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Explicit Instruction: Math Strategies

Principle · All principles
Evidence · 21 claims (17 for, 4 mixed) · 29 studies (14 causal, 5 review, 4 quant-synthesis, 4 theoretical, 1 associational, 1 qualitative), q1–q4 · 4 of 29 report an effect size · 14 claims rest on one study

Conditional relationship

This page owns one part of explicit teaching in mathematics: teaching problem-solving strategies, including when a strategy fits and why. For a learner who can already carry out the computation and holds the basic concept (the starting response to look for is a correct calculation on bare numbers but a stalled, keyword-driven or piecemeal attempt at a word problem or non-routine problem), explicit modelling of a strategy that names its decision points, side-by-side comparison of two or more valid methods on the same problem, and guided practice in which the learner chooses and justifies a method, are expected to produce more flexible and more accurate skill-performance on new problems of the same kind (immediate-performance, near-transfer) than practising one method by rote or being left to find a method alone. The target knowledge is a procedure together with the principle that tells the learner when to use it.

The relationship is conditional on four things: whether the learner already has the conceptual footing (without it, strategy teaching tends to become rule-following, and a short problem attempt before instruction may build the concept better for older learners); which outcome counts (procedural flexibility, conceptual understanding and correct solutions behave differently in the evidence here); how many strategies are introduced at once; and the horizon.

No claim in the wiki tests this page's whole relationship, explicit strategy instruction against an alternative without it, with the learners and outcomes above. The nearest evidence tests parts of it: comparing solution methods side by side against studying them one at a time (seventh graders, algebra); one whole-problem strategy against a phrase-by-phrase translation strategy for algebra word problems (ninth graders, 1972); and a single-classroom, pre/post-only run of explicit language-focused strategy instruction for word problems that found no significant gain. The default design below is therefore a set of labelled proposals.

How the converted siblings sit around this page: Direct Instruction owns the general relationship (explain, model and guide practice for a learner with no working method, against minimally guided discovery) and its qualifications by expertise, age, outcome and horizon; this page applies that model to mathematical strategies and adds what it does not cover (teaching when and why, comparing methods, choosing among them). Productive Failure owns whether a new concept is attempted first or taught first; this page takes its answer as the rule for when to place a problem attempt before strategy teaching. Cognitive Flexibility owns the general case of holding knowledge in several forms; comparing methods here is one way to build it. Worked Examples owns studying a solution before attempting one, and the Direct Instruction pattern holds the lesson sequence as a reusable policy.

Default design, while the relationship is untested

The earlier page's guidance (and that of the folded page on Polya-style instruction for verbal problems), kept as a concrete default a designer can act on and revise. Each step is a design proposal; its evidence status is stated beside it. Session lengths and thresholds are untested proposals unless a claim is linked.

