I apologize but I believe that at this age, children should learn by doing things with their hands. And by that I mean, the old fashioned way before computers. I know your question is specifically about electronic games and devices, but here are the tools that have worked for my 3 yr old dd:
Abacus-initially you are just playing with the beads, then counting, then learning place value as you start to count above single digit numbers. You can add, subtract. And a child can see touch and feel what the abstract numbers actually represent. That said I don't take out the abacus and sit her down to teach all this. But when she wondering what 2 apples plus 5 apples would be, I suggest if she could use the abacus to figure it out. So it is mostly self discovery.
Measuring cups and water buckets: good for understanding fractions without learning them
Magnets, compass, magnifying glasses
For language, we just make up silly willy songs that rhyme, read, drag the sounds of a word out so she can spell them if she wants to. I do use computer for language as she loves to type letters or make notes or grocery lists but does not want to write. We sometimes use the voice recorder function on iPad to say a word and see if the computer types it correctly or "goofs up".
I could go on and on but as I said initially at this age, IMHO, hands on play works wonders.
This works-- for
some kids and parents. Not all parents can really tolerate doing that kind of intensive/interactive toddler thing day after day. I would rouse myself to make the effort periodically, and I always felt horribly guilty for NOT doing all these great "hands-on" things with my daughter... but she didn't like them and neither did I. Every time I tried to be more like my "vision" of the perfect, Waldorfian mom, I wound up exhausted and cranky, and so did my daughter.
For kids with a non-tactile/kinesthetic bent, also, it is the very clumsiness of asynchronous physical development that frustrates them MOST. My DD was one of those children-- and she saw little point in the use of manipulative/concrete representations because by the time she COULD use them (physically), she already had moved past them into symbolic thinking. She also had (has) sensory issues and really couldn't stand to get her hands dirty/wet/messy. It was truly torture for her to play with play-do or paint or shaving cream (though I did try), and the pottery class that I signed her up for when she was three... oh my goodness, she was actually in TEARS much of the time. Talk about a BadMommy moment.
I say this with the understanding that we began parenting with EXACTLY this same outlook-- that technology (TV, computers, internet, etc) were not good for kids.
My daughter loved her LeapPad (she had one of the original ones when she was two). She hated being 'read to' but was very fond of reading "to herself" with the LeapPad.
One of the ways that I devised for getting DD to entertain herself for a short while was the "bean box." I bought bulk split peas and filled up an under-bed storage box with them and assorted old measuring cups, etc. DD liked to sit IN the box and pour the split peas (would also work with rice) and just play with them.
Other than that, she watched PBS and Blue's Clues, Bear-in-the-Big-Blue-House, and played Reader Rabbit (and similar software) on the computer. Computer chess. Oh, and talked to me. RELENTLESSLY. She loved Playmobil and before that, Fisher Price Little People. She also played elaborate games with her stuffed animals. She also enjoyed dress-up items (and really, scarves are terrifically versatile and VERY inexpensive if you buy at Goodwill or yard sales) and a gymnastics ribbon. We took out LOADS of books from the library. Puppets. She loved puppets.
Different things work for different parents and children. That's all.
