Friday, November 24, 2017

Anne of Green Gables

https://www.nytimes.com/2017/04/27/magazine/the-other-side-of-anne-of-green-gables.html


If you know only one thing about Anne Shirley, it is most likely that she has red hair. Maybe you can even picture it, braided into pigtails, sticking out from underneath an unfortunate straw hat, as when Matthew first sets eyes on her. In the 1890s, red hair was a symbol of witchiness, ugliness, passion. Anne’s hair immediately establishes her as an outsider, even as it intimates a kinship between her and Avonlea, with its startling red roads. Anne wants nothing so much as to be rid of it. “I can’t be perfectly happy,” she tells Matthew with characteristic drama on their first ride to Green Gables. “Nobody could who has red hair. ... It will be my lifelong sorrow.”
Anne longs to be beautiful. Not only does she wish for her hair to turn a more dignified auburn, she also tells her best friend, Diana Barry, “I’d rather be pretty than clever.” Praying at Marilla’s behest, she asks God to let her stay at Green Gables and to “please let me be good-looking when I grow up.” She loves pretty things, because she has had none, and swoons over cherry blossoms, an amethyst brooch and the possibility of one day having a stylish dress with puffed sleeves, which sensible Marilla refuses to make for her.
If “Anne of Green Gables” were written today, it is easy to imagine that over the course of the book, Anne would come to learn that none of these externalities matter: not the color of her hair, not the sleeves of her dress. Instead, in the novel, her hair mellows to the coveted auburn, and Matthew, in a moment of tremendous fatherly kindness, gives her a dress with puffed sleeves. Rather than dispense the message that it’s only what’s on the inside that counts, “Anne of Green Gables” conveys something more nuanced, that beauty can be a pleasure, that costumes can provide succor, that the right dress can improve your life — all things that adults know to be true, sometimes, but that we try to simplify for our children.
“Green Gables” is rife with complications like these; it’s an artifact from a different time that, instead of being outdated, speaks to ours in an uncanned, unpredictable voice. Anne has survived for so long because she is more sophisticated than she initially seems.
The book, in a manner that is rare for young-adult novels even now, is a celebration of Anne’s intelligence, which is ultimately cherished by her adoptive parents, her community and her future partner, Gilbert — who is also her closest academic rival and who instead of being threatened by Anne’s brain admires her for it. And yet at the end of “Anne of Green Gables,” Anne quits college and returns to the farm to care for an ailing Marilla, never becoming the writer she wanted to be as a child. This is, perhaps, a disappointing ending (and one that presages a string of follow-up novels in which Anne eventually becomes muted by family life), but it is an honest one: We still live in a world where a woman’s intellect does not preclude her from accruing vast domestic responsibilities.
“Anne of Green Gables” is also a romance, but a slow-burning one. Anne can’t stand Gilbert until the final pages of the first novel, her attentions turned less to love than her friendship with Diana. In the middle of the third novel in the series, “Anne of the Island,” Anne is still putting Gilbert off, rejecting his proposal of marriage. It is not until the beginning of the fifth book, “Anne’s House of Dreams,” by which point Anne has had other suitors and other proposals, that the two are finally married, going on to have seven children.
For all her curiosity, imagination and education, Anne eventually arrives at the traditional ending: a husband, a family and all the attendant duties, the nonconforming woman who conforms. But the novels do not present this as either a great tragedy or a great victory. Instead, these choices look a lot like the fraught and difficult compromises of adulthood, in which you might put aside personal desires to care for your mother or modify your career goals to accommodate children. We would flatter ourselves to think that there is anything particularly old-fashioned about Anne’s trajectory. She is a thoroughly modern girl.

Saturday, October 21, 2017

propositional logic

Some notes for the Stanford class "Introduction to Mathematical Thinking".

Propositional logic is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives, and how their value depends on the truth value of their components.

The following is an example of a very simple inference within the scope of propositional logic:
Premise 1: If it's raining then it's cloudy.
Premise 2: It's raining.
Conclusion: It's cloudy.
Both premises and the conclusion are propositions. 

This inference can be restated replacing those atomic statements with statement letters, which are interpreted as variables representing statements:


Premise 1: 
Premise 2: 
Conclusion: 

Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata range over all propositions and are commonly represented with Greek letters, most often φψ, and χ.

Conjunction is a truth-functional connective which forms a proposition out of two simpler propositions, for example, P and Q. The conjunction of P and Q is written P ∧ Q, and expresses that each are true.

Disjunction resembles conjunction in that it forms a proposition out of two simpler propositions. We write it P ∨ Q, and it is read "P or Q". It expresses that either P or Q is true.

Implication (also known as material conditional) also joins two simpler propositions, and we write P → Q, which is read "if P then Q". The proposition to the left of the arrow is called the antecedent and the proposition to the right is called the consequent. (There is no such designation for conjunction or disjunction, since they are commutative operations.) It expresses that Q is true whenever P is true. 

The material conditional can yield some unexpected truths when expressed in natural language. For example, any material conditional statement with a false antecedent is true (see vacuous truth). So the statement "if 2 is odd then 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement "if I have a penny in my pocket then Paris is in France" is always true, regardless of whether or not there is a penny in my pocket.

In the statement "it is necessary that Q in order for P", P is the antecedent and Q the consequent.

