1. “A Story of Ratios” Understanding the Structure of the Curriculum
7/17/2014
Erin Wheeler- Staff Development Specialist, Erie 2 BOCES

2. Textbook What do you think when you hear the word “textbook?”
What has your relationship with textbooks been like over the years.
Poll

3. A Different Vision of “Textbook”
Few U.S. textbooks paint mathematics as a dynamic, unfolding tale. They instead prioritize teaching procedures and employ a spiraling approach, in which topics are partially taught and then returned to— sometimes years later—with the unrealistic expectation that students will somehow connect the dots. But teaching procedures as skills without a rich context is ineffective. Students can too easily forget procedures and will fail if they do not have deeper, more concrete knowledge from which they can draw.
“How to Implement a Story of Units” page 4

4. “Knowing and Teaching Elementary Mathematics” Liping Ma
Interviews revealed that some teachers had a Profound Understanding of Fundamental Mathematics.
How do we develop this understanding?

5. A Different Vision of “Textbook”
“The textbook is the material on which Chinese teachers spend most of their time and devote most of their efforts to “study intensively.” They study it constantly throughout the school year when they teach it.
First of all, they work for an understanding of “what it is.” They study how it interprets and illustrates the ideas in the (Standards), why the authors structured the book in a certain way, what the connections among the contents are, what the connections are between the content of a certain textbook and its predecessors or successors, what is new in a textbook compared with an old version and why changes have been made, and so on.
(Knowing and Teaching Elementary Mathematics, Ma (1999, pg. 132)

6. A Different Vision of “Textbook”
At a more detailed level, they study how each unit of the textbook is organized, how the content was presented by the authors, and why. They study what examples are in a (topic), why these examples were selected and why the examples were presented in a certain order. They review the exercises in each section of a unit, the purpose for each exercise, section, and so on.
Indeed, they conduct a very careful and critical investigation of the textbook. Although teachers usually find the author’s ideas ingenious and inspiring, they also sometimes find parts of the textbook that from their perspective are unsatisfactory, or inadequate illustrations of ideas in the framework.”
(Knowing and Teaching Elementary Mathematics, Ma (1999, pg. 132)

7. What is the goal of my math class?
To be able to solve a certain type of problems by the end of today.
To use the problems we are solving today to learn mathematics.
(Phil Daro- Rationale of the Math Standards 5 min)

8. Textbooks that brings the world of mathematics research into the classroom.

9. A Trilogy of Mathematics
A Story of Units
PK-5
6-8
A Story of Ratios
A Story of Functions
9-12

10. A Misconception to Clear Up

11. Focus in the Standards
K–2
Addition and subtraction - concepts, skills, and problem solving and place value
3–5
Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving 6
Ratios and proportional reasoning; early expressions and equations 7
Ratios and proportional reasoning; arithmetic of rational numbers 8
Linear algebra and linear functions

12. Curriculum Map and Overview
Interactive Curriculum Map-

13. Curriculum Map and Overview
Module 2 provides an extra 25 days to build fluency around the multiplication and division with units of 2-5 and 10.
Note that module 6 is a post test module for grade 4. Consider the impact this module has to grade 5 being able to dive into their content.
Interactive version-
NYS Test Guides

14. Going Beyond the Map
Pg 16 is the start of the grade 1 overview.

15. Going Beyond the Map
Pg 16 is the start of the grade 1 overview.

16. Going Beyond the Map
Pg 16 is the start of the grade 1 overview.

17. Going Beyond the Map

18. Creating the Curriculum Map

19. Module Overview Topic A L1 L2 L3 Topic B L4 L5 Topic C L6 L7 L8
Module Focus
July 2013
Network Team Institute
TIME ALLOTTED FOR THIS SLIDE:
1 minute
MATERIALS NEEDED: X
Module Structure
Module Overview
Topic A L1 L2 L3
Topic B L4 L5
Topic C L6 L7 L8
Let’s take a minute to review the organizational structure of A Story of Ratios:
A Story of Ratios: A Curriculum Overview for Grades 6-8 provides a curriculum map and grade-level overview. The curriculum map provides an at-a-glance view of the entire story, making clear the coherence of the curriculum and the role that each module plays in that progression.
6-7 mods per grade
Mods comprised of topics (number varies per mod)
Topics vary in # of lessons
Lesson designed for 45 minute instructional period; Modules are comprised of topics, topics break into concepts, and concepts become lessons.
Graphic shows breakdown:
Each component, moving from the Overview to the Lesson, provides a more specific level of information. As you plan to implement A Story of Ratios, each of these components will be important to your understanding of the instructional path of the module.
The Standards, both Content and Practice, come to life through the lessons. Rigorous problems are embedded throughout the module. We will spend time in the sessions today and tomorrow exploring this further.

21. Overview Narrative Explains the sequence outlined on the title page.
Helps you see where we’ve been and where we’re going.

22. Progressions Documents
“…explain why standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics. This would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks.”

