Using the CCSS to Develop Mathematically Proficient Students

Using the CCSS to Develop Mathematically Proficient Students:

Fluency with Algebraic Thinking (6–8): Day 1

Professional Development

Facilitator Handbook

-SAMPLER-

Pearson School Achievement Services

Using the CCSS to Develop Mathematically Proficient Students: Fluency with Algebraic Thinking (6–8): Day 1

Facilitator Handbook

Published by Pearson School Achievement Services, a division of Pearson, Inc.

1900 E. Lake Ave., Glenview, IL 60025

Printed in the United States of America.

Facilitator Agenda

Using the CCSS to Develop Mathematically Proficient Students:

Fluency with Algebraic Thinking (6–8): Day 1

Section / Time / Agenda Items
Introduction / 15 minutes / Slides 1–4
Welcome
Agenda
Outcomes
Activity:Taxi problem and dialogue
1: Along the Way to Mathematical Proficiency / 60 minutes / Slides 5-18
Section Introduction/Big Question
Making Sense of Mathematics
Video:“John Van de Walle Teacher Workshop”
Reflection: Questions to debrief video.
Activity:Operations With Rational Numbers.
Activity: “History of CCSS”
Focus on Fluency
Activity:Mint problem
Focus on the Standards for Mathematical Practice
Reflection: Two statements and One Question
Break / 15 minutes
2: Teaching through Problem Solving in the CCSS-Based Mathematics Classroom / 105 minutes / Slides 19-28
Section Introduction/Big Question
Activity: Focal Areas for CCSSM
Task Selection and Focal Areas of the CCSS
Activity: Recycling Aluminum Cans .
Task Selection and Problem-Based Tasks
Activity: Three Key Features of a Math Problem
Task: Bacteria problem
Task: Wildflower problem
Reflection:Fluency in Algebraic Thinking
Lunch / 30 minutes
Section / Time / Agenda Items
3: Teaching as Learning in the CCSS-Based Mathematics Classroom / 90 minutes / Slides 30–35
Section Introduction/Big Question
Activity: What is it about teaching that influences student learning?
Introduce Three-Phase Lesson Structure
Activity: Model lesson – Percents
Break / 15 minutes
3: Teaching as Learning in the CCSS-Based Mathematics Classroom / 80 minutes / Slides 36-42

Debrief Lesson
Learning to Notice Differently

Attending
Interpreting
Deciding How to Respond
Activity:Evaluate student work
Reflection: Staying on Target
Reflection and Closing / 10 minutes / Slides 43–51
Final Reflection: What will I do differently?
Evaluation
References
Total / 6 hours

