Improving rational number knowledge using the NanoRoboMath digital game

Rational number knowledge is a crucial feature of primary school mathematics that predicts students’ later mathematics achievement. Many students struggle with the transition from natural number to rational number reasoning, so novel pedagogical approaches to support the development of rational number knowledge are valuable to mathematics educators worldwide. Digital game-based learning environments may support a wide range of mathematics skills. NanoRoboMath, a digital game-based learning environment, was developed to enhance students’ conceptual and adaptive rational number knowledge. In this paper, we tested the effectiveness of a preliminary version of the game with fifth and sixth grade primary school students (N = 195) using a quasi-experimental design. A small positive effect of playing the NanoRoboMath game on students’ rational number conceptual knowledge was observed. Students’ overall game performance was related to learning outcomes concerning their adaptive rational number knowledge and understanding of rational number representations and operations.

Second and fifth graders’ use of knowledge-pieces and knowledge-structures when solving integer addition problems

In this study, we explored second and fifth graders’ noticing of negative signs and incorporation of them into their strategies when solving integer addition problems. Fifty-one out of 102 second graders and 90 out of 102 fifth graders read or used negative signs at least once across the 11 problems. Among second graders, one of their most common strategies was subtracting numbers using their absolute values, which aligned with students’ whole number knowledge-pieces and knowledge-structure. They sometimes preserved the order of numbers and changed the placement of the negative sign (e.g., −9 + 2 as 9 – 2) and sometimes did the opposite (e.g., −1 + 8 as 8 – 1). Among fifth graders, one of the most common strategies reflected use of integer knowledge-pieces within a whole-number knowledge-structure, as they added numbers’ absolute values using whole number addition and appended the negative sign to their total. For both grade levels, the order of the numerals, the location of the negative signs, and also the numbers’ absolute values in the problems played a role in students’ strategies used. Fifth graders’ greater strategy variability often reflected strategic use of the meanings of the minus sign. Our findings provide insights into students’ problem interpretation and solution strategies for integer addition problems and supports a blended theory of conceptual change. Adding to prior findings, we found that entrenchment of previously learned patterns can be useful in unlikely ways, which should be taken up in instruction.

Measuring Emerging Number Knowledge in Toddlers

Recent evidence suggests that infants and toddlers may recognize counting as numerically relevant long before they are able to count or understand the cardinal meaning of number words. The Give-N task, which asks children to produce sets of objects in different quantities, is commonly used to test children’s cardinal number knowledge and understanding of exact number words but does not capture children’s preliminary understanding of number words and is difficult to administer remotely. Here, we asked whether toddlers correctly map number words to the referred quantities in a two-alternative forced choice Point-to-X task (e.g., “Which has three?”). Two- to three-year-old toddlers (N = 100) completed a Give-N task and a Point-to-X task through in-person testing or online via videoconferencing software. Across number-word trials in Point-to-X, toddlers pointed to the correct image more often than predicted by chance, indicating that they had some understanding of the prompted number word that allowed them to rule out incorrect responses, despite limited understanding of exact cardinal values. No differences in Point-to-X performance were seen for children tested in-person versus remotely. Children with better understanding of exact number words as indicated on the Give-N task also answered more trials correctly in Point-to-X. Critically, in-depth analyses of Point-to-X performance for children who were identified as 1- or 2-knowers on Give-N showed that 1-knowers do not show a preliminary understanding of numbers above their knower-level, whereas 2-knowers do. As researchers move to administering assessments remotely, the Point-to-X task promises to be an easy-to-administer alternative to Give-N for measuring children’s emerging number knowledge and capturing nuances in children’s number-word knowledge that Give-N may miss.