# Tangible Math

### Case Studies

The case studies presented in this section are from three high school classrooms: (1) a Principles of Engineering class designing and building ballistics devices, (2) a Digital Electronics class building digital circuits for a secure voting booth, and (3) an advanced Geometry class exploring properties of angles inscribed in circles using dynamic geometry software. The three cases illustrate how teachers and students manage the process of threading mathematical concepts across ecological contexts. Many aspects of curriculum and instruction observed across these cases of science, technology, engineering and mathematics (STEM) education exist in order to engage relation-production mechanisms to advance students’ perceptions of locally invariant properties (i.e. the stability of mathematical knowledge across contexts produced and enforced “locally” to the agents, time and situation), so they serve as a common thread of coherence through STEM activities.

Case #1: Theta in Symbols, Paper and Wood:
A Ballistic Device Design in a Precollege Engineering Classroom on Mechanical Engineering

The first case is of a high school Principles of Engineering classroom, part of the Project Lead the Way curriculum (www.pltw.org). The case study focuses on a portion of a multiday lesson on ballistics, where the students use mathematics (trigonometry and algebra) and physics (kinematics) to calculate the distance a projectile will travel under certain conditions. On Day One of the lesson, the students learned the mathematics and physics of calculating projectile motion, and the teacher highlighted for them the angle of ascent of the projectile - labeled theta. Theta was a key variable that students had to represent both in their sketches, and ultimately in the materials they assembled into a catapult, trebuchet, gun or other ballistic device of their own choosing and design. If these devices properly instantiate theta - that is, permit the adjustment of the angle of release while holding the other influential variables (e.g., initial velocity) constant - students will be able to predict the distance that the projectile will travel. Throughout the sequence of the lesson, knowledge of theta is inscribed in different modal engagements: symbols and diagrams on the whilte board during the initial lecture, paper and pencil during small group design meetings, and a collection of materials fashioned and assembled into a projectile device, which was ultimately manipulated and evaluated.

Case #2: Logic Enacted Through Boolean Algebra, Simulation and Silicone:
Designing a Digital Voting Booth

In a Digital Electronics class from the Project Lead the Way curriculum, high school students attempt to realize a set of logical relations for designing a voting booth privacy monitoring system. An effective monitoring circuit is indicated by two outputs, a green light-emitting diode (LED) that id activated whenever a voting booth is available for use, and a red LED that lights up whenever privacy is at risk and entry is being denied. The process unfolds sequentially across the following modal forms: a truth table with entries composed of 1’s and 0’s accounting for all of the possible states that the circuit (voting booth) can occupy; a spatial Karnaugh map (K-map); any of a class of Boolean algebraic expressions; hand-drawn Automated Optical Inspection (AOI) circuits; computer generated SIM diagrams; and a working electronic circuit made of a “bread board,” integrated circuits, resistors, and capacitors, wires, a power source and LEDs. Students use truth tables and K-maps to generate Boolean Algebraic expressions, which are then represented in hand-drawn AOI circuits and simulated using computer software that constructs virtual digital circuits and guides the wiring and testing of a working electronic circuit for monitoring occupancy in the voting booths.

Case #3: Circles in Action:
Proof in an Advanced High School Geometry Class

The final case is associated with a geometry lesson about properties of circles and their associated theorems. The observed lesson alternated between two contexts: the computer lab, where there was access to an interactive dynamic geometry software (DGS) environment; and a teacher-led discussion in the original classroom. In the lab, students worked together in a setting where each student had access to the DGS program Geometer’s Sketchpad (GSP). The direct manipulation interface of GSP allows students to experiment freely and have direct interactions with geometrical objects and their spatial relations on a computer screen. GSP also provides a tool for students to create, validate, and refine particular conjectures through their exploration and visualization of geometry. The teacher acted as a facilitator who guided and encouraged students to discover and construct knowledge for themselves. In the classroom context, free from the computer environment, the geometry teacher initiated conversations about the content, analyzed/evaluated student responses, and provided feedback that required more elaborate responses from students than had been possible during the group work in the computer lab. The classroom context often served as a way to summarize or formally state the geometric concepts that students discovered working in the dynamic geometry software environment.

### Modality Transitions Unfolding from the Case Studies

The analysis of the engineering and geometry classrooms indicates three important ways in which modality transitions unfold in classroom activities. First, participants make an ecological shift, which involves the reorientation of the activity context to include different modalities. Second, participants use projection over time to connect current modal engagements to past or future ecological contexts. Finally, participants engage in coordination, which is the juxtaposition and linking of co-occurring modal engagements.

Ecological Shift

At its surface, an ecological shift can simply appear as a reorientation of the activity; for example, when the geometry teacher called the class to stop their computer lab work and focus attention on the chalkboard, or when the digital electronics teacher told all students to stop their lab work and witness a conversation about checking a circuit for accuracy between the teacher and a student. Alternatively, the shift may be more dramatic and total, as when the engineering teachers took their students from the classroom to the wood shop or the digital lab, which altered the norms of proper (i.e., safe) conduct, the tools at their disposal, the ambient sounds, and the participant structures, while also placing what had been planned for the future into the present task of implementing the proposed designs. Thus, ecological shifts are not mere changes in context - they are transitions that can potentially alter all of the modal engagements with the mathematics.

Projection over Time

Projection builds a bridge between modal engagements of the present to past or future ecological contexts. Past projections link across an ecological shift that has already occurred, while future projections anticipate a coming shift in the instructional ecology (one that may be part of the curricular design). Teachers and students used the language and gestures of projection, along with representations, objects, and the environment itself, to both reflect upon a history of a concept as it unfolded in their classroom, and to plan for future manifestations of the concept in different ecological contexts. For instance, the dialogues and gestures produced during a discussion between the engineering teacher and a group of students on the second day of the ballistics lesson served to project toward a future context where the students will use their sketch to guide construction, and many aspects projected back to the past kinematics laws and mathematical relations that were presented in a prior lecture. Projections can take the form of brief utterances; for example, when the geometry teacher referenced the activities of the previous day’s lab. They can also be much more protracted, as when the digital electronics teacher spent the entire first day of the lesson planning the subsequent days’ lab work. Ecological shifts make it challenging for participants to preserve the coherence and continuity of mathematical ideas across modal engagements. Projections are an effort to identify connections over time and help students to establish that sense of coherence.

Coordination

The third modal transition process, coordination, is the task of weaving together the knowledge and activities of co-occurring modal engagements. When speakers integrate across modalities and time simultaneously—when they make a connection from a present device to an absent equation—we consider this as both coordination and projection. We also consider the possibility of intra-modal action as occurring within a single modality that is uncoordinated with any other modalities. One example of coordination is when the digital electronics teacher systematically connected entries in the truth table with the state of the digital circuit on the breadboard in conjunction with a student’s speech and gesture. The teacher models how the truth table maps directly to the circuit, running his finger from one row to the other. The student elicited coordination between the truth table and the state of the circuit by adjusting the input switches on the breadboard as he reported on the state of the inputs/output and moved his finger to each corresponding, successive row along the truth table. The student also glanced repeatedly between the truth table and the circuit. Another example is when the geometry teacher gestured over the computer screen to iconically simulate a key insight to the problem of why opposite angles in an inscribed quadrilateral are supplementary. This demonstrates how a social action like gesturing can be co-presented and juxtaposed with a computer-generated representation, and how coordinating these modal engagements manifests mathematical concepts.