Horizons Math uses an incremental or "spiral" approach to math instruction. Don't think of this in terms of those nearly immobile springs in your closet doors however—Horizons' spiral method is more like a Slinky, made of metal but flexible and able to move on its own with a little nudge. Not that this is primarily a self-guided program, but most students will be able to figure out a lot of the material on their own using the instructions in the student workbooks.
A common complaint is that Horizons only goes through grade 8, a good indication of its popularity. It is Alpha-Omega's "accelerated" math program to be used instead of the Lifepac Math, though it is comparable to other math curricula in its pace (A Beka, for example). Another common complaint (not so positive) is that there is little teacher support, though there are plenty who argue this is not the case. Teacher support does increase as the concepts become more advanced, and most K-3rd grade principles shouldn't be too hard for most parents to teach. Those who want their kids to learn wholly on their own may be frustrated at having to explain things periodically, but for those who take it for granted that younger kids need more direct instruction this program is surprisingly less teacher-intensive than many of the alternatives.
The spiral approach means that concepts are introduced a little at a time, reinforced by regular review, and reintroduced later in greater detail. This is not designed for instant mastery, but to familiarize students with ideas and then gradually build mastery as they see the same idea turn up in a variety of contexts. This is basically the approach of Saxon Math, though Horizons is faster paced, especially in the earlier grades. For those moving to Horizons from another curriculum, the Math Readiness Evaluation tests will help you determine what level your child is ready for.
Horizons covers grades K-8, with two 80-lesson student worktexts for each level (160 lessons per year) and a teacher's manual per grade. The consumable student texts are colorful and fun without being overpopulated by useless pictures of bunnies and kitty cats. Each lesson is between two and four pages, and includes straightforward instructions for completing the exercises; in the later grades (4-8), concepts are explained at the beginning of each lesson. Every tenth lesson is a test. There is generally enough information in the student text for kids to understand what they're supposed to do, though certain concepts (more as the course progresses) will definitely need elaboration by a parent/teacher.
Since these are "accelerated" texts, students moving into Horizons from another program may seem to be behind. Fortunately, you can start with an earlier grade and play catch-up if necessary, or you can simply fill in with a remedial course like the Key-To books until they're where they need to be. Students who do Horizons from the beginning, however, should have no trouble keeping up, as the progression of principles is logically-oriented. The kindergarten workbooks teach such basics as numeral identification, time, money, basic addition and subtraction, etc. By fifth and sixth grade, students are performing basic algebraic and geometric functions, while still being drilled on the basics; this approach is continued in Pre-Algebra. Some students find it difficult to keep pace with the introduction of new ideas, though students with low attention spans are excited by constantly learning fresh concepts.
The teacher's manuals are designed in two parts (in Grades 1-3, these are divided into seperate volumes). Part one contains suggestions for how to teach each concept as well as ideas for activities to reinforce important principles; part two contains reduced student pages with answers to problems, as well as drill sheet masters for further practice. (Though the texts include regular practice of old concepts, some find it important to further establish certain principles with more extensive review.)
Many parents find the K-3 teacher manuals unnecessary, as the concepts are such that even someone with little math background can easily teach them to young children. The 4-6 teacher manuals are by general consensus, however, indispensable, as they provide important information not just on how to solve basic algebra and geometry problems, but on why they must be solved in that way. The Pre-Algebra teacher's guide has less supplemental information, but complete answers and solutions to all written work. While many students could probably get by on their own, it is probably a good idea to provide some direct instruction just to make sure they know what they're doing; the teacher manuals offer help to do this, though it isn't as thorough as it could be.
Many parents appreciate the distinctly Christian nature of Horizons Math. Scripture references are frequently incorporated into lessons and word problems, and help kids remember they're learning math in order to better understand God's creation. While this is a welcome quality in any curriculum, it can be overdone, as when in the fourth grade text one word problem asks kids to record the length of the nails that were used to hang Jesus on the cross in feet and inches. This is just one lone example and there are very few like it, but questions like that can take an unsuspecting child by surprise.
The spiral approach doesn't work for everybody, but just because it is contrasted to the "mastery" approach (in which concepts are drilled over and over until the student "gets it," and then the concept is dropped altogether for a new one), doesn't mean that the spiral approach doesn't aim for mastery of skills and concepts. The main difference is that it takes longer to gain such mastery, but one advantage is that it is often better contextualized with related principles. This is an organic approach to learning math, not one that simply drills facts into kids' heads with no indication as to how their new knowledge will ever be useful.
Like so many other good math curricula, Horizons stops after the elementary grades, so you'll have to find another program for junior high and high school. If you stick with Horizons through sixth grade, we recommend moving to Bob Jones' Fundamentals of Math for seventh grade, and then on to Harold Jacobs' Elementary Algebra, Geometry, and Mathematics: A Human Endeavor for high school, as the approaches of those later texts are similar to that of Horizons. However, the authors of Horizons Math change after third grade, so the format and style of levels 4-6 are quite different from the first four; if you want to switch to something else after third grade, we recommend you start with Saxon 5/4 (or even 6/5).
Many parents who feel that Horizons is too concrete in its approach, find supplementing with Singapore (with its emphasis on abstract thought) a good solution. Choosing the right math curriculum is largely determined by what kind of learner your child is; for kids who need constant challenge and want to develop good critical thinking skills, this is one of the better choices.