Beware of mathematicians!
Sep. 20th, 2017 08:15 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
vit_ryesterday part
To discover the second word, please draw the second point.
How many dots did you draw?
1 + 1 = 2
This is the simplest thing about mathematics anybody learns at the primary school.
Now look at your dots. They are different!
We add second dot to the first and get "2". We add first dot to the second and again get "2".
A kid looks into a schoolbook and counts "One apple plus one apple equals two apples". Then it asks "Why can we add them? They are different! Why can we add a good apple and a rotten one and again write '2'? Why can we add one apple and one peach and get '2 fruits'?"
Teachers and parents almost always do not answer this questions. Even in rare cases they know the reasons they think first-graders are too young to know the true. Instead, they label this curious child with "You do not have the mathematical thinking". Most children believe them.
Classmates who have weak imagination do not have interest in questions teacher do not answer and blindly repeat what the elders say. They believe they understand mathematics.
This is terrible.
Mathematics does not add apples to peaches. It does not add apples to apples or dots to dots. It operates with mathematical objects.
Before you transform your dots into "1 + 1" you strip them from all distinguishing features.
The second word that connects our world with mathematics is "abstract".
Please write the number "1" below your first dot and the number "2" below your second dot.
Now your dots represent two objects from the set of natural numbers. Mathematicians usually identify this set with the letter "N".
Are "1" and "2" you have written the same numbers that stand in the equation "1 + 1 = 2"?
No. Numbers you have written under dots are called "ordinal numbers" and numbers you use in your equation are called "cardinal numbers". This means you could use the set of letters and label your dots with "a" and "b". Or you could use the set of words and write "first" and "second". If your only task is to force mathematics to distinguish your dots, all three methods are equivalent.
The next mathematical word for you to memorize is "context".
Things that look the same may have different mathematical meaning. Things that look different may have the same meaning. Mathematicians may use the same term for different concepts and label the same concept with different terms. Anything depends on context.
Note, the set of natural numbers may start with "1" and may include "0" before "1". It is printed in books usually not as "N" but as double-struck capital N "ℕ" however nobody bothers to draw two lines in "N" on paper or on a whiteboard. Even the simplest mathematical abstraction has misleading representations and two different definitions.
Thursday part.
To discover the second word, please draw the second point.
How many dots did you draw?
1 + 1 = 2
This is the simplest thing about mathematics anybody learns at the primary school.
Now look at your dots. They are different!
We add second dot to the first and get "2". We add first dot to the second and again get "2".
A kid looks into a schoolbook and counts "One apple plus one apple equals two apples". Then it asks "Why can we add them? They are different! Why can we add a good apple and a rotten one and again write '2'? Why can we add one apple and one peach and get '2 fruits'?"
Teachers and parents almost always do not answer this questions. Even in rare cases they know the reasons they think first-graders are too young to know the true. Instead, they label this curious child with "You do not have the mathematical thinking". Most children believe them.
Classmates who have weak imagination do not have interest in questions teacher do not answer and blindly repeat what the elders say. They believe they understand mathematics.
This is terrible.
Mathematics does not add apples to peaches. It does not add apples to apples or dots to dots. It operates with mathematical objects.
Before you transform your dots into "1 + 1" you strip them from all distinguishing features.
The second word that connects our world with mathematics is "abstract".
Please write the number "1" below your first dot and the number "2" below your second dot.
Now your dots represent two objects from the set of natural numbers. Mathematicians usually identify this set with the letter "N".
Are "1" and "2" you have written the same numbers that stand in the equation "1 + 1 = 2"?
No. Numbers you have written under dots are called "ordinal numbers" and numbers you use in your equation are called "cardinal numbers". This means you could use the set of letters and label your dots with "a" and "b". Or you could use the set of words and write "first" and "second". If your only task is to force mathematics to distinguish your dots, all three methods are equivalent.
The next mathematical word for you to memorize is "context".
Things that look the same may have different mathematical meaning. Things that look different may have the same meaning. Mathematicians may use the same term for different concepts and label the same concept with different terms. Anything depends on context.
Note, the set of natural numbers may start with "1" and may include "0" before "1". It is printed in books usually not as "N" but as double-struck capital N "ℕ" however nobody bothers to draw two lines in "N" on paper or on a whiteboard. Even the simplest mathematical abstraction has misleading representations and two different definitions.
Thursday part.
Tags:
no subject
Date: 2017-09-20 07:22 am (UTC)Так как очень замечательно.
no subject
Date: 2017-09-20 07:31 am (UTC)Хотел найти для детей правильное введение в математику - не нашёл. Пришлось придумывать самому. По физике что-то дельное обнаружилось в американских учебниках. Математика сразу поднимает на флаг "Нет царских путей в геометрию!" и рысью уносится в непролазные дебри.
no subject
Date: 2017-09-20 07:34 am (UTC)Я — существо изуродованное советско/русским преподаванием математики, причем не самыми лучшими учителями.
Поэтому до сих пор по крупице собираю вот такие вот красивые и полезные подходы, в надежде, когда нибудь всё таки исправить те логические заблуждения, что мне привили в детстве.
no subject
Date: 2017-09-20 09:24 am (UTC)То, что я видел в Германии и Швейцарии (включая учителей, школьные учебники, пособия для подготовительных курсов и подобное), не только бесполезно, но просто специально создано для того, чтобы дети не смогли ничего понять в точных науках.
no subject
Date: 2017-09-20 11:39 am (UTC)Он говорит примерно так же. В 60-е (и даже ранее) — был взят верный курс. И учебники и методология — всё было понятно, занятно, интересно.
Но где-то с середины 70-х годов, пошла мода на заумные объяснения и нарочитое усложнения предмета. Иногда, создавалось впечатление, что авторы рисуются перед публикой, мол посмотрите, как я классно владею терминологией.
И к 80-м годам, уровень доступности математики упал в разы.
no subject
Date: 2017-09-20 02:46 pm (UTC)no subject
Date: 2017-09-20 06:14 pm (UTC)no subject
Date: 2017-09-20 06:46 pm (UTC)Ох, ну откуда взялась помойка? А яблоки были в речи персонажа. Путать автора и героя - это грубейшая ошибка для критика.
Нет "важных" аспектов.
Математика - это 1 + 1 = 2.
Если 2 килограмма, то уже физика,
2 доллара - экономика,
2 яблока - складской учёт.
no subject
Date: 2017-09-20 08:40 pm (UTC)no subject
Date: 2017-09-20 11:48 pm (UTC)no subject
Date: 2017-09-21 06:16 am (UTC)Числа и количество - это не аспекты, а объекты, с которыми манипулирует математика. Они абстрактны. Любая привязка к символам и иллюстрациям - произвольна и может быть как общепринятой, так и просто распространённой в узких кругах, может даже быть индивидуальной. Синестетики связывают числа со звуком или цветом.
no subject
Date: 2017-09-22 10:05 pm (UTC)