Developing Mathematical Reasoning

Math is not rote-memorizable. Math is not random-guessable. Math is figure-out-able.

Author Pam Harris argues that teaching real math—math that is free of distortions–will reach more students more effectively and result in deeper understanding and longer retention. This book is about teaching undistorted math using the kinds of mental reasoning that mathematicians do.

Memorization tricks and algorithms meant to make math "easier" are full of traps that sacrifice long-term student growth for short-lived gains. Students and teachers alike have been led to believe that they’ve learned more and more math as they move through the content, but in reality students are not necessarily progressing in their ability to reason mathematically.

Using tricks may make facts easier to memorize in isolation, but that very disconnect distorts the reality of math. The mountain of trivia piles up until students hit a breaking point. Humanity's most powerful system of understanding, organizing, and making an impact on the world becomes a soul-draining exercise in confusion, chaos, and lost opportunities.

In her landmark book Developing Mathematical Reasoning: Avoiding the Trap of Algorithms, Pam emphasizes the importance of teaching students increasingly sophisticated mathematical reasoning and understanding underlying concepts rather than relying on set rules for solving problems. Now, in this next companion volume, Developing Mathematical Reasoning: The Strategies, Models, and Lessons to Teach the Big Ideas in Grades 3–5 equips educators with practical tools to move beyond rote memorization toward true mathematical thinking for students in upper elementary grades. Focusing on additive and multiplicative reasoning, the book introduces strategies designed to improve mathematical reasoning, Problem Strings, and strategic modeling to strengthen student understanding.

Highlights include:

• Reasoning-based strategies: Replace traditional algorithms with approaches that build critical thinking while ensuring understanding.
• Problem Strings: Step-by-step guidance on walking students through a sequence of problems that spark insight.
• Grade 3–5 focus: Comprehensive coverage of additive and multiplicative reasoning tailored for upper elementary learners.
• Practical tools: Ready-to-use routines, discussion prompts, and modeling techniques for immediate classroom application.

Help students learn to think mathematically rather than memorize. Build confidence, deep understanding, and an appreciation for the logic and beauty of math.