  1. Check the footing first, inside the work (a few minutes, session 1). Do not add a pre-test. Ask the learner to do one or two real items of the work in front of them and to say what the quantities mean: for a ratio or proportion strategy, ask what the ratio says ("for every 2 of this, 3 of that") and what happens to the other quantity when one doubles; for word problems, ask for the bare calculation and the same numbers in the story. If the brief says the concept must be checked first, do this before any strategy is modelled. If the calculation or the concept fails, teach it first and do not add strategies yet. Evidence: fourth graders scored much higher on computation than on word problems with the same numbers (Fourth-graders perform substantially better on computation tests than on word problems using the same numbers [+M]; one classroom of 24). Making this a task inside the work rather than a test is an untested proposal.
  2. For a new concept with secondary-age or older learners, let them try first (10–15 minutes). Give one problem the target strategy solves, let learners attempt it in pairs, and collect their methods, including wrong ones. Then teach the strategy by comparing it with their attempts. Evidence: Productive Failure Improves Conceptual Learning [~S] (conceptual knowledge and transfer, not procedural skill; reversed for second to fifth graders) and PS-I fosters learning only if specific design features (contrasting cases or building instruction on student solutions) are implemented [+M]. For a procedure, or for younger learners, skip this step and teach first; the decision is modelled in Productive Failure.
  3. Model one strategy, naming its decision points (10 minutes). Work a problem aloud and say, at each choice, what you looked at and why this strategy fits (for example, "the problem gives a total and asks for a part, so I write one equation for the total"). Annotate the written example with the reason, not only the step. Untested here; from the earlier page. Following the model with a closely related problem, rather than a problem first, has one supporting experiment (Example–problem sequences reduce cognitive load and improve learning outcomes [+M]; secondary-school novices, electrical circuits, carried to mathematics).
  4. For verbal (word) problems in algebra, teach a whole-problem strategy and keep translation as a sub-step. Teach learners to ask what is unknown, what relation links the quantities, to write one equation for the whole problem, solve it and check the answer against the story (a Polya-style approach: understand, plan, carry out, look back). Translating phrases into symbols is a step inside the plan, not the plan. Evidence: in one 1972 experiment, ninth graders taught the Polya method wrote correct equations more often than those taught the Dahmus phrase-by-phrase translation method (Polya Method students outscore Dahmus Method students on the equation criterion for verbal problems [+M]), and translation-method students often translated each phrase without combining them into one equation (Dahmus Method students translated phrases successfully but failed to combine them into a single solution equation [+W]); the two groups did not differ in solving the problems correctly (Polya and Dahmus methods do not differ on the problem solution criterion [~M]). The study's authors proposed a classroom synthesis of the two strategies, with the pure strategies kept as alternatives for students who struggle with the synthesis; that synthesis is an untested proposal.
  5. Compare methods side by side, and use the learners' own error as one of the cases (10–15 minutes, from the first session in which an error has been seen). Where an error has already been observed (a scaled recipe with one ingredient left unscaled, a ratio added instead of multiplied, one phrase translated without the others), put that attempt, anonymised, beside a correct method on the same problem and ask: where do they part, what in the problem shows which is right, and how would you check? Otherwise show two valid methods and ask which is quicker here and when you would use the other. Use at most two or three cases in one session. Evidence: Comparing cases side by side improves learning and transfer by a moderate average amount, though in one algebra study it raised procedural knowledge and flexibility but not conceptual knowledge [+S], Alternative-strategy practices are most effective for improving procedural flexibility, with weaker evidence for conceptual and procedural knowledge [+M], and, for comparing with learners' own attempts, PS-I fosters learning only if specific design features (contrasting cases or building instruction on student solutions) are implemented [+M] (a review of problem-first designs, carried to an error seen during ordinary teaching). Using the learners' error as a compared case, and the cap on cases per session, are untested proposals.
  6. Prompt the reason, not only the answer. In guided practice, ask learners to say or write why they chose the method ("which method, and what in the problem told you?"). Evidence: Prompting learners to self-explain improves understanding and problem solving on immediate tests by a moderate average amount, with little evidence yet on delayed or classroom gains. [+S] (self-explanation prompts in general; mathematics evidence is immediate and mostly not from classrooms).
  7. Set process goals early, answer goals later. In early sessions, mark and praise following the strategy's steps and stating a reason; move to accuracy and speed targets once the strategy is used reliably. Evidence: Process goals lead to better skill acquisition for novices than outcome goals. [+M] (high-school writing revision and self-regulation research, carried to mathematics).
  8. Fade the support across sessions and mix the problems. A default arc over about six to eight sessions of 20–30 minutes (an untested proposal): sessions 1–2, one strategy modelled with a written plan card and full worked examples; sessions 3–4, two methods compared, completion problems with the last steps left for the learner; sessions 5–6, mixed problems where the learner must choose the method, plan card available on request; sessions 7–8, mixed problems in the form of the target assessment, no card, followed by a look-back question ("would another method have been quicker?"). Evidence for fading worked steps: Fading support promotes the transfer of responsibility from instructor to learner. [+S] (near transfer; 7 distinct studies in its table, most outside school mathematics).
  9. Move a learner on when they choose and justify. A learner moves to the next strategy or to mixed independent work when they choose an appropriate method and give a correct reason on two of three new problems, unaided. Learners who already do this at the start skip to step 5 or 8. Untested threshold.
  10. Check with a short task inside later work, not with a test, and look at the plan as well as the answer. Add a test or a delayed check only if the brief asks for one. Otherwise, at the start of a later session, give one new problem as part of the work (a recipe to scale for tomorrow's service, a quantity to order) and look at the written plan or equation and the answer separately, because they can diverge. Evidence: in the 1972 study, solution scores on the more complex problems were higher at a four-week retention test than at the posttest (On the problem solution criterion, verbal problem solving scores do not decline over four weeks and rise on the more complex problems [+M]), and the equation and solution criteria gave different results (step 4); neither finding calls for a separate test.

