第1章
The Math Revolution in Your Pocket
Have you ever watched someone solve a complex math problem in their head while you were still reaching for your calculator? What if I told you that the difference between you and them isn't intelligence, but simply methodology? Bill Handley's "Speed Math for Kids" has become a quiet phenomenon in educational circles, with teachers across six continents implementing his techniques in their classrooms. Even celebrities like Bill Gates have praised the book's approach to demystifying mathematics. The book has sold over a million copies worldwide and transformed countless "math-haters" into confident calculators. What makes this approach revolutionary isn't just its effectiveness, but how it turns mathematics from a dreaded subject into an enjoyable game that children actually want to play.
第2章
Breaking the Mathematical Mindset Barrier
People don't truly hate mathematics-they hate failing at it. Success makes you love math. When Handley visits schools, students who claimed to hate math often beg to do it all day after learning his methods. This transformation isn't just about academic achievement; it fundamentally changes how students view themselves and their capabilities. A child who masters math often develops increased confidence across all subjects, and this newfound self-assurance extends beyond the classroom.
Many children avoid math not from laziness but because they don't know how to approach it. Telling them to "try harder" only causes frustration when they don't have the right tools. The traditional education system often teaches mathematics as a series of memorization tasks rather than logical processes. Students are expected to memorize multiplication tables and formulas without understanding the underlying patterns. This approach creates anxiety and resistance, leading many students to conclude they're "just not math people." Research shows that this fixed mindset about mathematical ability can persist into adulthood, limiting career choices and daily confidence with numbers.
The "mathematical geniuses" we admire don't have better brains-they have better methods. Eight-year-old Trudy demonstrated this when asked what 14 times 14 is. She immediately answered "196," calculating it while speaking. With just minutes of practice using Handley's techniques, you could outpace friends even when they use calculators. These methods aren't mysterious or complex; they're based on understanding number relationships and patterns that exist naturally in mathematics. For example, knowing that 15 is just 10 + 5 allows you to break down multiplication problems into simpler steps that feel natural and intuitive.
This approach isn't about memorizing more facts-it's about seeing mathematics differently. When you understand the patterns and relationships between numbers, calculations that once seemed impossible become intuitive. The methods in this book aren't isolated tricks but a comprehensive mathematical philosophy that transforms how you approach numbers. This new perspective reveals that mathematics is a language of patterns, and once you learn to recognize these patterns, complex problems become simple variations of familiar themes.
Think about how you currently solve a problem like 8 x 7. Do you recite memorized facts, count on your fingers, or feel a wave of anxiety? What if you could see this problem as a simple relationship to the number 10, making the answer obvious in seconds? By understanding that 8 x 7 is the same as (10 x 7) - (2 x 7), you can quickly calculate 70 - 14 = 56. This method works because it builds on your existing knowledge of base-10 numbers and simple subtraction, rather than requiring pure memorization.
The key to breaking free from math anxiety lies in understanding that mathematical ability isn't fixed-it's highly trainable. Just as athletes improve through practice with proper technique, mathematical fluency develops through exposure to effective methods and consistent practice. When students experience success with these approaches, their confidence grows, creating a positive feedback loop that transforms their relationship with numbers.
第3章
The Circle Method: Multiplication Reimagined
The speed mathematics method revolutionizes multiplication through an elegant approach called "the circle method," which eliminates the need to memorize hundreds of multiplication facts. Instead, students learn to work with strategic reference numbers like 10, 100, and 1000, making calculations more intuitive and manageable.
For basic single-digit multiplication like 8 x 6, the method employs a simple visual technique. Draw circles under each number and write how many more are needed to reach 10 inside them (2 under 8 and 4 under 6). Then perform two quick calculations: subtract crossways (8 - 4 = 4 or 6 - 2 = 4) for the first digit, and multiply the numbers in circles (2 x 4 = 8) for the second digit. The answer emerges as 48. This visual approach helps students understand the relationship between numbers and makes multiplication more concrete.
The method scales beautifully for larger numbers. Consider 96 x 97: write 4 and 3 in circles (representing distance from 100), subtract crossways to get 93, multiply circles to get 12, yielding 9,312. Students often discover they can solve these problems faster than friends using calculators, building confidence in their mathematical abilities. The method works equally well for numbers like 88 x 92 or 75 x 78, always using the closest convenient reference number.
When working with smaller numbers like 7 x 8, we use 10 as the reference point. Write the differences (3 and 2) in circles with minus signs, indicating how far below 10 each number falls. Subtracting crossways gives 5, which multiplied by the reference number (5 x 10 = 50) and combined with the product of circled numbers (3 x 2 = 6) yields 56. This approach helps students visualize number relationships and understand complement numbers.
