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Having better knowledge of fractions is causally related to the ability to learn algebra, so what happens when teaching and learning about fractions is disrupted, as was the case during the COVID-19 pandemic? In this study, we examine how educational disruption caused by a pandemic differentially impacted students' fraction learning relative to students who were learning other mathematics content during that time. This study provides results from a cross-sequential project examining various facets of mathematics knowledge for students in 4th-10th grades over three years (2021, 2022, 2023; N=903 students). We investigate differences in fractions and algebra knowledge based on students' grade levels across cohorts to determine if there are particular periods at which students' learning was differentially affected by the disruption. Individual differences in students' self-regulation, self-efficacy, and personality will also be explored as potential buffers.

Over the past decade, researchers have established a predictive relationship between students’ fraction knowledge and algebra performance. In middle school, fraction magnitude knowledge is predictive of algebra skills, however, fraction computation knowledge predicts algebra performance for college students. When do different facets of fraction knowledge become more or less important for algebra? The present study examines developmental and individual differences in fraction magnitude knowledge, fraction computation knowledge, algebra feature knowledge, algebra equation-solving, and the relationships between these measures for students (N=700) in grades 4-10 who were tested in two subsequent years as part of an ongoing study. Results will examine developmental changes in student performance on these individual measures and on the relations between the measures both cross-sectionally and longitudinally. Preliminary findings indicate fraction magnitude knowledge develops early and is most important for development of early algebra skills, while fraction computation remains important even for later algebra learning.

The current study assessed whether adding worked examples with self-explanation prompts focused on making connections between mathematical principles, procedures, and concepts of rational numbers to a curriculum focused on invented strategies improves pre-algebra students' fraction number line acuity, rational number concepts and procedures. Finally, the study assessed whether individual differences in prior knowledge of rational numbers moderate the effect of the textbooks. The experimental textbooks resulted in significantly greater improvements overall in fraction number line estimation skills than control textbooks. Further, example-supplemented textbooks were more effective than control textbooks at improving understanding of rational number concepts for those with low prior conceptual knowledge. Both textbooks were equally effective at improving procedural skills.

Both sketching and self-explanation are widely believed to be effective for problem-solving in science learning. However, it is unclear which aspects of these strategies promote learning and how they might interact. Compared to a read-only baseline, we examined the impact of instructing 11-year-old students to solve science problems to sketch, self-explain, or both. Problems were either high spatial or low spatial. We coded elements and relations in their sketches and think-alouds. Self-explanation led to greater accuracy on problems, but only when used alone. Sketching had no effect, although showing more elements and relations in sketches was associated with higher accuracy and occurred more often for high spatial problems. Students may be more comfortable with self-explanation than with sketching and require encouragement or specific instruction to use sketching routinely for the benefits hypothesized to be associated with sketching to appear.

Proportional reasoning failures seem to constitute most errors in probabilistic reasoning, yet there is little empirical evidence about its role for attaining probabilistic knowledge and how to effectively intervene with students who have less proportional reasoning skills. We examined the contributions of students' proportional reasoning skill and example-based practice when learning about probabilities from a reformed seventh grade curriculum. Teachers in their regular classrooms were randomly assigned to instruct with a reformed textbook (control) or a version revised to incorporate correct and incorrect example problems with prompts to explain (treatment). Students' prior knowledge in proportional reasoning skill separately predicted probabilistic knowledge at posttest, regardless of their prior knowledge in probability or minority status. Overall, students in the treatment condition improved more in their probabilistic knowledge, if they started with less proportional reasoning skills. Our findings suggest that example-based practice is beneficial for students with less prior knowledge of proportions, likely a key concept for developing probabilistic knowledge.

The present study examines the effectiveness of incorporating worked examples with prompts for self-explanation into a middle school math textbook. Algebra 1 students (N = 75) completed an equation-solving unit with textbooks either containing the original practice problems or in which a portion of those problems were converted into a combination of correct, incorrect, and incomplete examples. Students completed pre- and posttest measures of algebraic feature knowledge, equation-solving skills, and error anticipation. Example-based textbook assignments increased students' equation-solving skills and their ability to anticipate errors one might make when solving problems. Differences in students' anticipation of various types of errors are also examined. Error anticipation, a particular form of "negative knowledge," is a potentially important skill that relates to algebraic feature knowledge and equation-solving skills.

