Julie L. Booth

Developmental and individual differences in pure numerical estimation.

The authors examined developmental and individual differences in pure numerical estimation, the type of estimation that depends solely on knowledge of numbers. Children between kindergarten and 4th grade were asked to solve 4 types of numerical estimation problems: computational, numerosity, measurement, and number line. In Experiment 1, kindergartners and 1st, 2nd, and 3rd graders were presented problems involving the numbers 0-100; in Experiment 2, 2nd and 4th graders were presented problems involving the numbers 0-1,000. Parallel developmental trends, involving increasing reliance on linear representations of numbers and decreasing reliance on logarithmic ones, emerged across different types of estimation. Consistent individual differences across tasks were also apparent, and all types of estimation skill were positively related to math achievement test scores. Implications for understanding of mathematics learning in general are discussed.

Instructional Complexity and the Science to Constrain It

School-researcher partnerships and large in vivo experiments help focus on useful, effective, instruction. Science and technology have had enormous impact on many areas of human endeavor but surprisingly little effect on education. Many large-scale field trials of science-based innovations in education have yielded scant evidence of improvement in student learning (1, 2), although a few have reliable positive outcomes (3, 4). Education involves many important issues, such as cultural questions of values, but we focus on instructional decision-making in the context of determined instructional goals and suggests ways to manage instructional complexity.

Are diagrams always helpful tools? developmental and individual differences in the effect of presentation format on student problem solving.

BACKGROUND High school and college students demonstrate a verbal, or textual, advantage whereby beginning algebra problems in story format are easier to solve than matched equations (Koedinger & Nathan, 2004). Adding diagrams to the stories may further facilitate solution (Hembree, 1992; Koedinger & Terao, 2002). However, diagrams may not be universally beneficial (Ainsworth, 2006; Larkin & Simon, 1987). AIMS To identify developmental and individual differences in the use of diagrams, story, and equation representations in problem solving. When do diagrams begin to aid problem-solving performance? Does the verbal advantage replicate for younger students? SAMPLE Three hundred and seventy-three students (121 sixth, 117 seventh, 135 eighth grade) from an ethnically diverse middle school in the American Midwest participated in Experiment 1. In Experiment 2, 84 sixth graders who had participated in Experiment 1 were followed up in seventh and eighth grades. METHOD In both experiments, students solved algebra problems in three matched presentation formats (equation, story, story + diagram). RESULTS The textual advantage was replicated for all groups. While diagrams enhance performance of older and higher ability students, younger and lower-ability students do not benefit, and may even be hindered by a diagrams presence. CONCLUSIONS The textual advantage is in place by sixth grade. Diagrams are not inherently helpful aids to student understanding and should be used cautiously in the middle school years, as students are developing competency for diagram comprehension during this time.

Design-Based Research within the Constraints of Practice: AlgebraByExample.

Superintendents from districts in the Minority Student Achievement Network (MSAN) challenged the Strategic Education Research Partnership (SERP) to identify an approach to narrowing the minority student achievement gap in Algebra 1 without isolating minority students for intervention. SERP partnered with 8 MSAN districts and researchers from 3 universities to design and rigorously test AlgebraByExample, a set of 42 Algebra 1 assignments with interleaved worked examples that target common misconceptions and errors. In a year-long random-assignment study, students who received AlgebraByExample assignments had an average 7 percentage point boost on a posttest containing released items from state assessments, and students in the bottom half of the performance distribution where minority students are disproportionately concentrated had an average 10 percentage point boost on a researcher-designed assessment of conceptual understanding. AlgebraByExample is easily incorporated into any existing curriculum, and naturally serves as a launch point for mathematically rich discussion.

Persistent and Pernicious Errors in Algebraic Problem Solving.

Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed algebraic misconceptions using an in-depth error analysis on algebra students’ problem solving efforts at different points in the school year. Results indicate that different types of errors become more prominent with different content at different points in the year, and that there are certain types of errors that, when made during different levels of content are indicative of math achievement difficulties. Recommendations for the necessity and timing of intervention on particular errors are discussed.

Learning Algebra by Example in Real-World Classrooms.

