Home Toolkit Toolkit to Support Evidence-Based Algebra Instruction in Middle and High School

Introduction

Welcome educators! This Algebra Toolkit is a collection of professional development and implementation resources developed by the Regional Educational Laboratory Central through the Institute of Education Sciences, U.S. Department of Education. This toolkit is designed to support algebra educators in implementing evidence-based recommendations described in the What Works Clearinghouse (WWC) Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students practice guide.

The practice guide presents three evidence-based recommendations:

Use solved problems to engage students in analyzing algebraic reasoning and strategies.
Teach students to utilize the structure of algebraic representations.
Teach students to intentionally choose from alternative algebraic strategies when solving problems.

Toolkit Ingredients

In addition to school and district leader support, effective implementation of the Algebra Toolkit also requires the necessary resources. Learn more in:

Resources needed to implement the Algebra Toolkit PDF (141 KB)

The resources in this toolkit are designed to help you incorporate the three evidence-based recommendations from the practice guide into your classroom to help students succeed in algebra and set them up for success in future math courses. This toolkit includes a series of learning modules that will introduce you to the evidence-based recommendations from the practice guide, help you understand the key concepts and principles behind these recommendations, and provide specific instructional strategies to support your use of the recommendations in the classroom. You can integrate these instructional strategies into your current teaching and curriculum. That is, it is not intended to be a curriculum unto itself, but to help improve upon existing practices.

The Algebra Toolkit is intended primarily for use by in-service educators who want to improve their teaching of algebra. The instructional strategies supported by the toolkit can be useful for educators at all different levels: educators brand new to algebra instruction, those who have experience but have struggled in some aspect of instruction, and those whose practice is well-advanced but are just looking to sharpen their skills. This toolkit, through the framework of Plan-Do-Study-Act (PDSA) cycles, is designed to support each of you as you plan for and implement the instructional strategies in your classrooms, gather data to assess, and work to continually improve.

In developing this toolkit, we have designed it to facilitate the work of a group of educators, supported by a facilitator, working together in a professional learning community (PLC). We recognize that some educators may already be using some or all of these instructional strategies effectively. Others may be trying these instructional strategies but not getting results, so this toolkit is a way to examine why that might be so. Other educators might be working to make these instructional strategies a more common experience across their algebra courses, so they are interested in making them more routine. Still others might be struggling with the content itself, such as truly understanding what is meant by structure or being able to develop multiple representations for algebraic problems and solutions. Each of you brings individual strengths and can find value in collaborating with your colleagues in a PLC.

The toolkit is designed to be flexible and can be adapted to fit the needs and schedules of different groups of educators; for example groups of educators who work together within a school, across multiple schools in a district, or remotely across districts. Your team might also include administrators who wish to strengthen their knowledge and skills as instructional leaders for math.

The Algebra Toolkit includes several components to support you, as educators, as you implement instructional strategies to support the practice guide recommendations in your classrooms:

Facilitator's guides provide guidance and support for the facilitator who will lead your small group through the toolkit.
Participant workbooks provide an overview of the toolkit and resources you will need and guides you through the learning modules and activities.
Links to the What Works Clearinghouse practice guide, Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students, identifying the evidence-based recommendations, as well as aligned instructional strategies and examples that are the focus of the toolkit. This practice guide is research-based and provides a foundation for the learning modules and activities in the toolkit.
Data tools, including educator self-reflection tools, student surveys, classroom visitation tools, and student knowledge assessment tools, help you gather data on your use of the instructional strategies. These data tools can be used to inform the development of action plans (as part of the PDSA cycles) and guide the implementation of the instructional strategies in your classroom. By using these tools regularly, you can track your progress and make adjustments to improve your teaching over time.
The PDSA Tool provides a framework for applying the instructional strategies in your classroom and assessing their effectiveness.
Online videos provide guidance and support as you work through the learning modules and apply the instructional strategies in your classrooms. These videos feature educators who have successfully implemented the instructional strategies, as well as experts who provide insights and advice on key concepts and challenges.
Optional online knowledge checks help educators assess their comprehension and understanding of the material included in the practice guide and modules. They also provide educators opportunities to reflect and develop a greater awareness of their progression through the material.

Together, these components provide a comprehensive toolkit to support you in implementing the evidence-based recommendations and improving your teaching of algebra.

The overall structure of the toolkit is built around the three, evidence-based recommendations from the practice guide. We have developed three modules that focus on each of the practice guide recommendations. These modules will help you build an understanding of the recommendations, as well as guide you and your colleagues in implementing instructional strategies to support the recommendations in your classroom, supported by the structure of a Professional Learning Community (PLC). These three modules are sandwiched between an initial "Setting the Stage" session to orient you to the toolkit and help start your PLC, and a final module (Module 4) designed to support sustained and long-term use of the instructional strategies in your classroom, school, and district. In this final module, having completed Modules 1–3, you will reflect across the three modules to identify successes and work together to address challenges moving forward. Then, you will be guided to develop action plans to support sustained use of effective instructional strategies in your classrooms. Additionally, you will share your developed action plans with your administrators and begin discussions about how to expand the use of effective practices to additional educators and classrooms in your schools and/or districts.

This toolkit is designed for educators to complete as part of a PLC. Coming together as a community of learners around a shared interest brings coherence and continuous learning to professional development (Vescio et al., 2008), and learning collaboratively has many benefits, even if your peers are not in the same physical school building.

The phrase "professional learning community" has been used for many years, to mean many different things. For this toolkit, we mean:

Professional. Teaching is a profession, and professionals engage in ongoing improvement efforts, including participating in professional development such as the toolkit modules.
Learning. Educators are lifelong learners. This toolkit is for current algebra educators and is intended to help you learn more deeply about evidence-based recommendations for teaching algebra.
Community. The process of learning involves reasoning and communication, which are often facilitated by professionals learning as part of a community. Having reflective conversations and giving and receiving peer feedback can help educators improve their teaching.

Source: Vescio, V., Ross, D., & Adams, A. (2008). A review of research on the impact of professional learning communities on teaching practice and student learning. Teaching and Teacher Education 24(1), 80–91. http://eric.ed.gov/?id=EJ782410

A circle divided into four quadrants labeled plan, do, study, act. There are four connecting arrows on the outer edge of the circle.

A PDSA cycle is a four-phase cycle of inquiry used by practitioners—such as educators, school leaders, and school staff—to continually improve their practice. During PDSA cycles, practitioners initiate small changes to their practice; collect information (or data) about how well the changes are being implemented and whether they are contributing to intended outcomes; reflect on learnings from the data; and make decisions about whether to continue with the practice or initiate additional small changes.

Engaging in PDSA cycles is one way to improve mathematics teaching and learning continuously and iteratively. They are a component of improvement science approaches and have been used extensively in education improvement efforts.

The role of the facilitator in your team is to provide support, guidance, and coordination for the sessions within each of the three modules. When identifying the role of facilitator, you should be thoughtful about who makes the most sense in your context. In the least, the facilitator should be able to successfully coordinate the PLC sessions and guide participants through the content. In your school or district, this might be an experienced algebra educator or an instructional leader. In the case where a educator is serving as facilitator, it would likely be a heavy lift for that educator to also be a participant, so we recommend the facilitator focus on the role of facilitation, unless your PLC group is small and educators are sharing facilitation roles. Through the facilitator guide, we've provided details and examples, as well as scripts to support facilitators. The facilitator's main responsibilities are to:

Identify interested and motivated algebra educators (and possibly administrators) who will make up the team. Educators could be from a single school, multiple schools in the same district, or, if collaboration is easily feasible, in multiple schools across multiple districts.
Plan and prepare for the professional development modules and sessions based on the materials in the facilitator guide, and the calendars and agendas provided.
Schedule the professional development sessions around the participants' availability and coordinate communications sharing key information around session details and prework.
Facilitate the sessions and guide educators through the completion of each of the modules.
Provide support and guidance to the educators as they implement the instructional strategies in their classrooms, and as the educators collect and analyze data on the effects of the instructional strategies on their students' learning.
Continuously monitor and evaluate the effectiveness of the sessions and professional development modules and make adjustments as needed to improve their impact.

For implementation of the Algebra Toolkit to be successful, it is critical for school or district leaders to provide educators and facilitators with necessary support and resources. Some of the supports that might help make the toolkit process successful include:

Recruiting participants and facilitators. Identifying engaged and motivated participants who can support one another in completing the toolkit is critical. School and district leaders can help by identifying educators or administrators interested in taking on an instructional leader role as facilitator and educators who are ready to reflect on and improve their practice through the toolkit activities.
Encouragement and support. Administrators and school or district leaders can provide encouragement and support for educators as they work through the toolkit and implement the instructional strategies in their classrooms. This might involve providing regular updates and feedback on the progress of the group and recognizing and rewarding educators who successfully implement the instructional strategies.
Staff time and resources. Administrators and school or district leaders can provide the staff time necessary for participants to hold the PLC sessions and complete the toolkit's professional development modules as well as implement the instructional strategies in their classrooms. This might involve releasing participants from their regularly scheduled meetings, administrative responsibilities, or other professional development, or identifying summer or out-of-school time that can be used for toolkit activities supplemented with professional development funds. Remember that the toolkit is not intended to be a curriculum unto itself, but to help improve upon existing practices. As such, there may be places where the PLC sessions fit naturally as a replacement for other scheduled professional development.
Institutionalizing supports. Administrators with responsibility for or interest in algebra instruction in your school or district can take on a greater role in supporting the toolkit through the Algebra Toolkit Administrator's Guide. This set of materials outlines a briefer set of content and activities to guide administrators in understanding and supporting the work of the PLCs. With adequate understanding of the Algebra Toolkit content and processes, administrators can offer PLCs the support, structure, and resources to make the current year of toolkit implementation a success and can help lay groundwork to institutionalize the instructional practices and collaboration strategies to future years, as well as expand the use of effective instructional strategies to other educators in the school or district.

The Algebra Toolkit is intended for a group of in-service educators and a facilitator to work through over the course of a single school year. Each learning module includes independent prework, collaborative PLC sessions, and independent between-session work. Educators can usually complete the prework and between-session on their own time. Although the amount of time needed for this work may vary, it typically can be completed in one to two hours.

The length of time it takes to complete the Algebra Toolkit may vary depending on the group's schedule and the amount of time the group is able to devote to the learning modules and activities. The Facilitator's Guide includes a sample planning calendar. The facilitator and team should consider whether there are existing structures and activities, such as professional development or common planning time sessions, which can be used for the in-person PLC sessions. This can help minimize the impact of the meetings on educators' schedules and reduce scheduling conflicts.

GET STARTED

Begin your learning journey! Click the Next button below or select the Setting the Stage tab in the menu at the top of the page.

Introduction

A group of teachers sitting at a table with computers and notebooks.

Setting the Stage is an orientation to the Algebra Toolkit. Learning objectives include:

Orient participants to the toolkit experience, the structure and format of sessions, and professional learning community (PLC) groups.
Introduce the three recommendations and the practice guide.
Provide an overview of the PDSA cycles and the PDSA Tool.

Module overview

PRE-SESSION WORK
Explore toolkit content.
PLC SESSION 1
Establish PLC norms, introduce practice guide recommendations, and prepare for Module 1.
FOLLOW-UP WORK
Prepare for Module 1.

