangles;

What heights and widths can you find that will make a rectangle of area 48? 40? 28? 18? 19? 56? 42? 60? 64? 36? 72? 81? 100?

end of Activity Card One
[Remember to write in your maths notebook whatever you have learned so far. Insert the graph papers into the notebook.]
_______________________________________________________

Activity Card Two

You need a Pocket Calculator and the graph paper with the six rectangles you made in Activity One.

Now we ask, Is there any other way to arrange the six rectangles? [yes]
We can put them in order by Shape, from the widest to the tallest.

Let's do that now ... Which is the widest?
We would say that B (the 1 by 2) is the widest, because it is two times as wide as it is high, and none of the other rectangles has width two or more times its height.
[Check to make sure that is correct.]

The activity is to put all six rectangles in order
according to 'shape', from the widest to the tallest.
This is not easy to do just by looking at the rectangles.

Surely it is easy to see that the 'tallest' one is the square ___.
[Which letter is that?] but what about the other four?

Can you be sure of a way to arrange them in order by shape?

The good news is, there is a way to get a number with which to measure the shape.

The bad news is, these numbers are not numbers like the ones with which you count.

The good news is, you are going to find out that you can think about numbers that are not for counting; you will discover numbers that represent parts of a whole number.

More good news is, by doing one simple calculation for each rectangle, you can measure its shape.

Activities for Activity Card 2:

1. State which two of the six rectangles are the same shape.
See if you can see which ones they are before you calculate anything.
[Look and see if one of the rectangles is an exact scale model of another.]
The two rectangles which are the same shape are ___ and ___.

2. Here is how to find a number to measure the shape of any rectangle, when you know the height and width.
There is an operator button on the calculator for 'divided by'.
This button has a line and two dots, a dot above the line and a dot below the line.

So, to find the shape of a 3 by 5 rectangle, use your calculator like this:
Enter 5
Enter the operator "by"
Enter 3
Enter (=)
Read the answer: 1.6666666
We round off this number to the nearest 1000th -- 1.667
This number is the number that measures the shape of the 3 by 5 rectangle.

[Note: If you are not clear about the meaning of 1.667, you will be soon. If this is over your head for a moment, remember: You have a coach -- You won't drown!]

3. In Activity Card One you completed a chart with missing Sizes of the six rectangles.
In this activity you will find the Shape Measure:

Height Width Shape
A) 1 1 ?______
B) 1 2 ?______
C) 2 3 ?______
D) 3 5 ?______
E) 5 8 ?______
F) 6 10 ?______

Arrange the six rectangles in shape measure order, from widest to tallest.
The smaller the shape measure, the ________ the rectangle.

end of Activity Card Two
__________________________________________

Activity Card Three

You need some simple art materials -- colorful pens,
-- and of course, graph paper.
Also, you will need a ruler which can measure in centimeters.

The search for the golden rectangle begins with this activity --
but it doesn't end here!

This is the test for a Golden Rectangle:
When you remove the square from the rectangle, the shape of the smaller rectangle which is left over is the same as the shape of the rectangle with which you started.

To help you understand what this means, let's test the 3 by 5 rectangle to see if it is a golden rectangle, or not.
First, take a sheet of graph paper, and make a 3 by 5 rectangle.
Next, we mark off the largest possible square inside the rectangle.
[That would be a 3 by 3 square.]

So, with your crayon or felt-tip pen, color the 3 by 3 square, at the left, inside your 3 by 5 rectangle.
Color the 2 by 3 rectangle, at the right, with a different color.

Now we ask the question,
Is the shape of the smaller rectangle which is left over [2 by 3] the same as the shape of the rectangle with which we started [3 by 5] ?
[If you are lost then go back and read again the test for a Golden Rectangle.]

Let's see -- the shape of the smaller, left over rectangle [2 by 3] is 1.5 and the shape of the rectangle we started with [3 by 5] is 1.667
Are these shapes the same? [No.]
Is the 3 by 5 rectangle a golden rectangle? [No indeed.]

Your Activities:
1. Test the other rectangles we have worked with to see if one of them is golden.

2. Which of the rectangles is "almost golden"?
[To answer this question you will need to learn how to find the difference between numbers like 1.667 and 1.5.
You subtract to find the difference.]

3. Make several rectangles of size and shape that please you.
Use them as frames for artwork, or include them into the art itself.

4. Using a ruler which can measure in centimeters, measure the height and width of each rectangle you used in or around your artwork, and use the 'divided by' button to calculate their shapes.

