# Early Algebra Research Projects

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Assessment Studies

I. Mathematical Equivalence

II. Functions

III. Repeating Patterns

I. Preparing to Learn from Math Instruction by Solving Problems

II. When Conceptual Instruction Prior to Exploration Improves Mathematical Knowledge

III. Is the Benefit of Self-Explanation Simply Added Time on Task?

IV. Effects of Problem Solving Feedback: Prior Knowledge Matters

V. Importance of Executive Function for Learning About Patterns

I. Homework

# Assessment Studies

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I. Mathematical Equivalence*
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**Assessing Knowledge of Mathematical Equivalence **_{
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**Abstract
**Knowledge of mathematical equivalence, the principle that 2 sides of an equation represent the same value, is a foundational concept in algebra, and this knowledge develops throughout elementary and middle school. Using a construct-modeling approach, we developed an assessment of equivalence knowledge. Second through sixth graders (N = 175) completed the assessment on 2 occasions, 2 weeks apart. Evidence supported the reliability and validity of the assessment along a number of dimensions, and the relative difficulty of items was consistent with the predictions from our construct map. By Grade 5, most students held a basic relational view of equivalence and were beginning to compare the 2 sides of an equation. This study provides insights into the order in which students typically learn different aspects of equivalence knowledge. It also illustrates a powerful but underutilized approach to measurement development that is particularly useful for developing measures meant to detect changes in knowledge over time or after intervention.

**Instruments**

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Children's Understanding of the Equal Sign**

**Abstract
**Knowledge of the equal sign as an indicator of mathematical equality is foundational to children’s mathematical development and serves as a key link between arithmetic and algebra. This study extended prior efforts to use a construct modeling approach to unify diverse measures into a single assessment designed to measure knowledge of the equal sign. Children in Grades 2–6 (N = 224) completed the assessment. The current findings reaffirmed a past finding that diverse items can be integrated onto a single scale, revealed the wide variability in children’s knowledge of the equal sign assessed by different types of items, and provided empirical evidence for a link between equal-sign knowledge and success on some basic algebra items.

**Instruments**

### II. Functions

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**Assessing Functional Thinking Skills**

**Abstract
**Functional thinking is an appropriate way to introduce algebraic concepts in elementary school. We have developed a framework for assessing and interpreting students’ level of understanding of functional thinking using a construct modeling approach. An assessment was administered to 231 second- through sixth-grade students. We then developed a progression of functional thinking knowledge. This investigation elucidates the sequence of acquisition of functional thinking skills. This study also highlights the utility of a construct modeling approach, which was used to create criterion-referenced and ability-leveled assessment. This measure is particularly suited to measuring knowledge change and to evaluating instructional interventions.

**Instruments**

- Assessment with story-based function table section
- Assessment with indexical function table section
- Assessment with non-indexical function table section
- Script
- Help Script

### III. Repeating Patterns

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**Assessing Preschoolers' Knowledge of Repeating Patterns**

**Abstract
**Young children have an impressive amount of mathematics knowledge, but past psychological research has focused primarily on their number knowledge. Preschoolers also spontaneously engage in a form of early algebraic thinking—patterning. In the current study, we assessed four-year-old children’s knowledge of repeating patterns on two occasions (

*N*= 66). Children could duplicate and extend patterns, and some showed a deeper understanding of patterns by abstracting patterns (i.e., creating the same kind of pattern using new materials). A small proportion of the children had explicit knowledge of pattern units. Error analyses indicated that some pattern knowledge was apparent before children were successful on items. Overall, findings indicate that young children are developing an understanding of repeating patterns before school entry.

**Instruments
**Note: Assessment item information embedded in scripts

- Time1 Script (Version 1)
- Time1 Script (Version 2)
- Feedback Session and Time 2 Script (Version 1)
- Feedback Session and Time 2 Script (Version 2)

# Tutoring Studies

### I. Preparing to Learn from Math Instruction by Solving Problems

**Abstract
**Both exploration and explicit instruction are thought to benefit learning in many ways, but much less is known about how the two can be combined. We tested the hypothesis that engaging in exploratory activities prior to receiving explicit instruction better prepares children to learn from the instruction.Children (159 second – fourth graders) solved relatively unfamiliar mathematics problems (e.g., 3 + 5 = 4 + _) before or after they were instructed on the concept of mathematical equivalence. Exploring problems before instruction improved understanding compared to a more conventional instruct-then-practice sequence. Prompts to self-explain did not improve learning more than extra practice. Microgenetic analyses revealed that problem exploration led children to more accurately gauge their competence, attempt a larger variety of strategies, and attend more to problem features—better preparing them to learn from instruction.

