What the cubes actually are
A small wooden cube, hinged or opened like a box. Inside, the cube separates into eight pieces.
The pieces are not identical. They are colour-coded: three pieces are red (representing a³), three are blue (representing b³), and the remaining two are black (representing a²b and ab² in various orientations, actually three combined into colour groups; the standard AMI colouring distinguishes pieces carefully).
Precisely: for the Binomial Cube (a+b)³ = a³ + 3a²b + 3ab² + b³, the eight pieces are one red cube (a³), three black rectangular prisms (3a²b), three black prisms in a different proportion (3ab²), and one blue cube (b³). The colour coding distinguishes the identity's terms.
When the cube is correctly assembled, the outer faces show a 2×2 grid of coloured squares on each face: two red-family squares and two blue-family squares arranged in the same pattern. The cube looks deceptively simple from the outside; the inside mathematical identity is the point.
The Trinomial Cube extends the principle to (a+b+c)³ = a³ + 3a²b + 3a²c + 3ab² + 6abc + 3b²c + 3bc² + c³. Twenty-seven pieces. Harder puzzle; same concept.
The three-year-old's work
Invite the child. Open the box. Tip the pieces gently onto a work rug. Sit beside.
Take one piece. Find where it goes in the box, by looking at the colours on the top face of the box (which remains intact, showing the colour pattern). Place the piece. Take another. Fit it next to the first.
The child watches and then takes over. Their work is a puzzle: match the coloured face of each piece to its place in the grid on the top of the box, then fit the piece into the correct position underneath. With repetition the child learns the pattern; after several sessions they can rebuild the cube without reference to the box top.
No algebra is mentioned. The child is doing what a child does with a puzzle: matching, fitting, repeating, closing. The sensorial aim is the structure itself; the absorption happens below the level of language.
The Trinomial Cube is introduced a year or two later. Same work, more pieces, more demanding colour matching.
Why this works
Because the child's hands learn things that their words have not yet reached.
Maria Montessori observed that children who had worked with the Binomial and Trinomial Cubes as three- and four-year-olds showed, years later, a striking ease with the algebraic identities those cubes represented. When a Plane 2 child is asked to expand (a+b)² on paper, a child with Binomial Cube experience often responds with something like "oh, that's two squares and two rectangles" and draws the expansion. The concrete referent is there; the algebra becomes a re-naming of something the child already knows.
This is the logical extreme of the Montessori principle of indirect preparation. A sensorial material that looks like a colour-matching puzzle for a young child is, simultaneously, an embodied algebraic identity that will be called back in six years' time. The child's developmental path threads through both.
Presentation and extensions
The canonical presentation is short.
Invite. Carry the cube to the work rug, with both hands. Open the box. Tip the pieces out. Sit. Study the colour pattern on the top face of the box (which is a 2×2 grid of coloured squares, exposed when the box is closed). Pick up one piece; study its top face; locate the matching square on the box top; place the piece in that quadrant. Repeat for each piece. Close the cube. Pause. Invite the child.
Extensions.
Building outside the box. Once the child can rebuild inside the box reliably, introduce building the cube outside the box on a mat, using only the coloured-face relationships.
Combining with the Trinomial Cube. Both cubes side by side; the child notices the scaling.
Three-period lesson for colour-face names (Plane 2). "This face is red-red-blue-blue. What would be on the face opposite it?" Begins to formalise the structure.
Writing the expansion (Plane 2). The child, now ten or so, writes out (a+b)³ and connects each term to the coloured pieces.
Common home mistakes
Explaining the algebra to the three-year-old. The algebra is for Plane 2, not now. The child's work at three is sensorial puzzle assembly; naming the algebra would confuse rather than accelerate.
Leaving the cube permanently assembled as decoration. The work is in the assembly. A cube that sits as an ornament is not being worked with. Rotate off the shelf if unused.
Buying a cheap version where the precision is off. The pieces must fit precisely; sloppy manufacture produces pieces that fit "almost anywhere" and the colour-matching work fails. Buy from reputable UK suppliers (Absorbent Minds, Montessori Materials UK) or from reliable Etsy makers. Commercial Binomial Cubes cost £30-60; Trinomial Cubes £60-120.
Treating it as advanced material. It is sensorial, not advanced. A three-year-old who is doing Pink Tower work is ready. Many home families delay the Binomial Cube thinking it is "algebra work"; the delay misses the window.
A real family with both cubes
A mum we will call Thandi introduced the Binomial Cube to her daughter at three and a half, a month after the Pink Tower and Brown Stair were consolidated. She bought a reasonable-quality Binomial Cube new for £42.
Her daughter took to it immediately; the colour-matching puzzle was satisfying. Within two weeks she could rebuild the cube without looking at the box top. For the next year she returned to it every few weeks, sometimes combining it with the Pink Tower and Brown Stair in elaborate arrangements on the floor.
The Trinomial Cube was introduced at five, bought second-hand for £55. Harder; took longer to consolidate. By six her daughter could rebuild it, though less often than the Binomial.
Seven years later, her daughter at ten in Plane 2 was introduced to algebraic expansion by Thandi. The child's response to (a+b)² was to pick up the Binomial Cube pieces (still in a cupboard) and arrange them on the table in the pattern of the expansion. "Oh, it's two squares and two rectangles." Thandi has a photograph of that moment. The seven-year gap between the child's sensorial absorption and the adult's naming of the algebra is Montessori's signature arc in a single family.