Other methods for mental math?

[arvi]

What are the other ways to achieve high speed mental math? I know of Soroban. Are there any other methods? What are the pros and cons of each method? Somehow I feel that Soroban is highly mechanized and it doesn't foster love for numbers. Am I wrong?

Has someone researched on this topic. If so, can you share your findings.

[PokerDad]

I have a book "The Great Mental Calculators"... perhaps you were the one that suggested it. I haven't read it yet, it's next on my list.

Before my first post on BK, I went and started to look for kids that were successful in math. That was what I sought out. What I found was that the feats that blew my mind the most were performed by children trained in soroban and anzan. I have little experience with either and am trained traditionally in math - I was fairly good at it and never really struggled (I think much of the revamping of mathematics in US schools has come because people never did "get it" and did struggle with basic math).

I feel like I can somewhat grasp the nebulous concept of mental math. I read a Scott Flansburg book back in the 90s, but never applied his ideas.... just always felt as though math should be able to be done mentally.

I could be wrong, but your sentiment sounds a bit harsh on soroban (just based on what I've seen and the limited amount I've experienced for myself). I do think your concerns are legitimate though.

Posting here to subscribe to the thread and will give a little book report when I finish (which might be a month from now). I can say that the current book I'm reading, "Bounce" features a whole section on the mental calculators, and his tentative conclusion was that they loved numbers and spent a lot of time thinking about them and studying them. He then draws a comparison with experienced check out clerks that had to process change via mental calculation over years of time, and in calculating change, the clerks were as good or faster than the mental calculator whizzes.

EDIT: I wanted to add my uneducated opinion of the love of numbers. The point of Bounce regarding mental calculation is that practice was the key ingredient. The mental calculators loved numbers so much that they just spent loads of time with them; their expertise came from their love via the time spent and practice. So how to foster love of numbers? Well, I know how not to do it. Look at the mathpobic people - they struggle and then internalize their displeasure for math. It then creates a re-enforcement loop akin to the illiterate child that hates reading. Doman says to set your child up for success (he actually says to "always" set your child up for success and "never" set your child up for failure). While I think perhaps his view is off a bit, I do think he hits something very hard.

I've often thought about what caused me to become the swimmer that I became. Well, there are many factors and we don't need to discuss them here, but early on, my enjoyment of the water, coupled with praise and encouragement from my grandmother, were some early keys that led to my desire to join a team - and yada yada yada.

My uneducated opinion is that whatever you do, SET YOUR CHILD UP FOR SUCCESS with numbers early on. Make it something they can succeed at and that they'll enjoy learning (because they actually are learning instead of becoming frustrated) - and you have a decent foundation. The reason I like soroban as much as I do, is that if a 3 year old can learn how to multiply doing it, then it's probably a method that is geared around success. It may not be the end all be all for mathematics, but at least in my mind, it seems to do an awesome job of setting up success in math which can actually promote the love of numbers.

[arvi]

Thanks for sharing your thoughts, PokerDad. I should have probably added a disclaimer that I did not want to come out as harsh but that was my sentiment as well. I was just thinking aloud. I have an uncle who did not even cross 10th grade but he has amazing love for numbers that he can compute anything in his head. He used to question us with math problems when we were young and other kids who boasted that they were very good in math. None of us were ever able to win him. So my sentiment reflects my past experiences.

Maybe if I don't come across any other effective methods I might as well return to Soroban

[Soroban]

Hey Arvi,

I actually have a quick question for you.
What exactly do you mean when you said that Soroban felt mechanized?

I've heard this said a few times, but I'm still not 100% sure what people mean.

Thanks for your help.

Tom

[Mandabplus3]

Arvi, it's probably a bit ahead of your intended age group but Bill Handley has Speed math for Kids, which teaches tips and tricks for faster mental calculations. The book is not a complete program but is well worth the small expense, I am pretty sure he has a complete in school program, with results that are impressive. It will have something in it you yourself will find interesting
Also I got a two for one book including Teach your children tables, same author. It goes well beyond 12x12=! well beyond!
 google is your friend, unless in Australia where it was cheaper at big W 

[arvi]

Tom, by mechanized I  meant that there is only one way for computing a problem and if the child forgets the abacus formula then it is impossible for the child to arrive at a solution. But in traditional methods there are several ways for computing the problem.

Also, for example, to take the number 1729 (the famous Ramanujan-Hardy number) and to say "it is the smallest number expressible as the sum of two cubes in two different ways" needs a different mental workout which I am aiming for.

The irony here is I spent few hundred bucks to learn Soroban and Anzan myself soon after my son was born and now I'm rethinking if its a better fit for us.

Mandabplus, thanks for your suggestion and I'll look into it.

[PokerDad]

Hmmm. Arvi, when you say there are different ways of coming up with an answer - I do get what you're saying. For example, you want to figure out say 535x39 you could chunk the problem down in a couple different ways. Sounds a lot like TERC/Investigations type of talk (which my wife loves, and I'm no where near sold).

