Developing a learning progression in college-ready ...

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Developing a learning progression in college-ready algebraic thinking

through problem solvingJames Mason1, Amy E. Arneson1, Diah Wihardini2, & Mark Wilson1

1 Graduate School of Education, University of California, Berkeley2 Bina Nusantara University, Jakarta

National Council on Measurement in Education

2018 Annual Meeting15 April 2018

This work was supported by the National Math and Science Initiative (NMSI).

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Project Background• Multi-year project to design and build assessments of college

readiness○ Sponsored by the National Mathematics and Science Initiative

(NMSI)

• Algebraic thinking (this presentation), and

• Statistical thinking (next presentation)

• Allows both formative and summative uses

• Does not claim to cover the entire breadth, yet○ “Demonstration” constructs, on the way to laying out

a more complete framework

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Algebraic ThinkingNot just algebra, but knowing what algebra to use, and why...

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Background on Algebraic ThinkingWe found a diverse literature on the structure of algebraic thinking,

including:

• Classes of mathematical object, and how they are used:

○ Kaput (1995), Smith (2003) — generalizing and formalizing, the study

of functions, cluster of modeling languages;

• Algebraic Activities:

○ Kieran (1998, 2007) — generational, transformational, global-meta;

○ Gascon (1994) — modeling, exposing structure; abstracting

expressions to functions; abstracting functions to classes of function;

• Meaning-Making:

○ Mason, Graham, and Johnston-Wilder (2005) — expressing

generality, multiple expressions, freedom and constraint,

experiencing structure and generalized arithmetic

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Scope of College Ready Algebraic ThinkingThe algebraic thinking needed by students

• in their first two years of college,• majoring in a field other than mathematics

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Algebraic Thinking ConstructsThe problem solving process illustrated with a sample assessment task

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Algebraic Thinking through Problem SolvingHow do students use the algebraic resources they have learnedin order to solve problems in real-world contexts?

In the following example, we present:

• A cycle of problem solving

• Four algebraic thinking constructs

• A sample assessment task○ Contains a cluster of items○ Items can be on multiple constructs○ Follows the student’s problem-solving

AppliedMathematics

Modeling & Notation

Real-world Problem

1 4

2 3

TEST

1. _________

2. _________

3. _________

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Problem-Solving Process (1) 1. Start with a problem context for the task.

Real-world Problem

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Sample Assessment Task: “Home on the Range”1. Start with a problem context for the task.

Real-world Problem

A cowboy sees a drone hovering 5000 meters (m) directly above his ranch on a windless day and tries to destroy it by shooting his pistol perfectly straight up.

(Note that this is both illegal under federal law, and a very bad idea, but the cowboy doesn’t know that yet.)

The bullet leaves the pistol travelling at 300 meters per second (m/s), slowing down as it goes up, due to gravity. It misses the drone, and eventually, it stops and falls back down to the ground.

TEST1. ______2. ______3. ______

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Problem-Solving Process (2)

Modeling & Notation

Real-world Problem

Direction of increasing abstraction

MAP

2. Model the problem context mathematically.

Constructs

MAP:Modeling Applied Problems

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Modeling Applied Problems (MAP)

Applied Mathematics Domains

Modeling & Notation

Real-world Problem

MAP



Constructs


Modeling Applied Problems

Choosing models as a decision-making process

● Conceptual Model● Mathematical

Model

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Sample Assessment Task — Model Provided


Modeling & Notation

Real-world Problem

MAP



Constructs


TEST1. ______2. ______3. ______

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Modeling & Notation

Real-world Problem

MAP


FMO

3. Apply algebraic resources to work towards a solution

Constructs


FMO:Functions as Mathematical Objects

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Functions as Mathematical Objects (FMO)


Modeling & Notation

Real-world Problem

MAP


FMO


Constructs



Functions as Mathematical Objects

Understanding functions through multiple representations

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Sample Assessment Task — FMO items


Modeling & Notation

Real-world Problem

MAP


FMO


Constructs



TEST1. ______2. ______3. ______

Time (s) Height (m)

0 0

10 2500

20 4000

30

40 4000

50

60

4500

2500

0

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Modeling & Notation

Real-world Problem

MAP


FMO


Constructs



TEST1. ______2. ______3. ______

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Modeling & Notation

Real-world Problem

MAP


FMO


Constructs



TEST1. ______2. ______3. ______

How long, in seconds, will it take the bullet to return to the ground?

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Modeling & Notation

Real-world Problem

MAP


FMO

4. Sometimes multiple steps are needed...

Constructs



LRA:Location, Rate, and Acceleration

LRA ...

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Location, Rate, and Acceleration (LRA)


Modeling & Notation

Real-world Problem

MAP


FMO


Constructs




LRA ...

Location, Rate, and Acceleration

Connections involving slope and area-under-graph.Calculus not required.

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Sample Assessment Task — LRA item


Modeling & Notation

Real-world Problem

MAP


FMO


Constructs




LRA ...

TEST1. ______2. ______3. ______

Approximately how fast will the bullet be going when it returns to the ground?

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Problem-Solving Process (5)5. Interpret the results within the problem context

Constructs




IMR:Interpreting Mathematical Results


Modeling & Notation

Real-world Problem

MAP


FMO LRA ...

IMR

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Interpreting Mathematical Results (IMR)5. Interpret the results within the problem context

Constructs






Modeling & Notation

Real-world Problem

MAP


FMO LRA ...

IMR

Interpreting Mathematical Results

Connecting the resultsback to the problem, and to the real world.

