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Beyond the Common Core [Leader's Guide]

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Focus your curriculum to heighten student achievement. Learn 10 high-leverage team actions for mathematics instruction and assessment. Discover the actions your team should take before a unit of instruction begins, as well as the actions and formative assessments that should occur during instruction. Examine how to most effectively reflect on assessment results, and prepare for the next unit of instruction.

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Chapter 1 Before the Unit



Before the Unit

Teacher: Know thy impact.

—John Hattie

As a school leader, you are and always will be a teacher—of adults. Thus, the Hattie quote that opens this chapter is for you too. What will be your impact on the adults in your school or district, every month, every day, and on every unit of instruction? The ultimate outcome of before-the-unit planning is for your teachers to develop a clear understanding of the shared expectations for student learning during the unit. Do you expect a teacher learning culture that understands mathematics as an effort-based and not an ability-based discipline? Do you have high expectations that every teacher can ensure all students learn?

Your collaborative teams, in conjunction with district mathematics curriculum team leaders, prepare a roadmap that describes the knowledge students will know and be able to demonstrate at the conclusion of the unit. To create this roadmap, each collaborative team prepares and organizes work around five before-the-unit high-leverage team actions that you will need to monitor.


Chapter 2 During the Unit



During the Unit

The choice of classroom instruction and learning activities to maximize the outcome of surface knowledge and deeper processes is a hallmark of quality teaching.

—Mary Kennedy

Learning is experience. Everything else is just information.

—Albert Einstein

Much of the daily work of your teacher teams occurs during the unit of instruction. This makes sense, as it is during the unit that teachers place into action much of the team effort put forth in their before-the-unit work.

Your role is to support teachers’ efforts in data gathering, sharing, feedback, and action regarding student learning that forms the basis of an in-class formative assessment process throughout the unit. The teacher sharing of in-class formative assessment processes provides the platform that allows your collaborative teams to make needed adjustments to instruction, tasks, and activities that will better support student learning during the unit.


Chapter 3 After the Unit



After the Unit

You can’t learn without feedback…. It’s not teaching that causes learning. It’s the attempts by the learner to perform that cause learning, dependent upon the quality of the feedback and opportunities to use it. A single test of anything is, therefore, an incomplete assessment. We need to know whether the student can use the feedback from the results.

—Grant Wiggins

Teachers have just taught the unit and given the common end-of-unit assessment (developed through high-leverage team actions 3 and 4). What should happen next? Did the students reach the proficiency targets for the essential learning standards of the unit? As a school leader, how do you know? More important, what are the responsibilities for each of your collaborative teams after the unit ends?

The after-the-unit high-leverage team actions support steps four and five of the PLC teaching-assessing-learning cycle (see figure 3.1, page 102).


Epilogue: Taking Your Next Steps



Taking Your Next Steps

So now what? Your collaborative teams have moved through the five stages of the PLC teaching-assessing-learning cycle and should now be ready to start the process again with the next unit. Some of the considerations from this handbook relative to your teams’ work with the instructional unit include:

•   Was the size of the unit manageable within the teaching-assessing-learning cycle?

•   How did the team discussion of essential learning standards help support student understanding?

•   How did the design of the mathematical tasks and assessment instruments work? Were they aligned? Did they require demonstrations of student understanding?

•   How did the unit formative assessment plan fit with the end-of-unit assessment?

•   What was the student and teacher response at the end of the unit?

•   Did the team have the proper amount of time needed to complete its work?

The daily expectations of preparing class, scoring and grading student assessments, dealing with students who need extra help, meeting with parents, and having other school meetings often overwhelm teachers. Part of your responsibility as a leader within a PLC culture is to provide the team time necessary to support their work around the ten high-leverage team actions.


Appendix A: Standards for Mathematical Practice



Standards for Mathematical Practice

Source: NGA & CCSSO, 2010, pp. 6–8. © 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Used with permission.

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).


Appendix B: Standards for Mathematical Practice Evidence Tool



Standards for Mathematical Practice Evidence Tool

Source: © 2013 by Mona Toncheff & Timothy D. Kanold. All rights reserved.
Source for Mathematical Practices: NGA & CCSSO, 2010, pp. 6–8.

Mathematical Practice 1: “Make Sense of Problems and Persevere in Solving Them”


•   Check intermediate answers, and change strategy if necessary

•   Think about approaches to solving the problem before beginning

•   Draw pictures or diagrams to represent given information

•   Have the patience to complete multiple examples in trying to identity a solution

•   Start by working a simpler problem

•   Make a plan for solving the problem

In the classroom:

•   Student teams or groups look at a variety of solution approaches and discuss their merits.

•   Student teams or groups compare two different approaches to look for connections.

•   Students discuss with their peers whether a particular answer is possible in a given situation and explain their thinking.


Appendix C: Cognitive-Demand-Level Task-Analysis Guide



Cognitive-Demand-Level Task-Analysis Guide

Source: Smith & Stein, 1998. © 1998, National Council of Teachers of Mathematics. Used with permission.

Table C.1: Cognitive-Demand Levels of Mathematical Tasks

Lower-Level Cognitive Demand

Higher-Level Cognitive Demand

Memorization Tasks

•   These tasks involve reproducing previously learned facts, rules, formulae, or definitions to memory.

•   They cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use the procedure.

•   They are not ambiguous; such tasks involve exact reproduction of previously seen material and what is to be reproduced is clearly and directly stated.

•   They have no connection to the concepts or meaning that underlie the facts, rules, formulae, or definitions being learned or reproduced.


Appendix D: Sources for Higher-Level-Cognitive-Demand Tasks



Sources for Higher-Level-Cognitive-Demand Tasks

Common Core Conversation

Common Core Conversation is a collection of more than fifty free website resources for the Common Core State Standards in mathematics and ELA.

EngageNY Mathematics

The site features curriculum modules from the state of New York that include sample assessment tasks, deep resources, and exemplars for grades preK–12.

Howard County Public School System Secondary Mathematics Common Core

This site is a sample wiki for a district K–12 mathematics curriculum.

Illustrative Mathematics

The main goal of this project is to provide guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience upon implementation of the Common Core State Standards for mathematics.


Appendix E: How the Mathematics at Work High-Leverage Team Actions Support the NCTM Principles to Actions: Ensuring Mathematical Success for All



How the Mathematics at Work High-Leverage Team Actions Support the NCTM Principles to Actions: Ensuring Mathematical Success for All

The Beyond the Common Core: A Handbook for Mathematics in a PLC at Work series and the Mathematics at Work process include ten high-leverage team actions teachers should pursue collaboratively every day, in every unit, and every year. The goals of these actions are to eliminate inequities, inconsistencies, and lack of coherence so the focus is on teachers’ expectations, instructional practices, assessment practices, and responses to student-demonstrated learning. Therefore, the Mathematics at Work process provides support for NCTM’s Guiding Practices for School Mathematics as outlined in the 2014 publication Principles to Actions: Ensuring Mathematical Success for All (p. 5). Those principles are:

•   Curriculum principle—An excellent mathematics program includes a curriculum that develops important mathematics along coherent learning progressions and develops connections among areas of mathematical study and between mathematics and the real world.



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