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.

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.

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' 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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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).

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).

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.

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.

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.

It pays to compare: an experimental study on computational estimation.

Comparing and contrasting examples is a core cognitive process that supports learning in children and adults across a variety of topics. In this experimental study, we evaluated the benefits of supporting comparison in a classroom context for children learning about computational estimation. Fifth- and sixth-grade students (N=157) learned about estimation either by comparing alternative solution strategies or by reflecting on the strategies one at a time.

In pursuit of knowledge: comparing self-explanations, concepts, and procedures as pedagogical tools.

Explaining new ideas to oneself can promote learning and transfer, but questions remain about how to maximize the pedagogical value of self-explanations. This study investigated how type of instruction affected self-explanation quality and subsequent learning outcomes for second- through fifth-grade children learning to solve mathematical equivalence problems (e.g.

When generating answers benefits arithmetic skill: the importance of prior knowledge.

People remember information better if they generate the information while studying rather than read the information. However, prior research has not investigated whether this generation effect extends to related but unstudied items and has not been conducted in classroom settings. We compared third graders' success on studied and unstudied multiplication problems after they spent a class period generating answers to problems or reading the answers from a calculator.

Learning from explaining: does it matter if mom is listening?

The goal of the current study was to examine whether explaining to another person improves learning and transfer. In the study, 4- and 5-year-olds (N=54) solved multiple classification problems, received accuracy feedback, and were prompted to explain the correct solutions to their moms, to explain the correct solutions to themselves, or to repeat the solutions. Generating explanations (to selves or moms) improved problem-solving accuracy at posttest, and explaining to mom led to the greatest problem-solving transfer.

Promoting transfer: effects of self-explanation and direct instruction.

Explaining new ideas to oneself can promote transfer, but how and when such self-explanation is effective is unclear. This study evaluated whether self-explanation leads to lasting improvements in transfer success and whether it is more effective in combination with direct instruction or invention. Third- through fifth-grade children (ages 8-11; n=85) learned about mathematical equivalence under one of four conditions varying in (a) instruction on versus invention of a procedure and (b) self-explanation versus no explanation.