The Humean mother would give a lot to her child, moved by the sentiment rather than by an abstract sense of duty. She would support the son in everything he does and see his satisfaction as hers. She would be moved by sympathy.
But the human mother may not be the best type of mother when she has more than one child. If you have two children, you must be fair between them. And acting from sentiment may not be the best way to achieve fairness. Notice that treating your children as equals does not mean giving them equal amounts of resources (i.e. money).
Consider the following example: one of your children has a great musical talent, and she is accepted by a prestigious school. You need 2000 $ a month to support him and give him this chance. The other child goes to the university in the same city where the family lives. Suppose she has failed her applications to a better college in another city, or support she did not even apply. Does it follow that the mother should give 2000 $ to the other daughter, say, to live on her own, or, if she prefers to live with the family, to spend for holidays or other leisures? I do not think so.
Treating children as equals does not mean giving them equal share of resources. It requires to achieve more complex objectives such as for example, supporting their talents equally, giving them the same opportunities equally, and the like. And this is not done, in most cases, by giving them equal shares of resources, to spend as the two children like.
The humean mother can be a very good mother when she has only one child, but may have troubles when she has more than one child. For in order to treat her children as equals, she cannot simply be guided by her sentiments. If you let your action be guided by the sentiments you may very easily end up favoring a child over the other. This will especially be when the mother finds one of the two children easier to get along with, or nicer. (According to some psychologists even beauty could condition such responses.) Or, to make a more usual example, when you recognize yourself in the career of one child more than the other. The typical example is that of a parent who wanted to become a philosopher (or a pianist, or a novelist, or to get a degree), but could not do it, and projects this ambitions on her children. (I'm sure you are familiar with this from your life, or novels and films) Suppose that such a parent has two children: one that wants to study philosophy at the university (or to study piano, or to become a writer, or simply to get a degree), and another who does not like education and finds a job immediately after the high school. The first child is much lazier than the first, she does her degree very slowly and unsuccessfully; the second is hard-working, and has even chances of setting an activity on her own. But the Humean parent treats the first child much better than the second, he favors all of her aspirations, pays her summer schools, etc... The Humean parent, who follows the voice of her hearth, is very supportive with the one of the two children, but not enough supportive with the other, the one, let us say, who has not studied but has a good idea for a business and needs the support of her family to realize her dream.
The Kantian mother, by contrast, tries to deal with the requests of her child by applying principles. In this way she avoids being misled by sentiments and prejudices. The Kantian mother adopts the principle, for example, that she should not spend the family's money to support the living expenses of a child who does not want to find a job to support herself, or the principle that she should deprive herself of money, if one child has a real opportunity, but not simply to support the hobbies or leisure activities of the child. She tries to assess if she is following a reasonable principle, instead of listening to what her hearth tells her. If the principle sounds reasonable to her, she applies it. She can be consistent because she applies the same principles to all her children: hence she can be do what fairness requires, even in complicated situations. So she will use the money for her children , when the children really need them, and she will have the strength to say “no”, when the son asks for them, even if, had she followed her hearth, she would have give him money, just to see him happy. This mother knows that if she is unable to say “no” to a son, because she loves him a lot, she will be depriving all her other children, who do not ask so much, of resources that they are entitled to have. If any other child will be in a similar situation, she know that she will follow the same principle. In this way, she can assure herself that she is treating her children as equals, and that she is not being swayed by prejudices or emotions.
But even the Kantian parent is not a perfect parent, even if she may be in general fairer than the Humean parent. The problem of a Kantian mother is that there is no antecedent criterion to establish which concrete rules should be applied in a situation that presents itself in the novel way. The Kantian mother wants to apply the principle “treats your children as equals”. She understands this as “apply the same concrete rules in your decisions towards both of your children”. But the the rule “treats your children as equal” is silent on what specific behavior should be used to deal with the first child, when the mother faces a novel situation concerning his life. The only thing that fairness commands, is that, whatever rule the mother applies to the first child, she ought to apply the same rule when she deals with a similar situation involving the second child. But the imperative of treating your children as equals is silent on what counts as a good rule to apply in the first case. The Kantian mother risks to make twice as much mistakes as the Humean mother, and to do them for the sake of respecting her principle of fairness.
For, if the Kantian mother makes a wrong decision when dealing with the situation of the first child, for example because she does not support her ambitions enough, she is going to do a wrong decision when dealing with the situation of the second child by applying a similar principle,even if she realizes now, dealing with the second child, that she could have been more generous with the first as well as with the second.
The problem arises because the Kantian mother, as well as the Humean mother's behavior, is not free from the influence of (possibly uneducated) sentiments. Suppose that the Kantian mother loves the second child more than the first, or tends to favor the second child more because she identifies more easily with her. Suppose that this mother has failed to be appropriately supportive of the first child's ambitions (because she does not identify with her, or does not love her enough.) Now the situation materializes, in which the second child has an important opportunity. The mother realizes now that it would not be generous enough towards her child, if she does not do some extra sacrifice in order to support her ambitions. She also realizes, a fortiori, that she has not been as supportive with the ambition of the first child as she should have been. She realizes, that is, that the principle which she applied in the first case was not reasonable, or that it was not reasonably applied. But given that she wants to treat her children as equals, she must decide not to support the ambition of the second child as well. A humean mother would have acted unfairly, by supporting the ambitions of the second child even thought she behaved in a different way with the first. A Kantian mother acts with fairness, but in this way fails to be appropriately generous with both children.
The problem of the Kantian mother shows the limitations of the Kantian approach in ethics. Sticking to principles and applying them in an universalistic fashion is not a guarantee of acting in the right way, unless the decision made in the first case of the application of the rule was a reasonable one in the first place.
The Aristotelian mother should overcome the problems of both the Humean mother (who acted as her sentiments or feeling moved her to act) and of the Kantian mother. The Aristotelian mother understands that it is not enough to apply reasonable rules consistently, if she does not have the practical wisdom to know how a certain rule should be applied in the first place (for example knowing how to apply the rule “be generous and supportive of your children's ambitions when they face good opportunities" in practice). Aristotelian morality realizes that knowing how to apply such rules means also having well-developed and rational sentiments and dispositions. For the problem of the Kantian mother was that she was had not been generous enough with the first child, and had failed to support her adequately when she needed this support. What prevented her to act with the right degree of generosity was her insufficient degree of empathy towards the aspirations of the first child, or her (unconscious) lack of sympathy towards the first child (a problem that did not arise when the second child found herself in a similar situation.) Aristotelian ethics knows that without a therapy of this mother sentiments towards the first child, this mother cannot apply the right rule and make the right decision in the first case. The Aristotelian mother tries to act reasonably, but knows that reason cannot tell her how much to give to her children, if it is not accompanied by well-developed emotions.
These are roughly the reasons, which lead me to favor the Aristotelian picture of morality, in which reason penetrate and shapes the sentiments, to both the Humean, based upon the sentiments, and the Kantian one, based upon pure reason.