When a condition is not met: if learners cannot do the computation or lack the concept, teach it first; if learners already choose and justify methods, use only comparison and mixed practice; if no one can see attempts (self-paced online work), put the plan card and worked reasons in the materials and use problems with automatic checking and a worked solution after each. When the brief gives its own explicit sequence (explain, model, practise) or its own setting, keep both and fit these moves into its steps; do not add sessions, tests or checks it did not ask for.

Fitting the design to a situation

The three facts that most change the decision:

  • The learner's footing. Can they do the computation, and do they hold the concept the strategy relies on (for example, what a ratio means)? A learner without either needs the concept taught (or, if older and the goal is conceptual, a problem attempt first), not more strategies. If the brief omits it, ask: on one or two real items of the work, what does the learner do, and can they say what the quantities mean? If the brief already reports an observed error, that error is the first observation to use.
  • Which outcome counts. Choosing among methods (procedural flexibility), writing a correct equation or plan, getting the right answer, and understanding why a method works did not move together in the studies here. If the brief omits it, ask: on what problems, with what aids, and judged how, must the learner succeed, and when?
  • Who sees the learner's choice and reason. The design depends on someone or something reading the learner's stated reason for a method, not only the answer. If the brief omits it, ask: during practice, who will look at the method a learner chose and why, and how soon?
If the brief says… Then change… Basis
Learners cannot yet do the computation, or lack the concept Teach the computation or concept first; postpone strategy teaching; keep strategies to one, with a full worked example untested proposal; the page's earlier constraint ("a conceptual foundation strong enough..."); Example–problem sequences reduce cognitive load and improve learning outcomes [+M] for starting from a worked example (secondary-school novices, electrical circuits)
Learners are advanced or already fluent in one method Skip single-strategy modelling; go straight to comparing their method with an alternative on problems where the alternative is quicker; mixed problems requiring a choice Comparing cases side by side improves learning and transfer by a moderate average amount, though in one algebra study it raised procedural knowledge and flexibility but not conceptual knowledge [+S] (seventh graders in the algebra study); Fading support promotes the transfer of responsibility from instructor to learner. [+S]
The goal is conceptual understanding or transfer, learners are secondary-school age or older Open with a 10–15 minute problem attempt, then teach the strategy by comparing learners' solutions with it Productive Failure Improves Conceptual Learning [~S]; PS-I fosters learning only if specific design features (contrasting cases or building instruction on student solutions) are implemented [+M]
The goal is a procedure, or learners are child (early primary) Teach the strategy first, then practise; no problem-first phase Productive Failure Improves Conceptual Learning [~S] (no procedural advantage for problem-first; reversed for second to fifth graders)
The goal is flexibility, choosing the better method Make side-by-side comparison of two methods the core of each session, with "which is quicker here, and when would you use the other?" Alternative-strategy practices are most effective for improving procedural flexibility, with weaker evidence for conceptual and procedural knowledge [+M]; Comparing cases side by side improves learning and transfer by a moderate average amount, though in one algebra study it raised procedural knowledge and flexibility but not conceptual knowledge [+S]
Algebra word problems; learners translate phrase by phrase but do not form one equation Teach the whole-problem plan (unknown, relation, one equation, check); keep translation as a step inside it; offer the pure translation or pure planning version to learners who struggle with the combination Polya Method students outscore Dahmus Method students on the equation criterion for verbal problems [+M]; Dahmus Method students translated phrases successfully but failed to combine them into a single solution equation [+W]; the combination is untested (ninth-grade girls, 1972)
The brief reports an error the learners have made (a ratio applied to one quantity only, an operation chosen by keyword) First check the concept behind it on one real item; then make the error one of the compared cases: show the erroneous attempt (anonymised) beside a correct method on the same problem, ask where they part and how to check, then have learners redo a similar item with the correct method and a stated reason Comparing cases side by side improves learning and transfer by a moderate average amount, though in one algebra study it raised procedural knowledge and flexibility but not conceptual knowledge [+S] (compared methods, not errors); PS-I fosters learning only if specific design features (contrasting cases or building instruction on student solutions) are implemented [+M] (instruction built on students' own solutions, in problem-first designs); using an observed error this way is untested
