Frances Lawrence was speaking in public for the first time since her husband, head of St George's School in Maida Vale, west London, was fatally stabbed nine days ago. She said that when the young felt unloved and undervalued, they "grope out in the darkness to find something that will counteract this void they find inside them".
Often what they discovered was aggression, violence "and complete disregard for human life", she said.
"We all have to work together on this. It's not just the knives but what is inside someone who uses knives."
Mr Lawrence, 48, died as he tried to protect a pupil who was being attacked outside his school.
Public revulsion has led to Mrs Lawrence and her four children being "overwhelmed" with cards, gifts and messages of support. Her eight-year- old son Lucien had stirred national sympathy when his letter to Father Christmas asking for his Daddy back, "to stop my Mummy and sisters crying", was published in the press last week.
At the family home close to the school, Mrs Lawrence said: "Lucien's innocence helped because he can articulate more than any of us our loss. Out of the mouth of babes comes the ultimate truth."
She added: "We must no longer be content with a lily-livered approach to violence. We have to tackle it head-on and find the root causes of violence. It may take some courage but that is the only route forward."
The 47-year-old described her husband as a fearless man who was well aware that there was danger in the world.
"Like St George, he knew there were dragons out there, but he felt you must confront them." Mr Lawrence had "adored" children, and it "broke his heart" to see them suffer any harm or injustice.
Describing her feelings, she said: "Philip was my rock, he was my inspiration. Each day of our life together I admired him more and loved him more.
"For myself I cannot imagine any good would come from this, but for the world, good will come of this. I know that and I will try my hardest to ensure it does."
Police are still hunting the gang allegedly responsible for the killing.Reuse content