Words Do the Math: Language in Mathematical Thinking and Problem Solving

Introduction and Overview

• Host: Megan Bittner, Director of Marketing at William H. Sadlier.
• Presenter: Dr. Kateri Thunder, an educator, researcher, author, and coach focusing on equity and access at the intersection of literacy and mathematics.
• Core Theme: The critical role language plays in mathematical thinking and problem-solving, exploring research-backed strategies to merge literacy and mathematics in the classroom.
• Corporate Background: William H. Sadlier is an educational publisher with over $190$ years of experience, emphasizing research-based instruction.

Objectives of the Session

• Explain why mathematical proficiency requires more than computational skills.
• Demonstrate how contextualized language, multiple representations, and mathematical discourse deepen conceptual understanding for every learner.
• Identify research-backed routines where language serves as a tool for mathematical thinking.
• Provide specific language scaffolds to support all learners, particularly early readers and English Learners (ELs).

The Authentic Work of Mathematicians

• Language in the Community: Mathematics is used by a diverse range of professionals who rely on communication to convey ideas, including:
  • Mechanics working on vehicles.
  • Engineers building traffic circles.
  • Farmers caring for grass or working at markets.
  • Pharmacists, pilots, and meteorologists checking the weather.
• Applied vs. Pure Mathematics:
  • Applied Mathematicians: Use written and spoken language to solve real-world problems.
  • Pure Mathematicians: Use language to publish ideas, receive peer feedback, collaborate in meetings, and prove/represent abstract theories.
• Authenticity: Involving language in math is not just "school math"; it is the authentic practice of professionals in the field.

Mathematical Proficiency and Literacy Standards

• Five Strands of Mathematical Proficiency: Language facilitates all five strands, including engaging in a productive disposition, fluent calculation, applying strategies, and reasoning.
• Process Standards: The NCTM (National Council of Teachers of Mathematics) and Math Practice Standards explicitly highlight communication. Specifically, students must "construct viable arguments and critique the reasoning of others."
• The Reading Rope (Literacy Connection): According to the "Simple View of Reading," deep comprehension relies on both decoding and language comprehension. Language comprehension includes:
  • Vocabulary knowledge.
  • Understanding language structure.
  • Verbal reasoning.
  • Connecting to background knowledge.

Strategy 1: Contextualized Language

• Definition: Providing a story, real-life experience, or situation that gives meaning to mathematical language.
• Low Floor, High Ceiling Tasks: Contextualized math creates accessible yet deeply complex learning opportunities.
• Low Floor Attributes:
  • Natural language: Builds on the everyday language kids use during play (e.g., "How fast can I run?" versus "$\text{feet per second}$ ").
  • Prior knowledge: Validates what children already know.
  • Multiple entry points: Children can enter a task in various ways (e.g., describing volume in a water table with different containers).
• High Ceiling Attributes:
  • Open-ended/Open-middled: Multiple possible answers and multiple paths to find them.
  • Identifiable extensions: Easy to identify how to take thinking further (e.g., changing container shapes or materials like sand vs. water).
  • Unconstrained skills: Focuses on skills that can be deepened indefinitely (e.g., the concept of size or subtraction patterns).

Examples of Contextualized Language

• Abstract to Contextual Subtraction:
  • Abstract: $5 - 5 = 0$.
  • Contextual (Pre-K): Singing "Five Little Monkeys Sitting in a Tree" or "Five Little Monkeys Jumping on a Bed." As monkeys fall off, students feel the subtraction by putting fingers down or acting it out with toys.
  • High Ceiling Extension: Applying the pattern to diverse numbers ($8 - 8 = 0$) to see a universal mathematical rule.
• Robot Invention (Visual/Poetic Context):
  • Context: A poem about a robot with four wheels.
  • Thinking Change: Moving from "count the wheels" to "If you took off a wheel, how would the motion change?" This requires analyzing movement, not just counting.
• Textbook Contexts:
  • Shape attributes: "Dan has a solid figure. It rolls. It does not stack and it does not slide. Which figure is it?"
  • Comparative stories: "Sue counts $8$ bugs. Jim counts $2$ fewer than Sue. Jose counts $2$ fewer than Jim."

