How Many Are Hiding?¶

Strategy · All strategies
Evidence · 2 claims (2 for) · 7 studies (2 causal, 2 quant-synthesis, 2 review, 1 associational), q2–q4 · 1 of 7 report an effect size

Description¶

"How Many Are Hiding?" is a small-group number-composition activity for early childhood mathematics. A known total of counters or cubes (e.g., 7) is established, some are hidden under a cup, and the visible remainder (e.g., 3) is shown; children must infer the hidden quantity (4) by reasoning about the part–whole relationship. The activity is typically played in pairs or small groups, with children taking turns hiding and guessing, and explaining how they knew.

Design Implications¶

The activity targets the part–whole schema — the understanding that a whole number can be decomposed into parts that recombine to the same total — which underlies fluency with basic facts, missing-addend problems, and early algebraic reasoning. Because the total is fixed and visible counting of the hidden set is impossible, children must use derived strategies (counting on from the visible part, known complements to 5 or 10) rather than direct enumeration, pushing them beyond count-all [~M]. Verbalizing "how I knew" adds a self-explanation [+S] component that converts a guessing game into explicit reasoning about number relationships.

Context¶

Requirements¶

• Counters, cubes, or other countable objects plus an opaque cup or hand
• A total small enough to hold in working memory (typically 5–10 for kindergartners, extending to 20 as fluency grows) — chunking [+S] the total into known complements reduces load
• Norms for explaining reasoning aloud, not just stating an answer
• Repeated rounds with varied totals so children encounter many decompositions of the same whole

Constraints¶

• If totals exceed children's counting range or working-memory capacity, children revert to guessing, and the activity produces random answers rather than reasoning [-M]
• If the visible part is always shown first in the same way, children may learn a procedure ("subtract from ten") without connecting it to the part–whole structure [~W]
• Children who have not consolidated cardinality (understanding that the last count word names the set size) are not ready; the activity presumes reliable counting and one-to-one correspondence [-M]
• Overly competitive or fast-paced formats encourage answer-stating over strategy talk

Implementation Variability¶

• Finger version: children show a number of fingers, hide some behind their back — supports eventual use of finger patterns for arithmetic
• Ten-frame version: the visible part is arranged on a ten-frame, leveraging structural subitizing of 5-and-some-more patterns
• Reverse round ("Show me"): the hider announces the hidden amount and peers check by lifting the cup — shifts from inference to production
• Recording extension: children record the equation (3 + 4 = 7) after each round, linking the physical activity to symbolic notation

Target Learners¶

• Kindergarten through Grade 2 children developing number-composition fluency within 10 or 20
• Children who count reliably but still use count-all strategies; the hidden quantity forces more efficient strategies [~M]
• Less suitable for children still acquiring cardinality, who need activities with fully visible, countable sets first

Target Learning Goals¶

• Conceptual: part–whole relationships and decompositions of numbers within 10/20
• Procedural: fluency with complements (e.g., all pairs making 7) supporting later addition/subtraction fact mastery
• Strategic: choosing between counting on, known facts, and derived-fact strategies
• Discourse: explaining and justifying numerical reasoning

Instructions¶

1. Establish the total: count the set of cubes together so every child knows the whole (e.g., 7).
2. One child hides part of the set under the cup while peers close their eyes.
3. Reveal the visible part; peers determine the hidden amount and state it with justification ("I knew it was 4 because 3 and 4 make 7" or "I counted on from 3").
4. Lift the cup to check; the hider confirms or corrects.
5. Rotate roles and vary the total across rounds; after several rounds, ask the group to list all the ways to make the total, connecting rounds into a decomposition chart.
6. Extend by recording equations, moving to a ten-frame representation, or increasing the total as fluency develops.

Related Strategies¶

• Quick Look cards (subitizing routines) — builds the perceptual recognition of small quantities that children then combine in hiding games
• Ten-Frame flash routines — a complementary structural representation of the same complements
• Number Talks — the whole-class discourse format in which hiding-game strategies can be shared and compared

Patterns That Use This Strategy¶

• Cognitively Guided Instruction — the activity elicits children's informal strategies, which the teacher builds on rather than replacing

Examples¶

• Investigations in Number, Data, and Space (TERC) — Grade K/1 units include "how many are hiding"-style games (e.g., "Peek-a-Plus," "Counters in a Cup") as core number-composition activities. https://investigations.terc.edu
• DreamBox Learning — digital adaptation in which a known number of tokens is partially hidden and students drag to complete the set, with the software tracking which decompositions have been mastered. https://www.dreambox.com
• Everyday Mathematics (McGraw Hill) — "Disappearing Train" and similar hidden-quantity games in the Kindergarten and Grade 1 number-and-numeration units. https://www.mheducation.com/everydaymath

Key Sources¶

• Siegler, R. S., & Ramani, G. B. (2009). Playing linear number board games—but not circular ones—improves low-income preschoolers' numerical understanding. Journal of Educational Psychology, 101(3), 545–560. doi:10.1037/a0014239
• Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction. Heinemann.
• Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach (2nd ed.). Routledge. doi:10.4324/9780203520574
• Baroody, A. J. (1987). The development of counting strategies for single-digit addition. Journal for Research in Mathematics Education, 18(2), 141–157. doi:10.2307/749248