Wren Ed Instructional Routines

These ten instructional routines are structured, repeatable classroom activities designed to develop mathematical thinking, not just answer-getting. Each routine creates space for students to reason, discuss, make mistakes, and build understanding through consistent practice with a predictable format.

Think of them as your toolkit. You wouldn't use a hammer for every job, and you won't use the same routine for every lesson. But once students learn the structure of each routine, you can deploy them efficiently—no lengthy setup, no reinventing the wheel. Students know what to expect, which frees their cognitive energy for the actual mathematics.

These aren't worksheets or one-off activities. They're teaching structures you'll use again and again across different topics and grade levels. The routine stays the same; the mathematical content changes.

  1. Learn from Mistakes

    What it is: Students solve a problem individually, then review each other's work to identify and learn from errors. They create a shared "Discoveries List" describing both the mistakes and the reasoning behind them.

    When to use it: After initial instruction, when students have attempted a skill but are making predictable errors. Works particularly well for multi-step procedures (solving equations, simplifying expressions) and topics with common misconceptions.

    What it develops: Metacognition, error analysis, peer feedback skills. Students learn to recognize error patterns rather than just making isolated mistakes.

    Time: 25-30 minutes

  2. Test It, Tune It

    What it is: Students tackle a challenging problem—one that's meant to be hard—and record every strategy they try, whether it works or not. After seeing what strategies peers used and discussing which worked and why, students apply their learning to a second problem.

    When to use it: When introducing a new problem type or adding complexity to a familiar concept. Students need some foundation to build on, but the problem should stretch them beyond what they've practiced.

    What it develops: Productive struggle, strategy articulation, perseverance. Students build a toolbox of problem-solving approaches and learn when to use them.

    Time: 30-40 minutes

  3. Decide & Defend

    What it is: Students read a mathematical scenario and four statements about it. They choose which statement is correct and defend their reasoning. After reviewing arguments for all four statements (including wrong ones), students revise their thinking and discuss what makes mathematical arguments strong.

    When to use it: During or after instruction when students have some familiarity with concepts but need to develop precision. Exceptional for formative assessment—each wrong answer reveals a specific misconception.

    What it develops: Mathematical argumentation, critique of reasoning, precision in thinking. Also serves as real-time diagnostic: the wrong answers students choose tell you exactly what they misunderstand.

    Time: 20-30 minutes

  4. Graph-Table-Equation

    What it is: Students match graphs, tables, and equations that represent the same mathematical relationship. Working in pairs, they sort cards or digital slides to create "triples" of equivalent representations.

    When to use it: As introduction to a new type of relationship (linear, exponential, quadratic) or as consolidation after students have seen representations separately. Helps students see how symbolic, visual, and numerical modes connect.

    What it develops: Representation fluency, translation between modes, understanding how equation structure relates to graph features.

    Time: 30-40 minutes

  5. Graph Sort

    What it is: Students examine 30-40 graphs and sort them into groups based on visual patterns they notice. They create their own categories, label them, and explain what the graphs (and equations) in each group have in common.

    When to use it: As introduction to a function family (students discover patterns before learning vocabulary) OR as reinforcement (students formalize understanding by categorizing many examples). Works beautifully for linear, quadratic, and exponential functions.

    What it develops: Visual pattern recognition, connecting visual features to algebraic properties, understanding how changing equations affects graphs.

    Time: 35-45 minutes

  6. Fit or Misfit

    What it is: Students examine scatter plots with three different lines and decide which line best fits the data. After evaluating several examples, they draw their own line of best fit and articulate what makes it a good fit.

    When to use it: As introduction to linear modeling BEFORE teaching regression formulas. Students develop visual judgment about what "good fit" looks like—informal understanding that makes later formal methods meaningful.

    What it develops: Informal statistical reasoning, visual judgment about models, understanding that modeling involves approximation and judgment, not just calculation.

    Time: 30-40 minutes

  7. Data Stories

    What it is: Students create survey questions about topics that interest them, collect data from classmates, and each student creates a unique visualization of a dataset. After viewing how different students visualized the same data differently, they discuss how design choices affect the stories data tell.

    When to use it: As culmination of a data unit where students apply everything they've learned, or as introduction to data visualization where they experience design choices first. Can be used multiple times across a year as students' graph repertoire grows.

    What it develops: Data literacy, critical evaluation of visualizations, understanding that graphs are constructed arguments. Students see mathematics as a tool for understanding questions they care about.

    Time: 60-75 minutes (can split across two days)

  8. Always, Sometimes, Never

    What it is: Students evaluate general mathematical claims and decide if they are always true, sometimes true, or never true. If a claim is sometimes true, they must define the specific conditions under which it holds or fails.

