Mathematics is a good way to teach thinking skills and build an ability to reason logically. Younger children should master addition, subtraction, multiplication and division facts through at least the 12s. Older children should take algebra and geometry to master the abstract mathematical thinking required at the college level. The best math programs explain concepts well and use a repetitive method for drilling in the ability to understand and work problems.
Language is of primary importance in education at all levels since it is the basic way that humans communicate. This subject includes grammar, vocabulary and spelling at all levels in order to build a proficiency that will be of great value to the student. A good grammar curriculum should teach the basic parts of speech, their use in sentences, and at the higher levels sentence diagramming. As with math, grammar and spelling should be repetitive in order to refine and solidify language skills.
Reading is the key to learning all subjects and is critical to any homeschooling program. For the lower levels a good phonics program is best. While your children may not know every word they can read they can still read it and you can always have them look it up in the dictionary. Reading also allows your child to learn more about other subjects that you will need in your curriculum such as history and geography. A good reading program which includes real books is indispensable for any homeschooling program.
Handwriting is important to all other learning as it teaches neatness and discipline. It is not easy to form letters when you are 4 or 5 years old, but practice leads to improvement. Early learners should learn to print letters and then words, while fourth-grade and higher students should learn cursive. In both cases, the student should be encouraged to have good posture and hold pen and paper in the correct position. A positive benefit of good handwriting is that it makes the job of grading much easier for the parent.
As the old saying goes, those who do not learn from history are bound to repeat its mistakes. It is important that you give your child a good education on world history, your country's history, your state's history and your family's history. Lessons learned from each of these areas will help your child to establish her identity and even lend to a sense of purpose. Do not focus too much on dates but rather the whos and whys of history. This helps your child to learn to think, which is one of the primary goals of education.
Where you live is important and how your locale is part of the whole should be a part of your homeschooling program. In order to put historical events into perspective, a good understanding of geography is required. General geography should be taught in the early years with children learning about geographic features such as rivers, mountains, valleys and deserts. Older children should learn about world geography and the features of countries that have shaped world history.
Science is a broad subject which includes biology, chemistry and physics, all of which are based on both mathematics and language. It is also important to remember that history includes the history of science and that many scientific discoveries have profoundly affected human history. Basic science should be taught in the early years to give younger children a basis on which to build the more detailed and complex areas of science. Older children should take biology and chemistry at a minimum with physics as an optional challenge.