Strategy 2: Multiple Representations

To deepen conceptual understanding, educators must connect the five main categories of mathematical representations.

The Five Categories of Representation

1. Physical: Tangible objects, manipulatives (pattern blocks, base $10$ blocks), acting things out, using food, or measuring utensils.
2. Visual: Drawings, pictures, photographs, charts, tables, stickers, stamps, or $10$-frames.
3. Verbal: Spoken and written language; inundating receptive language and providing space for expressive language.
4. Contextual: Situations found in books, poems, or connections to science/social studies/STEM knowledge-building.
5. Symbolic: Numerals, equations (e.g., $35 + 38 = 73$), and letters as variables.

Connecting Representations

• Example: Concept Book "Our 123s": A page showing "$2$" may include the numeral $2$, the word "two," a sentence "Two eggs in a nest," a picture of two blue eggs, and a $10$-frame with two red dots. Discussion focuses on how the same idea ($2$) is shown differently.
• Example: Place Value: Connecting the symbolic equation $35 + 38$ with a physical/visual place value chart using base $10$ blocks.
• Example: Data Analysis: Comparing a tally chart to a bar graph based on a story about Tammy collecting toy animals.
• Example: Number Lines: The transcript emphasizes number lines as a high-ceiling tool. Transitioning from a $100$-chart to a number line is powerful because number lines extend to infinity and support future concepts like fractions and algebra.

Strategy 3: Mathematical Discourse

Discourse is more than talking; it involves a comprehensive approach to communication.

• The Modes of Discourse:
  1. Speaking.
  2. Listening.
  3. Viewing.
  4. Visually Representing (sketching).
  5. Reading (pausing to monitor comprehension).
  6. Writing (encoding mathematical thoughts).
• Integrative Practice: Reading and writing are often neglected in math but are vital for processing. Examples include:
  • Writing original story problems.
  • Reading verbal math instructions (e.g., "Start with $4$, double it, add $2$").
  • Representing probability on a "likelihood line" from $0$ to $1$.

MathTalks as a Routine

• Definition: A routine where language is treated as a tool for mathematical thinking.
• Routine Examples:
  • Chapter Openers: Predictable prompts (e.g., life cycle of a butterfly exploring ordinal numbers like first, second, third).
  • Notice and Wonder: Students observe an image (e.g., $10$ penguins) and ask, "What do you notice?" and "What do you wonder?"
  • Which Would You Pick? (Which Would You Rather?): Students choose between two scenarios and defend their choice using math (e.g., choosing between a zigzag road or a straight road on a map).
  • Estimation: Developing the "Notice, Wonder, Estimate" frame.

Scaffolding and Supporting Every Learner

Explicit instruction is required to ensure English Learners and early readers can access mathematical language.

1. Peer Talk Routines

• Explicit Teaching: Teach kids how to talk to each other, not just to the teacher.
• Knee-to-Knee, Eye-to-Eye: Physical positioning for turn-taking.
• Roles: Using speaker/listener icons or popsicle sticks to remind students of their current communication role.
• Routines: Think-Pair-Write-Sketch-Share (can be done in any order, such as Share-Think-Sketch).

2. Language Tools and Resources

• Word Banks: Providing specific vocabulary like "value," "length," "space," or "units."
• Language Frames: Scaffolding sentences (e.g., "Their answers are different because…" or "I agree/disagree because…").
• Mathematical Tools: Framing manipulatives as discourse resources (e.g., "Use the fraction pieces to defend your reasoning").

Conclusion and Upcoming Events

• Resources: References were made to Progress in Mathematics and Math Foundations from William H. Sadlier, and the Decode and Discover decodable math books.
• Sadlier Phonics Institute: A virtual event hosted by EdWebb from July $28$ to July $30$, featuring Dr. Kateri Thunder, Dr. Wiley Blevins, and Dr. Eugene Pringle.