Preparation and Background

Workshop Information

Big Ideas

The pressures that influence school mathematics have become much more complex with the rigorous problem solving required by the CCSSM. The ideas that follow create the foundation for teachers to effectively teach in a problem-solving environment, so that all students will be college and career ready.
Exploring what it means to know and do mathematics is at the very heart of teaching through problem solving—that is, teaching from the perspective of a student who must develop his or her own ideas and understanding. Teachers must come to believe that experts do not need to explain mathematics but the mathematics can emerge from carefully planned instruction.
If knowledge is the possession of concepts and ideas, then understanding is a measure of how well new concepts, facts, and procedures are integrated with or connected to other existing ideas. Another way to explain understanding of the content is by understanding the extent to which students demonstrate the CCSSM’s Standards for Mathematical Practice for a particular topic.
Teaching through problem solving means that teachers teach content by posing carefully planned problems through which students develop important concepts. Guidelines for selecting tasks are critical to teaching through problem solving.
Problem solving promotes active reflective thought and student involvement in mathematics and in turn enhances students’ understanding of the content.
Whether it is a ten-minute activity, a full-period problem, or a multiday task, a simple way to think about organizing instruction is to think of it in three phases: Before, During,andAfter. For each of these parts, there are separate agendas and corresponding teaching actions to meet these agendas. You cannot overemphasize the importance of the After portion of the lesson. In order to make teaching with problem-based tasks effective, there must be a significant period in which students share and discuss their results and justify them to the class (not to the teacher).
Planning a lesson based on the three-phase lesson structure helps teachers develop the mental processes for thinking through lesson planning while they keep students’ learning at the forefront. This process can serve as a guide for taking activities and turning them into effective lessons that integrate content and processes. The more experience teachers have with this process, the more likely they are to internalize the process of connecting objectives, assessment, and lesson activities.
Perhaps the most important benefit of a problem-solving approach to instruction is that it has the potential to meet the needs of a wide range of learners. In a problem-solving approach, every student can approach tasks with his or her own ideas and strategies.
Using video clips that show children’s mathematical thinking, model lessons, and carefully selected math activities in professional development contexts can motivate and promote growth in teachers’ content knowledge of mathematics. It can also hone their abilities to guide each child’s foundational knowledge development and deep understandings as outlined in the CCSSM.
Class discussion provides the opportunity for teachers and students to make explicit relationships between concepts and procedures in an intentional and public way and often provides clarification.
Teachers need to “dig below the surface” of correct answers to determine whether correct answers are associated with rich reasoning.
Teachers need to understand and expect that students can correctly answer a problem and still have mathematical misconceptions that teachers must address.
Teachers’ professional noticing of children’s mathematical thinking has three components (Jacobs, Lamb, and Philipp quoted in Philipp and Schappelle 2012, 8): attending, interpreting, and deciding how to respond.
Teachers can learn to focus closely on their students’ mathematical thinking and develop a “students’-thinking” lens to begin to see the mathematical content in two ways—as a teacher and as a student.
When teachers navigate student learning, they often transition from listening to students to determining whether they are right to listening to determine whether they have used correct reasoning.
Teachers are responsible for supporting students’ mathematical learning. To that end, teachers must focus on students’ reasoning and be flexible enough for it to make a difference in classroom instructional decisions.

Big Questions

What are the foundations and perspectives for developing mathematical proficiency through process and content? OR What is Fluency?
What is required for students to be fluent with Algebraic Thinking in Grade 8 according to CCSS?
What types of learning opportunities (tasks) do we need to get them there?
What does this look like in the classroom?

Outcomes

Articulate a structure for teaching through problem solving that incorporates the CCSSM and the Standards for Mathematical Practice.
Connect the design of a lesson to the opportunity for focused instruction and ongoing formative assessment.
Identify strategies for scaffolded instruction as a means of supporting students until they can apply new knowledge and skills independently.
Articulate daily classroom structures that build independence of learning.

Facilitator Goals

Use the three-phase lesson structure to effectively facilitate selected mathematical tasks, including the promotion of purposeful struggle.
Provide a forum for the discussion of the various solution strategies and results in an intentional and public way.
Support participants in understanding how their experience in working through the content models the development of mathematical proficiency in today’s classrooms.
Provide examples of tasks based on content standards and the features of problem-based tasks.
Provide examples of how the three-phase lesson structure plays out in the classroom.
Provide tools that participants can use to effectively navigate student learning.

Using the CCSS to Develop Mathematically Proficient Students:

Fluency with Algebraic Thinking (6-8)

Section 2: Teaching through Problem Solving in the CCSS-Based Mathematics Classroom (Slides 19-29)

Time: 105 minutes

Big Question

What is required for students to be fluent with Algebraic Thinking in Grade 8 according to CCSS?
What types of learning opportunities (tasks) do we need to get them there?

Training Objectives

Identify the primary focal areas and connection to the instructional support these areas provide to teachers.
Identify the three features of a problem-based task.

Materials per Section

Participant Workbook, pages 16–25
CCSSM (CCSSI_Math Standards.pdf found in the Additional Resources folder)
Chart paper
Progression to Algebra Continuum (Part of PW)
CCSS-M

Topic / Presentation Points / Presentation Preview
Section Introduction/Big Question /

Display Slide 19.
Use the question on the slide to introduce Section 2.
Review Van de Walle’s recommendation from the video to teach mathematics through problem solving and offering problems to students which provide for an appropriate degree of “struggle.”

