jfm DAY2 WEB

The Story of Standards: What We Know About Quality, Coherence, and Progression: 

The Story of Standards: What We Know About Quality, Coherence, and Progression Joan Ferrini-Mundy, Peter Bates, and the PROM/SE Associates PROM/SE Mathematics Associates Summer Institute, AUGUST 9-12, 2004

Slide2: 

Penny had a bag of marbles. She gave one-third of them to Rebecca and one-fourth of the remaining marbles to Aman. Penny then had 24 marbles left in her bag. How many marbles were in the bag to start with? DAY 2, TASK 1

The PROM/SE Process: 

Gather Data Implement Changes Design Solutions Conjecture Reasons Identify Challenges Analyze Data Interpret Data The PROM/SE Process

The PROM/SE Process: 

Gather Data Implement Changes Design Solutions Conjecture Reasons Identify Challenges Analyze Data Interpret Data The PROM/SE Process Day 2

The story ….: 

The story …. What Gets Taught: WHO DECIDES? The Case of Fractions: WHAT ARE THEY? Big Ideas and Trajectories: HOW CAN WE ACHIEVE COHERENCE?

What Gets Taught: WHO DECIDES?: 

What Gets Taught: WHO DECIDES?

The Many Aspects of Curriculum ….: 

The Many Aspects of Curriculum …. intended accessed implemented achieved assessed standards, benchmarks, indicators student learning MEAP, Ohio Proficiency and Achievement Tests what teachers do in classrooms student enrollments

The Many Aspects of Curriculum ….: 

The Many Aspects of Curriculum …. intended accessed achieved assessed standards, benchmarks, indicators student learning MEAP, Ohio Proficiency and Achievement Tests what teachers do in classrooms student enrollments

Slide9: 

The intended curriculum ……

High Achieving Countries’ Mathematics Standards: 

High Achieving Countries’ Mathematics Standards

NCTM PSSM with Top Achieving Countries’ Profile: 

NCTM PSSM with Top Achieving Countries’ Profile

Michigan’s (New) Mathematics GLCEs: 

Michigan’s (New) Mathematics GLCEs

Composite Mathematics Standards for MI PROM/SE: 

Composite Mathematics Standards for MI PROM/SE

Slide14: 

The “accessed” and “achieved” curriculum …..

National data, from CCSSO State Indicators of Science and Mathematics Education 2003: 

National data, from CCSSO State Indicators of Science and Mathematics Education 2003

Differing expectations leadto differing results: 

Math scores: Minority 12th graders vs. white 8th graders Differing expectations lead to differing results

Slide19: 

professional development AP syllabus textbooks MI GLCEs MEAP pacing guides NCTM standards

Slide20: 

professional development AP syllabus textbooks MI GLCEs MEAP pacing guides NCTM standards PROM/SE data national data research findings

Slide21: 

Ultimately, teachers determine the implemented curriculum -- with lots of inputs -- and part of what PROM/SE is about is helping to build capacity to do this in a way that supports mathematics learning and growth for all.

Slide22: 

COMMENTS ON MARBLES PROBLEM, PETER BATES

Slide23: 

Penny had a bag of marbles. She gave one-third of them to Rebecca, and then one fourth of the remaining marbles to John. Penny then had 24 marbles left in the bag. How many marbles were in the bag to start with? A. 36 B. 48 C. 60 D. 96 PROM/SE ITEM, GRADES 6,7,8 Grade 6 35.2% Grade 7 43.0% Grade 8 43.6%

Slide24: 

Fraction issues that arise in the Marble Problem: Meaning of fraction Equivalence of fractions Adding and subtracting fractions Multiplying and dividing by fractions Representing fractions …….. 1/4 = 8 marbles

The Case of Fractions: WHAT ARE THEY?: 

The Case of Fractions: WHAT ARE THEY?

Slide27: 

What does mean?

Slide28: 

What does mean?

Slide29: 

What does mean?

Slide31: 

What does mean?

Multiple Meanings and Uses of Fractions: 

Multiple Meanings and Uses of Fractions part of a collection part of a whole point on a number line measurement ratio/rate probability division of two numbers abstract number

What meanings of fraction came up in the marble problem?: 

What meanings of fraction came up in the marble problem? Fraction as division Number meaning (we calculated with fractions) Part of a collection Part of a whole

Slide34: 

Find the value of each of the following expressions when m = 4, p = 2, and q = 3. From Singapore Mathematics Curriculum NUMBER Grade 6

Slide35: 

From Connected Mathematics RATES Grade 7

Advantages & Limitations:part of a whole: 

Advantages & Limitations: part of a whole Advantages: emphasizes the unit (1 whole) familiar to children (sharing, splitting, pizza) Limitations: hard to calculate with fractions represented as part of a whole (1/3 + 1/5) in some representations (e.g., circles) Different size “wholes”

Advantages & Limitations:part of a collection: 

Advantages & Limitations: part of a collection Advantages: counting is easier leads to division meaning Limitations: the “whole” or unit is arbitrary computation is hard difficult to use to show fraction greater than 1

Advantages & Limitations:number line: 

Advantages & Limitations: number line Advantages: Leads to number meaning Allows for fractions greater than one Limitations: More abstract than part of a whole 0 1

Performance on a fractions problem: 

Performance on a fractions problem PROM/SE RESULTS: Grade 3 35.2% Grade 4 43.0% Grade 5 43.6% 1/4 1/2 8/8 10/8

Advantages & Limitations: division: 

Advantages & Limitations: division Advantages: builds on part of a collection idea relates to contexts Limitations: difficult to justify is the same as 2 3

Slide42: 

the same as 2 3 ? WHY IS

Review of Research: 

Review of Research Literature review yielded 73 research studies of student understanding and teaching of fractions Some of the authors: Behr, Bright, Borasi, Michaelsen, Davis, Kerslake, Mack, Middleton, Steffe, Olive, Wearne, Hiebert, and many others…. key findings….