Module resources

Overall
- Facilitator's Guide PDF (2 MB)
- Participant Workbook PDF (2 MB)

MODULE SEQUENCE

Continue your work in this module by moving through the tabs from left to right. When you are ready to move to the next section, Pre-session work, click the Next button below or select the tab in the menu at the top of the page.

Pre-session work

Complete the following prework to familiarize yourself with the Algebra Toolkit and the evidence-based recommendations in the Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students practice guide.

Explore toolkit content
- Read the toolkit introduction in the Participant Workbook PDF (2 MB).
- Read pages 1–3 in the Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students practice guide.
- Review the practice guide summary document developed by the Institute of Education Sciences to accompany the practice guide.
- Explore toolkit content through the website to get a sense of what's involved in the toolkit experience.
Complete the graphic organizer
Complete the graphic organizer in the Participant Workbook PDF (2 MB) in advance of PLC Session 1 to reflect on toolkit goals and content to make connections to your own practice. Come to the first facilitated session ready to share reflections from the graphic organizer.

MODULE SEQUENCE

Continue your work in this module by moving through the tabs from left to right. When you are ready to move to the next section, PLC Session 1, click the Next button below or select the tab in the menu at the top of the page.

By the end of PLC Session 1, participants will:

Establish norms for the toolkit PLC sessions.
Understand the three practice guide recommendations.
Understand the PDSA Tool and how it will be used throughout modules 1–3.

SESSION AGENDA

Establish PLC norms (10 minutes)
Reflect on the following questions, and be prepared to share with the group:
- What was the best professional learning community you have been part of, and what made it so good?
- What makes professional development most valuable to you?
- What kinds of structures or participation norms help you engage in professional learning?
Opening reflection (10 minutes)
Share one of your pre-session work reflections: A rose, thorn, or bud. Listen to your colleagues' reflections and be prepared to discuss common themes.
Overview of practice guide recommendations (20 minutes)
Use Handout S1: Overview of the practice guide recommendations in the Introduction Participant Workbook PDF (2 MB) to start thinking about the three recommendations and what you might ask students to do as you implement each of them.
PDSA cycle overview (15 minutes)
Explore the four phases of the PDSA cycle using Handout S2: PDSA tool for the Algebra Toolkit in the Introduction Participant Workbook PDF (2 MB).
Next steps (5 minutes)
Confirm the date and time for the first PLC session for Module 1 and independently complete the Module 1 pre-session work.

MODULE SEQUENCE

When you are ready to move to Module 1, select the Next button below.

Introduction

A cartoon educator displaying a correctly solved math problem.

By the end of Module 1, participants will:

1.1Understand the value of using solved problems to engage students in analyzing algebraic reasoning and strategies.
1.2Understand how to use solved problems effectively at different points within the instructional cycle.
1.3Be able to efficiently and strategically find or create solved problems that align with their lesson's instructional aim.
1.4Be able to design and facilitate discussions (educator-to-student and student-to-student) about solved problems that effectively support students to build a deeper understanding of algebra concepts and to make connections across concepts.
1.5Be able to design activities that encourage students to compare solved problems to gain insights about problem structures and strategies.
1.6Be able to create a plan for collecting data and select data tools that will give them information about the implementation and outcomes of their planned instructional strategies.
1.7Be able to discuss and analyze data collected during implementation and use data-driven insights to inform continued implementation of Recommendation 1.

Recommendation 1

Use solved problems to engage students in analyzing algebraic reasoning and strategies.

—Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students, p. 4

Module overview

PRE-SESSION WORK
Explore module content.
PLC SESSION 1
Develop an understanding of the recommendation and the related instructional strategies.
BETWEEN-SESSION WORK
Read and reflect about the recommendation and related instructional strategies.
PLC SESSION 2
Plan for implementing the recommendation.
BETWEEN-SESSION WORK
Learn about and review data tools.
PLC SESSION 3
Finalize plan for implementing the recommendation.
BETWEEN-SESSION WORK
Implement the recommendation and collect data.
PLC SESSION 4
Study results using data-driven dialogue and continue to implement the recommendation.

Compared to elementary math work like arithmetic, solving algebra problems often requires students to think more abstractly and to process multiple pieces of complex information simultaneously. This can challenge students' working memory and interfere with their ability to learn new content. Solved problems can minimize the burden of abstract reasoning by allowing students to see the problem and many solution steps at once—without executing each step—helping students learn more efficiently. Analyzing and discussing solved problems, including incorrectly solved problems, can help students develop a deeper understanding of logical strategies for problem solving.

Based on research evidence, the following instructional strategies may boost students' procedural skills and conceptual knowledge and improve algebraic problem solving:

Instructional Strategy A: Have students discuss solved problem structures and solutions to make connections among strategies and reasoning.
Instructional Strategy B: Select solved problems that reflect the lesson's instructional aim, including problems that illustrate common errors.
Instructional Strategy C: Use whole-class discussions, small-group work, and independent practice activities to introduce, elaborate on, and practice working with solved problems.

Educators can use solved problems in a variety of ways to engage students in analyzing algebraic reasoning and strategies. Consider the most appropriate strategies for using solved problems during each stage in the instructional cycle: Introducing a new topic, building understanding of the topic, and approaching proficiency (see sections below).

When introducing a new topic, it may be appropriate to begin by having students discuss correctly solved problem structures and solutions to make connections among strategies and reasoning. For example, you may facilitate a large-group discussion about a single, correctly solved problem to help students understand the logical problem-solving process and problem structure. You may also consider providing a correctly solved problem for students to analyze in small groups or pairs, and to reflect on their discussions with the large group. The following example demonstrates how an educator might use a solved problem along with guiding questions to introduce the topic of solving exponential equations.

A correctly solved problem showing the four solution steps. Solve for x in this equation, 3 to the power of 4 x plus 3 is equal to 81. The solution to the problem starts with an initial restarting of the problem, 3 to the power of 4 x plus 3 is equal to 81, then 3 to the power of 4 x plus 3 is equal to 3 to the fourth power, then 4 x plus 3 is equal to 4, then 4 x is equal to 1, and finally the correct solution of x is equal to one-fourth.

Guiding questions for student discussion – Problem-solving process:

What did the problem solver do first? Why?
What were the steps involved in solving the problem? Why do they work in this order? Would they work in a different order?
What are other problems for which this strategy will work?

Guiding questions for student discussion – Problem structure:

What quantities—including numbers and variables—are present in this problem?
What operations and relationships among quantities does the problem involve? For example, are there multiplicative or additive relationships?
Are parentheses or groupings used in the problem to indicate the problem's structure? How?

Notice that the guiding questions for student discussion ask students to think critically about the order of the problem-solving steps and the structure of the problem. These types of guiding questions are key to maximizing student learning in whole-class or small-group discussions because they help students understand the reasoning behind the solution strategy.

As students build their understanding of the topic, consider engaging them in discussion about problems that illustrate common errors. When displaying incorrectly solved problems, be sure to clearly mark them as incorrect. Ask students to analyze the solved problem, determine where the error occurred, and explain why the error may have occurred. Choose or create problems that show errors that your students are making so that you can correct their most common misconceptions. This strategy can help students avoid common errors and build a deeper understanding of correct problem-solving strategies.

One way to introduce incorrectly solved problems is to display them next to their correctly solved counterparts. The example below demonstrates how an educator might use correctly and incorrectly solved problems presented side by side to deepen students' understanding of solving rational equations.

Sample questions for student discussion:

What steps did Jasmin take to solve the problem? Prompt, if needed: What did she do first, second, third, and last?
What steps did Andrew take to solve the problem? How were Andrew's steps similar to Jasmin's?
What did Andrew do correctly?
Where did Andrew make his mistake?
Why do you think Andrew made this error? What do you think Andrew's reasoning may have been?
How can Andrew avoid this error in the future?

Notice that the guiding questions for student discussion ask students to compare the correct solution to an incorrectly solved problem that uses the same problem-solving strategy. (For example, both problem solvers used the distributive property as their second step rather than dividing both sides of the equation by 2). Displaying parallel problems with the same strategy can scaffold all learners' understanding of the concept. In addition, guiding questions support students to understand the reasoning that led to Andrew's error and how they might avoid the error.

As you consider additional examples to share with students, in the example above, Jasmin's correct strategy ends in an integer solution for x while Andrew's incorrect strategy does not. Students tend to expect integers when solving equations, so presenting a correct strategy that does not lead to an integer solution matched with an incorrect strategy that does lead to an integer solution is a way to challenge and build students' thinking.

As students approach proficiency, you may consider having them analyze and discuss multiple solved problems with varying degrees of complexity. This can help students recognize patterns in the solution steps across problems. As in the following example, you may arrange problems from simplest to most complex applications of the same concept and prompt students to notice similarities in structure and solution steps across problems.

Guiding questions for student discussion:

What do you notice about these three factored expressions? How are they similar and how are they different? Prompt, if needed: The problem solver's strategy needs to be slightly different for expression B than for A and C. Why? How is expression B different?
Why do expressions B and C take two steps to factor, when expression A takes only one step?
For each of the three expressions, create another expression that has a similar structure and solution steps. Show your expressions and their factored forms.

Notice the guiding questions ask students to observe similarities and differences across the three solved problems. In this example, it would be important for students to recognize that expression B does not start as a difference of two squares—the solver must factor 2x from both terms first. Prompting students to consider small, but important, differences across solved problem structures and solution strategies may help them reach proficiency.

As students near proficiency, you may also consider extending Strategy B by displaying problems with errors on their own without any correct counterpart. Ask students to identify the step that contains the error and why it is incorrect. Because students are approaching proficiency, you may ask them to answer guiding questions in pairs or independently (in writing) rather than as a large group. During a unit about linear equations, a teacher might provide the incorrectly solved problem, as in the following example, along with its guided questions, for students to reflect on independently before discussing their answers with a partner.

Problem: When Kendra joins a new gym, she spends $75 to buy her gym supplies, then pays $50 per month for her gym membership. Write an equation to show the overall cost, c, of her new gym plan for m months. How many months can Kendra continue to attend the gym to stay within her budget of $1000?

An incorrectly solved problem showing Jordan's solution. The equation to figure out the cost of gym is c is equal to 50m plus 75. If Kendra's budget is $1000, Jordan incorrectly solved the problem by first writing that c is equal to 50m plus 75, then c is equal to 50 open paren 1000 close paren plus 75, then c is equal to 50,000 plus 75, and lastly the incorrect solution of c is equal to 50,075.

Sample guiding questions for students' written responses and discussion:

Did Jordan write her equation correctly? How do you know?
What steps did Jordan take to solve the problem?
What error did Jordan make in solving this problem?
How can you show that Jordan's answer is incorrect?
What advice would you give to Jordan to help her avoid this error in the future?
Can anyone think of a different way to solve this problem? Show your work below and solve the problem correctly.

Module resources

Overall
- Facilitator's Guide PDF (2 MB)
- Participant Workbook PDF (2 MB)
Data tools
- Educator Self-Reflection Tool PDF (60 KB)
- Module 1 Visitation Tool PDF (219 KB)
- Module 1 Student Survey Part 1: Math Learning (pp. 2-3) PDF (160 KB) | Excel (105 KB)
- Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (160 KB) | Excel (101 KB)
- Module 1 Student Knowledge Assessment Tool Word (45 KB) | Excel (62 KB)

MODULE SEQUENCE

A cartoon teacher displaying a correctly solved math problem.

