a new generation of instructionally supportive assessment

A New Generation of Instructionally

Supportive Assessment: From Drawing

Board to the Classroom: Discussion

Jere Confrey

Joseph D. Moore Distinguished University Professor

The William and Ida Friday Institute for Educational Innovation

College of Education North Carolina State University

Raleigh, NC

CCSSO Assessment Meeting Orlando, FL June 21, 20111

Context—Then and Now

Common Core State Standards, 43 states

New designs for ESEA needed

Wireless networking and Cloud Computing

(shared services model)

Social Networking everywhere

Data-intensive empirical research changing

the way science is done, and different

demands on quantitative modeling and

literacy.

Analytics (SAS)

Knowledge Intensive Industries and STEM

Increasingly mobile student and teacher

populations

Increasing gaps associated with race and

poverty

NCTM Standards (1989) and AAAS

and NRC (Science)

Standards state-by-state

Beginning of NCLB

Disaggregation of data

Nascent technology applications,

graphing calculators

Early internet, minimal access in

schools

Increasingly mobile student and

teacher populations

A Fundamental Shift in Attention:

the Instructional Core

• Under No Child Left Behind, the emphasis was on accountability as

standards were linked to summative assessment. Virtually no

attention to what happened in the instructional core until the

introduction of the Common Core

• The Common Core Standards should focus attention on the

instructional core including curricular implementation, classroom

instructional practices and formative and diagnostic assessment.

• With eventual ESEA revision and reauthorization, more support

for the instructional core can be expected.

• Learning trajectories/progressions form the boundary object

among these ideas as a means to provide a scientific basis to the

instructional core.

Build to support learning trajectories/progressions

.

“One promise of common state standards is that over time,

they will allow research on learning progressions to inform

and improve the design of Standards to a much greater

extent than is possible today.”

CCSS 2010, p.5

Defining the Instructional Core

(Confrey and

Maloney, 2010)

A learning trajectory/progression is: (Simon (1995), Cobb et al. (2003), Confrey (2006), Lehrer and Schauble (2006), and others)

…a researcher-conjectured, empirically-supported description of the ordered network of constructs a student encounters through instruction (i.e. activities, tasks, tools, forms of interaction and methods of evaluation), in order to move from informal ideas, through successive refinements of representation, articulation, and reflection, towards increasingly complex concepts over time

(Confrey et al., 2009)

Learned

ideas

Conceptual

corridor

Landmarks

Obstacles

Key

Prior

knowledge

Constraints (borders of the

corridor)

Class’s actual

conceptual

trajectory

Confrey (2006) Design Studies Chapter

Cambridge Handbook of the Learning Sciences

5 characteristics of Learning Trajectories

1. Learning trajectories identify a particular domain and a goal level of

understanding.

2. Learning trajectories recognize that children enter instruction with

relevant yet diverse experiences that serve as effective starting points;

3. Learning trajectories assume a progression of cognitive states that

move from simple to complex; while not linear, the progression is not

random, and can be sequenced and ordered as “expected tendencies”

or “likely probabilities”.

4. Progress through a learning trajectory assumes a well-ordered set of

tasks (curriculum), instructional activities, interactions, tools, and

reflection.

5. Learning trajectories are based on synthesis of existing research,

further research to complete the sequences, and a validation method

based on empirical study.

Challenge for empirical study and validation…

• Movement through levels of a learning trajectory is not an

accumulation of partial states. Rather, understanding

undergoes multiple rounds of reorganization as students

encounter new challenges. The paths through those

reorganizations involve ephemeral intermediate states that

will “disappear” as understanding undergoes its

metamorphosis.

Value of Learning Trajectories to Teachers

• Know what to expect about students‟ preparation

• More readily manage the range of preparation of students in your class

• Know what teachers in the next grade expect of your students.

• Identify clusters of related concepts at grade level

• Clarity about the student thinking and discourse to focus on conceptual development

• Engage in rich uses of classroom assessment

Kindergarten Grade 1

Common Core Standards--

in Learning Trajectory Format

[K.NBT.1]

[1.NBT.6]

Content

Strand

Pla

ce

Va

lue

an

d D

ecim

als

[1.NBT.2]

[1.NBT.5]

[1.NBT.3]

Three posters-- K-

5, 6-8, and 9-12

CCSS-M standards,

available from

Wireless

Generation

The Common Core State

Standards for Mathematics,

in Learning Trajectory Format

Comments on Common Core State Standards for

Mathematics, in Learning Trajectory Format

• Abbreviated/abridged learning trajectories

• When formatted as learning trajectories,

• the sequences of standards are missing elements needed to support

fuller development.