Word-problem scores are far below computation scores; learners have low literacy or are learning in a second language Add reading work to each problem: read aloud, retell the story in own words, name the unknown and the relation before any numbers; do not expect a short strategy unit alone to close the gap; measure it Fourth-graders perform substantially better on computation tests than on word problems using the same numbers [+M]; Three weeks of explicit language-focused strategy instruction produced no significant word-problem gain in this study [~M] (one class of fourth graders, no comparison group)
Learners differ widely in ability or prior attainment Teach the same strategy to all, with more worked examples and more guided rounds for lower attainers; do not assign different strategies by ability on this evidence Algebra ability level differences appear on both criteria with no treatment-by-ability interaction [~M] (a non-significant interaction does not show the strategies suit all levels equally)
Learners are adult in work settings (apprentices, trainees, adult-workplace), or in adult basic education Keep the brief's own sequence, setting and materials (the kitchen, the recipe card, the shift); teach the strategy on the work's own quantities at the point the work needs it; ask which methods they already use and compare one of theirs with the taught method; check with a work task (scale tomorrow's batch), never a classroom test untested proposal; the earlier page's source is an adult-education practice guide, not a claim
Learners are anxious about mathematics or low in confidence Show two valid methods early so "more than one way" is visible; mark following the strategy and giving a reason in early sessions, not right answers; let learners answer privately (paper, device) before anything is shared Process goals lead to better skill acquisition for novices than outcome goals. [+M] (writing revision, carried to mathematics); the rest untested
online-self-paced, phone-only, or no teacher present Put the reason beside every worked step; use completion problems that leave the last steps to the learner; check each answer automatically and show the worked solution; give a one-screen plan card the learner can open Fading support promotes the transfer of responsibility from instructor to learner. [+S] (worked-example fading, carried to self-paced mathematics); Sixth graders scored significantly higher on posttests that included the seven-step problem-solving plan than on unformatted posttests [+M] (a plan printed with the problems)
Mixed modes: some learners remote, some in the room Show both methods on one shared screen; everyone, remote or not, posts the method they chose and one reason (chat, poll) before discussion; read out a remote learner's reason first untested proposal
One teacher for a large class Put two learner methods side by side on the board; everyone shows a method choice at once (cards, mini-whiteboards); discuss the commonest wrong choice; then go to those who chose by keyword untested proposal
Scarce facilitator time (volunteer tutor, short check-ins) Spend the check-in on one thing: ask the learner which method they chose on one problem and why; leave answer checking to an answer key untested proposal
A single session only Check the concept on one real item (5 min), model one strategy with its reason (10 min), one side-by-side comparison, using an observed error if there is one (10 min), one unaided item from the work with a stated reason (5 min); do not claim flexibility or retention untested proposal; a three-week unit showed no significant gain in one class (Three weeks of explicit language-focused strategy instruction produced no significant word-problem gain in this study [~M])
A fixed short time budget per session (for example 20 minutes) Keep one strategy or one comparison per session: 5 minutes model, 10 minutes guided choice with reasons, 5 minutes unaided item; spread the arc over more sessions rather than packing it untested proposal
The brief names its own routine, heuristic, categories or rubric (a school's problem-solving steps, a keyword chart, a seven-step plan) Keep it and its terms; add only the "when does this fit, and why" talk and one comparison with an alternative; judge with the brief's own rubric untested proposal; a printed plan raised scores when it appeared on the test (Sixth graders scored significantly higher on posttests that included the seven-step problem-solving plan than on unformatted posttests [+M])
The brief asks for no test, or names no assessment Add no pre-test, post-test or delayed check; judge progress from what learners do in the work (the method they choose on the next real item and the reason they give) untested proposal
Stakes are high, or the assessment is unaided Judge readiness only on problems without the plan card; score the equation or plan and the final answer separately untested proposal; scores differed with and without the printed plan (Sixth graders scored significantly higher on posttests that included the seven-step problem-solving plan than on unformatted posttests [~M]) and the equation and solution criteria diverged (Polya and Dahmus methods do not differ on the problem solution criterion [~M])
A whole real task the brief wants kept whole (a budget, a pricing decision, a measured plan for a job), or a product for a real audience Keep the task whole; teach the strategy at the point in the task where it is needed, then return to the task; do not split it into separate drill items untested proposal
The problems use learners' own private figures (pay, debts, benefits) Let learners use their own numbers privately, or offer realistic substitute figures; nobody else sees the numbers, only the method untested proposal