Learn about and reflect on the content that will be covered in Module 1 before your first PLC Session. This pre-session work will take approximately 60 minutes to complete.

Learn about Recommendation 1
Each of the resources listed below provides additional information about the recommendation. Complete one or more of the following to learn more about the recommendation:
- Read pages 4–15 in the Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students practice guide.
- Read the Module 1 Introduction.
- Watch the Module 1 video.
Complete the graphic organizer
Complete the graphic organizer in the Module 1 Participant Workbook PDF (2 MB) in advance of PLC Session 1 to reflect on the content you reviewed. Be prepared to share your reflections at the next PLC Session.
Watch a practice guide video (optional)
For more information about the levels of evidence in the practice guide, watch this video from the What Works Clearinghouse.

Download relevant resources (optional)
Navigate to the Module 1 Introduction Tab and download the resources you will need for Module 1.

MODULE SEQUENCE

By the end of PLC Session 1, teachers will:

Understand the value of using solved problems to engage students in analyzing algebraic reasoning and strategies.
Understand how to use solved problems effectively at different points within the instructional cycle.

SESSION AGENDA

Opening reflection (10 minutes)
Share one of your pre-work reflections from the graphic organizer. As a group, discuss any common themes.
Activity 1: Benefits of using correctly solved problems to engage students in analyzing algebraic reasoning and strategies (20 minutes)
Explore the benefits of using correctly solved problems in instruction by using Handout 1A: Using correctly solved problems to support student understanding in the Module 1 Participant Workbook PDF (2 MB).
Activity 2: Benefits of using incorrectly solved problems to engage students in analyzing algebraic reasoning and strategies (20 minutes)
Explore the benefits of using incorrectly solved problems in instruction by using Handout 1B: Identifying misconceptions in the Module 1 Participant Workbook PDF (2 MB).
Next steps (5 minutes)
Confirm the date and time for PLC Session 2 and complete the between-session work.

Two women sharing a desk at an adult education class look up.

MODULE SEQUENCE

Continue your work in this module by moving through the tabs from left to right. When you are ready to move to the next section, Between-session Work, click the Next button below or select the tab in menu at the top of the page.

Complete this between-session work before PLC Session 2. All steps will take approximately 60 minutes to complete.

Read Handout 1C
Read Handout 1C: Implementing Recommendation 1 in the Module 1 Participant Workbook PDF (2 MB) and highlight ideas that stand out as potentially useful in your classroom.
Collect or create solved problems
Collect or create at least five solved problems you can use in an upcoming lesson or series of lessons. Remember, solved problems can take many different forms. They can be correctly or incorrectly solved and can be displayed alone or in parallel with other solved problems. Please follow the steps below to collect or create your solved problems.
- Select an upcoming lesson or series of lessons. You may want to check whether the algebra topic in your lesson(s) of focus aligns with topics that are well-suited to Recommendation 1 (refer to Handout 1C, step 1).
- Look through curricular materials, such as textbooks and teacher guides, for solved problems. Consult other resources if needed, such as current or past student work. Refer to step 3 in Handout 1C for additional tips on finding and creating solved problems.
- Create a bank of at least five solved problems. This can be a physical folder with copies of solved problems or a digital folder with screenshots or other files containing solved problems.
Be prepared to share how you might use one or more solved problems in an upcoming lesson.
Complete a knowledge check
Reflect on the content you worked through during PLC Session 1 and complete the knowledge check below by clicking on “Start the knowledge check.” For all problems, consider the equation: 8 – 4b = 12.

Start the Knowledge Check

MODULE SEQUENCE

Continue your work in this module by moving through the tabs from left to right. When you are ready to move to the next section, PLC Session 2, click the Next button below or select the tab in the menu at the top of the page.

By the end of PLC Session 2, participants will:

Efficiently and strategically find or create solved problems that align with their lesson's instructional aim.
Design and facilitate discussions (educator-to-student and student-to-student) about solved problems that effectively support students to build a deeper understanding of algebra concepts and to make connections across concepts.
Design activities that encourage students to compare solved problems to gain insights about problem structures and strategies.

SESSION AGENDA

Opening reflection (10 minutes)
Share reflections from your between-session work. To prompt your reflection, think about the following:
- Which ideas from Handout 1C stood out to you as potentially useful in your classroom?
- How did you create a bank of solved problems? What types of solved problems did you choose? What was easy or challenging about this process? How do you think the algebra topic you were teaching made this easier or more challenging?
Activity 1: Example of Recommendation 1 in the classroom (25 minutes)
Examine an example of how an educator implemented Recommendation 1 into a lesson by using Handout 1D: In the classroom: Recommendation 1 in the Module 1 Participant Workbook PDF (2 MB).
Activity 2: Planning to implement Recommendation 1 in your classroom (20 minutes)
Complete the Plan phase portion (questions 1 and 2) of the PDSA Cycle Tool using Handout 1E: PDSA Cycle Tool – Plan phase part 1 in the Module 1 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
Next steps (5 minutes)
Confirm the date and time for PLC Session 3 and complete the between-session work.

MODULE SEQUENCE

Continue your work in this module by moving through the tabs from left to right. When you are ready to move to the next section, Between-session work, click the Next button below or select the tab in the menu at the top of the page.

Complete this between-session work before PLC Session 3. All steps will take approximately 60 minutes to complete.

Finalize plan
Finalize your plan for implementing the instructional strategies in your classroom based on the work and discussions from PLC Session 2.
Read Handout 1F
Read about how teachers can use data—in a practical, manageable way—to continuously learn about and improve their instructional practice in Handout 1F: Using data to continuously improve instructional practice in the Module 1 Participant Workbook PDF (2 MB).
Review the data tools
Review each of the data tools linked below. These tools are also included in appendix B of the Participant Workbook PDF (2 MB).
- Implementation data tools:
  - Teacher self-reflection tool PDF (60 KB): Complete a short, written reflection after using your planned instructional strategy in the classroom.
  - Visitation tool PDF (219 KB): Ask a colleague or administrator to visit a lesson in which you use an instructional strategy from the toolkit. The visitor records their observations and shares them with you after the lesson.
- Outcome data tools:
  - Module 1 Student Survey Part 1: Math Learning (pp. 2-3) PDF (160 KB) | Excel (105 KB): Administer this short survey to a class of students or to a smaller subset of students following a lesson in which you used an instructional strategy from the toolkit.
  - Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (160 KB) | Excel (101 KB): Administer this short survey to a class of students or to a smaller subset of students during a unit in which you used an instructional strategy from the toolkit.
  - Module 1 Student Knowledge Assessment Tool Word (45 KB) | Excel (62 KB): Administer this knowledge assessment to a class of students following a lesson in which you used an instructional strategy from the toolkit.
  - Data you already collect within your classroom, such as exit tickets, quizzes, or tests.
Complete Handout 1G
Complete questions 3 and 4 of the Plan phase portion of the PDSA Cycle Tool using Handout 1G: PDSA Cycle Tool – Plan phase part 2 in the Module 1 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
Bring materials to PLC Session 3
Bring your completed Handouts 1E and 1G to PLC Session 3 to discuss and get feedback from colleagues. For each data tool, be prepared to share (a) the extent to which you find the data tool useful and (b) how you might use the data tool in your classroom. In addition, bring the associated lesson plan(s) in which you will implement Recommendation 1.

MODULE SEQUENCE

Continue your work in this module by moving through the tabs from left to right. When you are ready to move to the next section, PLC Session 3, click the Next button below or select the tab in the menu at the top of the page.

By the end of PLC Session 3, participants will:

Create a plan for collecting data that will give them information about the implementation and outcomes of their planned instructional strategies.

SESSION AGENDA

Opening reflection (20 minutes)
Review the Plan phase of your PDSA Tool and the associated lesson plan during which you will incorporate the strategies. Record the following on sticky notes (either physical or virtual) and be ready to share your reflections with the group:
- One aspect of your plan that you are excited about and feel confident implementing.
- One aspect of your plan that you are unsure about and would like colleagues’ feedback.
Activity: Planning for data collection (35 minutes)
Gather with a colleague or group of colleagues who chose the same implementation data tool as you. Together, use the discussion questions that follow for your implementation data tool to discuss when and how you will use the data tool in your classroom. Document decisions you make for your classroom within Handout 1H: PDSA Cycle Tool – Plan phase part 2 (cont.) in the Module 1 Participant Workbook PDF (2 MB).
Discussion questions: Implementation data tools
- Educator self-reflection tool
  - On what days or class periods will you implement your planned instructional strategies?
  - When will you complete the educator self-reflection tool?
- Visitation tool
  - On what days or class periods will you implement your planned instructional strategies?
  - Who will your visitor be? If you don’t know yet, who might you consider asking to be your visitor?
  - When will you have the pre-visit conversation with your visitor? How will you prepare for your pre-visit conversation?

When you are finished with your implementation discussion, gather with a colleague or group of colleagues who chose the same outcomes tool as you. Together, use the discussion questions that follow for your outcome data tool to discuss when and how you will use the data tool in your classroom. Document decisions you make for your classroom within Handout 1H: PDSA Cycle Tool – Plan phase part 2 (cont.) in the Module 1 Participant Workbook PDF (2 MB).

Discussion questions: Outcome data tools

Student Survey
- Will you use all the items on the survey, or will you choose specific items? If you are choosing specific items, circle or highlight the items that you will use.
- You can choose to administer the survey twice (both before and after you implement your planned instructional strategies) or just once (after you implement your planned instructional strategies). Which option will you choose and why? What might the advantages and disadvantages of each option be?
- On what days or class periods will you administer the survey?
Student Knowledge Assessment
- Which template will you use and why?
- Which math problem will you use? If you don’t know yet, which problems might you consider using?
- On what day or class periods will you administer the knowledge assessment?
Exit ticket, quiz, or other classroom data
- What type of data will you use to assess student outcomes? Why did you choose this type of data?
- How will your classroom data align with the learning objectives for the lesson you are teaching or with the instructional strategies you’re planning to implement?
- Which math problems will you use? If you don’t know yet, which problems might you consider using?
- On what days or class periods will you administer the knowledge assessment?

Next steps (5 minutes)
Confirm the date and time for PLC Session 4 and complete the between-session work.

MODULE SEQUENCE

Complete this between-session work before PLC Session 4. All steps will take approximately 120 minutes to complete.

Review your plan
Review your plan for implementation and data collection based on colleagues' feedback from PLC Session 3.
Implement your plan
Implement your planned instructional strategies during a lesson or series of lessons according to the plan you laid out in the Plan section of the PDSA tool (Handouts 1E, 1G, and 1H).
Collect data and take notes
Collect data, record results, and take notes about happened when you implemented the instructional strategies in the first section of the PDSA Tool Handout 1I: PDSA Cycle Tool – Do phase in the Module 1 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB). Note: if you plan to complete the visitation tool, consider having a pre-visit conversation with your visitor before the visitation.
Independently study your data
Reflect on your implementation and/or outcome data for the instructional strategies aligned with Recommendation 1 in Handout 1J: PDSA Cycle Tool – Study phase in the Module 1 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB). Come to PLC Session 4 ready to share reflections and responses to these questions.

