主讲人： Jonah Luo

9 月 27 日下午 2.00 – 4.00, Jonah Luo 为中小学生的家长 作了 “给孩子有价值的数学教育”。

在报告中，Jonah Luo 介绍了中西数学教育的不同思路，并且通过具体的例子说明怎么样培养孩子学习的兴趣，启发探索和引导掌握方法。也回答了家长们关心的问题，对当下公立学校的 数学教育，流行的数学辅导机构的长处与问题做了具体说明。在谈到数学教育的意义时，既指出来随着社会发展数学教育也会更新内容，不仅为培养科学家和工程 师，而且也塑造全面发展的合格公民： 提升孩子分析解决问题的能力，掌握科学方法，欣赏数学和科学的美 — 美在和谐与严谨。

在指出数学和科学教育对于个人和社会发展的意义，他表示不赞成过分拔高，灌输式和突击式培养尖子的教育方式。主张从社会需要出发，让孩子真正掌握方法，为 未来成才打好基础。对于确有天赋，愿意挑战数学的孩子，可以在较高年级给他们吃一些小灶，鼓励他们参加竞赛活动。而参赛的态度是 “胜固可喜，败亦欣然” — 其实参赛本身就是开眼界，提高解题的实际能力； 把得失看淡一点，从中才会走出真正的人才。也只有如此，参赛才可以让更多学生受益。

在未来，为着进一步推广数学教育的正确理念，将会组织一些 Workshop，通过专题讲解实际问题，把好的数学教育理念落到实处。

请各位关心孩子数学与科学教育的家长继续关注。

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讲座中提到了一些例子。 读者有进一步兴趣，可以参考

]]>Sometimes you do not know the exact number, but you can guess at it by some clues (for example, the last digit, or the divisors of the number); and you can pin down the number if there are some estimate available to give you the upper / lower bound.

Let us look at a couple of problems as possibly to apply Guess and Estimate:

(1) Suppose n

^{2}= 57^{2}+ 28^{2}– 8^{2}

We know n is an integer, 57 < n < 65.

What is the value of n?

(2) Suppose that n is an integer,

42 sqrt 5 – 33 < n < (21 + 48 ⁄ 7) sqrt 5

What are the possible values of n?

The clues are as follows:

For (1), pay attention to the last digit (the units digit for n^{2} is 9).

For (2), to facilitate an estimate, it is helpful to know that sqrt 5 is slightly less than 9 ⁄ 4.

This estimate will require certain fraction calculations, though.

Another approach to (1) is to notice that:

n^{2} – 57^{2} = 28^{} – 8 ^{2} so

(n + 57) (n – 57) = (28+8) (28-8) = (36) (20)

The answers to the problems (1) and (2) are as follows: (your task is to quickly figure out WITHOUT USING A CALCULATOR; and better NOT USING INVOLVED CALCULATION)

(1) Integer n = 63; (2) Possible values of n is 62 or 63.

Lastly, you might find the following estimate on the radicals pretty interesting (and helpful in some situations):

sqrt 2 = (7 ⁄ 5) ^{+},

sqrt 3 = (7 ⁄ 4) ^{–}, or sqrt 3 = (5 ⁄ 3) ^{+}, and

sqrt 5 = (9 ⁄ 4) ^{–}

where ^{+} means slightly larger, and ^{–} means slightly smaller.

Below is a copy of one students’ work. From this copy, we find he kinds-of get three different answers: the area could be 24, 25 or 32.

But the area of a triangle should be the same under the given conditions. How come we get different answers?

]]>All us of think the following is correct:

Even + Even = Even

Odd + Odd = Even

Odd + Even = Odd

With the sum of three numbers, it holds that:

Odd + Even + Even = Odd

Odd + Odd + Odd = Odd

With a little bit more thinking, for any sum of integers, if the count of odd numbers is odd, then the sum is odd; and when the count of odd numbers is even, then the sum is even. The count of even numbers does not matter.

All these seem trivial, but it could be very useful too!

Just for convenience, let us introduce “/” for odd numbers, and ⋈ for even numbers.

Therefore, the sum of three numbers which is odd can be:

/ + / + / = /

or / + ⋈ + ⋈ = /

Consider the magic square problems: fill in the numbers from 1 to 9 into the square (of 3-by-3 cells):

so that the sum of each row, of each column and of each of the two diagonals, are equal.

Find the total of 1 + 2 + 3 + .. + 9: it must be odd (because there are five odd numbers: 1,3,5,7,9) ! So the sum of each row must be odd, and so is the sum of each column, and so is each of the diagonals.

Now try to guess at: of all the nine cells, which one is an odd number, and which one is an even number?

