Derived Facts¶

Strategy · All strategies
Evidence · 2 claims (1 for, 1 against) · 5 studies (2 causal, 2 review, 1 quant-synthesis), q2–q4 · 1 of 5 report an effect size

Description¶

The derived facts strategy asks learners to compute unknown facts by reasoning from known ones — for example, deriving 7 × 8 from 7 × 7 + 7, or 9 + 6 from 9 + 1 + 5. Rather than drilling every fact to immediate recall, learners build a small core of automatic facts and a repertoire of decomposition and compensation strategies to derive the rest. Over repeated use, derivation itself becomes fast, and many derived facts consolidate into direct recall.

Design Implications¶

Derived facts leverage existing knowledge as a scaffold for new knowledge, reducing the memory burden of learning large fact sets by brute repetition Chunking reduces working memory load. [+M]. The strategy builds number sense and flexibility alongside fluency: learners who derive facts understand why the answers are correct, which supports transfer and error correction [~S]. It also functions as formative assessment — the derivations a learner chooses reveal their conceptual understanding.

Context¶

Requirements¶

• A secure base of known facts (typically small doubles, tens facts, or 2s, 5s, 10s in multiplication)
• Explicit instruction in a small set of derivation strategies (near doubles, adding one, compensation, partitioning)
• Timely practice so derivations become efficient rather than laborious (Practice)

Constraints¶

• If the known-fact base is weak, derivation overloads working memory and produces errors Cognitive overload degrades learning. [-S]
• Learners who remain slow derivers under time pressure may experience frustration and avoid the strategy; fluency development must not be neglected [~M]
• Over-reliance on counting-based derivation (e.g., counting on by ones) predicts persistent low achievement and should be redirected toward decomposition [~S]

Implementation Variability¶

• Interview-based diagnosis: teachers ask "How did you figure that out?" to identify which facts are known and which strategies are available (the basis of Cognitively Guided Instruction)
• Strategy modeling: the instructor demonstrates a derivation aloud, then prompts learners to generate alternative derivations for the same fact
• Fact families: presenting related facts together (6 + 7, 7 + 6, 13 − 6, 13 − 7) so each known fact supports several others

Target Learners¶

• Elementary mathematics learners moving from counting strategies to reasoning-based recall
• Learners with gaps in fact knowledge who need a functional workaround while fluency develops
• Less appropriate for learners who already have automatic recall — derivation adds unnecessary steps [~M]

Target Learning Goals¶

• Fluency with basic facts grounded in understanding rather than rote memorization
• Conceptual structures: properties of operations (commutativity, distributivity, compensation)
• Strategic competence: selecting and adapting computation methods flexibly

Instructions¶

1. Diagnose the learner's known facts and current strategies through a brief oral interview; record which facts are automatic, derived, or counted.
2. Select an unknown fact that is near a known fact (e.g., 8 + 6 near the double 7 + 7, or 6 × 7 near 6 × 5).
3. Model a derivation with explicit reasoning (Think-Aloud), making the relationship between known and unknown facts visible.
4. Ask the learner to derive the same fact a second way, and compare the strategies (Comparing Cases) to build flexibility.
5. Repeat the derived fact in varied contexts over subsequent sessions until it is recalled directly; log progress and retire mastered facts.

Related Strategies¶

• Activating Prior Knowledge — derivation is a within-task application of the same principle: new knowledge built from retrieved knowledge
• Spaced Retrieval Practice — the consolidation mechanism that converts derived facts into automatic recall

Examples¶

• Cognitively Guided Instruction (CGI) — teachers use children's derived-fact strategies as the basis for instruction in number and operations; see Cognitively Guided Instruction.
• Number Talks (Parrish, 2010) — daily classroom routines in which students mentally compute a fact or problem and share their derivation strategies, making multiple derivation paths public.
• Math Recovery — one-to-one intervention programs that assess whether a child derives or counts, and target instruction at moving them toward derived-fact strategies.

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
• Siegler, R. S. (1988). Strategy choice procedures and the development of multiplication skill. Journal of Experimental Psychology: General, 117(3), 258–275. doi:10.1037/0096-3445.117.3.258
• Dowker, A. (2005). Individual differences in arithmetic: Implications for psychology, neuroscience and education. Psychology Press. doi:10.4324/9781315755526-2
• Parrish, S. D. (2010). Number talks build numerical reasoning. Teaching Children Mathematics, 18(3), 198–206.
• Baroody, A. J. (2006). Why children have difficulties mastering the basic number combinations and how to help them. Teaching Children Mathematics, 13(1), 22–31. doi:10.5951/tcm.13.1.0022