MODULE SEQUENCE

Continue your work in this module by moving through the tabs from left to right. When you are ready to move to the next section, PLC Session 4, click the Next button below or select the tab in the menu at the top of the page.

By the end of PLC Session 4, participants will:

Be able to discuss and analyze data collected during implementation and use data-driven insights to inform continued implementation of Recommendation 1.

SESSION AGENDA

Opening reflection (10 minutes)
Using Handouts 1I and 1J, share how your implementation and data collection went for you.
Consider the following reflection questions:
- Which instructional strategies did you use?
- What was challenging about using solved problems to engage students in analyzing algebraic reasoning and strategies?
- What possible explanations do you have for any findings or trends in your data?
- Did you observe improvement in outcomes? How do you know?
Data discussion (25 minutes)
In this activity, you will work in pairs and take turns presenting so that each of you has equal opportunity to share. Spend half the time focusing on the first presenter's data, then switch. The presenter should briefly share an overview of their implementation. Then, collaboratively examine the data and the presenter's responses to the discussion questions in Handout 1J: PDSA Cycle Tool – Study phase in the Module 1 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
- What do the data suggest about aspects of the instructional strategies that are being implemented effectively?
- What do the data suggest about how you should revise implementation of the strategies?
- What do the data suggest about student learning outcomes that are being achieved?
- What do the data suggest about how you should revise implementation to improve student outcomes?
Summary discussion (5 minutes)
Share out about overall success and challenges:
- What went well when you used solved problems in instruction? Discuss details about what went well with your instructional planning; how your instructional strategies played out in the classroom; and how students responded.
- What was challenging about using solved problems in instruction?
Reflection – Act phase (15 minutes)
Create a high-level plan for continuing to refine your implementation strategies for Recommendation 1 by using Handout 1K: PDSA Cycle Tool – Act phase in the Module 1 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
Next steps (5 minutes)
Review the follow-up work and confirm the date and time for the first PLC session for Module 2.

Follow-up work: Begin a new cycle of implementation

Congratulations on completing your first PDSA cycle! Now you can refine your instructional strategies regarding using solved problems in instruction. Please find the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB) or available for download here PDF (116 KB). Using your completed Act phase of Handout 1K, create a new plan for implementation using the PDSA Tool based on the data-informed revisions you noted above and continue a new cycle of implementation.

As you continue to integrate Recommendation 1 into your teaching, consider what supports you need from your PLC members and other members of your school community. This might involve scheduling collaborative planning or visitation time with PLC members to discuss how to incorporate Recommendation 1 into lessons, share successful strategies, and observe each other's implementation. You also might solicit feedback from administrators, colleagues, and students on the effectiveness of your implementation of Recommendation 1, using some of the data tools in the Participant Workbook PDF (2 MB) appendices B1, B2, and B3, as well as structuring more uses of the student assessment tools in appendix B4. Continue to draw from the guidance for implementation shown in Handout 1C to identify approaches to using solved problems in instruction.

Please note that you should aim to complete multiple PDSA cycles for Recommendation 1 within this current school year. Feel free, however, to complete these cycles when it is feasible for you considering your schedule, curriculum, and progression through the other toolkit modules.

Five persons celebrating.

MODULE SEQUENCE

When you are ready to move to Module 2, select the Next button below.

Introduction

A cartoon student raising their hand to ask a question.

By the end of Module 2, participants will:

2.1 Understand what is meant by the structure of algebraic representations.
2.2 Understand the three recommended instructional strategies to teach students to utilize the structure of algebraic representations.
2.3 Be able to design activities that support students to utilize the structure of algebraic representations: promoting the use of language that reflects mathematical structure; encouraging students to use reflective questioning to notice structure as they solve problems; and/or teaching students that different algebraic representations can convey different information about an algebra problem.
2.4 Be able to create a plan for collecting data and select data tools that will give them information about the implementation and outcomes of their planned instructional strategies.
2.5 Be able to discuss and analyze data collected on implementation and outcomes and use data-driven insights to inform continued implementation of Recommendation 2.

Recommendation 2

Teach students to utilize the structure of algebraic representations.

—Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students, p. 16

Module overview

PRE-SESSION WORK
Explore module content.
PLC SESSION 1
Develop an understanding of the recommendation and the related instructional strategies.
BETWEEN-SESSION WORK
Read and reflect about the recommendation and related instructional strategies.
PLC SESSION 2
Plan for implementing the recommendation.
BETWEEN-SESSION WORK
Learn about and review data tools.
PLC SESSION 3
Finalize plan for implementing the recommendation.
BETWEEN-SESSION WORK
Implement the recommendation and collect data.
PLC SESSION 4
Study results using data-driven dialogue and continue to implement the recommendation.

Utilizing the structure of algebraic representations can simplify solving algebra problems. Paying attention to algebraic structure helps students make connections among problems, solution strategies, and representations that may seem different but are mathematically similar, as shown in Example 2.1. Recognizing structure helps students understand the characteristics of algebraic expressions and problems regardless of whether the problems are presented in symbolic, numeric, verbal, or graphic forms.

Structure refers to an algebraic representation's underlying mathematical features and their logical relationships to each other. Examples of mathematical features and their logical relationships include:

The number, type, and position of quantities, including variables.
The number, type, and position of operations.
The order in which operations are done.
The presence of an equality or inequality.
The relationships between quantities, operations, and equalities or inequalities.
The range of complexity among expressions, with simpler expressions nested inside more complex ones.

Examining the underlying structure of an algebra problem (the algebraic representation), regardless of how the problem itself is communicated (for example, symbolic, numeric, verbal, or graphic), can help students see similarities among problems and solution paths. It also leads students to develop an understanding about algebraic expressions. Based on research evidence, use of reflective questioning and graphical representations may boost students' conceptual and procedural knowledge and improve algebraic problem solving. The use of appropriate language and multiple representations might further support students. Educators can use each of these instructional strategies to teach students to utilize the structure of algebraic representations:

Instructional Strategy A: Promote the use of language that reflects mathematical structure.
Instructional Strategy B: Encourage students to use reflective questioning to notice structure as they solve problems.
Instructional Strategy C: Teach students that different algebraic representations can convey different information about an algebra problem.

This instructional strategy extends beyond just using precise mathematical language. This strategy helps us contrast between precise mathematical language that accurately communicates the structure of the problem (but that requires knowledge of the meaning of the mathematical terms), and more commonplace language that is more accessible to students but does not as clearly describe the structure of the problem (and therefore might lead to confusion as students choose solution strategies). As educators, we know there are times when students use more accessible language to help them describe or understand a math task. The goal here is not to always correct students, but to guide them from this informal language to more precise language. This will help ensure the language that educators and students use does not cause confusion but helps students understand the structure of the problem so they can choose appropriate solution strategies for this structure. Review the table below for examples of how imprecise language might cause confusion for students as they choose solution strategies for a given problem.

Using precise language to understand structure: (adapted from practice guide Example 2.3)

Imprecise language	Precise mathematical language	How does the imprecise language cause confusion for students regarding structure and appropriate solution strategies?
Move the 5 over.	Subtract 5 from both sides of the equation.	Students might move the quantity from one side of an operation to the other, rather than conducting the inverse operation to both sides of the equation to maintain equality.
Solve an expression.	Solve an equation. Rewrite an expression.	Students might attempt to solve an expression by using the operation(s) as the equality sign and manipulate around that sign.
The numbers cancel out.	The numbers add to zero. The numbers divide to one.	Students might not realize they need inverse operations to "cancel the numbers out" (e.g., 5+5=10 vs. 5+-5=0).

Below, you'll see a list of reflective questions for noticing structure. The list presents the kinds of questions educators can use to help students grasp the structure of algebra problems. You can use these questions to help students think about the structure of a math problem and the potential strategies they could use to solve the problem in a variety of ways in the classroom. You may choose to write down questions you consider for a particular math problem and how you might answer them, modeling for students the process of considering reflective questions. Then you might present a problem during whole-class instruction and ask students to write down what questions they might ask themselves to help solve the problem. Students can practice this process in pairs or on their own to help internalize what to consider when approaching a problem.

Using reflective questioning to notice structure

Reflective questions for noticing structure (Practice Guide Example 2.5)

What am I being asked to do in this problem?
How would I describe this problem using precise mathematical language?
Is this problem structured similarly to another problem I've seen before?
How many variables are there?
What am I trying to solve for?
What are the relationships between the quantities in this expression or equation?
How will the placement of the quantities and the operation impact what I do first?

Recognizing and explaining corresponding structural features of two representations of the same problem can help students understand the relationships among different algebraic representations, such as equations, graphs, and word problems. In the table below, you'll see an example of how a educator asks students to compare different forms of equations for the same line. Students may come up with different equations, or educators can present students with equations in different forms so you can ask students to identify similarities and differences. As an alternative, educators can ask students to connect the information provided in various algebraic representations of a math task. Working in pairs, students can then discuss the similarities and differences they identified or the connections they made.

Multiple algebraic representations

Equations of the same line in different forms (Practice Guide Example 2.6)
Compare different forms of equation for the same line.
	Similarities	Differences
Slope-intercept form y=mx+b y=2x-3	Both are equations of straight lines. In both equations, you can identify the slope without any calculation or manipulation. You cannot identify the x-intercept without manipulating the equations.	Slope-intercept form allows you to identify the y-intercept without manipulation.
Point-slope form y-y₁=m(x+x₁) y-5=2(x-4)		Point-slope form allows you to identify a point on the line without manipulation.

Module resources

Overall
- Facilitator's Guide PDF (1 MB)
- Participant Workbook PDF (1 MB)
Data tools
- Educator Self-Reflection Tool PDF (60 KB)
- Module 2 Visitation Tool PDF (168 KB)
- Module 2 Student Survey Part 1: Math Learning (pp. 2-3) PDF (132 KB) | Excel (99 KB)
- Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (132 KB) | Excel (101 KB)
- Module 2 Student Knowledge Assessment Tool Word (45 KB) | Excel (59 KB)

MODULE SEQUENCE

Learn about and reflect on the content that will be covered in Module 2 before your first PLC session. This pre-session work will take approximately 60 minutes to complete .

Learn about Recommendation 2
- Read pages 16–25 in the Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students practice guide.
- Read the Module 2 Introduction.
- Watch the Module 2 video.
Complete the pre-work reflection
Complete the graphic organizer in the Module 2 Participant Workbook PDF (1 MB) in advance of PLC session 1 to reflect on the content you reviewed. Be prepared to share your reflections at the next PLC session.
Watch a practice guide video (optional)
For more information about the levels of evidence in the practice guide, watch this video from the What Works Clearinghouse.

Download relevant resources (optional)
Navigate to the Module 2 Introduction Tab and download the resources you will need for Module 2.

MODULE SEQUENCE

By the end of PLC Session 1, participants will:

Understand what is meant by the structure of algebraic representations.
Understand the three recommended instructional strategies to teach students to utilize the structure of algebraic representations.

