Generalisation is a skill that enables learners to acquire knowledge in general, and mathematical knowledge in particular. It is a core aspect of algebraic thinking and, in particular, of functional thinking, as a type of algebraic thinking. Introducing primary school children to functional thinking fosters their ability to generalise, explain and…

To efficiently solve mathematical expressions and equations, students need to notice the systemic structure of mathematical expressions (e.g., inverse relation between 3 and 3 in 3 + 5 - 3). We examined how symbols--specifically variables versus numbers--and students' algebraic knowledge impacted seventh graders' problem-solving strategies and use…

In issue 31(2) of the "Australian Senior Mathematics Journal", Kok (2017) describes a useful four-step process for investigating number patterns and identifying the underlying function. The process is demonstrated for both linear and quadratic functions. With respect to the quadratic example, I provide an additional idea relevant to step…

The purpose of this study was to examine the effects of algebraic equation solving intervention for sixth graders with mathematics learning difficulties (MD). A total of 48 students with MD were randomly assigned to either the algebraic equation solving intervention, "Mystery Math" (n = 24) or control condition (n = 24). The…

We present results of a grounded analysis of individual interviews in which students play Vector Unknown -- a digital game designed to introduce visualizing vectors, scaling vectors, vector addition, and vector equations, attending to the geometric and algebraic representations of vectors. The game was designed to be used at the beginning of a…

Randomized evaluations of educational technology produce log data as a bi-product: highly granular data on student and teacher usage. These datasets could shed light on causal mechanisms, effect heterogeneity, or optimal use. However, there are methodological challenges: implementation is not randomized and is only defined for the treatment group,…

This study investigates prospective elementary and secondary school mathematics teachers' ways of reasoning about differentiability at a point and corner points while working on a mathematical modelling activity. Adopting a multiple-case study design, the participants of the study were 68 prospective elementary school mathematics teachers enrolled…

Many literature studies point out how the process that moves the student from personal reasoning to the arrangement of fruitful argument could help them in the development of mathematical argumentative thinking. Starting from the above assumptions, in this paper we investigate the interaction among teacher/researcher and students in a Digital…

Faded worked examples have been used to promote problem solving performance, such as mathematics performance in many laboratory studies and short-term classroom studies. However, few studies have examined the ways educators may use fading in their own classroom on more accessible platforms that do not require programming experience. Further, few…

Perseverance is vital for the process of learning mathematics with understanding. To do so, students must wrestle with obstacles and amend their problem-solving plan when necessary. I collected qualitative data to investigate the nature of perseverance for 11 ninth-grade algebra students working on a challenging task for which they affirmed…

Solving systems of linear equations is of central importance in linear algebra and many related applications, yet there is limited literature examining the symbolizing processes students use as they work to solve systems of linear equations. In this paper, we examine this issue by analyzing final exam data from 68 students in an introductory…

Students in secondary school often struggle with symbol sense, that is, the general ability to deal with symbols and to recognize the structure of algebraic formulas. Fostering symbol sense is an educational challenge. In graphing formulas by hand, defined as graphing using recognition and reasoning without technology, many aspects of symbol sense…

A selection of secondary school mathematics textbooks from twelve countries on five continents was analysed to better understand the support they might be in teaching and learning mathematical problem solving. Over 5700 tasks were compared to the information provided earlier in each textbook to determine whether each task could be solved by…

Cognitive load can play a key role in learners' abilities to solve complex problems like mathematics. Many factors can affect the presence of cognitive load in learning including instructional strategy, task difficulty and prior knowledge. To understand the interaction of above factors and their influence on learner cognitive load and performance,…

This study explores high school students' and their mathematics teachers' mathematical discourses on the equal sign and variables in the classroom context. The participants in this case study were ninth graders and their mathematics teachers. The data were analyzed in terms of participants' routines, specifically ritual and explorative routines in…