Looking out over a sea of hands on a recent school visit, I felt a warm rush of elation at the sight of every single pupil raising their arm to affirm that they were a feminist.

Except that’s not quite what happened.

In fact, when I asked everybody who was a feminist to put their hand up, the result was a paltry scattering or hands – 20% of the assembly hall at best. So I asked the pupils to raise their hands if they thought everybody should be treated equally regardless of their sexuality, and every hand in the room went up. I asked them if everybody should be treated equally regardless of skin colour and, again, the response rate was 100%. Finally I asked them to put their hand up if they thought everybody should be treated equally regardless of their sex. Everybody in the room raised an arm.

“If you have your hand up now”, I explained, “then you’re a feminist. That’s what feminism means.”

Apart from a few horrified boys who snatched their hands down in dismay, the general reaction was one of bemusement. Several kids asked if boys were allowed to be feminists, and others protested that they couldn’t possibly be, since feminism meant wanting women to defeat, overtake, or generally beat men into submission.

It’s not surprising that these outdated and false stereotypes persist, given their stubborn repetition in the media and across the internet. In fact, there seems to be a huge amount of anxiety about the current resurgence of feminism and what it might mean for men. In the past week alone we have seen wails that the sacking of Jeremy Clarkson points to an “emasculated” BBC, articles proclaiming that UN statistics on sexual violence unfairly malign men, comment pieces that declare the “real” everyday sexism in the UK to be against men; even Russell Crowe mourning “the loss of traditional masculinity”.

What’s strange is that often at the heart of this panic is an entirely false dichotomy. First, such arguments suggest that tackling issues such as sexism, street harassment or domestic violence somehow precludes action on problems that disproportionately affect men.

The idea that the fight for gender equality somehow erases masculinity or disempowers men seems to be strangely insulting to any man whose sense of identity doesn’t come from being offensive to women. Feminism doesn’t mean doors can’t be held open any more,or the end of flirting, or that men should never again pay a woman a compliment. That’s simple human kindness we should all show one another regardless of gender.

The idea that feminism must somehow result in either deliberate or collateral damage to men is simply not true. Almost every issue that feminists campaign about would have a positive knock-on effect for men. The entries to the Everyday Sexism Project reveal this with brilliant clarity – in the same week, we’ll receive one entry from a man refused parental leave and ridiculed in the office for asking for it, and one from a woman who has been refused a promotion because she is considered a “maternity risk”. We hear from girls who aren’t allowed to join in football games and boys who are bullied for wanting to take “girly” subjects such as art or drama. We learn of fathers congratulated for “babysitting” their own children and mothers criticised for “taking a night off”. Many of these problems are so obviously two sides of the same coin.

The same is true for many of the issues that men’s rights activists raise as exclusively “male” concerns, with the suggestion that feminism ignores these problems. Invariably, these include accusations of gender imbalance in the allocation of custody, or the fact that the male suicide rate is several times higher than it is for women. What they don’t seem to realise is that these too are closely linked to the inequality that feminism seeks to address. If there is an unfair bias towards female carers, it likely stems from stereotypes about women being family-oriented and men being career-focused. It seems sensible to assume that at least some part of the gender disparity in suicide rates may be connected to the pervasive idea that men must be tough and strong, that boys don’t cry and it’s shameful for men to talk about their feelings or reach out for help. Tackling these stereotypes would be good for everybody.

It will slow us all down if people persist in peddling this outdated dogma that sets men and women up against each other. Of course, not all men are sexist, and not every woman will necessarily face sexism. Gender inequality has a negative impact on men as well as women, though its structural and ingrained nature (politically, economically, socially and culturally) does mean that women tend to experience its effects more frequently and more severely. There is a vital role for men to play in this battle, and it isn’t as detractors or naysayers, but as allies, agents of change and beneficiaries. This isn’t about men against women, it’s about people against prejudice, and everybody needs to get on board.