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Math tends to be the least favourite subject of many kids in North America, particularly girls, and my daughter feels bored at schools.
I thought about perplexed questions: What are the most imperative and essential subjects in elementary, middle and high schools? Is only English important? Do only Asians learn math? I told her Google co-founder Sergey Brinn is a math prodigy. And Bill Gates surpassed all of his peer's abilities in nearly all subjects, especially math and science. Actually math serves as a great training ground for a student in any academic subject: biology, physics, chemistry, economics, communication and engineering etc. It really aids in the study of any kind of logic. It helps with argumentation and debate too, especially for bright students. Kids may not understand how work in math now may apply later. Without math foundation, a student can hardly successfully study in universities. An English philosopher and scientist Roger Bacon wrote: "Mathematics is the door and key to the sciences. ……Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world."
An article about home school gave me an enlightening guidance. About 2 percent of U.S. students are home schooled. Reid Barton, the first American student who ever won four-time gold medals at the International Math Olympiad, was a Massachusetts home schooler. Yet, in the geography bee, 22 percent of the national finalists and 40 percent of the final 10 students were home schoolers. And home schoolers swept the top three spots in the same spelling bee in 2000.
I, therefore, am taking matters into my own hands. My love and exceptional understanding of math provide me the quality of home school teaching. Although it involves a lot of time, energy and effort, I really enjoy it. I am always there to support her. I am very happy that home school make her feel learning math is full of fun and she fell in love of math. I would like to share my experience.
I use a heuristic method to guide her to learn and understand math. Everything I explain in a way that can make sense rather then confuse. She has attained understanding of mathematical concepts, refined the concepts as she covered new work, and progressed with deeper understanding. I remembered she used to make a very common mistake like –3 –2 = -1. I brought in an example from real life to demonstrate the signed number concept: “I borrowed $3 from you, and then borrowed $2 from you again. How much do I owe you? ” Just $1? What a deal! She told me even now she still clearly remembered this analogy.
I value math fundamentals. Further math ability improvement largely relies on them. Learning math is a systematically proceeding procedure, cannot be slapdash. I would like to use a question of Math League Contests to illustrate how important the math fundamentals are. "What is the one’s digit in the number 319911991?" It seems hard at first glance, but actually it is an ingenious composition of 3 basic questions. Firstly, the one’s digit changing rule: 3*3=9, 9*3=27, 7*3=21, 1*3=3, 3*3=9, …… every 4 numbers repeat themselves: 3, 9, 7, 1, 3, 9, 7, 1…… It is just a question of multiplication table. Secondly, it is a question what the remainder of 1991 divided by 4 is. Lastly, due to any number that is a multiple of 100 is surely a multiple of 4 and 25, the remainder of 1991 divided by 4 can be simplified as the remainder of 91 divided by 4. If disdaining the basics, it is impossible to master math.
How to have solid math foundation? Practice makes perfect! I do not think that intelligence alone is the key to success. A kid who is able to focus and who has a good work ethic are more vital. I am developing software, which consists of a variety of math drills for her practicing. It takes her just a couple of minute everyday, but it is really helpful for her to lay a solid math foundation. While there were about half of her classmates failed in a math fraction test, she got a perfect mark. So I gave her teacher the fraction drills, which was composed of 66 step-by-step different kinds fraction drills. She told me it was a precious gift and she liked it very much. She immediately gave them to her students to practice.
Math enrichment and math fundamentals are two phases of math learning. They complete each other, and of equal importance. No phases can be ignored. I know what she needs is far beyond the basic stuff. Even for addition, there is much space for math enrichment, for example: “Each letter a, b, c, d, e and f represents a one-digit number. The sum of three-digit number abc and def is 1995. What is the sum of a, b, c, d, e and f? ” Can any kid who knows how to do addition certainly solve the question? I am afraid I don’t think so. These questions are really useful for logical thinking training. Math enrichment is the different phase of math learning, and cannot be neglected.
