Our Learning Journey

ContinuesShelly R. Rider

College and Career-Ready Standards for Mathematics

The Practice Standards and Content Standards define what students should understand and

be able to do in their study of mathematics. Asking a student to understand something

Means asking a teacher to assess whether the student has understood it. But what does

Mathematical understanding look like? One hallmark of mathematical understanding is the

ability to justify, in a way appropriate to the student’s mathematical maturity, why a

particular mathematical statement is true or where a mathematical rule comes from. There

is a world of difference between a student who can summon a mnemonic device to expand

a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes

from. The student who can explain the rule understands the mathematics, and may have a

better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y).

Mathematical understanding and procedural skill are equally important, and both are

assessable using mathematical tasks of sufficient richness.

Mathematical Understanding

The Overarching Habits of Mindof a Productive Mathematical Thinker

Quality Instruction Scaffolding Professional Development Process

- Talk Moves- Conceptual Learning- Environment [physical & emotional]

- Productive Math Discussions- Task Selection- Quality Questioning

PLT 2012-2013

PLT 2013-2014

Levels of Cognitive Demand

High LevelDoing MathematicsProcedures with Connections to Concepts,

Meaning and Understanding

Low LevelMemorizationProcedures without Connections to

Concepts, Meaning and Understanding

Hallmarks of “Procedures Without Connections” Tasks Are algorithmic Require limited cognitive effort for completion Have no connection to the concepts or meaning that

underlie the procedure being used Are focused on producing correct answers rather

than developing mathematical understanding Require no explanations or explanations that focus

solely on describing the procedure that was used

Procedures without Connection to

Concepts, Meaning, or Understanding

Convert the fraction to a decimal and percent38

3.008 .375 = 37.5%2 4

60

.375

564040

Hallmarks of “Procedures with Connections” Tasks

Suggested pathways have close connections to underlying concepts (vs. algorithms that are opaque with respect to underlying concepts)

Tasks often involve making connections among multiple representations as a way to develop meaning

Tasks require some degree of cognitive effort (cannot follow procedures mindlessly)

Students must engage with the concepts that underlie the procedures in order to successfully complete the task

“Procedures with Connections” Tasks

Using a 10 x 10 grid, identify the decimal and percent equivalent of 3/5.

EXPECTED RESPONSE

Fraction = 3/5

Decimal 60/100 = .60

Percent 60/100 = 60%

Hallmarks of “Doing Math” Tasks There is not a predictable, well-rehearsed pathway

explicitly suggested Requires students to explore, conjecture, and test Demands that students self monitor and regulated

their cognitive processes Requires that students access relevant knowledge

and make appropriate use of them Requires considerable cognitive effort and may

invoke anxiety on the part of students

Requires considerable skill on the part of the teacher to manage well.

“Doing Mathematics” Tasks

Shade 6 squares in a 4 x 10 rectangle. Using the rectangle, explain how to determine each of the following:

a) Percent of area that is shaded

b) Decimal part of area that is shaded

c) Fractional part of the area that is shaded

a) Since there are 10 columns, each column is 10% . So 4 squares = 10%. Two squares would be 5%. So the 6 shaded squares equal 10% plus 5% = 15%.

b) One column would be .10 since there are 10 columns. The second column has only 2 squares shaded so that would be one half of .10 which is .05. So the 6 shaded blocks equal .1 plus .05 which equals .15.

c) Six shaded squares out of 40 squares is 6/40 which reduces to 3/20.

ONE POSSIBLE RESPONSE

The Importance of Student Discussion

Provides opportunities for students to:

Share ideas and clarify understandings Develop convincing arguments regarding why

and how things work Develop a language for expressing

mathematical ideas Learn to see things for other people’s

perspective

Quality Instruction Scaffolding Professional Development Process

- Talk Moves- Conceptual Learning- Environment [physical & emotional]

Grade Level Teachers 2013-2014

Classroom Impact

Type ofTraining

KnowledgeMastery

SkillAcquisition

ClassroomApplication

Theory 85% 15% 5-10%

PLUS

Practice 85% 80% 10-15%

PLUS

PeerCoachingStudy TeamsClass Visits

90% 90% 80-90%

PLT TEAMS

Peer to Peer Coaching

Peer to Peer Coaching is a confidential process through which two or

more professional colleagues work together

to reflect on current practices.

Immediate Next Steps of the CCRS Journey1) Peer-to-Peer Coaching

Process

2) Vertical Math PLT Process

Talk Move PD August Part 1

Talk Move Facilitator Notes to guide the PD

Talk Move Participant PacketTalk Move PowerPointTalk Move Video(s)Talk Move Research Article

These resources will be located at http://amsti-usa.wikispaces.com at the close of Monday, August 5th.

The Journey Ahead Form

&Peer-to-Peer

Coaching Form

PD Structures to Facilitate Learning

Teacher Professional Learning Teams PLT Facilitator Coaching Communities PLT Facilitator Side-by-Side Coaching Administrator Professional Learning

Teams Peer-to-Peer Coaching

Our Learning Journey

ContinuesShelly R. Rider