Educators' Learning from Lesson Study

Mathematics for Ages 5-13

Akihiko Takahashi, Thomas McDougal, Shelley Friedkin, Tad Watanabe, Akihiko Takahashi, Thomas McDougal, Shelley Friedkin, Tad Watanabe

• Publisher: Routledge
• Year: 2022
• eBook ISBN: 9781000589535
• Edition: 1
• Topic: Education
• Subtopic: Education General

Offering voices from the field – the first of its kind outside of Japan – this guide to teaching and learning elementary mathematics highlights real case examples from teachers and educators who share what they have learned through Lesson Study.

The teachers' reports provide vivid examples of new insights and ideas about mathematics, about pedagogy and lesson design, about student learning, and about professional collaboration gained through Lesson Study. Each report includes an abbreviated plan of the specific research lesson that led to the new insights, which readers can draw from to replicate the powerful learning in their own community. The case examples of this book are from Lesson Study in mathematics, elementary to lower secondary grade levels, focused on what teachers and educators have learned about improving mathematics teaching and learning; but many ideas from each report can be applied to other subjects and different grade levels.

This unique book will be an excellent resource for mathematics teachers in training and practice who seek to improve mathematics teaching and learning in their own and others' classrooms, including researchers and school administrators who lead professional development.

II What we learned about the contents we teach

DOI: 10.4324/9781003230915-5

"Doing the math"

Word problems in the primary grades — solve "take-from with change unknown" story problems with grade 1 (6- and 7-year-old) students

Brigid Brown

DOI: 10.4324/9781003230915-6

One might think it would be hard to find a topic of common interest for a Lesson Study group spanning three grade levels, two languages of instruction, brand new teachers and veterans, and both a general- and a special-education-inclusion context. That was not the case for this team of primary grade teachers at Acorn Woodland Elementary School, for which I was a facilitator. With resounding conviction, they agreed: our students struggle with subtraction word problems. What causes this struggle? And what learning experiences and tools can we provide to help them make sense of this work? In our research, we would come to discover that a powerful way to answer these questions was to explore subtraction word problems from our students’ perspectives, attempting to solve as we imagined they might.

Across these different contexts and among students aged 5 to 8 years old, the teachers had noticed that when our students encounter a word problem, they often rush into calculations without taking time to comprehend the situation in the problem. Often this meant that students would scan the problem, take the two numerals they saw, add them, and name the total as the answer. The team wondered how they could slow students down enough for them to make sense of a problem first and then choose an operation that matched the context of the problem. To do so, students would need to attend to the meanings of the numbers in the problem, especially to the unknown. More broadly speaking, team members wanted to better understand their students’ mathematical thinking and to teach their classes in a way that would allow students to experience agency in their learning.

To ground our research in our students’ own experience of solving word problems, I led the team in a hands-on activity for exploring word problem types and structures, an activity that the team called “doing the math.” Using Elementary and Middle School Mathematics: Teaching Developmentally (Van de Walle, Karp, & Bay-Williams, 2016) as our guide, we first read about the different types of addition and subtraction problem types and structures. Then we went through each one and attempted to solve the way we thought our students might, using counters, diagrams, and equations. For reference, a chart similar to the one our team used is available in the Mathematics Glossary, Table 1, of the Common Core State Standards, available online at www.corestandards.org/Math/Content/mathematics-glossary/Table-1/.

Very quickly in the process of “doing the math” ourselves, the team noticed differences in the degrees of difficulty in the problem types and structures. Problems in which the result was unknown were easy to model with counters, and the equations looked familiar. But in problems where the unknown quantity represented the change or the start, we found it was difficult to use the counters to show our work. We were surprised to see that we came up with different equations to represent certain problems as well. For example, in a “take-from” subtraction situation where the change was the unknown, some team members were certain that an equation showing the total minus the known result was the proper representation (T − R =?), whereas others were convinced that the known result should fall to the right of the equal sign, with the total minus the unknown change to the left (T −? = R).

Team members reflected on how they’d never have guessed that discussing solution methods for addition and subtraction word problems would have led to such rigorous debate (and frankly, confusion) among a group of adults – these are kindergarten and first grade standards after all! The exercise left team members with a deeper appreciation for the complexities of these problems, and how significantly that complexity can vary among problem types and structures.

The team began to think of word problems as complex texts. Drawing on their literacy work, the team decided to try explicitly teaching students to use visualization with their word problems, just as they did in their reading. As a bridge to this strategy, the team would dedicate a series of lessons to unpacking one word problem at a time, treating the word problem like a storybook. The teachers designed four illustrations to align with the three quantities in their problem plus the question at the end, hypothesizing that this might serve as a productive support for relating the context to the corresponding equation.

