See the Entire Picture.

Today was my first time to attach an image to my blogs.
I'm going to solve one simple math problem:

2x+1=7-x.



Please take a look at this ugly math picture.

This shows the solution to a simple linear equation : 2x + 1 = 7 - x.
We learn how to solve this equation in our first algebra course. It is like this:

2x + 1 = 7 - x

2x + x = 7 -1

3x = 6

x= 2

I would call it "Just Calculate It" method.

But at precalculus level (I guess), we learn to use graphs to solve the same problem.
The blue line is y = 2x + 1, showing the value of the LHS.
The red line is y = 7 - x, showing the value of the RHS.
We can see that LHS = RHS at (x,y)=(2,5).
This means that LHS = RHS when x = 2 and both sides become 5 at that time.

This "Graph" method requires a higher point of view than "Just Calculate It" method.
What's good about this?

Well, think about the Google Map.
Nowadays more people use "GPS", but I don't have one. So I still rely on Google Map.
When I want to go from Point A to Point B, Google Map can give me the Direction and Map.

"Just Calculate It" method is "Just Rely on Direction."
"Graph" method is "Use Direction and Map" .


Look at the following. I don't recommend "Just Rely on Direction." method.
Try to view the problem at the higher point of view.

By the way, did you notice that I wrote "blogs" in my first sentence?
It's not a mistake.
Please check my profile if you can read Japanese.

Driving directions to Boston, MA

This route has tolls.- more info »

New York, NY

	1. Head southwest on Broadway toward Chambers St
		0.2 mi
	2. Turn right onto Barclay St
		0.3 mi
	3. Turn right onto West St
		1.7 mi
	4. Continue onto 11th Ave
		0.6 mi
	5. Continue onto New York 9A N/12th Ave Continue to follow New York 9A N
		7.4 mi
	6. Take exit 14 for I-95/George Washington Bridge toward Cross Bronx Expy/W 178 St
		0.4 mi
	7. Follow signs for I-95 N/Cr Bronx Expy
		0.2 mi
	8. Keep left at the fork, follow signs for Interstate 95 N/Cross Bronx Expressway and merge onto Interstate 95 Lower Level N/George Washington Bridge
		0.7 mi
	9. Continue onto I-95 N Partial toll road Entering Connecticut
		69.7 mi
	10. Take exit 48 on the left to merge onto I-91 N toward Hartford
		36.8 mi
	11. Take exit 29 to merge onto CT-15 N/US-5 N toward I-84 E/E Hartford/Boston
		0.5 mi
	12. Continue onto CT-15 N
		1.5 mi
	13. Merge onto I-84 E Partial toll road Entering Massachusetts
		40.8 mi
	14. Keep right at the fork, follow signs for I-90 E/N.H. - Maine/Boston and merge onto I-90 E Partial toll road
		55.9 mi
	15. Take exit 24 on the left for Interstate 93 S toward Quincy Toll road
		0.1 mi
	16. Take exit 24A toward S Station Toll road
		0.1 mi
	17. Keep left at the fork and merge onto Atlantic Ave Partial toll road
		0.6 mi
	18. Keep right at the fork
		0.4 mi
	19. Turn left onto State St
		0.4 mi
	20. Slight left onto Tremont St
		371 ft

Boston, MA

Make It Short!

Admonishment to myself:

Simple is the Best.
Make It Short!

Many of my previous posts are too loooooooooooooooooooooooooooooooooooooong!

Less Freedom.

Sometimes, steal freedom away from others can help us.
Here I have three situations for examples.

Situation: Deciding a Restaurant with friends.
1
Suppose my friend (let's call him Bob) and I have to decide which restaurant to visit.
I used to follow Bob's decision.
I tried not to show my preference because I thought then Bob would have more freedom to choose.
I was wrong!
Bob didn't have a freedom; instead, he had a work to do.
He has to choose the restaurant from the scratch.
He might like such work or such authority.
But many of us don't like it.

2
One day, my colleage asked me to go out for a dinner.
He asked me what kinds of food I like with "List"
He gave me a "List": Japanese, Korean, Chinese, Italian, and French.
It made me very easy to answer.

Situation: Seating in Class.

I teach math at high school.
In my first year, I didn't preassign seat for students.
So my students can choose any seats and they could change seat everyday.
But some of them had difficulty choosing a seat. Certainly they didn't like this freedom.
Also, most of them didn't change their seat since first day. They don't want to think about it anymore.

From second year, I stole their freedom of choosing a seat and it didn't cause any problems.
I learned that some people don't like freedom.
Situation: Making a Questionnaire.
Actually I should've learned this strategy long time ago in sociology course in college.
I had to make a questionaire to conduct research.
My teacher advised
"Free Response type questions are more difficult to answer than Multiple Choice type."


