The main focus has got to be curriculum, but that's a difficult question. Many of the pointedly homeschool curricula are drill and kill to the extreme. For example, one homeschooler who reported 132 practice problems per lesson. This post is not going to be for those who are looking for that.
What to teach?
The core standards are definitely the wave of the next decade. Many states are going to have exactly these as content objectives and most states will be close. There are still too many objectives, and too little attention to process, but it's a place to start. The NCTM standards are practically venerable, now, but still the best guide to worthwhile content that there is. The NCTM Focal Points would have made a better start to a common core, but still could help a homeschooler know what to center on.
- Problem Solving. The big one. Working on finding answers to questions when you not only don't know the answer, but you also don't know how to find the answer. The trying and finding of methods to investigate is the heart of mathematics.
- Representing. Making, interpreting, translating between and choosing ways of showing and displaying mathematical information. Numbers, equations, functions, tables, charts, mind maps, written descriptions, pictures, diagrams, physical enactments, etc. Typically this is the biggest support to problem-solving, and one of the key means of differentiating in mathematics.
- Connecting. Seeing how what you know, mathematical information, and the real world relate and what they can tell us about each other. Many problem solving strategies boil down to
- Reasoning. Following connections in a particular direction (forward or backwards) or examining the strength of those connections. Answering why and how does this work questions. One framework for this is a) Making sense b) Conjecturing c) Arguing (which includes Proving).
- Communicating. Sharing or recording what you know. Strive to be clear, coherent, complete, correct and consolidated.
My favorite curriculum, bar none, is Contexts for Learning Mathematics. Excellent activities, literacy integration, and student and teacher support. I use these materials with my preservice elementary teachers. Issues are price (though they are almost half off right now), and focus on number to the exclusion of geometry. So a little bit of supplementary materials might be needed. Strong on pre-algebraic thinking and reasoning though.
Second choice, or for the whole span of content objectives would be Investigations in Number, Data, and Space. There's an option for online access to games for practice, too. Unfortunately, it can be a daunting task teaching it for someone who might feel shaky on the math themselves. If you want quality activities, and are willing to explore along with your student(s), this might be the series for you. Unfortunately it is priced like college textbooks, $420 for a year's worth of teacher books. You can buy by unit to supplement another curriculum.
An interesting option is the relatively recent translation of the Japanese curriculum. They're available by grade level or the overview. It does not cover all of our objectives, because the Japanese have a more reasonable curriculum, but it definitely covers what's important. There are workbooks, too.
The cheapest option I've seen that has some value is the Jump Math program. It's a bit worksheet centered, so you would have to supplement activities and problem solving, but there's a try at conceptual focus and it's a start.
Sue Van Hattum thinks it's worth looking at the Mathematics Enhancement Programme (it's English) - and I'll be looking at it. (You can always trust the Math Mama.)
- A membership in NCTM gets you access to the standards and the journal of your choice. Probably the single greatest resource for learning about the math, the teaching of the math, and finding worthwhile activities. The Calculation Nation games are free to all, though.
- NRich. Best single source for math problems I know. Searchable by topic, filterable by grade level and challenge level.
- Let's Play Math. Terrific blog of Denise, a homeschooling mother. Resources, links, activities, discussion. Also maintains a collection of Mostly Free Math Resources.
- Homeschool Math is run by Maria and is also worth a look.
- Me - write with questions and I'll help however I can.
Good math teaching is the same wherever it happens - it's supporting worthwhile and significant student learning. My main advice would be to try to transfer from the areas where you feel your teaching is strongest. Science, reading, etc. What's making that work, and how would it look in math? Emphasize making sense. Don't you and don't let your student do anything just because. Connect with other teachers (blogging, commenting, or tweeting) and discuss your teaching.