5. Using graph paper, make rectangles with various heights and widths, and see if you can find a rectangle which is "almost" a golden rectangle.

end of Activity Card Three


More activities will be sent to those who go as far as they can
with these three activities, and let me know how it was for you.
[Coach me on being the best possible coach for you.]

James F. Holwell
mathscoach@gmail.com

Search for the Golden Shape

There are two faces of mathematics in elementary and secondary schools:

1. Calculation and manipulation of symbols

2. Using mathematical models to interpret and solve problems

Most of our time and evaluation is centered on number 1.

But which is the more essential in this age of calculators, that we know how to do long division, and to divide a mixed number by a mixed number, and deal with expressions like 2x(5a + 7) -- or that we take away from school something we can use to deal with real situations?

Here is an excerpt from something I sent to the list as a reply to someone working with black teen-agers in Chicago. I believe these considerations are relevant everywhere.

What needs to be talked about is why some students believe they are unable to succeed at doing what they 'should' do.
Isn't this at the basis of defiance of authority in nearly all cases? Are we beating them to death with 'remedial' stuff in what they are not good at, instead of expanding upon their strengths?
In math, specifically, what I see happening almost everywhere, is the holding down of a child who might excel at concepts and applications, because she or he hasn't yet memorized times tables or learned to divide.
We need to distinguish Arithmetic skills from Mathematics, just as we distinguish Grammar and Spelling from Creative Writing!


A REPLY:
Thank God someone is talking common sense about arithmetic maths!
Mathematics was ruined for me at school because I could not do the calculations quickly enough, however hard I tried, and it was assumed that this marked me indelibly as 'no good at maths'.
In later life I have seen some of the pleasures which can be obtained from mathematics as a science, and though I understand it very imperfectly, I also know that if I had had a calculator at school (they were not invented until more than a decade later), I might well have derived serious intellectual benefit from mathematics.

As it was, I stumbled along and failed miserably, as much as anything because the sheer drudgery of calculating meant that I was never really alert to what the teacher was saying.

Mathscoach responds:
Chris, that is so well put!
Here, for example, is the beginning of a unit on the Golden Mean.
I use this with children who barely know how to add, using a calculator as needed.
Of course I teach them Arithmetic also, and then we subtract, multiply, and divide with whole numbers and with 'numbers in-between numbers' whether expressed as fractions, decimal fractions, or percents.
When teaching the calculation part of math, of course calculators are not used.

HOW CAN A RECTANGLE BE "GOLDEN"?
Good, got your attention.
So here, may it lead to an even greater level of joy in your life, the joy that arises from the clear awareness that you found out something new today.

What is the Golden Rectangle?
Think of a rectangle ... the edge of a 3 by 5 index card.
Go ahead,
Get a 3 by 5 index card.
Hold it so it stands up in front of you.
Notice it is five inches wide and three inghes high.

We measure the shape of it by dividing the width by the height.
We say that idea this way: "The Ratio of Width to Height".
We agree that this ratio is the measure of the SHAPE of the rectangle.

So we say that the shape of the edge of the 3 by 5 card is measured by the VALUE OF THE RATIO 5 to 3, or 5 : 3.
This value is found by doing the division problem 5 / 3.
Find this measure as a decimal fraction.
You will need to find the result when 5 is divided by 3.
Do this division any way you like; here is how to do it with a calculator:

HOW TO DIVIDE 5 by 3 WITH A CALCULATOR
1. Enter the 5
2. Press "Divided By" (the key with two dots and a line between them)
3. Enter the 3
4. Press the Equal Sign (=)
Do you get an answer?
Depending on how your calculator is set up, it may be 1.6666666 or 1.67

HOW TO DIVIDE 5 by 3 WITH PENCIL AND PAPER
___
1. Make a division holder like this: 3 ) 5
2. Ask yourself what you can multiply 3 by to get 5.
3. Notice that 3 times 1 is less than 5, but 3 times 2 is more than 5,
so the answer starts with 1.
4. Multiply the 1 times the 3 and subtract that from the 5.
5. This gives you 5 - 3, which is 2.
6. The remainder is 2 then, and now make a fraction with the remainder over the divisor.
7. This gives you the answer 'one and two-thirds' which we can write as: 1 2/3.

Notice that all of these answers are EQUIVALENT,
that is, these are all ways of stating the value of the ratio 5 : 3.
The value of the ratio 5 : 3 is EXACTLY 1 2/3 and that is APPROXIMATELY 1.667.
(When you study decimal fractions, you learn why that is so.)