• the intervals between standards (granularity) are of varied sizes

• some trajectories have strong empirical support

• others are logical thought experiments by mathematicians and need

proper empirical investigation

• We will demonstrate the work that remains to be done to

extend CCSS into a fuller treatment of LTs over time.

Features of the Chart

• Each Common State Standard displayed horizontally shows

how the standards progress in sophistication over time

• All standards for each grade level displayed in a single

column to support coherence across content domains or

conceptual categories

Transition to Hexagon Map

• While the Common Core Standards represented in Learning Trajectories format helps to articulate the development over time, there are limitations to the chart format.

• Learning trajectories are not strictly linear and often support multiple forms of connections among standards and proficiency levels.

• Led us to build a hexagon map to display the standards K-12.

Convert measures and

transform units when

multiplying or dividing

by applying ratio

reasoning.

6.RP.3d.iv

Understand the

concept of ratio as a

relationship between

2 quantities

6.RP.1

Solve ratio problems

using tables of values

with coordinate plots to

find missing values and

compare ratios.

6.RP.3.a.i

Understand the

concept of a unit ratio

such that a:b :: a/b:1

6.RP.2

Solve real world

problems involving

ratios by using tables,

tape diagrams, and

number lines. Extend

to include percents.

6.RP.3.c.iii

Solve real world

problems including

unit pricing and

constant speed.

6.RP.3.b.ii

6.G.3

Find unit ratios

equivalent to complex

fractions (a/b : c/d ::

a/b / c/d : 1)

7.RP.1

6.G.3

Recognize and

represent proportional

relationships between

quantities using

tables, graphs, and

unit ratios or rates

7.RP.2.ad.i

6.G.3

Represent proportional

relationships by

equations, tables,

graphs, diagrams, and

verbal descriptions

including identifying

proportionality constant

7.RP.2bc.ii

6.G.3

Use proportional

relationships to solve

multistep ratio

problems (including

percent increase and

decrease)

7.RP.3

Examine proportional

relationships using

multiple representations

(graphs, tables, and

equations) and interpret

slope

8.EE.5

6.G.3

Define slope using ratios

and similar triangles and

derive the equation y =

mx and y = mx+b

8.EE.6

Recognizes ratio

equivalence when 2

quantities are both

doubled or tripled,

etc.

1

6.RP.1

Can use ratio unit to

increase or decrease

in table or graph

5

6.RP.3a.i

Given ratio

relationship and one

additional value,

identifies missing

value

8

6.RP.3b.iiFor a given ratio

identifies unit ratio or

unit rate

7

6.RP.2

Uses equivalence

between ratio and

percentage in context

9

6.RP.3c.iii

Applies ratio

reasoning to convert

measurement units

and explain

10

3.RP.d.ivCompare ratios using

multiple methods,

tables, graphs, and

number displays

11

8.EE.5

Solves applied

problems in context

for any rational values

12

7.RP.1 Relates ratio relations

to direct variation,

y=kx

13

7.RP.2bc.ii

Distinguishes direct

variation from non-

proportional relations

14

8.EE.6

Relate percent


to ratio equivalence to

solve problems

15

7.RP.3

Recognizes ratio

relations in table or

graph based on rate

of change

6

7.RP.2ad.i

2

Creates table of

values for equivalent

ratios extended by

scaling

3

Recognizes ratio

equivalence when 2

quantities are both

are split

2

Recognizes ratio

equivalence when 2

quantities are both

doubled or tripled,

etc.

