Many people regard Picasso as the greatest artist of the 20th century, a man with a prodigiously creative mind and a zest for life in all its variety. We know much about his paintings, sculptures, drawings, ceramics and prints, but there was also a special, little-known kind of draughtsmanship that he practised for his young son, Claude, while shaving.

As his friend Lee Miller recalled it, “In Picasso’s special children’s performance, he lathers everything from Adam’s apple to the crown of his head, leaving a horrifying blank. Then with his fingernails ploughing through the soap to the brown skin underneath, he draws a clown’s face. He continues, covering with the brush, gouging with his fingers through a hundred subtle expressions until the audience falls back into the bathtub.” According to Miller, Claude was quiet only when his father was shaving.

Picasso adored children. He had two older children from previous relationships, but probably his happiest, most settled and fulfilled time as a father and husband was spent with Françoise Gilot in the South of France with their two small children between the years 1949 and 1953.

As a new show of ceramics, photographs, sculptures and mementoes opens at the National Museum of Scotland, Picasso: Fired with Passion, we can reassess this period immediately after the war as one of the key turning points in the private life of the artist.

Picasso had met Gilot in 1943. She was an artist, too, 40 years younger than him, beautiful, determined and fiercely intelligent. In the summer of 1947 they moved from Paris with their baby son Claude into a house known as La Galloise on the hillside above Vallauris, not far from Cannes. Vallauris had been a pottery town since Roman times. Good local clay and plentiful wood from the Aleppo pine trees for wood-burning furnaces had led to huge growth in the industry over the centuries, providing traditional clay cooking pots and tableware. Its fortunes had declined with the 20th-century use of metal cooking pots, but soon after the war a revival began under the leadership of two ceramicists, Suzanne and Georges Ramie. Picasso and Gilet had visited Vallauris in 1946 and Picasso had spent time at the Ramies’ studio experimenting with clay. It was when they moved to live there that Picasso’s passion for ceramics blossomed.

Conditions were good for him. The war was over, he had a beautiful and inspiring partner, a large light studio in a disused scent factory, plenty of space and time to experiment, a young son and, by 1949, a daughter, named Paloma after the Spanish word for dove.

You can see in the photographs of the period from this exhibition that their life together appeared to be one of perfect family contentment. There was a boxer puppy named Yan, and they were never short of visitors who provided friendship and inspiration, many of them artists such as Georges Braques, Jean Cocteau, Roland Penrose and his wife Lee Miller.

Roland and Lee’s son Tony remembers visiting Picasso and Gilot with his parents. “Even as a small boy I could see that he was happy. He had this lovely studio where he made fantastic sculptures out of junk – masses of them. He seemed to have a great sense of playfulness. You can see it in the slightly whimsical ceramics he was making. He was comfortably off and famous without the huge demands that came later with his expanded fame.”

The first ceramics he made at Vallauris on an early visit with Gilot show the playfulness that would develop later. Picasso, she says, decorated “two or three plates made of red clay that [have] already been fired – what is called biscuit – with a few drawings of fish, eels and sea urchins”. Another friend, the poet Jaime Sabartés, recalls him modelling small objects – the head of a faun ( above) and a bull in clay.

By 1947 he was decorating plates with birds, fish and bunches of parsley and experimenting with sculptural shapes, pigments and glazes. He was also producing sculptures made from whatever he found in a nearby rubbish tip.

Women and children were favourite subjects. For Woman with Pushchair in 1950 he used a real pram, while the baby was made of broken crockery.

He also devoted hours to making toys for his own young children and their visiting friends. “I have a lovely little wooden sculpture he made, which I call Mrs Noah,” says Tony. “It’s made from a little offcut of wood, about four inch-es high, and the shape suggests a woman, nicely tubby, in a motherly way. He had very simply given her arms, a face and a dress with a blue crayon.”

Picasso had a natural way with children. “He accepted us as we were. Of course he never had to tick us off because there were others to do that, but he had a particularly magnetic quality. He was tremendously generous, too. He allowed us to go to his studio and touch things. He was 19 years older than my dad, but he seemed absolutely ageless. I remember going to the beach at Cannes with him. I used to make little things and give them to him, and he was always so receptive and kind. He was great fun. And he was clearly so very happy.”

Picasso: Fired with Passion, National Museum of Scotland, Chambers St, Edinburgh (www. nms.ac.uk 0131-247 4422), until Oct 28; Picasso on Paper, Dean Gallery, 75 Bedford Road, Edinburgh (www. nationalgalleries.org 0131-624 6200), until Sept 23

Editor's choice

• U2 at Nou Camp Stadium, Barcelona
• The 50 Best Movie Villains
• The 15 golden rules of theatre etiquette