It's been said that finger counting will slow down memorization of math facts and calculations in general. But I personally feel that finger counting as a toddler will not affect any of that in the long run. I mean, when the kids get to school they will be taught new ways to count and stuff. They'll slowly be weaned off using the fingers as math questions get harder.
I don't why not. I was taught to count and do maths with my fingers, and I teach my daughter the same way. She gets it, albeit little slow because she likes to do the count in rythem like they do in kids show. I guess it's okay for younger kids, but as they grow older, they will stop relying on their fingers and start doing mental sums.
If you are not favor in using finger counting method, you can use manipulative to teach toddler how to count. E.g. bear counters. You can place a number card and ask your kid to place the amount of bears on the number card that correspond to it to get a concrete concept on number counting.
I personally think counting fingers is a good way to introduce the concept of counting. Extend the same concept to counting different objects so that the baby to learn to fully grasp the idea rather than limiting it to just counting his/her fingers.
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thanks