SESSION AGENDA

Opening reflection (10 minutes)
Share one of your pre-session work reflections: a rose, thorn, or bud. Listen to your colleagues' reflections and be prepared to discuss common themes.
Activity 1: Utilizing structure with an everyday example (20 minutes)
Think about structure using an everyday example: a cooking recipe. Imagine it is your turn to cook dinner, and you would like to try a new recipe for spaghetti and meatballs from the cooking blog Natasha's kitchen. Read through the recipe and answer questions 1-4 in writing using Handout 2A: Using an everyday example to illustrate structure in the Module 2 Participant Workbook PDF (1 MB).
Activity 2: Utilizing the structure of a tile growth pattern (25 minutes)
Imagine you are an Algebra 1 student and are asked to complete a math task using Handout 2B: Applying structure to a math task in the Module 2 Participant Workbook PDF (1 MB). Your teacher has let you know that in addition to writing an equation, you should answer each question on the assignment.
Next steps (5 minutes)
Confirm the date and time for PLC Session 2 and complete the between-session work.

Two men in a school library looking at something on a computer screen.

MODULE SEQUENCE

Complete this between-session work before PLC Session 2. All steps will take approximately 60 minutes to complete.

Read Handout 2C
Read Handout 2C: Guidance for implementing Recommendation 2 in the Module 2 Participant Workbook PDF (1 MB). Highlight ideas that stand out as potentially useful in your classroom.
Complete a knowledge check
Reflect on the content you worked through during PLC Session 1 and complete the knowledge below by clicking on “Start the knowledge check.”

Start the Knowledge Check

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MODULE SEQUENCE

By the end of PLC Session 2, participant will:

Be able to design activities that support students to utilize the structure of algebraic representations: promoting the use of language that reflects mathematical structure; encouraging students to use reflective questioning to notice structure as they solve problems; and/or teaching students that different algebraic representations can convey different information about an algebra problem.

SESSION AGENDA

Opening reflection (10 minutes)
Share reflections from your pre-session work: which ideas from Handout 2C stood out to you as potentially useful in your classroom?
Activity: Applying the recommendation to math problems (25 minutes)
Apply the instructional strategies of Recommendation 2 to a math problem you used in a recent lesson or will use in an upcoming lesson using Handout 2D: Recommendation 2: Application to a math problem in the Module 2 Participant Workbook PDF (1 MB).
Planning: Recommendation 2 in your classroom (20 minutes)
Complete the PLAN Phase portion (questions 1 and 2) of the PDSA Cycle Tool using Handout 2E: PDSA Cycle Tool – Plan phase part 1 in the Module 2 Participant Workbook PDF (1 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB)).
Next steps (5 minutes)
Confirm the date and time for PLC Session 3 and complete the between-session work.

A woman writing in a notebook.

MODULE SEQUENCE

Complete this between-session work before PLC Session 3. All steps will take approximately 60 minutes to complete.

Finalize plan
Finalize your plan for implementing the instructional strategies in your classroom based on the work and discussions from PLC Session 2.
Review the data tools
Review each of the data tools linked below or in the Participant Workbook PDF (1 MB).
Implementation data tools:
- Educator self-reflection tool PDF (60 KB). Complete a short, written reflection after using your planned instructional strategy in the classroom.
- Visitation tool PDF (168 KB). Ask a colleague or administrator to visit a lesson in which you use an instructional strategy from the toolkit. The visitor should record their observations and share with you after the lesson.
- Evidence/artifacts demonstrating implementation such as lesson plans, student work, other classroom observation or walkthroughs.
Outcome data tools:
- Module 2 Student Survey Part 1: Math Learning (pp. 2-3) PDF (132 KB) | Excel (99 KB): Administer this short survey to a class of students or to a smaller subset of students following a lesson in which you used an instructional strategy from the toolkit.
- Module 1–3 Student Survey Part 2: Math class engagement (pp. 4) PDF (132 KB) | Excel (101 KB): Administer this short survey to a class of students or to a smaller subset of students during a unit in which you used an instructional strategy from the toolkit.
- Module 2 Student Knowledge Assessment Tool Word (45 KB) | Excel (59 KB): Administer this knowledge assessment to a class of students following a lesson in which you used an instructional strategy from the toolkit.
- Data you already collect within your classroom, such as exit tickets, quizzes, or tests.
Complete Handout 2F
Complete the Plan Phase portion (questions 3 and 4) of the PDSA Cycle Tool using Handout 2F: PDSA Cycle Tool – Plan phase part 2 in the Module 2 Participant Workbook PDF (1 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
Bring materials to PLC Session 3
Bring your completed Handouts 2E and 2G to PLC Session 3 to discuss and get feedback from colleagues. For each data tool, be prepared to share (a) the extent to which you find the data tool useful and (b) how you might use the data tool in your classroom. In addition, bring the associated lesson plan(s) in which you will implement Recommendation 2.

MODULE SEQUENCE

By the end of PLC Session 3, participants will:

Be able to create a plan for collecting data and select data tools that will give them information about the implementation and outcomes of their planned instructional strategies.

SESSION AGENDA

Activity: Leveraging insights on learning from data from past PDSA cycles (25 minutes)
Reflect on the successes and challenges you experienced using PDSA cycles during Module 1 of the Toolkit. As part of the activity, you’ll engage in a group discussion, reflecting on the following questions.
- What common successes did we have as we used data during the Module 1 PDSA cycles?
- What common challenges did we encounter?
- What questions do you have for colleagues about their successes or challenges using data?
- What responses can you offer colleagues regarding their own successes and challenges using data?
- What suggestions do you have for how colleagues can overcome some of the challenges they encountered?
Critical friends activity: Giving and receiving feedback (30 minutes)
Continue to work on your plan for implementing Recommendation 2 by giving and getting feedback from colleagues. You will:
- Write down any pressing questions you have about your plan focused on where you would like feedback.
- Hear from a colleague and record notes detailing their feedback.
- Discuss revisions to you plan for implementing the instructional strategies and record any questions you would like feedback on from the whole group.
Next steps (5 minutes)
Confirm the date and time for PLC Session 4 and complete the between-session work.

MODULE SEQUENCE

Complete this between-session work before PLC Session 4. All steps will take approximately 120 minutes to complete.

Review your plan
Review and finalize your plan for implementation and data collection based on colleagues' feedback from PLC Session 3.
Implement your plan
Implement your planned instructional strategies during a lesson or series of lessons according to the plan you laid out in the Plan phase of the PDSA Tool (Handouts 2E and 2F).
Collect data and take notes
Collect data, record results, and take notes about happened when you implemented the instructional strategies in the first section of the PDSA Tool Handout 2G: PDSA Cycle Tool – Do phase in the Module 2 Participant Workbook PDF (1 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB). Note: if you plan to complete the visitation tool, consider having a pre-visit conversation with your visitor before the visitation.
Independently study your data
Reflect on your implementation and/or outcome data for the instructional strategies aligned with Recommendation 2 and complete questions 6-10 in Handout 2H: PDSA Cycle Tool – Study phase in the Module 2 Participant Workbook PDF (1 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB). Come to PLC Session 4 ready to share reflections and responses to these questions.

A woman writing in a notebook.

MODULE SEQUENCE

By the end of PLC Session 4, participants will:

Be able to discuss and analyze data collected on implementation and outcomes and use data-driven insights to inform continued implementation of Recommendation 2.

SESSION AGENDA

Opening reflection (5 minutes)
Consider the following reflection questions:
- Which instructional strategies did you use?
- What did you notice about your data?
- What possible explanations do you have for any findings or trends in your data?
- Did you observe improvement in outcomes? How do you know?
Data discussion (25 minutes)
In this activity, you will work in pairs. In your pair, you will take turns presenting so that each of you has equal opportunity to share. Spend half the time focusing on the first presenter's data, then switch. The presenter should briefly share an overview of their implementation. Then, collaboratively examine the data and the presenter's responses to the Study handout (Handout 2H) to answer the discussion questions.
- What do the data suggest about aspects of the instructional strategies that are being implemented effectively?
- What do the data suggest about how you should revise implementation of the strategies?
- What do the data suggest about student learning outcomes that are being achieved?
- What do the data suggest about how you should revise implementation to improve student outcomes?
Summary discussion (5 minutes)
Share out about overall success and challenges:
- What went well when you focused on teaching students to utilize the structure of algebraic representations? Discuss details about what went well with your instructional planning, how your instructional strategies played out in the classroom, and how students responded.
- What was challenging about teaching students to utilize the structure of algebraic representations?
Reflection: Act phase (20 minutes)
Create a high-level plan for continuing to refine your implementation strategies for Recommendation 2 by using Handout 2I: PDSA Cycle Tool – Act phase in the Module 2 Participant Workbook PDF (1 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
Next steps (5 minutes)
Review the follow-up work and confirm the date and time for the first PLC session for Module 3.

Follow-up work: Begin a new cycle of implementation

Congratulations on completing your Module 2 PDSA cycle! Now you can refine your instructional strategies regarding using the structure of algebraic representations. Please find the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB) or available for download here PDF (116 KB). Using your completed Act phase of Handout 2I, create a new plan for implementation using the PDSA Tool based on the data-informed revisions you noted above and continue a new cycle of implementation.

As you continue to integrate Recommendation 2 into your teaching, consider what supports you need from your PLC members and other members of your school community. This might involve scheduling collaborative planning or visitation time with PLC members to discuss how to incorporate Recommendation 2 into lessons, share successful strategies, and observe each other's implementation. You also might solicit feedback from administrators, colleagues, and students on the effectiveness of your implementation of Recommendation 2, using some of the data tools in the Participant Workbook PDF (1 MB) appendices C1, C2, and C3, as well as structuring more uses of the student assessment tools in appendix C4.

Continue to draw from the guidance for implementation shown in Handout 2C to identify approaches to integrating the structure of algebraic representations effectively into your lessons and classroom environment. This might include adding visual supports to your classroom to promote the use of precise language that reflects algebraic structure, such as word walls or color-coded mathematical expressions, that you can then refer to across many lessons. It might include displaying standard reflection questions that you return to and structure student use of in considering different problems. It might include displays of the various representations one could use to show algebraic structure (e.g., written, verbal, pictorial, manipulatives, graphs, diagrams), with prompts for some of the advantages and disadvantages of each. Understanding algebraic structure involves applying a complex set of skills that educators must approach and reinforce across topics, assessments, lessons, and units, in many different ways.

Two teachers high-fiving.

MODULE SEQUENCE

When you are ready to move to Module 3, select the Next button below.

Introduction

A cartoon student with a though bubble showing an arrow pointing to a check mark.

By the end of Module 3, participants will:

3.1 Understand the three instructional strategies to teach students to intentionally choose from alternative algebraic strategies when solving problems.
3.2 Be able to design activities that support students to intentionally choose from alternative algebraic strategies when solving problems in their classrooms: teach students to recognize and generate strategies for solving problems; encourage students to articulate the reasoning behind their choice of strategy and the mathematical validity of their strategy when solving problems; and/or have students evaluate and compare different strategies for solving problems.
3.3 Be able to define what successfully teaching students to intentionally choose from alternative algebraic strategies when solving problems will look like.
3.4 Be able to create a plan for collecting data and select data tools that will give them information about the implementation and outcomes of their planned instructional strategies.
3.5 Be able to discuss and analyze data collected during implementation and use data-driven insights to inform continued implementation of Recommendation 3.

Recommendation 3

Teach students to intentionally choose from alternative algebraic strategies when solving problems.

—Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students, p. 26

Module overview

PRE-SESSION WORK
Explore module content.
PLC SESSION 1
Develop an understanding of the recommendation and the related instructional strategies.
BETWEEN-SESSION WORK
Read and reflect about the recommendation and related instructional strategies.
PLC SESSION 2
Plan for implementing the recommendation.
BETWEEN-SESSION WORK
Learn about and review data tools.
PLC SESSION 3
Finalize plan for implementing the recommendation.
BETWEEN-SESSION WORK
Implement the recommendation and collect data.
PLC SESSION 4
Study results using data-driven dialogue and continue to implement the recommendation.

Students benefit from learning multiple solution strategies to apply to algebraic problem solving. Strategies are more general and abstract than memorized algorithms. Learning multiple solution strategies, and learning how to compare and choose between them, helps students develop flexibility in solving problems. This recommendation can help students develop their problem-solving skills beyond the memorization of one approach, allowing them to extend their knowledge and think more abstractly. Instructional strategies within the recommendation have been shown to improve students' procedural flexibility, while also helping them develop procedural and conceptual knowledge.

To achieve this purpose, the practice guide includes three instructional strategies educators can implement in their classrooms:

Instructional Strategy A: Teach students to recognize and generate multiple strategies for solving problems.
Instructional Strategy B: Encourage students to articulate the reasoning behind their choice of strategy and the mathematical validity of their strategy when solving problems.
Instructional Strategy C: Have students evaluate and compare different strategies for solving problems.

Based on the research evidence, these strategies are most effective when implemented once students have some fluency with algebra procedures and strategies. Although this recommendation promotes the understanding and use of multiple solution strategies, the recommendation does not advocate that all students be fluent in all possible strategies for solving a given problem type. Learning alternative strategies can empower students to select from different options when they encounter a problem. However, to avoid overwhelming students, be sure to introduce one solution strategy at a time and provide sufficient time to practice before introducing new strategies.

Solution strategy. A general approach for accomplishing a task or solving a problem that may include sequences of steps to execute, as well as the rationale behind the use and effectiveness of these steps.
Algorithm. A sequence of steps that, when executed in the prescribed order, lead to the desired or expected outcome.

A strategy involves a general approach for accomplishing a task or solving a problem. It may require students to make choices based on the specifics of the problem. Strategies are general and broadly applicable, making them useful in solving a variety of problems. Conversely, an algorithm is a sequence of steps that are intended to be executed in a particular order with little or no flexibility. Students are often taught algorithms designed to accomplish specific algebraic tasks and, as a result, may think of solving algebraic problems as merely executing a set of algorithms to arrive at the correct answer. However, this process does not support developing their conceptual understanding nor their procedural flexibility.

To intentionally choose between alternative solution strategies, students should consider the strategy's validity, efficiency, and fit.

Validity: Students must be able to determine if a strategy is mathematically valid, or correct, given the problem at hand. For example, when solving an algebraic equation, if a student subtracted a non-zero integer from only one side of the equation, the solution strategy would not be mathematically valid.
Efficiency: Students can observe that strategies vary in their efficiency. A solution strategy is more efficient if it can be executed relatively easily, quickly, and/or without error, as compared to another strategy that may be more difficult or more tedious to implement or that may be more likely to lead to error.
Fit: Considering fit asks students to judge if a given solution strategy is appropriate, given the structure of the problem at hand. This skill builds on students' ability to notice the structure of a given problem, developed throughout Module 2. For example, when solving a quadratic equation, a student may notice that the values on both sides of the equation are perfect squares and determine that "taking the square root"—as a solution strategy—is structurally fit.

Module resources

Overall
- Facilitator's Guide PDF (2 MB)
- Participant Workbook PDF (2 MB)
Data tools
- Educator Self-Reflection Tool PDF (50 KB)
- Module 3 Visitation Tool PDF (191 KB)
- Module 3 Student Survey Part 1: Math Learning (pp. 2-3) PDF (134 KB) | Excel (100 KB)
- Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (134 KB) | Excel (101 KB)
- Module 3 Student Knowledge Assessment Tool Word (95 KB) | Excel (60 KB)

MODULE SEQUENCE

Learn about and reflect on the content that will be covered in Module 3 before your first PLC session. This pre-session work will take approximately 60 minutes to complete.

Learn about Recommendation 3
- Read pages 26–36 in the Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students practice guide.
- Read the Module 3 Introduction.
- Watch the Module 3 video.
Complete the pre-work reflection
Complete the graphic organizer in the Module 3 Participant Workbook PDF (2 MB) in advance of PLC Session 1 to reflect on the content you reviewed. Be prepared to share your reflections at the next PLC session.
Watch a practice guide video (optional)
For more information about the levels of evidence in the practice guide, watch this video from the What Works Clearinghouse.

Download relevant resources (optional)
Navigate to the Module 3 Introduction and download the resources you will need for Module 3.

MODULE SEQUENCE

By the end of PLC Session 1, participants will:

Understand the three instructional strategies to teach students to intentionally choose from alternative algebraic strategies when solving problems.

SESSION AGENDA

Opening reflection: Identifying multiple solution strategies (15 minutes)
Solve the math problem provided in Handout 3A: Identify multiple solution strategies in the Module 3 Participant Workbook PDF (2 MB) multiple times, using as many different solution strategies as you can think of. Then, take turns sharing solution strategies until your group has exhausted all of them and reflect on the process.
Reflect and discuss: Accessing educators' experience (10 minutes)
Recall the three instructional strategies involved in implementing Recommendation 3 and discuss the following questions.
- How do you already incorporate these instructional strategies in your classroom? What did you have in the "rose" column of your brainstorm?
- What about this recommendation seems promising or interesting to you? How might students benefit from these instructional strategies? What did you have in the "bud" column of your brainstorm?
- What questions or concerns do you have about these instructional strategies? What challenges have you faced, or do you anticipate facing, in teaching students to use multiple strategies? What did you have in the "thorn" column of your brainstorm?
Activity: Exploring the instructional strategies (30 minutes)
To get a sense of what the instructional strategies could look and feel like in your classroom setting and to begin thinking about different ways you could incorporate each into your existing instructional strategies and lesson material, consider one of the following handouts.
- Handout 3B: Recognize and generate strategies in the Module 3 Participant Workbook PDF (2 MB).
- Handout 3C: Explain and justify chosen strategies in the Module 3 Participant Workbook PDF (2 MB).
- Handout 3D: Evaluate and compare strategies in the Module 3 Participant Workbook PDF (2 MB).
Next steps (5 minutes)
Confirm the date and time for PLC Session 2 and complete the between-session work.

MODULE SEQUENCE

Complete this between-session work before PLC Session 2. All steps will take approximately 60 minutes to complete.

Read Handouts 3E & 3F
Read Handout 3E: Guidance for implementing Recommendation 3 in the Module 3 Participant Workbook PDF (2 MB) and Handout 3F: Discussion questions and sentence starters: Recommendation 3 in the Module 3 Participant Workbook PDF (2 MB). Highlight ideas that stand out as potentially useful in your classroom.
Complete a knowledge check
Reflect on the content you worked through during PLC Session 1 and complete the knowledge below by clicking on "Start the knowledge check."

Start the Knowledge Check

MODULE SEQUENCE

By the end of PLC Session 2, participants will:

Be able to design activities that support students to intentionally choose from alternative algebraic strategies when solving problems in their classrooms: teach students to recognize and generate strategies for solving problems; encourage students to articulate the reasoning behind their choice of strategy and the mathematical validity of their strategy when solving problems; and/or have students evaluate and compare different strategies for solving problems.

SESSION AGENDA

Opening reflection (5 minutes)
Share reflections from your pre-session work.
- For which algebra topics do you see the most opportunities for incorporating the instructional strategies for this recommendation? Why?
- How might you have to adjust your implementation of the recommendation based on the topic?
Activity 1: Unpacking an example of Recommendation 3 in the classroom (25 minutes)
Review and discuss an example of how a fictional educator, Mr. Ruiz, implemented Recommendation 3 in his classroom using Handout 3G: In the classroom: Recommendation 3 in the Module 3 Participant Workbook PDF (2 MB).
Activity 2: Planning to implement Recommendation 3 in your classroom (25 minutes)
Complete the Plan Phase portion—questions 1 and 2 of the PDSA Cycle Tool—using 3H: PDSA Tool – Plan phase part 1 in the Module 3 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
Next steps (5 minutes)
At the end of the session, confirm the date and time for the next PLC session (PLC 3). Before that, proceed to the next PLC Session tab (i.e., Between-session work) to complete the prework.

MODULE SEQUENCE

Complete this between-session work before PLC Session 3. All steps will take approximately 60 minutes to complete.

Review the data tools
Review each of the data tools linked below or in the Participant Workbook PDF (2 MB).
- Implementation data tools:
  - Educator self-reflection tool PDF (50 KB): Complete a short, written reflection after using your planned instructional strategy in the classroom.
  - Visitation tool PDF (191 KB): Ask a colleague or administrator to visit a lesson in which you use an instructional strategy from the toolkit. The visitor should record their observations and share with you after the lesson.
- Outcome data tools:
  - Module 3 Student Survey Part 1: Math Learning (pp. 2-3) PDF (134 KB) | Excel (100 KB): Administer this short survey to a class of students or to a smaller subset of students following a lesson in which you used an instructional strategy from the toolkit.
  - Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (134 KB) | Excel (101 KB): Administer this short survey to a class of students or to a smaller subset of students during a unit in which you used an instructional strategy from the toolkit.
  - Module 3 Student Knowledge Assessment Tool Word (95 KB) | Excel (60 KB): Administer this knowledge assessment to a class of students following a lesson in which you used an instructional strategy from the toolkit.
  - Data you already collect within your classroom, such as exit tickets, quizzes, or tests.
Complete Handout 3I
Complete the Plan Phase portion (questions 3 and 4) of the PDSA Cycle Tool using Handout 3I: PDSA Tool – Plan phase part 2 in the Module 3 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
Bring materials to PLC Session 3
Bring your completed Handouts 3H and 3I to PLC Session 3 to discuss and get feedback from colleagues. For each data tool, be prepared to share (a) the extent to which you find the data tool useful and (b) how you might use the data tool in your classroom. In addition, bring the associated lesson plan(s) in which you will implement Recommendation 3.

MODULE SEQUENCE

By the end of PLC Session 3, participants will:

Be able to define what successfully teaching students to intentionally choose from alternative algebraic strategies when solving problems will look like.
Be able to create a plan for collecting data and select data tools that will give them information about the implementation and outcomes of their planned instructional strategies.

SESSION AGENDA

Opening reflection (20 minutes)
Review the Plan phase of your PDSA Tool and the associated lesson plan in which you will incorporate the instructional strategies. Reflect on the following questions.
- What is one aspect of your plan you are excited about and feel confident implementing?
- What is one aspect of your plan you are unsure about and would like colleagues’ feedback on?
Activity: Data tools reflection (25 minutes)
Think back to your process from the first two modules and reflect on your experience selecting a data tool and collecting data.
- Which data tools did you choose to use in Modules 1 and 2?
- How did your data collection process go? What was challenging and what was successful?
Closing reflection (10 minutes)
Reflect on the following questions.
- Has your perspective on the data collection tools shifted or evolved since Module 1? If so, how?
- What have you learned from the data tools or from the PDSA process?
- What is one pressing question or realization you have after today’s session?
Next steps (5 minutes)
Confirm the date and time for PLC Session 4 and complete the between-session work.

