Counting On¶

Strategy · All strategies
Evidence · 1 claim (1 for) · 3 studies (2 review, 1 causal), q2–q3 · 0 of 3 report an effect size

Description¶

Counting On is a foundational mental arithmetic strategy for addition in which the learner begins at the value of one addend and counts forward the number of steps indicated by the second addend (e.g., for 5 + 3: "6, 7, 8"). It replaces the less efficient "counting all" strategy, in which children count both sets from one. It is typically taught in early elementary mathematics, often through finger counts, number lines, or counters, and serves as a bridge between concrete counting and derived-fact or recall-based addition.

Design Implications¶

Counting On reduces working memory demand relative to counting all by shortening the count sequence Chunking reduces working memory load. [+M]. Its instructional value depends on learners first understanding that the starting addend represents a completed count — a conceptual insight, not just a procedure. Research on children's strategy development shows that explicit strategy instruction is most effective when learners understand why the shortcut works and when it applies [~S].

Context¶

Requirements¶

• Learners can already count forward fluently from arbitrary starting points, not just from one
• Learners understand cardinality — that the last count word names the size of the set
• Concrete or visual support (number lines, counters, fingers) during initial instruction
• Explicit modeling of the strategy with reasoning made visible (Think-Aloud)

Constraints¶

• Inefficient for addends larger than about 3–4; counting on 8 from 12 taxes working memory and invites errors [-M]
• Can become a procedural crutch that delays movement toward decomposition and fact retrieval if practiced exclusively [-M] — children who remain at the counting level past grade 2 show slower arithmetic development
• Requires solid cardinality; teaching it before learners grasp that the first addend is a set size produces rote, error-prone counting [-M]

Implementation Variability¶

• Count on from the larger addend ("counting on from the greater number") — more efficient, but requires learners to recognize commutativity
• Count on the number line vs. with fingers vs. with rhythmic tapping — each shifts the representational load
• Embedded in word problems, where learners choose the strategy themselves, as in Cognitively Guided Instruction

Target Learners¶

• Early elementary learners (typically ages 5–7) transitioning from counting all to more efficient strategies
• Learners who persist in counting all despite having the counting skills needed for the shortcut
• Less appropriate for learners who have already automated basic facts; continued counting practice is redundant and may interfere with retrieval [~M]

Target Learning Goals¶

• Procedural fluency: efficient single-digit addition
• Conceptual foundation: implicit grasp of commutativity and the relationship between counting and addition
• Pathway toward Automaticity with basic facts

Instructions¶

1. Diagnose current strategies — observe whether learners count all; do not teach the shortcut to learners still building cardinality.
2. Model the strategy with Think-Aloud on a visible quantity: "I have 5. I keep 5 in my head and count 3 more: 6, 7, 8."
3. Practice with small addends (1–3) using fingers, counters, or a number line, fading supports as fluency grows (Fading).
4. Introduce counting on from the larger addend to build toward commutativity.
5. Push toward decomposition and derived facts (e.g., 5 + 3 as 5 + 5 − 2) so counting remains a transitional, not terminal, strategy.

Related Strategies¶

• Counting All — the precursor strategy Counting On replaces
• Doubles Plus One — a derived-fact strategy that typically develops after Counting On
• Make Ten — a decomposition strategy that supersedes counting for addends near 10
• Teach struggling students a counting-up strategy for calculation and missing-number word problems

Examples¶

• Cognitively Guided Instruction (CGI) classrooms treat Counting On as one of several observed strategies children invent; teachers elicit and build on it rather than imposing it, following research by Carpenter and colleagues on children's mathematics (CGI for Math).
• Number Talks (Parrish, 2010) — teachers record Counting On alongside other student strategies on the board, making strategy choice itself an object of discussion.
• First-grade addition units in most core curricula (e.g., Investigations in Number, Data, and Space) sequence counting all → counting on → derived facts across the year.

Key Sources¶

• Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.-P., & Loef, M. (1989). Using knowledge of children's mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26(4), 499–531. doi:10.3102/00028312026004499
• Fuson, K. C. (1992). Research on whole number addition and subtraction. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 243–275). Macmillan.
• Siegler, R. S., & Jenkins, E. (1989). How children discover new strategies. Lawrence Erlbaum Associates.
• 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