Invented Strategies Before Instruction¶

Strategy · All strategies
Evidence · 2 claims (2 for) · 6 studies (3 causal, 3 quant-synthesis), q3–q4 · 2 of 6 report an effect size

Description¶

Learners first grapple with a novel problem type using whatever informal, self-generated strategies they can devise, before the teacher presents the standard or expert method. The invention phase is typically done in pairs or small groups, produces multiple (often incorrect) solutions, and is followed by explicit instruction that consolidates and extends what learners discovered.

Design Implications¶

Struggling to generate solutions before instruction prepares learners to learn deeply from the subsequent explanation — a finding established by Schwartz & Bransford's contrasting-cases work [~S] and replicated across mathematics classrooms Productive failure outperforms direct instruction on deep conceptual measures despite worse performance during the invention phase. [+S]. The mechanism is preparatory: invention activates prior knowledge, surfaces misconceptions that instruction can then target, and gives learners a "need" that the canonical method visibly resolves Activation before instruction improves subsequent learning. [+M]. Critically, the benefit appears on conceptual understanding and transfer measures, not on immediate procedural fluency.

Context¶

Requirements¶

• Problems that are solvable at some level with informal knowledge but whose canonical solution requires new concepts
• Time for learners to generate, compare, and discuss multiple invented approaches
• A subsequent instruction phase that explicitly connects canonical methods to learners' invented ones
• A classroom norm that treats failed attempts as productive rather than penalizing them

Constraints¶

• Invention without high-quality follow-up instruction produces confusion and no benefit [-S] — the strategy is "before instruction," not a substitute for it
• Low-guidance invention with weak prior knowledge can overload working memory and leave learners floundering [~M]; the expertise-reversal effect means experts gain little or are harmed by the extra phase
• Benefits accrue to conceptual and transfer measures; invention-first sequences can reduce short-term procedural accuracy and efficiency [-M]
• Requires substantial class time and skilled facilitation of the comparison discussion; poorly moderated sharing of invented methods can reinforce errors [-W]

Implementation Variability¶

• Contrasting cases (Schwartz & Bransford): learners compare worked cases that isolate key features before a lecture
• Productive failure (Kapur): collaborative invention on complex problems, then structured consolidation of canonical solutions
• Individual short-cycle invention: brief individual attempts (5–10 minutes) before teacher demonstration, suitable when collaboration time is limited
• Problem-solving prior to instruction (DeCaro & Rittle-Johnson): individual attempts with feedback on correctness but no method instruction, before formal teaching

Target Learners¶

• Learners with enough informal or everyday knowledge to attempt a solution, even incorrectly [+S]
• Mixed-ability groups benefit particularly: lower-knowledge learners gain activation and diagnostic feedback, higher-knowledge learners gain flexibility from comparing methods
• Less suitable for complete novices with no relevant prior knowledge, who cannot generate any strategy [~M]

Target Learning Goals¶

• Conceptual understanding of why a procedure works, not just how to execute it
• Transfer to novel problem variants and unfamiliar contexts [+S]
• Adaptive expertise: flexibility in choosing and combining methods
• Not well suited to routine procedural fluency or safety-critical procedures where errors during invention are costly [-M]

Instructions¶

1. Select a problem that is accessible through informal reasoning but whose canonical solution requires new learning; avoid problems where any strategy trivially succeeds.
2. Have learners attempt solutions in pairs or small groups, generating multiple approaches and recording them, even incomplete or incorrect ones (Activation).
3. Orchestrate a comparison discussion: display contrasting invented solutions and ask learners to evaluate what each accomplishes and where each breaks down (Cognitive Conflict).
4. Deliver explicit instruction on the canonical method, explicitly linking it to the invented strategies ("Notice that Maria's method is the same idea, organized differently") (Direct Instruction).
5. Follow with Practice on the canonical method, including problems that reward comparing it with invented alternatives.

Related Strategies¶

• Productive Failure — the most researched formalization of this strategy (Kapur)
• Contrasting Cases — a variant where learners compare prepared cases rather than inventing solutions
• Worked Examples First — the complementary approach; evidence suggests the two trade off depending on learner expertise and target outcome
• Use examples of another student's complete failed problem-solving process prior to instruction as an implementable alternative to productive failure
• Adapt productive failure principles to large university lectures by having the lecturer discuss preparatory exercises and common misconceptions at the start of the lecture

Examples¶

• Productive Failure in Singapore mathematics classrooms (Kapur's research program): students invent methods for statistics or average-speed problems before formal teaching; invention-phase performance is worse, but post-test conceptual understanding and transfer are better. See https://www.nie.edu.sg for the research program.
• Cognitively Guided Instruction (CGI) — children solve word problems with their own strategies before teachers introduce standard algorithms; see CGI for Math.
• "A Time for Telling" (Schwartz & Bransford, 1998): students analyzed contrasting descriptive cases of psychological phenomena before a lecture, dramatically improving the lecture's effectiveness.

Key Sources¶

• Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475–522. doi:10.1207/s1532690xci1604_4
• Kapur, M. (2008). Productive failure. Cognition and Instruction, 26(3), 379–424. doi:10.1080/07370000802212669
• Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning. Educational Psychologist, 51(2), 289–299. doi:10.1080/00461520.2016.1155457
• DeCaro, M. S., & Rittle-Johnson, B. (2012). Exploring mathematics problems prepares children to learn from instruction. Developmental Psychology, 48(3), 892–900. doi:10.1016/j.jecp.2012.06.009
• Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students' learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Information Age Publishing.