During his years working as an instructional coach for a national network of schools, Geoff Krall had the chance to witness several inspirational moments when math class comes alive for middle or high school students - when it is challenging but also fun, creative, and interactive. In Necessary Conditions: Teaching Secondary Math with Academic Safety, Quality Tasks, and Effective Facilitation, Krall documents the essential ingredients that produce these sorts of moments on a regular basis and for all students. They are Academic Safety, Quality Tasks, and Effective Facilitation.

  • Academic Safety: Krall implements equitable classroom experiences that help fight stigmas associated with race and gender in schools. This allows students to feel socially and emotionally secure while nurturing their identities as mathematicians and increasing engagement during classroom discussions
  • Quality Tasks: Teachers can adapt or create dynamic, student-centered lessons that break down math into small, manageable sections, removing the frustrations felt by students who aren't considered math people
  • Effective Facilitation: This book shows how to incorporate teaching moves and math routines designed for engagement, persistence, and interactivity. Teachers can allow students to explore safely while maintaining consistent classroom expectations.

"My work as a math instructional coach for a network of schools has afforded me the unique opportunity to visit exceptional teachers across the country, documenting their tasks, teaching moves, and academically safe learning environments. You'll experience dispatches from these effective classrooms in which we'll observe how teachers attend to all three elements that make up the ecosystem." - Geoff Krall from his book, Necessary Conditions.

chapter Chapter 1|13 pages


Welcome to Math Class

part |34 pages

Academic Safety

chapter Chapter 2|9 pages

Math and the Mathematician

chapter Chapter 3|22 pages

Math and Academic Safety

part |58 pages

Quality Tasks

chapter Chapter 4|19 pages

Defining Quality Tasks

chapter Chapter 5|35 pages

Task Models and Designing, Finding, and Adapting Tasks

part |44 pages

Effective Facilitation

chapter Chapter 6|18 pages

Norms, Structures, and Routines

chapter Chapter 7|20 pages

Facilitation Within a Task

part |76 pages

Putting the Pieces Together

chapter Chapter 7|20 pages

Planning the Lesson

chapter Chapter 9|13 pages

Problem Solving and Pedagogy

chapter Chapter 10|26 pages


chapter Chapter 11|14 pages

The Physical Environment

part |37 pages

Zooming Out

chapter Chapter 12|16 pages

Structuring the Year

chapter Chapter 13|19 pages

Aligning the System, Crafting the Story