I have found `Trachtenberg Speed system of basic Mathematics' an illuminating approach to basic maths. I am 39 yrs old. I have not needed a great deal of maths throughout my life and extensively used calculators or spreadsheets for when I did. After working through this book, I can do exactly what it claims a reader will be able to achieve; mental basic arithmetic involving multiplication, addition, division, squaring and square roots, along with being introduced to algebra once again. This last point is the main reason I have returned to maths at this point in my life.
The book is presented in a logical format, not always obvious until you have progressed to division.
The first part deals with multiplication, using numbers between 1 and 12. It [the book] claims that it is not required for the reader to know their tables. However, it states that knowledge of them is very useful.
To explain, the system demonstrates a method of multiplying single numbers by single numbers - the times tables. With this knowledge, all other calculations within the book can be completed. However, with a solid understanding of times tables, many calculations, such as squaring 2 digit numbers and multiplication of up to 4 digits by numbers between 1 and 12, can be completed in less than 10 seconds, in your head, once the system is fully understood.
The multiplication sequence to learn is broken down into groups;
a. 11 & 12
b. 5,6 & 7
c. 8 & 9 (I suggest learning 9x before 8x)
d. 4 & 3
The reason for this is that each group uses extensions of a given method, such as:
a. Multiplication by 11:
11 x 45 = first digit + right hand digit= 5 + 0 = 5
Next digit + right hand digit = 4 + 5 = 9
Next digit + right hand digit = 0 + 4 = 4
b. Multiplication by 12:
12 x 45 = first digit x 2 + right hand digit = 5 x 2=10 + 0 = 10 which is 0 carry 1
Next digit x 2 + right hand digit + carry = 4x2=8 + 5+carry(1)= 14 which is 4 carry 1
Next digit x 2 + right hand digit + carry = 0x2=0 + 4+ carry(1)= 5 which is 5
A fictitious zero is used to facilitate the method at either end of the number to be multiplied so that the number would resemble 045.0 for calculation purposes.
I hope my examples illustrate the point that both 11x & 12x use a system that takes the first digit and adds it to the digit immediately to its right to find each digit of the answer in turn. However, the method for multoiplying by 12 also doubles the digit before it adds the neighbour (right hand digit), which is an extension to the method of multiplying by 11. It is not my intent to reproduce any part of the book, nor demonstrate rigorously any method taught. My example is to illustrate that the methods are broken into groups, as each group method is a successive extension of the last within that group.
One point I should like to mention at this point is that the methods for multiplying any given number by 4 and by 3 are the most complex of all the methods given. As such, it is quicker to mentally multiply each digit from right to left, writing down the product as you go. The methods for 3 & 4 are given for completeness but are slower than the conventional method of multiplication. Likewise, the method for multiplying by 1 & 2 are also given for completeness.
After basic multiplication, the multiplication of larger numbers is taught. This is a single, simple method for multiplication of any number by any number. It is at the end of this chapter that I found my first surprise - how to quickly and accurately check the sum of any calculation without having to re-calculate the sum. The logic is very easy to pick up and became instantly and constantly used by me once I had learnt it.
Chapter 3 - "Two Finger" method was the most confusing chapter of the book as I could not understand why I should learn another way of multiplication, which is longer, when I had just learnt how to mentally multiply utilising rules learnt in the previous chapter. However, this chapter is essential to understand how to carry out division, covered in Chapter 5.
After the, initial, struggle of Ch 3, Ch-4 Addition is covered, and comes as a nice break. I thought this would be rather dull but was again surprised at the simplicity, logic and ability of the system. Furthermore, the way of checking the answers of long addition without re-calculation is demonstrated. I would say that Chapter 4 is the most immediately useable chapter of the book for daily living.
Division is covered in Chapter 5 and where the skills of Chapter 3 come to the fore, along with an extension to them. This is almost as intense as Chapter 3 but once the system was fully understood, I was left, as with every chapter covered, wondering what the fuss was about. I was able to carry out all basic arithmetic either in my head or by writing just the sum down, followed by the answers, as they evolved and in a record time, even with answer checking, which I now do as a matter of course.
Chapter 6 deals with the very simple methods of squaring 2 and 3 digit numbers. After this, the method of finding the square root is dealt with and warns that the method is unlike any method so far learnt. This method is the most intense to learn as there are numerous steps to remember. However, once I set out the work in front of me, I began to understand the sequence and finally learnt it.
Lastly, algebra is discussed. This chapter is an addition and not required unless the reader wishes to articulate the logic of the systems or intends to go further into mathematics. For me, it was a nice gentle way of brushing the cobwebs from my brain as I intend to embark upon a technical degree (Astronomy), which is heavily maths based.
It is not my intent to comment on the extraordinary life of the author. Nor discuss whether this is a better system than the currently conventional one taught in state schools. I will say, however, that having been given a foundation in mathematics from the state methods taught within school and with my mature approach to re-visiting mathematics, I found this system fascinating, fun and exciting. The buzz of learning and maximising mathematical work with the least amount of effort (excluding the effort of learning the methods) and being able to carry out much greater mental arithmetic than I have ever been previously able to do, was a very enjoyable experience.
Helping my daughter sees me using speed maths for her homework, along with answer checking. She does not know how I do it but also does not see me reaching for calculators. I would hope that, through observation and time together, and once she has reached a suitable level of understanding with conventional methods, she will be curious to know how I did what I did so much quicker than her. This may lead her to `discover' the speed method for herself. I am not a teacher and could never expect her to learn another system whilst dealing with the conventional one. I prefer, and believe in, leading by example and action, although I will probably show her how to verify her answers in the near future.
Finally, I should like to say that I was greatly assisted in practising the methods with a purchased downloadable program based directly on this book but not associated with it. This program greatly speeded up my understanding of the systems and provided me with 1000's of random sums to calculate. With both, the book and program, I was able to learn and cement the system into my understanding within 10 days, spending about an hour an evening on it. I am currently revising on GCSE maths in preparation for an Astronomy degree next year. I have also purchased the book `Vedic Mathematics', which is similarly system based but, in my opinion, is for applied mathematical procedures using algebraic equations. It is a much heavier book and very hard going due to its terminology and premise.