skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.


Title: Interactive Tools for Teaching Fourier Transforms
Fourier transforms (FT) are universal in chemistry, physics, and biology. Despite FTs being a core component of multiple experimental techniques, undergraduate courses typically approach FTs from a mathematical perspective, leaving students with a lack of intuition on how an FT works. Here, I introduce interactive teaching tools for upper-level undergraduate courses and describe a practical lesson plan for FTs. The materials include a computer program to capture video from a webcam and display the original images side-by-side with the corresponding plot in the Fourier domain. Several patterns are included to be printed on paper and held up to the webcam as input. During the lesson, students are asked to predict the features observed in the FT and then place the patterns in front of the webcam to test their predictions. This interactive approach enables students with limited mathematical skills to achieve a certain level of intuition for how FTs translate patterns from real space into the corresponding Fourier space.  more » « less
Award ID(s):
1847199
PAR ID:
10171185
Author(s) / Creator(s):
Date Published:
Journal Name:
The Biophysicist
Volume:
1
Issue:
1
ISSN:
2578-6970
Page Range / eLocation ID:
4
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. According to NASEM (2018), data science has foundations in computing, mathematics, and statistics. However, at the K-12 level, these foundations are usually taught as standalone courses that are unconnected with each other. Students may struggle to see their connections. We proposed a framework unifying those foundations using mathematical logic. A core concept in mathematical logic is function. A general function has one or more possibly non-number inputs and an output. Data science motivates a comprehensive understanding of functions and provides extensive culturally relevant, real-world, and data-rich problems and applications for students to practice their understanding. It is interesting to know how well students understand functions. We developed a six-lesson online module with more than 100 in-lesson questions. Initial analysis of the students’ answers to the questions shows that students can understand the basics of the general functions but have more difficulties in involved applications of functions. 
    more » « less
  2. Fourier transform-fluorescence recovery after photobleaching (FT-FRAP) using a diffractive optical element (DOE) is shown to support distance-dependent diffusion analysis in biologically relevant media. Integration of DOEs enables patterning of a dot array for parallel acquisition of point-bleach FRAP measurements at multiple locations across the field of view. In homogeneous media, the spatial harmonics of the dot array analyzed in the spatial Fourier transform domain yield diffusion recovery curves evaluated over specific well-defined distances. Relative distances for diffusive recovery in the spatial Fourier transform domain are directly connected to the 2D (h,k) Miller indices of the corresponding lattice lines. The distribution of the photobleach power across the entire field of view using a multidot array pattern greatly increases the overall signal power in the spatial FT-domain for signal-to-noise improvements. Derivations are presented for the mathematical underpinnings of FT-FRAP performed with 2D periodicity in the photobleach patterns. Retrofitting of FT-FRAP into instrumentation for high-throughput FRAP analysis (Formulatrix) supports automated analysis of robotically prepared 96-well plates for precise quantification of molecular mobility. Figures of merit are evaluated for FT-FRAP in analysis for both slow diffusion of fluorescent dyes in glassy polymer matrices spanning several days and model proteins and monoclonal antibodies within aqueous solutions recovering in matters of seconds. ■ INTRODUCTION 
    more » « less
  3. This article presents a mathematical modeling activity for students related to the process of memorization in which students collect their own data to drive their model development, parameterization, and validation. Engaging in the data collection gives them insight to critique and evaluate various models. This task is a low-floor high ceiling problem that offers both a relatable context and a window to quantitative approaches in cognitive science. Experimental results of students’ participation in this activity are discussed. This article also includes pedagogical recommendations with a focus on fostering equitable teaching practices and a detailed analysis of the situation comprised of several mathematical approaches to model the memorization process that highlight the richness of the problem. Instructors can adapt and implement this modeling exploration for use in various undergraduate courses, from introductory to advanced, depending on the emphasis of the lesson. 
    more » « less
  4. Ecological succession explored at the secondary and postsecondary level is often limited to terrestrial ecosystems. The emphasis is traditionally placed on how deforestation leads to ecological succession. However, aquatic ecological succession is just as important and allows for many connections to be made with other ecological concepts. Succession initiated by beavers (Castor canadensis) in particular links both aquatic and terrestrial ecosystems over time. We present a guide to an inquiry-based lesson for AP Environmental Science and undergraduate ecology courses that explores the effects of aquatic and terrestrial ecological succession initiated by deforestation and beavers. Specifically, the focus is ecological succession and its effects in both terrestrial and aquatic ecosystems. In this lesson, students (1) engage with a preassessment and broad overview of ecological succession, (2) explore authentic research data representing secondary succession in beaver ponds, (3) explain data using detective activities, (4) elaborate with a mystery pond, and (5) evaluate their new understanding by comparing a pre- and postassessment. This lesson plan meets the objectives for AP Environmental Science Biology courses as well as the core concepts and competencies for undergraduate biology education from the Vision and Change report by the American Association for the Advancement of Science in 2011. 
    more » « less
  5. We propose a novel educational gamification approach that employs and reinforces otherwise abstract concepts currently taught in graduate-level courses to become a standard part of undergraduate communications courses in the future. In particular, we develop software tools that visually demonstrate relevant wireless communications parameters and processes, including those already taught in undergraduate communications courses (e.g., power, modulation type, data rate and bandwidth, coding rate) to students in an interactive way. We exploit our Internet-accessible wireless communication testbed X to enable students to experience wireless communication challenges and learn different solutions in realistic environments via an ordinary web browser. This paper presents our approach to authentic STEM learning and compares it with other past and ongoing initiatives. We conclude that there is a gap of practical teaching tools and methods for educating students as well as radio engineers and researchers about new trends in wireless communications with a focus on spectrum sharing. 
    more » « less