Thursday, April 26, 2018

NCTM 2018: Building meaning into algebra equations

Raymond Johnson, David Webb, and I facilitated a session at NCTM 2018 in Washington DC.

In the session, we introduced two models for algebra equations, discussed how each model reveals a different structure, and how each model can be used develop meaningful strategies for algebraic manipulations. 

We also discussed how the models can be developed through progressive formalization of informal contexts.

Download the activity sequence for the arrow chain model


A number line is also a great model for algebra equations.

Download an activity sequence for the number line model

Friday, October 21, 2016

MEA/MFT 2016 - Beyond Rise over Run!

Slope is more than just “steepness” or “rise over run.” Slope has five—count 'em, five—faces. Students shouldn’t focus on just one or two, and in this session, neither will we! We'll explore a sequence of learning activities that guides students to invent and connections all of slope’s five faces through engagement in realistic and meaningful activity. The sequence is grounded in research literature, tested in classrooms, and aligned with the Montana Common Core.  
(This is a revised 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!

Thursday, May 26, 2016

Reinventing fractions and division as the are used in Algebra: The power of preformal productions


In this paper, Michael Matassa and I explore a problem of practice that we experienced as algebra teachers. Namely, we noticed that students in algebra often struggled with division when the quotient is not an integer (e.g., 7÷9). Furthermore, even though division is always represented as fractions in algebra courses, we noticed that students rarely, if ever, represented quotients as fractions. We therefore set out to explore student thinking around fractions and division. Subsequently, we engaged in a design study—oriented around RME design principles—to guide students to reinvent the relationship between fractions and division.

Our major finding in the study concerned the role of so-called "preformal" mathematical productions. These are: 
Mathematical models, tools, and strategies that embody historic activity and social interaction. They are simultaneously general and specific, and as such they exist between students’ informal realities and formal mathematics. Through activity, preformal productions can be made general enough so as to be applicable to a wide variety of problems, but they retain contextual cues to specific situations (Peck & Matassa, 2016, p. 272)
For example, the "bar model" for fractions is a preformal production. So is the "partition, distribute, iterate" strategy for fair sharing. In the figure below from the paper (p. 255), one of the students in our study is using both of these preformal productions to find the equal share when 5 people share 4 sandwiches equally.

We found that preformal productions played two key roles for students.

(1) Preformal productions help students do math. 

(2) Preformal productions help students learn formal mathematics

In the paper, we document how preformal productions emerge in the classroom, and we argue that preformal productions should be considered cultural artifacts: shared and durable features of communities, not simply individual cognitive possessions.

With respect to our initial problem of practice, we provide detailed descriptions of students' understanding of fractions and division, and we provide an activity sequence that can help guide students to reinvent the relationship between fractions and division.  

Full text (approved manuscript)


Peck, F. A., & Matassa, M. (2016). Reinventing fractions and division as they are used in algebra: The power of preformal productions. Educational Studies in Mathematics, 92(2), 245–278.

Monday, April 18, 2016

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

Sunday, April 17, 2016

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

Sunday, September 20, 2015

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.


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.

Friday, September 18, 2015

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!