# Early Algebra Research Projects

## Early Algebra Research Projects

Jump to:

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

VI. Beyond numeracy in preschool: Adding patterns to the equation

VII. Easy as ABC: Abstract language facilitates performance on a concrete patterning task

I. Homework

II. Delaying instruction improves mathematics problem solving

III. "Just tell me how to solve it." The impact of including procedural instruction in conjunction with conceptual instruction

Integrative Self-Explanation Literature Review

Note: A complete list of publications available for download can be found here.

# 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**

- Time 1 Assessment
- Time 1 Assessment KEY
- Time 2 Assessment (Version 1)
- Time 2 Assessment (Version 2)
- Time 2 Assessment KEY
- Explanation Coding Scheme
- Script

**
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
- Coding Scheme
- 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)
- Time 1 Scoring Guide
- Time 1 Strategy Coding Scheme
- Feedback Session and Time 2 Script (Version 1)
- Feedback Session and Time 2 Script (Version 2)
- Feedback Session Coding Scheme

# 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

### 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

*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.

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

**Abstract
**Children’s knowledge of repeating patterns (e.g., ABBABB) is a central component of early mathematics, but the developmental mechanisms underlying this knowledge are currently unknown. We sought clarity on the importance of relational knowledge and executive function (EF) to preschoolers’ understanding of repeating patterns. 124 children between the ages of 4 and 5 years were administered a relational knowledge task, 3 EF tasks (working memory, inhibition, set shifting), and a repeating pattern assessment before and after a brief pattern intervention. Relational knowledge, working memory, and set shifting predicted preschoolers’ initial pattern knowledge. Working memory also predicted improvements in pattern knowledge after instruction. The findings indicated that greater EF ability was beneficial to preschoolers’ repeating pattern knowledge, and that working memory capacity played a particularly important role in learning about patterns. Implications are discussed in terms of the benefits of relational knowledge and EF for preschoolers’ development of patterning and mathematics skills.

**Instruments
**

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

### VI. Beyond Numeracy in Preschool: Adding Patterns to the Equation

**Abstract
**Patterns are a pervasive and important, but understudied, component of early mathematics knowledge. In a series of three experiments, we explored (a) growth in children’s pattern knowledge over the pre-K year (

*N*= 65), (b) the frequency of pattern activities reported by parents (

*n*= 20) and teachers (

*n*= 5) relative to other mathematical activities, and (c) changes in 4-year-old children’s pattern knowledge after brief experience generating or receiving explanations on patterns (

*N*= 124). Together, these experiments illustrate the types of experiences preschool children are receiving with patterns and how their pattern knowledge changes over time and in response to explanation. Young children are able to succeed on a more sophisticated pattern activity than they are frequently encouraged to do at home or at school.

**Instruments**

*Experiment 1*

*Experiment 2*

*Experiment 3*

- Pretest Script
- Intervention script with instructional explanation
- Intervention script with self-explanation
- Intervention script with both instructional and self-explanations
- Posttest script
- Self-explanation coding scheme

### VII. Easy as ABC: Abstract Language Facilitates Performance on a Concrete Patterning Task

**Abstract
**We tested the impact of concrete versus abstract labels in reference to physical manipulatives on preschool children’s patterning skills. Sixty-two preschoolers solved and described eight pattern abstraction tasks (i.e., recreated the model pattern using novel materials). Some were exposed to concrete labels (e.g., red, blue, red, blue), and others were exposed to abstract labels (e.g., A, B, A, B). Children exposed to abstract labels solved more pattern problems correctly than children exposed to concrete labels. Children’s correct adoption of the abstract language into their own descriptions was particularly beneficial. Thus, using physical manipulatives in combination with abstract representations can enhance their utility for children’s performance. Further, abstract language may play a key role in the development of relational thinking.

**Instruments**

# Classroom Studies

### I. Homework

### 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.

- 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 presented at the Cognitive Development Society, Memphis, TN.

