Probably the most important Standard for Mathematical Practice in the Common Core classroom is to "construct viable arguments and critique the reasoning of others." That's because argumentation is at the heart of math as a discipline.
An argument in math isn't really the same as an argument your children may have in the back seat of the car. The back seat argument is likely to devolve quickly into name‐calling, spitballs, and fisticuffs if you as a parent don't intervene.
This behavior shouldn't happen in a mathematical argument. What these two kinds of arguments have in common is disagreement. However, in math, the disagreement is usually around whether something is true.
Although many people perceive that math is about computing with numbers and variables, it's really about making particular kinds of arguments. Mathematicians base their arguments on things they already know are true and on the rules of logic. If something is true in math, you can know it in a much more certain and timeless way than in any other subject.
You can help your child practice making arguments by asking one simple question on a regular basis: "How do you know?" If your 7-year-old says that she ate four pistachios because she has eight pistachio shells on her plate, ask her how she knows eight shells mean four pistachios. If your 9-year-old claims that
scoops of
cup of flour is
cup, ask him how he knows. Make a habit of asking your children how they know so that they can start making a habit of building mathematical arguments.
If you have ever tried to explain to a child exactly why he can't do something, you have likely noticed that children are very good at critiquing the reasoning of others. Even toddlers can be like little lawyers when there is candy or some other privilege or liberty at stake, which can be tremendously frustrating as a parent, but it's a highly useful mathematical technique. Encourage it when you can and continue to lay down the law when you need to.