The arc
Laid out explicitly, from three to twelve.
Age three to four. Binomial Cube introduced. The child rebuilds it as a colour-matching puzzle. They absorb the structure of (a+b)³ physically without any mathematical language.
Age four. Trinomial Cube. Same principle, three terms.
Age four to five. Golden Bead material. The child holds one unit, ten units on a bar, a hundred units on a square, a thousand units on a cube. The decimal-system structure of place value is in their hands. This is algebraic in the sense that they are manipulating base-10 expressions: 3a + 4b where a = 100 and b = 10.
Age five to six. Bead Chains. The short chains for each number 1 to 10. The child walks the 5-chain (five 5-bars), folds it into the 5-square, counts 25 beads. They physically build n² for each n from 1 to 10. They also build cubes by stacking squares: five 5-squares stacked is 5³ = 125 beads.
Age six to seven. Bead Chain extensions. The 100-chain folds into the 100-square; the 1000-chain into the 1000-cube. The relationships between numbers and their squares and cubes are physical.
Age seven to eight. Metal Fraction Insets. The child learns fraction operations physically. These are linear expressions in fraction form; the algebraic structure (a/b + c/d = (ad + bc)/bd) is what they are discovering empirically.
Age eight to nine. The Checkerboard for multi-digit multiplication. A grid with coloured squares representing place values and powers of ten. The child does multi-digit multiplication on this board, and the grid structure prepares for polynomial multiplication (multiplying two algebraic expressions).
Age nine to ten. The hierarchical material. Very large and very small numbers laid out in graduated cubes, from units through thousands and millions to billions and beyond. Powers of ten made visual.
Age ten to eleven. Named algebra. The child is introduced to letters standing for numbers. (a+b)² = a² + 2ab + b² is written on paper. The Binomial Cube comes off the shelf; the child recognises the expansion as the pieces of the cube they have known for seven years.
Age eleven to twelve. Solving equations. Beginning with balance puzzles ("what number plus 3 equals 7?") and progressing to formal equations. The child who has held a ten-bar in one hand and three unit-beads in another already understands that adding and subtracting the same quantity from both sides of an equation keeps them equal.
What the concrete materials are teaching, in algebraic terms
Most home parents are not comfortable articulating the algebraic content of the sensorial materials. This is not a problem. The child absorbs the structure without naming it. The article below is for the parent who wants to see the algebra under the materials, not for the child.
The Binomial Cube is a visual exploded form of (a+b)³. One red cube (a³), three long blue-red pieces (3a²b), three shorter blue-red pieces (3ab²), one blue cube (b³). Eight pieces; eight terms in the expansion.
The Trinomial Cube is the same for (a+b+c)³ = a³ + 3a²b + 3ab² + b³ + 3a²c + 3b²c + 3ac² + 3bc² + 6abc + c³. Twenty-seven pieces.
The Golden Bead material is the base-10 representation of numbers. The ten-bar is 10¹; the hundred-square is 10²; the thousand-cube is 10³. The child exchanging ten units for a ten-bar is doing 10 × 10⁰ = 10¹.
The Bead Chains short squares are the squares of integers 1² through 10². The short cubes are the cubes.
The Bead Chains long chains are larger powers of ten. 100-chain is 10² in bead form; 1000-chain is 10³.
The Metal Fraction Insets are linear expressions in fractional form. The operations the child learns are the operations on rational numbers.
The Checkerboard is a polynomial-multiplication grid. Multiplying 23 × 45 on the checkerboard is structurally the same as multiplying (2a + 3) × (4a + 5) symbolically, where a = 10.
The hierarchical material is a place-value representation of very large numbers. Powers of 10 from 10⁰ to 10⁹ or beyond.
What the home parent does
Sustain the materials. Present each at the right moment. Allow the child to work with them. Not explain the algebra until Plane 2.
The crucial parental move is not to accelerate. A seven-year-old who has done deep Binomial Cube work is better prepared for algebra than a seven-year-old who was told "this represents (a+b)³" too early. The hands know before the mouth does.
When algebra arrives at ten or eleven, the parent's role shifts. Now the naming matters. "Let me show you how to write down what the Binomial Cube is." "Let me show you how to add terms using the pieces." The child who has done the physical work for seven years accepts the naming as a reframing rather than as a lesson.
A realistic home approach
Not every home family will reach Plane 2 algebra with their child at home. Many will transfer to school or to mixed educational routes by eight or nine, and algebra will come from a different source. The Montessori pre-algebra preparation still pays off; the child's number sense and concrete understanding of structure makes secondary-school algebra easier.
Families who do continue to home-educate into Plane 2 have the full arc available. The materials are expensive (a full secondary-Montessori mathematics set can cost £500-1,500) and the adult's mathematical comfort matters more at this stage. Families without a strong maths background often combine Montessori concrete materials with a textbook-based algebra programme (Singapore, Saxon or Art of Problem Solving) for the written work.
The combination is usually stronger than either alone.
Common mistakes
Teaching algebra too early. The sensorial materials do the preparation; naming the algebra at four interrupts the absorption.
Skipping the Binomial Cube because "we're not doing algebra yet". This is precisely when the Binomial Cube does its most important work.
Rushing through the Bead Chains. The squaring and cubing work is foundational for pre-algebra; short cuts here cost you later.
Not connecting the materials when algebra arrives. When the child meets (a+b)² at ten, the parent's job is to pull the Binomial Cube back off the shelf and lay it out. The concrete-to-abstract link needs to be made visible; the child is unlikely to make it unprompted.
Replacing all Montessori maths with a school scheme at eight. The pre-algebra arc depends on continuing Montessori materials into Plane 2. A hard switch at eight loses the Checkerboard, the hierarchical material and the named-algebra transition.
A real family with a ten-year-old
A family we will call the Okazakis kept their son in home Montessori through Plane 2. He had done Binomial Cube at four, Trinomial at five, full bead chain work at six, fraction insets at seven, checkerboard at eight, hierarchical material at nine. At ten his dad introduced formal algebra.
The first lesson was the Binomial Cube. His dad said: "This is (a+b)³. Let me show you why." He wrote the expansion on a sheet of paper. Then he took out the cube, laid out the eight pieces, and said: "This piece is a³. These three are a²b. These three are ab². This one is b³." The son looked, looked back at the paper, and said "oh, that's the same".
Over a fortnight, the son worked through multiplying (a+b)(a+b) on paper, getting a² + 2ab + b², and then doing the same with the Trinomial Cube. By the end of the term, he was doing basic quadratic expansion and factoring on paper.
The Okazakis' son entered secondary school at eleven two full years ahead of the typical UK maths curriculum on algebra. He was not a "maths-talented" child in any obvious way; he had simply done seven years of the concrete preparation.