Pedagogical Reference

Pedagogical Foundations of Illustrative Mathematics

A guide to IM's learning philosophy — and how Math As A Second Language (MASL) scaffolding integrates within it.

Math As A Second Language Project · Algebra 2 · Unit 4

Foundation

The Core Principle: Activity Before Definition

Illustrative Mathematics (IM) is built on a single organizing conviction, drawn from Hiebert et al. (1996):

"Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving."

— Hiebert et al. (1996), as cited in IM's design principles

This inverts traditional instruction. Conventional math class runs: teach concept → students practice. IM runs: students encounter a problem → grapple → class synthesizes → definition emerges. Vocabulary and formalisms are labels given to things students have already experienced — not prerequisites for experiencing them.

Why this matters for Algebra 2

A student told "i² = −1, therefore i is the square root of −1, therefore complex numbers exist" has been given a fact. A student who first computes what happens when they square imaginary numbers and discovers the rotation pattern then receives the definition has earned it. The definition names something real to them. IM consistently chooses the second path.

Theoretical Roots

Constructivism (Piaget): Learners actively construct knowledge through experience. IM is structurally constructivist — every lesson is designed so that student thinking, not teacher demonstration, is the engine of learning.

Sociocultural theory (Vygotsky): Learning happens in the Zone of Proximal Development — tasks designed to be reachable with support. Productive struggle is the pedagogical mechanism. Mathematical understanding also develops through discourse; community and language are not supplementary but constitutive.

NRC's Five Strands (Adding It Up, 2001): IM targets all five strands simultaneously — conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition — rather than treating them as sequential stages.

Structure

Lesson Architecture: The Four-Part Structure

Every IM lesson follows the same four-part structure. The order is not arbitrary — it enacts a learning arc: activate and orient → encounter and struggle → consolidate → assess.

Warm-Up
5–10 min
Activates prior knowledge; builds community
→
Activities (1–3)
15–20 min each
Problem encounter, grappling, sharing
→
Lesson Synthesis
5–10 min
Teacher formalizes; integrates all activities
→
Cool-Down
5 min
Independent formative assessment

Warm-Up (5–10 min)

Serves one of two functions: (1) preparing students for that day's specific mathematics by reminding them of prior context, or (2) strengthening number sense and flexible reasoning as a daily practice. Warm-ups are also community-building — predictable, low-stakes rituals that communicate, by daily repetition, that mathematics is about reasoning and noticing, not speed.

Key warm-up routines include: Notice and Wonder (what do you observe? what do you wonder? — no wrong answers), Which One Doesn't Belong? (all four can be argued; precision of language required), Number Talk (mental math with shared strategies; builds structural reasoning), and Co-Craft Questions (see a context without a question; write your own — builds meta-awareness of problem structure).

Activities (15–20 min each)

The mathematical core of the lesson. Activity 1 is typically more accessible, closer to prior knowledge. Activity 2 deepens, extends, or formalizes. Each follows the same Launch → Work Time → Synthesis sub-structure (see next section).

Crucially: new vocabulary and formal definitions are introduced during or after activities, not before them. IM explicitly recommends against writing learning targets on the board at the start of class — doing so can reveal the mathematical concept prematurely and undermine the inquiry.

Lesson Synthesis (5–10 min)

This is the teacher's primary "teaching moment" — but it comes last. The lesson synthesis integrates the learning from all activities and makes explicit what students should take away from the lesson. It is where the teacher most directly delivers instruction — but only after students have the experiential foundation to receive it.

"A well-executed lesson synthesis is essential to accomplishing the learning goals of a problem-based lesson, where teachers make explicit what students should take away from the lesson."

— IM curriculum documentation

Cool-Down (5 min)

Independent work on a brief task. Formative data for the teacher: if most students understood → continue; if some struggled → plan to revisit; if many did not → create time the next day. The cool-down tests whether the lesson synthesis landed.