The method's versatility shines when handling more complex calculations. For 92 x 94, using 100 as reference, write the differences (8 and 6) in circles. Then follow three steps: subtract crossways (94 - 8 = 86), multiply by the reference number (86 x 100 = 8,600), and multiply circled numbers (8 x 6 = 48). The final sum of 8,648 demonstrates how breaking down complex problems makes them manageable.
Beyond multiplication, this method reinforces fundamental mathematical concepts. Students naturally learn about complements, negative numbers, and the relationship between operations. They develop number sense by understanding how numbers relate to convenient reference points like 10, 100, or 1000. The circle method transforms multiplication from rote memorization into a logical, systematic process that students can visualize and understand. This deeper comprehension leads to greater mathematical confidence and improved problem-solving abilities across all areas of mathematics.
The method also proves valuable for mental math, as students can visualize the circles and perform calculations without writing anything down. With practice, they develop the ability to solve complex multiplication problems quickly and accurately in their heads, a skill that serves them well in both academic and real-world situations.
第4章
Beyond the Basics: Advanced Multiplication Techniques
The speed math method extends far beyond simple multiplication. It works for numbers above the reference number, below it, or a combination of both.
For multiplying teen numbers, circles are drawn above the numbers instead of below. Using 13 x 15 as an example with 10 as the reference number, we place +3 and +5 in circles above the numbers. Adding crossways (13 + 5 = 18), then multiplying by the reference number (18 x 10 = 180), and finally multiplying the circled numbers (3 x 5 = 15) gives us 195.
The pattern works consistently: when numbers are above the reference, we add diagonally; when below, we subtract. The method works equally well for larger numbers like 113 x 102. We place +13 and +2 in circles above the numbers, add crossways (113 + 2 = 115), multiply by the reference (115 x 100 = 11,500), then add the product of the circled numbers (13 x 2 = 26) to get 11,526.
When multiplying numbers on opposite sides of the reference number, like 97 x 125 with reference 100, place -3 below 97 and +25 above 125. Calculate crossways by either adding or subtracting (125 - 3 = 122), multiply by the reference (122 x 100 = 12,200), then multiply the circled numbers. Since one number is negative, the product is negative (-3 x 25 = -75), so subtract 75 from 12,200 for a final answer of 12,125.
The rule for multiplying numbers in circles is simple: when both circles are above or both below the numbers, add the result to your answer. When one circle is above and one below, subtract. Mathematically speaking, multiplying two positive or two negative numbers yields a positive result, while multiplying a positive and negative number gives a negative result.
Mental calculation becomes more efficient by streamlining what you say in your head. For 16 x 16, you'd think "16 plus 6 equals 22, times 10 equals 220, plus 6 times 6 (36) equals 256." With practice, this shortens to just "220...256." For simpler problems like 7 x 8, you'd visualize the differences from 10 (3 and 2), subtract crossways to get 5, multiply by 10 to get 50, add 3 x 2 (6) to get 56-all while just thinking "fifty...six."
第5章
The Art of Verification: Casting Out Nines
What if you could always find the right answer to every math problem? The digit sum method, known to mathematicians for about 1,000 years, transforms any size number into a single-digit number for easy verification, even for calculations too large for calculators. This ancient technique, developed during the golden age of mathematics in India and later refined by Arab mathematicians, provides a reliable way to catch calculation errors.
To check calculations, we use substitute numbers-smaller, easier numbers that replace the originals. For example, to verify 12 x 14 = 168, add the digits of each number: 1+2=3 (substitute for 12) and 1+4=5 (substitute for 14). Multiply these substitutes: 3x5=15, then add those digits: 1+5=6. Now add the digits of our answer 168: 1+6+8=15, then 1+5=6. Since both equal 6, our answer is correct. This process works equally well for addition, subtraction, and division problems, making it a versatile verification tool.
A shortcut to this checking method is "casting out nines"-crossing out any 9s or digits that sum to 9 in your calculation. For example, with 168, the 1 and 8 sum to 9 and can be crossed out, leaving just 6. For larger numbers like 137 x 456 = 62,472, we can quickly find substitutes by eliminating 9s: 137 becomes 2 (1+3+7=11, 1+1=2), 456 becomes 6 (4+5+6=15, 1+5=6), and the answer's digits reduce to 3. Multiplying 2x6=12, which reduces to 3, confirming our answer is correct. This method is particularly useful in accounting and financial calculations where accuracy is crucial.
This checking method works brilliantly for numbers too large for calculators. For instance, to verify 12,345,678 x 89,045 = 1,099,320,897,510, we simply cast out nines. The digits in the first number sum to 3, the second to 8, and all digits in the answer cancel out to 0. Since 3x8=24, which reduces to 6, and our answer reduces to 0, we've made an error. This demonstrates how even massive calculations can be verified quickly and efficiently.
