Math programs that cover all the bases do exist, but so do kids who just don't get it all right away. Not everyone is born with a head for numbers, and even those who are will come across difficult concepts from time to time that require more attention and study. Some parents just try to push on through, without really assessing the child's retention or understanding, hoping eventually they'll just get it.
While there is something to be said for continuity, sometimes it's essential to stop and evaluate, maybe even to camp out on one topice for awhile and really make sure your kids know what's going on before resuming regular lessons. After all, one of the benefits of homeschooling is that you can take your time when needed, adapt teaching methods to your child's needs, and generally exercise liberty in the education of your kiddos.
Probably since Adam and Eve undertook the education of their sons, humans have used physical objects to convey in concrete terms the abstract concepts of mathematics. Before a child can understand that "2+2=4," they have to see two apples put beside two more apples (or coconuts, or pennies, or whatever). We call these physical objects manipulatives.
Manipulatives come in all types and sizes and quantities. From plastic bear counters, to working scales, to Cuisenaire rods, to colored pattern blocks, these objects lie somewhere between the realm of toys and educational tools. Kids like handling things, but when those things actually impart important information, parents are happy too. You might be able to overdo manipulatives, but then again, entire curriculum lines are built around their use.
Starting with manipulatives as early as possible is a good idea. Older kids will have no interest in what they see as "little kid stuff," and anyway, they're probably past the age where such methods are effective. Younger kids can begin to learn to think mathematically before they're even "doing school" properly, just by having access to math-oriented toys, games, and gadgets.
Older kids are probably better off with flashcards and drill books, simply learning through repetition to do the basic, as well as the more advanced, functions of math. Part of being good at math is learning the reason behind the functions, but mastery often comes simply through doing the same kinds of problems over and over and over till they become second nature.
A lot of mathematics, after all, is simply facts to be memorized. These facts have the advantage of being discoverable and provable in the physical realm, but there aren't shortcuts, and there isn't a single secret that unlocks the whole scheme of things. Math study takes concentration and commitment, and both parents and students need to be aware of this before getting started.
At the same time, math can be fun, and to a great extent our kids will be enthusiastic about the same things that excite us. If we moan when the math books get dragged out, they'll moan too; if we treat arithmetic, algebra, geometry, and trigonometry like adventurous terrains to be explored, they'll gladly follow wherever we lead.
Some resources are better suited to cultivating this attitude than others. Occasionally there's no alternative to simply setting kids down with a drill book and making them do division or graphing until they comprehend it, but if we make math exciting from the beginning there's a strong chance kids won't ever get hung up on a single topic in the first place.
For little kids, you might start by reading them picture books like the Sir Cumference Series by Cindy Neuschwander and Wayne Geehan, beautifully illustrated stories that impart mathematical concepts and information through fun and imaginative stories. There are many more math-based picture books available as well, some as basic as identifying numbers and shapes, others more advanced and complex.
Middle and high school students typically need to see how math relates to real life in order to get really excited about it. Again, there are plenty of options, but our favorites are Dr. Stan Schmidt's Life of Fred series. While more of a curriculum than simple resources, these books show through narrative and humor how even the most advanced math problems have real-life application and value.
You'll find all kinds of math resources below, from manipulatives to drill books to picture books and games. Math is an essential and invaluable subject, and we can't afford to give our kids half-hearted exposure to its principles, functions, and theories: these products will help you fill in gaps, reinforce important concepts, and generally guide your kids to thorough understanding of a field of study too often seen as confusing and frustrating.