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You could consider using inducement which means:

something that gives you a reason for doing something and makes you want to do it

a motive or consideration that leads one to action or to additional or more effective actions

Example:

English Language and Usage offers reputation points and badges as an inducement for users to keep visiting the site.

[Merriam-Webster]

Edit: Your example makes me wonder if inducement is the right word. In that case, you could consider using reinformcement which means:

a strengthening of a specific behavior due to its association with a stimulus. Reinforcement is an important part of operant or instrumental conditioning. A reinforcer is the stimulus that strengthens the behavior, in contrast to punishment that weakens the behavior.

Your new example:

Our parents and teachers used reinforcement to reinforce our routines, rewarding every useful contribution (as positive reinforcement or reinforcer) and penalizing unproductive actions (as punishment).

You could consider using inducement which means:

something that gives you a reason for doing something and makes you want to do it

a motive or consideration that leads one to action or to additional or more effective actions

Example:

English Language and Usage offers reputation points and badges as an inducement for users to keep visiting the site.

[Merriam-Webster]

Edit: Your example makes me wonder if inducement is the right word. In that case, you could consider using reinforcement which means:

a strengthening of a specific behavior due to its association with a stimulus. Reinforcement is an important part of operant or instrumental conditioning. A reinforcer is the stimulus that strengthens the behavior, in contrast to punishment that weakens the behavior.

Your new example:

Our parents and teachers used reinforcement to reinforce our routines, rewarding every useful contribution (as positive reinforcement or reinforcer) and penalizing unproductive actions (as punishment).

You could consider using inducement which means:

something that gives you a reason for doing something and makes you want to do it

a motive or consideration that leads one to action or to additional or more effective actions

Example:

English Language and Usage offers reputation points and badges as an inducement for users to keep visiting the site.

[Merriam-Webster]

Edit: Your example makes me wonder if inducement is the right word. In that case, you could consider using positive reinforcer / positive reinforcing stimulus which means:

a reinforcing stimulus that serves to increase the likelihood of the response that produces it

Your new example:

Our parents and teachers used a positive reinforcer, rewarding every useful action. Today, it's known as gamification!

You could consider using inducement which means:

something that gives you a reason for doing something and makes you want to do it

a motive or consideration that leads one to action or to additional or more effective actions

Example:

English Language and Usage offers reputation points and badges as an inducement for users to keep visiting the site.

[Merriam-Webster]

Edit: Your example makes me wonder if inducement is the right word. In that case, you could consider using reinformcement which means:

a strengthening of a specific behavior due to its association with a stimulus. Reinforcement is an important part of operant or instrumental conditioning. A reinforcer is the stimulus that strengthens the behavior, in contrast to punishment that weakens the behavior.

Your new example:

Our parents and teachers used reinforcement to reinforce our routines, rewarding every useful contribution (as positive reinforcement or reinforcer) and penalizing unproductive actions (as punishment).

You could consider using inducement which means:

something that gives you a reason for doing something and makes you want to do it

a motive or consideration that leads one to action or to additional or more effective actions

Example:

English Language and Usage offers reputation points and badges as an inducement for users to keep visiting the site.

[Merriam-Webster]

You could consider using inducement which means:

something that gives you a reason for doing something and makes you want to do it

a motive or consideration that leads one to action or to additional or more effective actions

Example:

English Language and Usage offers reputation points and badges as an inducement for users to keep visiting the site.

[Merriam-Webster]

Edit: Your example makes me wonder if inducement is the right word. In that case, you could consider using positive reinforcer / positive reinforcing stimulus which means:

a reinforcing stimulus that serves to increase the likelihood of the response that produces it

Your new example:

Our parents and teachers used a positive reinforcer, rewarding every useful action. Today, it's known as gamification!