katie mceldoon, kelley durkin & bethany rittle-johnson 1

Katie McEldoon, Kelley Durkin

& Bethany Rittle-Johnson

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Instructional Time Push to spend more time on topics to increase

depth of learning (Hu, 2010)

Instructional time is limited

Need to utilize this limited time with the most effective learning activities

In math classrooms, students spend a lot of time practicing skills (Hiebert et al., 2003)

How is this instructional time best used? Scaffolding the practice with a conceptually

oriented learning activity

Completing additional practice2

Conceptually Oriented Activity:Self-Explanation

Prompting students to generate explanations to themselves in an attempt to make sense of new information (Chi, 2000)

Many domains: e.g. Biology, reading, computer programming, electrical engineering

Mathematics (e.g., Atkinson, Renkl, & Merrill, 2003)

Within mathematics, self-explanation has been shown to increase both learning and transfer of knowledge to novel tasks (e.g., Rittle-Johnson 2006; Atkinson, Derry, Renkl & Wortham, 2000)

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Conceptually Oriented Activity:Self-Explanation

Mechanisms of Self-Explanation

Integrates new and existing knowledge (Chi et al. 1994)

Correction of current mental model (Chi et al. 1994)

Inference rules proceduralized into usable skills (Chi et al. 1989)

Fosters generalization (Lombrozo 2006; Rittle-Johnson, 2006)

Procedural and conceptual knowledge help each other grow (Rittle-Johnson & Alibali, 1999; Rittle-Johnson, Siegler, & Alibali, 2001)

Competence in mathematics (Hiebert, 1986)

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Benefit of Extra Practice

- Greater skill at applying initial problem solving strategy (Chi, Glaser, Farr 1988; Ericsson, Krampe, Tesch-Romer, 1993)

- Strengthen correct strategy application, weaken incorrect strategies (Seigler, 2002)

- Problem solving procedure becomes more automatized - Leaving more working memory free to acquire

new and more efficient strategies (Logan 1990; Schneider, Shiffrin, 1977; Anderson, 1982, 1983, 1987; Rosenbloom & Newell, 1987)

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Research Questions

1. What is the learning benefit of completing self-explanation prompts?

2. What is the learning benefit of solving additional practice problems?

3. Which use of this additional instructional time is the most beneficial for student learning?

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Hypotheses1. What is the learning benefit of

completing self-explanation prompts?

1. What is the learning benefit of solving additional practice problems?

1. Which use of this additional instructional time is the most beneficial for student learning?

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Self-explanation prompts will result in greater procedural knowledge in familiar and novel problem types

Additional practice problems will result in greater procedural knowledge in familiar problem types

Self-explanation will be most beneficial for student learning

Study Design

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Control Self-Explanation

Additional Practice

Practice Problem 1 Practice Problem 1 Practice Problem 1

Practice Problem 2 Self-Explain Practice Problem 2

Practice Problem 3 Practice Problem 2 Practice Problem 3

Practice Problem 4 Self-Explain Practice Problem 4

Practice Problem 5 Practice Problem 3 Practice Problem 5

Practice Problem 6 Self-Explain Practice Problem 6

Practice Problem 4 Practice Problem 7

Self-Explain Practice Problem 8

Practice Problem 5 Practice Problem 9

Self-Explain Practice Problem 10

Practice Problem 6 Practice Problem 11

Self-Explain Practice Problem 12

Additional Instructional

Time

Learning Domain – Math Equivalence

The notion that the equal sign means that two sides of an equation are equivalent

4 + 2 + 3 = ___ + 6 (McNeil, 2008)

Many children view the equal sign operationally, as a command to carry out arithmetic operations (Baroody & Ginsburg, 1983; Carpenter, et al., 2003; McNeil & Alibali, 2005)

4 + 2 + 3 = _9_ + 6 9

Design Participants: 75 students in grades 2, 3, and 4

Procedure

Pre Test (paper & pencil)

Inclusion Criterion: <80% on pretest

Intervention (one on one)

Procedural Instruction

Practice Problems

Post Test (immediate)

Retention Test (two weeks)

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Manipulation Here!

• Pretest, Immediate Posttest, & Retention Test (Rittle-Johnson, Matthews, Taylor & McEldoon, 2011)

1.Procedural Knowledge Section

•Solving Open Equations 2 + 5 + 8 = ___ + 8

• Learning Items 3 + 4 + 6 = ___ + 4

• Transfer Items 8 + ___ = 8 + 6 + 4

6 - 4 + 3 = ___ + 3

2.Conceptual Knowledge Section•Meaning of the Equal Sign

•Recognizing Valid Equation Structures

Assessments

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Intervention: Procedural Instruction

6 + 4 + 9 = 6 + __

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• Instructed students on Add-Subtract strategy (Perry, 1991; Rittle-Johnson, 2006)

• Students asked to solve

• Accuracy Feedback

• Two Instructional Problems

Intervention: Practice Problems

What number goes in the box?

3 + 4 + 8 = + 8

How did you get your answer?

Right/Actually, 7 is the right answer.

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Study Design

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Control Self-Explanation

Additional Practice

Practice Problem 1 Practice Problem 1 Practice Problem 1

Practice Problem 2 Self-Explain Practice Problem 2

Practice Problem 3 Practice Problem 2 Practice Problem 3

Practice Problem 4 Self-Explain Practice Problem 4

Practice Problem 5 Practice Problem 3 Practice Problem 5

Practice Problem 6 Self-Explain Practice Problem 6

Practice Problem 4 Practice Problem 7

Self-Explain Practice Problem 8

Practice Problem 5 Practice Problem 9

Self-Explain Practice Problem 10

Practice Problem 6 Practice Problem 11

Self-Explain Practice Problem 12

Additional Instructional

Time

Intervention: Self-Explanation Prompts

3 + 4 + 8 = 15 + 8Jacob got 15, which is a wrong answer.

