Emily Fyfe

A longstanding debate concerns the use of concrete versus abstract instructional materials, particularly in domains such as mathematics and science. Although decades of research have focused on the advantages and disadvantages of concrete and abstract materials considered independently, we argue for an approach that moves beyond this dichotomy and combines their advantages. Specifically, we recommend beginning with concrete materials and then explicitly and gradually fading to the more abstract. Theoretical benefits of this “concreteness fading” technique for mathematics and science instruction include (1) helping learners interpret ambiguous or opaque abstract symbols in terms of well-understood concrete objects, (2) providing embodied perceptual and physical experiences that can ground abstract thinking, (3) enabling learners to build up a store of memorable images that can be used …

Recent studies have suggested that educators should avoid concrete instantiations when the goal is to promote transfer. However, concrete instantiations may benefit transfer in the long run, particularly if they are “faded” into more abstract instantiations. Undergraduates were randomly assigned to learn a mathematical concept in one of three conditions: generic, in which the concept was instantiated using abstract symbols, concrete in which it was instantiated using meaningful images, or fading, in which it was instantiated using meaningful images that were “faded” into abstract symbols. After learning, undergraduates completed a transfer test immediately, one week later, and three weeks later. Undergraduates in the fading condition exhibited the best transfer performance. Additionally, undergraduates in the generic condition exhibited somewhat better transfer than those in the concrete condition, but this …

This study examined whether practice with arithmetic problems presented in a nontraditional problem format improves understanding of mathematical equivalence. Children (M age = 8;0; N = 90) were randomly assigned to practice addition in one of three conditions: (a) traditional, in which problems were presented in the traditional “operations on left side” format (e.g., 9 + 8 = 17); (b) nontraditional, in which problems were presented in a nontraditional format (e.g., 17 = 9 + 8); or (c) no extra practice. Children developed a better understanding of mathematical equivalence after receiving nontraditional practice than after receiving traditional practice or no extra practice. Results suggest that minor differences in early input can yield substantial differences in children’s understanding of fundamental concepts.

Feedback can be a powerful learning tool, but its effects vary widely. Research has suggested that learners’ prior knowledge may moderate the effects of feedback; however, no causal link has been established. In Experiment 1, we randomly assigned elementary schoolchildren (N= 108) to a condition based on a crossing of 2 factors: induced strategy knowledge (yes vs. no) and immediate, verification feedback (present vs. absent). 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. In Experiment 2, we induced strategy knowledge in all children (N= 101) and randomly assigned them to 1 of 3 conditions: no feedback, immediate correct-answer feedback, or summative correct-answer feedback. Again, feedback had negative effects relative to no feedback. Results provide evidence for a causal role of prior …

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 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.

Early mathematics knowledge is a strong predictor of later academic achievement, but children from low‐income families enter school with weak mathematics knowledge. An early math trajectories model is proposed and evaluated within a longitudinal study of 517 low‐income American children from ages 4 to 11. This model includes a broad range of math topics, as well as potential pathways from preschool to middle grades mathematics achievement. In preschool, nonsymbolic quantity, counting, and patterning knowledge predicted fifth‐grade mathematics achievement. By the end of first grade, symbolic mapping, calculation, and patterning knowledge were the important predictors. Furthermore, the first‐grade predictors mediated the relation between preschool math knowledge and fifth‐grade mathematics achievement. Findings support the early math trajectories model among low‐income children.

Children often struggle to gain understanding from instruction on a procedure, particularly when it is taught in the context of abstract mathematical symbols. We tested whether a “concreteness fading” technique, which begins with concrete materials and fades to abstract symbols, can help children extend their knowledge beyond a simple instructed procedure. In Experiment 1, children with low prior knowledge received instruction in one of four conditions: (a) concrete, (b) abstract, (c) concreteness fading, or (d) concreteness introduction. Experiment 2 was designed to rule out an alternative hypothesis that concreteness fading works merely by “warming up” children for abstract instruction. Experiment 3 tested whether the benefits of concreteness fading extend to children with high prior knowledge. In all three experiments, children in the concreteness fading condition exhibited better transfer than children in the other …