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Sample Assessment Task — IMR item5. Interpret the results within the problem context

Constructs






Modeling & Notation

Real-world Problem

MAP


FMO LRA ...

IMR

TEST1. ______2. ______3. ______

Why might the cowboy need to know these things?

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Problem-Solving Process

Constructs






Modeling & Notation

Real-world Problem

MAP IMR


FMO LRA ...

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ResultsHighlights from Qualitative Data Analysis

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Findings about assessment tasksFrom cognitive labs, we learned about issues with particular assessment tasks which allow for a cycle of iterative revision.

Successive rounds of cognitive labs for “Home on the Range” revealed:

1. Student answers to “which representations were useful” follow-ups were repetitive and uninformative, as was their thought processes described in the interview.

2. Filling in the entire table involved repetitive computation, taking too much time without revealing algebraic thinking.

3. Some students used the pre-completed rows in the table to check their logic.

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Findings about assessment tasksIn “Give me a Brake” (an LRA-focused cluster, with emphasis on graph interpretation), cognitive labs revealed that students can use an unintended solution strategy.

• Particulars of the problem setup lead to a discontinuity in the “rate” graph, which students interpreted instead of the overall shape.

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Student responses to open-ended items during field tests also revealed occasional response patterns that appeared to contain algebraic thinking, but not on the intended construct.

• For example, for [3](c) (why the cowboy needs to know speed and time-in-flight): “so he doesnt miss, but i doubt he can do this math”

Findings about assessment tasks

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Findings about constructsFrom cognitive labs, we also learned about construct issues.

• Upper levels of MAP and FMO require “non-school” problems, where the solution strategy is not obvious to them.○ In interviews, some students went a long way down an unfruitful solution

path, before realising the error.○ Acceptable, even desirable, in an instructional task, in an assessment task

problems can arise in terms of testing time and motivation.○ Only partly addressable by scaffolding in the task structure, without

removing the possibility for students to demonstrate their ability to work with unfamiliar contexts.

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DiscussionWhere do we go from here?

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What we learned...• Our item clusters with extended problem scenarios did allow

students to reveal algebraic thinking processes, however○ More work needed to balance the authenticity of problem contexts with

accessibility in terms of domain knowledge and testing time.○ It is challenging to invoke real-world problem contexts without requiring

too much domain knowledge (e.g., physics in “Home on the Range”)○ Contexts that are novel (for the students) can allow them to reveal higher

levels of algebraic thinking, but can also allow them to spend time on a wrong path.

• Iterative assessment development is crucial○ In addition to item paneling, multiple rounds of cognitive labs and initial

pilot testing were required to refine the assessment tasks and our understanding of the constructs.

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Future Steps• Quantitative Analysis (fit a measurement model)

○ This may reveal issues with particular items that were not apparent from individual responses

• Next iteration of assessment development○ Revisions to constructs, changes to items, revisions to outcome space…

• Collect more data with the revised instrument○ Both qualitative and quantitative

• Multidimensional Analysis○ We want to learn how the constructs in this learning progression relate to

each other

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Thank [email protected]

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Shruti BathiaAlexander BlumJonathan BosticRich BrownMichelle CarneyDavid ConleyKaren DraneyPerman GochyyevEmiliano GomezJerred Jolin

Ana Maria Albornoz ReitzeRebecca FreundShazi KhanChanel QuisolDavid Torres Irribarra

ACKNOWLEDGEMENTS

Advisory Panel BEAR IT

Tzur KarelitzMaureen LahiffLinda MorrellRich PatzJulia PhelanLaura PryorWeeraphat SuksiriJosh SussmanYukie ToyamaDanhui Zhang

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Appendices

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Neglecting air resistance, the height of the bullet in flight is given by the function:

where t is the time in flight (in seconds), d(t) is the height above the ground (in meters), and v0 is the initial speed, 300m/s .

We will use a=−10m/s2, as the acceleration due to gravity.

The bullet does not travel horizontally, because it is shot straight up and there is no wind.

Sample Assessment Task — Stimulus MaterialA cowboy sees a drone hovering 5000 meters (m) directly above his ranch on a windless day and tries to destroy it by shooting his pistol perfectly straight up.

(Note that this is both illegal under federal law, and a very bad idea, but the cowboy doesn’t know that yet.)

The bullet leaves the pistol travelling at 300 meters per second (m/s), slowing down as it goes up, due to gravity. It misses the drone, and eventually, it stops and falls back down to the ground.

TEST

1. _________

2. _________

3. _________

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Modeling Applied Problems (MAP)

Conceptual ModelMathematical Model

Choosing models as a decision-making process

5 - General Framework

4 - Justified Comparison

3 - Unjustified Comparison

2 - Uncritical

1 - Inaccurate

5 - General Framework

4 - Justified Comparison

3 - Unjustified Comparison

2 - Uncritical

1 - Inaccurate

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Functions as Mathematical Objects (FMO)Understanding functions through multiple representations

5 - Critical

4 - Connected

3 - Multirepresentational

2 - Representational

1 - Functional

0 - Prefunctional

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Location, Rate, and Acceleration (LRA)Connections involving slope and area-under-graph

5 - Connected

4 - Slope and Area

3 - Slope only

2 - Vague

1 - No Connection

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Interpreting Mathematical Results (IMR)Connecting the results back to the problem, and the real world.

4 - Step Beyond

3 - Full Connection

2 - Unconnected

1 - Simple

0 - No Interpretation

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