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In Paper 2 I argue that prior research in online learning has tested the impact of online courses on measures such as student retention rates, satisfaction scores, and GPA but that research is needed to describe the meanings students construct for mathematical ideas researchers have identified as critical to their success in future math courses and other STEM fields. This paper discusses the need for a new focus in studying online mathematics learning and calls for cognitive researchers to begin developing a productive methodology for examining the meanings students construct while engaged in online lessons.
Paper 3 describes the online Precalculus course intervention we designed around measurement imagery and quantitative reasoning as themes that unite topics across units. I report results relative to the meanings students developed for exponential functions and related ideas (such as percent change and growth factors) while working through lessons in the intervention. I provide a conceptual analysis guiding its design and discuss pre-test and pre-interview results, post-test and post-interview results, and observations from student behaviors while interacting with lessons. I demonstrate that the targeted meanings can be productive for students, show common unproductive meanings students possess as they enter Precalculus, highlight challenges and opportunities in teaching and learning in the online environment, and discuss needed adaptations to the intervention and future research opportunities informed by my results.
Many other studies have researched the benefits of digital manipulatives and digital environments through student completion of tasks and testing. This study intends to research students’ use of the digital tools and manipulatives, along with the students’ interactions with the digital environment. To this end, I conducted exploratory teaching experiments with two calculus I students.
In the exploratory teaching experiments, students were introduced to a GeoGebra application developed by Fischer (2019), which includes instructional videos and corresponding quizzes, as well as exercises and interactive notepads, where students could use digital tools to construct line segments and circles (corresponding to the physical straight-edge and compass). The application built up the students’ foundational knowledge, culminating in the construction and verbal proof of Euclid’s Elements, Proposition 1 (Euclid, 1733).
The central findings of this thesis are the students’ interactions with the digital environment, with observed changes in their conceptions of radii and circles, and in their use of tools. The students were observed to have conceptions of radii as a process, a geometric shape, and a geometric object. I observed the students’ conceptions of a circle change from a geometric shape to a geometric object, and with that change, observed the students’ use of tools change from a measuring focus to a property focus.
I report a summary of the students’ work and classify their reasoning and actions into the above categories, and an analysis of how the digital environment impacts the students’ conceptions. I also briefly discuss the impact of the findings on pedagogy and future research.
This dissertation reports results of an investigation into the ways of thinking that support and inhibit students from constructing and reasoning about graphs in terms of covarying quantities. I collected data by engaging three university precalculus students in asynchronous teaching experiments. I designed the instructional sequence to support students in making three constructions: first imagine representing quantities’ magnitudes along the axes, then simultaneously represent these magnitudes with a correspondence point in the plane, and finally anticipate tracking the correspondence point to track how the two quantities’ attributes change simultaneously.
Findings from this investigation provide insights into how students come to engage in covariational reasoning and re-present their imagery in their graphing actions. The data presented here suggests that it is nontrivial for students to coordinate their images of two varying quantities. This is significant because without a way to coordinate two quantities’ variation the student is limited to engaging in static shape thinking.
I describe three types of imagery: a correspondence point, Tinker Bell and her pixie dust, and an actor taking baby steps, that supported students in developing ways to coordinate quantities’ variation. I discuss the figurative aspects of the students’ coordination in order to account for the difficulties students had (1) constructing a multiplicative object that persisted under variation, (2) reconstructing their acts of covariation in other graphing tasks, and (3) generalizing these acts of covariation to reason about formulas in terms of covarying quantities.
This thesis is an extension of previous research done by Roh and Lee (2018). Their research involved the design and implementation of a survey to analyze students’ cognitive inconsistencies. This thesis expands upon this research to interview students who demonstrated logical inconsistencies and evaluates the kinds of struggles students faced while evaluating statements and validating arguments. Three students who demonstrated logical inconsistencies were interviewed and asked to answer questions originally pulled from Roh and Lee’s (2018) survey. This thesis found that there were many aspects of each section of the survey that students had struggled with, including use of intuition, analyzing a proof-by-contradiction that utilized a negated statement, and distrust of alternate proving methods. Overall, these techniques the students used while evaluating statements and validating arguments gives interesting insight into the pedagogy of teaching proofs.
