Firstly, I think you meant "Cooperation" for the title!
Be careful with your tenses, they jump around a lot.
The role of fathers in child rearing is essential but unfortunately fewer men pay attention to this matter. In this situation, women shouldered the burden of household management in spite of they work outside the home. Their families face with some problems.
Corrections: The role of fathers in child rearing is essential, but unfortunately fewer men are paying attention to this matter. In this situation, women shoulder the burden of household management in spite of their work outside the home. Their families are faced with some problems.
Also be careful about plurals.
I will highlight changes in red.
In the past, in most societies, men were the breadwinners and financial providers of families, while women were responsible for house chores, such as upbringing the children. We witnessed (<-- awkward word choice) husbands sometimes telling their wives irrational statements such as "You wanted to have a child so you should raise him alone" in their argument. In this condition (awk), men made women stay at home and spend their time cooking food, cleaning the house and raising the children. They did not let women attend social activities.
But these days, due to financial problems of families in one hand, and the tendency of most women to work outside the home on the other hand, husbands should help their wives in raising their children and other domestic chores as well as they collaborate with each other to provide the family with money and facilities.
From another point of view, during the process of upbringing of a child, the child needs the attention and emotion of both his mother and father. None of them can play the role of another. Some emotional problems of the child can be solved by mother and some of them can be handled by father. So if husband does not properly play his role, his child may be faced with some disappointments and emotional shortages in the future. For example, many adults say they wanted to be more with their fathers when they were a child. It is obviously clear that love can never be replaced by money and welfare.
In summary, the necessity for cooperation between mothers and fathers to raise children is undoubtedly more vital than before. Men should pay more heed to the emotional and spiritual requirements of their children as well as the financial ones.
You bring up a lot of good points here that you could elaborate on (such as "some emotional problems of a child"...), but I'm not sure how long your essay is supposed to be. You've only touched the surface of the issue, but one can only write so much. :) It's good so far, minus little grammar issues! Hope this helps.