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I'm a nanny and I've looked after a few thumb suckers. Not a lover of it - as well as sores on thumbs and potential teeth/speech problems there's also the hygiene aspect. You can't stop the thumb going in after they've been climbing on the climbing frame that everyone's muddy shoes have been on, after they stroked the friendly dog at the park, after feeding the animals at the farm etc (yes, we wash hands after such things but the thumbs can be sneaked in pretty darned quickly)
At your dd's age I would start to discourage it apart from sleep times. If she's sucking it while you are in close proximity (eg reading stories or doing puzzles) I discreetly remove the thumb from their mouth trying not to draw attention to them that I am actually doing so. Remove the thumb and tickle her palm or stroke her fingers while carrying on reading the story or remove the thumb and put a piece of puzzle in that hand etc. She's also old enough to understand what 'no thumbs' means. When I say this to them I always use a bright and breezy voice with a big smile. I also tell them 'thumbs are for sleeping' 'you're a big boy/girl now, you don't need your thumb when you're awake' I would never chastise a child for putting their thumb in their mouth - I laugh and say things like 'caught you', 'can I have a taste' and 'are you going to sleep'
As they get a bit older (maybe 2-3) I gently introduce some rules - no thumbs at the table, no thumbs while you are watching tv and no thumbs in the car. I don't make a big deal of it - I keep it light and smiley and generally they enjoy the challenge to see if they can do it. The one I am a little firmer on is no thumbs whilst watching tv - tv is a treat and only watched for short periods so I tell them if they can't watch tv without thumbs then we will be doing another activity.
It's all about breaking the habit and eventually they just 'forget' to suck their thumbs at times that they would have before. I always keep it light so that they don't see thumb sucking as naughty and if they use their thumbs for sleeping then I have no problem with that. I start work on that one at around the age of 5 before they loose baby teeth and get their adult teeth.
I've been successful with all but one little boy who was a finger sucker. He resisted all my little tricks and actually asking him not to suck or trying out the rules made him more determined to do it. His fingers were in his mouth about 75% of his day! We resorted to this and in the first 2 weeks we met huge resistance we stuck to the full 4 weeks of wearing it 24/7 and it has worked, he no longer sucks his fingers. He was 4 years old, I'm not sure I'd be keen on using it on an under 3.
Sorry, but I disagree with nearly all the above. For a start, nuts to hygiene - the more germs in the better the immune system. You will probably find in any case that as she starts to use her hands more during toddlerdom then she isn't sucking as much anyway. The fact that she's got a sore right now may just mean that she's going through a developmental spurt and needs a bit more comfort at the moment. Dentists don't worry for ages - until they're about 5, I think, by which time it will probably be greatly reduced anyway. DD has sucked her thumb since 9mo (now 3.4) and her teeth and palate are fine. Self-soothing is great - what's the point in trying to stop it if it's not a problem? - and the thumb is always there.
In any case, loads of adults suck their thumbs (including at least three friends of mine), and so what?
I'm not against the self soothing, particularly for sleep which is why I wouldn't discourage it for night time (or daytime naps) until much older. But often a child that sucks their thumbs through waking hours is not doing it for self soothing but through habit - why would they be sucking their thumb while listening to stories or watching a favourite tv programme? They are doing it through habit not because they are upset and need it, which is why I use various techniques to discourage it. Breaking the habit at a younger age is easier and can be done more gently than with an older child.
Thumb sucking doesn't always lead to dental problems, but it certainly can and can lead to tongue thrust which affects speech developement. There are different ways that children suck thumbs and fingers - different positions in the mouth, different strengths of suction.
As to the hygiene - yes I think it can be beneficial to be exposed to germs and not live in a sterile environment, but I do believe in washing hands after going to the toilet and before meals, so would want to limit a child putting their hands in their mouth after petting farm animals or playing in the mud at the park.
From personal experience I would try to nip it in the bud sooner rather than later. DS was 6 when we finally tackled the issue. The dentist said it was really affecting his teeth, plus he had sores and his thumb was staring to look deformed. He also started to get teased at school by other children and it did delay his speech (although he has caught up now). IMO the longer you leave it the harder the habit is to break. Have a look online for the thumb guards you can buy, especially to use at night, and during the day try to keep his hands busy!
DS2 (4.8) was a demon thumb-sucker from a very young age - incessant, really. I'd go in to check on him last thing at night and hear this vigorous sucking from his cot/bed. It has pushed his teeth forward a little, but not badly. And it has started to wane as he has got older - since going to pre-school it has been less, and he's now at school so even less, just when he is tired/at bed-time. I think he will do it to that extent for some time, though, and I am fine with it. Just hoping that he doesn't do it for as long as I did (14 !)