“I used to think boys were just better at maths,” said Linah. “But now, it’s like, we go to the same school, we do the same subjects – so if you can do it, I can do it.”

There’s a reason for the renewed confidence in the 15-year-old from Dagenham and it’s sitting next to her in a swanky office in the City of London. Elaine McLoughlin is a business control manager for Bank of America Merrill Lynch. She is also a volunteer tutor on a programme to help children from disadvantaged backgrounds and get more girls studying sciences at top universities.

“[Maths] used to be one of my favourite subjects but over the years I’ve gone into autopilot,” said Linah, who lives with her two sisters, her mum, who works as a carer and her dad, who is between jobs. “I would freak out if I didn’t understand a topic in class but now I know we can go over it here.”

Since last year, her predicted GCSE maths grade at Lister community school in Newham has jumped from a 4 (the equivalent of a C), to a 7 (equivalent to an A). “Practice makes perfect so it’s going back and looking back at those topics that she’s no longer studying in class,” said McLoughlin.

I know if I was falling back a bit, I’ve got someone who will support me Melissa Blakemore, 15

In many ways, Linah is archetypal of her age group: she counts editing Instagram photos for friends among her hobbies and is relieved at getting to skip an extra physics class in order to be interviewed on a Friday evening. She has hopes, though, to attend one of the top-tier UK universities to study forensic psychology. Visiting McLoughlin’s office each week is a source of inspiration.

Linah said: “I thought it was going to be really intimidating but you walk in and feel like you’re a different person. You see people who have their lives together and you’re like: ‘I’m so gonna be like that guy’, even if you don’t know what their job is.”

Quick guide What is the Upside? Show Hide Ever wondered why you feel so gloomy about the world - even at a time when humanity has never been this healthy and prosperous? Could it be because news is almost always grim, focusing on confrontation, disaster, antagonism and blame? This series is an antidote, an attempt to show that there is plenty of hope, as our journalists scour the planet looking for pioneers, trailblazers, best practice, unsung heroes, ideas that work, ideas that might and innovations whose time might have come. Readers can recommend other projects, people and progress that we should report on by contacting us at theupside@theguardian.com

Although the divide is steadily shrinking, there is still a gender gap within Stem (science, technology, engineering and maths) subjects. Women make up 14.4% of people working in Stem professions in the UK, which falls far short of the country’s goal of 30%.

On top of this, Britain’s booming private tuition sector, worth an estimated £2bn, is making the playing field even less level for disadvantaged children. Last July, an annual survey by The Sutton Trust revealed that 27% of secondary students in England and Wales have had home or private tuition. The figure rises to 41% in London.

The Access Project, founded by teacher Alex Kelly in 2008, works to address this inequality in 28 schools. It has more than 1,000 tutors supporting 1,250 students across London and the Midlands.

To Mcloughlin, who graduated with a maths and economics degree from University College Dublin and revises Linah’s topics over the weekend and offers her ad-hoc career guidance, diversifying the workforce is about more than equality. “Having more women on board and more diverse teams leads to more successful output,” she said, adding that her boss is a working mum who has “turned a lot of things around”.

There is a stigma attached to Stem for women, according to Karishma Honap, 23, an assistant engineer at WSP and Access Project tutor in Birmingham. In the engineering sector, fewer than 9% of professionals are women. “When I applied to do engineering everyone was sort of like: ‘Oh, don’t guys do that? That’s rare for a girl,’” she said.

Computer says yes: how tech is a force for good Read more

But, according to her student, Melissa, a year 10 pupil at Wood Green Academy in Wednesbury, in the West Midlands, mindsets are already shifting. The 15-year-old says there’s a 50-50 split between boys and girls in her triple science class, despite the fact that the percentage of female A-level physics students has stagnated at about 21% for the past decade.

Although she’s always enjoyed science at school, since starting physics mentoring with Honap in October, Melissa’s confidence has blossomed. “I know if I was falling back a bit, I’ve got someone who will support me and understand what I’m facing,” she said.

Melissa, who lives in Wednesbury with her mum, a retail assistant, her younger brother and stepdad, wants to study at Oxford University. She will visit the campus with the charity next week. There’s still a way to go, including finishing her GCSEs and starting A-levels before she can apply but in some ways it feels like doors are already being opened to spaces she isn’t accustomed to.

When she visits Honap at the Mailbox, an upmarket shopping and office development in Birmingham’s city centre, which she hasn’t visited since she was eight, she feels welcomed. “I was nervous to begin with but now I find it cool how the receptionist knows my name, knows who to contact and just tells me to carry on up to the office,” she added.

This article is part of a series on possible solutions to some of the world’s most stubborn problems. What else should we cover? Email us at theupside@theguardian.com