    When to use it: When you want students to confront a generalization they hold loosely and articulate the conditions under which it actually works. Excellent for addressing common overgeneralizations.

    What it develops: Mathematical reasoning, constructing and testing counterexamples, defining domains/conditions of claims, and mathematical precision.

    Time: 20-30 minutes

  9. Rank and Reason

    What it is: Students examine four mathematical objects and rank them along a specified dimension (e.g., from greatest to least rate of change). Working in pairs, they explain the criteria and mathematical features they used to determine the order.

    When to use it: When you want students to identify which feature actually answers a question, especially when competing salient features might mislead them or when ordering can be justified in different ways.

    What it develops: Comparative reasoning, distinguishing competing mathematical features, and articulating criteria for ordering.

    Time: 15-25 minutes

  10. Which One Doesn't Belong

    What it is: Students examine exactly four mathematical objects of the same category (e.g., four equations, four graphs) and find mathematically defensible reasons why each object could be the outlier that doesn't belong.

    When to use it: When you want students to attend to more features of mathematical objects than they normally do, and to see that grouping is defensible in multiple ways. Great for warm-ups or introducing new concepts.

    What it develops: Observation of multiple features, justifying classification choices, flexibility in mathematical reasoning, and vocabulary usage.

    Time: 15-20 minutes

If you want students to...

  • Analyze and learn from errors → Learn from Mistakes
  • Develop problem-solving strategies → Test It, Tune It
  • Build and critique mathematical arguments → Decide & Defend
  • Connect different representations → Graph-Table-Equation
  • Discover patterns across many examples → Graph Sort
  • Develop informal understanding of modeling → Fit or Misfit
  • See math as personally meaningful → Data Stories
  • Evaluate claims and conditions → Always, Sometimes, Never
  • Compare and order objects → Rank and Reason
  • Find outliers and justify classifications → Which One Doesn't Belong

If your content is...

  • Procedural (equations, simplifications) → Learn from Mistakes
  • Challenging problem types → Test It, Tune It
  • Conceptual with common misconceptions → Decide & Defend
  • Function relationships → Graph-Table-Equation, Graph Sort
  • Data and scatter plots → Fit or Misfit
  • Statistics and visualization → Data Stories
  • General claims and rules → Always, Sometimes, Never
  • Comparing multi-feature objects → Rank and Reason
  • Classification and vocabulary → Which One Doesn't Belong

If your timing in the unit is...

  • Introduction to new content → Graph Sort, Fit or Misfit, Data Stories, Which One Doesn't Belong (introduce-then-formalize)
  • During instruction → Decide & Defend, Test It, Tune It, Always, Sometimes, Never, Rank and Reason
  • After initial practice → Learn from Mistakes, Graph-Table-Equation
  • Review/consolidation → Any routine works; choose based on content type

Research shows that certain teaching practices matter more than others. These routines are built on practices that consistently improve student learning:

  • Productive struggle before instruction (Test It, Tune It; Graph Sort)
  • Learning from errors and explaining mistakes (Learn from Mistakes)
  • Comparing multiple examples or approaches (all routines use comparison)
  • Peer discussion structured around specific tasks (all routines include partner work)
  • Making thinking visible (all routines require articulation of reasoning)
  • Student agency and choice (particularly Data Stories, Graph Sort, Test It Tune It)

These aren't just engaging activities—they're evidence-based practices that develop mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition.

New to these routines? Start with one and use it multiple times with different content until students (and you) become fluent with the structure. We recommend:

  • Learn from Mistakes if your students make predictable errors you want to address
  • Decide & Defend if you want formative assessment built into instruction
  • Graph-Table-Equation if you're teaching functions and relationships

Ready to build your routine repertoire? Add routines gradually. Most teachers find they can effectively use 3-4 routines regularly and pull in others as needed for specific content.

Want detailed guidance? Each routine has a comprehensive resource that includes:

  • Step-by-step facilitation guide
  • What to look for during student work
  • Common challenges and solutions
  • Research foundation
  • Connections to Mathematical Practices and NCTM standards

The Wren Education activity engine uses AI to generate mathematical problems based on exemplars created by experienced math teachers. But here's what's important: the AI generates the problems; the routines, the pedagogy, and the learning structures are entirely human-designed and research-based.

Every routine in this collection is grounded in decades of mathematics education research. The quality and effectiveness depend on:

  • The routine structure (designed by educators)
  • Your facilitation (irreplaceable teacher expertise)
  • The classroom culture you create (human relationships and norms)
  • The mathematical thinking students develop (the real work of learning)

AI is a tool for producing content at scale. These routines are teaching practices that help students learn. The combination gives you research-based structures with fresh, appropriate problems—but you remain the pedagogical decision-maker.