Knowing what they know: Preschool teachers' knowledge of math skills and its relation to instruction.

Math experiences during the preschool years play an important role in children's later math learning. Preschool teachers exhibit considerable variability in the amount and types of mathematics activities they engage in with their students; one potentially important source of these individual differences is adults' knowledge of early math development. The current study aimed to describe preschool teachers' knowledge of numeracy, patterning, and spatial/geometric skills developed in preschool and its relation to their reported mathematics instruction. Participants (N = 83) completed a survey in which they judged whether particular early math skills could be observed in typically developing 4-year-olds in the United States and reported their frequency of engaging in different math instructional activities. Pre- and in-service preschool teachers' knowledge varied across the different domains (i.e., numeracy, patterning, and spatial/geometric) of mathematical thinking, but their reported frequency of instruction did not. Teachers who were found to be more accurate in their knowledge of early math development were more likely to report higher frequency of math instruction; looking specifically at the domains, the strength of association between knowledge and instruction was the strongest for numeracy. Such findings highlight the possibility that supporting preschool teachers' knowledge of the range of math skills their students can be developing may be one component of improving early math teaching and learning.

What counts as STEM, and does it matter?

BACKGROUND: To accurately measure students' science, technology, engineering and mathematics (STEM) career interest, researchers must get inside the 'black box' to understand students' conceptualizations of STEM careers. AIMS: The aim of Study 1 was to explore whether students' conceptualizations of STEM included medical careers. The aim of Study 2 was to explore whether predictors of STEM career interest (e.g., gender and motivation) varied by STEM definition (inclusion/exclusion of medical careers). SAMPLES: In Study 1, the sample was US college students (N = 125) who were mostly White (80%). In Study 2, the sample was US 10th-grade high school students (N = 455) who were mostly Black (79%). METHODS: In Study 1, students completed an online questionnaire. In Study 2, students completed various measures of math achievement, motivation (science and math expectancies of success, interest and importance value) and career interest with an importance. RESULTS: In Study 1, medical careers were less often classified as STEM careers than traditional STEM careers, but more often classified as STEM than non-STEM careers. In Study 2, science importance value was the only motivational predictor of students' STEM+Medicine career interest, and no motivation constructs predicted traditional STEM career interest. Boys expressed greater interest in traditional STEM careers, while girls expressed greater interest in STEM+Medicine careers. CONCLUSIONS: Students' conceptualizations of STEM are not binary. Thus, we recommend researchers are explicit about their definition of STEM with study participants, in their coding and in their publications.

Developing and validating a measure of parental knowledge about early math development.

Parents' knowledge about the math skills that most preschool-aged children can develop might be an important component of the Home Math Environment (HME) as it might shape their math beliefs and efforts to support their preschoolers' math development. This study aimed to systematically develop measures of parents' knowledge about two critical early math topics, numeracy, and patterning, across five studies conducted with a total of 616 U.S. parents of 3- to 5-year-olds (66% mothers, 54% sons, 73% White, 60% college-educated). Parents were recruited via CloudResearch or a university database. Study 1 focused on item generation to revise a previous measure to capture a wider set of children's early math skills and analysis of the psychometric properties of the measure after it was completed by 161 parents via a survey. Study 2 included an analysis of a new sample of parents (n = 21) who responded to the measures twice across two weeks to explore test-retest reliability. The measures were iteratively revised, administered to new samples, and analyzed in Studies 3 (n = 45), 4 (n = 46), and 5 (n = 344). The measures demonstrated adequate internal consistency and validity (construct, convergent, and discriminant) in Study 5 such as being positively related to parents' numeracy and patterning beliefs about their children. Overall, the newly developed measures satisfy standards for the development of an adequate measure and can be used to better understand what parents know about early math development and how this relates to the HME that they facilitate.

Parents' numeracy beliefs and their early numeracy support: A synthesis of the literature.

