Emily R. Fyfe

Emerging Understanding of Patterning in 4-Year-Olds

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 4-year-old childrens 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.

The Influence of Relational Knowledge and Executive Function on Preschoolers’ Repeating Pattern Knowledge

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. One hundred twenty-four children aged 4 to 5 years old 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.

An Alternative Time for Telling: When Conceptual Instruction Prior to Problem Solving Improves Mathematical Knowledge.

BACKGROUND The sequencing of learning materials greatly influences the knowledge that learners construct. 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 instruction remains unclear. AIMS We tested the impact of conceptual instruction preceding or following mathematics problem solving to determine when conceptual instruction should or should not be delayed. We also examined the learning processes supported to inform theories of learning more broadly. SAMPLE We worked with 122 second- and third-grade children. METHOD In a randomized experiment, children received instruction on the concept of math equivalence either before or after being asked to solve and explain challenging equivalence problems with feedback. RESULTS Providing conceptual instruction first resulted in greater procedural knowledge and conceptual knowledge of equation structures than delaying instruction until after problem solving. Prior conceptual instruction enhanced problem solving by increasing the quality of explanations and attempted procedures. CONCLUSIONS Providing conceptual instruction prior to problem solving was the more effective sequencing of activities than the reverse. We compare these results with previous, contrasting findings to outline a potential framework for understanding when instruction should or should not be delayed.

Wait for It . . . Delaying Instruction Improves Mathematics Problem Solving: A Classroom Study.

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. Correspondence: Abbey Marie Loehr, 230 Appleton Place, Peabody #552, Vanderbilt University, Nashville, TN 37203; Phone: (615) 343-7149. Email: abbey.loehr@vanderbilt.edu

Assessing Formal Knowledge of Math Equivalence among Algebra and Pre-Algebra Students.

A central understanding in mathematics is knowledge of math equivalence, the relation indicating that 2 quantities are equal and interchangeable. Decades of research have documented elementary-school (ages 7 to 11) children’s (mis)understanding of math equivalence, and recent work has developed a construct map and comprehensive assessments of this understanding. The goal of the current research was to extend this work by assessing whether the construct map of math equivalence knowledge was applicable to middle school students and to document differences in formal math equivalence knowledge between students in pre-algebra and algebra. We also examined whether knowledge of math equivalence was related to students’ reasoning about an algebraic expression. In the study, 229 middle school students (ages 12 to 16) completed 2 forms of the math equivalence assessment. The results suggested that the construct map and associated assessments were appropriate for charting middle school students’ knowledge and provided additional empirical support for the link between understanding of math equivalence and formal algebraic reasoning.

The benefits of computer-generated feedback for mathematics problem solving.

The goal of the current research was to better understand when and why feedback has positive effects on learning and to identify features of feedback that may improve its efficacy. In a randomized experiment, second-grade children received instruction on a correct problem-solving strategy and then solved a set of relevant problems. Children were assigned to receive no feedback, immediate feedback, or summative feedback from the computer. On a posttest the following day, feedback resulted in higher scores relative to no feedback for children who started with low prior knowledge. Immediate feedback was particularly effective, facilitating mastery of the material for children with both low and high prior knowledge. Results suggest that minimal computer-generated feedback can be a powerful form of guidance during problem solving.

Feedback influences children's reasoning about math equivalence: A meta-analytic review

ABSTRACT Decades of research have focused on childrens reasoning about math equivalence problems for both practical and theoretical insights. Not only are math equivalence problems foundational in arithmetic and algebra, they also represent a class of problems on which childrens thinking is resistant to change. Feedback is one instructional tool that can serve as a key trigger of cognitive change. In this paper, we review all experimental studies (N = 8) on the effects of feedback on childrens (ages 6–11) understanding of math equivalence. Meta-analytic results indicate that feedback has positive effects for low-knowledge learners and negative effects for high-knowledge learners, and these effects are stronger for procedural outcomes than conceptual outcomes. Findings highlight the variable influences of feedback on math equivalence understanding and suggest that models of thinking and reasoning need to consider learner characteristics, learning outcomes, and learning materials, as well as the dynamic interactions among them.

Improving conceptual and procedural knowledge: The impact of instructional content within a mathematics lesson

BACKGROUND Students, parents, teachers, and theorists often advocate for direct instruction on both concepts and procedures, but some theorists suggest that including instruction on procedures in combination with concepts may limit learning opportunities and student understanding. AIMS This study evaluated the effect of instruction on a math concept and procedure within the same lesson relative to a comparable amount of instruction on the concept alone. Direct instruction was provided before or after solving problems to evaluate whether the type of instruction interacted with the timing of instruction within a lesson. SAMPLE We worked with 180 second-grade children in the United States. METHODS In a randomized experiment, children received a classroom lesson on mathematical 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). RESULTS Children who received two iterations of conceptual instruction had better retention of conceptual and procedural knowledge than children who received both conceptual and procedural instruction in the same lesson. Order of instruction did not impact outcomes. CONCLUSIONS Findings suggest that within a single lesson, spending more time on conceptual instruction may be more beneficial than time spent teaching a procedure when the goal is to promote more robust understanding of target concepts and procedures.

Predicting success on high-stakes math tests from preschool math measures among children from low-income homes.

State-mandated tests have taken center stage for assessing student learning and for holding teachers and students accountable for achieving adequate progress. What types of early knowledge predict performance on these tests, especially among low-income children who are at risk for poor performance? We report on a longitudinal study of 519 low-income American children ages 5–12, with a focus on mathematics performance. We found that nonsymbolic quantity knowledge and repeating pattern knowledge at the end of preschool were reliable predictors of performance on standards-based high-stakes tests across three different grade levels (4th–6th grade), over and above other math and academic skills. Further, these effects of preschool math knowledge were partially mediated through symbolic mapping and calculation knowledge at the end of 1st grade. These findings suggest that nonsymbolic quantity knowledge and repeating pattern knowledge prior to formal schooling are valuable indicators of low-income children’s performance on high-stakes state math tests in the middle grades.

Making “concreteness fading” more concrete as a theory of instruction for promoting transfer

Abstract To promote learning and transfer of abstract ideas, contemporary theories advocate that teachers and learners make explicit connections between concrete representations and the abstract ideas they are intended to represent. Concreteness fading is a theory of instruction that offers a solution for making these connections. As originally conceived, it is a three-step progression that begins with enacting a physical instantiation of a concept, moves to an iconic depiction and then fades to the more abstract representation of the same concept. The goals of this paper are: (1) to improve the theoretical framework of concreteness fading by defining and bringing greater clarity to the terms abstract, concrete and fading; and (2) to describe several testable hypotheses that stem from concreteness fading as a theory of instruction. Making this theory of instruction more “concrete” should lead to an optimised concreteness fading technique with greater promise for facilitating both learning and transfer.