### Effects of Extended Homework Use

**Abstract
**Generating explanations, particularly for another person, is associated with greater learning (Teasley, 1995; Rittle-Johnson et al., 2008). Additionally, Van Voorhis (2011) found that family involvement in homework increased student motivation and achievement. Using weekly homework assignments, we investigated the effectiveness of second graders solving word problems and explaining their thinking to a family partner, compared to independently solving and explaining. Requested family involvement improved accuracy and explanation quality on homework, but had no effect on independent performance on an in‐class posttest. Results from an exploratory follow‐up on a classroom that continued to use the family homework assignments for another semester showed that extended use of the homework greatly improved explanation quality but did not improve word-problem accuracy.

**Instruments**

- See above

### II. Delaying Instruction Improves Mathematics Problem Solving

**Abstract**

Engaging learners in exploratory problem-solving activities prior to receiving instruction (i.e., explore-instruct approach) has been endorsed as an effective learning approach. However, it remains unclear whether this approach is feasible for elementary-school children in a classroom context. In two experiments, second-graders solved mathematical equivalence problems either before or after receiving brief conceptual instruction. In Experiment 1 (n = 41), the explore-instruct approach was less effective at supporting learning than an instruct-solve approach. However, it did not include a common, but often overlooked feature of an explore-instruct approach, which is provision of a knowledge-application activity after instruction. In Experiment 2 (n = 47), we included a knowledge-application activity by having all children check their answers on previously solved problems. The explore-instruct approach in this experiment led to superior learning than an instruct-solve approach. Findings suggest promise for an explore-instruct approach, provided learners have the opportunity to apply knowledge from instruction.

**
Instruments
**

*Experiment 1*

- Pretest
- Script for explore-instruct condition
- Script for instruct-solve condition
- Intervention problem-solving student packet (includes midtest) for explore-instruct condition
- Intervention problem-solving student packet (includes midtest) for instruct-solve condition
- Posttest

*Experiment 2*

### III. The Impact of Including Procedural Instruction in Conjunction with Conceptual Instruction

**Abstract
**Students, parents, teachers and theorists often advocate for direct instruction on procedures. At the same time, instruction on concepts is critical for supporting understanding. Is it best to combine the two? The current study focused on if and when procedural instruction was provided in addition to conceptual instruction in a lesson on mathematical equivalence. Second-grade children (N = 180) received a classroom lesson on equivalence in one of four conditions that varied in instruction type (conceptual or combined conceptual-and-procedural) and in instruction order (instruction before or after solving problems). Children who received only conceptual instruction had better retention of their conceptual and procedural knowledge than children who received combined instruction. Order of instruction did not impact results. Findings suggest that providing procedural instruction in addition to conceptual instruction guided attention away from reflecting on the conceptual instruction, decreasing learning.

**
Instruments
**

- Pretest
- Script for instruct (conceptual)-solve condition
- Script for instruct (conceptual-and-procedural)-solve condition
- Script for solve-instruct (conceptual) condition
- Script for solve-instruct (conceptual-and-procedural) condition
- Intervention problem-solving student packet (includes midtest) for instruct-solve conditions and solve-instruct conditions
- Posttest and Retention test

# Integrative Self-Explanation Review

### When and Why Self-Explanation Matters: An Integrative Review

**Abstract
**Self-explanation is a powerful learning mechanism that has been shown to improve understanding and transfer in a broad range of domains. This integrative literature review provides a brief history of self-explanation research, organizes past experimental research on self-explanation into four primary research paradigms, and discusses dimensions of self-explanation that influence its effectiveness. The four research paradigms vary in what is being explained – worked-out examples, exemplars, text, or one’s own problem solving. Multiple studies within each paradigm indicate that prompts to self-explain can improve learning. However, self-explanation prompts do not always aid learning and can have a double-edge effect, increasing one type of knowledge at the expense of another type. Evidence for dimensions of self-explanation suggest how self-explanation quality and subsequent learning can be improved, such as matching the type of prompts to characteristics of the learning content. Guidelines for effectively supporting self-explanation are provided.

**Submitted Paper
**Rittle-Johnson, B. & Loehr, A. M. (under review). When and why self-explanation aids learning: An integrative review.

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