CCTM 2014: Beyond Rise over Run: Activities to invent and connect slope's five faces

Slope has five—count 'em, five—faces. Students shouldn’t focus on just one or two, and in this session, neither will we! Instead, we'll explore realistic and meaningful activities and a learning progression designed to help students invent and make connections between all of slope’s five faces. 
(This is a shortened version of the session I facilitated at NCTM 2014.) 

Below are links to the handout for the session, a research paper that describes the approach in more detail, and a link to the complete unit for teachers and others to use. Please download, modify, and use the tasks with your students!

• Session handout (1 MB PDF file)
• Session presentation (22 MB PDF file)
• Research paper (2 MB PDF file)
• Download the complete unit to use in your classroom (34 MB .zip file; MS Word, MS PowerPoint, and a PDF reader are required to view and edit files)

NCTM 2014: Beyond rise over run!

Research talk and gallery workshop for teachers

   

Again, similar titles to my previous presentations, but again, a lot has changed in my thinking around our study of slope in 2012. At NCTM 2014, I presented the updates in a research talk and a gallery workshop for teachers. Here are all of the resources for those presentations, as well as a link to the complete unit for teachers and others to use. Please download, modify, and use the tasks with your students!

Gallery workshop for teachers

• Handout (1 MB PDF file)
• Slides (16 MB PDF file)
• Download the complete unit to use in your classroom (34 MB .zip file; MS Word, MS PowerPoint, and a PDF reader are required to view and edit files)

Research session talk:

Abstract:

I present a local instructional theory for slope that emerged during a design experiment in a high-school Algebra I classroom. In the design experiment, students explored situations related to making predictions. As students engaged with these situations, they reinvented and made-meaningful multiple sub-constructs of slope. I show that this process involved the assemblage and coordination of mathematical artifacts, and I introduce the notion of a cascade of artifacts to describe this process. I suggest that artifacts are inextricably bound with activity, and I discuss the nature of the classroom activities that promoted the development of the cascade of artifacts.

Resources:

• Paper (2 MB PDF file)
• Slides (20 MB PDF file)

RME 4: Beyond rise over run: Contexts, representations, and a learning trajectory for slope.

I was excited to facilitate this workshop at RME 4 in Boulder on September 29, 2013. Even thought the title, abstract, and slide design is very similar to the ICME talk, a lot has changed in my thinking around our study of slope in 2012. In this workshop we looked deeply at a couple of tasks and explored how the context and representations used in the tasks mediated the reinvention process. 

Below the abstract, there are links to the complete unit in editable form for teachers and others to use. Please download, modify, and use the tasks with your students!

Abstract:

Despite its foundational nature in secondary and post-secondary mathematics, student understanding of slope is often formulaic and underdeveloped. To explore how students learn slope in a more robust way, we designed a curriculum for slope in which students mathematize situations involving rates of change.  We designed the curriculum using RME principles, and tested and refined it in a design experiment. In this workshop, participants will engage in activities from the curriculum and I will discuss our design process. I will also discuss key findings, including how contexts and representations mediated student learning of slope.

Resources:

• Download the complete unit to use in your classroom (34 MB .zip file; MS Word, MS PowerPoint, and a PDF reader are required to view and edit files)
• Download our presentation (8 MB PowerPoint file)

ICME-12: Beyond "Rise over Run"

ICME-12 Workshop and sharing group on slope and linear functions

Michael Matassa and I were honored to conduct a workshop and sharing group at the ICME-12 conference in Seoul, South Korea on July 11, 2012.  During the workshop, participants explored tasks that Michael and I created during a design experiment focused on helping students develop a robust understanding of slope and linear functions.

We are happy to make our presentation, paper, and complete unit available for download.  We welcome your feedback (via email or in comments), and we hope that you use and adapt the activities with your students!

Abstract:

Student understanding of slope and rate of change is often formulaic and underdeveloped. This presents problems for students in secondary and post-secondary mathematics where slope and rate of change are key foundational concepts. To study how students develop robust understandings of slope and rate of change, we conducted a design experiment in a U.S. high school Algebra I classroom that focused on developing versatile and adaptable knowledge of slope using rate of change as a foundational concept for slope. In this workshop, participants will contribute to an international perspective on the teaching of slope, engage in key activities that were used in the design experiment, and explore student work generated from these activities.

Resources:

• Download the complete unit to use in your classroom (34 MB .zip file; MS Word, MS PowerPoint, and a PDF reader are required to view and edit files)
• Download our presentation (8 MB PowerPoint file)
• Read our paper (PDF)

RME 2011 presentation and resources

At the RME 3 conference, we gave a presentation on a quadratic functions unit that we designed using principles of RME.

Title: "Length times width equals area" and "Line times line equals parabola": Incorporating two RME models into a cohesive learning trajectory for quadratic functions.

Abstract: RME researchers have discussed two alternatives to the projectile motion model for
quadratic functions: (1) “line times line equals parabola” (Kooij, 2000), and (2) the area
model (Drijvers et al., 2010). Our learning trajectory incorporates and connects these
models in a cohesive unit that begins with a motivating contextual problem, guides students
to construct both of the above models, and concludes with formal algebraic representations.
We have used this unit for the past two years in our Algebra I courses. During this time, we
have collected a body of evidence that supports the approach: 1) Assessment results
suggest that students learn quadratic functions at or above the level expected in a typical
algebra course; 2) Student work suggests that learning multiple models leads to deeper
understanding and flexible problem solving; 3) Student feedback suggests that the models
help students solve problems and understand formal mathematics. Participants will receive
our complete unit in electronic form. Furthermore, we will discuss our lessons learned, and
avenues for extensions and future research.

Resources:

• View our presentation online
• Download our presentation (PowerPoint file)
• Download the complete unit to use in your classroom (98 MB .zip file, MS Word and PDF reader required to view and edit files)
• Links to the applets we demonstrated: 

• Applet home page
• The "geometric algebra 2D" applet that we demonstrated