Engaging Tasks to Gauge Student Learning · Uncovering Student Thinking About Mathematics in the Common Core: 25 Formative Assessment Probes, Cheryl Rose Tobey and Emily R. Fagan

Engaging Minds to Gauge

Student Learning

NCTM Annual Meeting New Orleans, Louisiana

Thursday, April 10, 2014

Diana L. Kasbaum

ASSM Immediate Past-President

WI Department of Public Instruction

[email protected]

Michelle Butturini

Fifth Grade Teacher, District Math Leader

Reedsville Elementary/Middle School

[email protected]

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Goals for Today’s Session

Identify characteristics of rich and

informative tasks

Discuss rigor

Experience tasks that engage

students

Look at student work and identify

resources for rich tasks

Let’s Do a Little Math…..

Five friends ordered 3 large sandwiches.

James at 3/4 of a sandwich

Katya ate 1/4 of a sandwich

Ramon ate 3/4 of a sandwich

Sienna ate 2/4 of a sandwich

How much sandwich is left for Oscar?

Smarter Balanced Released Item 43051

How did you solve the problem?

Could you approach the problem

another way?

How could you scaffold the problem?

How could you extend the problem?

Vision for school mathematics

Imagine a world where students …..

Are actively engaged with worthwhile tasks

Become mathematical problem solvers

Learn to reason mathematically

Are confident in their ability to learn and use

mathematics

Principles to Actions Draft, NCTM 2014

Short tasks are rich in mathematics learning opportunities provide valuable information so that teachers can meet the needs of their students.

Tasks provide opportunity to influence classroom discourse around the task and make the task accessible to all kids.

Tasks provide an invisible formative assessment opportunity for teachers to gauge learning while students are involved in rich mathematics.

NCTM ‘10 Minute Tasks’ Strand

What the difference between what

we traditionally refer to as math

‘problems’ and rich math tasks?

Problems are generally non-contextual and procedural in nature.

Problems may require little or no justification.

Rich tasks engage students in problem solving and have multiple entry points.

Rich tasks naturally evoke persistence.

Rich tasks are deepened through discourse and collaboration.

Rich tasks provide much more information about student understanding.

Effective teaching of mathematics

engages students in solving and

discussing tasks that promote

mathematical reasoning and problem

solving and allow multiple entry points

and varied solution strategies.

Principles to Actions, p. 17, NCTM 2014

Effective teaching of mathematics

engages students in solving and

discussing tasks that promote

mathematical reasoning and problem

solving and allow multiple entry

points and varied solution strategies.

AND provides information to inform

instruction. (Kasbaum, 2014)

What do we know from research?

1. Not all tasks provide the same

opportunities for student thinking and

learning.

Hiebert et al, 1997; Stein et al, 2009



2. Student learning is greatest in

classrooms where the tasks consistently

encourage high-level student thinking

and reasoning and least in classrooms

where the tasks are routinely procedural

in nature.

Boaler and Staples, 2008; Hiebert and Wearne, 1993;

Stein and Lane, 1996



3. Tasks with high cognitive demands are

the most difficult to implement well and

are often transformed into less

demanding tasks during instruction.

Stein, Grover and Henningsen, 1996;

Stigler and Hiebert, 2004


This matrix from the Smarter Balanced Content Specifications for Mathematics draws from both Bloom’s (rev.)

Taxonomy of Educational Objectives and Webb’s Depth-of-Knowledge Levels below., Based on Cognitive Rigor Matrix

from Karen Hess, 2009, [email protected].

Co

gnit

ive

Rig

or

Mat

rix

Higher Level Demands (doing mathematics)

Require complex, non-algorithmic thinking.

Demand self-monitoring or self-regulation.

Require students to analyze the task.

Require considerable cognitive effort.

Higher level Demands (procedures with connections)

Focus on use of procedures to develop deeper levels of understanding of concepts and ideas.

Usually represented in multiple ways.

Require cognitive effort

Lower Level Demands (memorization)

Reproduces facts

No connection to concepts or

meaning.

Lower Level Demands (procedures w/o connections)

Algorithmic.

Focused on producing correct

answers.

Require no explanations or

explanations that focus on

describing a procdure.

Leve

ls o

f D

eman

d

Smith and Stein, 1989

• Reason abstractly and quantitatively. (MP 2)

• Construct viable arguments and critique the reasoning of others. (MP 3)

Reasoning and Explaining

• Model with mathematics. (MP 4)

• Use appropriate tools strategically. (MP5)

Modeling and Using Tools

• Look for and make use of structure. (MP 7)

• Look for and express regularity in repeated reasoning. (MP 8)

Seeing Structure and Generalizing

• Make sense of problems and persevere in solving them. (MP 1)

• Attend to precision. (MP 6)

S

ta

nd

ard

s fo

r M

ath

em

atic

al P

ra

ctic

e • Make sense of problems and persevere in

solving them. (MP 1)

• Attend to precision. (MP 6)

Snow Day Chores

I have to shovel ½ of the driveway. Before lunch I shoveled 3/8 of the whole driveway. What fractional part of my share of the driveway did I shovel before lunch?

