
We did a lot of things with blocks, and beads (in 5's and 10's, etc), and wooden pie pieces and toy grocery stores with play money when I was in kindergarten.
"Before entering on an understanding, I have meditated for a long time, and have foreseen what might happen. It is not genius which reveals to me suddenly, secretly, what I have to say or to do in a circumstance unexpected by other people; it is reflection, it is meditation."  Napoleon I





Long past beads and wooden pieces. We're dealing with numbers in the thousands and tens of thousands, especially with negative numbers.





The real question here is how can I have 2 pieces of apple. And if multiply by 1 does nothing, how can I draw 2 apples out of thin air after multiply by 1.





The "1" is a "transform" (translate?) ... you've simply changed the coordinate system; the "2" is still a 2.
"Before entering on an understanding, I have meditated for a long time, and have foreseen what might happen. It is not genius which reveals to me suddenly, secretly, what I have to say or to do in a circumstance unexpected by other people; it is reflection, it is meditation."  Napoleon I





I was terrible, terrible at math as a high school student.
I didn't know why.
I was a fixed mindset on math too. I had a very smart honorsstudent friend and I believed he was just born knowing math. That fixed mindset was bad for me learning.
I went to college and took an algebra class.
I really wanted to get past the issues I had in high school.
I decided to do every odd problem at the end of each chapter.
I started thinking deeply about math just so I could hack my way into understanding math.
Then something happened.
I discovered that if you learned the rules of math and just applied those rules you could take on basically any math. You just simply memorize the rules because math is all made up by humans and someone down the line made up the rules for how things work  for how we calculate.
So, for example I started really looking at the simplest rules:
Commutative property of addition:
1+2 = 2 + 1
No matter the order you add numbers, you get the same answer.
Associative property of addition:
(2+3) + 4 = 2 + (3+4)
Now the student has a base of understanding from simple principles.
Next, explain how the commutative property doesn't work on subtraction
3  2 != 2  3
Allow them to see that this is just a madeup way of dealing with numbers but it works out.
It's not some huge subject that only geniuses can learn but learning these little things provide confidence.
Next, the = property really helped me understand algebra.
I had to really think about what it meant that two sides of = have to be the same.
5 = 5
5 = 7  2
7  2 = 5
So then if you do something to one side you have to do something to the other side to keep the = sign true.
step 1. 7 = 7
step 2. 7(+2) = 7
Step 3. what do you need to do to the 7 on the right to get it to be equal?
Later all of this helped me when we got to exponents for example
I was like why can you just add exponents when multiplying exponents ?
for example x^2 * x^4 = x^6
But, why?
Well, because if I multiply out x*x * (x*x*x*x) that would be the same as X^6.
So now I knew the rule and knew "ok, just add up the exponents when you multiply them".
But what about if someone gives you this?
x^2 + x^4
Well, you have to remember that is:
(x*x) + (x*x*x*x) which is 2x + 4x = 6x and 6x is different than x^6.
Now replace using a simple number:
(2*2) + (2*2*2*2)
4 + 16 = 32
(2*2) * (2*2*2*2)
4 * (16) = 64
This was a lot, but I had to see these rules to really get them into my head.
It's a first principle to learn the very basic in a solid way to provide the student with confidence and then from that confidence they can learn and grow much faster.
Build from simple rules.
Good luck.





Thanks for the very detailed reply. All this is to be done keeping maths as one of the six subjects which the child has to study.
Will certainly use your ideas.





Glad it might help.
I think the mental frame I had to get to was:
1. Math isn't just nebulous thing done by geniuses.
2. If you learn the very basic rules
3. and learn to recognize when to apply the specific rule
4. Then you will no longer just sit and stare at the problem but instead (like any puzzle) you will just apply the rule(s) and complete the work.
This gave me freedom from thinking that math was just "magic that some geniuses understand.
Instead, it is just a game like any other. Learn the rules and apply them.





i was a math major but a poor student . i was excited at start of each course but quickly became frustrated . somehow i graduated . many years later it occurred to me mathematics is discovered not as was often stated by my most hated math Perfessor invented like boring tax laws just as the laws of nature are of course discovered and not invented by a bunch of boring accoutants . i then restudied Calculus and was able to solve every problem in the book . for a demonstration of mathematical discovery i direct you here viz. HACKENBUSH: a window to a new world of math  YouTube[^]





Great story, thanks for sharing. I too like the idea of "discovery of math".
Once my mindset changed (from fixed mindset to growth) I discovered that what we see in schools is often the filtereddown results of bunches of people who summarize everything.
It makes most subjects boring and distant.
As soon as you discover that mathematical concepts are just really great patterns that super smart people have learned then you begin to see that you can:
1. go back to the roots of it and discover what the original people discovered  it will just take a long while and you may not get a huge breadth of understanding.
2. understand that those smart people have shortcut a lot of learning and discovery to give us what we have but we all need to remember that even those giants who we stand on the shoulders of were picking their way through the challenges too.
It's kind of like seeing into the subject and being a part of it instead of just "another thing to memorize."





