Forging Dynamic Mathematical Connections

This site supports the NSF project on Technology-Rich Units for Future Secondary Teachers: Forging Mathematical Connections Between Geometry and Functions, which is part of the NSF's program on Improving Undergraduate STEM Education (IUSE).

Work in Progress

Our IUSE project team has already made considerable progress in designing and implementing improved technology for lesson delivery, using a single cleanly-organized lesson web page that provides easy access to the resources that support each lesson. The resulting lesson web-page template also serves as a guide to the various features of the lessons we are working on. Those features include:

- The lesson's table of contents is in full view.
- A click on any of the lesson’s activity titles shows or hides that activity.
- Activities engage students in creating and manipulating the mathematical objects of study.
- Many of the websketches are in the form of challenging mathematical games.
- The dynamic tools in each websketch are customized for the purpose of that sketch.
- Hint videos provide students with addiitional scaffolding.
- Performance-based assessment games are built into the lesson.
- Lesson objectives are clearly spelled out and linked to the relevant part of the lesson.
- The teacher can optionally view the teacher notes directly on the lesson webpage.
- Each lesson pays explicit attention to important pedagogical issues: encouraging student collaboration, maintaining cognitive demand, guiding productive discussions, leveraging progressive abstraction, and above all having students enact the mathematics they are learning.

In tandem with our work on the lesson template, we’ve created a prototype lesson page (The Reflect Family) that showcases these features.

Worksheets and websketches for the first unit below have already been drafted, piloted in classrooms, and revised. Some teacher support materials are already available, and we are adding more as we bring these earlier draft lessons up to the level of our prototype.

The third and fourth units are in development and undergoing testing with students.

Unit 1: Understand Geometric Transformations as Functions

- Identify Functions: Students distinguish independent from dependent variables, describe the behavior of dependent variables as they drag the independent variable, distinguish between functions and non-functions, and arrive at their own definition of
*function.* - Reflect Family: Students construct, investigate, and formalize the reflect family, accomplishing objectives similar to those described above for the Rotate Family (which is more fully developed).
- Identify Families: Students drag variables to solve “Which one is different” puzzles involving four functions, three of which are from the same family. They begin by comparing the shapes of traces left by the variables, analyze the relationship between the traced shapes and the rate of change of the variables, and learn to distinguish functions based on rate of change. Students have their first encounter with function notation, as a way of indicating which dependent variable is related to which independent variable.
- Rotate Family: Students create rotate functions, formulate informal descriptions of the rate of change that distinguishes rotate functions, use function notation, and restrict the domain of a rotate function to observe the corresponding range. Students perform physical and virtual function dances, they solve puzzles requiring them to rotate one shape to superpose it upon another congruent shape, and they compose two rotations and figure out how to adjust the second rotation to turn the composition into an identity function. Students play a video game in which they're given three of the four elements of a rotate function (independent variable, center of rotation, angle of rotation, dependent variable), and their job is to find the value or location of the missing element.
- Dilate, Translate, and Glide Reflect Families: (Not yet available) Students work in small groups to construct, analyze, and formalize their understanding of these function families. Each group specializes in one family, and they share their results (including a choreographed dance) with other groups.

Unit 2: Experiment with Composition and Congruence

- Mystery Transformations: The student chooses a triangle transformation tool and adjusts the transformation to superpose one triangle on another. Should this also include a sketch in which the student chooses a point transformation tool and matches one shape polygon to another by restricting the independent variable to the polygonal domain?
- Mirror, Mirror: The student uses the Reflect tool and adjusts the result to superpose one triangle on another. The five pages of the sketch include triangles that have been reflected, rotated, dilated, translated, and glide-reflected. The sketch includes the tools needed to construct mirrors for each problem instead of adjusting them. Should this web page include two sketches, one with only the Reflect tool for students to adjust, and one with the additional tools asking students to construct the necessary mirror?
- The Segment Congruence Theorem: The student first adjusts one or two mirrors to superpose the segments, and then constructs the mirrors that put the segments into superposition. As an extension, students are challenged to construct a single rotation that superposes two general segments, and then a single glide reflection that superposes two general segments.
- The Side-Side-Side Theorem: The student's first task is to use SSS to construct triangles: given three segments of various lengths, construct a triangle with side lengths that match the given segments, clearly establishing what we mean by SSS. In the second task, the student constructs one, two, or three mirrors to superpose two triangles that have their three sides related by SSS. The four pages of the sketch have four possible cases (corresponding to reflection, translation, rotation, and glide reflection). In a third optional task, the student is challenged to construct a single transformation for each case.
- Explore Similarity: In the first activity students adjust some combination of dilation and reflection to superpose one triangle on another. In the second activity, students dilate triangles and investigate the similarities and differences between the pre-image triangle and its image.
- SAS, SSA, and ASA: (Not yet available) Students work either sequentially or in small groups to solve three tasks, each of the following form: Are triangles connected by ___ always congruent? If so, prove it by construction; if not, construct a counter-example. Fill in the blank with SAS, SSA, or ASA to describe each task.

