Counting to 120 Without Place Value
Aiden, 4 years 9 months, rote-counts to 120, recognizes numerals to 100, and completes the teen board quickly, saying “ten-one, eleven” as he goes. When the guide gives him the large number cards and asks him to build 67 with the golden beads, he lays out six ten-bars and seven units, then counts every piece aloud and announces “thirteen.” He can point to 13 when asked but cannot bring 13 beads on request. A substitute gave him one stamp game lesson last week, and he now asks for it daily. The guide’s records show that the golden bead quantity presentation happened once in November and that the exchange game has not been given. The class has 26 children and a student teacher who is eager to “keep him moving.”
1. Diagnose what Aiden does and does not understand, and decide whether he should have the stamp game now. Justify the decision.
Read the worked answer
Aiden has the language and the symbols but not the quantity underneath them. Rote counting to 120 and rapid teen board work demonstrate names and recognition; counting “thirteen” after laying out six tens and seven units shows he is counting pieces rather than reading place value, and that the numeral 67 is not yet attached to an amount. The teen board teaches the names of the teens; it does not teach the decimal system, and it is not a test of it.
So the stamp game should wait, even though he asks for it daily. Moving on now would give him another procedure to perform without understanding, which is how gaps become habits. Go back to the golden beads and rebuild the link between quantity, symbol, and place: bring-me games (“bring me 34”), the three-period lesson on quantity, the large card layout with 1000, 100, 10, and 1, then the 45 layout, then the exchange game (“ten units for one ten”). Only then the stamp game, which will then be a pleasure rather than a trick.
2. How would you sequence the next six weeks, and how would you bring the student teacher and the substitute into line without a confrontation?
Read the worked answer
Six weeks, sequenced and documented. Weeks 1–2: golden bead quantity and the bring-me game daily, ten minutes, recorded each time. Week 3: large number cards and the 45 layout, building numbers from cards and beads together. Week 4: the exchange game, with the child making the trades. Week 5: return to 67 and similar numbers, asking him to read the quantity before counting anything. Week 6: reassess with the same task he failed, and only if he reads value correctly introduce the stamp game.
With the adults, be direct and private. Show the student teacher the “thirteen” transcript and explain the sequence rule: a material is given when the child has the prerequisite, not when the child asks or when the adult is enthusiastic. Ask her to record every math presentation with a date and a one-line observation so you can see what he does afterward. With the substitute, send a short written note rather than a correction in front of the children: “Please hold the stamp game until I introduce it; he is still consolidating the golden beads.”