Parents' academic beliefs influence the academic support they provide to their children. In this chapter, we review the published literature on empirical studies conducted with parents of preschoolers and propose a conceptual model for how different parental numeracy beliefs uniquely and differentially influence parents' early numeracy support and vary with their demographic characteristics. Parents' numeracy beliefs about their children were more consistently related to their numeracy support than their other numeracy beliefs but were inconsistently related to demographic characteristics. Parents' numeracy beliefs about themselves were significantly related to their socioeconomic status and the extent to which their numeracy support focused on advanced early numeracy skills, but were not significantly related to how often they provided numeracy support. We also discuss parents' beliefs regarding where and how children should learn numeracy. Overall, evidence to date highlights the role of parents' beliefs about their children as well as their socioeconomic status. We discuss several future directions.

Finding patterns in objects and numbers: Repeating patterning in pre-K predicts kindergarten mathematics knowledge.

Both recent evidence and research-based early mathematics curricula indicate that repeating patterns-predictable sequences that follow a rule-are a topic of major importance for mathematics development. The purpose of the current study was to help build a theory for how early repeating patterning knowledge contributes to early math development, focusing on development in children aged 4-6 years. The current study examined the relation between 65 preschool children's repeating patterning knowledge (via a fast, teacher-friendly measure) and their end-of-kindergarten broad math and numeracy knowledge, controlling for verbal and visual-spatial working memory (WM) skills as well as end-of-pre-K (pre-kindergarten) broad math knowledge. Relations were also examined between repeating patterning and specific aspects of numeracy knowledge-knowledge of the count sequence to 100 and the successor principle. Children's repeating patterning knowledge was significantly predictive of their broad math and general numeracy knowledge, as well as one specific aspect of their numeracy knowledge (counting to 100), even after controlling for verbal and visual-spatial WM skills. Further, repeating patterning knowledge remained a unique predictor of general numeracy knowledge and counting to 100 after controlling for end-of-pre-K broad math knowledge. The relation between repeating patterning and mathematics may be explained by the central role that identifying predictable sequences based on underlying rules plays in both. Theories of math development and early math instruction standards should thus give even greater attention to the role of children's repeating patterning knowledge.

The role of recalling previous errors in middle-school children's learning.

BACKGROUND: Committing errors is a common part of the learning process, and adults are more likely to correct errors that they can recall. However, preadolescent children's recall of previous errors (i.e., memory for errors) may be limited. AIMS: We examined children's ability to recall their past errors and tested whether recalling an error aids error correction. SAMPLES: We worked with 102 (Study 1) and 173 (Study 2) middle-school children in the United States. METHODS: In Study 1, children studied and were tested on their memory for math definitions. After reviewing the correct answers, children recalled their initial test answers and then took a final test. Reminders of past errors were provided for some children in Study 2. In two other conditions, children either recalled their past errors or studied the correct answers only. RESULTS: Children's recall of their past errors was poor, and errors that were recalled were no more likely to be corrected than errors that were not recalled. Across children, there was a positive association between memory for errors and error correction even after controlling for covariates. Being reminded of past errors and recalling past errors reduced error correction relative to studying the correct answers only. CONCLUSIONS: Preadolescents' memory for errors is very limited, their ability to recall past errors predicts error correction overall, and recalling an error or being reminded of an error does not facilitate error correction.

Preschoolers' broad mathematics experiences with parents during play.

The current study broadens our understanding of preschoolers' early math experiences with parents, recognizing that math knowledge and experiences are inclusive of numeracy as well as non-numeracy domains. Parents and preschoolers (N = 45) were observed exploring three domains of early mathematics knowledge (i.e., number, space, and pattern) during play in three activities (playing cards, building with blocks, and stringing beads, all with activity suggestions). Children were administered a broad math and numeracy measure and individual measures of spatial and patterning skills concurrently and 7 months later. Dyads explored math broadly across most activities but emphasized number more than space or patterning. In addition, there was more overall math exploration during card and bead play than during block play, with the greatest parent support during card play. Parent support was not linked to children's skills, although children's exploration of space and patterns related moderately to their concurrent spatial and pattern skills. Overall, parents and young children explored a variety of early math domains in guided play contexts, with an emphasis on numeracy. Future work should aim to increase the breadth and rigor of individual concepts that parents and preschoolers explore during play.