NOW WE ARE READY TO DISCOVER THE GOLD!
Is the edge of the 3 by 5 card a "Golden Rectangle"?
We are going to find out if it is, and why, or why not!

First we are going to make a smaller rectangle from the 3 by 5 card, like this:
1. Measure a 3 inch square at one end of the card.
2. Draw a line on the card so that there is a 3 by 3 square at one end of the card.
3. Cut off the square with scissors.
4. Throw away the square.
5. See the little rectangle you are holding in the other hand.

If you followed the five steps correctly, the Height and Width of this new smaller rectangle are 2 inches by 3 inches.
We call this a '2 by 3' rectangle, and we ask the question,
WHAT IS THE SHAPE of a 2 by 3 rectangle?

We will find the shape just as we did with the edge of the 3 by 5 card.
The Shape of a 2 by 3 rectangle is the value of the ratio 3 : 2.
The shape is the result when 3 is divided by 2,
so follow the steps given above, this time using the numbers 2 and 3.

This should turn out to be either 1.5 (with a calculator) or 1 1/2 when you divide 3 by 2 with a pencil and paper.

Now, here is the test to find out if the edge of a 3 by 5 card is a Golden Rectangle:
The shape of the 3 by 5 rectangle would have to be EQUIVALENT TO the shape of the new smaller rectangle (2 by 3) you had in your hand after you cut off the 3 by 3 square.

Are these two shapes EQUIVALENT?
Here is how you can tell: The value of the ratio of 5 to 3 is 1.667,
while the ratio of 3 to 2 has the value 1.5 or 1.500

Since these values are not equal we would say,
"NO, the edge of the 3 by 5 card is NOT a Golden Rectangle."

Is the edge of the 3 by 5 card NEARLY a Golden Rectangle?
We would say yes, because the DIFFERENCE between 1.667 and 1.500 (obtained by Subtracting 1.500 from 1.667) is 0.167 and that is not very large.

However, if you are IN SEARCH OF THE GOLDEN RECTANGLE,
then you will want to find one which is more nearly golden than 3 by 5, and guess what?

YOU NOW GET TO DO A DISCOVER - ACTIVITY!

Using only the numbers from 1 to 10 as the height and width of rectangles, you are going to find dimensions (height and width) of a rectangle that is MORE NEARLY GOLDEN than the edge of the 3 by 5 card!

So try the numbers from 1 to 10 as the height and width of rectangles,
and go through all the steps we did above with the 3 by 5 card.


HINT: It is easy to do this experiment if you have paper with squares already printed on it (graph paper).
(Get the graph paper with four squares to the inch.)


In other words, say you are experimenting with the numbers 4 and 7; you will then draw a 4 by 7 rectangle on the graph paper, calculate the shape of it (7 divided by 4), and then use the five steps given above to obtain a new smaller rectangle.

The only difference is that now you are not working with a card,
you are doing this on a piece of graph paper.

So here are the same five steps, adapted for use with graph paper instead of a card:
After you have made a diagram of a 4 by 7 rectangle on graph paper,
Do this:
1. Mark off 4 units along the top and bottom edges of your rectangle.
2. Draw a line on the graph paper so that there is a square (4 by 4) at one end of the rectangle.
3. Notice that you now have a 4 by 4 square and a 4 by 3 rectangle inside your 4 by 7 rectangle.
4. Put a big "X" inside the 4 by 4 square, so you will not look at it again.
5. See the new smaller rectangle and find its shape.
[Draw the 4 by 3 rectangle turned about so that it is a 3 by 4 rectangle. Its shape will then be 4 : 3, and the value of that ratio is found by dividing 4 by 3.]

You may now find the value of each shape...
the 4 by 7 rectangle you started with... 7 divided by 4 equals __________
the new 3 by 4 rectangle... 4 divided by 3 equals __________
[For convenience, show the values as a decimal fraction rounded to three decimal places.]

Are the measures of the two shapes equal?
Is the 4 by 7 rectangle "Golden"?
Is the 4 by 7 rectangle CLOSER to being golden than the edge of the 3 by 5 card?

[Hint: You will find the DIFFERENCE between the two shapes and see whether this difference is more or less than 0.167]

Your Activity now is to find the height and width (the two dimensions) of a rectangle which is CLOSER to being golden than the 3 by 5 rectangle.
As we said, there is such a pair of whole numbers, and the numbers are less than 10!

'bye for now, and Happy Searching for the Golden Shape!