**Instruments**

Scripts

- Pretest script
- Intervention script with self-explanation
- Intervention script with extra practice
- Posttest script
- Retention test script

Assessments

DeCaro, D., DeCaro, M., & Rittle–Johnson, B. (under review). Achievement motivation and knowledge development during exploratory learning. Click here for abstract

### II. When Conceptual Instruction Prior to Exploration Improves Math Knowledge

**Abstract
**The sequencing of learning materials is often as important as the content itself. Recently, learning theorists have focused on the sequencing of instruction in relation to solving related problems. The general consensus suggests explicit instruction should be provided; however,

*when*to provide different types of instruction remains unclear. In this experiment, we tested the impact of conceptual instruction preceding or following mathematics problem solving. Specifically, elementary school children (

*N*= 122) received instruction on the concept of math equivalence either before or after being asked to solve and explain challenging problems with feedback. Providing conceptual instruction first resulted in greater learning, transfer, and knowledge of problem structure than delaying instruction until after problem solving. Prior conceptual instruction enhanced problem solving by increasing the quality of explanations and attempted procedures. We compare these results with previous, contrasting findings to outline a potential framework for understanding when instruction should or should not be delayed.

**Instruments**

- Pretest
- Intervention script with problem solving first
- Intervention script with instruction first
- Midtest
- Posttest
- Retention test

**Submitted Paper
**Fyfe, E. R., DeCaro, M. S. & Rittle-Johnson, B. (under review). An alternative time for telling: When conceptual instruction prior to exploration improves mathematical knowledge. Click here for submitted abstract

### III. Is the Benefit of Self-Explanation Simply Added Time on Task?

**Abstract**

*Background*: Self-explanation, or generating explanations to oneself in an attempt to make sense of new information, can promote learning. However, self-explaining takes time, and the learning benefits of this activity need to be rigorously evaluated against alternate uses of this time.

*Aims*: In the current study, we compared the effectiveness of self-explanation prompts to the effectiveness of solving additional practice problems (to equate for time on task) and to solving the same number of problems (to equate for problem-solving experience).

*Sample*: Participants were sixty-nine children in grades 2 through 4.

*Methods*: Students completed a pretest, brief intervention session, and a post and retention test. The intervention focused on solving mathematical equivalence problems such as 3+4+8=__+8. Students were randomly assigned to one of three intervention conditions: self-explain, additional-practice or control.

*Results*: Compared to the control condition, self-explanation prompts promoted conceptual and procedural knowledge. Compared to the additional-practice condition, the benefits of self-explanation were more modest and only apparent on some subscales.

*Conclusions*: The findings suggest that self-explanation prompts have some small unique learning benefits, but that greater attention needs to be paid to how much self-explanation offers advantages over alternative uses of time.

**Instruments**

- Pretest
- Intervention script with self-explanation
- Intervention script without self-explanation
- Posttest
- Retention test

### IV. The Effects of Feedback During Exploratory Mathematics Problem Solving: Prior Knowledge Matters

**Abstract
**Providing exploratory activities prior to explicit instruction can facilitate learning. However, the level of guidance provided during the exploration has largely gone unstudied. In this study, we examined the effects of one form of guidance, feedback, during exploratory mathematics problem solving for children with varying levels of prior domain knowledge. In two experiments, second- and third-grade children solved 12 novel mathematical equivalence problems and then received brief conceptual instruction. After solving each problem, they received (a) no-feedback, (b) outcome-feedback, or (c) strategy-feedback. In both experiments, prior knowledge moderated the impact of feedback on children’s learning. Children with little prior knowledge of correct solution strategies benefitted from feedback during exploration, but children with some prior knowledge of a correct solution strategy benefitted more from exploring without feedback. These results suggest that theories of learning need to incorporate the role of prior knowledge and that providing feedback may not always be optimal.

