If there is one thing you insist your children complete, it would be Algebra and preferably by Grade 8. The magic in passing Algebra is the ability to think with logic. Logic allows your child to make important choices for themselves and their family, understand and participate in their community and have self efficacy. A lack of Algebra is a life of substantially lower quality. http://www.algebra.org/whoweare.php
The nature of Algebra is such one can not learn it in one year...this is why there are so many pre-algebra math concepts. A child needs to master many different aspects of math in order to succeed in Algebra. It is not enough to add, subtract, multiply and divide. It is not enough to understand order of operations and what all those 'symbols mean'. Part of Algebra is the systematic format of problem solving (grinding the numbers). Another part is being able to do error analysis and correct (none of us are perfect and we need to know how to fix many different types of errors). Still another part is the way one has, in a sense, 3-D problem solving thoughts.
It is invaluable to know the various guide points your child needs to pass through to get up to Algebra. http://www.greatschools.org/students/academic-skills/531-K-5-benchmarks.gs You can look at the specific standards for your state. It is important to work with a tutor who knows and understands math syntax and what are common mistakes students make as they learn and how to work out of the problem.
Have a discussion with the tutor(s) you are interested in to find out their belief about learning area models of multiplication and division, memorization or both. Can they explain to you the idea of multiplication being shorthand for addition? Can they explain why one does not write math across the page and they should solve downwards?
There are many things to know about a good algebra (or any maths) tutor and the best way to ascertain who is right for your child is to ask. It is important your philosophy is in alignment. A good or great tutor will be pleased to have this discussion and answer your questions!