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:

# RME in the Classroom

Mathematics is a human activity

## Wednesday, March 10, 2021

## Friday, October 18, 2019

### 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)

*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.*

## Friday, April 5, 2019

### 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)

*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.*

## Thursday, October 18, 2018

### 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

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

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.

Download the activity sequence for the arrow chain model

Download an activity sequence for the number line model

### Extension:

A number line is also a great model for algebra equations.Download an activity sequence for the number line model

## 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.

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

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

Download an activity sequence for the number line model

### Extension:

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!

- 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)

## Thursday, May 26, 2016

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

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)

#### 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
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