Glossary · Montessori Math

Absorbent mind: Maria Montessori's name for how a young child (roughly birth to six) soaks up the world — language, order, number — without effort or drilling. It's why a four-year-old can take in the whole decimal system through beads, joyfully, while the same content taught later takes real work.
Abstraction, passage to: The long bridge from doing math with objects to doing it with written symbols alone. On this site it's an entire strand: stamp game, then bead frames, then checkerboard and racks & tubes — each material a little less concrete than the last, until pencil and paper is all the child needs.
Album: A Montessori teacher's handbook: every lesson written out step by step, in sequence, with its aims and the exact words to say. The lessons here are album pages rewritten for parents.
Bead bar: Beads strung on a stiff wire in a fixed quantity from 1 to 10, colored by quantity (see colored bead stair). A bar of six is one object and six units at the same time — which is exactly the idea multiplication will need.
Bead cabinet / bead chains: Long chains of bead bars used for skip counting: the short chain of 5 folds up into five 5-bars — a square of 25 — and the long chains do the same for cubes. In a classroom they hang in a tall glass "bead cabinet"; here they live at bead chains.
Bird's-eye view: A layout that lets the child take in a whole system at a glance — most famously the number cards from 1 to 9,000 laid out in columns by place, which shows in one look that after nine of anything you must move to the next place.
Colored bead stair: The bead bars 1 through 9 in their fixed colors: 1 red, 2 green, 3 pink, 4 yellow, 5 light blue, 6 lavender, 7 white, 8 brown, 9 dark blue (and 10 is always golden). Once the colors are learned, a child reads "eight" off a brown bar without counting — see the bead stair introduction.
Control of error: The way a material lets the child check their own work with no adult verdict needed: leftover beads at the end of the snake game, a skittle that came up short, number cards that don't stack flush. Correction stays impersonal, and the child stays in charge. Every lesson on this site names its control of error.
Decimal system: Our base-ten place value system: ten of anything makes one of the next thing up. Montessori gives it concretely, thousands and all, with the golden beads at age four or five — years before written arithmetic depends on it.
Direct aim / indirect aim: Every lesson has both. The direct aim is what the child is visibly learning now — say, adding four-digit numbers. The indirect aim is the seed planted for later: the logic of carrying, met in the hands years before it shows up in a written algorithm.
Dynamic vs. static: Montessori's words for operations with or without exchanging. Static addition never needs a carry (3,241 + 1,325); dynamic addition does. Every operation is presented static first, so the operation itself is the only new thing — see isolation of difficulty.
Exchange: The ten-for-one trade at the heart of the decimal system: ten unit beads back to the bank for one ten-bar, ten ten-bars for a hundred-square. Carrying and borrowing are simply exchanges written down — watch it happen in golden bead addition.
Extension: A follow-up that stretches a material past its first presentation — bigger numbers, a memory game, a new question asked of the same beads. Lessons here list their extensions so the material keeps earning its shelf space.
Golden beads: The flagship material of the decimal system: single golden beads for units, ten-bars, hundred-squares, and thousand-cubes. A thousand is a cube you need two hands to carry — the child feels how big numbers are before ever writing them. Start with the golden beads introduction.
Isolation of difficulty: The design rule that each new lesson adds exactly one new hard thing. Quantity is taught before symbol; static operations before dynamic ones. If a child struggles, the fix is usually a step back to isolate the piece that's actually new.
Materialized abstraction: Montessori's own description of her math materials: an abstract idea given a body. Place value is an abstraction; a thousand-cube is place value you can hold.
Planes of development: Montessori's four six-year stages of childhood. For math, the first plane (0–6) wants to touch and count; the second plane (6–12) wants reasons, patterns, and enormous numbers — which is why the same beads keep returning with bigger questions.
Point of interest: The detail in a presentation that catches and holds the child's attention — a bead clicking into its hole, cards stacking to reveal a number. Lessons here list their points of interest so you know exactly where to slow down.
Practical life: The pouring, spooning, and buttoning work of early childhood. It looks like housekeeping, but it is math preparation: exact sequences, left-to-right and top-to-bottom order, sustained concentration, and a hand under fine control.
Presentation: A Montessori lesson: the adult shows the material slowly, with few words, then hands it over. Each lesson on this site scripts one presentation step by step — our how to present a lesson guide covers the general craft.
Quantity vs. symbol: The deliberate split between an amount itself (six beads on a wire) and its written sign ("6"). Montessori teaches each separately, then joins them — cards and counters is the classic union for the numbers 1 to 10.
Seguin boards (teen board & ten board): Two board sets named for Édouard Séguin. The teen board builds 11–19 by sliding a digit card over a printed 10, showing that "fourteen" is ten-and-four; the ten board does the same for 10 through 99.
Sensitive period: A window when a child is spontaneously hungry for one kind of learning — order, counting, huge numbers — and absorbs it with unusual ease. This is why the ages on this site are readiness ranges, not deadlines.
Skittle: A small pawn-shaped figure that stands for a person in division work. Twelve beads dealt fairly to three skittles, one round at a time — each skittle's share is the answer. See the division board.
Snake game: Colored bead bars laid end to end in a winding "snake," then counted and traded, ten by ten, into golden bars. It's addition facts and making-ten disguised as a game — the snake game lesson shows the whole ritual.
Stamp game: Place value shrunk down to small tiles the child moves alone: green 1s, blue 10s, red 100s, green 1,000s — the same colors as everywhere else in the sequence. It's the first step of the passage to abstraction, covering all four operations; start at the stamp game introduction.
Three-period lesson: The Montessori way to teach any new name, in three steps: "This is nine." (naming), "Show me nine." (recognizing — where most of the time is spent), and only last, "What is this?" (recalling). If period three wobbles, you simply return to period two another day. Details in how to present a lesson.
Variation: A different way to do the same work at the same level of difficulty — new numbers, swapped roles, a changed layout. Variations keep repetition fresh, and repetition is where the learning actually happens.
Work cycle: The full arc of one work session: the child chooses the work, carries it to the table or rug, works and repeats as long as interest lasts, then returns everything to order. Protecting that cycle — above all, not interrupting it — matters more than the math on any particular day.

Two more good starting points: the Montessori math overview for the big picture, and the scope & sequence for where every lesson fits.