Andrzej Sokolowski

Modeling with Exponential Decay Function

Understanding rates of change sets the foundations for differential equations that are central to modeling dynamic phenomena in science and engineering. While modeling with a constant rate of change is well understood, modeling an exponential change requires a more detailed approach due to a diversity of its computing. An exponential model can be characterized by a constant ratio of change of the quantity, a constant percent rate, decay or growth factor, decay or growth rate, and so forth. The purpose of this STEM activity was to create a STEM environment that would enable the learners to discover a constant ratio as a departing block to construct an exponential model. An experiment of investigating maximum heights of a bouncing ball was used as a scientific context. This phenomenon was selected due to providing observable and measurable data. A pretest analysis was used to identify students’ weaknesses, and the posttest analysis has been used to discuss students’ changes of perception on modeling exponential behavior. The STEM context has disclosed that the commonly applied term decay denoting a decreasing exponential function did not adequately describe the virtue of decaying processes. Conclusions and suggestions for further studies are discussed.

STEM Education: A Platform for Multidisciplinary Learning

This chapter summarizes general purposes, the learning settings, and the outcomes of using the STEM as a platform for multidisciplinary learning. By encompassing several disciplines, mathematics, physics, chemistry, biology, technology, and engineering, exercising STEM activities posit specific challenges. These challenges are especially visible in high school where students learn contents of STEM subjects in uncorrelated manners. While exercising multidisciplinary STEM activities during extra designed instructional units would be the most efficient, this approach might be problematic to put in practice. Therefore, alternative routes of exercising STEM learning experiences are sought. In this chapter, a framework for an alternative route is suggested and its general theoretical underpinnings discussed. Attention is given to research findings on ways of exercising scientific inquiry and mathematical reasoning in STEM practice. These ideas will also be further discussed in the next chapters.

Integrating Mathematics and Science Within STEM Paradigm

This chapter discusses research findings on integrating mathematics and science within STEM platform. It summarizes the premises that benefit the learners as seen from the perspective of mathematics and science content knowledge and highlights the areas that need more attention. In the traditional curriculum, science provides the contexts and mathematics and offers the tools to quantify the contexts. While students do use the tools of mathematics to solve problems in science and use scientific contexts in mathematics, research shows that the methods applied are often disconnected. STEM activities can be seen as offering opportunities to create a new, unique knowledge rooted in merging these two disciplines into an integrated inquiry. In the attempt to merge these two learning disciplines in such a way, this chapter also provides an analysis of the primary phases of scientific investigation and its possible fit to STEM mathematics activities. A draft of the general theoretical framework on merging scientific inquiry with mathematical reasoning and its relation to STEM competencies is also brought to the reader’s attention in this chapter.

Formulating Conceptual Framework for Multidisciplinary STEM Modeling

This chapter builds on the accumulated research findings of previous chapters and proposes a theoretical framework to design multidisciplinary STEM activities whose main learning goal is to develop students’ mathematical and scientific reasoning skills. Research showed that mathematics in the STEM is underrepresented; therefore, enhancing mathematical reasoning was given a priority. Also, the theoretical framework aims to have students mathematize phenomena and use the structures to support problem-solving techniques. This chapter explains the structure of the framework (Fig. 6.1) along with how the links between mathematical and scientific concepts will mediate. While the framework is proposed for mathematics classes, it can also be used to design activities in other STEM disciplines. The framework was used to develop activities that are described in Chaps. 7, 8, 9, and 10.

Survey of the Field of Empirical Research on Scientific Methods in STEM

Research shows that being versed in scientific modeling is a precursor to succeed in engineering modeling and might be a factor attracting students to engineering. This finding suggests that STEM activities that develop the skills of modeling should not only focus on students’ mathematical and scientific reasoning skills but also provide an environment where students would feel comfortable and encouraged to continue these enterprises in their college and professional careers. One of the main obstacles in adopting inquiry-based learning projects in mathematics is the gap between problem-solving in mathematics and inquiry in science. It appeared worthy to search the literature on STEM education to determine what the research findings in this domain are. The synthesis of the survey findings will support multidisciplinary STEM framework developed in Chap. 6.

