Repertoire of Strategies¶

Strategy · All strategies
Evidence · 3 claims (2 for, 1 against) · 8 studies (4 causal, 3 quant-synthesis, 1 associational), q2–q4 · 2 of 8 report an effect size · 1 claim rests on one study

Description¶

Procedural fluency requires having a repertoire of strategies. Before learners can flexibly choose an appropriate strategy, they must have strategies from which to choose. Strategies are flexible ways to solve a problem; algorithms are step-by-step procedures. Students should be able to flexibly use and adapt strategies and switch to a different strategy when their first choice is not working well. Every student must have the opportunity to learn more than one method — limiting students to only one method puts them at a disadvantage, denying them access to more intuitive methods and the opportunity to flexibly choose a method that fits the problem at hand.

Design Implications¶

Building a strategy repertoire shifts instruction from single-procedure mastery toward comparison and adaptive choice. Comparing multiple solution methods for the same problem improves procedural flexibility and conceptual knowledge relative to studying methods in isolation Comparing solution methods supports flexible procedural knowledge. [+M]. The goal is not maximal breadth but a curated set of methods whose trade-offs (speed, reliability, transparency) learners can articulate and justify.

Context¶

Requirements¶

• Educators who provide varied methods, promote strategy comparisons, and cultivate a learning environment that values exploration and adaptation
• Deliberate sequencing: introduce a second strategy only after the first is reasonably secure, then compare (Comparing Cases)
• Tasks that genuinely favor different strategies — if one method always wins, learners will not develop adaptive choice
• Opportunities to explain and justify strategy choice (Self-Explanation, Class Discussion)

Constraints¶

• Requires significant preparation and instructional time; teaching multiple strategies effectively is slower than teaching one
• Presenting too many methods at once overwhelms novices and increases extraneous load Example–problem sequences reduce cognitive load for novices. [-M] — breadth must be built incrementally
• Premature comparison before any method is understood produces shallow, confusable knowledge rather than flexibility
• Weaker students can fixate on an inefficient preferred strategy if comparison is left unstructured [~M]

Implementation Variability¶

• Comparison-first: present two worked solutions side by side and discuss trade-offs before practice
• Invention-first: let learners attempt problems, then share and compare student-generated methods (as in Cognitively Guided Instruction)
• Faded repertoire: begin with one anchor strategy, add alternatives as expertise grows, consistent with the expertise-reversal principle [~S]

Target Learners¶

• Learners who have secured at least one workable method and are ready to compare alternatives Comparing solution methods supports flexible procedural knowledge. [+M]
• Intermediate learners moving from accuracy toward efficiency and adaptivity; complete novices benefit first from a single clear method
• Students prone to rigid procedure-following, who need explicit permission and practice in switching strategies

Target Learning Goals¶

• Flexible problem-solving: selecting and adapting methods to problem context
• Strategic thinking: articulating why a method fits a problem
• Adaptive expertise: switching strategies when the first choice stalls
• Conceptual grounding: understanding why different methods work Self-explanation improves conceptual understanding. [+M]

Instructions¶

1. Establish one reliable anchor strategy through Practice and Provide Guidance
2. Introduce an alternative method via Comparing Cases — two worked solutions to the same problem, side by side
3. Prompt learners to explain when each method is advantageous (Self-Explanation)
4. Assign mixed tasks where the "best" method varies, so learners must choose, not just execute
5. Assess Performance on strategy selection and justification, not only on answers

Related Strategies¶

• Use Worked Examples — worked examples are the vehicle for presenting alternative methods for comparison
• Think-Aloud Modeling — makes the reasoning behind strategy choice visible
• Teach students to intentionally choose from alternative algebraic strategies when solving problems, after they have procedural fluency

Related Elements¶

• Comparing Cases — the core mechanism for building the repertoire
• Practice — mixed practice is where adaptive selection develops
• Cognitive Strategies — the individual strategies that populate the repertoire
• Assess Performance — assessment must target choice and justification, not just execution

Examples¶

• Number talks (K–6 mathematics): a teacher poses one computation, students solve mentally with different methods, and the class compares strategies (e.g., decomposition vs. standard algorithm), making the repertoire public and evaluated.
• Comparison in algebra (Rittle-Johnson & Star's studies): students studying two worked examples of the same problem side by side, explaining similarities and differences, showed greater procedural flexibility than students studying the examples sequentially.
• Cognitively Guided Instruction — teachers elicit and map children's informal strategies (direct modeling, counting, derived facts) rather than imposing a single algorithm.

Key Sources¶

• Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574. doi:10.1037/0022-0663.99.3.561
• Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.
• Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24(3), 335–359. doi:10.1007/BF03174765
• National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. NCTM.