We start by introducing shapes. Shapes they can put in their mouths and manipulate are best, but shape books are also included in this. We introduce what most teachers consider the basic shapes - circle, square, rectangle, triangle, pentagon, hexagon, octagon, trapezoid, parallelogram, rhombus, star, and oval.
Climbing on a shape toy
There are a few things you should notice about this list. There is no heart listed. It's not a geometric shape and most kids will learn it through society (Valentine's Day, on signs, clothing, etc). The other thing that will jump out at most adults is that they have never heard of a rhombus. This is what we were told was a diamond. Technically, a diamond is a gem and not a shape. When children get to school, they will be taught rhombus. It's much easier to learn something right the first time than to have to re-learn it. Proof of this is seen when trying to switch, as an adult, to calling it a rhombus.
The other thing many adults have difficulty with is pentagon, hexagon, and octagon. It's amazing how many adults have difficulty identifying an octagon when it's not red. Make a point of showing your child 8 sided shapes that aren't stop signs. Yes, you could also teach heptagon and nonagon, but we are holding off on those until our kids learn all 12 of these more basic shapes.
As babies this is simply letting the kids play with the shapes. We also mention 3 dimensional shapes and give them their proper names - a ball is not a circle, for example. By 18 months old my son was able to distinguish between all of the shapes except the octagon and hexagon. We had a lot of discussions about those.
Do I want the cube or the sphere?
As far as counting, we don't count much with the babies. We sang songs with numbers in them (in three different languages), but didn't focus on counting. Counting is a very abstract concept and while many kids can recite their numbers, most have no clue what counting actually is and how to use it. We will introduce counting as they get older, but this was not important to us in the first year.
We also introduce other important vocabulary that will later apply to Math. Words like more, nothing, and half seem to come up every day. There are also other phrases that we use that many people wouldn't even think of relating to math. For example, positional words - on top, beside, etc. Teaching fractions and order of operations will rely on an understanding of these words.
I'm on top.
Beside each other
We also talk about patterns we see. We don't create patterns at this age, but will point them out when we see them. For example, house, mailbox, house, mailbox or flower, flower, tree, flower, flower, tree.
Dog park - dog, dog, human, human pattern
Other math concepts include sizes (big/little) and some comparative measurement (that book is 2 puppets tall). These are also concrete and easy to demonstrate and talk about with babies. We also spend a lot of time talking about differences. Sizes, shapes, color, weight, and on and on. This helps babies notice things they might have over looked. With young children it is easier to only have 1 difference to help them focus. A big red circle and a small red circle would better teach big/small than a big red circle and a small blue triangle. As they demonstrate they understand a concept more than 90% of the time you can move on to presenting two at a time and so forth.
She's reading Big and Little
One fun activity for older babies, is to go to a home improvement store and pick up some paint sample cards. You cut them up and let the kids sort by shade of color. For example, place 5 different shades of blue from lightest to darkest. You would want to start with 2-3 shades and work your way up. This helps focus and trains the brain to notice small differences.
The great thing about this age is it doesn't matter if they "get" it. It's really awesome when they do, but if they don't it's not a big deal. This is just an introduction and they will be taught all of these things again later. Have fun with it and remember to give them as many things to hold and manipulate as possible at this age.
Cooking (even pretend) can involve lots of Math