Right up front the book says they will not give you the seven or three or whatever magic steps you have to do to make "good" children. Instead in this book they set out how you should give your children grace despite the fact that you do not have good children. They say this because they believe in Original Sin. There is no one who does good, all have turned away from God. If this is true for adults, why is that not true for children? Your child is not perfect, in fact they are a fallen sinful human, just like you. You would think this line or reasoning would have a detrimental effect on a kid, but that is not the whole story. The issue raised at the end of the book is that the kid will feel too loved.
Coupled with the idea of Original Sin is also the idea of grace. If you are a Christian, God has has shown you grace despite your sinful nature, remembering this fact should help you live in light of that. Like in some counselling/pastoral care books, these authors turn other people's problems (your child's) back on to you for you to reflect on why you get upset and frustrated with them. Are you embarrassed/frustrated/upset about your child's behavior? Are you trying to build an identity as a good parent, or on Christ?
The book stresses the sovereignty of God. God is in control of everything, so relax and let God do His thing. Stop trying to make your child a Christian, that is not your job. You can not do that. Instead show your children grace and tell them about the grace of God.
On the issue of reward and punishment or a carrot and a stick, this book completely destroys that model. Christians hold that Jesus took our punishment (or stick) and gave us the reward (or carrot). Only after that are we asked to obey Him. It is not our works that get us a reward or God's love, it is because we are loved that we get the rewards. The book then challenges you to parent in light of that model.
This book isn't devoid of practical advice. It provide a framework for you to point your child to repentance and better behavior. They suggest a framework of Management, (Gospel) Nurturing, (Gospel) Training, (Gospel) Correcting and (Reminder of Gospel) Promises or MNTCP which can be remembered by Mum's Need To Constantly Pray (you can guess from this acronym who the target audience of the book is). You need to assess what the situation calls for and respond to your kid in some way that points to what God has done for them.
In the back they present a bunch of scenarios where this framework might work (the example in brackets is a very cut down version on the issue of disobedience).
Does your kid just need to be stopped or Managed ("I need you to stop now")?
If the situation is calm can you Nurture gospel truths to them ("I see you want something and don't want to obey, Christ will give you grace to help you obey as He understands obedience in hard situations")?
Can you Train them to act out the gospel ("Jesus obeyed in hard circumstances because he loves you. He did it perfectly and you can pray to him for help")?
Is there any Correction that needs to be done? ("When you disobey you are saying what you want is more important than anything else, but there is something more important than why you think, Jesus' obedience on the cross has redeemed you, look to His obedience as something greater" ).
Is there a need for you to remind them about the Promises of the gospel ("Living for yourself will not help. Repent" or if a Christian "God sees you as obedient as He sees you in Christ. He submitted to His parents as well as to people who hated Him. He did this for you, so you can be seen as obedient")?
Elyes is honest as she says you don't have to follow this - she didn't have this framework when she was raising her own children. (In the book it is Jessica's perspective that brings things to a practice side as she is on the ground with the children.) The authors point out that you will misjudge a situations and get it wrong, but again the point is the rest in God's faithfulness and not in your own behavior. The concept of grace isn't just for your child, it is also for you.
One of the biggest negatives I think is the pretend conversations the parent has with the child in the book. Pretty much in the example the upset kid is able to take in a 750 word dissertation form you that moves form what the child has done wrong to Jesus and the forgiveness of the cross. Some how I think the struggle is for the parents of a young one is to somehow reflect that, but in 15 words, and I am not sure that is possible. Nor do parents have theological degrees that are able draw typological situations or examples on their feet in any given situation.
The book is a good challenge to Christians to move away from trying to make moral children and calling that "Christian". There is nothing wrong with moral children, but the authors plead with you, if that is your intent, to not call it "Christian". This is the only parenting book I have read, so therefore it is the best in that category that I can recommend to someone...
The Gospel Coalition - These guys have a good review of this book
Give Them Grace - The books website
Parenting Audio - A list of audio talks I have listened to (or meant to listen to) on parenting