We figured out there are only two possibilities so as to satisfy the “odd-sum” condition of all rows, columns, and both diagonals:

// Pattern 1

/ | ⋈ | / |

⋈ | / | ⋈ |

/ | ⋈ | / |

// Pattern 2

⋈ | / | ⋈ |

/ | / | / |

⋈ | / | ⋈ |

One can go one step further to colour the cells to get a nice pattern!

This is an example to demonstrate how much we can progress using the roperty of odd/even number ONLY. Although the solution is still not at hand, we have significantly reduce the space for searching!

]]>Try the following questions, and see how many can you answer properly?

A. To represent a two-digit number with different tens-digit and units-digit, what is the best?

∇ ∗ ,

(∇ ∗) ,

10 x ∇+ ⋈ ,

10 (∇) + ∗,

10 (∇) + ⋈ ,

10*w* + *m*, (where *w*, *m* are digits)

B. Write the following numbers in integers or decimals, then arrange them in order (from largest to smallest):

37, 3 x 4, 3 ½

C. To represent three times of a variable x: which is the one that we prefer not to use:

3x, 3 x *x*, 3(*x*)

D. The following expressions contain signed (negative) numbers. Which of them will you recommend to use:

3 + (-2), 3 + – 2, 3 – 2,

3 x (-2), 3 x -2

All these seems to be small stuffs. But it is in these small stuffs that we shall pay attention, to keep the right usage consistently.

]]>There are cards numbered 1, 2, 3, … , 2000 face down on a table. If I choose one at random, what is the probability the number on it will contain a 5?

Do not be scared away by the word “probability”; in this question: it’s just the ratio of the count of numbers that contain digit “5” to the total count of given numbers (which is 2000). — For your warm-up: if you cannot solve the problem right away, can you work on a smaller-scaled version? Say, the cards are numbered 1,2,3, .. until 200.

The question below is a truly mind blower. It has shown you some of the most amazing facts through division of a shape which seems to create something from nothing!

[This link leads to an external video.]

This video is a rich discussion about infinity. It starts from countable infinity, to uncountable infinity, to Cantor’s diagonalization system and comes to Hilbert’s hotel, and the Banach-Tarski Paradox.

]]>Proof Without Words: If x >0, x + 1/x >=2 A picture is worth a thousand words, even in mathematics. One example is the sum of which when multiplies equals 1.

A function is a special relation where the vertical line test is passed (you might be familiar with this result). But what is a horizontal line test. It is a test to decide whether the inverse of the function exists.

What is the horizontal line test? We have learned that if a vertical line intersect a graph more than once, then that graph is not a function. In this post, we learn about the horizontal line test and its relation to inverse functions.

]]>The simple point (and typical way) to introduce square root is to find the side of a square when the area A is given. The side must be √̅A — it’s just an integer! Or we can introduce radicals when studying Pythagorean Theorem for the right triangles. If wo straight edges of the triangle are *a* respectively *b*, and the hypotenuse *c*, then *a ^{2} + b^{2} = c^{2}*, or c = square-root of (a

We have collected a few articles related to the radicals /square roots. Take a look as you like.

(i) Irrational Roots. You’re used to working with quadratics with integer roots, but what about when the roots are irrational?

(ii) Staircase Sequences. Can you make sense of these unusual fraction sequences?

(iii) Nested Surds.

For a quick peep, (i) discusses the irrational roots resulted from solving a quadratic equation, esp. conjugate roots when coefficients of the equation are all integers; In (ii) we consider a staircase sequence, each one is a usual fraction (like 1/2 oe 2/3), but the sequence leads to irrationals; and in (c) we eventually go into double radicals: that is, we have square root under the square root, like square-root of (2+√̅3): for some of them, it is possible to simplify so as to use ONLY ONE layer of square-root!

All those are fun and thought-provoking!

]]>Let us consider this question: how does the value and the methods in math education fit in these traits.

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Now check with your friend about his / her answer. If the two answers (yours and your friend’s) does not agree, you are not alone.

Say, Jack insisted that in the middle part (by dropping two vertical lines from the two ends of the top to the bottom) there is a square (or at least it’s a rectangle), and the colored part – the green triangle is one-quarter of the total area of the square / rectangle. Do you agree with him?

Since Mary cannot find agreement with Jack, she suspect that the middle part should not be a square / rectangle. — Well, the diagram might not be that perfect to be a square, or not even a rectangle. Do you think that Mary has a point?

Even though this seems easy — only calculation of the area, but we find people cannot agree on one answer. Should we rely on the accurate drawing of a figure to work on a geometric problem?

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