Draw and label a picture representing the fractional parts of the shoveled driveway.

Milwaukee Mathematics Partnership Constructed Response Item

http://www4.uwm.edu/Org/mmp/CR2_2012-13/GR6CR2Anch.pdf

Student Work Sample A





Student Work Sample B





Student Work Sample C





Student Work Sample D





http://www4.uwm.edu/Org/mmp/_resources/CR_Items.htm

What are teachers doing?

Providing opportunities for exploration and

problem solving that build on and extend

current understanding.

Select tasks that provide multiple entry points.

Pose tasks that require high level of cognitive

demand.

Support students in exploring tasks without

taking over student thinking.

Encourage a variety of approaches and

strategies to make sense of and solve tasks.

Imp

lem

enti

ng

Task

s


Persevering in exploring and reasoning through

tasks.

Taking responsibility for making sense of tasks by

drawing on and making connections with prior

understanding.

Using tools and representations to support thinking

and problem solving.

Using a variety of solution approaches and

discussing and justifying their strategies.

Imp

lem

enti

ng

Task

s What are students doing?


For each figure, estimate the shaded portion

as a fraction of the whole figure. Explain how

you got each of your estimates.

Uncovering Student Thinking About Mathematics in the Common Core: 25 Formative

Assessment Probes, Cheryl Rose Tobey and Emily R. Fagan

Uncovering Student Thinking About Mathematics in the Common Core: 25 Formative

Assessment Probes, Cheryl Rose Tobey and Emily R. Fagan

Finally, we know that with

good tasks students will demonstrate

mathematical understanding through

the use of:

concrete illustrations

mathematical representations

example applications.

Phil Daro, 2011

Go

od

Tas

ks

Co

nte

nt

Sta

nd

ard

s

“Understanding” standards are the points of

intersection between the Standards for

Mathematical Content and the Standards for

Mathematical Practice”

Teaching Student-Centered Mathematics, Grades 3-5 Volume

3, John A. Van de Walle and LouAnn H. Lovin

Uncovering Student Thinking About Mathematics in the

Common Core: 25 Formative Assessment Probes, Cheryl

Rose Tobey and Emily R. Fagan

Cognition-Based Assessment and Teaching of Fractions,

Michael T. Battista

Learning Mathematics in the Intermediate Grades, Madison

Metropolitan School District

https://mathweb.madison.k12.wi.us/files/math/LMIGcomplete.pdf

Math Reasoning Inventory, Marilyn Burns

www.mathreasoninginventory.com

NCTM Illuminations: http://illuminations.nctm.org/

Figure This: http://www.figurethis.org/index.html

Re

sou

rce

s

Educational Development Center: http://mathpractices.edc.org/

Illustrative Mathematics Project:

http://www.illustrativemathematics.org/

Mathematics Assessment Project:

http://map.mathshell.org/materials/index.php

Georgia Department of Education - CCGPS Frameworks

https://www.georgiastandards.org/Common-Core/Pages/Math-K-

5.aspx and https://www.georgiastandards.org/Common-

Core/Pages/Math-6-8.aspx

Engage NY Modules - Sprints from each lesson

http://www.engageny.org/resource/grade-5-mathematics

CGI Problem Solving and Reasoning Assessment Questions

Re

sou

rce

s

http://www.illustrativemathematics.org/

SBAC Sample Performance Tasks:

http://sampleitems.smarterbalanced.org/itemprevie

w/sbac/index.htm

PARCC Sample Items:

http://www.parcconline.org/samples/math

Milwaukee Mathematics Partnership:

http://www4.uwm.edu/Org/mmp/_resources/CR_Ite

ms.htm

Re

sou

rce

s

Implementing the CCSSM Through Problem Solving, Grades 3-5 By Mary Foote, Darrell Earnest, Shiuli Mukhopadhyay, Frances Curcio

http://www.nctm.org/catalog/product.aspx?id=14446

Even an engaging and approachable mathematical

task is not sufficient to support student achievement, if

it is not also accompanied by a classroom atmosphere

that values all students, their contributions, and the

knowledge they bring.

Along with providing interesting and appropriately

challenging tasks, it is critical to engage learners in the

examination and discussion of mathematical ideas

and processes. Students who do not participate are

unlikely to learn (Lave and Wenger, 1991).

Thank you

Diana L. Kasbaum Association of State Supervisors of Mathematics,

Immediate Past-President

Mathematics Education Consultant

WI Department of Public Instruction

[email protected]

Michelle Butturini Fifth Grade Teacher, District Math Leader

Reedsville Elementary/Middle School

[email protected]

Documents

Engaging Tasks to Gauge Student Learning · Uncovering Student Thinking About Mathematics in the Common Core: 25 Formative Assessment Probes, Cheryl Rose Tobey and Emily R. Fagan