Abstract Math and science textbook chapters invariably supply students with sets of problems to solve, but this widely used approach is not optimal for learning; instead, more effective learning can be achieved when many problems to solve are replaced with correct and incorrect worked examples for students to study and explain. In the present study, the worked example approach is implemented and rigorously tested in the natural context of a functioning course. In Experiment 1, a randomized controlled study in ethnically diverse Algebra classrooms demonstrates that embedded worked examples can improve student achievement. In Experiment 2, a larger randomized controlled study demonstrated that improvement in posttest scores as a result of the assignments varies based on students’ prior knowledge; students with low prior knowledge tend to improve more than higher knowledge peers.

Student Magnitude Knowledge of Negative Numbers

Numerous studies have demonstrated the relevance of magnitude estimation skills for mathematical proficiency, but little research has explored magnitude estimation with negative numbers. In two experiments the current study examined middle school students’ magnitude knowledge of negative numbers with number line tasks. In Experiment 1, both 6th (n = 132) and 7th grade students (n = 218) produced linear representations on a -10,000 to 0 scale, but the 7th grade students’ estimates were more accurate and linear. In Experiment 2, the 7th grade students also completed a -1,000 to 1,000 number line task; these results also indicated that students are linear for both negative and positive estimates. When comparing the estimates of negative and positive numbers, analyses illustrated that estimates of negative numbers are less accurate than those of positive numbers, but using a midpoint strategy improved negative estimates. These findings suggest that negative number magnitude knowledge follows a similar pattern to positive numbers, but the estimation performance of negatives lags behind that of positives.

Relation of Spatial Skills to Calculus Proficiency: A Brief Report

ABSTRACT Spatial skills have been shown in various longitudinal studies to be related to multiple science, technology, engineering, and math (STEM) achievement and retention. The specific nature of this relation has been probed in only a few domains, and has rarely been investigated for calculus, a critical topic in preparing students for and in STEM majors and careers. We gathered data on paper-and-pencil measures of spatial skills (mental rotation, paper folding, and hidden figures); calculus proficiency (conceptual knowledge and released Advanced Placement [AP] calculus items); coordinating graph, table, and algebraic representations (coordinating multiple representations); and basic graph/table skills. Regression analyses suggest that mental rotation is the best of the spatial predictors for scores on released AP calculus exam questions (β = 0.21), but that spatial skills are not a significant predictor of calculus conceptual knowledge. Proficiency in coordinating multiple representations is also a significant predictor of both released AP calculus questions (β = 0.37) and calculus conceptual knowledge (β = 0.47). The spatial skills tapped by the measure for mental rotation may be similar to those required to engage in mental animation of typical explanations in AP textbooks and in AP class teaching as tested on the AP exam questions. Our measure for calculus conceptual knowledge, by contrast, did not require coordinating representations.

Evidence for Cognitive Science Principles that Impact Learning in Mathematics

Abstract Numerous issues with mathematics education in the United States have led to repeated calls for instruction to align more fully with evidence-based practices. The field of cognitive science has identified and tested a number of principles for improving learning, but many of these principles have not yet been used to their fullest to improve mathematics learning in US classrooms. In this chapter, we describe eight principles that may have particular promise for mathematics education: abstract and concrete representations, analogical comparison, feedback, error reflection, scaffolding, distributed practice, interleaved practice, and worked examples. For each principle, we review laboratory and classroom evidence related to benefits for mathematics learning and identify priorities for future research.

Simple Practice Doesn't Always Make Perfect: Evidence From the Worked Example Effect

Findings from the fields of cognitive science and cognitive development propose a variety of evidence-based principles for improving learning. One such recommendation is that instead of having students practice solving long strings of problems on their own after a lesson, worked-out examples of problem solutions should be incorporated into practice sessions in Science, Technology, Engineering, and Mathematics (STEM) classrooms. Research in scientific laboratories and real-world classrooms has also identified a number of methods for utilizing worked examples in lessons, including fading the examples; prompting self-explanation of the examples, including incorrect examples; and providing opportunities for students to compare multiple examples. Each of these methods has been shown to lend itself well to particular types of learning goals. Implications for education policy are discussed, including rethinking the ways in which STEM textbooks are constructed, finding ways to support educators in recognizing and implementing effective cognitive science–based pedagogical techniques, and changing the climate in classrooms to include the perception of errors as a functional part of the learning process.