1.1 TEACHING FOR UNDERSTANDING Mathematical Thinking for Instruction K-2 ©CDMT 2008

Negative Positive ©CDMT 2008

I. Welcome, Announcements, Housekeeping, etc. II. Lizzie Problem III. Teaching for Understanding & PD Goals IV. Measurment V. Wrap-Up Session 1: Agenda July 16 3 © DMTI

Solve each problem 3 mathematically different ways. Use any notation you’d like ©CDMT 2008

Building Mathematical Understanding Take Students’ Ideas Seriously Press Students Conceptually Encourage Multiple Strategies Address Misconceptions Focus on the Structure of the Mathematics ©CDMT 2008

Example: Lizzie Problem Lizzie collects lizards and beetles. She has 8 creatures in her collection so far. All together they have 36 legs. How many of each kind of creature does she have in her collection? ©CDMT 2008

Lizzie Problem Strategies: Drawing

Lizzie Problem Strategies: Table ©CDMT 2008

Lizzie Problem Strategies: Equation

Modes of Representation July 16 © CDMT 10 EnactiveIconicSymbolic

Extension of ‘Lizzie’-Joey’s Problem Joey is in science class and his group is given 8 packages of seeds. They dump them on the table and found there were 76 seeds. Pumpkin seeds come in packages of 12 and sunflower seeds come in packages of 8. How many pumpkin and sunflower packages does Joey’s group have? ©CDMT 2008

Lizzie Problem: Guided Reinvention Trajectory

Instructional Model: TFU Teaching For Understanding Connecting related pieces of knowledge together Knowing how to do something and why Knowing how...  Enables us to flexibly use procedures and understand the relationship of these procedures within the structure of mathematics Knowing why...  Enables us to use concepts flexibility, extend our knowledge to new situations, and connect it to the world outside of school ©CDMT 2008

Teaching for Understanding Structural Perspective  Knowledge is structured through web-like/ hierarchal connections.  Mental representation as part of a network of representations.  Claim: The stronger and the greater number of connections there are in this complex structure, the higher degree of understanding. ©CDMT 2008

Teaching for Understanding Functional Perspective  Maintains that students need to actively integrate incoming information with existing knowledge through social interactions.  So, students must interact and share knowledge with others.  Claim: By being in situations in which students are communicating with others, they build lasting and coherent concepts and skills, which leads to an increased degree understanding. ©CDMT 2008

What Does This Mean For Teaching? From a functional perspective, we focus on providing the types of tasks and activities that place students in situations where, through articulation, they are able to reflect on how they solve problems and construct relationships – a structural perspective. Functional Perspective Structural Perspective Under- standing ©CDMT 2008

But HOW Does One Do This? Realistic Mathematics Education (Gravemeijer & van Galen, 2003; Freudenthal, 1973, 1991; Treffers, 1987)  Guided Reinvention  Mathematizing

©CDMT 2008 Instructional Practice: Guided Reinvention Is the process of first allowing students to develop informal strategies for solving problems, and then, by critically examining those strategies, encouraging students to develop more sophisticated, formal, conventional and abstract strategies and algorithms Students are encouraged to make connections between existing knowledge (informal ideas) and new knowledge (more formal mathematical ideas) – structural aspect of understanding.

Guided Reinvention: Trajectory ©CDMT 2008

Guided Reinvention: Arithmetic “By thinking and talking about similarities and differences between arithmetic procedures, students can construct relationships between them. … the instructional goal is not necessarily to inform one procedure by the other but, rather, to help students build a coherent mental network in which all pieces are joined to others with multiple links.” ©CDMT 2008

Mathematizing (Treffers, 1987) Horizontal Mathematization  Occurs when students represent a contextualized problem mathematically in order to find a solution strategy. Vertical Mathematization  Involves taking the mathematical matter to a higher level, and occurs when students make their representations and strategies objects of mathematical examination.

Mathematizing & Guided Reinvention Guided reinvention ‘happens’ when students mathematize. Students should go through a similar process as mathematicians go through themselves. Objective: “…for students to experience their mathematical knowledge as the product of their own mathematical activity” (Gravemeijer & van Galen, 2003, p. 117) ©CDMT 2008

Building Mathematical Understanding Take Students’ Ideas Seriously Press Students’ Conceptually Encourage Multiple Strategies Address Misconceptions Focus on the Structure of the Mathematics ©CDMT 2008

Student Achievement Data IDMT (2009)

· 79.2% of students in DMT teachers’ classrooms scored proficient or advanced compared to 65.7% in comparison teachers’ classrooms · These differences are statistically significant (z=2.603, p <.01) RMC (2008) IDMT (2009)

Comparison of student ISAAT Scores over 5 years IDMT (2009) RMC (2009)

Poverty and ESL IDMT (2009)

2007 TIMSS Results – 4 th Grade Math Poverty Effect IDMT (2009)

DMT Schools Free and Reduced Lunch IDMT (2009) Years teaching DMT

DMT and Non-DMT (in district) Average School Proficiency IDMT (2009)

Caldwell Schools LEP Proficiency: Grades IDMT (2009)

Caldwell Schools LEP Proficiency: Grades 3 – 5 IDMT (2009)

ISAT Student Achievement DMT Compared to Other Students:

ISAT Student Achievement DMT and Other Student Bins:

2009 IDMT (2009)

2009 IDMT (2009)

Professional Development Model IDMT (2009)

Professional Development Plan Professional DevelopmentDEVELOPING MATHEMATICAL THINKING Year 1Year 2Year 3 Focus AreaNumber & Algebra Measurement & Geometry Probability & Statistics Summer PD 5 Days (45 hours) In-depth topics XXX Ongoing PD 18 days Work Study (4 X semester) Observations (monthly) Demonstrations (monthly) XXX IDMT (2009)

PRETEST – POSTTEST FOCUSING CALENDAR UNIT LESSONS ANALYSIS OF DATA Work Study Unit (Example) IDMT (2009)

Professional Development: Work Study Show and Tell  Task and student responses  15 min Data discussion  Focus on the past unit  Teacher’s pre and post results  Teachers’ pre and post results (common issues)  Artifacts – what do we keep and what needs to change Building of the next unit  Pre and post test  Table of contents (lesson outline)  Lesson planning IDMT (2009)

Teacher Levels Level 1: Resistant (7%) Level 2: Developing (58%) Level 3: Proficient (31%) Level 4: Reflective and Refined (4%) IDMT (2009)

Teacher Level Grade-level ISAT Proficiency Average Examples: 4 th grade Level 4 Teacher: 83% Level 3 Teacher: 79% Level 2 Teacher: 66% 3 rd Grade Level 4 Teacher: 92% Level 3 Teachers: 92% and 83% Level 2 Teacher: 74% IDMT (2009)

BUILDING MATHEMATICAL UNDERSTANDING Take Students’ Ideas Seriously Press Students Conceptually Encourage Multiple Strategies Address Misconceptions Focus on the Structure of the Mathematics D EVELOPING M ATHEMATICAL T HINKING IDMT (2009)