Getting Your Hands Dirty in Integral Calculus

The landscapes of many elementary, middle, and high school math classrooms have undergone major transformations over the last half-century, moving from drill-and-skill work to more conceptual reasoning and hands-on manipulative work. However, if you look at a college level calculus class you are likely to find the main difference is the professor now has a whiteboard marker in hand rather than a piece of chalk. It is possible that some student work may be done on the computer, but much of it contains the same type of repetitive skill building problems. This should seem strange given the advancements in technology that allow more freedom than ever to build connections between different representations of a concept. Several class activities have been developed using a combination of approaches, depending on the topic. All activities use student note outlines that are either done in a whole group interactive-lecture approach, or in a group work discovery-based approach. Some of the activities use interactive graphs designed on desmos.com and others use physical models that have been designed in OpenSCAD and 3D-printed for students to use in class. Tactile objects were developed because they should provide an advantage to students by enabling them to physically interact with the concepts being taught, deepening their involvement with the material, and providing more stimuli for the brain to encode the learning experience. Web-based activities were developed because the topics involved needed substantial changes in graphical representations (i.e. limits with Riemann Sums). Topics covered in the activities include Riemann Sums, Accumulation, Center of Mass, Volumes of Revolution (Discs, Washers, and Shells), and Volumes of Similar Cross-section. Assessment techniques for these topics include online homework, exams, and online concept questions with an explanation response area. These concept questions are intended to measure students’ ability to use multiple representations in order to answer the question, and are not generally computational in nature. Students are also given surveys to rate the overall activities as well as finer grained survey questions to try and elicit student thoughts on certain aspects of the models, websites, and activity sheets. We will report on student responses to the activity surveys, looking for common themes in students’ thoughts toward specific attributes of the activities. We will also compare relevant exam question responses and online concept question results, including common themes present or absent in student reasoning. Introduction We don’t pay attention to boring things. Stimulate more of the senses. We are powerful and natural explorers. These are just a few of the dozen statements describing how our brains work in John Medina’s book “Brain Rules” [1]. These also provide insight into the fundamental reasons for our Integral Calculus reforms developed over the last year. We don’t pay attention to boring things. Math courses are notorious for being boring, lectureheavy periods of time that are unavoidable since they are necessary for earning a degree. Over the years, several interventions have described ways to transform math classrooms into a place where students are actively engaged and not drooling on their desks or intently gazing at the back of their eyelids. Flipped Classrooms [2], Inquiry Based Learning [3], Interactive Lectures with Clicker Questions or visualization software [4] [5], and Active Learning [6] [7] are just some of the more recent interventions a select number of math faculty are trying. Watson, et al., provide an excellent overview of calculus course interventions that have been reported through ASEE from 2005 to 2018 [8]. Most of these interventions are designed to help improve student success or retention in future engineering courses. Some of these interventions include direct changes within calculus courses, while others add extra experiences outside of the usual classroom period. A key feature of our intervention in Calculus II is the flexibility of implementation. Changes to courses were two-fold. The first is physical models or websites that enhance student interaction with the concepts being explored. The second is student note/activity sheets that can be used in a variety of ways, depending on the topic and how the instructor wants to structure the course. These can be used as guides for students to fill out independently in groups, or as guided lecture notes to be completed as a class. These will be described in more detail in the next section, with samples provided and a link to the rest of the guided notes. We will also explore student reactions to the interactive websites, models, and activity sheets. Stimulate more of the senses. There is a reason that so many elementary school classrooms have physical manipulatives to help teach math. Not only do they counteract boredom, but they also give a concrete representation of a more abstract concept. The student that uses these tactile tools is in a better position to encode the learning experience in a deeper way. Yet we rarely find tactile manipulatives in Calculus courses. Some of this may be due to a lack of availability of models at this level, and some may be due to a lack of interest from faculty. We are powerful and natural explorers. How our brains learn and encode things into memory is complex, and often different in various individuals [1]. To complicate matters further, the way we are presented information can also be a help or a hindrance. Graphs, pictures, diagrams, equations, and language are all important aspects of mathematical and general STEM ideas. Our ability to “see through” the surface features of a representation to the underlying concept has been referred to as representational competence (RC) in chemistry education literature [9]. The construct of RC has also been applied in education research in other science fields [10] [11], engineering [12] [13], and mathematics [14]. Experts demonstrate the ability to freely move between different representations of an idea, knowing which ones are most useful, and which do not provide relevant insights. Novices can sometimes translate from one representation to another, but they typically have trouble integrating the information from these various sources into a larger cohesive framework. Studies in chemistry education demonstrate the promise of tactile models and computer simulations to help scaffold students’ development of RC [15], [16]. We have leveraged the relatively recent boom in 3D printing technology to create physical manipulatives to use in the integral calculus classroom. These tactile tools are designed to be used in conjunction with activity sheets, and help provide a concrete focal point for students to “get their hands dirty” while investigating the concepts being learned. Our intervention aims to help students become more aware of the different representations in ideas through the use of the activity sheets, technology, physical models, and different types of assessment questions. Several “concept check” questions were given to students as a means of testing their conceptual understanding and representational competence, rather than just rote procedural calculations. Some of these questions were given online as a pre-test “readiness check”, and others were given on the exams. Questions typically consisted of a multiple choice portion, followed by an explanation portion where students justified their responses. Topics covered in the activities include Riemann Sums, Accumulation, Center of Mass, Volumes of Revolution (Discs, Washers, and Shells), and Volumes of Similar Cross-section. Assessment techniques for these topics include online homework, exams, and online concept questions with an explanation response area. In this paper, we provide examples of activities and assessment data primarily from the Volumes of Revolution topic that is in integral calculus, mainly to keep things concise and thematic. Links to activity sheets and associated .stl files for 3D printing are available here: https://graspthemath.wordpress.com/integral-calculus/ [17]. Although all of the examples in this paper are from integral calculus, a similar approach of using activity sheets, physical manipulatives, and concept check questions could be applied to other STEM courses. Guided Notes with Electronic Aides Some topics lend themselves to visual representations that involve real time animations or manipulations to change specific aspects of the image. For instance, Figure 1 shows a Desmos page created for the purpose of investigating Riemann Sums. When investigating Riemann sums in terms of area under a curve, we often estimate the area using rectangles with either righthanded sums, left-handed sums, or midpoint sums, and the limit of these sums as the number of rectangles increases. It is quite time consuming to do many of these by hand, and yet it is beneficial for students to physically draw a sample of these by hand to get a sense of how they work and how the graphs connect to more symbolic representations. Creating graphical representations through Desmos or other software can provide a way to quickly manipulate the different aspects of the rectangles typically used to explain Riemann sums. Although the “area under a curve” interpretation of an integral is the most commonly used graphical representation, it gives only limited insight into the Fundamental Theorem of Calculus and the Net Change Theorem. An accumulation approach to Riemann sums is another graphical approach that more closely ties together the ideas of sums and antiderivatives, and gives a much richer insight into these theorems. Figure 2 shows an interactive Desmos webpage for an accumulation approach to Riemann sums, which is often left out of calculus textbooks. Figure 2: Interactive Desmos site for interpreting Riemann Sums as accumulation. This interactive graph is ava