How do I enrich her math experience? Challenge her, motivate her, and develop more potential of her. AMC and CMC are very good for math enrichment, but sometimes it contains quite hard questions. Without guidance, a kid is very easy to lose interest, even if she is good at math. So I share my exuberance for problem solving and passion for mathematics with her. I guide her solving thought-provoking problems. I talk about topics involving discrete mathematics as well as probability. Although these areas are usually systematically learned at universities, they are worthy of study even in middle school. Normally there are often probability questions in math competitions, such as “2 identical bags contain different numbers of white and black balls. There are 2 white balls and 3 black balls in bag A. And there are 3 white balls and 4 black balls in bag B. A bag was chosen at random and a single ball remove from it, the ball was white. What was the probability that the bag, which was chosen, was bag A?” This probability question is not easy. What are tricks behind the question? Addition principle and multiplication principle. This question is an agile application of these two principles! I guided her solve the probability questions from easy to hard. I gave her many examples and different solutions, compared these solutions, and showed her how to nimbly apply these principles. After she knows how to deal with probability problems, question like “We can express 4 as the sum of one or more natural numbers in eight different ways if order matters: 4, 1+3, 2+2, 3+1, 1+1+2, 1+2+1, 2+1+1, 1+1+1+1. How many such expressions are there for the number 7? ” is not tough.
My way of teaching math is much more than just math materials. I am using math as a way to teach her how to learn, for learning how to learn is crucial than learning the materials. Patterning is excellent for math enrichment, but some patterning questions are not easy, such as,
2, 3, 7, 25, 121, 721, 5041, 40321, ( )
53, 42, 26, 15, 17, 6, 14, ( )
503, 248, 121, 58, 27, 12, ( )
-3, 2, 1, 0, 5, 22, 57, 116, ( )
-3, 2, 1, 0, 3, 4, 7, ( )
1, 4, 11, 29, 76, ( )
So I classified patterning into different kinds, and told her 5 strategies to deal with them. These strategies worked quite well. And patterning questions seem easy if she uses these strategies.
Is math hard to learn? Yes and no. Without math foundation, it is hard, there is no shortcut to learn math. If there is a good teacher, it is not hard. What I pursuit for is using the easiest way to solve complicated questions. I guide her to learn to use different thoughts to analyze and solve problems. I want to broaden her math experience. And I want to let her discover the joy of learning. I would like to use following example,
2222220/4444441 and 33333334/66666667, which fraction is bigger?
How to solve this question? The general way to compare two fractions is:
Firstly, make denominators the same using equivalent fractions.
Secondly, compare the numerators.
It is very hard if using the general way to solve this question, because 66666667 is a prime number and calculator might be overflow. Even if getting a common denominator, it takes longer time.
How do I do?
Firstly estimate. Then know the two fractions all near ½.
Secondly, compare the two fractions with ½. It takes less than 1 minute to solve it.
2222220/4444441 < ½, and 33333334/66666667 > ½
Therefore, 2222220/4444441 < 33333334/66666667.
When she was learning Divisibility, I asked her a serial of following questions:
1.Which number: 123986719999999999 or 774579321999999999, is a multiple of 3?
2.Why is a number divisible by 3 only if the sum of a number’s digits is divisible by 3?
3.With the same principle, can you infer what kind of number is a multiple of 9?
4.How to quickly know 7+7+4+5+7+9+3+2+1+9+9+9+9+9+9+9+9+9+9 is a multiple of 3?
Just mechanically get the sum? No. Look at 7+7+4+5+7+9+3+2+ 9+…+9=7*3 + (4+5) +9 + 3 +(2+1) + 9+…+9, each item 7*3, (4+5), 3, (1+2), 9 is a multiple of 3, so the sum is surely a multiple of 3. Needless to calculate what the sum is.
5.What is the remainder of 12398671999999999999¸3?
