Mathematical Exploration Research Articles

A Fibonacci Simple Ecosystem: Prey and Predator

Predators and prey in the population of an ecosystem can be evaluated and investigated with mathematical explorations. Intermediate elementary students explore Fibonacci sequences in relation to prey and discuss models to maintain a balance between the species through mathematics by modifying variables. iSTEM (Integrating Science, Technology, and Engineering in Mathematics) authors share ideas and activities that stimulate student interest in the integrated fields of science, technology, engineering, and mathematics (STEM) in K-grade 6 classroom.

Call for Manuscripts: Mathematical Explorations

This call is soliciting manuscripts for the student activity department in Mathematics Teaching in the Middle School called Mathematical Explorations.

A modern tool for a modern student. Video games in the exploration and learning of mathematics

Solving equations is one of the central topics in mathematics curricula. Teachers would like to pay more attention to building formulas and formulating algebraic models because modern students dare to translate a problem situation into algebra, but very often their students are helpless because of the lack of practice. There is not much time for exercises. The problem is how to help students with acquiring algebraic skills, like transforming expressions and solving equations. There is a new medium in education that can be used for this – video games. Could they help? I designed a special learning arrangement using a special game that was built for learning algebra. The results suggest that This can be a really relevant didactic medium for teaching mathematics

A mathematical exploration into manipulation and control of a bifurcative dreaming process

Mathematical models derived from the concepts of nonlinear dynamics and chaos theory for representing the highly nonrational transformations and bifurcations in a dreaming process may be highly useful for the study of dreams. In this work, three chaotic dynamical models known as the logistic map, sine map and cubic map have been adopted as a mathematical analogy for representing bifurcative dreaming processes. Further, the effect of a dream potentiator has been incorporated into the adopted models. For studying the effects of the dream potentiator on the actual dreaming process, computer simulations were performed. Results demonstrate that the dream potentiator may enhance the actual dream by elevating the dream states, by expanding the bifurcations and by reducing the number of successive steps required for attaining a dream state of deterministic chaos.

Mathematical Explorations: Exercise Away the Big Mac: Ratios, Rates, and Proportions in Context

An activity centers around an authentic context: the nutritional content of a Big Mac and how to exercise the calories away. The task, which focuses on ratios, percentages, rates, and proportion concepts, also includes activity sheets.

Mathematical Explorations: Finding Pi with Archimedes's Exhaustion Method

An analysis of Archimedes' contributions to the field of mathematics and the meaning behind C = pd are explored. Activity sheets are included.

Mathematical Explorations: Understanding Geometry and Measurement through Service-Learning

A service-learning activity that is tied to geometry concludes with a donation to a community food drive.

Technology Tips: Investigating Extrema with GeoGebra

Technology can be used to manipulate mathematical objects dynamically while also facilitating and testing mathematical conjectures. We view these types of authentic mathematical explorations as closely aligned to the work of mathematicians and a valuable component of our students' educational experience. This viewpoint is supported by NCTM and the Common Core State Standards for Mathematics (CCSSM).

TEACHING AND LEARNING MODES FOR DESIGN ENGINEERING

Important learning outcomes include students’ ability for effective communication (verbal and written, but also graphical for sketching and data interpretation) and teamwork. The ability to apply a systematic engineering design method to design or re- design technical systems is not stated, yet that capability distinguishes engineers from scientists, and artistic designing from design engineering. It is to some extent related to creativity. Applying a systematic engineering design method includes the ability to use the engineering sciences heuristically for ‘order-of-magnitude’ and ‘what-if’ estimates of future configurations. This requires understanding the physical behavior of phenomena, individually and in their relationships, both by ‘visual feel’ and by mathematical exploration, in a student’s development of expertise. These requirements indicate a need for change in the teaching and learning procedures and methods, departing from the mainly science-oriented lecturing and examination assessment that has been conventional, and therefore changes in the curriculum. A guiding premiss, also valid for design engineering, is formulated by Klaus: ‘Both theory and method emerge from the phenomenon of the subject’.

Mathematical Explorations: The Bridge to Mathematics and Literature

Various works of literature are explored in light of the math that can be gleaned from their pages; includes activity sheets.

