This research aimed to determine how preservice secondary mathematics teachers supported students learning with multiple representations in their lesson plans. Participants were nine preservice secondary mathematics teachers who prepared lesson plans at the end of a 14- week mathematics methods course at a public university in Türkiye. The primary data source was the participants...
The aim of this research was to analyze the level of mathematical understanding based on the mathematical connections made by a group of Mexican High School students when solving mathematical tasks related to the concepts of equation and linear function. The study employed the thinking aloud method to collect data, whereby students verbalized their thought processes while solving...
Students’ development of reasoning through algebraic symbols is a crucial component of most mathematics courses. This paper reports the journey of one second-semester calculus student, Cedric, as he attempted to reason about and create algebraic representations for arbitrary partial sums and infinite series through two exploratory teaching interviews. We report Cedric’s...
Graph literacy, the ability to interpret and create graphical representations, is an important skill for students to learn mathematics and to succeed in STEM coursework and careers. Additionally, with the rapid development of technological devices and media, students are encountering increasingly more situations in which graph literacy is needed to make sense of and respond to...
While slope is a topic in the algebra curriculum, having a robust understanding of slope is needed for students to truly understand several single and multivariable calculus topics with any depth. We begin with a review of the topic of slope and present what is known from its existing corpus of literature. We then outline the tenets of APOS theory. Building from there, we suggest...
This article delves into an intervention designed to enhance a precalculus student’s understanding in constructing linear equations and in approximating function values. Before the intervention, the student primarily relied on the algebraic formula y2-y1/x2-x1 to determine slope, knew slope geometrically as Δy/Δx solely for integer values, and struggled to construct a linear...
Connecting algebraic and graphical representations encompasses a large portion of mathematical activity for students in grades 8-14. In Calculus, the ability to represent distances on graphs using algebraic expressions is foundational for a wide range of results. However, research has shown that students may struggle to make such connections. In this article, we seek to answer...
A Review of: 1. Alan Schoenfeld, Heather Fink, Alyssa Sayavedra, Anna Weltman, & Sandra Zuñiga-Ruiz’s (2023) Mathematics Teaching on Target, Routledge, 164pp. ISBN (HB): 978-1-0324-4167-2; 2. Alan Schoenfeld, Heather Fink, Sandra Zuñiga-Ruiz, Siqi Huang, Xinyu Wei, & Brantina Chirinda’s (2023) Helping Students Become Powerful Mathematics Thinkers, Routledge, 272pp. ISBN (HB): 978...
Consider a 1 x 3 grid whose top-left vertex is connected to the bottom-right vertex of each of the unit squares. What is the sum of the three acute angles formed by the connecting segments with the unit squares
Undergraduate research in tertiary education offers mathematics students a pathway to engage in high-impact practices. Using a simple sum equals product identity from number theory as a motivator, we build a series of inclusion-exclusion identities for convex polygons using the symmetry inherent in the tangent function. The techniques used are simple and accessible, illuminating...
Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students’ arguments...
This study investigates 40 preservice middle school teachers’ problems posed for two given fractions. Pre-service teachers were asked to pose three problems (farea, length and set model) for each of the operations of addition, subtraction, multiplication and division with these fractions. 12 problems posed by each pre-service teacher were examined in terms of their suitability...
In response to student evaluations I revised my undergraduate course in real analysis to a slides-and-worksheets model. This is the story of that revision, including why and how it was done, together with the results.
Current mathematics students are members of Generation Z, a generation proving to be quite different than previous ones. Generation Z has never known a time without Google, nor a time of safety. Generation Z has a declining tendency to even attend college. If they do attend college, their expectations need to be met to keep them engaged in mathematics. Professors will need to...
By Francesco Beccuti, Published on 02/01/24
Professional noticing of mathematical thinking, as defined by Jacobs, Lamb, and Philipp (2010) can be broken down into three components: attending to relevant cues, interpreting the mathematical understanding, and deciding the next best instructional steps. Most research on this topic has been conducted with elementary children. However, there is a gap in the research on...
This article gives an overview of questions on diagnostics and procedures of high mathematical talent. Various methods such as intelligence tests, school achievement tests and checklists are presented and discussed. The conclusions favor multidimensional and multi-step approaches with a focus on special mathematical tests. As an example the approach of identifying children with a...
The authors consider a conjecture by Chebyshev in 1853 on the distribution of odd primes among those that are one more than a multiple of four and those three more than a multiple of four—and use technology to explore the cardinality of these subsets. Generalizations are presented for student exploration along with several sources for more in-depth research.
Relational and instrumental understanding of the Chain Rule can help teachers provide students a deeper and more meaningful calculus experience.
Separation of variables is a common method for producing an analytical based solution to partial differential equations. Despite the wide application of this method, often the physical phenomena described by the differential equations are not adequately involved in the discourse over the appropriate methods to solve a given problem, particularly in mathematics curricula. However...
We mark the 50th anniversary of mathematical modeling education by reviving the term the spirit of mathematical modeling (SoMM), which idealistically reflects core aspects of mathematical modeling. The basis of our analysis is the notion of bildung, which is an educational philosophy that strives for harmonizing heart, mind, social life and culture. We built SoMM on five...
This article describes the development of a coding scheme for analyzing mathematical tasks in Primary Source Projects (PSPs), curriculum materials based on primary historical sources designed for teaching standard topics from today’s undergraduate mathematics curriculum. Our scheme attends to social-cultural aspects of mathematical learning while focusing on the student actions...
Figured numbers are sequences of natural numbers that, when represented by dots, form geometric configurations. The immediate sequence of figured numbers presents in its configuration the equilateral triangles, being the basis for finding several others, as long as a relationship between them is discovered. This article shows the construction of polygonal, stellar and three...
When solving problems, mathematically gifted individuals tend to internalize intuitive ideas and approaches, and to shorten their reasoning. Consequently, for teachers it is difficult to observe gifted students’ mathematical reasoning in the context of problem solving. In this paper we investigate nine gifted Swedish 9th grade students’ mathematical reasoning during problem...
