The multiplication principle (MP) is foundational for combinatorial problem-solving. From a units-coordination perspective, applying the MP with justification entails establishing unit relationships between the number of options at each independent stage of a counting process and the total number of combinatorial outcomes. Existing research…

The application of decomposition strategies (i.e., associative or distributive strategies) in two-digit multiplication problem solving supports algebraic thinking skills essential for later complex mathematical skills like solving algebra problems. Use of such strategies is also associated with improved accuracy and speed in mathematical problem…

There is broad consensus on the assumption that adults solve single-digit multiplication problems almost exclusively by fact retrieval from memory. In contrast, there has been a long-standing debate on the cognitive processes involved in solving single-digit addition problems. This debate has evolved around two theoretical accounts. Proponents of…

Several studies have shown that children do not only erroneously use additive reasoning in proportional word problems, but also erroneously use proportional reasoning in additive word problems. Traditionally, these errors were contributed to a lack of calculation and discrimination skills. Recent research evidence puts forward an additional…

Background: Adaptive expertise is a highly valued outcome of mathematics curricula. One aspect of adaptive expertise with rational numbers is adaptive rational number knowledge, which refers to the ability to integrate knowledge of numerical characteristics and relations in solving novel tasks. Even among students with strong conceptual and…

This document includes instructional tips on: (1) Building on students' informal understanding of sharing and proportionality to develop initial fraction concepts; (2) Helping students recognize that fractions are numbers that expand the number system beyond whole numbers, and using number lines to teach this and other fraction concepts; (3)…

The probability is exceptional in the teaching of mathematics because students often have difficulties to understand the basic terms and the problem solving strategies. Understanding lacks of the probability concept and various types of misconceptions arise from the misleading intuition and misinterpretations of experience with the stochastic…

We examined the type of errors on multiplication and division computation problems of 326 rising fifth graders enrolled in four elementary schools in Northern Portugal. We further examined whether there was a difference in the number of errors across age and whether there was an association between students' performance on number knowledge and…

The present study is the first to examine the computation estimation skills of dyscalculics versus controls using the estimation comparison task. In this task, participants judged whether an estimated answer to a multidigit multiplication problem was larger or smaller than a given reference number. While dyscalculics were less accurate than…

Preschool education focuses on the efforts to provide fun and meaningful learning opportunities to children aged four to six years old. The main focus is the process of teaching and learning which is children-centered, emphasis on the findings inquiry concept, the use of integrated teaching and learning, thematic learning, learning through…

Why might it be (at least sometimes) beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study…

There is a renewed debate about whether educated adults solve simple addition problems (e.g., 2 + 3) by direct fact retrieval or by fast, automatic counting-based procedures. Recent research testing adults' simple addition and multiplication showed that a 150-ms preview of the operator (+ or ×) facilitated addition, but not multiplication,…

To further understand student thinking in the context of combinatorial enumeration, we examine student work on a problem involving set partitions. In this context, we note some key features of the multiplication principle that were often not attended to by students. We also share a productive way of thinking that emerged for several students who…

While previous studies mainly focused on children's additive and multiplicative reasoning abilities, we studied third to sixth graders' "preference" for additive or multiplicative relations. This was investigated by means of schematic problems that were "open" to both types of relations, namely arrow schemes containing three…

The move from additive to multiplicative thinking requires significant change in children's comprehension and manipulation of numerical relationships, involves various conceptual components, and can be a slow, multistage process for some. Unit arrays are a key visuospatial representation for supporting learning, but most research focuses on 2D…