The mathematics begins with counting. Because as you can see from the basic concepts math is the study of numbers and all we do with numbers is count. How do we teach counting? Start with patterning. Just say the numbers from one to five, then one to ten then ten to twenty. Repeatedly. Start with your fingers, at bed time is a good time but any time is fine.
With most beginning students I make them write out the five basic concepts. With older students counting is redundant, they can all count, and you can make a joke of it. After they write down "Math is the study of numbers and all we do with numbers is count", or some variation thereof, ask them to count to twenty. Then ask them to count backwards from twenty.
“See, I can teach you math, we are one fifth of the way there...you can even count backwards.” Then ask them to recite the ABC's backwards. So far I have only had a very few kids that could.
“Hmm, looks like you're better at math.”
Younger students (and many older students) need to be taught the patterns whether it's counting or addition with addends, multiplication or what have you. When it's counting start with the simple concept: the highest number we count to is nine. The numbers tell us how many the places tell us what kind. After that it's just vocabulary, we have English names for all the numbers, even really big numbers. One, two...seven, eight, nine and then ONE of the next kind, one ten. One ten and one unit is called eleven. One and one is just two, you may have to explain this several times to a three year old even if you are using manipulatives. 11 looks like two to them and this looks like three: 111. Several explanations may be required to
Then just pattern: Ten, twenty, thirty...seventy, eighty, ninety ONE hundred.
One hundred, two hundred...eight hundred, nine hundred, ONE thousand...it never gets past nine.
Two tens and three units has a name, twenty three...what's important is understanding the concepts place value is easy and almost visually obvious and becomes clear after just a few explanations.
We count the big ones first. One hundred, one ten and one unit are one hundred eleven. 111. With manipulatives this IS visually obvious but when writing 111, little kids often think THREE...can you see how much easier it is to teach place value?
You will at Crewton Ramone's House of Math!
Teaching a child to count properly takes MONTHS, you can teach them to pattern and memorize by counting from one to 20 over and over again, which is fine; however this can lead to some confusion. Ever hear, "my kid can count to 10 but gets a little confused in the teens"? I have. A lot. Make sure they see all the teens are, are just ten and some more, or specifically ten and one through nine units more, and that the pattern repeats with two tens and one through nine units, and three tens and so on. This way they get the concept and the pattern and link the vocabulary to the numbers.
I can't tell you how many times I got a student that was failing algebra that had to use their fingers to add numbers. Simple numbers like nine and three. Asked to multiply they didn't have their tables memorized either without thinking hard about it. OF COURSE THEY WERE FAILING ALGEBRA: they had no foundation upon which to stand. Back to basics. Addends. No magic. Then again, when the basics are mastered correctly the magic begins. First counting, then addition with addends, then multiplication: seems obvious.
“Obvious” is the most dangerous word in mathematics.” ~Anon
So first thing: get them “off their fingers.”
Addition and multiplication are just way of counting very quickly. In order to get there you start off crawling then you walk then you can run, then you can hit the hyperspace button...or at leaste turn on the after burners, the easy way, visit Crewton Ramone's House of Math for help with teaching counting and much more.