MODULE SEQUENCE

Complete this between-session work before PLC Session 4. All steps will take approximately 120 minutes to complete.

Review your plan
Review and finalize your plan for implementation and data collection based on colleagues' feedback from PLC Session 3.
Implement your plan
Implement your planned instructional strategies during a lesson or series of lessons according to the plan you laid out in the Plan section of the PDSA tool.
Collect data and take notes
Collect data, record results, and take notes about happened when you implemented the instructional strategies in the first section of the PDSA Tool Handout 3J: PDSA Tool – Do phase in the Module 3 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB). Note: if you plan to complete the visitation tool, consider having a pre-visit conversation with your visitor before the visitation.
Independently study your data
Reflect on your implementation and/or outcome data for the instructional strategies aligned with Recommendation 3 and complete questions 6-10 in Handout 3K: PDSA Tool – Study phase in the Module 3 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB). Come to PLC Session 4 ready to share reflections and responses to these questions.

A man working on a laptop and writing in a notebook.

MODULE SEQUENCE

By the end of PLC Session 4, participants will:

Be able to discuss and analyze data collected during implementation and use data-driven insights to inform continued implementation of Recommendation 3.

SESSION AGENDA

Opening reflection (5 minutes)
Consider the following reflection questions:
- How did your instructional strategies play out in the classroom? How did students respond?
- What went well? What was challenging?
Data discussion (25 minutes)
In this activity, you will work in pairs and will take turns presenting so that each of you has an equal opportunity to share. Spend half the time focusing on the first presenter's data, then switch. The presenter should briefly share an overview of their implementation. Then, collaboratively examine the data and the presenter's responses to the Study handout—Handout 3K—to answer the discussion questions.
- What do the data suggest about aspects of the instructional strategies that are being implemented effectively?
- What do the data suggest about how you should revise implementation of the strategies?
- What do the data suggest about student learning outcomes that are being achieved?
- What do the data suggest about how you should revise implementation to improve student outcomes?
Summary discussion (5 minutes)
Share out about overall success and challenges:
- What went well when you focused on teaching students to intentionally choose from alternative algebraic strategies? Discuss details about what went well with your instructional planning; how your instructional strategies played out in the classroom; and how students responded.
- What was challenging about teaching students to intentionally choose from alternative algebraic strategies?
Reflection: Act phase (20 minutes)
Create a high-level plan for continuing to refine your implementation strategies for Recommendation 3 by using Handout 3L: PDSA Tool – Act phase in the Module 3 Participant Workbook PDF (2 MB). You may also refer to the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB).
Next steps (5 minutes)
Review the follow-up work and confirm the date and time for the first PLC session for Module 4.

Follow-up work: Begin new cycle of implementation

Congratulations on completing your Module 3 PDSA cycle! Now you can refine your instructional strategies regarding using alternative solution strategies in your instruction. Please find the full PDSA tool in Appendix A of the Toolkit Introduction Participant Workbook PDF (2 MB) or available for download here PDF (116 KB). Using your completed Act phase of Handout 3L, create a new plan for implementation using the PDSA Tool based on the data-informed revisions you noted above and continue a new cycle of implementation.

As you continue to integrate Recommendation 3 into your teaching, consider what supports you need from your PLC members and other members of your school community. This might involve scheduling collaborative planning or visitation time with PLC members to discuss how to incorporate Recommendation 3 into lessons, share successful strategies, and observe each other's implementation. You also might solicit feedback from administrators, colleagues, and students on the effectiveness of your implementation of Recommendation 3, using some of the data tools in the Participant Workbook PDF (2 MB) appendices D1, D2, and D3, and structuring more uses of the student assessment tools in appendix D4. Continue to draw from the guidance for implementation shown in Handouts 3E and 3F to identify approaches to teaching students to intentionally choose from alternative algebraic strategies.

Three teachers celebrating.

MODULE SEQUENCE

When you are ready to move to Module 4, select the Next button below.

Introduction

A group of people around a table raising their hands in the air in celebration.

By the end of this final module, participants will be able to:

4.1 Identify key successes, lessons learned, and roadblocks to implementation from Modules 1 through 3.
4.2 Set goals and develop action plans for sustaining and institutionalizing change.
4.3 Celebrate, reflect on, and acknowledge the efforts of the group.

Module overview

PRE-SESSION WORK
Complete a cumulative self-reflection and an optional activity to collect students' perspectives.
PLC SESSION 1
Set goals for sustaining change.
BETWEEN-SESSION WORK
Draft action plans.
PLC SESSION 2
Share and refine action plans.

Module resources

Overall
- Facilitator's Guide PDF (1 MB)
- Participant Workbook PDF (1 MB)

MODULE SEQUENCE

A student writing an equation on a whiteboard.

Now that you have completed Modules 1 through 3, you (and your students) will use these materials to complete an integrated self-reflection and goal-setting activity. This pre-work is designed to help you reflect on your use of the algebra teaching instructional strategies you learned about in Modules 1 through 3 and set goals for sustaining the practices for the remainder of the school year and beyond.

To help you identify areas of strength and areas for improvement, this pre-work also provides an activity to facilitate with students to get their perspectives on the instructional strategies. This is not designed as a test or an evaluation. Rather, it is meant to support you in deepening and sustaining the changes embarked upon in Modules 1 through 3.

Complete the cumulative self-reflection
Using Handout 4A: Cumulative self-reflection in the Module 4 Participant Workbook PDF (1 MB), briefly consider the extent to which you made use of the recommended instructional strategies learned about in Modules 1 through 3.
Collect students’ perspectives (optional)
If time permits, use the following activity in one or more of your algebra classrooms to get students’ perspectives on the instructional strategies of the Algebra Toolkit. Gathering students’ perspectives on the teaching practices can provide you with information about students’ experiences with and perceptions of the practices that will be helpful as you set learning goals for sustaining the toolkit practices over the course of the remainder of the school year and beyond (see part 3). An example of how to facilitate this activity with students is provided below. You can adapt this activity as you see fit (including using digital rather than paper cards and using available resources such as Jamboard, Google Slides, or Desmos).

Refer to Handout 4B: Cards for card sort in the Module 4 Participant Workbook PDF (1 MB).

Algebra teaching instructional strategies card sort activity (30 minutes).

Explain that you are participating in a PLC with other math educators. In this PLC you have been learning about different teaching strategies and have tried out some of them with your classes. The purpose of these instructional strategies is to more effectively help students gain algebra knowledge and skills. The goal of this activity is to gather students’ opinions about the instructional strategies.
Emphasize that there are no right or wrong answers to this activity and that you are interested in learning more about students’ understanding of and opinions of the instructional strategies. Their honest insights will help you as you continue to set goals around improving your teaching.

Pass out a set of cards (shown in Handout 4B) to each group. There are several ways you can decide which cards to use for this activity. Options include:
- Giving each small group the full set of cards for one module. For example, group A would use the Module 1 cards, group B would use Module 2, and group C would use Module 3.
- Choosing targeted cards from each module and using this set with each small group.
Give students 15 minutes to sort the cards into meaningful categories. You can pre-define the categories for students, give students options from a list of pre-defined categories, or let students come up with their own categories for sorting. Example categories include:
- Strategies I have used before/have not used before.
- Strategies I enjoy using/don’t enjoy using.
- Strategies I understand a great deal/somewhat/not at all.
- Strategies that really helped me learn algebra more easily or better/didn’t help me learn algebra.
- Strategies I want my teacher to use more frequently/the same amount/less frequently.

Ask each small group to share with the class how they sorted their cards and why they placed certain cards in different categories. (The share-out could also be facilitated as a gallery walk, where students look at how other groups categorized their cards and explain what made them choose similarly or differently in their categorization.)
As students share, capture notes on what you learned from students using the prompts below. The goal here is to continue to focus on interactions with students and them sharing out. You might use a clipboard as you walk around or a whiteboard up front to capture students’ sharing so you can continue to facilitate the sharing out. (Note: You could also complete this as a summary after facilitating the activity. If you choose this approach, you could take photos of the card arrangements students lay out and review them later as you complete the summary table.)
- Looking at students’ responses, which instructional strategies are they…
- Familiar with (i.e., they have used before or understand a great deal)?
- Not familiar with? (i.e., they have not used before or do not understand)?
- Looking at students’ responses, which instructional strategies do they….
- Perceive positively (i.e., they enjoy using or want their teacher to use more frequently)?
- Perceive negatively (i.e., they do not enjoy using or want their teacher to use less frequently)?
Bring your notes with you to PLC session 1. Thank students for sharing their opinions with you and share several insights that you have learned from the activity.

Download relevant resources (optional)
Navigate to the Module 4 Introduction Tab and download the resources you will need for Module 4.

MODULE SEQUENCE

PLC Session 1: Set goals for sustaining change

By the end of PLC Session 1, participants will:

Identify key successes, lessons learned, and roadblocks to implementation from Modules 1 through 3.
Set goals and develop action plans for sustaining and institutionalizing change.

SESSION AGENDA

Activity: Set classroom goals for sustaining practices (25 minutes)
Looking across the instructional strategies in the self-reflection and that were part of the student feedback, which do you:
- Implement most frequently?
- Implement least frequently?
- Feel the most confident implementing?
- Feel the least confident implementing?
- Think were most effective for improving student learning?
- Think were least effective for improving student learning?
For those instructional strategies you implemented least frequently and feel the least confident implementing, consider why that is.

For those instructional strategies you feel most confident implementing, what were some of the times that you felt especially successful with one of these instructional strategies? Can you provide an example of a key success story that you experienced when using this instructional strategy? What were the pieces of evidence that helped you see this as a success story?

For those instructional strategies you think are the best for improving student learning, can you share an example to help demonstrate students’ improvement as a result of using one of the strategies? What is some of your evidence for coming to this conclusion?

Now that you have reflected on your own practice, revisit your summary of how students responded to the card sort activity. Share with colleagues your responses to the two questions below:
- What instructional strategies are students not familiar with or do they perceive negatively, and why do you think that is?
- Is there a connection between the instructional strategies that you have implemented most often and feel confident implementing, and what your students are familiar with and perceive positively?
Discussion
- Which instructional strategies is the group interested in sustaining or strengthening over the current school year and beyond?
- Why are these instructional strategies the ones you want to focus on?
- What do you hope to see change in students’ perceptions or their algebra skills in relation to these instructional strategies over the remainder of the school year? What about in future years of your teaching career?
- What are key supports from school leaders, fellow educators, other professionals, or parents that would help you more effectively implement these instructional strategies?
Activity: Create an action plan (30 minutes)
Using Handout 4C: School- or PLC-level Action Plan in the Module 4 Participant Workbook PDF (1 MB), brainstorm potential action steps to achieve the goals you identified in the prior activity.
Next steps (5 minutes)
Confirm the date and time for PLC Session 2 and complete the between-session work.

MODULE SEQUENCE

Between-session work

Complete this between-session work before PLC Session 2.

Create an action plan
Complete Handout 4C (started during PLC Session 1) to identify who should take the action steps, what resources will be needed, approximate timelines, and measures of success.

The words action plan and numbers 1 through 5 written in an open diary on a wooden table.