**Instruments
**

*Experiment 1*

- Pretest
- Intervention script no feedback condition
- Intervention script outcome feedback condition
- Intervention script strategy feedback condition
- Midtest
- Posttest
- Retention test

*Experiment 2*

- Pretest
- Intervention script no feedback condition
- Intervention script outcome feedback condition
- Intervention script strategy feedback condition
- Midtest
- Posttest
- Retention test

**Published Paper
**
Fyfe, E. R., Rittle-Johnson, B., & DeCaro, M. S. (2012). The effects of feedback during exploratory

mathematics problem solving: Prior knowledge matters.

*Journal of Educational Psychology,104,*1094-1108. doi: 10.1037/a0028389

Fyfe, E. R., DeCaro, M. S. & Rittle–Johnson, B. (under review). The role of feedback type and working memory capacity during problem solving. Click here for abstract.

*Experiment 3 *

**Abstract
**Feedback can be a powerful learning tool, but its effects vary widely. Research suggests that learners’ prior knowledge may moderate the effects of feedback; however, no causal link has been established. In this study, we recruited elementary school children who could not solve the target math problems correctly. Children were then randomly assigned to condition based on a crossing of two factors: induced strategy knowledge (yes vs. no) and feedback (present vs. absent). Prior to problem solving, some children were taught a correct strategy, while others were not. During problem solving, some children received feedback while others did not. Feedback had positive effects for children who were not taught a correct strategy, but negative effects for children with induced knowledge of a correct strategy. Results provide evidence for a causal role of prior knowledge and indicate that feedback can hinder learning.

**Instruments**

- Pretest
- Intervention script high knowledge feedback condition
- Intervention script high knowledge no feedback condition
- Intervention script low knowledge feedback condition
- Intervention script low knowledge no feedback condition
- Posttest
- Retention test

**Submitted Paper
**Fyfe, E. R., & Rittle–Johnson, B. (under review). Feedback both helps and hinders learning: The causal role of prior knowledge. Click here for abstract

### V. Importance of Executive Function for Learning About Patterns

**Abstract
**Relational thinking is fundamental to children’s knowledge of repeating patterns (e.g., ABBABB), a central component of early mathematics knowledge. We sought clarity between 3 competing theories (Relational Primacy, Relational Shift, Relational Complexity) differing on the importance of relational thinking and executive function (EF) to preschoolers’ understanding of repeating patterns. 124 children between the ages of 4 and 5 years were administered a Match-to-Sample task (relational thinking), 3 EF tasks (working memory, inhibition, cognitive flexibility), and completed a repeating pattern assessment before and after a brief pattern intervention. Working memory and cognitive flexibility predicted preschoolers’ pattern knowledge at pretest, controlling for age and relational thinking. Working memory also predicted improvements in pattern knowledge after instruction. Findings support the Relational Complexity theory, suggesting that greater EF capacity is beneficial to preschoolers’ repeating pattern knowledge, and that working memory capacity plays a particularly important role over and above relational thinking.

**Instruments
**

- Pretest script
- Intervention script with instructional explanation
- Intervention script with self-explanation
- Intervention script with both instructional and self-explanations
- Posttest script

**Paper in Preparation
**Miller, M., Rittle-Johnson, B., & Loehr, A. (in preparation). Relations Between Preschoolers’ Repeating Pattern Knowledge and Individual Differences in Cognitive Ability.

# Classroom Studies

### I. Promoting Mathematical Problem Solving and Explanation at Home

**Abstract
**Generating explanations, particularly for another person, is associated with greater learning (Teasley, 1995). In fact, Rittle-Johnson et al. (2008) found that students who explained correct solutions to their moms had greater problem-solving transfer compared to those who explained to themselves. Additionally, Van Voorhis (2011) found that family involvement in homework increased student motivation and achievement. To harness the benefits of both parent involvement and explanation, we investigated the effectiveness of second graders solving word problems and explaining their thinking to a family partner, compared to independently solving and explaining in writing. Requested family involvement improved accuracy and explanation quality on homework, and homework accuracy and explanation quality were predictive of performance on an in-class posttest. However, requested family involvement did not directly impact posttest performance, although it did increase attempts to provide an explanation. Overall, explaining homework to a parent shows some potential for improving aspects of student learning.

**Instruments**

- Vanderbilt Story Problems 2 Assessment (Fuchs & Seethaler, 2008)
- Sample Student Homework Assignment (homework problems taken from Singapore Math Word Problems)

Loehr, A. M., Rittle-Johnson, B., & Rajendran, A. (2013, October). Promoting mathematical problem solving and explanation at home. Poster to be presented at the Cognitive Development Society, Memphis, TN.

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