What I want to say?
Freedom means that one has choices, hence freedom requires thought and decision.
Less freedom, less thinking....and things often go more smoothly.
So sometimes I should deliberately take away one's freedom...for better.
But be careful.
No freedom, people dont' have to think anything. But they will complain!

Name It

Name is very important.
You earn nothing if your test is nameless.

Well, that's not my point.

Naming is a process of abstraction.

For each stuff  around us, there is a name if it is relevant to us.
With name we can think and discuss about the object.

Example 1:
In arithmetic there is a fundamental property called "Associative Property"
In mathematical expression, it is:

(A+B)+C=A+(B+C) or (AB)C=A(BC).

When algebra students solve equations, they often need this property to solve their problem.
But do they really "Use" the property?
If I ask them to show all the steps or show more details, they can't properly answer.
They are lack of the concept "Associative Property" because they don't have the word for it.
No wonder why every sections of every math books start with new terminologies.

Example 2:
When I read math books written by "Jyuku instructors" for high school students, I find that they give a name to certain process or technique. The name is not known worldwidely. But the name they use is memorizable. Indeed this is my start point to think about naming.

Example2:
Proverbs are names that compress some concepts for life.
With their short and easy to memorize name, we can input its concepts smoothly into ourselves.

Example 3:
Think about Beethoven's piano sonatas. He wrote 32 of them.
But ordinary people are probably familiar to only a few of them.
They are probably with special names such as "Moonlight" "Passionata" and "HammerKlavier".
Most of the 32 have no special names. To refer to them, we say "No. ##" and/or "in X major/minor".
Of course, the one with special names are probably a finer piece of musics.
But I think their popularity is partly due to their name.

Example 4:
In programming, I have to give names to routines, functions, variables, and constant.
With easy to understand names, my program becomes easier to read and it help readers (include me) to grasp what's the script is about.

Also in MS. Excel application, I can define name to ranges, print areas, etc. With names, I can manipulate the workbook quickly and simply.

.....
"Naming" is a technique to enhance our understanding about it (can be things/objects/concepts etc).

Know Yourself

According to Art of War by Sun Tzu (孫子）,
"If you know your enemies and know yourself, you will not be imperiled in a hundred battles."
(知彼知己、百戦不殆) "From en.wikiquote.org"
(I wasn't sure how to translate them from Japanese....So I searched.)

Until very recently I didn't show much interest to other people.
That's must be the reason that my commuinication skill was not improving.
Opponent is more suitable word for enemy in the original context.
At least I don't see other people as my enemy.
Actually opposite, I think people should help each other.

On the other hand, I'm getting better at knowing my feeling.
Here I would like to describe about one of my weakness.

When I took a "Personal Development" course several years ago, one of the assignment was to rank what gives me the motivation to work.
There were 10 things such as "money", "fame" and "skill up".
The Top One I chose was "Praise".
I clearly remember that a student sitting next to me scorned when he saw my ranking.
Nobody else in the class thought that "Praise" as the most important thing for work, not even the top 3.
But I'm certain that I feel happier when I receive "praise" than when I receive "money".
I also feel happy when I solve some difficult math problem or accomplish something even no one pay attention. Though I would take "praise" over "challenge".
I think that is vulgar, dirty, horrible, disgusting mind set. It is far away from my ideal mind.
However I must understand and accept the reality first.
That's my weakness.

Dear readers, now you know about me more. So please help me out.
Then study yourself, too. Be honest with your feeling.

Thanks.

NOTE: my vulgar heart loves praises however, it will rejects praises that doesn't make sense.
Also I'm happy to take all sorts of critism because they give chance to think more!

Replacing a Car Battery

This happened in the early afternoon of the last Saturday.
I replaced my car's battery with help of three nice guys.
Originally I planned to replace it by myself.
The idea is
"I will learn by doing it by myself and I will gain confidence" .

But my neighbors were really nice, they came to help me.
Well, here I would like to write about how they helped me and the problem solving skill they used.

Mr. A

• He helped me removing the old battery.
• He advised me to disconnect the battery from the car when I don't drive for long.
  He told me that it will extend the battery's life.
• He gave me a ride to go to an auto parts shop.
• He paid the new battery for me. (I forgot to bring my wallet! Of course, I paid back)He paid the new battery for me. (I forgot to bring my wallet! Of course, I paid back)

His help was huge. But I can't find what kind of problem solving technique he used.


Mr. A and I got the new battery, but we had difficulty putting it in the hood. There was a part for stablizing the battery at one point. This had two poles with a hook on one side. Our problem was where to hook it. When we face this problem, Mr. B came.

