Instructional Demands of the 2011 Mathematics Frameworks Challenges and Opportunities Pioneer Valley Regional School District October 28, 2011 Farshid Hajir

Outline Structure of 2011 Frameworks Guiding Principles Standards For Mathematical Practice Tools For Teachers The Cookie Jar Problem

Guiding Principle 1: Learning Mathematical ideas should be explored in ways that stimulate curiosity, create enjoyment of mathematics, and develop depth of understanding. Students need to understand mathematics deeply and use it effectively. The Standards for Mathematical Practice describe ways in which students increasingly engage with the subject matter as they grow in mathematical maturity and expertise through the elementary, middle, and high school years.

To achieve mathematical understanding, students should have a balance of mathematical conceptual procedures & understanding. Students should be actively engaged in doing meaningful mathematics, discussing mathematical ideas, and applying mathematics in interesting, thought-provoking situations. Student understanding is further developed through ongoing reflection about cognitively demanding and worthwhile tasks.

Tasks should be designed to challenge students in multiple ways. Short- and long-term investigations that connect procedures and skills with conceptual understanding are integral components of an effective mathematics program.

Activities should build upon curiosity and prior knowledge, and enable students to solve progressively deeper, broader, and more sophisticated problems. (See Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them.) Mathematical tasks reflecting sound and significant mathematics should generate active classroom talk, promote the development of conjectures, and lead to an understanding of the necessity for mathematical reasoning. (See Standard for Mathematical Practice 2: Reason abstractly and quantitatively.)

The Standards for Mathematical Practice “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.” (Frameworks, 2011) Multi-Million Dollar Question: How?!

Implementing SMPs A few Ideas 1)Modeling by the Instructor 2)Creation of Classroom Norms and Cultural Mores 3)Inter-weaving the SMPs with the content standards 4)Mathematical Learning as Mathematical Research 5)Understanding the SMPs in the context of Guiding Principles 6)Assessments must include conscious SMP self-evaluation

Implementing SMPs Prior to implementing SMPs, educators in each district need to develop their own shared understanding and appreciation for the meaning of SMPs, a highly non-trivial task! Opportunities for doing so are limited -- but educators are each other’s best resources. By visiting our colleagues’ classrooms, analyzing model lessons together, and sharing resources found on the web and in collaboration with Universities, teachers, principals and curriculum directors will change the way math is taught in the USA over the next decade.

Some Resources Frameworks: Inside Mathematics: Linking Content and Practice:

An Elementary Problem In the next slide, we’ll see a mathematics problem suitable for fifth graders. As you work in your group to solve the Problem, monitor the SMPs you and your teammates employ. Record your observations regarding usage of SMPs alongside your notes for solving the Problem.

The Cookie Jar Problem There was a jar of cookies on the table. Katie was hungry because she hadn't had breakfast, so she ate half the cookies. Then Myron came along and noticed the cookies. He thought they looked good, so he ate a third of what was left in the jar. Gina came by and decided to take a fourth of the remaining cookies with her to her next class. Then Nancy came dashing up and took a cookie to munch on. When Chelsea looked at the cookie jar, she saw that there were two cookies left. "How many cookies were there in the jar to begin with?" she asked Katie.

Next Tasks Choose one of these to work on now: 1.If there is a method that you are stuck on, try to figure out whether and how it can work. 2.Are there approaches other than those your group members have done? 3.Is the answer unique? How do you know? 4.Are there ways to map the correspondences among apparently different solution methods?