Structure

Within an Activity: Launch → Work Time → Synthesis

Launch

Teacher ensures students understand the context — not the solution path. The problem itself is not explained. MLRs like Three Reads (MLR6) may scaffold access.

Work Time

Students work individually, then in pairs or groups. Teacher circulates, monitors, and selects/sequences work to share. Does NOT lecture or demonstrate.

Activity Synthesis

Teacher orchestrates sharing of selected student approaches. Makes the "mathematical punch line" explicit. Introduces formal vocabulary. Connects representations.

The Launch: Context Without Solution

The teacher ensures students can read and understand what the problem is asking — not that they know how to solve it. IM is explicit:

"This is not the same as making sure the students know how to do the problem — part of the work that students should be doing for themselves is figuring out how to solve the problem."

— IM How to Use the Materials

Work Time: Protected Productive Struggle

The teacher uses the 5 Practices framework (Smith & Stein) during work time: anticipate student approaches (before class), monitor (circulate and listen), select student work to share, sequence the sharing purposefully (typically concrete → abstract), and connect the approaches during synthesis.

The "Not Rescuing" Principle

Teachers are explicitly instructed not to answer student questions that amount to "how do I do this?" during work time. The productive struggle is the learning. The teacher may clarify context, ask what the student has tried, or pose an advancing question — but does not demonstrate the solution path.

Activity Synthesis: The Mathematical Punch Line

This is where the teacher orchestrates a discussion that ensures all students understand the key insight of the activity. Selected student strategies are shared in a purposeful order. The teacher asks connecting questions. Formal vocabulary enters here, as a name for something students have already experienced through the activity.

Teacher Practice

The Teacher's Paradox: Teaching Through Not Telling

The counterintuitive heart of IM: the teacher talks most during the phases that come after student mathematical work. Teacher talk is minimal during work time — the phase where the most significant learning is happening.

Phase Teacher Role Teacher Talk
Warm-Up Facilitator — invite contributions, ask "did anyone see it differently?" Moderate; orchestrating
Launch Clarifier — context comprehension, not mathematical guidance Brief; targeted
Work Time Monitor and observer — listens, notes approaches, selects for sharing Minimal; circulating
Activity Synthesis Orchestrator — selects, sequences, connects student approaches Substantial; guided
Lesson Synthesis Synthesizer — makes explicit the lesson's mathematical insight Most direct instruction
Cool-Down Silent — students work independently None

"The teacher has many roles in this framework: listener, facilitator, questioner, synthesizer, and more."

— IM Design Principles
Advancing vs. Funneling Questions

During work time, teachers use advancing questions (help students think further without giving direction) rather than funneling questions (lead students to the teacher's desired answer). The distinction: "What have you tried so far?" (advancing) vs. "Have you thought about combining like terms?" (funneling). The first honors the student's mathematical agency; the second removes it.

Language Development

Math Language Routines (MLRs)

The eight Math Language Routines were developed with Stanford's Understanding Language/SCALE project (Zwiers et al., 2017). Their purpose: create structured conditions for mathematical understanding and language competence to develop simultaneously. They do not simplify content for language learners — they amplify access without reducing cognitive demand.

"Amplifying means anticipating where students might need language support to understand concepts or mathematical terms, and providing multiple ways to access them."

— IM Access for English Language Learners documentation
Core Design Principle of MLRs

Mathematical understanding and language competence develop interdependently. Students cannot fully possess a mathematical concept without language to articulate it; they cannot fully develop mathematical language without meaningful content to use it on. MLRs create structured conditions for both to develop together.

MLR 1
Stronger and Clearer Each Time

Students draft a response, then meet with 2–3 partners successively. Each partner gives feedback; the speaker revises. Final drafts are both mathematically stronger and linguistically clearer.

MLR 2
Collect and Display

Teacher transcribes student informal language onto a public display during work time. Formal vocabulary is added over time. Bridges student idiom to mathematical terminology. Creates a living record of the informal-to-formal progression.