Mr. B

• At first, he asked me "Do you have a repair manual?"
  Wow! A very basic strategy that I totally forgot. Indeed, I had the repair manual.
  I would call this strategy as "READ INSTRUCTION!"
  Unfortunately, this didn't solve our issue because the part had a different shape from the picture in the manual.
• He checked how the battery is set in Mr. A's car. Another basic strategy: "COMPARE".
  I must be brainless at this time. How could I not come up with this idea!
  Unfortunately this didn't work out, too. The part used in Mr. A's car was different from mine.

Mr. C

• He came after Mr. B. I had met him before. We played a baseball with kids one time before the summer break. But only once if I remember correctly.
• One thing he did was to "OBSERVE" the space that the battery must be installed. He removed the battery and the pad under the battery, then he took a close look inside. He didn't solve the problem at first, but he did find the holes to hook the poles at the second observation.
• He brought his own tool kit. I had mine, but it was a mini set. His tool kit was substantial.
  I thought that I should "EQUIP" better.

Anyway they helped me a lot, not only about replacing the battery, but also about teaching PS skills.
The only PS skill I unconsciously applied is "RIGHT TIMING"
If I decided to replace the battery on weekday or in the evening, I probably didn't get any help.

More Check Points II

During this summer, I practice playing piano.
I'm just a beginner, but that's good for experiencing problem solving application.

At first, I just try to follow the sequence and length of tones.
But as I get better, I find points that needs improvement.

1. How hard or how fast to hit each piano key.
2. Position of hand. It took me a while to notice that the piano key is not arranged linearly, but it is arranged on a plane. I mean there is some freedom to move my hands vertically.
3. How much exeggeration is required for "forte","piano","crescendo" ,"decrescendo"...
4. How to play each chord for better sounding. Should I push them simultaneously or at a slight time difference.

I think if I keep practicing, I will find more check points of my own.


By the way, I learned the idea of checklists when I was reading about baseball articles one on Cliff Lee and another on Ichiro Suzuki.  Both of them does tremendous amount of preparation and they have large number of checklists for their motion (pitching and batting.) Before reading these, I didn't have such concepts. 

More Check Points

Some of my students does algebraic computation rapidly, but they are prone to make mistakes.
My only advice for them was to do it more carefully.
But I wonder what "carefully"  really means. What a clever advisor I am!

I can do computations more accurately than them. Of course! But what makes me so?
Here are the reason I come up:
1. I practiced much more.
2. I have more checkpoints for the validity of my answer.

"1" is the biggest reason and "2" is the consequence of it. I going to explain about "2".
The ability to compute more carefully depends on how many checkpoints ones have.
For example, when I solve some equation. I have the following ways to check.
A. Substitute my answer value into the original equation.
B. If it is a polynomial equation of degree N, I would check whether I already got exactly N solutions.
It's an advantage of knowing Fundamental Theorem of Algebra.
C. Check the answer graphically.
D. Make sure the sign of the answer makes sense.
E. Make sure the size of the answer makes sense. (size as a number of digit.)
F. Solve the same problem in different method or in different order.
G. Guess or Estimate the answer first, and determine whether my final answer is as expected or not.
    If not, I would think why not.
H. If it is an integer problem, I would consider parity or divisibility.
I. When recheck my calculation steps, I pay extra attention to where I know I often make mistakes.
    Experience is required to know such points.

Well, what I wanted to say is I have various techniques and point of view to test the validity of my answer.  Hence I can do "carefully". I think the students who caculates rapidly with poor accuracy don' t know how to do "carefully" because they don't know how to verify their answer.
Know I think that's the main problem. Now I have to think how to solve this.


PS. If you have checkpoint that I didn't list above, please let me know. Thanks.

Act Quickly + Take Notes

I thought of myself as a lazy person.
I read books but don't apply knowledge to my life.
I got some idea but quickly forget about it or just using my head and don't move my body.

To improve such behavior into more active one, I began to use memos more frequently.
I felt that the key attribute to active people is their promptness.
The memos remind me and gave me an awareness that I have to do this and do that.
Well, taking memos and awareness of act fast are like chicken or egg dilemma: which causes which?
But either way,  taking memos is very effective so far.
At this moment, I don't feel I'm lazy
Well...somewhat lazy still, but as lazy as I used be.

Problem Solving for Life.

This blog will be my note on practicing Problem Solving (PS).
In year 2010-2011 school year, I taught a course "Advanced Problem Solving".
My aim for this course was a mere preparation for the AMC 12.
While teaching this course, my interest shifted from PS for math to PS for life.
It took me while to realize that I do not possess problem solving skills.
What a shame! I used to write "Problem Solving" as one of skills in my resume.

Starting a blog is one of strategies that I come up to improve my PS.
Let's see how much I can improve from now on.