Mason’s approach to arithmetic is pretty common sense, concrete, and focused on reasoning.
She emphasizes word problems with small numbers, and using manipulatives such as dry beans. For multiplication, for example, she would have the child find 4 x 3 by setting out three rows of four beans each and count them. This is exactly how Montessori does multiplication, too. Children should be able to use the beans for all of their math problems, and then to do mental math with imagined beans, before being asked to do math in writing. They can even use the beans to set up a whole addition table such as 1 + 1, 1 + 2, etc, and likewise for subtraction.
Mason doesn’t work on the notion of the decimal system until after all of this bean work. Her method is similar to Montessori’s, but she is critical of Montessori’s bead material, arguing that it puts more importance on the material than on the ideas it represents.
Here I prefer Montessori and the bead material. It so concretely gives the idea of tens. There are single beads for units, bars of ten beads strung together for tens, squares of ten ten-bars for hundreds, and cubes of ten hundred-squares for thousands. Children learn to exchange, to understand place, that a number in the tens place is a number of tens, and so on. Montessori also involves number cards, so that kids can work with symbols and relate the symbols to the bead material, even before they can write figures well.
Mason addresses some other areas of math, too. Kids should have tools for measuring and weighing things, and should have opportunities also to estimate weights and measures of things in their environment. Weighing and measuring can also introduce fractions, she says — half a pound, a quarter of a yard, for example.
When a child gets a math problem wrong, Mason argues for the parent or teacher to simply pronounce it wrong (never “nearly right”) and let it sit, not having (or allowing) the child to do it over again, but moving on to the next. I’m not sure why she doesn’t allow trying the problem again, unless it has something to do with getting more muddled with reworking. Maybe one could try that particular problem again a few days later or something, without calling attention to it being a repeat.
She says it’s silly to have pictures of geometric shapes displayed everywhere, as if sacramentally “the continual sight of the outward and visible signs… should beget the inward and spiritual grace of mathematical genius” (263). Instead, let children engage with real things that have such shapes, and later let them take interest in those underlying shapes. Here again she is implicitly criticizing Montessori, I think, with the boxes of constructive triangles and the geometric solids and so on. And here again I think that the concrete workable Montessori materials far surpass posters of shapes, and can be used alongside plenty of experience of real household or natural objects in a variety of shapes.