NCTM 2016: Modeling your way to understanding with Realistic Mathematics Education

Raymond Johnson and I facilitated a session on RME for high school teachers at NCTM 2016.

In this session we discussed the unique way that "modeling" is conceptualized in RME (as organizing rather than translating), and we  exemplified the RME approach by engaging participants in an activity sequence that guides students to reinvent formal operations with quadratics and polynomials.

Download the slides

NCTM 2016 research session: The intertwinement of activity and artifacts in Realistic Mathematics Education

In RME, mathematics is understood as both an activity and as the product of that activity. In other words, it's a unidirectional relationship: activity produces products. Often, the activity is understood as private, mental activity, and the product is understood as mental objects:

In this talk, I argued for a cultural perspective, in which activity produces artifacts, but also, those artifacts mediate and transform activity. The upshot is that activity and artifacts become intertwined:

From this perspective, mathematics artifacts are not private mental objects, but rather they are things in the world. They are durable and shared, and they exist across people and time (within a social group). Furthermore, activity is not private mental activity, but rather it is the accomplishment of a system composed of both persons and artifacts.

As I detail in the paper, adopting such a perspective resolves some internal tensions in RME. It also has implications for the activity principle, reality principle, and interaction principle. Finally, it leads to a new principle: the producer principle, which states that people are produced as particular kinds of people as they engage in activity with artifacts:

To learn more, you can download the presentation and/or the paper. 

RME-5: Emergent modeling: From chains of signification to cascades of artifacts

I gave this talk at the Fifth International Conference on Realistic Mathematics Education in Boulder CO.

It's a somewhat theoretical talk.

Abstract:

Emergent modeling is a key design principle in RME. The process of emergent modeling is often described using a construct from semiotics called a chain of signification. I show that chains of signification are inadequate to describe both the process and product of emergent modeling. To overcome these inadequacies, I introduce a new construct called the cascade of artifacts. I conclude with implications for research and design.

In English: 

Often, in RME, we see the process of emergent modeling as a hierarchical process in which progressively more formal versions of a model emerge in linear sequence. This is described as creating a "chain of signification:"

In this talk, I argued that the construct of a chain of signification is an inadequate representation of both the process and product of emergent modeling.

• As a representation of the process of emergent modeling, a chain of signification is inadequate because it requires a strict one-to-one mapping between signifier and signified. 
• As a representation of the product of emergent modeling (viz. the students’ mathematical worlds), a chain of signification is inadequate because it implies that the mathematical world is hierarchical and siloed. 

Given the above shortcomings, I argued that emerged modeling is a process where students create new models, tools, and strategies as coordinated assemblies of existing mathematical objects. This happens as students engage in mathematical activity that is not accessible using their existing mathematics and social interaction.

The metaphor shifts: from a chain of signification to a cascade of artifacts:

Visual representations of the cascade, like the one above, give researchers and teachers a representation of both the process of emergent modeling (how objects get assembled and coordinated to make new objects), and the product (the mathematical world). With respect to the latter, the cascade shows how the mathematical world is relational and web-like, and not hierarchal and siloed.

To learn more, download the presentation, and see the posts on slope (NCTM 2014, CCTM 2014) which exemplify the use of the cascade of artifacts.

RME-5: Opening plenary: An orientation to Realistic Mathematics Education

My colleagues and I from the Freudenthal Institute US gave this opening plenary at the Fifth International Conference on Realistic Mathematics Education. If you're wondering what RME is all about, look no further!

CTR 2015: Using models to do and learn math

At the Courage to Risk 2015 conference, Mark Semmler, Cindy Ritter and I gave three talks about how models help students do and learn mathematics. CTR is a conference for special education professionals. These presentations are designed to be practical, and they link research in special education to research in math education to real student work and classroom ideas. Each presentation discusses a specific model, and then shows how that model evolves from elementary to middle to high school, with examples from each level.

Click on the images below to download the slides

  

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)