Jemma Leech listens in her wheelchair as people – even a trainee paediatrician – talk about her as if she can understand nothing. Then she laboriously spells out messages to her mother such as: “Please tell him that I am not an imbecile and I can outperform his expectations of me without breaking sweat.” And, unlike many seven-year-olds, she spells every word right first time.

At Palmerston Primary School, in Barry, South Wales, Jemma is fascinated by history, geography and science lessons, but in literacy and numeracy classes she is frequently uncooperative out of boredom. At the family home in Penarth, she giggles as she spells out: “The best thing to do is to give no sign of life at all or refuse to have eye contact.”

It is hardly surprising that she does not enjoy the basics of spelling and comprehension or subtraction and multiplication when her maths, reading and general knowledge have been found by an educational psychologist to be comparable to a 15-year-old’s.

She remembers being able to read at 3 (though, because she cannot talk, no one realised until she was almost 5), can do long division in her head and solve the T2 crossword, adores The Railway Children, and composes her own stories, which include lines such as: “I wished I had brought a torch as I crept over to investigate. Even in the dark, I could sense that the figure was not aware of the presence of another person.”

Yet she has been able to write these stories only at home because severe athetoid cerebral palsy — caused by a lack of oxygen for several minutes at birth — means that she cannot walk or sit unaided, and has no fine-motor control. (Her parents, Caroline and Perryn, who both work with the Welsh National Opera and also have twins — Rory and Kirsty, aged 3 — are pursuing a compensation case against St George’s Hospital in Tooting, London.)

In a breakthrough just before her fifth birthday, Jemma and her mother devised a way for her to communicate. By sitting on the floor, her slight body supported by an adult, she can swipe at an alphabet chart, using two movements, denoting column and row, to pick each letter.

But this time-consuming and very physical process is not practical in a busy classroom, so in her first three years at school, Jemma, who is full of ideas and mischief (she told her mum: “I don’t think your voice is rubbish when you’re singing but it isn’t the most tuneful at times and when you are singing opera it sounds like you are murdering a cat”) has been able to respond only to yes/no questions. What is particularly painful is her inability to communicate with the other children around her at playtime, who sometimes treat her like a favourite pet rather than a person in her own right. Her eyes fill with tears and her pretty face is distorted as she spells out: “I find it difficult to make them understand that I am not a muppet but someone with whom they could have a real friendship.”

The problem with communication has not been funding — the Vale of Glamorgan, Jemma’s local education authority, has pledged to provide a system for her — but finding a method that works for Jemma. In the past few weeks she has started to use an electronic toy alphabet belonging to the twins. She can’t press the letters but she can point to the one she requires, as she sits in her chair with a helper supporting her elbow. This should transform her communication at school and make it easier at home. But the question of how best to stimulate and develop her intellect and keep her engaged remains.

It would be a problem with such a gifted child even without the immense complication of her physical disability, but for Jemma it feels even more important because her life is so much the life of the mind.

For almost four years, her parents worked for hours every week with a team of volunteers to reprogramme Jemma’s brain by moving her body in exercises devised for her by the charity Brainwave in Somerset. They also took her to a centre for conductive education at Craighalbert, near Stirling, in Scotland, and are willing to do whatever it takes to enable her to exploit her potential fully.

“We want to know what to do for the best,” says Caroline. “The school has been wonderful at including her, but she is becoming frustrated in lessons that she finds too easy.

“How do we keep her inspired in the subjects she excels at while giving her a proper education in the things that are new to her? It feels so important to get it right and to do it now, while she is so enthusiastic and absorbs information so quickly.”