Fourteen of the twenty-two rows cite a claim; three of those cite one only for its limit (the single-session, high-stakes and named-routine rows), and several carry a finding to learners or settings it did not test, as each row says; the observed-error row carries comparison evidence to a learner's own error. The adult, mixed-mode, large-class, facilitator-time, time-budget, no-test, whole-task and privacy rows are untested proposals.

Observation, state and explanation

What can be observed is an attempt under stated conditions: the problem, the aids available (plan card, worked example, calculator), the method the learner chose, the reason they gave, the written equation or plan, the answer, and the time taken. "Has a strategy", "chooses by structure" and "lacks the concept" are inferred states. A correct answer does not show which method was used or why, and a method chosen during guided practice with the card visible does not show what the learner chooses alone.

Response Plausible explanations Observation that could change the interpretation
Correct on bare calculations, wrong on word problems with the same numbers Reading or language demand; no plan for representing the story; the learner picks an operation by keyword Read the problem aloud and ask the learner to retell it, then ask for the unknown before any numbers; compare with a version with the keyword removed.
Translates each phrase into symbols but writes no single equation (or several unconnected ones) Translation taught as the whole strategy; the relation linking quantities not identified; the learner solves by arithmetic and treats equations as a formality Ask what single relation the story describes; give a problem where arithmetic alone is hard but one equation is easy.
Always uses the same method, even where another is much quicker Only one method known well; the alternative not trusted; the learner values safety over speed Show both methods on a problem where they differ greatly in effort and ask which the learner would choose and why; check whether the learner can carry out the alternative at all.
Chooses a method correctly but cannot say why The choice rests on surface features that happen to align; the learner lacks words for the reason; the reason is held but not verbalised Give a pair of problems with the same surface but different structure; allow a reason to be shown by pointing or drawing.
Right equation, wrong answer (or the reverse) Algebra execution errors; checking skipped; the answer reached by a different route than the written plan Score plan and answer separately on a few items; ask the learner to check the answer against the story aloud.

These rows are proposals for local observation, untested with learners. A probe is also practice and can change what it measures, so record each one, and keep open the explanations a probe has not ruled out.

Evidence and qualifications

No claim in the wiki compares explicit strategy instruction with practice that lacks it after the same concept teaching, or varies the number of strategies taught. The claims below test parts of the relationship.

The limiting claim. Three weeks of explicit language-focused strategy instruction produced no significant word-problem gain in this study [~M]: in one classroom, three weeks of daily 20-minute explicit strategy instruction raised the word-problem mean from 47.08 to 55.42, not a significant gain (t = −1.26), with no comparison group. It is weak evidence (one class, coded r1), and a non-significant gain does not show no effect, but it is the only claim in the wiki that runs explicit word-problem strategy instruction and measures the result, and it did not find a reliable gain. Do not promise that a short strategy unit will move word-problem scores.

