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What are some polite ways to assert myself in a situation such as this, without coming across as pushy or self centered?

This depends a bit on what you consider pushy or self-centered. Sometimes, especially in informal or animated conversations, it's push or be pushed! Some people are more confident than others and take the floor more frequently. A few people are rude. If you're considerate and reserved, "pushing" to be heard will feel like you're sinking to their level.

Being talked over is so significant a problem that in some discussion groups, people have a limited number of tokens representing minutes they can talk. That way everyone gets a chance to speak uninterrupted by the more eager-to-share or dominant members. It's frustrating and discouraging, but like many things, it's most likely not about you (so please don't take it personally.) It's more likely to be "about" the other participants.

It's good that your voice is strong. That's a big hurdle you overcame. I would suggest you pick a few phrases that you can say quickly (slower, more deliberate speakers risk being talked over more often unless they rank high in the hierarchy) as soon as you pitch in, perhaps,

I'd like to say something here.
Here's a thought... One option worth considering... What about this?...

If someone cuts you off, be polite but confident.

Give me a minute here.
Hold on, let me finish my thought. Just a second, please.

If someone is being rude, you can repeat a variation on the above.

Group dynamics are tricky. Groups can have different standards (very polite and formal to excited, competitive, and rowdy) and you may need to match what you see to be heard.

The Harvard Business Review has a lot of helpful articles devoted to just this thing: breaking into a conversation gracefully, holding your own, etc. How to Interject in a Meeting

What are some polite ways to assert myself in a situation such as this, without coming across as pushy or self centered?

This depends a bit on what you consider pushy or self-centered. Sometimes, especially in informal or animated conversations, it's push or be pushed! Some people are more confident than others and take the floor more frequently. A few people are rude. If you're considerate and reserved, "pushing" to be heard will feel like you're sinking to their level.

Being talked over so significant a problem that in some discussion groups, people have a limited number of tokens representing minutes they can talk. That way everyone gets a chance to speak uninterrupted by the more eager-to-share or dominant members. It's frustrating and discouraging, but like many things, it's most likely not about you (so please don't take it personally.) It's more likely to be "about" the other participants.

It's good that your voice is strong. That's a big hurdle you overcame. I would suggest you pick a few phrases that you can say quickly (slower, more deliberate speakers risk being talked over more often unless they rank high in the hierarchy) as soon as you pitch in, perhaps,

I'd like to say something here.
Here's a thought... One option worth considering... What about this?...

If someone cuts you off, be polite but confident.

Give me a minute here.
Hold on, let me finish my thought. Just a second, please.

If someone is being rude, you can repeat a variation on the above.

Group dynamics are tricky. Groups can have different standards (very polite and formal to excited, competitive, and rowdy) and you may need to match what you see to be heard.

The Harvard Business Review has a lot of helpful articles devoted to just this thing: breaking into a conversation gracefully, holding your own, etc. How to Interject in a Meeting

What are some polite ways to assert myself in a situation such as this, without coming across as pushy or self centered?

This depends a bit on what you consider pushy or self-centered. Sometimes, especially in informal or animated conversations, it's push or be pushed! Some people are more confident than others and take the floor more frequently. A few people are rude. If you're considerate and reserved, "pushing" to be heard will feel like you're sinking to their level.

Being talked over is so significant a problem that in some discussion groups, people have a limited number of tokens representing minutes they can talk. That way everyone gets a chance to speak uninterrupted by the more eager-to-share or dominant members. It's frustrating and discouraging, but like many things, it's most likely not about you (so please don't take it personally.) It's more likely to be "about" the other participants.

It's good that your voice is strong. That's a big hurdle you overcame. I would suggest you pick a few phrases that you can say quickly (slower, more deliberate speakers risk being talked over more often unless they rank high in the hierarchy) as soon as you pitch in, perhaps,

I'd like to say something here.
Here's a thought... One option worth considering... What about this?...

If someone cuts you off, be polite but confident.

Give me a minute here.
Hold on, let me finish my thought. Just a second, please.

If someone is being rude, you can repeat a variation on the above.

Group dynamics are tricky. Groups can have different standards (very polite and formal to excited, competitive, and rowdy) and you may need to match what you see to be heard.

The Harvard Business Review has a lot of helpful articles devoted to just this thing: breaking into a conversation gracefully, holding your own, etc. How to Interject in a Meeting

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What are some polite ways to assert myself in a situation such as this, without coming across as pushy or self centered?

This depends a bit on what you consider pushy or self-centered. Sometimes, especially in informal or animated conversations, it's push or be pushed! Some people are more confident than others and take the floor more frequently. A few people are rude. If you're considerate and reserved, "pushing" to be heard will feel like you're sinking to their level.

Being talked over so significant a problem that in some discussion groups, people have a limited number of tokens representing minutes they can talk. That way everyone gets a chance to speak uninterrupted by the more eager-to-share or dominant members. It's frustrating and discouraging, but like many things, it's most likely not about you (so please don't take it personally.) It's more likely to be "about" the other participants.

It's good that your voice is strong. That's a big hurdle you overcame. I would suggest you pick a few phrases that you can say quickly (slower, more deliberate speakers risk being talked over more often unless they rank high in the hierarchy) as soon as you pitch in, perhaps,

I'd like to say something here.
Here's a thought... One option worth considering... What about this?...

If someone cuts you off, be polite but confident.

Give me a minute here.
Hold on, let me finish my thought. Just a second, please.

If someone is being rude, you can repeat a variation on the above.

Group dynamics are tricky. Groups can have different standards (very polite and formal to excited, competitive, and rowdy) and you may need to match what you see to be heard.

The Harvard Business Review has a lot of helpful articles devoted to just this thing: breaking into a conversation gracefully, holding your own, etc. How to Interject in a Meeting