Time(inseconds)

Distancefrom

starto

froad(in

feet)

A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.

What observations can you make about the Bike and Truck based on the graph?

Bike and Truck Task (purple handout)

Effective Teaching Practices: The Key to Supporting Students’ Learning or Ambitious Standards

PEGSMITH

UNIVERSITYOFPITTSBURGH

Engaging Students in Learning: Mathematical Practices An NCTM Interactive Institute for Grades 9-12

July 14 - 16, 2016

Agenda

•  Overview of Principles to Actions

•  Discuss a set of eight effective teaching practices and relate them to a classroom episode

•  Discuss how to get started in making these practices central to your instruction

•  Learn about new resources that can help you in this effort

Principles to Actions: Ensuring Mathematical Success for All

The primary purpose of PtA is to fill the gap between the adoption of rigorous standards and the enactment of practices, policies, programs, and actions required for successful implementation of those standards.

Guiding Principles for School Mathematics

1.   Teaching and Learning

2.   Access and Equity

3.   Curriculum

4.   Tools and Technology

5.   Assessment

6.   Professionalism

Essential

Elements

of Effective

Math

Programs

Guiding Principles for School Mathematics

1.   Teaching and Learning

2.   Access and Equity

3.   Curriculum

4.   Tools and Technology

5.   Assessment

6.   Professionalism

Essential

Elements

of Effective

Math

Programs

Effective Mathematics Teaching Practices

1.  Establish mathematics goals to focus learning. 2.  Implement tasks that promote reasoning and

problem solving. 3.  Use and connect mathematical representations. 4.  Facilitate meaningful mathematical discourse. 5.  Pose purposeful questions. 6.  Build procedural fluency from conceptual

understanding. 7.  Support productive struggle in learning

mathematics. 8.   Elicit and use evidence of student thinking.

The Bike and Truck Context

School: Tyner Academy Principal: Carol Goss Teacher: Shalunda Shackelford Class: Algebra 1 Curriculum: Creating and Interpreting Functions (IFL) Class Size: 26 Shalunda Shackelford was a teacher at Tyner Academy in the Hamilton County School District. The lesson occurred in April in an Algebra 1 class. She has been working on facilitating mathematical discussions and targeting the Common Core standards for content and mathematical practice in very deliberate ways. (Version 1 – blue handout)

Establish Mathematics Goals To Focus Learning

Learning Goals should: •  Clearly state what it is students are to learn

and understand about mathematics as the result of instruction;

•  Be situated within learning progressions; and

•  Frame the decisions that teachers make during a lesson.

Formulating clear, explicit learning goals sets the stage for everything else.

(Hiebert, Morris, Berk, & Janssen, 2007, p. 57)

Goals to Focus Learning

Students will understand that: •  The language of change and rate of change

(increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values.

•  Context is important for interpreting key features of a graph portraying the relationship between time and distance.

•  The average rate of change is the ratio of the change in the dependent variable to the change in the independent variable for a specified interval.

Connections to CCSSM Content Standards

Interpre7ngFunc7ons F–IFInterpretfunc7onsthatariseinapplica7onsintermsofthecontext.

F-IF.B4 For a func7on thatmodels a rela7onship between twoquan77es, interpretkeyfeaturesofgraphsandtablesintermsofthequan77es,andsketchgraphsshowingkeyfeaturesgivenaverbaldescripConoftherelaConship.Keyfeaturesinclude: intercepts; intervals where the func6on is increasing, decreasing,posi6ve, or nega6ve; rela6ve maximums and minimums; symmetries; endbehavior;andperiodicity.★

F-IF.B5 Relate the domain of a func7on to its graph and, where applicable, to thequan7ta7verela7onshipitdescribes.Forexample,ifthefunc6onh(n)givesthenumber of person-hours it takes to assemble n engines in a factory, then theposi6veintegerswouldbeanappropriatedomainforthefunc6on.★

F-IF.B6 Calculate and interpret the average rate of change of a funcCon (presentedsymbolicallyorasatable)overaspecifiedinterval.Es7matetherateofchangefromagraph.★

★IndicatesthatthestandardisalsoaMathemaCcalModelingstandard.

Implement Tasks that Promote Reasoning and Problem Solving

Mathematical tasks should: •  Provide opportunities for students to engage

in exploration or encourage students to use procedures in ways that are connected to concepts and understanding; •  Build on students’ current understanding; and •  Have multiple entry points.

There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.

(Lappan & Briars, 1995)

Tasks that Promote Reasoning and Problem Solving

•  Whatfeaturesofthebikeandtrucktaskstandouttoyou?(considerbothversions)

•  Howdothesefeaturessupportreasoningandproblemsolving?