Fraction knowledge and algebraic skill are closely linked. Algebra is a gatekeeper for advanced courses (Booth & Newton, 2012; Brown & Quinn, 2007). This study uses the person-centered approach of latent profile analysis to examine individual differences in middle schoolers' (N = 350) algebra performance at the end of the year (EOY). The relative importance of a range of fraction skills is considered in predicting the likelihood of displaying a particular profile of EOY algebra knowledge, measured by feature knowledge, equation encoding, equation solving, and word-problem skills. Notably, fraction number line estimation did not predict algebra performance on any of the four measures. Fraction arithmetic at the beginning of the year is most predictive of displaying the highest performing EOY algebra profile.

Numerical magnitude knowledge is related to current math abilities and predictive of future math learning. However, this relationship has previously been shown only for knowledge of positive magnitudes; the present study is the first, to our knowledge, to examine the connection between magnitude knowledge and math ability with negative integers. In the current study, 8th Grade Algebra I students (n = 154) were given assessments of their algebra and negative sign knowledge and 2 number line tasks, an all negative (-10,000 to 0) and a bidirectional scale (-1,000 to 1,000) as part of a pretest and posttest surrounding their unit on 2-step equation solving. Results show that students' negative sign and numerical magnitude knowledge are associated with their current, or pretest, algebra performance. However, only negative magnitude knowledge from the bidirectional number line was predictive of improvement in students' overall algebra knowledge. These findings illustrate differences between knowledge of positive and negative magnitudes and provide support for the integrated theory of numerical development.

Psychologists and mathematics educators have long viewed flexibility as critical to students' mathematical development. In this paper, we focused on the multidimensional nature of flexibility to better understand how preference, knowledge, and use of effective methods for solving algebra problems are related. In Study 1, we identified research-based aspects of flexibility with algebra and assessed students on them following a two-step equations unit. Results indicated that certain aspects of flexibility develop prior to others and that prior knowledge of algebra plays a significant role in that development. Study 2 confirmed and elaborated on these results using a larger sample size. Implications for theory and for supporting flexibility in classrooms are discussed.

Although findings from cognitive science have suggested learning benefits of confronting errors (Metcalfe, 2017), they are not often capitalized on in many mathematics classrooms (Tulis, 2013). The current study assessed the effects of examples focused on either common mathematical misconceptions and errors or correct concepts and procedures on algebraic feature knowledge and solving quadratic equations. Middle school algebra students (N = 206) were randomly assigned to four conditions. Two errorful conditions either displayed errors and asked students to explain or displayed correct solutions and primed students to reflect on potential errors by problem type. A correct example condition and problem-solving control group were also included. Studying and explaining common errors displayed in incorrect examples improved equation-solving ability. An aptitude-by-treatment interaction revealed that learners with limited understandings of algebraic features demonstrated greater benefits. Theoretical implications about using examples to promote learning from errors are considered in addition to suggestions for educational practice.

Success in Algebra I often predicts whether or not a studentwill pursue higher levels of mathematics and science.However, many students enter algebra holding persistentmisconceptions that are difficult to eliminate, thus, hinderingtheir ability to succeed in algebra. One way to address thesemisconceptions is to implement worked-examples and self-explanation prompts, which have been shown to improvestudents’ conceptual knowledge. However this effect seems tobe greater after a delay. The current study sought to exploresuch time-related effects on algebra conceptual knowledge. Ina year-long random-assignment study, students either studiedworked-examples and answered self-explanation prompts (n =132) or solved typical isomorphic problems (n = 140). Athree-way mixed ANCOVA (pre-algebra knowledge xcondition x time) found a significant condition by time effect.The growth of algebra conceptual knowledge was greater forstudents studying worked-examples than for those solvingtypical problems.

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.

The current study examined the effectiveness of self-explanation prompts, visual signaling cues, and a combination of the two features on middle school students' (N = 202) algebra learning. Also explored were the differential effects of features for students with faulty conceptual knowledge (evidenced by a higher prevalence of making errors during problem solving) on learning. That is, we assessed whether students who make prevalent conceptual errors predictive of algebra performance differentially benefit from design features. Participants were randomly assigned to complete 1 of 4 sets of worked example assignments supplemented with self-explanation prompts (n = 51), visual signaling cues (n = 49), both features (n = 51), or neither feature (n = 51). Worked examples supplemented with either self-explanation prompts or signaling cues led to greater learning from pre- to posttest in comparison to the worked example control, with practically meaningful effects. The effect of assignments supplemented with signaling cues was moderated by error prevalence. Those who made errors more frequently demonstrating misunderstanding of algebraic concepts (e.g., the meaning of a coefficient) benefited significantly more from visual signaling cues alone than self-explanation prompts alone or control assignments. These findings highlight the importance of considering differential effects of design features when used in combination or in isolation, particularly for struggling students.

