The present study explored 285 11th-grade students' preconceptions, misconceptions, and errors in solving mathematics tasks by graphical method. A descriptive-explorative study design was adopted. Cluster sampling was used to select students from sampled secondary schools in eastern and central Uganda. Students' paper and pen solution sketches…

To fulfill the Standards for Mathematical Practice (National Governors Association Center for Best Practices & Council of Chief State School Officers [NGA Center & CCSSO], 2010), it is important for teachers to empower students to critique, revise, and expand their ideas as they solve problems. However, students often face challenges such…

The goal of this design-based research study was the creation and evaluation of a mini-unit intended to foster perceptually grounded understandings of the concept of slope in middle-school students. Central to this unit was an innovative device designed to create a productive pedagogical space between student intuition for steepness and formal…

While one branch of literature is replete with investigations of problem solving and another branch frequently investigates student use of dynamic mathematics environments (DMEs), most of the studies in both of these fields consider whether or not students can solve problems. Far fewer number of studies consider the cognitive processes associated…

Multiplicative thinking is widely accepted as a critically important 'big idea' of mathematics that underpins much mathematical understanding beyond the primary years. It is therefore important to ensure not only that children can think multiplicatively, but that they can do so at a conceptual rather than procedural level. This paper reports on a…

The National Council of Teachers of Mathematics (NCTM) advocates for all students to have inspiring and positive mathematics experiences. This article surveys four key recommendations that can serve as catalysts for achieving an equitable mathematics education for each and every student. The recommendations are official positions of NCTM and are…

Developing multiplicative reasoning is an important milestone for elementary school students, which influences their learning of later mathematical concepts (Hackenberg and Tillema 2009). For children to conceptually understand multiplication, one should move beyond merely counting by ones to dealing with composites (twos, fives, etc.) and other…

The learning process of constructing concepts in mathematics can be supported by visualization in graphical, numerical, and symbolic representations. However, previous research suggests that students have not fully taken advantage of the opportunities provided by these technologies. Two key factors, including mathematics conception and mathematics…

Students in upper secondary education encounter difficulties in applying mathematics in physics. To improve our understanding of these difficulties, we examined symbol sense behavior of six grade 10 physics students solving algebraic physic problems. Our data confirmed that students did indeed struggle to apply algebra to physics, mainly because…

This article provides an example of how one sixth-grade classroom teacher and an instructional coach built a bridge for students in a grade 6 self-contained classroom to support their conceptual understanding and procedural fluency with dividing fractions. Baseline data showed that most of the students in this class had mastered only the…

Visual thinking plays an essential role in solving problems and in learning mathematics. Many students do not understand how to graphically or geometrically represent problems and solve algebra problems. Visual thinking is the ability, process, and results of creating, interpreting, using, and imagining images and diagrams on paper or with…

In this article, physical quantities related to the motion of a parachute system in the framework of Newton's law of mechanics guide an informal approach to the mathematical concept of function. Furthermore, a graphical simulator that enables visualization of the fall of object-parachutes system is presented. The article is aimed at students and…

In this paper, we analyse the results of an experience designed and implemented to evaluate the epistemic and cognitive conflicts identified in the study of proportionality, carried out by a group of 21 students in their last year of primary education. Initially the students exhibit difficulties to recognize situations where proportionality can be…

This study is to reveal the construction process of student representation in solving of ill-structured geometry problems. The three phases involved in the construction process are interpreting the problem, giving opinions, and concluding. A total of four high school students participated in Surabaya and Sidoarjo in this qualitative study. They…

One does not have to teach for very long to see students applying the wrong formula in the wrong situation (e.g., Kirshner and Awtry 2004; Tan-Sisman and Aksu 2016). Students can become overreliant on the power of the formula instead of thinking about the relationships it describes. It is not surprising that students can see formulas as a way to…