Slide44: 

Number line at 4th grade is difficult Translations between various representations Need a lengthy readiness period for fractions Fraction/ratio confusion It helps to encourage children to talk about their interpretation of fraction Greater emphasis is needed on division interpretation Need to recognize the limitations of the “part to whole” model Need to transition better from realm of counting numbers to rational numbers Knowing different meanings and interpretations strengthens understanding

Slide45: 

Students bring experience with part-to-whole interpretation, and fair shares Need variety of models for the “whole” (circles, rectangles, irregular shapes) In part-to-whole, not only do partitioning, but complete the whole

Big Ideas and Trajectories: HOW CAN WE ACHIEVE COHERENCE?: 

Big Ideas and Trajectories: HOW CAN WE ACHIEVE COHERENCE?

Coherence: A Worthy Goal: 

Coherence: A Worthy Goal Content standards are coherent if they reflect a sequence of topics and performances organize along the logical and hierarchical structure of mathematics Schmidt, Wang, and McKnight, in press.

From NCTM’s Principles and Standards for School Mathematics: 

From NCTM’s Principles and Standards for School Mathematics The Curriculum Principle: A curriculum is more than a collection of activities; it must be coherent, focused on important mathematics, and well articulated across the grades.

From NCTM’s Principles and Standards for School Mathematics: 

From NCTM’s Principles and Standards for School Mathematics The Table of Standards and Expectations in the appendix highlights the growth of expectations across the grades. It is not expected that every topic will be addressed every year. Rather, students will reach a certain depth of understanding of the concepts and acquire certain levels of fluency with the procedures by prescribed points in the curriculum, so further instruction can assume and build on this understanding. (p. 30)

Emphasis Across the Grades: 

Emphasis Across the Grades Number Algebra Geometry Measurement Data Analysis and Probability Pre-K–2 3–5 6–8 9–12

Defn of big ideas (knowledge package, Liping Ma): 

Defn of big ideas (knowledge package, Liping Ma)

Slide52: 

part of a whole Big idea: number meaning part of a collection number line Big idea: division meaning measurement

Working with the Michigan Grade Level Expectations: 

Working with the Michigan Grade Level Expectations

Finding coherence amidst clutter?: 

Finding coherence amidst clutter? Looking for trajectories Aiming for big ideas

Fraction Meaning Excerpts from the Michigan Grade Level Content Expectations: 

Fraction Meaning Excerpts from the Michigan Grade Level Content Expectations

Slide56: 

Number line meaning

Slide57: 

Number line meaning Part of a whole meaning

Slide59: 

0 1 1/2

Slide60: 

EMPHASIZING A DIVISION MEANING OHIO STANDARDS OHIO STANDARDS

Identify Challenges: 

Identify Challenges Fraction is a central concept and our students have difficulty with it There are many meanings of fraction, and the transitions and connections among them are not easy to see

Conjecture Reasons: 

Conjecture Reasons There are many meanings of fraction The intended and implemented curricula might not integrate those meanings effectively We may not all agree on a common target of understanding (e.g., division and number meanings) Teachers may need to coordinate better from grade to grade in moving a target

The PROM/SE Process: 

Gather Data Implement Changes Design Solutions Conjecture Reasons Identify Challenges Analyze Data Interpret Data The PROM/SE Process fall 2004

Slide64: 

WE CAN LEARN FROM DATA WE CAN LEARN FROM EACH OTHER WE CAN ALWAYS IMPROVE TEACHING AND LEARNING, AND…..

Slide65: 

Tyler Baker Becky Murthum Lee Cogan Richard Houang Mary Bouck Gail Burrill Jan Eberhardt Bill Schmidt Kelly Rivette Neelam Kher Betty Bonney Kathy Cilluffo Jamie Tenney Vicki Nedelman Amy Wettingill Stephanie Myer Sandy Denison Karen Rego Beth Joe Adri Aaron Kris, and Max!

Task for Next Session….: 

Task for Next Session…. Geometry -- basic shapes and properties. what do you teach about this topic at your grade level?

Slide67: 

Look at your school DRAFT report from yesterday. Identify one challenge that you can find in the area of fractions. Write on a post-it note and post on a piece of flip chart paper somewhere around the room. DAY 2, TASK 2

The PROM/SE Process: 

Gather Data Implement Changes Design Solutions Conjecture Reasons Identify Challenges Analyze Data Interpret Data The PROM/SE Process Day 2

Afternoon: 

Afternoon 1:00 - 1:30 mathematical reflections, Peter 1:30 - 2:00 conjecturing causes, Joan Say more about what the “big ideas” are in an area Similarity ought to look different across the grades 2:00 - 2:15 break 2:15 - 3:15 Taking Stock Worksheets, gail 3:15 - 3:45 What did we learn today, joan

2:15 - 3:30 session: 

2:15 - 3:30 session Pink sheet: TAKING STOCK 2 Complete, keep for tomorrow Go to assigned room Find a facilitator Discuss these questions, make notes on your sheets for tomorrow Others will be circulating; remember the “Parking Lot” sheet