Just that simple: (1+2+8+7+1) Mod 3 = ((1+2) + (8+7) +1) Mod 3 = 1
6.Why is the above rule always true? Can you prove it?
The first question is learned at school, but the other 5 questions are for math enrichment. And it is quite helpful to broaden thoughts. No matter how many questions a kid has solved, the key of learning is that not only does she need know how to do, but also should know why she does so. If a kid can infer from what is already learned, her problem solving skills is definitely improved.
I place her in a challenging environment, an environment in which she experience the satisfaction of accomplishing things that she did not think she could do, for example, “There are 9 balls, identical in appearance. One of the balls is either heavier or lighter than the others. If I give you a balance and allow you to make 3 weightings, how could you discover which ball is the odd one out, and whether it is heavier or lighter than the others? ” This is a very tough question. So I simplified the question: “There are 9 balls, identical in appearance. One of the balls is heavier than the others. If I give you a balance and allow you to make 2 weightings, how could you discover which ball is the odd one out? ” I let her do the simpler one first. Then I showed her a straightforward and elegant solution, which helped her to learn more and discover more, and also gave her newfound confidence to tackle questions that before were simply impossible. The result was her wisdom was enlightened.
Why learn math? Do I expect my kid grow up to be a mathematician or a math professor? No, I never think so. And why learn math? I think the pith of learning math is to nurture a resourceful brain and relaxed learning. Just like learning Taiji is for the purpose of keeping in good health, NEITHER for being a Taiji instructor NOR for being a champion of martial art. The obvious benefit of learning math is relaxed learning. If it takes an hour for others to finish homework, she only needs 10 minutes. When math gets harder and harder in higher grade, she feels solving math problems is a piece of cake. The same thing happens in other subjects. The secret is she masters a key – math!
I feel I do have an impact. Sometimes my daughter is not aware of it until later. Once she told me: “Mom, you are really the greatest teacher I have ever met.” My hard work paid off when her keen interest in math is developed, her potential is explored, her mind get motivated, her confidence is inspired, and her genius is released. I believe these will lead her to achieve a variety of life skills.
I thought about perplexed questions: What are the most imperative and essential subjects in elementary, middle and high schools? Is only English important? Do only Asians learn math? I told her Google co-founder Sergey Brinn is a math prodigy. And Bill Gates surpassed all of his peer's abilities in nearly all subjects, especially math and science. Actually math serves as a great training ground for a student in any academic subject: biology, physics, chemistry, economics, communication and engineering etc. It really aids in the study of any kind of logic. It helps with argumentation and debate too, especially for bright students. Kids may not understand how work in math now may apply later. Without math foundation, a student can hardly successfully study in universities. An English philosopher and scientist Roger Bacon wrote: "Mathematics is the door and key to the sciences. ……Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world."
An article about home school gave me an enlightening guidance. About 2 percent of U.S. students are home schooled. Reid Barton, the first American student who ever won four-time gold medals at the International Math Olympiad, was a Massachusetts home schooler. Yet, in the geography bee, 22 percent of the national finalists and 40 percent of the final 10 students were home schoolers. And home schoolers swept the top three spots in the same spelling bee in 2000.
I, therefore, am taking matters into my own hands. My love and exceptional understanding of math provide me the quality of home school teaching. Although it involves a lot of time, energy and effort, I really enjoy it. I am always there to support her. I am very happy that home school make her feel learning math is full of fun and she fell in love of math. I would like to share my experience.
I use a heuristic method to guide her to learn and understand math. Everything I explain in a way that can make sense rather then confuse. She has attained understanding of mathematical concepts, refined the concepts as she covered new work, and progressed with deeper understanding. I remembered she used to make a very common mistake like –3 –2 = -1. I brought in an example from real life to demonstrate the signed number concept: “I borrowed $3 from you, and then borrowed $2 from you again. How much do I owe you? ” Just $1? What a deal! She told me even now she still clearly remembered this analogy.
