San Francisco Field Site

Current Research Collaborations

• How do math teachers unpack the range of student thinking present in classrooms so they can help students develop rich understanding of the mathematics?

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A SERP collaboration between SFUSD practitioners and researchers from the University of California at Berkeley is developing innovative instructional approaches designed to improve the capacity of middle school mathematics teachers and math teacher leaders to explore students’ ways of thinking about mathematics in an effort to increase their understanding and skill.

Timeline: November 2007 to present

District co-developers:
Jeanne D’Arcy, Supervisor, Mathematics and Science
Kirstin Hernandez, Instructional Support Specialist, Superintendent Zone-Mission
Ricki Tai, Instructional Support Specialist, Middle School Mathematics

Teacher co-developers:
Rena Frantz, 7th grade, Roosevelt Middle School
Norman Mattox, 6th grade, James Lick Middle School
Jennifer Gallardo Payne, 7th / 8th grade, Herbert Hoover Middle School
Shauna Poong, 8th grade, Marina Middle School
Elaine Tam, 6th grade, Alice Fong Yu Alternative Elementary School
Jonathan Woahn, 6th/7th grade, Willie L. Brown Jr. Academy College Preparatory
School

Researcher co-developers:
Evra Baldinger, Graduate student UC Berkeley
Phil Daro, Site Director, SERP
Tina Cheuk, Assistant Director, SERP
Yan Liu, Assistant Director, SERP
Nicole Louie, Graduate student, UC Berkeley
Alan Schoenfeld, Professor of Education, UC Berkeley
Kim Seashore, Graduate student, UC Berkeley
Niral Shah, Graduate student, UC Berkeley

Meaningful engagement with word problems can improve students’ performance later in Algebra I and beyond. Success with word problems is of critical importance in terms of students’ mathematical development and understanding plays an important role in student success on high-stakes tests. In designing new instructional approaches, this collaborative team has developed a set of ‘diagnostic lessons’ that have a powerful impact on the way teachers think about their teaching and about student thinking and learning in the context of word problems. The lessons can be used with existing curricula regardless of teaching styles and support the Standards for Mathematical Practice as stated in the Common Core State Standards.

The purpose of formative or “diagnostic” teaching is to build on what students know in order to help them develop rich mathematical understandings. This is very different from classic “remediation,” which focuses on what students don’t know and tries to “fix” it, often by repeating narrow drills the students have failed to learn from before. Students are not blank slates; they bring to the classroom the partial understandings they have developed through the years. The goal is to identify what understandings the students do have, and to build on them in a robust way.


A diagnostic lesson begins with a well-designed mathematical task that focuses on a central mathematical topic. The task is carefully designed to: (1) be engaging – motivation matters! (2) provide multiple access points, so all students can get involved in meaningful ways, and (3) reveal typical understandings and misunderstandings.

Students are given time to respond to the task, and discuss their thinking with their peers. As they do, the teacher often circulates through the classroom, identifying issues for discussion. Teachers are prepared with a series of questions that reveal the varied ways students approach the task.

In whole class discussion, the teacher highlights typical misconceptions, indicating where student thinking might go wrong. But – and at least as important – they show how student ideas can be made to work. Starting from the way of thinking that is easiest to begin with, teachers lead students progressively through ways of thinking that wind up connecting all the student approaches and building a solid understanding of the chapter’s target mathematics.

Such lessons take time. It might take two or three class periods to build all the connections related to a problem that can be “solved” in just 10 or 15 minutes using techniques from the chapter. But, teaching in this way can be stunningly efficient. Formative or diagnostic lessons can pull things together for all students, building foundations for those who need them and helping more advanced students make multiple mathematical connections, strengthening their understandings. In that way diagnostic lessons can replace the huge amount of time spent on unsuccessful remediation, and pull together content not only from various parts of a unit, but across units as well.