Investigating Function Extreme Value: Case of Optimization

Optimization is a central process in engineering designs. Its core idea is rooted in applying mathematics and calculus techniques to finding a maximum or minimum value of a function, often of several variables, subject to a set of constraints. This study investigated how calculus students formulated and analyzed functions that led them to find dimensions of a rectangle that produced a maximum area enclosed by a string of a fixed length. A group of 23 high school calculus students was immersed in an activity that involved hypothesizing possible outputs, direct measurements, data collecting, model formulating, and optimizing the model values. While typical textbook problems on optimization focus students’ attention on determining unique dimensions that maximize an enclosed area, this activity extended the exploratory part and underpinned not only the behavior of the function of interest but also the behavior of the constraint functions. This phase helped to disclose potential effects of the constraint functions on absolute maximum or minimum. Posttest analysis revealed that STEM activity not only deepened and helped understanding of underlying optimization processes but also challenged students’ mathematical reasoning skills regarded the interpretation of the behavior of the derivative function.

Modeling in STEM

Knowledge in sciences is traditionally derived inductively, whereas in mathematics and engineering it is derived deductively. One of the purposes of STEM projects is to blend science with mathematics and engineering into a coherent learning experience. While this book is an attempt to develop and put in practice a theoretical framework for exercising multidisciplinary STEM learning experiences, a need to discuss a strategy that would link all learning methods of the component STEM subjects emerged. Research shows that modeling is the most common approach exercised in all of these disciplines. However, the phases of modeling in mathematics might not parallel with the phases of modeling in science. To design a learning environment that would support multidisciplinary learning, a need for a modeling cycle that would integrate the features of all types of STEM modeling appeared as a necessary step before the multidisciplinary projects could be designed. The purpose of this chapter is to synthesize the characteristic features of currently applied modeling cycles in each of the STEM disciplines and select these features that can support STEM learning objectives exemplified in this book.

Teaching and Learning Representations in STEM

Context in the STEM is a critical factor in learning. Context can be delivered in various ways depending on the form of the final product. Research shows that representations are very effective in conveying knowledge because they help learners visualize abstract ideas and diversify the forms of information. This chapter discusses the effects of representations on learning and attempts to answer a question why representations support knowledge acquisition and retention. Representations can function in two primary capacities: as provided by the instructor or produced by the learners. Being able to gain understanding using representations and constructing representations is one of the most critical factors in supporting knowledge retention. What are the features of well-designed representations and how they affect knowledge processing are other questions that this chapter also attempts to answer.

Applying Function Transformations to Model Dynamic Systems

Being able to transform functions due to given conditions is an essential math skill. A preliminary survey of research on teaching transformations has shown that majority of assessment items gravitated toward predicting new graphs due to assigned shifts, compressions, or reflections. Real-life applications of these concepts were rarely discussed. The activity focuses on identifying possible transformations of trajectories of projected objects and constructing new functions. STEM context was provided in the form of a physics simulation Projectile Motion that is available for free at http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html. Parabolic trajectories were generated by varying the parameters of the object’s initial velocity and its relative position. A group (N = 25) of pre-calculus students mathematized a parent-simulated trajectory and then used it to formulate algebraic functions of other trajectories. It was hypothesized that situating the concept of function transformation in an environment that related to students’ prior experience would enhance the purpose of formulating algebraic representations and explicate on applicability of transformation. Analysis of posttest results revealed that situating the learning in realistic contexts brought another dimension to understanding function transformations and infused a deeper understanding of the techniques of constructing transformed functions.

Exploring Function Continuity in Context

Although function analysis is widely applied in science, there are some areas, like function limits or continuity that are underrepresented. The purpose of this study is to model motion and support this process by applying function continuity. Students will model the principles of continuity by formulating position functions for objects moving along a horizontal path with multiple rules. The scientific context will be supplied by an interactive simulation called Walking Man that is available for free at http://phet.colorado.edu/en/simulation/moving-man. This simulation allows designing a movement that can be mathematized using piecewise polynomial functions. Function continuity and sided limits will be used as tools to support the construction of these functions. The activity was conducted with a group of 20 calculus students. It was hypothesized that by applying the principles in context, the students would realize that function continuity is a critical condition that functions representing motion must satisfy. Posttest results supported the hypothesis.