MLR 3
Clarify, Critique, Correct

Students analyze an intentionally flawed mathematical statement — in reasoning, notation, or language — and write a corrected version. Normalizes error as a learning tool without requiring students to defend their own work.

MLR 4
Information Gap

Two partners hold complementary information. Neither can succeed without the other. Creates authentic communicative need — students must formulate precise questions and justify why they need specific information.

MLR 5
Co-Craft Questions

A context is shown without a question. Students write what questions could be asked, then compare. Builds meta-awareness of problem structure. Provides time with vocabulary before solution pressure. Often used as a warm-up.

MLR 6
Three Reads

A problem is read three times: (1) What is the situation? (2) What quantities or relationships are present? (3) What strategies might we use? Structures the sense-making that proficient readers do automatically.

MLR 7
Compare and Connect

Students display their work and examine peers' approaches, identifying similarities and differences. Teacher presses: "How does this representation connect to that one?" Develops meta-awareness of mathematical structure.

MLR 8
Discussion Supports

A toolkit for inclusive discourse: sentence frames, revoicing, pressing for elaboration, choral response, thinking aloud, gestures. The most flexible MLR — embedded in almost every lesson's activity synthesis and lesson synthesis.

Language Development

MLR Quick Reference: Where They Appear in Algebra 2

MLR When to Use Lesson Phase Appears in L12?
MLR1 Stronger & Clearer After initial student writing; revise through peer interaction Work time → synthesis Optional
MLR2 Collect & Display During work time — capture informal language Work time Implicitly (board)
MLR3 Clarify, Critique, Correct When analyzing a flawed argument or notation Activity synthesis No
MLR4 Information Gap When authentic communicative need should be created Work time No
MLR5 Co-Craft Questions Before an exploratory activity Warm-Up Yes — warm-up
MLR6 Three Reads Complex multi-part word problems Launch No
MLR7 Compare & Connect When multiple student strategies should be compared Activity synthesis Optional
MLR8 Discussion Supports Any time mathematical discourse happens Throughout Yes — synthesis
Language Development

The Informal → Formal Language Progression

This progression is not advice to teachers — it is an architectural feature of every IM lesson. Formal vocabulary enters as a name for a concept students already understand through activity, not before.

Stage Student Language Teacher Action MLR
1. Informal "The number without the i" · "the rotating part" · "when i is squared it becomes negative" Accept and collect informal expressions; write them on the board MLR2
2. Bridging Informal terms sit alongside emerging formal terms on the board Draw explicit connections: "What you called 'the number without the i' — mathematicians call that the real part" MLR2, MLR8
3. Formal Students begin using "real part," "imaginary part," "standard form" Introduce formal term in synthesis; mark it in the lesson as "new vocabulary introduced here" MLR8
4. Precision Students revise their own explanations to use formal language with precision Press for precision: "Can you say that again using the phrase 'real part'?" MLR1, MLR3, MLR8

IM's language development framework targets not just vocabulary but mathematical rhetoric — the language moves of mathematical thought: "I notice that…," "I claim that…," "This is different from X because…," "A counterexample would be…" Sentence frames scaffold these moves before students can produce them independently.

MASL Integration

Where MASL Belongs in an IM Lesson

MASL (Math As A Second Language) scaffolding fits naturally into IM's framework as a language formalization layer — but only after IM's inquiry cycle is complete for the relevant concept. The cardinal rule:

The MASL Rule

MASL cards should never appear before students have encountered the mathematics they name. A student who has not yet grappled with complex number addition cannot meaningfully receive the card. A student who has just synthesized what "real part" means in context is fully ready to receive the MASL trio that names it.

