Research on parental influence in adolescent development consistently finds that behaviour modelled in the household predicts outcomes better than stated attitudes or explicit encouragement.
This is worth taking seriously, because encouragement is what most parents concentrate on and modelling is what most parents do without noticing.
The household division of labour finding
Several studies have examined the relationship between how domestic work is distributed in a household and children's later attitudes and aspirations.
The most cited found that the distribution of household work between parents predicted daughters' occupational aspirations, with daughters in households with a more equal distribution reporting less traditionally gendered aspirations.
Notably, the association with the distribution of work was stronger than the association with parents' stated attitudes about gender roles.
This is a single line of research and the effect sizes are modest, so it should not be overstated. But the direction is consistent with a large body of work on modelling, and the mechanism is intuitive: a child observes the arrangement daily and hears the stated attitude occasionally.
The mathematics conversation
The transmission research on maths anxiety, discussed elsewhere on this site, is directly relevant here.
Parental maths anxiety was found to predict children's maths learning, but only among parents who frequently helped with homework. The transmission required contact.
The related finding is that the specific statement matters. "I was never any good at maths" delivered as reassurance communicates that ability is fixed, heritable and absent — three separate unhelpful claims in one sentence.
The alternative is not enthusiasm about mathematics, which is not available to most people. It is describing the difficulty as a property of the problem rather than of the person: this bit is hard, let us work out what it is asking.
Risk, permission and the supervision asymmetry
The observational research on playgrounds, discussed in this site's article on risk-taking, found differing intervention thresholds by child gender — with girls more likely to be cautioned or assisted and boys more likely to be coached through.
The relevant point for parents is that this is not a conscious decision. Studies find it in populations with strong stated commitments to equal treatment, which suggests it operates below deliberate choice.
The practical response is to notice the threshold rather than to adopt a policy. The useful question in the moment is whether you would be intervening if this were a different child doing the same thing.
Most people who ask themselves this honestly find the answer is sometimes no.
What encouragement does and does not do
Verbal encouragement is the third of four sources of self-efficacy, behind direct experience and observation of similar others.
It has an effect. It is not the main lever, and it can be undone quickly by contradicting experience.
What appears to matter more is what is arranged rather than what is said: whether the subject is available, whether the activity is attended, whether the difficulty is tolerated rather than resolved by an adult, and whether failure is survivable.
A parent who says a daughter can do anything and also completes her difficult homework for her has delivered the message and contradicted it in the same evening.
The specific interventions with some support
Making prerequisites explicit at subject choice. Parents are frequently the only people who will tell a fifteen-year-old what dropping physics closes off, because schools often do not.
Discussing money in the terms usually reserved for sons. The financial socialisation research finds conversations with daughters weighted towards saving and spending restraint and with sons towards earning and investing. This is correctable at no cost.
Involving daughters in household financial and technical decisions rather than only in domestic ones, since observation of a decision is how the assumption that one is entitled to make it forms.
Tolerating visible struggle. The ability-praise research suggests the most damaging pattern is a child who succeeds constantly and has no evidence that difficulty is survivable, and the fastest way to produce that is to remove all the difficulty.
The limits of the parental influence literature
It is worth being honest that this research is difficult and its effect sizes are generally small.
Genetic confounding is a real problem in any study comparing parental characteristics with child outcomes. Peer influence is substantial and in some analyses exceeds parental influence during adolescence. And reverse causation — children shaping parental behaviour — is plausible in most of these designs.
The claims that survive these criticisms best are the ones about specific behaviours in specific moments: what is said while helping with homework, whether a prerequisite is explained, where the intervention threshold sits.
Those are small, actionable and unglamorous, which is a fair description of most of what is well supported in this entire field.