I value math fundamentals. Further math ability improvement largely relies on them. Learning math is a systematically proceeding procedure, cannot be slapdash. I would like to use a question of Math League Contests to illustrate how important the math fundamentals are. "What is the one’s digit in the number 319911991?" It seems hard at first glance, but actually it is an ingenious composition of 3 basic questions. Firstly, the one’s digit changing rule: 3*3=9, 9*3=27, 7*3=21, 1*3=3, 3*3=9, …… every 4 numbers repeat themselves: 3, 9, 7, 1, 3, 9, 7, 1…… It is just a question of multiplication table. Secondly, it is a question what the remainder of 1991 divided by 4 is. Lastly, due to any number that is a multiple of 100 is surely a multiple of 4 and 25, the remainder of 1991 divided by 4 can be simplified as the remainder of 91 divided by 4. If disdaining the basics, it is impossible to master math.
How to have solid math foundation? Practice makes perfect! I do not think that intelligence alone is the key to success. A kid who is able to focus and who has a good work ethic are more vital. I am developing software, which consists of a variety of math drills for her practicing. It takes her just a couple of minute everyday, but it is really helpful for her to lay a solid math foundation. While there were about half of her classmates failed in a math fraction test, she got a perfect mark. So I gave her teacher the fraction drills, which was composed of 66 step-by-step different kinds fraction drills. She told me it was a precious gift and she liked it very much. She immediately gave them to her students to practice.
Math enrichment and math fundamentals are two phases of math learning. They complete each other, and of equal importance. No phases can be ignored. I know what she needs is far beyond the basic stuff. Even for addition, there is much space for math enrichment, for example: “Each letter a, b, c, d, e and f represents a one-digit number. The sum of three-digit number abc and def is 1995. What is the sum of a, b, c, d, e and f? ” Can any kid who knows how to do addition certainly solve the question? I am afraid I don’t think so. These questions are really useful for logical thinking training. Math enrichment is the different phase of math learning, and cannot be neglected.
How do I enrich her math experience? Challenge her, motivate her, and develop more potential of her. AMC and CMC are very good for math enrichment, but sometimes it contains quite hard questions. Without guidance, a kid is very easy to lose interest, even if she is good at math. So I share my exuberance for problem solving and passion for mathematics with her. I guide her solving thought-provoking problems. I talk about topics involving discrete mathematics as well as probability. Although these areas are usually systematically learned at universities, they are worthy of study even in middle school. Normally there are often probability questions in math competitions, such as “2 identical bags contain different numbers of white and black balls. There are 2 white balls and 3 black balls in bag A. And there are 3 white balls and 4 black balls in bag B. A bag was chosen at random and a single ball remove from it, the ball was white. What was the probability that the bag, which was chosen, was bag A?” This probability question is not easy. What are tricks behind the question? Addition principle and multiplication principle. This question is an agile application of these two principles! I guided her solve the probability questions from easy to hard. I gave her many examples and different solutions, compared these solutions, and showed her how to nimbly apply these principles. After she knows how to deal with probability problems, question like “We can express 4 as the sum of one or more natural numbers in eight different ways if order matters: 4, 1+3, 2+2, 3+1, 1+1+2, 1+2+1, 2+1+1, 1+1+1+1. How many such expressions are there for the number 7? ” is not tough.
My way of teaching math is much more than just math materials. I am using math as a way to teach her how to learn, for learning how to learn is crucial than learning the materials. Patterning is excellent for math enrichment, but some patterning questions are not easy, such as,
2, 3, 7, 25, 121, 721, 5041, 40321, ( )
53, 42, 26, 15, 17, 6, 14, ( )
503, 248, 121, 58, 27, 12, ( )
-3, 2, 1, 0, 5, 22, 57, 116, ( )
-3, 2, 1, 0, 3, 4, 7, ( )
1, 4, 11, 29, 76, ( )
So I classified patterning into different kinds, and told her 5 strategies to deal with them. These strategies worked quite well. And patterning questions seem easy if she uses these strategies.