/ PW page 16: Big Question
Focal Areas of the CCSS /

Instruct participants to form groups according to the grade levels that they teach (Grades 6, 7 or 8).
Display Slide 20 and point out to participants that the introduction to each grade level in the CCSSM begins, “In Grade X, instructional time should focus on . . .” (National Governor Association Center for Best Practices (NGA Center), Council of Chief State School Officers (CCSSO). 2010b, 13).
In grade-level groups, have participants review the Progress to Algebra Continuum on page 17 of their workbook and the Introduction of the CCSSM for their grade level. Remind them that the primary focal areasof each grade are noted in the introductions for each of the grade levels. Ask participants toexamine the focal areas through the lens of instructional time. Be sure to point out that this requires them to consider the following:
Prior knowledge from what they have taught in the past
The concept areas that they typically have spent a significant percentage of time developing
The content standards for their grade-specific CCSSM relative to Operations and Algebraic Thinking and Number.
Give participants time to discuss the following two questions:
What is different about your current content standards/areas of focus and the expectations of the CCSSM relative to instructional time?
How are they different?
Display Slide 21. Based on their grade-level discussions, ask each participant to summarize the details of the focal concepts as they are developed through 6th, 7th and 8th grade. Use the table in the PW on page 18

Examples of how focal concepts develop include the following:
Ratio and proportions develop as follows:
6th grade: Problem solve with ratios and unit rates
7th grade: Analyzing proportional relationships and solve problems with proportional reasoning including unit rates and percent.
8th grade: Understand connections between proportional relationships and linear equations. There is not a domain in 8th grade for ratio and proportion, however the connections are there.
The number system develops as follows:
6th grade: Apply/extend understanding of numbers to rational number system
7th grade: Apply/extend understanding of operations to rational numbers.
8th grade: Real numbers are introduced including how to approximate them.
Expressions and Equations develops as follows:
6th grade: Expressions, algebraic problem solving approach
7th grade: Generate equivalent expressions, algebraic problem solving approach.
8th grade: Analyze and solve linear equations and inequalities,

Mention that in a few minutes participants will examine a task and determine how the critical areas provide instructional guidance for teachers using the tasks.
Ask participants to share their thinking relative to the tables they completed for the activity.

/ PW page 17: Progress to Algebra Continuum
PW page 18: Focal Areas of the CCSS-M
Task Selection and Focal Areas of the CCSS /

Display Slide 22, and direct participants to the problem in the Participant Workbook on page 19. Provide participants with a few minutes to work through the problem.

Answer Key:

Carl earns $0.80 per pound of aluminum cans.
2.5 pounds is worth $2.00. 15 pounds is worth $12.00.
$36 is equal to 45 pounds of recycled aluminum.
1.25 pounds is worth $1.

Direct groups to the instructions on the slide, and ask for volunteers to summarize the answers to the question, taking into account the previous discussions by grade level and grade bands (Slide 23):

Using the CCSS, identify the grade-specific focal areas for the task selections above.
Explain how the focal areas provide instructional guidance for you in using the selected task.

Potential answers:

This is a 7th grade problem. The key focal area is “applying proportional relationships.” This problem could also be classified as a 6th grade problem based on the first three questions because students could use “reasoning about multiplication and division” to answer the first three questions. The fourth question is what qualifies this problem as a 7th grade problem because it is not so easily answered with simple multiplication/division.
Focal areas suggest instruction should include the “identification of the constant of proportionality (unit rate) in tables...and verbal descriptions of proportional relationships” as stated in 7.RP.2b in the CCSSM.

Wrap up this discussion by asking participants how their ideas might impact future grade-level and vertical team meetings.