3 + 4 + 8 = 7 + 8Hannah got 7, which is the right answer.

HOW do you think Jacob got 15?

WHY do you think 15 is a wrong answer?

HOW do you think Hannah got 7?

WHY do you think 7 is the right answer?

15(Siegler, 2002; Rittle-Johnson 2006)

Posttest & Retention Test

Immediate Posttest Paper & pencil Approx. 25 minutes

Retention test Average of two weeks after intervention session Paper & pencil Approx. 25 minutes

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Instructional Time Intervention Total Problem Solving Time

Average Problem Solving Time per Problem: 26s

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Intervention Accuracy

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No differences by condition

No gains during additional practice problems

Low strategy Invention

Procedural Knowledge Analysis

Procedural knowledge: Correct action sequences or strategies for

solving problems (Rittle-Johnson & Alibali, 1999; Anderson 1993)

Assessment Procedural Knowledge Section

Solve equations with operations on both sides

4 + 5 + 8 = + 8

Students asked to show their work

Coded for strategy use

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Coding Examples Correct Codes (5)

Equalizer: Sets up the two sides as equal 4 + 5 + 8 = 9 + 8

4 + 5 + 8 = 17

9 + 8 = 17

Incorrect Codes (3)

Add to Equal- adds up all numbers before equals sign and puts that number in blank

4 + 5 + 8 = 17 + 8

4 + 5 + 8 = 17

Blank: 4 + 5 + 8 = + 820

Results Roadmap

Procedural Knowledge Items Learning Items Transfer Items

Student Performance Correct Strategy Use Incorrect Strategy Use Unattempted Items

Means for post and retention test scores

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Procedural Learning Items

Same equation structure as the intervention items

Same learned problem solving strategies can be applied to solve

7 + 6 + 4 = 7 + ___

3 + 6 + 5 = __ + 5

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Procedural Learning- Correct

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Procedural Learning- Not Correct

24No Significant Differences

Procedural Learning Summary

There is a benefit of both self-explanation and additional practice

Increased correct strategy use

Decreased incorrect strategy use

No differential performance between additional self-explanation and additional practice

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Procedural Transfer Items

Items that are unlike those in the intervention session

different equation format

includes subtraction

Require a modification of the learned strategy in order to correctly solve

8 + __ = 8 + 6 + 4

6 - 4 + 3 = __ + 326

Procedural Transfer- Correct

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Procedural Transfer- Not Correct

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Procedural Transfer Summary Self-Explanation benefitted procedural

transfer Increased correct strategy use Decreased incorrect strategy use

Self-Explanation leave as many items blank as the control, but they are getting more of the attempted items correct

Additional Practice increases the number of novel problems attempted, even if they may not get them correct

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Assessment Results Summary

Procedural Learning Self-explaining and additional practice

conditions had better performance than control More correct, less incorrect strategy use

Procedural Transfer Self-Explanation group had the best

performance More correct, less incorrect strategy use

Additional practice students attempted more novel items

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Benefits of Additional Instructional Time

1. What is the learning benefit of completing self-explanation prompts?

1. What is the learning benefit of solving additional practice problems?

1. Which use of this additional instructional time is the most beneficial for student learning?

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Self-explanation prompts resulted in greater procedural learning and transfer

Additional practice problems resulted in greater procedural learning

Self-explanation is the most beneficial for student learning

Conclusions Self-explaining during math learning increases

both procedural learning and transfer

This benefit is not just due to the additional time on task (Aleven & Koedinger, 2002; Matthews & Rittle-Johnson, 2009)

Same amount of practice as Control

Same amount of time as Additional Practice

Goal of instruction is to allow students to transfer their knowledge to novel problems Inert Knowledge Problem (Bransford, Brown, & Cocking,

2001)

Self-Explanation is a worthwhile use of instructional time 32

http://peabody.vanderbilt.edu/earlyalgebra.xml

Laura McLean

Marci DeCaro

Kristin Tremblay

Maryphyllis Crean

Maddie Feldman

The Children’s Learning Lab

The first author is supported by a predoctoral training grant provided by the Institute of Education Sciences, U.S. Department of Education, through Grant R305B040110 to Vanderbilt University. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education.

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Conceptual Strategy Use

Perhaps one mechanism is the early adaptation of a conceptually oriented problem solving strategy

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Explanation Quality

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All Procedural Items- Correct

36No Significant Differences

All Procedural Items- Not Correct

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All Procedural Items Summary

No differences in correct strategy use

However, self-explanation decreased the amount of incorrect strategy use

They were leaving more items unattempted instead

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Conceptual Knowledge

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Compared to What Same number of problems, same amount

of time

E.g. Atkinson, Renkl, Merrill, 2003; Hilbert Renkl, Kessler & Reiss, 2008; de Bruin, Rikers & Schmidt, 2007; Grosse & Renkl, 2003; Mwangi & Sweller, 1998

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Control Self ExplainPractice Problem 1 Practice Problem 1

Self-ExplainPractice Problem 2 Practice Problem 2

Self-ExplainPractice Problem 3 Practice Problem 3

Self-ExplainPractice Problem 4 Practice Problem 4

Self-ExplainPractice Problem 5 Practice Problem 5

Self-ExplainPractice Problem 6 Practice Problem 6

Self-Explain