Unit 3: Formalize Congruence and Similarity

- Prove Segment Congruence: The student constructs transformations that superpose one segment on another of equal length so that the construction remains valid no matter how the segments are repositioned. The student then uses the construction to formulate a proof of the Segment Congruence Theorem.
- Prove the SSS Theorem: After reviewing the definition of reflection and the Perpendicular Bisector Theorem, the student transforms one of two triangles having the same side lengths. The transformation superposes the image of one triangle on the other triangle, and the construction forms the basis for the student's proof.

Unit 4: Connect Geometry and Algebra Through Functions

- Flatland to NumberLand: In this lesson, you’ll restrict the domain of each geometric transformation family (reflect, rotate, translate, glide reflect, dilate) to a number line, and decide which of these families can be arranged to have their range on the same nummber line. For each family, you’ll consider how the numbers corresponding to the independent and dependent variables are related.
- Construct a Dynagraph: In this lesson, you'll compose two geometric functions, using one number line for the independent variable and putting the composition’s dependent variable on a separate parallel number line. Using parallel lines, and constructing a segment to connect the independent and dependent variables, will make it particularly easy to observe and analyze the relative motion of both variables!
- Dynagraphs in Algebra: In this lesson, students use dynagraphs without numbers to practice their observation skills, comparing the relative direction and speed of the independent and dependent variables. Students then use dynagraphs with numbers to relate these same speed-and-direction behaviors to functions expressed algebraically and to determine the input or output values designated by expressions using function notation.
- Construct a Cartesian Graph: In this lesson, you'll compose two geometric functions, using a horizontal number line for the independent variable and using rotation to put the dependent variable on a separate perpendicular number line. Using perpendicular lines will make it particularly easy to a trace a point that shows the relative motion of both variables!

Tools: The Web Sketchpad Tool Library and Sketch Viewer

The Web Sketchpad Tool Library and the WebSketch Viewer are important resources for both teachers and students. See these short introductory videos to learn what they can do and how they can be useful for you.

- WSP Tool Library: The Websketchpad Tool Library offers over 50 tools you can include with your websketch. You can create new sketches or modify existing sketches by adding or removing tools, adding or removing pages, and changing the size of the sketch window. The resulting sketch, saved as a .json document, can then be used on any desired WSP-enabled webpage.
- WebSketch Viewer: On this web page you can load many sketches, with all of them being completely live. Students can show off their work, students can compare different approaches to a problem, and teachers can use them to facilitate a class discussion.

Project Description

Forging Mathematical Connections Through the Geometry of Functions

This Engaged Student Learning Exploration and Design project seeks to capitalize on the power of Web Sketchpad technology-enabled curriculum units to deepen pre-service teachers’ knowledge of mathematics topics and rich connections among them. This project aims to create and test five web-based mathematics units for pre-service teachers that promote this highly geometric approach, cultivating in teachers a robust conception of the secondary mathematics courses they will teach.

**Project Goals**

The curriculum units we will develop and implement will build on foundations of cognitive science, technology, and pedagogy to achieve the following goals:

- strengthen the mathematical background of future mathematics teachers, using a geometric approach to functions as an overarching framework for technology-rich investigations of geometric transformations, analytic geometry, trigonometry, complex numbers, and rates of change with connections to calculus;
- lead with geometry rather than algebra, grounding teachers’ understanding in a visual, sensorimotor, and dynamic approach to mathematics throughout the units;
- capitalize on the ability of Web Sketchpad, a new web-based software funded in part by our prior DRK–12 NSF grant, to design one-of-a-kind mathematical tools tailored to the needs of each particular lesson;
- focus on student thinking and pedagogy through discussions of student interview videos and modeling the pedagogy that teachers will bring to their classrooms;
- construct measures aligned with each unit to assess what preservice teachers learn;
- solicit and utilize feedback from experts to make modification to the units and measures;
- design teacher support materials for implementation; and
- pilot test each unit and make revisions based on data collection.

**Intellectual Merit**

The intellectual merit of the project is to advance knowledge about how visual, kinesthetic technology-based instructional activities grounded in theories of embodiment can support pre-service teachers’ knowledge of school mathematics topics and their connections from an advanced perspective. Findings from our research will be shared with other mathematics educators to inform their preparation of teachers and our materials will be distributed to mathematics teacher educators at other universities that prepare secondary mathematics teachers.

**Broader Impacts**

Strong preparation of middle and high school mathematics teachers is needed to support future generations of students prepared to enter STEM fields. By assuring teachers are prepared to teach the rigorous mathematics content in middle and high school, we can better prepare students to pursue STEM careers.