What is the Stamp Game, physically?
A shallow wooden tray or box divided into four compartments. Each compartment holds a stack of small square tiles, about 2.5cm on a side, in one of four colours.
Green tiles marked with a "1" are units. Blue tiles marked with "10" are tens. Red tiles marked with "100" are hundreds. Larger green tiles marked with "1,000" are thousands.
The colours match the Golden Beads exactly (green units, blue tens, red hundreds, green again for thousands). The child who has worked with the beads recognises the colour system on first sight. The stamps themselves are smaller and, crucially, each tile is one tile; the child does not count anything on it. "10" on a blue tile is just a symbol for ten. The child trusts it.
This is the conceptual leap. The Golden Beads were literally ten unit-beads strung together; the child could count them. The stamp is a symbol standing in for that quantity. The bridge from literal to symbolic is happening on this tray.
The same operations, done with symbols
The child works on a mat, laying out stamps in four columns corresponding to the four place values (thousands, hundreds, tens, units, left to right). A number like 2,453 is laid out as two thousand-stamps, four hundred-stamps, five ten-stamps and three unit-stamps.
Addition is the combining of stamps across two numbers. When the unit column exceeds ten, the child exchanges ten unit-stamps for one ten-stamp. The logic is identical to the Bank Game; the execution is now in symbols.
Subtraction is the removal of stamps. When the unit column does not have enough stamps to remove, the child "borrows" from the ten column: one ten-stamp is exchanged for ten unit-stamps, the subtraction continues. This is column-subtraction with full underlying understanding.
Multiplication is the combining of multiple copies of a number. Four times 2,453 is four layouts of 2,453 combined into a single row. The result, with exchanges, is read off.
Division is the sharing of stamps equally into rows representing "how many shares". 6,842 divided by three is six thousand-stamps shared into three rows (two each), eight hundred-stamps shared into three rows (two each with two left over; the two become twenty ten-stamps and join the ten column), and so on.
The Stamp Game takes the child through all four operations of arithmetic with dynamic exchanging in symbolic form. By the end of Stamp Game work, a six-year-old is doing four-digit column arithmetic on paper with ease, because the paper operations are the same as the stamp operations with different-looking tokens.
The bead frames
After the Stamp Game, two further materials continue the passage.
The small bead frame. A wooden frame with four horizontal wires. Each wire holds ten beads. The wires represent, top to bottom, units, tens, hundreds, thousands. Beads are slid left-to-right to form a number: three beads on the unit wire and two beads on the ten wire represents 23.
The small bead frame is narrower than the stamps; it introduces the idea of column-notation. A number has a specific horizontal position on the frame. The child records the same number on paper in column form.
The large bead frame. A bigger frame with seven wires: units, tens, hundreds, thousands, ten thousands, hundred thousands, millions. This is where the child meets numbers beyond the Golden Bead range. Operations continue, with the same exchanging mechanism.
The large bead frame is also where the child starts to record calculations on paper with accompanying stamp or bead work. By the end of this sequence the "paper" is just a record of what has already happened on the frame or tray; the paper calculation and the physical operation move in step.
Why the bridge cannot be skipped
Parents sometimes see a child doing well with the Golden Beads and move directly to written column arithmetic. The child stalls; the written columns look foreign after beads.
The Stamp Game is the specific bridge. It keeps the four-column layout, the colour coding and the exchange logic, but reduces the physical representation to symbols. A child who has done months of stamp work sees a written "26 + 14" in columns and does not need to be told where to start; the columns are already familiar as the stamp columns.
Skipping this step is the single most common reason a Golden-Bead-proficient child then struggles with primary-school column arithmetic. The physical understanding was solid; the abstract representation had not been built.
The dot game
A smaller material, sometimes introduced alongside the Stamp Game. A gridded paper with columns for units, tens, hundreds and thousands; the child marks dots in the correct column when adding.
For addition: "three thousand, four hundred and fifty-two plus two thousand, seven hundred and sixty-four". The child marks three dots in the thousand column, four in the hundred column, five in the ten column, two in the unit column. Then two dots in the thousand column, seven in the hundred column, six in the ten column, four in the unit column. They count the dots in each column. When a column exceeds nine, they cross off ten dots and add one dot to the column to the left. The final result is read off.
The dot game is the closest to written column arithmetic but still uses a dot (a quantity) rather than a digit. It is often the last step before pure paper-and-pencil column operations.
Long division with the division boards
The culmination of the passage to abstraction for 3-6 maths. The long and short division boards (sometimes called the racks and tubes in more elaborate sets) are frames on which the child lays out long-division operations with coloured beads representing the divisor and the dividend.
The child performs multi-digit division with remainders on the board, recording the work on paper alongside. By the end of this material the child has done long division with multi-digit divisors (twelve divided into one thousand three hundred and twelve; three thousand four hundred and fifty-six divided by twenty-three), with full underlying understanding.
Most home Montessori families reach the short division board in the later part of Plane 1 and do the long division board in Plane 2, often alongside or after transitioning to a mixed maths approach.
Common home mistakes
Moving to the Stamp Game before the Golden Beads are consolidated. The Stamp Game assumes the child already understands place-value exchange physically. A child who has not yet done sustained Bank Game work tries to use the Stamp Game as if the stamps were countable quantities, which confuses them.
Skipping the Stamp Game altogether. Covered above. The single most expensive skip in the passage to abstraction.
Treating the bead frames as toys. The small and large bead frames are specific materials with specific progression. Sliding beads around randomly without the paper-recording is not the work.
Not recording operations on paper as the child progresses. The Stamp Game to bead frames to paper sequence depends on the child seeing the same operation in three forms. If only the physical is ever done, the paper version remains foreign.
Rushing through the stages. Each stage typically takes months, not weeks. The child who is given enough time through Stamp Game and bead frames never has to "learn column arithmetic" as a separate topic later; they have already done it.
A real family's year through the Stamp Game
A mum we will call Leah started the Stamp Game with her six-year-old son after about a year of Golden Bead work. She bought a Stamp Game second-hand for £45, which included a small bead frame.
Months one to two: introducing the stamps and their colour coding. He found the shift from physical beads to symbolic tiles briefly disconcerting but within a few sessions was laying out four-digit numbers confidently.
Months three to five: addition and subtraction with exchange, following exactly the Bank Game logic he had internalised from the Golden Beads. By the end of month five he was doing four-digit dynamic addition and subtraction with stamps faster than with beads.
Months six to eight: multiplication and introduction of the dot game. The dot game moved him one step closer to paper arithmetic; he began recording stamp-game operations on lined paper in column form.
Month nine: introduction of the small bead frame. He spent several sessions simply exploring the frame before beginning recorded operations.
By the end of the year, Leah's son was doing four-digit column addition and subtraction on paper directly, without beads or stamps, with reliable accuracy. The multiplication had moved to pencil-and-paper with the Stamp Game still available as a backup. Division was in early stages.
Cost across the year: £45 for the Stamp Game and small bead frame. Time investment: about three sessions per week, twenty minutes each.