Features of Bike and Truck

•  Alignedwithgoalsforlesson•  SignificantmathemaCcscontent•  HighcogniCvedemand•  Low-threshold,highceiling(version2–purplehandout)•  DifferentrepresentaCons•  Familiarcontext

Pose Purposeful Questions

Effective Questions should: •  Reveal students’ current understandings; •  Encourage students to explain, elaborate, or

clarify their thinking; and •  Make the mathematics more visible and

accessible for student examination and discussion.

Teachers‘ questions are crucial in helping students make connections and learn important mathematics and science concepts. Teachers need to know how students typically think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students deepen their understanding.

(Weiss & Pasley, 2004)

Facilitate Meaningful Mathematical Discourse

Mathematical Discourse should: •  Build on and honor students’ thinking; •  Provide students with the opportunity to

share ideas, clarify understandings, and develop convincing arguments; and •  Advance the mathematical learning of the

whole class.

Discussions that focus on cognitively challenging mathematical tasks…are a primary mechanism for promoting conceptual understanding of mathematics (Hatano and Inagaki 1991; Michaels, O’Connor and Resnick 2008).

(Smith, Hughes, Engle and Stein 2009, p. 549)

Bike and Truck Lesson

As you watch the video, identify any of the effective mathematics teaching practices (NCTM, 2014) that you notice Ms. Shackelford using, with a particular emphasis on pose purposeful questions and facilitate meaningful discourse. [The yellow handout is the transcript for the video.]

Bike and Truck Lesson

NCTMmemberscanaccessthevideoforthislessonisavailableat:hUp://www.nctm.org/Conferences-and-Professional-Development/Principles-to-AcCons-Toolkit/The-Case-of-Shalunda-Shackelford-and-the-Bike-and-Truck-Task/

Purposeful Questions

•  TheteacherselectedataskthatincludedquesConsthat

targetedthemathemaCcalideasshewantedstudentstolearnandrequiredstudentstoexplaintheirthinking.

•  Theteacherintroducedherimaginaryfriend,Chris,whohadmisinterpretedthegraphandaskedstudentsiftheyagreedordisagreedwithhim(line4).

•  Theteacheraskedstudentstoexplainwhattheywerethinking(lines46,68,82),andwhytheywerethinkinginaparCcularway(lines23,88).

•  TheteacherusedquesConsinplaceoftelling–shewasmakingsureto“neversayanythingakidcouldsay”(Reinhart,2000).

Facilitate Meaningful Discourse

T supports students in sharing and defending their own ideas: •  T introduces the idea that the truck was “moving in a straight path” and asks students to defend whether they agree or disagree: Lines 1-6. • Students are given opportunities to explain their thinking: Lines 12-17, 24-26, 33-35, 47-52, 56-63, 76-79.

T provides students with the opportunity to develop convincing arguments: • Developing convincing arguments was an expectation: “Make sure you justify your reasoning”: Lines 8-9. • Students are asked to re-explain ideas to the class: Line 33-35, 76-79.

Facilitate Meaningful Discourse

T provides students with the opportunity to clarify understandings: •  T tells class to “listen, because if you disagree, say something.”– students are expected to listen for understanding and ask questions if they disagree: Line 9-10. •  T uses a scenario to help Jacobi understand that time is moving even though she is standing still. She then asks Jacobi to explain his new understanding: Lines 42-52. •  Following a students’ explanation, the class seems to agree, but the teacher then presses for questions (line 66-68). Two students question how “the truck got there first” (lines 69-73), and the teacher asks for explanations (lines 76-79). •  T asks students what they are thinking now or why they agree/disagree: Lines 82-90.

Elicit and Use Evidence of Student Thinking

Evidence should: •  Provide a window into students’ thinking; •  Help the teacher determine the extent to

which students are reaching the math learning goals; and

•  Be used to make instructional decisions during the lesson and to prepare for subsequent lessons. Formative assessment is an essentially interactive process, in which the teacher can find out whether what has been taught has been learned, and if not, to do something about it. Day-to-day formative assessment is one of the most powerful ways of improving learning in the mathematics classroom. (Wiliam 2007, pp. 1054; 1091)

Ms. Shackleford Elicits and Uses Evidence of Student Thinking

The teacher elicits student thinking by asking them to agree or disagree with something that has been said and uses the thinking of students to reach consensus. A student explains which vehicle reaches 300 ft. first (lines 56-63), the teacher elicits the thinking of several students (lines 69-80), and continues to press students until they understand; lines 82-90). Lines 69-71 (MaKayla) See, my question is probably about the same as Stephanie’s because if the truck stopped, and the bike kept going, the bike should have got there before the truck did.

Use and Connect Mathematical Representations

Different Representations should: •  Be introduced, discussed, and connected; •  Focus students’ attention on the structure or

essential features of mathematical ideas; and •  Support students’ ability to justify and explain

their reasoning.

