Number Relationships and Reasoning Strategies¶

Strategy · All strategies
Evidence · 2 claims (2 mixed) · 8 studies (3 causal, 3 quant-synthesis, 2 review), q2–q4 · 2 of 8 report an effect size

Description¶

Basic facts should be taught using number relationships and reasoning strategies, not rote memorization. Learners derive unknown facts from known ones using structured strategies — Making 10 (7 + 5 → 7 + 3 + 2 → 10 + 2), near doubles, compensation, and benchmark numbers — so that fluency emerges from understanding rather than preceding it. Because these strategies rest on number relationships and benchmarks, they support emerging conceptual understanding and flexibility, and they generalize: Making 10 with whole numbers becomes Make-a-Whole with fractions and decimals, and compensation strategies extend to multidimensional and mental computation.

Design Implications¶

Strategy-based fact instruction produces better retention and transfer than drill-only approaches because each fact is connected to a web of relationships rather than stored as an isolated verbal association Baroody et al. argue retrieval practice alone produces fragile, quickly forgotten fact knowledge. [~M]. The approach treats fluency as the outcome of reasoning about relationships, not the prerequisite for it, which reverses the traditional memorize-then-apply sequence. Strategy instruction must be paired with substantial practice in applying and choosing among strategies, or learners may know the strategy names without being able to execute them fluently.

Context¶

Requirements¶

• Thorough teacher preparation: instructors must understand the strategy landscape (doubles, Making 10, compensation, near doubles) and typical developmental progressions well enough to recognize and name strategies learners invent
• Instructional materials that make number relationships visible — ten-frames, number bonds, open number lines, Cuisenaire rods — rather than flashcards organized by fact family
• Extended time for exploration and discussion of multiple solution paths before efficiency is expected
• A classroom culture where explaining one's reasoning is routine (Class Discussion, Articulation)

Constraints¶

• Instruction is time-consuming and resource-intensive relative to timed-drill routines; teachers without deep number-sense training often default to memorization pressure [-M]
• If strategies are taught as procedures to memorize ("when you see 8+, make 10") rather than as reasoning tools, the approach collapses into a second layer of rote learning [-M]
• Learners with weak subitizing or part-whole understanding of numbers within 10 lack the prerequisites for Making 10 and need earlier work on composing and decomposing before strategy instruction can take hold [-M]
• Untimed, discussion-heavy instruction can leave some learners without the retrieval automaticity needed to free working memory for complex tasks; brief, low-stakes practice after strategies are established is still necessary Chunking known facts reduces working-memory load during computation. [~S]

Implementation Variability¶

• Invented strategies first: learners generate their own methods, which the teacher names and connects to canonical strategies (the Cognitively Guided Instruction stance)
• Explicit strategy teaching: the teacher demonstrates a strategy with manipulatives, then learners apply it (Direct Instruction sequencing)
• Game-based practice: strategy application embedded in games (e.g., "Make 10" card games) so retrieval practice follows reasoning rather than replacing it
• Adaptive sequencing: digital tools select facts for practice based on which relationships a learner has already secured (Adaptive Difficulty)

Target Learners¶

• Elementary learners in grades 1–4 acquiring addition and multiplication facts, the primary audience for this approach
• Older learners with persistent fact difficulties, for whom strategy-based reteaching addresses the relational gaps that caused memorization to fail [-M]
• Less appropriate as written for learners who already retrieve facts automatically; for them the value shifts to extending relationships to fractions, decimals, and algebra

Target Learning Goals¶

• Conceptual understanding of part-whole and benchmark relationships
• Procedural fluency with automaticity as an emergent outcome, not a gate
• Adaptive reasoning: selecting among strategies based on the numbers involved
• Foundation for later strategies (Make-a-Whole with fractions and decimals; compensation in algebra)

Instructions¶

1. State objectives — frame the goal as "use relationships to figure out facts," not "get faster."
2. Recall prior knowledge — secure prerequisites: part-whole relationships within 10, doubles, and benchmarks of 5 and 10, using ten-frames and number bonds.
3. Introduce one strategy through a context or manipulative, making the relationship visible (e.g., 8 + 6 on a ten-frame shows 2 moving from the 6 to the 8).
4. Have learners apply the strategy to a string of related problems, articulating each step (Articulation).
5. Compare strategies on the same problem and discuss which is most efficient and why (Class Discussion).
6. Follow with brief, distributed practice that requires strategy selection, not just retrieval Spreading practice over time improves retention. [+S].
7. Extend the relationship to new domains — Make-a-Whole for 0.7 + 0.4, compensation for 3.98 + 2.17 — to build transfer.

Related Strategies¶

• Activating Prior Knowledge — strategy instruction builds directly on secured relationships (doubles, benchmarks), so activating them before new facts is essential
• Analogies — Making 10 in whole numbers is the analogy for Make-a-Whole in fractions and decimals

Examples¶

• Making 10: 8 + 5 → take 2 from the 5 to make 10, leaving 10 + 3 = 13. Later: 0.8 + 0.5 → 1 + 0.3 = 1.3.
• Near doubles: 6 + 7 as 6 + 6 + 1, extending to 25 + 26 as 25 + 25 + 1.
• Compensation: 53 − 29 as 53 − 30 + 1, extending to 4.6 + 2.9 as 4.6 + 3 − 0.1.
• Cognitively Guided Instruction classrooms (Carpenter et al., University of Wisconsin) build fact fluency from learners' own invented strategies, with teachers eliciting and naming them rather than imposing procedures.

Key Sources¶

• Baroody, A. J., Bajwa, N. P., & Eiland, M. (2009). Why can't Johnny remember the basic facts? Developmental Disabilities Research Reviews, 15(1), 69–79. doi:10.1002/ddrr.45
• 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
• National Research Council. (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press. doi:10.17226/9822
• Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2015). Children's Mathematics: Cognitively Guided Instruction (2nd ed.). Heinemann.
• Fosnot, C. T., & Dolk, M. (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Heinemann.