Human Papillomavirus, or HPV, is a viral pathogen that most commonly spreads through sexual contact. HPV strains 6 and 11 normally cause genital warts, while HPV strains 16 and 18 commonly cause cervical cancer, which causes cancerous cells to spread in the cervix. Physicians can detect those HPV strains, using a Pap smear, which is a diagnostic test that collects cells from the female cervix.
Johann Gregor Mendel studied patterns of trait inheritance in plants during the nineteenth century. Mendel, an Augustinian monk, conducted experiments on pea plants at St. Thomas’ Abbey in what is now Brno, Czech Republic. Twentieth century scientists used Mendel’s recorded observations to create theories about genetics.
In the 1930s, George Beadle and Boris Ephrussi discovered factors that affect eye colors in developing fruit flies. They did so while working at the California Institute of Technology in Pasadena, California. (1) They took optic discs (colored fuchsia in the image) from fruit fly larvae in the third instar stage of development. Had the flies not been manipulated, they would have developed into adults with vermilion eyes. (2) Beadle and Ephrussi transplanted the donor optic discs into the bodies of several types of larvae, including those that would develop with normal colored eyes (brick red), and those that would develop eyes with other shades of red, such as claret, carmine, peach, and ruby (grouped together and colored black in the image). (3a) When implanted into normal hosts that would develop brick red eyes, the transplanted optic disc developed into an eye that also was brick red. (3b) When implanted into abnormal hosts that would develop eyes of some other shade of red, the transplanted optic discs developed into eyes that were vermilion. Beadle and Ephrussi concluded that there was a factor, such as an enzyme or some other protein, produced outside of the optic disc that influenced the color of the eye that developed from the disc.
This illustration shows George Beadle and Edward Tatum's experiments with Neurospora crassa that indicated that single genes produce single enzymes. The pair conducted the experiments at Stanford University in Palo Alto, California. Enzymes are types of proteins that can catalyze reactions inside cells, reactions that produce a number of things, including nutrients that the cell needs. Neurospora crassa is a species of mold that grows on bread. In the early 1940s, Beadle and Tatum conducted an experiment to discover the abnormal genes in Neurospora mutants, which failed to produce specific nutrients needed to survive. (1) Beadle and Tatum used X-rays to cause mutations in the DNA of Neurospora, and then they grew the mutated Neurospora cells in glassware. (2) They grew several strains, represented in four groups of paired test tubes. For each group, Neurospora was grown in one of two types of growth media. One medium contained all the essential nutrients that the Neurospora needed to survive, which Beadle and Tatum called a complete medium. The second medium was a minimal medium and lacked nutrients that Neurospora needed to survive. If functioning normally and in the right conditions, however, Neurospora can produce these absent nutrients. (3) When Beadle and Tatum grew the mutated mold strains on both the complete and on the minimal media, all of the molds survived on the complete media, but not all of the molds survived on the minimal media (strain highlighted in yellow). (4) For the next step, the researchers added nutrients to the minimal media such that some glassware received an amino acid mixture (represented as colored squares) and other glassware received a vitamin mixture (represented as colored triangles) in an attempt to figure out which kind of nutrients the mutated molds needed. The researchers then took mold from the mutant mold strain that had survived on a complete medium and added that mold to the supplemented minimal media. They found that in some cases the mutated mold grew on media supplemented only with vitamins but not on media supplemented only with amino acids. (5) To discover which vitamins the mutant molds needed, Beadle and Tatum used several tubes with the minimal media, supplementing each one with a different vitamin, and then they attempted to grow the mutant mold in each tube. They found that different mutant strains of the mold grew only on media supplemented with different kinds of vitamins, for instance vitamin B6 for one strain, and vitamin B1 for another. In experiments not pictured, Beadle and Tatum found in step (4) that other strains of mutant mold grew on minimal media supplemented only with amino acids but not on minimal media supplemented only with vitamins. When they repeated step (5) on those strains and with specific kinds of amino acids in the different test tubes, they found that the some mutated mold strains grew on minimal media supplemented solely with one kind of amino acid, and others strains grew only on minimal media supplemented with other kinds of amino acids. For both the vitamins and amino acid cases, Beadle and Tatum concluded that the X-rays had mutated different genes in Neurospora, resulting in different mutant strains of Neurospora cells. In a cell of a given strain, the X-rays had changed the gene normally responsible for producing an enzyme that catalyzed a vitamin or an amino acid. As a result, the Neurospora cell could no longer produce that enzyme, and thus couldn't catalyze a specific nutrient.