Teaching a 4 year old maths Im 18 years old and getting to grips with advanced mathematics (pre-university) and I have a younger brother of 4 years old (quite an age gap). I want to get him interested in learning (and away from the iPad/tv), especially through maths. He hasn't formally started school, but went to a nursery where they may have taught numerics but that's about it. I have printed worksheets such as "add one to this number" kind of thing but he fails to understand. I also don't know how to convey the idea of counting. Is there a certain protocol to teaching kids of such an age mathematics, however basic? Is he too young to spend at least 10-20 mins doing some maths with me, or should I let him turn 5 and progress from school?
My aim is for him to develop a passion for maths as I currently do, and not take the subject as a burden as many people do.
thanks (sorry if this is not to be discussed on this site)
 A: Kids that age don't learn too well from worksheets; they're too abstract and not engaging enough. There's a huge conceptual leap between "two marbles in one hand and three in the other" and "2+3=5" that's difficult to see when you're on the other end of it. At this stage I'd focus on 'concrete' counting and visual math, like Möbius strips and Platonic solids. If you want him to get interested in math, then the flashy attention grabbing stuff like that is probably your best bet.
A: Physical objects and comparisons. Take objects that there are plenty of that the child is interested, I will use building blocks as an example.
Place a pile of three blocks, one at a time, in front of the child. Count each block as you place it. Don't make complex shapes with them, make the focus on the counting, not the presentation.
Create a second pile of four blocks. Count out loud again as you place them. Emphasize the new number you're adding, FOUR.
Do this again for a third pile.
Then with the child, count the first pile but stop at the number before the last block in the pile. "One," point at a block. "Two," point at the second block, then without speaking point at the third block. You may have to prompt the child.
Once the child is able to identify the number of blocks in the group, and the next number in the sequence, you can start with moving blocks from one pile to another. At this point it's all about teaching via comparison, and giving them something to put their hands on.
Did this with both of my two boys who are now going in to (US) first and second grade and they are able to handle math on 3rd and 4th grade levels respectively and have no problem with common tasks such as determining how many more of something they need or how many minutes are left between one time and the next, etc.
A: Enthusiasm
One thing to keep in mind when trying to pass on an enthusiasm for mathematics to a child who is very young is to remember that the thing that will get a kid to fall in love with math is the feeling of working hard on a problem and coming up with their own solution.  Do not take this away from them!
If you're talking about a problem that's just beyond the child's ability, but not getting anywhere, there are a few ways to avoid becoming frustrated.  You can try to give some hints, you can try acting things out, or you can say something like, "This is a tough problem, maybe we should think about it for a while," and then move on.  This can help to teach perseverence.
Please don't teach a child that math is all about worksheets with arrays of addition facts on them.  Do teach them their addition facts by asking them to add things they encounter every day:  "Mario has 3 lives, but if I play this level I will get 2 free lives.  How many will I have after this level?"
Sense-making
Try to get in the habit of asking the child to explain why they think their answer is correct both when they actually are correct and when they are incorrect.  Try even harder to avoid giving them clues to whether they are correct in the way you ask the question.
Act like you don't understand how to do something and do really silly things, like mis-count, so they explain to you the correct thing.  When they are explaining, try to think like a genie and ask them questions to make them clarify their explanation:

CHILD:  "A bench is a thing you sit on"
ADULT:  "Like this chair?  Is this a bench?"
CHILD:  "Noooo! A bench is long."

Things you probably forgot you learned
At one point it was not obvious that 3+2 is the same as 2+3.  It did not become obvious because someone pointed it out to you, but rather because you computed 3+2 and 2+3 the hard way many times.  Expect a young child not to realize this, too, and give them lots of chances to compute those sums.
Once, when you were very young, to compute 5+2 you counted "1, 2, 3, 4, 5" and then "6, 7" while looking at your fingers.  Preschool age children usually don't learn the strategy called counting on until they are around 5 or so.  Expect this, and give them lots of chances to practice, so that they will get tired of all that counting.  They will "invent" counting on
At one point, the thing that made a square a square was the fact that it looked like a square.  Turn it 45 degrees and that's not a square, it's a diamond.  To young children, what makes a shape is how it looks.  Try taking some drinking straws cut to various lengths and making different looking triangles.  A fun game is to each take matching pieces and see if you can make different triangles.  It's fun if you make your triangle in secret and then reveal them together.
Building number sense
One tool that teachers of young children employ in helping their students visualize numbers is called a ten frame and it is a grid with two rows and 5 columns.  To use a ten frame to visualize a number, you put that many dots in the grid, one per cell.  Make several of these.  Then pull one out and ask the child to tell you how many dots are there.  Then ask them how they see it.  Then say a different way to see it.

"There are seven dots.  I see five on the top row and two more on the bottom"
"Oh, that's interesting.  I see that there are three dots missing to fill this up"

Finally
Be yourself around your brother.  You seem like a caring and thoughtful older sibling.  You like mathematics.  Your brother looks up to you.  When you show that you care about math, he will learn to care about math.
PS
Avoid at all costs people who like to say things like "I was never good at math", or "I'm just not a math person."  And when you hear a person say such things around a young child, you absolutely must have a conversation about it with the child, the same way you would if the child witnessed a murder.  Because if you don't, you might let that random comment murder the child's nascent love of math.
A: There is a great article by Alexander Zvonkin 

Zvonkin, Alexander, "Mathematics for Little Ones", Journal of Mathematical Behavior, 1992.

and the very helpful report that contains almost all his sessions published as a book and recently translated into English Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers.
I recommend both very much!
A: Here is how I taught my children at this age:


*

*First, I tried to count things with them, like steps if we were walking up and down steps together, or parked cars as we walked along the footpath.

*I would make up "stories" about characters they liked (e.g. Thomas the tank engine), of the form "Say there are two trucks in the yard, and Thomas brings in one more; how many trucks are in the yard now?".  

*I made them "number snakes" (something I learned from Joe Silverman): draw a snake on piece of paper, and break up its body (just by drawing lines) into a three of our segments.  In each segment put a (very small, to begin with) number.  Now your brother has to add up the numbers to see what total number the snake wants to say.  It helps to be a bit theatrical about the whole thing.  
One thing to remember is that at this age, even if your brother knows how to count "1, 2, 3, 4, ...", if you ask him to find $3 + 1$ (directly, or via a story or number snake), it probably won't be obvious that this is the same as just counting on one step from 3.  So be patient, make the questions very easy and interesting, and don't overdo it.  If you really want to do this, it has to be something of a long term project.  (In particular, 10-20 minutes of studying with you may not be feasiable; going for a 10 minute walk around the neighbourhood, and using that time to incidentally count cars, or trees, or cracks in the pavement, or tell counting stories, might be better.)

You might also find this MO question helpful.   I like this answer in particular.
A: Why don't you teach him names of geometric shapes, like cubes, spheres, rectangles, ovals. These are tangible shapes they can relate to, lots of kids toys have those shapes. Do you remember that wooden cubic box through which you have to push the correct object through its holes on the sides? That's an excellent toy for a kid, maybe to easy for a 4 year old. Geometric skills develop earlier in toddlers than arithmetic skills for reasons I mentioned.
A: I have some online mathematics exercises / tutorials (autocorrected), ranging from kindergarten to college, at: http://www.public-domain-materials.com/folder-student-exercise-tasks-for-mathematics-language-arts-etc---autocorrected.html