I never understood that last part but you made it sound easy.
Bastard Programmer from Hell
"If you just follow the bacon Eddy, wherever it leads you, then you won't have to think about politics."  Some Bell.





Game my mother played with me and I played with my children.
Usually while driving somewhere.
just have a conversation
25 divided by 5 times 4 times 5 minus 1 divided by 11 is what?
Nothing about order of operands in this. Just in linear order. But it makes one think instead of using a calculator or sheet of paper. You can make it as easy or hard as you want. But it causes the person to work thru things in their head. It worked wonderfully for math.
9
To err is human to really elephant it up you need a computer






I apologize beforehand that my reply is colored by my personal experience. Why not try to teach him something else than basic arithmetic? There are so many branches of mathematics you can choose from: geometry, sets theory, algebra.
For me, the first 5 years were absolutely horrible. I had (and still have) no inclination for arithmetic and struggled to get mediocre grades. Luckily, in grade 6 I started basic synthetic geometry (triangles and stuff) and a bit of algebra. All of a sudden I discovered that math is about reasoning and first "reductio ad absurdum" proofs seemed like intellectual martial arts. Got hooked for life.
Mircea





Mircea Neacsu wrote: in grade 6
Mircea Neacsu wrote: a bit of algebra
Were you in a class for gifted students?
The difficult we do right away...
...the impossible takes slightly longer.





Richard Andrew x64 wrote: Were you in a class for gifted students? No, that was the standard curriculum at that time in Romania. We were starting school at 7 so they probably had to cram more stuff in our little heads. Anyway, at 13, the idea that you use a letter for a number that you don't know how much it is, didn't seem too difficult to grasp.
A year latter I was reading George Gamow's "One, two, three, infinity..."[^] and discovering Cantor's diagonal method and Moebius bands. As I said: I've got hooked. Once a nerd, always a nerd
Mircea





The issue is that we've to adhere to the school syllabus schedule, because tests and homeworks are as per their schedule. And Integers, Addition, Subtraction come first.





Tell him that there is really only one operator: addition. The other three just all reduce down to that.
"One man's wage rise is another man's price increase."  Harold Wilson
"Fireproof doesn't mean the fire will never come. It means when the fire comes that you will be able to withstand it."  Michael Simmons
"You can easily judge the character of a man by how he treats those who can do nothing for him."  James D. Miles





David Crow wrote: Tell him that there is really only one operator: additionIncrementation. The other threefour just all reduce down to that.
FTFY
(In fact, incrementation, addition, subtraction, and multiplication all reduce to set operations. I'm not sure how to represent division using set operations, other than as repeated subtraction.)
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.
 6079 Smith W.





Exactly, that's what I'm doing. Transforming a Subtraction problem to an Addition problem. Need to continue this with Multiplication and Division also.





If your student has learnt a bit of programming, why not write a calculator with him/her that takes two numbers represented as strings (one digit per character), and operates on them using the basic multidigit arithmetic operations?
I did that with my daughter, and it was fun for both of us.
Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.
 6079 Smith W.





Yes, perhaps need to introduce Programming to this kid.





I try to explain things in a way that the kid can see it as something to understand, not to memorize. For example:
Kid trying to count to 100 and you hear "forgetting" numbers. Explained:
You only need to repeat from 0 to 9 (and you already know that)
Once you get to 9, you move back to 0 and add 1 to the tens
then I told the name of all the tens and once I got to 100 I just said,
and now we repeat again, but with 3 numbers instead of 2,
and told the name of the hundreds until 1000
and when the numbers behind the first one are not zero... you go back to the previous "scale" and add it
so
234 = 200 + 30 + 4
The kid learned how to count to ten thousand in one afternoon.
For arithmetic I use examples on everyday routines, i.e. preparing the meal box...
How many pieces of apple do you want? 5, Ok... If I only pack 2 how many are you missing? And if I pack 9, how many additional pieces do you have?
Multiplications... writing down the equivalent in times added on the side and saying, multiplication just save time and pencil. Question back, but daddy, 3x1 you need more pencil than writing only 3... me
If the kid is curious (and that's almost by definition of Kid), you only has to find the way that particular kid needs. Playful, logical, by heart, laws of nature... whatsoever. If you find it, the kid learns way faster.
Just try to find it.
M.D.V.
If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about?
Help me to understand what I'm saying, and I'll explain it better to you
Rating helpful answers is nice, but saying thanks can be even nicer.





Thanks. Could do many of these things if the kid were in front of me. This kid is in a different city, and I am doing Skype sessions, so need to adapt this method to an online teaching session.





A very long time ago I found a book in the school library and read it. It was called "Quick and Easy Math" by Isaac Azimov (1964). Give it a read.
David Wright





Great book. Thanks for sharing.