Mathematical Explorations: Interpreting Box Plots with Multiple Linked Representations

A box plot activity is driven by a TI-Nspire calculator.

Analytic evaluation of the electronic self-energy in theGWapproximation for two electrons on a sphere

The GW approximation for the electronic self-energy is an important tool for the quantitative prediction of excited states in solids, but its mathematical exploration is hampered by the fact that it must, in general, be evaluated numerically even for very simple systems. In this paper I describe a nontrivial model consisting of two electrons on the surface of a sphere, interacting with the normal long-range Coulomb potential, and show that the GW self-energy, in the absence of self-consistency, can in fact be derived completely analytically in this case. The resulting expression is subsequently used to analyze the convergence of the energy gap between the highest occupied and the lowest unoccupied quasiparticle orbital with respect to the total number of states included in the spectral summations. The asymptotic formula for the truncation error obtained in this way, whose dominant contribution is proportional to the cutoff energy to the power -3/2, may be adapted to extrapolate energy gaps in other systems.

Searching for Tomorrow's Innovators: Profiling Creative Adolescents

Profiling may be a viable means of identifying those creative adolescents who can benefit from specialized guidance and exploration of science, technology, engineering, and mathematics (STEM) fields, arts, and human services. The experimenters developed 1 general and 5 specific profiles including interest, personality, and achievement variables based on the profiles of eminent people in five domains of creative endeavor. Educators of gifted students at schools throughout a Midwestern state identified 485 students to attend a research through service counseling laboratory. One cohort received the Vocational Preference Inventory, the Personality Research Form, and the Tellegen Absorption Scale, and a second cohort received the VPI, the Revised NEO Personality Inventory (NEO- PI-R), and Tellegen Absorption Scale. For each cohort, descriptive data were gathered and principal components analyses were performed on scales of interest and personality inventories. In addition, a cluster analysis was performed for the second cohort. The finding supported the hypothesis that profiling could be used to identify creative adolescents for career development programs. Both principal components analyses and cluster analyses revealed profiles of fine and performing arts students: one or two profiles of interpersonally talented groupings: and an engineering/technical profile. Creative students were more agreeable than those in previous studies, and there was strong evidence for crossover arts/science profiles.

Making Connections in Practice

This study examines the ways prospective elementary teachers (PSTs) made connections to children’s mathematical thinking and children’s community funds of knowledge in mathematics lesson plans. We analyzed the work of 70 PSTs from across three university sites associated with an instructional module for elementary mathematics methods courses that asks PSTs to visit community settings and develop problem solving mathematics lessons that connect to mathematical practices in these settings (Community Mathematics Exploration Module). Using analytic induction, we identified three distinct levels of connections to children’s mathematical thinking and their community funds of knowledge evidenced in PSTs’ work (emergent, transitional, and meaningful). Findings describe how these connections reflected different points on a learning trajectory. This study has implications for understanding how PSTs begin to connect to children’s mathematical funds of knowledge in their teaching, a practice shown to be effective for teaching diverse groups of children.

Classroom Elections

Each month, elementary teachers receive a problem, along with suggested instructional notes and often a student activity sheet. Teachers are to use the problem in their own classrooms and report solutions, strategies, reflections, and misconceptions to the journal audience. November is election month in the United States. Every four years, opportunities arise for classroom discussions related to selecting the next U.S. president. In elementary school classrooms, mock elections can provide data to use for many mathematical explorations and graphical representations.

Mathematical Explorations: Developing the Common Denominator Fraction Division Algorithm

Division of fractions, a troublesome area for students, is looked at anew using a common denominator algorithm.

Mathematical Explorations: Rethinking Dr. Seuss's The Lorax—Mathematically

A mathematical analysis of Dr. Seuss's classic literature is explored.

Mathematical Explorations: The Electoral College

A mathematical analysis of the electoral college is coupled with activity sheets.

Call for Manuscripts: Mathematical Explorations – April 2012

This call for manuscripts is requesting articles for the Mathematical Explorations department.

Reader Reflections – April 2012

Readers react to published articles or submit their own mathematical explorations.