Where does Alyce show personal growth in The Midwife's Apprentice?

Alyce shows personal growth throughout the entire book.  For me though, there is one specific location where I see significant growth for Alyce.    

The time is when Alyce actually takes on the name Alyce.  Jane has been injured, and she is not capable of going to the Saint Swithin's Day Fair to buy supplies.  Alyce goes instead, but at this point in the story, her name is Beetle.  Jane originally found Alyce sleeping on a pile of dung and decided to call the girl Beetle.  While at the fair, somebody mistakes Beetle for somebody else.  He calls her Alyce, and Beetle decides to take the name as her own because she likes it.  The change in name is symbolic of Alyce's growing confidence.  She no longer feels like the girl found on a pile of poop.  She is gaining confidence in herself, her education, and her skill set, and Alyce feels that "Beetle" doesn't fit with who she is becoming.  

What a day! She had been winked at, complimented, given a gift, and now mistaken for the mysterious Alyce who could read. Did she then look like someone who could read? She leaned over and watched her face in the water again. "This face," she said, "could belong to someone who can read. And has curls. And could have a lover before nightfall."

From this point in the story, Alyce begins showing a great deal of confidence in herself.  Sure, she still makes mistakes, but this new confidence is what allows her to continue working hard for personal improvement.  Beetle would not have returned to Jane at the end of the story to get her job back.  Alyce would though, and she did.  

How does Alyce demonstrate intelligence in The Midwife's Apprentice?

Alyce shows herself to be smart by quickly picking up the rudiments of midwifery. Not everyone can be a midwife, but Alyce shows herself more than capable of mastering this age-old profession. Although there are setbacks on the way, such as when Alyce is unable to deliver a mother's baby, she's smart enough to learn from her mistakes, as well as tenacious in picking herself up and dusting herself off.

Like most people of her time, Alyce is illiterate. Nevertheless, she has the intelligence to learn how to read. While working at the Darks' inn, a patron teaches her how to read. As well as giving her an important new skill, this boosts Alyce's self-respect. She's gone through her whole life being treated like dirt, and yet now she feels that she matters and that she can handle just about anything that life throws at her.

How does Alyce gain pride in The Midwife's Apprentice?

It takes many years for Alyce to have any self-esteem, but slowly, she is able to gain some self-esteem and confidence in her abilities as a midwife.  She does encounter some "failures," but she does not give up.  She does not let those problematic experiences prevent her from becoming a successful midwife.  Despite her horrible and difficult childhood, she is able gather her courage and face her struggles head-on.  Alyce has confidence and is proud of herself and her abilities:

Alyce now realizes that she has a calling and purpose in life if she will only sieze it; being a midwife's apprentice is her place in the world and nothing else can give her what she wants and needs. (eNotes)

In what ways does Alyce demonstrate intelligence in The Midwife's Apprentice?

Although she's often called a "lackwit fool" by others, Alyce is actually rather an intelligent young lady. Alyce may be illiterate as with most people at that time, and she may be an abandoned orphan from a dirt-poor background, but she has a certain sharpness about her. This is a young lady with the medieval equivalent of street smarts. And without them, she'd find it hard to survive in such a tough world.

But Alyce's intelligence also helps others to survive. When Will's close to drowning, Alyce thinks fast and crawls onto the tree branch so that it dips in the water and Will can grab hold of it. Although the other boys insist on calling Alyce a nitwit, Will now knows better and calls Alyce by her name.

Alyce further shows her intelligence through her remarkable ability to learn by observing other people's behavior. For instance, closely watching Gilbert Gray-Head carve wood allows Alyce to pick up the trade herself, something that few other girls of her time were able to do. And of course, Alyce's intelligence and capacity to learn quickly allows her to become a skilled, caring midwife, which greatly improves her sense of worth and self-respect.