23. This seems strange…

24. Module Overview Topic A L1 L2 L3 Topic B L4 L5 Topic C L6 L7 L8
Module Focus
July 2013
Network Team Institute
TIME ALLOTTED FOR THIS SLIDE:
1 minute
MATERIALS NEEDED: X
Module Structure
Module Overview
Topic A L1 L2 L3
Topic B L4 L5
Topic C L6 L7 L8
Let’s take a minute to review the organizational structure of A Story of Ratios:
A Story of Ratios: A Curriculum Overview for Grades 6-8 provides a curriculum map and grade-level overview. The curriculum map provides an at-a-glance view of the entire story, making clear the coherence of the curriculum and the role that each module plays in that progression.
6-7 mods per grade
Mods comprised of topics (number varies per mod)
Topics vary in # of lessons
Lesson designed for 45 minute instructional period; Modules are comprised of topics, topics break into concepts, and concepts become lessons.
Graphic shows breakdown:
Each component, moving from the Overview to the Lesson, provides a more specific level of information. As you plan to implement A Story of Ratios, each of these components will be important to your understanding of the instructional path of the module.
The Standards, both Content and Practice, come to life through the lessons. Rigorous problems are embedded throughout the module. We will spend time in the sessions today and tomorrow exploring this further.
Simple Complex

25. Making the Connections
Concrete
Pictorial
Abstract
9 + 6 = 15
As students work on solving problems, it important to make an intentional connection from the concrete, to the pictorial, to the abstract. Students need concrete models when building number sense, but as they develop their understanding it is important that they transition to more efficient modeling tools.

31. Introduction to Tape Diagrams and Double Number Lines https://www
“How to Implement a Story of Units” Mathematical Models pages

32. Universal Design for Learning
Multiple Means of Representation
Multiple Means of Action and Expression
Multiple Means of Engagement
“How to Implement a Story of Units” Differentiating Instruction pages

35. Types of Lessons Problem Set Socratic Exploration Modeling
Header
July 2013
Network Team Institute
TIME ALLOTTED FOR THIS SLIDE:
5 minutes
MATERIALS NEEDED:
Types of Lessons
Problem Set
Students and teachers work through examples and complete exercises to develop or reinforce a concept or procedure.
Socratic
Teacher leads students in a conversation to develop a specific concept or proof.
Exploration
Independent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge
Modeling
Students and teacher practice part of the modeling cycle with problems that are ill-defined and have a real world context.
5 min
This slide will teach participants about the four types of lessons.
Say: Number off at your tables 1, 2, 3, 4, 1, 2, 3, 4, etc.. If you are a 1 you will read out #1 out loud and then the next number 1 will summarize it with your table. The 2’s will read number 2 and the next number 2 will summarize it, etc.. For tables with more than 8 people, the extra persons can add any additional thoughts or clarifications on any of the lesson types. Before you begin reading or summarizing when it is your turn, briefly introduce yourself to your table partners.
Pause for 3-4 minutes while participants discuss the types of lessons at their tables.
Say: Does anyone have any questions? (pause and answer or move on)
Say: Module 1 for this grade only contains Problem Set and Exploration lessons. Socratic lessons will only make up about 10% of our entire curriculum.
Now that we have discussed the Topic Opener, let’s take a look at lesson components and how we will make this all come together for 7th grade students.
NOTE TO FACILITATOR: See this background knowledge to help you address any questions.
Problem Set Lesson – Teacher and students work through a sequence of 4 to 7 examples and exercises to develop or reinforce a concept. Mostly teacher directed. Students work on exercises individually or in pairs in short time periods. The majority of time is spent alternating between the teacher working through examples with the students and the students completing exercises.
Exploration Lesson – Students are given 20 – 30 minutes to work independently or in small groups on one or more exploratory challenges followed by a debrief with the goal of clarifying, expanding upon or developing a concept, definition, theorem or proof. Typically a challenging problem or question that requires students to collaborate (in pairs or groups) but can be done individually. Class discussion on the problem for a period of time (10 minutes might be appropriate) to draw conclusions and consolidate understandings.
Socratic Lesson – Teacher leads students in a conversation with the aim of developing a specific concept or proof. Only about 10% of lessons fall under this category minutes devoted to student/teacher conversation. Useful when conveying ideas that students cannot learn/discover on their own. The teacher asks guiding questions to make their point and engage students. The remaining time could include a fluency activity to open the lesson or there may be a debrief or application problem at the end of the lesson.
Modeling Cycle Lesson –At this level, students are involved in practicing part of the modeling cycle. The problem students are working on is ill-defined and has a real world context. Students are likely to work in groups on these types of problems, but teachers may want students to work for a period of time individually before collaborating with their group members. Students and teacher work through the modeling cycle in a reduced form to complete an application problem over 1 or 2 days.

36. Topic Opener- Narrative

37. Math Publisher’s Criteria
Rigor: in major topics, pursue with equal intensity
conceptual understanding
procedural skill and fluency
applications
Publisher’s Criteria-
This document outlines the expectations for materials that are aligned to the common core standards.

38. Fluency follows instruction!
Fluency is designed to promote automaticity by engaging students in practice in ways that get their adrenaline flowing.
Automaticity is critical so that students avoid using up too many of their attention resources with lower-level skills when they are addressing higher-level problems.
Set a timer and stop when your fluency time is up!
Fluency follows instruction!

39. Examples- Developing the Concept

40. Exercises- Practicing the Concept

41. Closing- Debrief the patterns observed

42. Exit Ticket
Follows the closing after the student have had an opportunity to share and hear classmates observations.
Provides a daily gauge of student understanding.

43. Problem Set Practice applying the day’s learning.
Consider a time frame vs. a task frame.

44. Assessments

45. Evidence of the Standards

46. Evidence of the Standards

47. Accessing PDF’s of the Modules