Humans represent numbers on a mental number line with smaller numbers on the left and larger numbers on the right side. A left-to-right oriented spatial-numerical association, (SNA), has been demonstrated in animals and infants. However, the possibility that SNA is learnt by early exposure to caregivers' directional biases is still open. We…

This commentary reviews each of the three content chapters in the integers section and offers questions to promote further discussion. In addition to the themes raised in the three chapters, I introduce the role of formal mathematical structure in generalizing systems of number, from natural numbers to integers, and analogously, from real numbers…

Writing and evaluating contextual problems is an important task in the work of teaching, and thus is part of the knowledge that prospective teachers must develop. In dealing with word problems posed both by children and themselves, prospective teachers will need to attend to the realism of the context and the consistency between the operation and…

The typical trigonometry, precalculus, or calculus student might not agree that logarithms are hot stuff, but we drew motivation from chili peppers to help students get a better taste for logarithms. The Scoville scale, which ranges from 0 to 16,000,000, has been the sole quantitative metric to measure the pungency (spiciness) of peppers since its…

Representing the base-10 structure of numbers is a challenging cognitive ability, unique to humans, but it is yet unknown how precisely this is done. Here, we examined whether and how literate adults represent a number's full syntactic structure. In 5 experiments, participants repeated number-word sequences and we systematically varied the order…

As students develop understanding and fluency in single-digit operations such as addition, they develop sophisticated strategies and number sense (magnitude, number order, and composition). Deficits in number sense and reliance on inefficient approaches can lead to struggle in mathematics. Intervention research in this area reported effects on…

The aim of this study was to investigate second-grade elementary school students' performance related to their counting skills, place value understanding, and addition operation in natural numbers. A total of 205 second-grade elementary school students from Turkey participated in this study. The data were collected through a 'Personal Information…

Many studies have used fraction magnitude comparison tasks to assess people's abilities to quickly assess fraction magnitudes. However, since there are multiple ways to compare fractions, it is not clear whether people actually reason about the holistic magnitudes of the fractions in this task and whether they use multiple strategies in a flexible…

We conducted a multiyear project across three institutions to develop an instructional tutorial that supports student understanding of change of basis in quantum mechanics. Building from our previous work, we identified learning goals to guide activity development. The tutorial makes an analogy between spin-1/2 states and a Cartesian coordinate…

Neuropsychological case-studies suggested that dates and encyclopedic numbers may be processed differently than unknown numbers. However, this issue was seldom investigated in healthy participants. Therefore, it is unclear whether known dates are read like words (as lexical items), or like numbers (each position strictly defines digits' values in…

The mathematical abilities of children with autism spectrum condition have been understudied. Magnitude representation is a fundamental numerical ability that emerges early in development and is linked to children's learning of formal mathematics. It remains unclear about whether children with autism spectrum condition differ from their peers…

Background: South African learners face the double disadvantage of living in low socioeconomic conditions with access to few resources and attending schools with challenging learning conditions. Mathematics performance reflects such conditions with extreme performance gaps between wealthier and poorer learners. The need for early intervention is…

In an undergraduate analysis course taught by one of the authors, three prompts are regularly given: (i) What do we know? (ii) What do we need to show? (iii) Let's draw a picture. We focus on the third prompt and its role in helping students develop their confidence in learning how to construct proofs. Specific examples of visual models and their…

This study evaluated the effectiveness of a teaching procedure derived from an analysis of joint control to teach math skills to students with autism spectrum disorder. Given the limited amount of evidence on teaching procedures derived from the analysis of joint control to teach students with autism, additional investigations on this topic were…

This study investigated how six Year Four (9-10 years old) students interacted with a dynamic (zoomable) digital number line to demonstrate knowledge and understandings of decimal fractions. Results from a task-based interview indicated that the zoomable number line is able to assist students in developing conceptual understandings of decimal…