Here is a link to Cathy Duffy's article on teaching math without a textbook in kindergarten through third grade.
If you know what concepts you need to teach and how to teach them, it is possible to teach through sixth grade without a textbook.
Charlotte Mason had a lot to say about math instruction. Here are some of my favorite quotes (in bold) from Home Education (Volume 1) pages 253-258.
Of all his early studies,
perhaps none is more important to the child as a means of education than that of arithmetic.
The chief value of arithmetic, like that of the
higher mathematics, lies in the training it affords the reasoning
powers, and in the habits of insight, readiness, accuracy, intellectual
truthfulness it engenders.
Mason believed children needed to be able to apply math skills. She also believed in scaffolding - not a word she used, but a word that, translated from "teacherese" means giving children problems they can solve, and increasing difficulty little by little.
Care must be taken to give the child such
problems as he can work, but yet which are difficult enough to
cause him some little mental effort.
The young [inexperienced] governess delights to set a noble 'long
division sum,'––, 953,783,465/873––which shall fill the child's
slate, and keep him occupied for a good half-hour; and when it is finished, and the child is
finished too, done up with the unprofitable labour, the sum is not
right after all: the last two figures in the quotient are wrong, and
the remainder is false.
Mason did not think children in elementary school should be made to spend long periods of time, working independently, doing math problems they could get wrong. It seems common sense, but this is the opposite of what happens in most public school classrooms. Textbooks are written with two main parts - guided instruction and independent practice. Guided instruction is where the teacher explains the lesson's concept and demonstrates how to solve problems, and students complete independent practice after they've shown they can be "released." (The phrase "gradual release" is another teacherese term. When I hear it, I can't help but imagine a fisherman pulling a hook out of a fish's mouth and letting it back into the water, or an animal being let out of a cage into the wild.) The teacher walks around, helping students, or sits with a small group who need more help understanding the concept. But at the end of the day, the children go home with their homework, homework that some of them still cannot do on their own. If this is the case, why has it been assigned?
Because school districts create pacing plans so their teachers will cover certain concepts by certain dates.
But back to Miss Mason.
[A child] must not be
discouraged by being told [his answer] is wrong; so, 'nearly right' is the
verdict, a judgment inadmissible in arithmetic.
Mason did not believe children should be told things like "Well...you're close, but..." and she didn't think telling a child his answer was incorrect would damage his self-concept.
The next point is to demonstrate everything
Use manipulatives. There are
lots of manipulatives you can buy - like Unifix cubes and pattern blocks
- but there are also lots of manipulatives you already have on hand,
like macaroni noodles, beans, buttons, and pebbles.
Mason believed a child should be able to use manipulatives as needed, and that before a child learns facts, such as the multiplication table, he or she should demonstrate the concept. For example, when exploring the concept of multiplication, a child can build arrays with counters.
A bag of beans, counters, or buttons should be used in all the early
arithmetic lessons, and the child should be able to work with these
freely, and even to add, subtract, multiply, and divide mentally,
without the aid of buttons or beans, before he is set to 'do sums' on
This philosophy is quite different than what happens in many public school classrooms, where students who do not know multiplication facts are given a multiplication chart as a "cheat sheet." The rationale behind this is that students can learn the process of solving problems involving multiplication (such as 43x12 or 768x56), and complete independent practice pages, even if they don't know multiplication facts. Mason would very much disagree with this.
More math talk later.