Strengthening the ability to move between and among these representations improves the growth of children’s concepts.

(Lesh, Post & Behr, 1987)

Connecting Representations

What connections did she make in the brief clip we watched?

Connecting Representations Bike and Truck

Connections were made between the contextual situation and the graph. Specifically:

•  what the horizontal line on the graph represents in terms of what the truck was doing; and

•  how you can tell whether the bike or truck reached 300 feet first.

Support Productive Struggle in Learning Mathematics

Productive Struggle should: •  Be considered essential to learning

mathematics with understanding; •  Develop students’ capacity to persevere in the

face of challenge; and •  Help students realize that they are capable of

doing well in mathematics with effort. By struggling with important mathematics we mean the opposite of simply being presented information to be memorized or being asked only to practice what has been demonstrated.

(Hiebert and Grouws 2007, pp. 387-88)

Bike and Truck Productive Struggle

•  How did Ms. Shakelford support students when they struggled?

Bike and Truck Productive Struggle

The teacher: •  provided students with time to make

sense of explanations given; •  checked on students to see what they

understood; •  Never told students they were wrong or

what was correct; she let them figure it out

When Jacobi still does not understand the meaning of the horizontal portion of the graph of truck, she had Charles explain it again and then actually did a role play of what it would mean for time to pass and not to move (lines 29-52).

Build Procedural Fluency from Conceptual Understanding

Procedural Fluency should: •  Build on a foundation of conceptual

understanding; •  Result in generalized methods for solving

problems; and •  Enable students to flexibly choose among

methods to solve contextual and mathematical problems.

Students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results. (Martin 2009, p. 165)

MindlessUseofProcedures

Solvingsystemsofequa6ons

Rulesforexponents

Procedural Fluency

This was not the focus on the lesson but the

teacher was working on developing students intuitive understanding of slope and what it means in context.

In the transcript for the second video clip (green handout), the segment begins with the teaching asking, “Between what two seconds did the truck drive the fastest? Shh. What do you think and tell me why.” (Lines 1-2)

So how do the practices fit together?

Establishmathgoalstofocuslearning

Elicitanduseevidenceofstudent

thinking

PosepurposefulquesCons

UseandconnectmathemaCcalrepresentaCons

SupportproducCvestruggleinlearning

mathemaCcs

FacilitatemeaningfulmathemaCcaldiscourse

Effec7veMathema7csTeachingPrac7ces“BuildingaTeachingFramework”

Implementtasksthatpromotereasoningand

problemsolving

Buildproceduralfluencyfromconceptualunderstanding

Effective Mathematics Teaching Practices

Okay, Peg. Nice set of practices. But this institute is suppose to be about Mathematical Practices. Yes, I know. But I want to argue that IF you engage in these teaching practices you will be providing your students with the opportunity to engage in most of the mathematical practices.

Getting Started

•  Learn more about the effective teaching practices from reading the book, exploring other resources, and talking with your colleagues and administrators. • Engage in observations and analysis of teaching (live or in narrative or video form) and discuss the extent to which the eight practices appear to have been utilized by the teacher and what impact they had on teaching and learning. • Co-plan lessons with colleagues using the eight effective teaching practices as a framework. Invite the math coach (if you have one) to participate. • Observe and debrief lessons with particular attention to what practices were used in the lesson and how the practices did or did not support students’ learning.

“If your students are going home

at the end of the day less tired

than you are, the division of labor

in your classroom requires some

attention.” Wiliam (2011)

Peg Smith

pegs@pitt.edu

PrinciplestoAc.onsProfessionalLearningToolkit:TeachingandLearningPurpose - Develop materials to support teacher learning of the eight Effective Mathematics Teaching Practices. Each grade-band module engages teachers in analyzing artifacts of teaching (e.g., mathematical tasks, narrative and video cases, student work samples) to strengthen teaching that effectively supports the learning of all students.

NCTMPtAMaterials&ResourceshTp://www.nctm.org/PtAToolkit/

Components of the PtA Video Module

•  PowerPoint slides (w/facilitation notes)

•  Math Task featured in the video

•  Video clips

•  Transcript of the video

•  Focused Discussion of selected Effective Mathematics Teaching Practices

•  And more ... Lesson guide, Readings, CCSSM connections (in selected modules)

Comingin2017fromNCTMThreeBookSeries:•  TakingAc6on:Implemen6ngEffec6veMathema6csTeaching

Prac6cesinGradesPreK-5(Huinker&Bill)

•  TakingAc6on:Implemen6ngEffec6veMathema6csTeachingPrac6cesinGrades6-8(Smith,Steele,&Raith)

•  TakingAc6on:Implemen6ngEffec6veMathema6csTeachingPrac6cesinGrades9-12(Boston,Dillon,Miller,&Smith)

PegSmith,SeriesEditor