<h3 class="gsh_h3">Background</h3> 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. <h3 class="gsh_h3">Aims</h3> 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. <h3 class="gsh_h3">Sample</h3> We worked with 122 second‐ and third‐grade children. <h3 class="gsh_h3">Method</h3> 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. <h3 class="gsh_h3">Results</h3> Providing …

This experiment tested if a modified version of arithmetic practice facilitates understanding of math equivalence. Children within 2nd-grade classrooms (N= 166) were randomly assigned to practice single-digit addition facts using 1 of 2 workbooks. In the control workbook, problems were presented in the traditional “operations= answer” format (eg, 4+ 3= __) and were organized pseudorandomly throughout the workbook. In the modified workbook, problems were presented with operations on the right side (eg, __= 4+ 3), the equal sign was sometimes replaced by relational words (eg,“is equal to”), and problems were organized by equivalent sums such that several problems in a row would all have the same sum. Children who used the modified workbook constructed a better understanding of math equivalence than did children who used the control workbook. This advantage persisted approximately 5–6 months …

Patterns are a pervasive and important, but understudied, component of early mathematics knowledge. In a series of three studies, 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 studies 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.

This experiment tested the hypothesis that organizing arithmetic fact practice by equivalent values facilitates children's understanding of math equivalence. Children (M age= 8 years 6 months, N= 104) were randomly assigned to 1 of 3 practice conditions:(a) equivalent values, in which problems were grouped by equivalent sums (eg, 3+ 4= 7, 2+ 5= 7, etc.),(b) iterative, in which problems were grouped iteratively by shared addend (eg, 3+ 1= 4, 3+ 2= 5, etc.), or (c) no extra practice, in which children did not receive any practice over and above what they ordinarily receive at school and home. Children then completed measures to assess their understanding of math equivalence. Children who practiced facts organized by equivalent values demonstrated a better understanding of math equivalence than children in the other 2 conditions. Results suggest that organizing arithmetic facts into conceptually related groupings …

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 …

Feedback is generally considered a beneficial learning tool, and providing feedback is a recommended instructional practice. However, there are a variety of feedback types with little guidance on how to choose the most effective one. We examined individual differences in working memory capacity as a potential moderator of feedback type. Second- and third-grade children (N = 64) solved unfamiliar math problems prior to receiving instruction. Children received verification feedback on their answers (outcome-feedback) or on their strategies (strategy-feedback). Working memory capacity moderated the effect of feedback type on procedural transfer—the ability to solve novel problems. Children with lower working memory capacity benefitted less from strategy-feedback than outcome-feedback, whereas children with higher working memory capacity benefitted similarly from the two types of feedback. Results …

The labels used to describe patterns and relations can influence children's relational reasoning. In this study, 62 preschoolers (M_age = 4.4 years) solved and described eight pattern abstraction problems (i.e., recreated the relation in a model pattern using novel materials). Some children were exposed to concrete labels (e.g., blue‐red‐blue‐red) and others were exposed to abstract labels (e.g., A‐B‐A‐B). Children exposed to abstract labels solved more 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 concrete learning materials in combination with abstract representations can enhance their utility for children's performance. Furthermore, abstract language may play a key role in the development of relational thinking.

Homework is transforming at a rapid rate with continuous advances in educational technology. Computer-based homework, in particular, is gaining popularity across a range of schools, with little empirical evidence on how to optimize student learning. The current aim was to test the effects of different types of feedback on computer-based homework. In the study, middle school students completed a computer-based pretest, homework assignment, and posttest containing challenging algebraic problems. On the homework assignment, students were assigned to different feedback conditions. In Experiment 1 (N = 103), students received no feedback or correct-answer feedback after each problem. In Experiment 2 (N = 143), students received (1) no feedback, (2) correct-answer feedback, (3) try-again feedback, or (4) explanation feedback after each problem. For students with low prior knowledge, feedback resulted in …