The small bead frame and large bead frames (sometimes called the "Montessori abacus", which is  not entirely accurate, but it sure looks like one!) are a necessary yet often overlooked piece of Montessori mathematics material.
 Large Bead Frame - goes up to 10 million

They each allow for addition, subtraction and long multiplication and are a step towards abstraction in the process of carrying into new categories and in exchanging.

While this material can be used for 3 operations, the main focus of this material is its teaching of *Long Multiplication* - this IS the Montessori material that teaches all the steps in proper order. Other materials either prepare for it, or expect a child to have this experience tucked away already.

The small bead frame has four bars, going into the thousands (with 10 thousand-beads, it can go to 10-thousand).

The large bead frame, with 7 bars, is the child's first experience going into the MILLIONS. And they LOVE it!

The children also utilize lightness of touch, visual organization, proper order of steps, which is preparation for the division with racks and tubes (test tube division) which starts with short division (1-divisor with a 4-7 digit dividend) and goes into long division (2, 3, or 4-digit divisor into a 4, 5, 6, or 7-digit dividend).

Many primary level children will get to this work and the short division with racks and tubes - generally between 5 1/2 and 6. Whether they get in primary or not, this material is heavily utilized and crucial in lower elementary on their work with long multiplication.

The main thing with this material is the connection to the writing. There is long bead frame paper that goes along with this material, for the children to begin their writing out of long multiplication - one sized for the short bead frame; the other sized for the large bead frame.

They are introduced to the bead frame with nomenclature first; then addition and subtraction just to get a feel for how it works and how the writing on the paper works.

Then they are introduced basic multiplication - 52 taken 3 times for example. They are writing the steps on paper and doing it on the bead frame, so they understand what they are writing and when they are exchanging, WHY they are exchanging or carrying. At this point, they might do a number such as 54 taken 3 times.

Once they are comfortable, they are given longer and longer multiplication examples, until they are making up their own (our goal is for them to create their own numbers, so that they own those numbers and will pay more attention to how they work, what are the patterns, what are the results).

And they begin to add zeroes into their writing.

Before you know it, they are writing out long multiplication problems without the use of the bead frame or a calculator.

AND they are prepared for the greater task of long division!

UPDATED TO ADD due to a small number of private messages received: Yes, there are a couple (?) of album sets you can purchase that do not introduce this material until the 3rd year of lower elementary and seem to indicate that it is an upper elementary work. AMI albums sit this material firmly at 5-8 years of age for normally functioning children, with the children working with it as long as they need (through any age). The fact is, upper elementary children are getting "less fiddly with the beads" (as quoted by one of my AMI trainers) and should really be far beyond the introductory stages to this material if they have had Montessori math since at least 5 1/2; still working with it as needed, but not JUST receiving it at age 8. If just getting it at age 8, they won't be doing LONG division until at least age 9, perhaps even 10, long after kids in the public schools have already started it; and Montessori has prepared them from the time they were 3 if they've been with Montessori that long - even starting at 5 1/2, there will be minimal slowing down in the early mathematics presentations. The big question is, if they're not doing long multiplication stages and long division stages at age 6 and 7, what have they been doing?
NEW children (starting Montessori after age 6 1/2 or 7) and children with severe learning impairments will follow a later schedule. THAT IS OKAY - but they are the exception and their needs should be considered when "Following the Child".