1

6.RP.1

Can find smallest

whole number ratio

equivalent

4

Can use ratio unit to


in table or graph

5

6.RP.3a.i

Given ratio

relationship and one

additional value,

identifies missing

value

8

6.RP.3b.iiFor a given ratio

identifies unit ratio or

unit rate

7

6.RP.2

Uses equivalence

between ratio and

percentage in context

9

6.RP.3c.iii

Applies ratio

reasoning to convert

measurement units

and explain

10

3.RP.d.ivCompare ratios using

multiple methods,

tables, graphs, and

number displays

11

8.EE.5

Solves applied

problems in context

for any rational values

12

7.RP.1 Relates ratio relations

to direct variation,

y=kx

13

7.RP.2bc.ii

Distinguishes direct

variation from non-

proportional relations

14

8.EE.6

Relate percent


to ratio equivalence to

solve problems

15

7.RP.3

Recognizes ratio

relations in table or

graph based on rate

of change

6

7.RP.2ad.i

Ratio and Proportion:

Abridged and Unabridged

Learning Trajectories vs. “Standards Progressions”

• Following comparison illustrates how learning

trajectories, as we define them, are distinct from

„standards progressions‟ as prepared by CCSSO

writers.

Learning Proficiency Common Core Standards

5 Can find and use ratio unit to increase and

decrease using the ratio unit in a table or graph.

6.RP.3a.i Make tables of equivalent ratios relating

quantities with whole-number measurements, find

missing values in the tables, and plot the pairs of

values on the coordinate plane. Use tables to

compare ratios.

4 Can find the smallest whole number ratio that is

equivalent to a given ratio.

3

Can create a table of values for pairs of equivalent

ratios and extend the table through splitting and

scaling for indefinite number of points

2

Recognizes that for situations in which there are

two quantities covarying, that splitting each

quantity produces an equivalent relation between

the quantities.

1

Recognizes that for situations in which there are

two quantities covarying such as fair shares or

mixtures, that doubling, tripling, etc. the amount

of each of the two quantities by the same factor

produces an equivalent relationship between the

quantities and that these processes can be

continued indefinitely still maintaining

equivalence.

6.RP.1 Understand the concept of a ratio and use

ratio language to describe a ratio relationship

between two quantities. For example, “The ratio of

wings to beaks in the bird house at the zoo was 2:1,

because for every 2 wings there was 1 beak.” “For

every vote candidate A received, candidate C

received nearly three votes.”

Learning Proficiency Common Core Standards

10

Applies ratio reasoning to convert

measurement units and to explain unit

conversions.

6.RP.3d.iv Use ratio reasoning to convert measurement units;

manipulate and transform units appropriately when

multiplying or dividing quantities.

9

Identifies equivalence between a ratio relation

and its same value written as a percentage and

can solve simple percentage problems.

6.RP.3c.iii Use ratio and rate reasoning to solve real-world and

mathematical problems, e.g., by reasoning about tables of

equivalent ratios, tape diagrams, double number line

diagrams, or equations.

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a

quantity means 30⁄100 times the quantity); solve problems

involving finding the whole, given a part and the percent.

8

For a given ratio relationship ,given one

additional value of the proportion, identifies

the missing value of the proportion and

justifies the reasoning for it.

6.RP.3b.ii Solve unit rate problems including those involving

unit pricing and constant speed. For example, if it took 7

hours to mow 4 lawns, then at that rate, how many lawns

could be mowed in 35 hours? At what rate were lawns being

mowed?

7

For a given ratio relationship , identifies the

value that corresponds to per one (unit ratio or

unit rate).

6.RP.2 Understand the concept of a unit rate a⁄b associated

with a ratio a:b with b ≠ 0, and use rate and slope language in

the context of a ratio relationship.

6

Can recognize ratio relations when displayed

in a table or a graph based on rate of change

(as related differences or based on slope and

similarity).

7.RP.2ad.i Recognize and represent proportional relationships

between quantities. a. Decide whether two quantities are in a

proportional relationship, e.g., by testing for equivalent ratios

in a table or graphing on a coordinate plane and observing

whether the graph is a straight line through the origin.

What Instructional Practices Follow From the Use

of Learning Trajectories ?

(Confrey and

Maloney, 2010)

Teacher‟s Math

Knowledge

Fostering Discourse

Examination of

Curricular Materials

Formative Assessment

Practices

Selection of

Instructional Tasks

Interactive Diagnostic

Assessment System

Teacher Practices Affected by LTs

1. Teachers‟ Math Knowledge for Teaching

2. Examination of Curricular Materials

3. Selection of Instructional Tasks

4. Fostering Discourse

5. Formative Assessment Practices

6. Implementing Interactive Diagnostic System

a new generation of instructionally supportive assessment

Documents