The method works because of how numbers relate to 9. Each place value (10, 100, etc.) equals a multiple of 9 plus 1. So 10 is 1x9+1, 20 is 2x9+2, and so on. The remainder when dividing by 9 equals the sum of the digits. When you multiply any number by 9, the digits of the result will always sum to 9 (or a multiple of 9). Whatever operation you perform on a number-adding, multiplying, halving-you do the same to its remainder when divided by 9. This mathematical property extends to all arithmetic operations, making it a fundamental tool in number theory and practical mathematics.
The technique also has practical applications in computer programming and digital systems, where error checking is essential. Many programming languages use similar modular arithmetic principles for data validation and error detection. Modern applications include check digits in credit card numbers and ISBN codes, showing how this ancient method remains relevant in today's digital world.
第6章
Flexible Reference Points: Multiplication Beyond 10 and 100
In previous chapters, we learned multiplication using reference numbers 10 and 100. But what about multiplying numbers around 30 or 60? The same method works with any reference number, though we must choose numbers that are easy to multiply by.
We can use 20 as a reference number because it's easy to multiply by-just multiply by 2 and then by 10. For example, with 23 x 21, both numbers are just above 20, so we put +3 and +1 in circles above them. Adding diagonally (23+1 or 21+3) gives 24, which we multiply by the reference number 20 (24x2=48, 48x10=480). Then multiply the circled numbers (3x1=3) and add to the previous result: 480+3=483.
For numbers below 20, we can still use 20 as a reference number. With 18 x 17, we place -2 and -3 in circles below the numbers. Subtracting diagonally (17-2=15), we multiply by 20 (15x20=300), then multiply the circled numbers (-2x-3=6) and add: 300+6=306.
Alternatively, we can use 10 as a reference number, placing +8 and +7 in circles above, adding diagonally (18+7=25), multiplying by 10 (25x10=250), then adding the product of circled numbers (8x7=56) to get 306. Either reference number works-choose whichever is easier.
For larger numbers, 50 works well as a reference number. To multiply by 50, multiply by 100 then divide by 2. With 45 x 47, place -5 and -3 below the numbers, subtract diagonally (45-3=42), multiply by 50 (42x100=4,200, 4,200/2=2,100), then multiply the circled numbers (-5x-3=15) and add: 2,100+15=2,115.
For even larger numbers, we can use reference numbers like 200, 500, or 1,000. To multiply by 200, multiply by 2 then by 100; for 500, multiply by 1,000 then halve; for 1,000, just add three zeros.
To use reference numbers like 20 and 50 effectively, we need to double and halve numbers easily. For halving odd-tens numbers like 78, think of it as 80-2; half of 80 is 40, half of -2 is -1, so half of 78 is 39. Similarly, to double numbers with high units digits like 38, think of it as 40-2; double 40 is 80, double -2 is -4, so double 38 is 76.
第7章
Simplifying Addition and Subtraction
Addition can be made easier through strategic approaches. While adding small numbers (1, 2, 10, 20, 100, 200) is straightforward, larger numbers benefit from specific techniques. For example, to add 90, add 100 and subtract 10 (46 + 90 = 46 + 100 - 10 = 136). This approach transforms difficult additions into combinations of easier calculations.
When adding mentally, work from left to right instead of right to left as taught in school. For example, to add 5,164 + 2,938, first add 5,000 + 2,900 = 7,900, then add 40 and subtract 2 to get 8,102. This makes it easier to track calculations and hold numbers in your head.
When adding a column of numbers, look for pairs that make 10 first (like 6 + 4), then add remaining digits. For multi-digit numbers like 26 + 32 + 14, group strategically - combine 26 + 14 first (as 6 + 4 = 10) to get 40, then add 32 for 72. This approach simplifies mental calculations considerably.
Subtraction can be made easier through similar strategic approaches. Most find addition easier than subtraction, but with the right techniques, subtraction becomes simple. The key is knowing number combinations that add to 10 and using rounding methods to simplify calculations.
When subtracting a number just below 100 from one just above 100, use a simple method: draw a circle below the number you're subtracting and write the amount needed to make 100. Then add this amount to whatever the first number exceeds 100 by. For example, 123 - 75 = ? Since 75 is 25 below 100, add 25 to 23 for an answer of 48.
When subtracting from numbers ending in zeros (like 1,000 or 10,000), use this shortcut: subtract the units digit from 10, each successive digit from 9, then subtract 1 from the digit to the left of the zeros. For example, 1,000 - 368 = 632. You're essentially subtracting from 999 and adding 1 back. This method eliminates carrying and borrowing, making calculations like 40,000 - 3,594 = 36,406 possible to do mentally with impressive speed.
