This approach attempts to remove the restrictive atmosphere found in most schools.
Interestingly enough, this approach (as well as the Montessori method) is often used in preschool centers for teaching young children, and, yet, once the child gets put into a private or government school, the learning environment is completely opposite to how they had been learning.
Most reviewers I have read like to emphasize that this approach attempts to educate the child as a whole person, rather than to just focus on the academics.
Learning science through exploration of the outdoors, math through cooking, and history through story form are just a few examples of how this teaching method attempts to educate.
Generally, the testing in the curricula who use this approach attempt to make the questions less "test" like and more narrative form. Meaning the child tells the teacher/parent what they have learned, rather than answering a form question on a test.
This approach has the advantage of making school less mundane and getting the child more active in the world around them.
It also focuses on building strong relationships with people as a way to truly know their culture.
This approach generally has a disdain for rote memorization. While that is understandable, the reality is that all children (and adults) generally require repetition for permanent memory retention.
Multiplication tables are an excellent example.
Repetition is required for some things; there is no way around this.
But, the question is whether individual curriculums manage to find new and engaging forms of repetition rather than the standard, boring forms seen in traditional school settings.
The main disadvantage to this curriculum is that the focus is so fluid, flowing from one idea to another and so forth, that there are times when foundational concepts are missed altogether.
Not necessarily because they are not there, but because the approach to this style of teaching can lend itself to being carried by the wind in most assignments or lessons.
In this way, whether or not a foundational concept was bypassed may go unnoticed until that skill or knowledge nugget is needed for another course or career down the road in the child's education journey.
Special care to prepare ahead for what your child will want to do when they grow up (college, technical training, straight to a job, housewife, etc) will assist in making sure you have checked all the necessary skills and core knowledge boxes.

In Summary:

Positive:

• Very different, non-constraining approach to learning.
• Treats the child as a whole person beyond academics.
• Provides for many methods of learning through discovery.

Negative:

• Does not encourage repetition even when this is a vital form of memorization.
• Can be easy to skip or bypass what may be important concepts later on.
• Some students may find it hard to stay focused on what is actually being taught because it is such a fluid form of teaching.

My Father's World

Simply Charlotte Mason

A Gentle Feast

The Peaceful Press

Beautiful Feet Books

Brighter Day Press

Charlotte Mason Simple Studies

My Compass Classroom

Ambleside Online
(Free curriculum)

Living Books

Wildwood
(Secular based)

Nautilus Homeschool
(Secular Based)