Monday, October 3, 2022

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.

Wednesday, March 10, 2021

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:

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 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.

Activity sequences for your classroom

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

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

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!