Placement Decisions in Lesson 12

❌ Not here
Warm-Up / Launch: Vocabulary before experience. Students cannot receive what they haven't earned. Showing the a+bi MASL card before Activity 12.2 short-circuits the inquiry.
❌ Not here
Work Time: Students are solving — this is the protected struggle zone. Scaffolding that names the concept removes the productive difficulty.
✅ Best placement
After Activity 12.2 Synthesis: Students have added complex numbers, connected to the complex plane, and heard "real part" and "imaginary part" introduced. Now the MASL card trio for a + bi, real part, and imaginary part lands on earned understanding.
✅ Also valid
During Lesson Synthesis: After all activities are complete. Less time pressure; can connect MASL cards to the general rules. Second-best option if Activity 12.2 synthesis runs long.
❌ Not here
Cool-Down: Independent assessment time. No new scaffolding — students should be consolidating independently.

Why MASL + IM Works Together

IM's inquiry sequence ensures that when MASL vocabulary arrives, it names something students genuinely understand. This makes the MASL trio — Math symbol (★), We Say spoken form (▲), Meaning sentence (●) — a consolidation and precision tool rather than a transmission tool. Students receive formal language after informal understanding is established, which matches IM's designed trajectory exactly.

The card sort activity that follows MASL introduction then becomes a language consolidation activity, reinforcing what students discovered in the inquiry — not a substitute for the inquiry. This distinction is essential.

MASL Integration

MASL as an Extension of IM's MLRs

MASL can be understood as extending two MLRs that IM includes implicitly but does not make fully explicit for multilingual math learners:

MASL extends MLR2: Collect and Display

MLR2 instructs teachers to collect student informal language and display it alongside formal terms. MASL's three-card structure (Math / We Say / Meaning) makes this bridge explicit and permanent — it's a portable, structured version of the Collect and Display board, one that students can revisit, sort, and manipulate.

MASL adds what MLR8 gestures toward

MLR8 (Discussion Supports) includes sentence frames for mathematical language moves. MASL's "We Say" card is a specialized sentence frame for the specific spoken form of mathematical notation — something IM's MLR8 calls for but never scaffolds explicitly. For multilingual learners, the gap between seeing a mathematical symbol and saying it aloud in English is significant. MASL closes that gap systematically.

The Four Multilingual Learner Principles — MASL Alignment

IM Principle MASL Alignment
Support Sense-Making
Amplify, don't simplify		 MASL adds spoken-language access without reducing mathematical demand. The card sort requires the same reasoning; it adds linguistic scaffolding on top.
Optimize Output
Students produce language		 The "We Say" card requires students to produce the spoken form — not just receive it. The card sort requires verbal explanation of matches.
Cultivate Conversation
Authentic back-and-forth		 The partner card sort creates authentic communicative need: students must explain why a card matches, using mathematical language to justify.
Maximize Meta-Awareness
Students notice how language works		 The explicit three-card structure (symbol ↔ spoken ↔ meaning) develops metalinguistic awareness: students see the gap between symbol and spoken form as a meaningful, bridgeable distance.
MASL Integration

Implementation Checklist: MASL in an IM Lesson

Use this checklist when building or reviewing a slide deck that combines IM lesson structure with MASL scaffolding.

Before you add MASL cards to a slide

☐ Has the student encountered the mathematical concept through an IM activity?
☐ Has there been an activity synthesis where informal language was collected?
☐ Does the synthesis include the mathematical vocabulary the MASL card will name?
☐ Is the MASL card positioned after the synthesis, not before or during work time?

Card sort placement

☐ The MASL card sort (drag-and-drop or print) comes after the relevant activity synthesis
☐ The Math card (given, pre-placed) names a symbol the student has already worked with
☐ Students can complete the sort using understanding from the inquiry — not prior knowledge
☐ The card sort is followed by a sentence frame for spoken production

Slide deck structure for IM + MASL

☐ Warm-Up: IM routine only (no MASL cards)
☐ Activity Launch: context and question only (no MASL cards)
☐ Work Time: problems only (no MASL cards)
☐ Activity Synthesis: color-coded solutions + vocabulary introduction
☐ MASL Moment: card trio appears here, clearly labeled "Language & Notation"
☐ Next activity or Lesson Synthesis follows
☐ Cool-Down: problem only (no MASL cards)

References

Sources