Previous research has shown sketching to be useful tostudents solving math problems. The present studyexamines which aspects of middle school students’sketching are related to, or predict, successfullyanswering math problems. The effects of individualdifferences in cognitive factors – working memory,spatial ability, and prior math knowledge – on answeraccuracy are also analyzed. Stepwise regressionanalysis indicates that prior math knowledge and theinclusion of numerical representations of key problemrelationships in sketches positively predict answeraccuracy, whereas including irrelevant relationships ina sketch is associated with lower answer accuracy.Methodological implications for future research arediscussed.

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 (ß = 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 (ß = 0.37) and calculus conceptual knowledge (ß = 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.

Researchers have extensively documented, and math teachers know from experience, that algebra is a "gatekeeper" to more advanced mathematical topics. Students must have a strong understanding of fundamental algebraic concepts to be successful in later mathematics courses. Unfortunately, algebraic misconceptions that students may form or that deepen during middle school tend to follow them throughout their academic careers. In addition, the longer that a student holds a mathematical misconception, the more difficult it is to correct. Therefore, it is imperative that teachers attempt to address these algebraic misconceptions while students are still in middle school. One tool commonly used to do such a task is the combination of worked examples and self-explanation prompts. This article will describe not only the benefits of using this strategy but also how it connects to the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010). It will also provide instruction on creating worked-example and self-explanation problem sets for students.

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.

For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the "activity of generating explanations to oneself" (Chi 2000, p. 164), especially "in attempt to make sense of new information" (p. 163) as one reads or studies. A "worked example problem," to be differentiated from "working an example problem," shows students an already completed problem and directs their attention to certain steps of the task as the focus of questioning. Self-explanation, then, specifically encourages students to identify the reasoning behind the steps that they see carried out and to explain why these steps were completed. This strategy of providing worked example problems coupled with prompts for self-explanation has recently been shown to influence students' learning positively in both traditional (Booth, Koedinger, and Paré-Blagoev 2011) and computer-based classrooms (Booth et al. 2013). A unique and powerful aspect of using worked examples in the classroom occurs with the inclusion of examples of both correct and incorrect solutions (subsequently referred to as correct and incorrect examples). Using incorrect examples forces students to think about the steps that have been carried out and the reasons why these actions are wrong and then to confront their own possible underlying misconceptions. The desired result is a deeper understanding of mathematics for all students, regardless of prior skill level. By using probing questions that require students to explain a previously worked example, teachers can ensure that students are making sense of what solving equations really entails. Students also engage in reasoning while constructing explanations and strengthen critical thinking skills while critiquing the correct or incorrect solutions. These tasks, carried out in conjunction with the Common Core State Standards, serve to promote deeper understanding of solving equations, which will help students of all ability levels prepare for higher-level mathematics. This article explores using examples in a computer-based activity, using examples in a traditional classroom, the student experience, and targeting student misconceptions.

The National Mathematics Advisory Panel (NMAP, 2008) asserts that a foundational knowledge of fractions is crucial for students' success in algebra; however, empirical evidence for this claim is relatively nonexistent. In the present study, we examine the impact of middle school students' fraction and whole number magnitude knowledge on various components of their algebra readiness. Results suggest that fraction knowledge is related to algebra readiness, more so than number magnitude knowledge in general; students' magnitude knowledge of unit fractions (i.e., those with a numerator of 1) appears particularly important. Findings confirm the intuition of the NMAP (2008) and support the recommendation of the Common Core Standards (National Governors Association Center for Best Practices (NGA Center), 2010) that students' fraction knowledge should be cultivated using number lines. (Contains 3 tables.)

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 diagram's 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. (Contains 3 footnotes, 5 figures and 3 tables.)

This study examined whether the quality of first graders' (mean age = 7.2 years) numerical magnitude representations is correlated with, predictive of, and causally related to their arithmetic learning. The children's pretest numerical magnitude representations were found to be correlated with their pretest arithmetic knowledge and to be predictive of their learning of answers to unfamiliar arithmetic problems. The relation to learning of unfamiliar problems remained after controlling for prior arithmetic knowledge, short-term memory for numbers, and math achievement test scores. Moreover, presenting randomly chosen children with accurate visual representations of the magnitudes of addends and sums improved their learning of the answers to the problems. Thus, representations of numerical magnitude are both correlationally and causally related to arithmetic learning.

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.