Sunday, September 24, 2017

math books


"How the Brain Learns Mathematics" by Sousa


"What is the name of this book?" by Raymond Smullyan for logic puzzles.

https://www.amazon.ca/Parrots-Theorem-Novel-Denis-Guedj/dp/0312303025
https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

Journey Through Genius by William Dunham

John Mighton's books "The Myth of Ability" and "The End of ignorance"
The Talent Code, by Daniel Coyle
Daniel Willingham's "Why Don't Students Like School?"

Sunday, September 17, 2017

a: the sum of odd numbers

Three solutions:


Solution 1 is geometric


square divided into odd numbers




Solution 2 uses pairwise addition:


Add the 1st and last number in the sequence, then the 2nd and next to last, etc. 


For example, if you have 4 numbers
1 + 3 + 5 + 7, 
then you add 1 + 7 = 8 and 3 + 5 = 8.

Each pair adds to 8, twice the value of the number of numbers (4).

On average, each member of the pair is 4 and you have 4 members so the answer is 4^2.

This can be done for any length of a sequence of odd numbers starting from 1.


Here is the proof:
The number of terms is "n".

When n is even, there are n/2 pairs.

Each pair's value is the same as the sum of the first and last term. The first term is 1, the last term is 2n - 1

The sum of the terms is the number of pairs times the value of the pairs.

The total is then n/2 * (1 + (2n - 1)) = n/2 * 2n = n^2

When n is odd, there are (n-1)/2 pairs plus the middle term whose value is n.

Each pair's value is the same as the sum of the first and last term. The first term is 1, the last term is 2n - 1

The sum of the terms is the number of pairs times the value of the pairs plus the middle term.

The total is then (n-1)/2 * ( 1+ (2n - 1)) + n =
(n-1)/2 * 2n + n =

(n-1)*n + n = n^2



Solution 3 uses induction:



  ie,


squaring the next number adds the corresponding odd number.



q: the sum of odd numbers

The sum of sequential odd numbers starting from 1 is a square number.
Examples:

1+3=4 (or 2 squared)
1+3+5=9 (or 3 squared)
1+3+5+7=16 (or 4 squared)
1+3+5+7+9=25 (or 5 squared)
1+3+5+7+9+11=36 (or 6 squared)

Question 1:
Is this always true?

Question 2:
Why?

Friday, August 18, 2017

mathematical education

Mark Taylor Nice summary. As an educational psychologist, I have a strong interest in this topic. Since the 1990s we have known definitively what causes dyslexia and we know clearly how to treat it (not that it is easy), but we still know little about math disabilities. 

I used to organize the major annual training institute for the Montana Association of School Psychologists. Our institutes last for two-and-a-half days, all on a single topic, and all presented by just one speaker. I tried to get David Geary to come to present to us, having had a chance to visit with him at a major conference in Chicago in the late 1990s, but he declined, on the grounds that no one in the entire world knew enough about math disabilities to present for that much time. Finally, in 2011, after calling a bunch of top researchers, I was able to bring Michele Mazzocco from Johns Hopkins to talk to us. We still do not know much, but we are finally getting some good research. 


David Geary was lead author on the National Panel report on math learning processes in - I think - 2009, and that is available for free on line. A book that I like a lot is Dehaene, S. The Number Sense: How the Mind Creates Mathematics. (OUP - not sure what the current edition is). He is working in France but writes brilliantly in English and for a geek like me the book is laugh-out-loud funny in parts. He also had a terrific article in Science on sources of mathematical thinking. It is a little bit older information but it is solid. If you are a member of AAAS you can find it on line.



http://www.newyorker.com/magazine/2008/03/03/numbers-guy

Tuesday, June 6, 2017

Russian Empress Alexandra Fedorovna on children

Our children naturally bring along with them a multitude of cares and concerns, and for this reason there are people who look upon the appearance of children as a misfortune. But it is only cold egotists who can look upon children in such a manner.

It is a momentous thing to take upon oneself the responsibility for these tender young lives, which can enrich the world with beauty, joy, and power, but which can also easily perish; it is a momentous thing to nurture them, form their character, – this is what one should think about when establishing a home. It should be a home in which children will grow up to a sincere and noble life, grow up for God.

No treasures in the world can replace for man the loss of truly incomparable treasures – his own children.

There are things which God gives often, and others that are given only once. The seasons of the year pass and return again, new flowers bloom, but youth never comes twice. Childhood and all its possibilities are given only once in a lifetime. Whatever you can do to adorn it, do it quickly.

Parents should be what they wish their children to be – not in words, but in deed. They should teach their children by the example of their own life. The greatest treasure that parents can leave their children is a happy childhood, with tender memories of father and mother. It will lighten the forthcoming days, it will preserve them from temptation, and it will help them face the harsh realities of life after they leave the parental roof.

May God help each mother understand the majesty and glory of her forth-coming endeavor, when she holds at her breast her infant, whom she must nurture and bring up. As far as children are concerned, the parents’ duty is to prepare them for life, for any trials that God may send them. While the parents are alive, the child will always remain a child for them and should treat his parents with love and respect. The children’s love for their parents is expressed in complete trust in them. A real mother finds importance in everything in which her child is interested. She listens just as willingly to his adventures, joys, disappointments, achievements, plans, and dreams as other people listen to a romantic narrative.

vision problems in elementary school aged children

There are vision problems which can impair a child's ability to read which are not easily caught by general eye exams. There is a special field called "developmental optometry".

What areas of vision are assessed by developmental optometrists?

Binocularity: the ability of the eyes to work together to transmit information to the brain

Ocular Motility (tracking): the ability to smoothly and accurately move the eyes, which is especially important for reading

Accommodation: the ability to rapidly re-focus the eyes when looking at something up close, then from a distance, and back again