Data on preschool children׳s math, patterning, and spatial knowledge.

Initial participants were 79 children who were recruited from six preschool programs in the U.S. Full assessment data was available for 73 children (average age of 4 years 7 months), including demographic data (gender, ethnicity, financial need, language(s) spoken at home and special education status). Children׳s math, repeating patterning, spatial and verbal skills were assessed at the beginning of the pre-kindergarten year. Assessments included the brief version of the Research-Based Early Mathematics Assessment, two measures of repeating patterning skills, three measures of spatial skills (the Block Design subtest of the Wechsler Preschool and Primary Scale of Intelligence, the Position in Space subtest of the Developmental Test of Visual Perception, and a Corsi Block Tapping Task), the Picture Vocabulary Test from the NIH Toolbox app and a backward letter span task. Near the end of the school year, their math knowledge was re-assessed using the same math measure, as was their memory span (forward and backward digit span task from the Wechsler Intelligence Scale for Children). Findings on the relations between patterning, spatial and math skills are published elsewhere (Rittle-Johnson et al., 2018) [1].

Diagrams benefit symbolic problem-solving.

BACKGROUND: The format of a mathematics problem often influences students' problem-solving performance. For example, providing diagrams in conjunction with story problems can benefit students' understanding, choice of strategy, and accuracy on story problems. However, it remains unclear whether providing diagrams in conjunction with symbolic equations can benefit problem-solving performance as well. AIMS: We tested the impact of diagram presence on students' performance on algebra equation problems to determine whether diagrams increase problem-solving success. We also examined the influence of item- and student-level factors to test the robustness of the diagram effect. SAMPLE: We worked with 61 seventh-grade students who had received 2 months of pre-algebra instruction. METHOD: Students participated in an experimenter-led classroom session. Using a within-subjects design, students solved algebra problems in two matched formats (equation and equation-with-diagram). RESULTS: The presence of diagrams increased equation-solving accuracy and the use of informal strategies. This diagram benefit was independent of student ability and item complexity. CONCLUSIONS: The benefits of diagrams found previously for story problems generalized to symbolic problems. The findings are consistent with cognitive models of problem-solving and suggest that diagrams may be a useful additional representation of symbolic problems.

Early Math Trajectories: Low-Income Children's Mathematics Knowledge From Ages 4 to 11.

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.

Eliciting explanations: Constraints on when self-explanation aids learning.

Generating explanations for oneself in an attempt to make sense of new information (i.e., self-explanation) is often a powerful learning technique. Despite its general effectiveness, in a growing number of studies, prompting for self-explanation improved some aspects of learning, but reduced learning of other aspects. Drawing on this recent research, as well as on research comparing self-explanation under different conditions, we propose four constraints on the effectiveness of self-explanation. First, self-explanation promotes attention to particular types of information, so it is better suited to promote particular learning outcomes in particular types of domains, such as transfer in domains guided by general principles or heuristics. Second, self-explaining a variety of types of information can improve learning, but explaining one's own solution methods or choices may reduce learning under certain conditions. Third, explanation prompts focus effort on particular aspects of the to-be-learned material, potentially drawing effort away from other important information. Explanation prompts must be carefully designed to align with target learning outcomes. Fourth, prompted self-explanation often promotes learning better than unguided studying, but alternative instructional techniques may be more effective under some conditions. Attention to these constraints should optimize the effectiveness of self-explanation as an instructional technique in future research and practice.

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.

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.

Easy as ABCABC: Abstract Language Facilitates Performance on a Concrete Patterning Task.