Is math hard to learn? Yes and no. Without math foundation, it is hard, there is no shortcut to learn math. If there is a good teacher, it is not hard. What I pursuit for is using the easiest way to solve complicated questions. I guide her to learn to use different thoughts to analyze and solve problems. I want to broaden her math experience. And I want to let her discover the joy of learning. I would like to use following example,
2222220/4444441 and 33333334/66666667, which fraction is bigger?
How to solve this question? The general way to compare two fractions is:
Firstly, make denominators the same using equivalent fractions.
Secondly, compare the numerators.
It is very hard if using the general way to solve this question, because 66666667 is a prime number and calculator might be overflow. Even if getting a common denominator, it takes longer time.
How do I do?
Firstly estimate. Then know the two fractions all near ½.
Secondly, compare the two fractions with ½. It takes less than 1 minute to solve it.
2222220/4444441 < ½, and 33333334/66666667 > ½
Therefore, 2222220/4444441 < 33333334/66666667.
When she was learning Divisibility, I asked her a serial of following questions:
1.Which number: 123986719999999999 or 774579321999999999, is a multiple of 3?
2.Why is a number divisible by 3 only if the sum of a number’s digits is divisible by 3?
3.With the same principle, can you infer what kind of number is a multiple of 9?
4.How to quickly know 7+7+4+5+7+9+3+2+1+9+9+9+9+9+9+9+9+9+9 is a multiple of 3?
Just mechanically get the sum? No. Look at 7+7+4+5+7+9+3+2+ 9+…+9=7*3 + (4+5) +9 + 3 +(2+1) + 9+…+9, each item 7*3, (4+5), 3, (1+2), 9 is a multiple of 3, so the sum is surely a multiple of 3. Needless to calculate what the sum is.
5.What is the remainder of 12398671999999999999¸3?
Just that simple: (1+2+8+7+1) Mod 3 = ((1+2) + (8+7) +1) Mod 3 = 1
6.Why is the above rule always true? Can you prove it?
The first question is learned at school, but the other 5 questions are for math enrichment. And it is quite helpful to broaden thoughts. No matter how many questions a kid has solved, the key of learning is that not only does she need know how to do, but also should know why she does so. If a kid can infer from what is already learned, her problem solving skills is definitely improved.
I place her in a challenging environment, an environment in which she experience the satisfaction of accomplishing things that she did not think she could do, for example, “There are 9 balls, identical in appearance. One of the balls is either heavier or lighter than the others. If I give you a balance and allow you to make 3 weightings, how could you discover which ball is the odd one out, and whether it is heavier or lighter than the others? ” This is a very tough question. So I simplified the question: “There are 9 balls, identical in appearance. One of the balls is heavier than the others. If I give you a balance and allow you to make 2 weightings, how could you discover which ball is the odd one out? ” I let her do the simpler one first. Then I showed her a straightforward and elegant solution, which helped her to learn more and discover more, and also gave her newfound confidence to tackle questions that before were simply impossible. The result was her wisdom was enlightened.
Why learn math? Do I expect my kid grow up to be a mathematician or a math professor? No, I never think so. And why learn math? I think the pith of learning math is to nurture a resourceful brain and relaxed learning. Just like learning Taiji is for the purpose of keeping in good health, NEITHER for being a Taiji instructor NOR for being a champion of martial art. The obvious benefit of learning math is relaxed learning. If it takes an hour for others to finish homework, she only needs 10 minutes. When math gets harder and harder in higher grade, she feels solving math problems is a piece of cake. The same thing happens in other subjects. The secret is she masters a key – math!
I feel I do have an impact. Sometimes my daughter is not aware of it until later. Once she told me: “Mom, you are really the greatest teacher I have ever met.” My hard work paid off when her keen interest in math is developed, her potential is explored, her mind get motivated, her confidence is inspired, and her genius is released. I believe these will lead her to achieve a variety of life skills.
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