My question is, while soroban is mechanized in that you push beads up or down instead of breaking numbers down directly, why couldn't a soroban user find multiple ways to an answer if he or she wanted to? I would surmise the rejoinder would be that there's no need to, since the soroban user could do it correctly just using the standard abacus methods. Why break the problem down when you can just get the answer faster without breaking it down?.... but, the soroban user COULD break it down if they wanted to, just like any other math student COULD break this problem down if they wanted to. Therefore, I'm guessing that this notion of seeing multiple ways to break down a number or break down a problem is one of THOUGHT, not a function of method.

Your thoughts?

[arvi]

PokerDad, you are providing an interesting perspective to use Soroban just for the method and not for the THOUGHT. I have to work out myself to see how practical it is. I have read in some articles that Soroban/Anzan experts discourage us to do like that because speed of calculation decreases in it. So it is likely possible to do as you suggested but just that there will be a tradeoff in the speed.

[Soroban]

Hi Arvi,

Thanks for your reply.

So what I think you're saying is that the method for Soroban you learned was very formula baed. For example it was more similar to 5+5 = 10, not 5+5 has to be 10 because counting 5 more from 5 causes you to have 10.

Is that right?

Or, do you mean that Soroban practices the use of one method of solving problems?

If it's the Former, then it's just the way some Soroban programs are designed. I know that many Soroban programs don't emphasize understanding why calculations work, but that's simply the philosophy of the teachers. It's a more traditional approach to teaching that I think program like Kumon shares.

I know that as a Soroban teacher I've had to make tradeoffs in how much understanding I teach vs practice, not because of time constraints but because of a child's development. I've found that children just don't understand numbers the same way an adult does.

An example might be addition. As an adult we've seem how addition works through the context of arithmatic and algebra so that the concept of adding is fundamentally different from what a child might know. So, I think it's important to treat understanding as just part of the skill necessary for math. Almost like scaffolding that gives kids a starting point to build further math skills, but may eventually be replaced with better and deeper understanding as they grow up.

Let me know if that makes sense.

Tom

[Mandabplus3]

Personally I would happily trade out some speed for deeper understanding, both in myself and my kids. But I guess most people who sign up their kids for these classes want speed.hens Soroban saying sometimes it's a tradeoff.
 Arvi, I can't see why you can't have both. I can't see why you can't learn/teach soroban for the speed and treat the understanding separately or in conjunction. It's just the same as teaching times tables by using visual groups of object and then having kids memorize them by rote..it's been done for years successfully. Use two programs side by side if it's easier. Soroban ( for the mental)  and rightstart or... ( for the manipulatives and understanding)
Spending time just playing with numbers and actively encouraging your kids to come to the same answer in different ways for at least some of their soroban practice would do the same trick too.

[Chris1]

These general techniques for multiplying promote familiarity with the properties and relationships of numbers that remain hidden with the rote methods encountered at school. These methods are easy to understand and can be taught to children. 

Starting with easy tricks teaches children that there is room for creative thinking and that math is fun. I can still remember the day that my father taught me this simple trick to multiply any two digit number by eleven.

23 x 11.  You simply add the digits 2+3=5 and put this result between the 2 and the 3 to get your answer:  253

If the addition exceeds nine you simply place the units digit in the middle as before and add the tens digit to the number on the left.

Here is an example. 67 x 11:  6+7 = 13 so you place the 3 between the digits 6 and 7 to get 637 and then add 1 to the 6 to arrive at the answer 737.

Squaring two digit numbers ending in 5 is another easy trick. To calculate you start by multiplying the first digit with the next higher digit and the answer will always end in 25. (Calculate left to right)

15 x 15:    1 x 2 = 200 then add 25 to get 225
25 x 25:    2 x 3 = 600 then add 25 to get 625
65 x 65:    6 x 7 = 4200 then add 25 to get 4225

When one of the multipliers is near a multiple of 10 we can simplify the problem by multiplying by the nearest round number followed by a minor adjustment.

Example 19 x 32 = 20 x 32 – 32          20 x 30    + 20 x 2     - 32                                                             
                                                                 = 600      
                                                                                   = 640
                                                                                                    = 608

Multiples of 9 and 11 share this interesting property

11, 22, 33 ……..99      10% more than a multiple of 10

09, 18, 27 ……..99      10%   less than a multiple of 10

Two examples-

22 x 43:  calculate 20 x 43 and add 10%

20 x 43 = 860, then add 10%    860 + 86 = 946

18 x 61: find 20 x 61 = 1220, then 1220 – 122 = 1098

This technique can be extended 15%, 20% …

Multiplying by 5 (calculated from left to right is insightful)

This is calculated by halving each number or even grouping in the first number and then appending a zero

6212032241 x 5 = 31060161205 

Dividing even groups of numbers is faster but it is easier to understand the process if we take one number at a time-
Half of 6 = 3
Half of 2 = 1
Half of 1 = 0 requiring a carry of 5 to the right
Half of 2 = 1 plus the carry = 6
Half of 0 = 0
Half of 3 = 1 again requiring a carry of 5
Half of 2 = 1 plus the carry = 6
Half of 2 = 1
Half of 4 = 2
Finally, half of 1 = 0 with the carry of 5 added to the appended zero.