MODULE SEQUENCE

PLC Session 2: Share and refine action plans

By the end of PLC Session 2, participants will:

Set goals and develop action plans for sustaining and institutionalizing change.
Celebrate, reflect on, and acknowledge the efforts of the group.

SESSION AGENDA

Activity: Share with school and/or district leaders and formalize action plan (40 minutes)
Using Handouts 4A and 4C, summarize for the leader(s) joining the PLC today.
- What were the key successes, challenges, and evidence of progress the group has encountered in completing the toolkit work over the year?
- What goal did you select as key to sustaining this work?
Review and finalize action planning in Handout 4C:
- What are some steps or resources you all agreed were key for achieving this goal?
- Are there any resources or steps that the group is missing?
- Do these steps hang together as a coherent plan? Are there some steps or goals that will make sense for individuals to follow through on but are not as important to the group? How feasible are each of these steps within the expected timeframe?
Key questions to guide discussion:
- What pattern of meeting or collaboration will be most effective in supporting one another in carrying out action plans and sustaining use of the recommended instructional strategies? Will that pattern continue for the remainder of this school year? What about the following school year?
- What resources and support from school and district leaders will be most important in carrying this out?
- How will you incorporate the PDSA worksheets, data tools, and structure as part of this plan?
- Are there any other audiences (including district or state-level associations, conferences, or publications) with whom we could share our work?
Activity: Celebrating success, reflecting on lessons learned (20 minutes)
Use Handout 4D: Loved, learned – and shout-outs in the Module 4 Participant Workbook PDF (1 MB) to share feedback on your overall experience, as well as appreciation and shout-outs for members of the PLC (by name or anonymously). Reflect on the whole journey and jot down things that you’ve loved and/or learned. As you reflect, also feel free to celebrate any of your colleagues for something they have done. After you’ve had time to add your thoughts, share them with your group.
Next steps (5 minutes)
Review the follow-up work.

Follow-up work

Congratulations on completing the toolkit! This challenging and complex process of professional development hopefully has left you with new insights into evidence-based algebra instruction, new tools for instructional improvement and collaboration, and goals and action steps for continued improvement and professional growth. Now you move forward with the action plans and collective goals that you and the other team members have developed and refined.

As you move forward with these plans and goals, consider what supports you need from your PLC members and other members of your school community. This might involve scheduling next step meetings—during the school year, over the summer, or early in the coming school year—to identify progress on the action plans, share successful strategies in continuing to implement the recommended instructional strategies, and observe each other’s implementation, move ahead with additional PDSA cycles, or collaboratively plan. You also might continue to solicit feedback from school leaders, colleagues, and students on the effectiveness of your implementation of the recommendations and the steps you have taken to move ahead with action plans. Consider continuing to return all the tools, handouts, and resources found in the toolkit as support and sources of feedback.

Handout 4E: Reflections for school leaders supporting toolkit team action plans in the Module 4 Participant Workbook PDF (1 MB) provides guidance for reflection to share with school leaders as they support the team in moving forward to implement their goals and plans. Take some time to fill in sections you feel are appropriate or simply use these as prompt to share your thoughts and reflections with your school leader about the process of implementing the Algebra Toolkit this year.

Group of five teachers give high five at office table.

Teacher and Facilitator Materials

Access all toolkit materials

Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students Practice Guide

Toolkit Facilitator Guides

Introduction and Setting the Stage PDF (2 MB)
Module 1 PDF (2 MB)
Module 2 PDF (1 MB)
Module 3 PDF (2 MB)
Module 4 PDF (1 MB)

Toolkit Participant Workbooks

Introduction and Setting the Stage PDF (2 MB)
Module 1 PDF (2 MB)
Module 2 PDF (1 MB)
Module 3 PDF (2 MB)
Module 4 PDF (1 MB)

School Leader Guide: Institutionalizing Supports PDF (1 MB)

Ingredient List PDF (141 KB)

PDSA Tool PDF (116 KB)

Implementation Data Tools

Educator Self-Reflection Tools

Module 1 Educator Self-Reflection Tool PDF (60 KB)
Module 2 Educator Self-Reflection Tool PDF (60 KB)
Module 3 Educator Self-Reflection Tool PDF (50 KB)

Classroom Visitation Tools

Module 1 Visitation Tool PDF (219 KB)
Module 2 Visitation Tool PDF (168 KB)
Module 3 Visitation Tool PDF (191 KB)

Outcome Data Tools

Student Surveys

Module 1 Student Survey Part 1: Math Learning (pp. 2-3) PDF (160 KB) | Excel (105 KB)
Module 2 Student Survey Part 1: Math Learning (pp. 2-3) PDF (132 KB) | Excel (99 KB)
Module 3 Student Survey Part 1: Math Learning (pp. 2-3) PDF (134 KB) | Excel (100 KB)
Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (160 KB) | Excel (101 KB)

Student Knowledge Assessment Tools

Module 1 Student Knowledge Assessment Tool Word (45 KB) | Excel (62 KB)
Module 2 Student Knowledge Assessment Tool Word (45 KB) | Excel (59 KB)
Module 3 Student Knowledge Assessment Tool Word (95 KB) | Excel (60 KB)

Facilitator's Guide PDF (2 MB)

Participant Workbook PDF (2 MB)

Facilitator's Guide PDF (2 MB)

Participant Workbook PDF (2 MB)

Data Tools

Educator Self-Reflection Tool PDF (60 KB)
Module 1 Visitation Tool PDF (219 KB)
Module 1 Student Survey Part 1: Math Learning (pp. 2-3) PDF (160 KB) | Excel (105 KB)
Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (160 KB) | Excel (101 KB)
Module 1 Student Knowledge Assessment Tool Word (45 KB) | Excel (62 KB)

Facilitator's Guide PDF (1 MB)

Participant Workbook PDF (1 MB)

Data Tools

Educator Self-Reflection Tool PDF (60 KB)
Module 2 Visitation Tool PDF (168 KB)
Module 2 Student Survey Part 1: Math Learning (pp. 2-3) PDF (132 KB) | Excel (99 KB)
Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (132 KB) | Excel (101 KB)
Module 2 Student Knowledge Assessment Tool Word (45 KB) | Excel (59 KB)

Facilitator's Guide PDF (2 MB)

Participant Workbook PDF (2 MB)

Data Tools

Educator Self-Reflection Tool PDF (50 KB)
Module 3 Visitation Tool PDF (191 KB)
Module 3 Student Survey Part 1: Math Learning (pp. 2-3) PDF (134 KB) | Excel (100 KB)
Module 1–3 Student Survey Part 2: Math Class Engagement (pp. 4) PDF (134 KB) | Excel (101 KB)
Module 3 Student Knowledge Assessment Tool Word (95 KB) | Excel (60 KB)

Facilitator's Guide PDF (1 MB)

Participant Workbook PDF (1 MB)

School Leader Supports

A principal holding a clipboard on a high school campus with students walking in the background.

The resources in the School Leader Guide: Institutionalizing Supports ( PDF (1 MB)) are designed to provide school leaders with an overview of the Algebra Toolkit along with guidance on how to support educators in working through the toolkit modules. You, as school leaders, are crucial for the uptake and success of school-level changes, and the resources in this guide play a key role in providing you with the necessary information to understand how, when, and with whom to implement the toolkit.

Toolkit Ingredients

In addition to school and district leader support, effective implementation of the Algebra Toolkit also requires the necessary resources. Learn more in:

Resources needed to implement the Algebra Toolkit PDF (141 KB)

HOW TO USE THE SCHOOL LEADER RESOURCES FOR INSTITUTIONALIZING SUPPORTS

These resources are intended to be reviewed and used on your own schedule. As the Algebra Toolkit is intended for a group of in-service teachers and a facilitator to work through over the course of a single school year, it will be useful to review these resources before the school year begins, and then to return to them again throughout the year as needed. As suits your schedule, you can review them all at once to get a full picture of the ways a school leader can support changes in algebra instruction. Alternatively, you could spread them out over several weeks or months, but we recommend engaging with the resources before you and the toolkit facilitator launch your school's professional learning community (PLC). Your familiarity with these resources will help you set up an effective toolkit experience alongside the facilitator.

Download the School Leader Guide for answers to:

What support does the toolkit ask from school leaders?
What resources are needed to implement the toolkit?
How can I engage with these supports?
Who is the toolkit for and what's in it?
Why suggest educators work as a Professional Learning Community?
How long should the toolkit take to complete?
How can I support the implementation of algebra practices in the classroom?
Which toolkit data tools involve school leaders?
What PLC sessions should I participate in?
Should I collect feedback from facilitators and participants?

The School Leader Guide also includes a reflection activity. As your teachers get started, you can complete the reflections in this activity to think about how to effectively support them in this journey.

Thank you

for incorporating the Algebra Toolkit resources in your school's professional development and your commitment to excellence in algebra education.

Introduction

Toolkit Ingredients

Who is the Algebra Toolkit for?

What's in the Algebra Toolkit?

How is the Algebra Toolkit organized?

What is a professional learning community (PLC)

What are PDSA cycles?

What is the role of the facilitator?

What support does the Algebra Toolkit ask from administrators?

How long should the Algebra Toolkit take to complete?

GET STARTED

Setting the Stage

Introduction

Module overview

Module resources

MODULE SEQUENCE

Pre-session work

MODULE SEQUENCE

PLC session 1: Setting the stage

SESSION AGENDA

MODULE SEQUENCE

Module 1: Use solved problems to engage students in analyzing algebraic reasoning and strategies

Introduction

Recommendation 1

Module overview

Purpose

Summary of recommendation

Introducing a new topic

Building understanding of the topic

Approaching proficiency

Nearing proficiency

Module resources

MODULE SEQUENCE

Pre-session work: Explore module 1 content

MODULE SEQUENCE

PLC Session 1: Develop an understanding of the recommendation

SESSION AGENDA

MODULE SEQUENCE

Between-session work: Understand how to implement the recommendation

MODULE SEQUENCE

PLC Session 2: Plan for implementation of the recommendation

SESSION AGENDA

MODULE SEQUENCE

Between-session work: Planning how to use data tools to improve instructional practice

MODULE SEQUENCE

PLC Session 3: Finalize plan for implementation of the recommendation

SESSION AGENDA

MODULE SEQUENCE

Between-session work: Implement the recommendation and collect data

MODULE SEQUENCE

PLC Session 4: Study results using data-driven dialogue

SESSION AGENDA

Follow-up work: Begin a new cycle of implementation

MODULE SEQUENCE

Module 2: Utilize the structure of algebraic representations

Introduction

Recommendation 2

Module overview

Purpose

Summary of recommendation

Promote the use of language that reflects mathematical structure

Encourage students to use reflective questioning to notice structure as they solve problems

Teach students that different algebraic representations can convey different information about an algebra problem

Module resources

MODULE SEQUENCE

Pre-session work: Explore module 2 content

MODULE SEQUENCE

PLC Session 1: Develop an understanding of the recommendation

SESSION AGENDA

MODULE SEQUENCE

Between-session work: Understanding how to implement the recommendation

MODULE SEQUENCE

PLC Session 2: Plan for implementation of the recommendation

SESSION AGENDA

MODULE SEQUENCE

Between-session work: Planning how to use data tools to improve instructional practice

MODULE SEQUENCE

PLC Session 3: Finalize plan for implementation of the recommendation

SESSION AGENDA

MODULE SEQUENCE