Learners, tasks and outcomes differ across these claims (seventh-grade algebra, ninth-grade word problems in 1972, fourth graders, circuit troubleshooting, writing revision), so do not rank them by effect label. None measures delayed transfer of a taught strategy against no strategy teaching.

The evidence by question

Further evidence, not yet read against this model

Objective and learner-valued goal

The designer's objective is usually that the learner chooses an appropriate method for a new problem, writes a correct equation or plan, and reaches the right answer, unaided, on a stated set of problems at a stated horizon. Elicit separately what the learner wants: to pass a test, to stop feeling lost in front of word problems, to work out a payslip or a discount, to help a child with homework. Agreement and divergence are both observations. A learner who values getting answers quickly may resist explaining a method; one who values understanding may resist a plan card.

For example, an adult in a basic-skills class may want to check whether a "30% off, then a further 10%" offer is really 40% off, while the designer's objective is writing one equation for a percent problem. One worked comparison of two methods on that offer, then an unaided attempt on a different offer, serves both; record whether the learner can now judge such offers, separately from whether they write the equation.

What would revise this model?

The expectation should weaken if comparisons of explicit strategy instruction (with when and why) against practice without it, after the same concept teaching and with comparable learners, show no advantage on unaided new problems, or if gains at the end of a unit disappear at a delay. If comparing methods raises flexibility but never accuracy or solution rates, the page should say comparison serves flexibility only. If whole-problem planning keeps matching phrase translation on solving problems, as in the one study here, the case for it rests on equation writing alone. If problem-first designs keep beating strategy-first teaching on the outcomes a design targets, the default sequence should start with the attempt more often. Do not protect the model by calling every failure "not enough conceptual footing" after the fact.

Choosing a method, writing a correct equation, solving correctly, explaining why, and delayed transfer are separate claims. The present evidence sets no number of strategies, no dose and no move-on criterion.

Claims

  • Worked Examples — strategy instruction often begins with explicit modeled solutions
  • Error Analysis — comparing failed and successful strategies can sharpen strategic judgment
  • Real-World Math — strategy flexibility matters most when problems are varied and contextualized
  • Explaining Their Thinking — strategy use becomes durable when learners justify choices aloud or in writing
  • Direct Instruction — the general model this page applies to mathematical strategies (explain, model, guide practice; qualified by expertise, age, outcome and horizon)
  • Productive Failure — decides whether learners attempt a problem before the strategy is taught
  • Cognitive Flexibility — comparing methods is one way to hold the same knowledge in several forms
  • Direct Instruction — the explain–model–guide–release lesson sequence as a reusable policy

Examples

Key Sources

  • Durkin, K., Star, J. R., & Rittle-Johnson, B. (2017). Using comparison of multiple strategies in the mathematics classroom. ZDM, 49(4), 585-597. doi:10.1007/s11858-017-0853-9
  • 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
  • Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41(2), 75-86. doi:10.1207/s15326985ep4102_1
  • U.S. Department of Education, Office of Career, Technical, and Adult Education. (2014). TEAL Math Works! Guide.
  • Bassler, O. C.; Beers, M. I.; Richardson, L. I. (1972). Comparison of Two Instructional Strategies for Teaching the Solution to Verbal Problems. Final Report. George Peabody College for Teachers. https://eric.ed.gov/?id=ED075260
Page metadata
Field Value
type principle
id explicit-instruction-math-strategies
aliases recommend-polya-style-instruction-for-verbal-problems
title Explicit Instruction: Math Strategies
description For a learner who can compute but not yet choose a method for word or non-routine problems, explicitly modelling when and why a strategy fits, comparing methods side by side and guiding justified choice are expected to build flexible, accurate problem solving, though no claim in the wiki tests strategy instruction against its absence.
status review
canonical True
generated by claude/unspecified on 2026-10-14
trust unverified — no one has confirmed this page's content
sources 4 — durkin-2017, rittle-johnson-2007, kirschner-2006, bassler-1972

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