This idea can be extended to other numbers like 15, 25, 50, 75,100 and 125. To multiply by 15 you would start by appending a zero followed by adding half of each number or even grouping to itself working from left to right.

244366122 x 15 = (24+12) (4+2) (36+ 18) (6+3) (122+61)   0
                           =    36            6       54            9      183        0

These methods require you to look at the whole number rather than the individual digits and it is this shift in attention that develops number sense.

Chris.

[Soroban]

Ok I think I wasn't clear enough.

The tradeoff I meant was between the understand I wanted the kids to have and what was possible given their level of understanding and skill. I don't think you can teach very deep understanding (the level that adults have) first.
Instead understanding and skill depend on each other. For instance when teaching addition, I don't start by trying to teach a deep understanding. Instead I start by getting a student to understand just enough to feel comfortable solving problems. The just enough would be to understand that addition is a story about how someone is giving you some amount of stuff, and you're trying to figure out how much stuff you have. This way they don't feel like they're memorizing random facts (ie 1+1 = 2) but don't feel overwhelmed by trying to over-understand addition (ie.  knowing that repeating addition would be multiplication)

As for what Chris is saying about the tricks leading to creativity is counter to my experiences. For most students I feel that the tricks create a belief that math is just a series of memorized facts. I know some students like yourself may become interested by the tricks, but I think you're in the minority. I think its much better to teach robust fundamental techniques that work across all numbers, so that students realize that nothing in math is arbitrary.

Tom

[Mandabplus3]

Well I pulled out that book I mentioned above. Bill Handley Teach Your Children Tables. Wow should have read it earlier 
This afternoon in one sitting I taught my daughter how to get the answer for any times table, mentally!!!!!! Not by rote learning but by simple breakdown steps. She was CALCULATING 7x8 in her head in less than 2 minutes of teaching her the technique. I repeat this was not rote learning, we had fun! She calculated 97x95 quickly and easily. I was fascinated she was absorbed! She needed to know her 2 times tables to use this technique, but she knows them and my other girl used her fingers both got the answer before I could get the calculated app up on my iPad.
  All you mums and dads who are interested in math, buy this book. It will be useful. You will find a way to use it over the years ahead. It is not a complete program ( well I can't say that for sure yet, i only read the first chapter!) we will definately be using it again and again, judging by my flick through. This book is not for beginners but your average 7-8 year olds math ability, plus curious parents 
Perhaps math tricks arnt the best way to learn, perhaps rote is needed (eventually) but I think my girls will prefer having some fun with these math tricks and in the process will be reinforcing their rote learning. If they work out 7x8 enough times they will remember it. I also am not likely to substitute tricks for solid math education and thorough understanding. I liked the particular trick we were using today because it was explained after the kids were fasinated by it. Very sneaky 
I havnt even looked at the second book yet ( speed math for kids) but I got them as a two in one book and the times tables one is at the front.

[Chris1]

I agree that children should be taught techniques that work on all numbers and that   mathematical concepts are generally easier to understand after the student has acquired procedural skill in using the concept. The soroban is an excellent tool to teach the symbolic rules.

Using the addition, subtraction or factoring methods of multiplication with practice will enable a child to master 2-by-2 multiplication.

Addition method:

36 x 42 can be seen as 40 x 36 + 2 x 36 which is 1440 + 72   

36 x 42 can be seen as 30 x 42 + 6 x 42 which is harder to solve.

Subtraction method:

This method simplifies the problem whenever one of the numbers ends in 8 or 9.

69 x 17 can be seen as (70-1) x 17 which is 1190 – 17

The first example above could have been solved using the Factoring Method:

36 x 42 = (36 x 7) x 6 

Made even easier if you notice this (72x7) x 3

Memorising square numbers or being able to calculate them quickly can simplify problems.

Looking at the first example again 36 x 42 we can see that it can be expressed as
39 x 39 – 9 which when simplified is 40x38-8

To square 39 you simply calculate 40x38+1

Multiplication can be used as a shortcut to calculating a repeated addition sum but it is not repeated addition.

Chris.

[joyb]

Interesting posts here...is it worth getting the book Bill Handley Teach Your Children Tables for a 2 and 4 year old or should I wait?  I am not familiar in soroban math at all, what is the best way to get started?  Teaching my children how to read was easy, I'm not a math person myself (although I would love my children to be) so I'm not sure where to begin.

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