第8章
Division Demystified
Division is essential for everyday situations like splitting money, calculating sports statistics, or comparing product values. Though many find it difficult, division can be made much easier with the right approach.
For simple division like 33 / 4, ask what multiplied by 4 gives 33 or as close as possible without going over. The answer is 8 with remainder 1, written as 8 r1 or 814. This approach makes division intuitive by connecting it to multiplication.
When dividing larger numbers like 3,721 by 4, work from left to right. Since 3 is less than 4, combine it with the next digit to make 37. Divide 37 by 4 (getting 9 with 1 remainder), carry the remainder to make the next digit 12, continue the process, and express any final remainder as a fraction over the divisor.
The circle method offers an alternative approach for division, especially useful for dividing by 7, 8, and 9. For example, with 56 / 8, draw a circle below the divisor (8) and write the number needed to make 10 (2). Add this to the tens digit (5) to get 7. Draw a circle below 7 and write the number needed to make 10 (3). Multiply the circled numbers (2 x 3 = 6) and subtract from the units digit (6 - 6 = 0). The answer is 7 with no remainder.
A clever shortcut for dividing two-digit numbers by 9: the first digit becomes your answer, and the sum of both digits gives the remainder. For example, 42 / 9 = 4 r6 (4+2=6), and 61 / 9 = 6 r7 (6+1=7). If the digit sum exceeds 9, like in 65 / 9 where 6+5=11, add 1 to your answer (6+1=7) and take the remainder from 11 (11-9=2), giving 7 r2.
Most people feel uneasy about long division, but using factors makes it manageable without memorizing large multiplication tables. Instead of dividing directly by numbers like 36, you can break the division into smaller, more manageable steps.
Factors are numbers that multiply together to equal another number. For example, 2 and 10 are factors of 20, as are 4 and 5. When dividing by larger numbers like 36, you can divide by its factors sequentially (like 6 then 6) instead of dividing by 36 directly. For instance, to calculate 2,340 / 36, first divide 2,340 by 6 to get 390, then divide 390 by 6 to get 65. This approach makes complex division much simpler.
第9章
The Practical Impact of Mathematical Fluency
The methods in this book aren't mere shortcuts but a comprehensive mathematical philosophy that transforms how you approach calculations. Unlike isolated tricks that are easily forgotten, these strategies become part of your everyday thinking process, applicable to all mathematical operations you encounter.
Mathematical ability is equated with intelligence. By using these methods, students will gain a reputation for being smart among parents, teachers, and peers. This positive perception often leads to more intelligent behavior and willingness to tackle difficult problems. Making these methods part of your thinking process can positively change your life's direction.
Being skilled at mathematics makes others perceive you as highly intelligent, but handle this reputation carefully. Don't explain every method-when Isaac Asimov once explained his mathematical approach, the listener dismissed it as "just a trick." These aren't tricks but legitimate alternative methods. People often respond with "anyone could do that" once they understand the method-which is precisely what makes these approaches valuable: they're accessible to everyone.
Teaching these methods creates high-achieving students and makes you a successful teacher. Students will perform better and develop deeper mathematical understanding. You'll teach more content in less time with a motivated, better-behaved class, making every lesson an adventure that benefits everyone.
Parents should encourage children to learn and play with these methods, allowing them to demonstrate their skills. Never criticize mistakes or slow learning-instead, offer support and focus on improvement rather than comparing them to others. These methods effectively boost self-esteem for children who think they're "dumb" at math, with improvements often extending to other academic areas.
The language you use inside your head during mental calculations dramatically impacts speed. Say only what's necessary-just subtotals and totals, or ideally just the answer. For multiplication by 11, don't recite the procedure; simply see that 8+4=12 means saying "Nine hundred and twenty-four" in one breath. When carrying numbers, name them appropriately (thirty for 30, three hundred for 300). Practice repeatedly until calculations that initially seemed difficult become effortless, which will simultaneously build your concentration skills.
Estimation is often more practical than exact answers. When buying 58 computer screens at $399 each, quickly estimating 400 x 60 = 24,000 gives a rough idea of the cost (actual: $23,142). When estimating travel time for a 450km journey at 100km/h, consider factors beyond the mathematical 412 hours-like fuel stops and traffic. The general rule for estimation is to balance rounding up and down. Good estimation develops mathematical intuition and helps verify if answers make sense.
We all think and learn differently. While some teachers believe students should adapt to their teaching style, the better approach is finding multiple ways to explain concepts until everyone understands. When struggling with a textbook explanation, don't assume you lack intelligence-seek alternative explanations from different books, libraries, or friends who understand the concept.
The revolution in mathematical thinking isn't about being faster or smarter-it's about seeing patterns and relationships that make numbers dance in your mind. With these methods, mathematics transforms from a dreaded subject into a playground of possibilities where anyone can excel.
