Earlier this year, the Department for Education issued new statutory guidance for all schools and colleges in England on how to keep children safe online.
Graham Stuart's committee is looking into whether the guidance is adequate.
The government says it has taken measures to tackle online abuse.
But it continues to be a widespread problem. Only last week, athlete Jessica Ennis-Hill was reportedly sent a tweet about rape after she said she would want her name removed from a stand at Sheffield United if convicted rapist Ched Evans was allowed to play football for the club again.
In January, Isabella Sorley and John Nimmo, both from Tyneside, were jailed for sending abusive messages on Twitter to feminist campaigner Caroline Criado-Perez.
BBC Radio 5 live went with Sorley to visit Dyke House College in Hartlepool and talk to pupils about the dangers of getting caught up in online abuse.
Sorley, 24, explained to pupils that she sent tweets - including one telling Ms Criado-Perez to "go kill yourself" - after a heavy night of drinking but said that was no excuse.All children in local authority schools must be taught about internet safety from the age of five
She said: "There's banter and then there's bullying, and even bullying doesn't cover what I did. I went to the extreme - it was absolutely disgusting."
Sorley told the students: "Ultimately I'm here to warn you that anyone can land in prison. It's not just going to be the stereotypical person from a broken home. I do have a degree, I have 13 GCSEs, so please just be careful what you do put on social media because what you write down, there's always a record.
"Even if you delete it, someone might have retweeted it or taken a screenshot of it. You can always be traced back and ultimately it can land you in a lot of trouble."
Sorley added: "I will always be labelled a troll. However, I had never done anything like this before and I will never do it again. My aim is hopefully that something good can come out of what happened to me."'I was terrified'
Ms Criado-Perez told 5 live: "I was terrified, I lost about a stone in about a week. I still get abuse on a daily basis and I often feel the people sending it to me just aren't aware that I'm a human being and I don't think they'd say it to me if they could see my face and could see my reaction.
"Young people need to have these things explained to them. We're all getting used to social media and I think when you're young you're less likely to really think about long-term consequences, and be able to empathise fully with other people because you're still learning about yourself.Continue reading the main story Protecting yourself and your family Top 10 tips for online safety What can parents do to protect their children? Parents on social media by Louise Minchin
"Obviously education on social media has a role to play, the police have a role to play and social media companies have a role to play. But most important is that we tackle a society where we are bringing up children to hate women."
Mr Stuart's committee's investigation is part of an inquiry into the teaching of personal, health, social and economic education (PHSE).
The Conservative MP told 5 live Breakfast he did not believe teachers were getting it right: "What's clear from Ofsted and other reports is that schools are failing to provide young people with the guidance and support they need to be safe online.
"Schools have a part to play in ensuring young people are safe and are kept away from the misery and depression which online abuse can bring about."
All children in local authority-maintained schools must be taught about internet safety from the age of five as part of the national curriculum, under changes brought in last year.
Parliament is also considering tougher penalties for internet trolls, with those found guilty of making violent threats over the web liable to a maximum two-year jail sentence.
The Department for Education said: "We expect schools to take firm action to tackle victimisation both on and offline. As with all forms of bullying, the best schools create an ethos to stop it from happening in the first place.
"We have given schools the power to search for, and if necessary delete, inappropriate images on electronic devices and are working with social networking sites and internet providers to make the internet a safer place for young people."
A YouGov survey in September of just over 700 teachers suggested two-thirds had seen pupils abusing and bullying each other on the internet.Chloe Madeley, daughter of TV presenters Richard Madeley and Judy Finnigan, received online threats last month
Just over 40% said they had never taught e-safety, while a third said they would feel out of their depth tackling it in class.
Graham Stuart thinks teacher training must "evolve" with the needs of pupils: "It's quite hard to keep up. It's therefore an important part of continuing professional development of teachers that they are kept up to date with these issues so they can help the youngsters.
"A lot of us with children feel that their children are far ahead of us in understanding the technology and the software. This is another reason, I think, why parents would be keen that teachers should be trained in these issues and be able to communicate with the children," he said.
Christine Blower, general secretary of the NUT, the largest teachers' union, said the DfE guidance "has been adopted in many schools" and "the document sets the tone by saying that cyberbullying is unacceptable and gives advice as to how it can be tackled".
She said: "Using the curriculum - in PHSE, citizenship as well as IT - to educate children and young people about the possibilities and potential pitfalls of social media and the internet is an effective way of ensuring that the issues relating to cyberbullying are discussed more openly within the school community."