The labels used to describe patterns and relations can influence children's relational reasoning. In this study, 62 preschoolers (Mage  = 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.

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.

Is self-explanation worth the time? A comparison to additional practice.

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 alternative 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 69 children in grades 2-4. METHODS: Students completed a pre-test, 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.

Exploring mathematics problems prepares children to learn from instruction.

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- to fourth-grade students) 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 with 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.

Developing procedural flexibility: are novices prepared to learn from comparing procedures?

BACKGROUND: A key learning outcome in problem-solving domains is the development of procedural flexibility, where learners know multiple procedures and use them appropriately to solve a range of problems (e.g., Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). However, students often fail to become flexible problem solvers in mathematics. To support flexibility, teaching standards in many countries recommend that students be exposed to multiple procedures early in instruction and be encouraged to compare them. AIMS: We experimentally evaluated this recommended instructional practice for supporting procedural flexibility during a classroom lesson, relative to two alternative conditions. The alternatives reflected the common instructional practice of delayed exposure to multiple procedures, either with or without comparison of procedures. SAMPLE: Grade 8 students from two public schools (N= 198) were randomly assigned to condition. Students had not received prior instruction on multi-step equation solving, which was the topic of our lessons. METHOD: Students learned about multi-step equation solving under one of three conditions in math class for about 3 hr. They also completed a pre-test, post-test, and 1-month-retention test on their procedural knowledge, procedural flexibility, and conceptual knowledge of equation solving. RESULTS: Novices who compared procedures immediately were more flexible problem solvers than those who did not, even on a 1-month retention test. Although condition had limited direct impact on conceptual and procedural knowledge, greater flexibility was associated with greater knowledge of both types. CONCLUSIONS: Comparing procedures can support flexibility in novices and early introduction to multiple procedures may be one important reason.

Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge.

Competence in many domains rests on children developing conceptual and procedural knowledge, as well as procedural flexibility. However, research on the developmental relations between these different types of knowledge has yielded unclear results, in part because little attention has been paid to the validity of the measures or to the effects of prior knowledge on the relations. To overcome these problems, we modeled the three constructs in the domain of equation solving as latent factors and tested (a) whether the predictive relations between conceptual and procedural knowledge were bidirectional, (b) whether these interrelations were moderated by prior knowledge, and (c) how both constructs contributed to procedural flexibility. We analyzed data from 2 measurement points each from two samples (Ns = 228 and 304) of middle school students who differed in prior knowledge. Conceptual and procedural knowledge had stable bidirectional relations that were not moderated by prior knowledge. Both kinds of knowledge contributed independently to procedural flexibility. The results demonstrate how changes in complex knowledge structures contribute to competence development.

Iterating between lessons on concepts and procedures can improve mathematics knowledge.

BACKGROUND: Knowledge of concepts and procedures seems to develop in an iterative fashion, with increases in one type of knowledge leading to increases in the other type of knowledge. This suggests that iterating between lessons on concepts and procedures may improve learning. AIMS: The purpose of the current study was to evaluate the instructional benefits of an iterative lesson sequence compared to a concepts-before-procedures sequence for students learning decimal place-value concepts and arithmetic procedures. SAMPLES: In two classroom experiments, sixth-grade students from two schools participated (N=77 and 26). METHOD: Students completed six decimal lessons on an intelligent-tutoring systems. In the iterative condition, lessons cycled between concept and procedure lessons. In the concepts-first condition, all concept lessons were presented before introducing the procedure lessons. RESULTS: In both experiments, students in the iterative condition gained more knowledge of arithmetic procedures, including ability to transfer the procedures to problems with novel features. Knowledge of concepts was fairly comparable across conditions. Finally, pre-test knowledge of one type predicted gains in knowledge of the other type across experiments. CONCLUSIONS: An iterative sequencing of lessons seems to facilitate learning and transfer, particularly of mathematical procedures. The findings support an iterative perspective for the development of knowledge of concepts and procedures.