Learning through play and movement, observations in life, and experiences are how children learn in a developmentally appropriate manner. Of course they shouldn't be expected to do worksheets and drills. Those aren't even developmentally appropriate for a 2nd grader, though school systems choose to often teach them in this manner.
Early math introduction has far-reaching benefits for children, and many cross-over aspects into reading and other subjects.
There is no expectation on the part of the child here. The expectation is on me to provide the teaching, the exposure to the concepts. The child either embraces it, ignores it, masters it, manipulates it, or just stores bits of it away for later. Often much later. But the point is that the information is there for them when they want or need it and more of it is retained, earlier, due to the exposure through casual, fun opportunities.
This is what people often view as math learning. Number order and number recognition. This is such a blip on the radar of math learning, though.
Rote counting is done daily here with our 0-10 chant. They love it. Rote counting is done more for introduction of number vocabulary, patterning/sequencing, and the concept of consistency, rather than getting them to be able to count, at this age.
Since we begin in infancy with numeracy introduction, the 2's know their number names. Can they count? Not really. They have numbers they like and usually put them in numerical order. Mr. R likes "1,2,5!" Mr. H likes "1,3,4!" Usually they can pop up with the next number in a sequence, and they recognize some numbers. We say things like, "One, two, threeeee, GO!" to help with beginning counting. They are often observed counting in play.
From birth we teach one-to-one correspondence. This is SO important for math and reading. We spend a little time on rote counting, but we spend a LOT of time counting THINGS. Numbers by themselves have little meaning, it is numbers as representations of quantities that have meaning, and that concept is as vital as the concept that words have meaning. We count things as least a dozen times a day: animals on a page, cups on the table, rocks in a bucket, balls we pick up, grapes on a plate, etc.
Shapes & Geometry
We work on identification of: circle, square, triangle, heart, star, crescent, with diamond, rectangle and oval added in as needed. Shapes are not something we actively work on, but casually discuss in environmental and book experiences. Just like colors, they seem to just pick them up.
Even more casually, I introduce 3D shape vocabulary, keeping in mind that they are TWO. The moon is a circle. They lack the abstract thinking to interpret it as a sphere, so for now, it is a circle or a crescent.
More actively, we work with shapes. Often this is not a shape learning activity, but a geometric manipulation activity and/or logic/reasoning activity. For instance, at 2 years and 3 months, the three in the picture below, all born within days of one another, are able to do tanagrams, shape sorters, shape puzzles, Wedgits, etc. that work with shapes, but work with much more than just learning shape identification. Even puzzles are working on perspective, movement across a plane, translation, rotation, etc.
We also actively work with coordinate geometry terminology: up/down, back/forward, in/out, etc. I call this positional work. We do this as physial activities, casual observations, and purposeful book work. such as, "What is BELOW the bird?"
If I ask the 2's to get one or two of something, they can do that. We are working on more/less, a lot, big/small high/low and various other quantification concepts.
Much of this type of learning is done in the sandbox and water table and playing with loose parts. Filling, pouring, dumping into a bigger or smaller container, trying to fit items into other items, all teaching proportion, volume, weight, length, height, concepts.
Even jumping off of things, stepping between pavers, reaching up for items, all teach measurement and proportion concepts.
Logic & Reasoning
Logic and reasoning are in play throughout the day. When I ask, "Is that okay?" I can see their young minds running through the reasoning. "Do you need to spend some time in time out?" "No." "Then what do you need to do?" Off they go making a better choice. Much of their logic and reasoning comes through playing with loose parts. We have a ton of stuff in our outdoor play area/classroom that they can use as they will. They often come up with uses for items that I never would have or could have thought up. Much of logic and reasoning comes from simply enabling them to experience a vast amount of different situations and the outcomes of their decisions pertaining to those situations.
Patterning & Patterns
At 2, pattern learning is continued from infancy as a physical and musical activity. We are currently working on "Pat-pat-clap, pat-pat-clap, 1-1-2, 1-1-2, pat-pat-clap, I Love You, 1-1-2, 1-1-2."
Last month it was "Jump-down, jump-down, jump-down, spin around. [repeat]" They would jump up then squat down, and turn around after three rounds of that.
There are patterns sprinkled throughout our play and routine. "1-2-3 GO!" is even a pattern. The order we put out beds, is a pattern.
They have their counting bears to color group as an acitivity center choice.
We also do this in pick-up, "Put all the cars away. Now, put all the baby doll stuff away." It may not seem like a mathematical activity, but it is purposefully intended to be such. They begin early on seeing that like goes with like and making inferences about grouping. This is also a science aspect - characterization, identification of traits.
This is casual through observation. "Oh, look, you have three, 1-2-3, and he has three, 1-2-3. TOGETHER you have 1-2-3-4-5-6!" "He has one, can you give him another one so that he has two?" "You have one car and one more car, so you have two cars. one and one more is 1-2."
Not enough attention is paid to the early introduction of math vocabulary and concepts. Not just my opinion, finer minds than mine are saying this. Children learn many of these through their play, and we can scaffold that learning to a higher level through purposeful teaching. We can also provide them with a much broader base of concepts and vocabulary through casual observations of their activities.
I think I do a pretty good job at this.