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If a parent tells me their child is not yet readily imitating, I think one of several things. Are they imitating anything? Do they understand what it means to imitate? Are they hearing everything accurately? Are they being given a simple and clear model of speech to imitate? Some children automatically begin imitating or mimicking actions, sounds and speech at a fairly young age.
For others, this is a more difficult task to master. If a child doesn't really imitate actions like clapping or waving, he probably isn't going to be imitating words because this is much more difficult. If the child doesn't understand what imitating is, we have to teach him. The best way to do this is to do just that. I like to use my musical instrument toys for this. If the child bangs their hand on the drum, I bang my hand on the drum. He does it again. So do I. When he shakes the tambourine, I shake the tambourine.
The child should notice this mimicking behavior and possibly increase their eye contact with you. It sort of grabs their attention. You may go through several back and forth turns at this imitation game (now you're taking turns as well!). This repetitive play is teaching the child what imitating is and soon they will be initiating the game by imitating you. If the child learns the game quickly, challenge them and you start the game this time. Bang your hand on the drum to see if the child imitates you. Do this several times and with several different variations to make sure they understand what it means to imitate.
Add a sound in and see if they can pair a sound with the action (boom!). Just as any new skill we learn, we have to practice. Practice imitating. After a child has begun to imitate actions, movements, and gestures, they are then ready to begin imitating verbalizations. If the child is already imitating actions, but not imitating speech, I model simple and short speech. Depending on their current expressive language level, I start out with simple animal sounds (moo, woof, quack, meow, sss), environmental sounds (vroom), and exclamations (wee, wow, yay, whoa, ouch, uh-oh, mmm), and words that are commonly used in their everyday life (hi, bye bye, more, doggy, night night, yum yum,).
Make sure the child can see your mouth when you're making a sound or saying a word for them to imitate. The child may need that extra visual cue of what it actually looks like. If you're playing with a puzzle or a toy that has pieces, you can bring the piece up to your mouth to encourage the child to look at your face and then prompt the sound or word while they're looking at you, pause and give them time to respond and then give them the piece they wanted.