/ PW page19: Recycling Aluminum Can
Task Selection and Problem-Based Tasks /

Display Slide 24: Task Selection and problem-based tasks.
Explain to participants that when they consider problems for use in their mathematics lessons, in addition to the content standards, they should also look for problems with the following three features (See bold print below)(Van de Walle, Karp, and Bay-Williams 2013, 35):
Allow 5-10 minutes for participants to read through the passage from Van de Walle’s book on page 20 of the PW then and list the key components of each feature of a math problem in the table on page 21 in the PW.

1. It must beginwhere the students are.The design or selection of the task must take into consideration the students’ current understanding. They should have the appropriate ideas to engage and solve the problem, and yet still find it challenging and interesting.
2. The problematic or engaging aspect of the problem must be due to the mathematics that the students are to learn. In solving the problem or doing the activity, students should be concerned primarily with making sense of the mathematics involved and thereby developing their understanding of those ideas. Although it is desirable to have contexts for problems that make them interesting, these aspects should not be the focus of the activity. Nor should nonmathematical activity (for example, cutting and pasting, coloring graphs, and so on) detract from the mathematics involved.
3. It must require justification and explanation for answers and methods. Students should understand that the responsibility for determining if answers are correct and why they are correct rests within themselves and not with the teacher. Justification should be an integral part of doing mathematics.

Ask participants to share their thoughts on the passage referred to above (Three features of a problem-based task) and their summaries from the PW.
Show slide 25 and allow participants a moment to read the quote from John Van de Walle below,

“There is no other decision that teachers make that has a greater impact on students’ opportunity to learn and on their perceptions about what mathematic is than the selection or creation of the tasks with which the teacher engages the students in studying mathematics” (Lappan and Briars quoted in Van de Walle, Karp, and Bay-Williams 2010, 32).

Show slide 26 and prompt participants to decide which of the three key features of task selection they think is most challenging to satisfy when selecting a suitable task for students? Why?

/ PW page 20–21: Three features of problem-based task.

Show Slide 27. Explain to participants that their groups will now examine two grade-band tasks for consideration of the three features that Slide 24 previously described.
Display Slide 28. Provide participants 10-15 minutes to work through each task independently, and then talk to a partner or in table groups to answer the questions about each task.
Participants should keep in mind the two questions on their slide as they work through the tasks.

The two questions are:

Is the applicable task above a vehicle for each of the three features of a problem? Explain your thinking.
Suppose that you were to use this task with a group of students. Describe any changes you would make to the task. Explain the changes you are considering.

Show Slide 28 as participants work through the first and second problems in their workbook. Allow participants 15-20 minutes to work through the problems and respond to the questions presented in the slide and workbook.

Answer key to the Grade 8 problem:
a)Answers will vary. Some participants will guess, “The number of cells double every minute.”
b)The rate of production for the green bacteria changes over time (This is a non-linear function). This can be described by # bacteria = # minutes^2 when the # minutes 2. The population of blue bacteria increases by 1 bacteria every minute. This can be described by # bacteria = # minutes + 1.
c)99 minutes
Answer key to the Grade 7 problem:
The number of pounds of wildflower seeds is 800 pounds.

Give participants time to solve both problems individually and then time to compare their answers in partners.

Direct groups to the instructions on the Slide 28, and facilitate a discussion in which participants describe how the assigned tasks are, or are not, vehicles for each of the three features of a problem-based tasks and what changes could be made to the tasks. / PW pages 22-23: Grade 8 Problem – Bacteria
PW pages 24–25: Grade 7 Problem – Wildflowers
Reflection: /

Display Slide 29
Activity: Provide a few minutes for participants to complete the table then facilitate a brief discussion on the participant’s responses.
Possible responses:

1. Focal areas can be identified from the introduction to the CCSSM for each grade level and from the Progress to Algebra Continuum.
2. Three primary features of an effective task: See content from Section 2 / PW page 26: Fluency with Algebraic Thinking

Using the CCSS to Develop Mathematically Proficient Students: