NCTM 2022: Making meaning of systems of equations with contexts and representations

 

Students often learn techniques for solving systems of equations, like substitution and elimination. However, these are often learned as procedures without meaning. Students may also see representations of systems of equations, like a Cartesian plane with two lines crossing, or a symbolic representation with two equations. Often, however, these are disconnected. Students may not be able to articulate the connection between the techniques and the representations (e.g., what does two lines crossing have to do with elimination?)

We think students need more contexts and representations to make systems of equations meaningful. In this session we discussed contexts and representations beyond tables, graphs, and symbols to help students make systems of equations meaningful.

Session resources

• Slides

Complete unit

• Student book
• Teacher guide

Related research

• Webb et al (2008) - Beneath the tip of the iceberg
• Webb and Abels (2011) - Restrictions in Algebra 

Making algebra meaningful: The area model from arithmetic to algebra (and calculus!) [NCTM 2021 virtual conference]

We'll offer this session on Saturday 4/24/2021 as part of the NCTM Virtual Annual Meeting, 2021. Here are the slides for the session: 

Montana Educator's Conference 2019: Making logarithms meaningful through progressive formalization

(This is similar to the session that David Webb, Raymond Johnson, and I facilitated at NCTM, 2019)

Logarithms are often seen as one of the most complicated and abstract concepts in secondary mathematics.

Consider these problems. Many people—even those who can solve the problems using abstract procedures—would struggle to find meaning in the problem.

In this session, we shared a unit that makes logarithms meaningful. The unit uses realistic contexts and multiple representations to help guide students towards an understanding of progressively more formal logarithms.

You can download the slides and the complete unit below.

Resources from the session

• Download the slides from the session
• Download the complete unit to use with your students (in Word format, so it is easily modifiable)
• Part A: exponential growth
• Part B: exponential functions
• Part C: logarithmic functions

The unit was initially developed by Henk van der Koij, with input and further development from Monica Geist and David Webb. The unit has been taught at many high schools, community colleges, and universities. 

NCTM 2019: Making logarithms meaningful through progressive formalization

Logarithms are often seen as one of the most complicated and abstract concepts in secondary mathematics.

Consider these problems. Even for those who can solve the problems using abstract procedures, most would struggle to find meaning in the problem.

In this session, we shared a unit that makes logarithms meaningful. The unit uses realistic contexts and multiple representations to help guide students towards an understanding of progressively more formal logarithms.

You can download the slides and the complete unit below.

Resources from the session

• Download the slides from the session
• Download the complete unit to use with your students (in Word format, so it is easily modifiable)
• Part A: exponential growth
• Part B: exponential functions
• Part C: logarithmic functions

The unit was initially developed by Henk van der Koij, with input and further development from Monica Geist and David Webb. The unit has been taught at many high schools, community colleges, and universities. 

Montana Educator Conference 2018: Making Algebra Meaningful

This session is based on my work as an algebra teacher and research. I also drew on ideas that my colleagues Raymond Johnson and David Webb presented at NCTM.

The session focused on how to make algebraic equations and manipulations meaningful for students. The theme of the session is the relationship between models, structure, and strategy.

I introduced two models for algebra equations: a balance model and an arrow chain model, discussed how each model produces a different structure for algebra equations, and how each model can be used develop meaningful strategies for algebraic manipulations. 

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

Resources from the session

Download the slides

Download the problems from the session

Activity sequences for your classroom

Download the activity sequence for the balance model

Download the activity sequence for the arrow chain model

Extension: 

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

Download an activity sequence for the number line model

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 slides

Download the problems from the session

Download the activity sequence for the balance model

Download the activity sequence for the arrow chain model

Extension: 

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

Download an activity sequence for the number line model

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!

• Session handout (1 MB PDF file)
• Session presentation (7 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)

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.  

Publisher (paywalled) version
Read only version

Full text (approved manuscript)

Citation

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. http://doi.org/10.1007/s10649-016-9690-y

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)

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