Finally, the team considered what discussion questions could guide students to think carefully about the story context and the missing information: “What do we already know? What are we trying to figure out?” These questions could be applied in many lessons throughout the year in order to reinforce the concept of the “unknown” (or “mystery number”) in a word problem as a more precise way to think of the “answer.”

This process of study left a strong impression on our team of how mathematics teaching in the primary grades can be deceptive in its apparent simplicity. Working through specific word problems as we thought our students might allowed us to experience the “sense-making” required of our students, which was often in contrast to the “answer-getting” we remembered being emphasized in our own early schooling. We saw that even for what seems like a simple problem, “doing the math” could take us a long way towards teaching our students more effectively.

Lesson plan

• Title of the lesson: Solve Take-From with Change Unknown Story Problems
• Students’ school: Acorn Woodland Elementary School, Oakland, CA
• Student ages: Grade 1
• Instructor: Malia Vitousek
• Co-authors: Kristen Brett, Brigid Brown, Shelley Friedkin, Jayme Kritzler, Francisco Llaguno, Maira Lopez, Natalia Ruiz, Tala Sullivan, Malia Vitousek
• Date: October 24, 2018
• Goal of the lesson: Students will be able to use their chosen strategy effectively to solve the problem on their own. Students will be able to articulate what each number represents.

Learning standards

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem (CCSS Math 1.OA.A.1) ( Common Core State Standards Initiative, 2010).

Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8 (CCSS Math 1.OA.B.4) (Common Core State Standards Initiative, 2010).

Flow of the lesson

Introduction

TEACHER: “We have been working on math story problems and today I have a real math storybook to read with you!”

Posing the problem

Teacher reads aloud the story problem: “There are 6 bears swimming. Then some bears went home. Now 2 bears are swimming. How many bears went home?”

Teacher then rereads the story, posting pictures on the board as she goes.

Anticipated responses

Note: Students may use any combination of the following strategies to reach their answer.

• manipulatives
• drawings
• picture bond
• number bond
• equations

• R1: Students combine 6 and 2 and name 8 as their answer.
• R2: Students use manipulatives or diagrams to separate 6 into 4 and 2, but choose 6 as the answer or are unsure of which number is the answer.
• R3: Students use manipulatives or diagrams to separate 6 into 4 and 2 and correctly identify 4 as the answer.
• R4: Students use either an addition or subtraction equation but choose 6 or 2 as their answer.
• R5: Students use an addition equation to identify 4 as the missing addend. 2 + _ = 6
• R6: Students use subtraction to identify 4 as the difference. 6 – 2 = _
• R7: Students use subtraction to identify 4 as the missing subtrahend. 6 − _ = 2

Comparing and discussing

Teacher presents two to three strategies that correctly depict the story, so the emphasis in discussion is on understanding how students’ strategies connect to the story, not on choosing the correct answer. Ideally one strategy uses manipulatives and one involves an equation, so as to have both concrete and abstract strategies represented in the discussion.

Teacher will ask questions such as:

• What does the (name specific part of diagram or equation) represent? Where’s this in the story?
• Six whats? (Six bears swimming at first)
• Which numbers did we know? What was the “mystery number”?
• How are these two strategies similar? Different? Where’s the (name specific part of diagram or equation) in this strategy . .. and in this one?”

Teacher may push students’ thinking by

• adding labels according to students’ responses
• prompting students to add equations to match their models
• “playing the fool,” asking “Ok, so 2 went home?” in order to push students to explain, no, we don’t know how many went home.

Teacher may offer the equation 6 − _ = 2 under the story book as a way to represent the problem.

Summing up

Planned summary: A picture bond [or number bond] can help us find the mystery number.

Extension problem: There are 9 birds in a tree. Some birds flew away. Now there are 3 birds in the tree. How many birds flew away?

Board plan

Figure 2.1 Board plan for the lesson

Observations from the lesson

During the lesson, I immediately noticed the routines and norms Malia had been building in her classroom. Malia invited the students into the work of the lesson, asking them “are you up for the challenge?” to which they chorally responded in the affirmative. Students ea...

Table of contents

• Cover
• Half Title
• Series
• Title
• Copyright
• Dedication
• Table of Contents
• List of figures
• List of tables
• Foreword
• Preface
• Acknowledgements
• List of contributors
• I How we expect Lesson Study to contribute to the quality of teaching and learning
• II What we learned about the contents we teach
• III What we learned about lesson design and pedagogy
• IV What we learned about student
• V What we learned about teacher collaboration and leadership
• VI Ideas for establishing sustainable Lesson Study - what we learned from US schools
• Index