Get Close Support™ for Your Algebra Teachers

By Raymond Rose and Beryl Jackson

Algebra is a gateway course, serving as a prerequisite to advanced math and science courses. It’s also the most frequently taken math course in high school and is required for college admission.

In this age of tougher standards and rigorous instruction, algebra is often offered at the seventh and eighth grade levels. To meet the needs of in-service and pre-service teachers of algebra, Concord Consortium’s Seeing Math project and PBS TeacherLine have created an innovative series of online professional development modules to support mathematics teachers. Teachers can register now for fall pilot tests of these materials.

Ready to Teach Algebra has been developed and tested over the past year and will be offered in its entirety for the first time beginning in the fall. This unique online professional development program is built around ten major themes (see “RTT Algebra Modules,” page 5) presented in today’s popular algebra texts and aligns with the sequence of a typical algebra curriculum. This alignment is a hallmark of our Close Support™ approach.

Close Support connects professional development activities quickly and directly to the teacher’s classroom experience by providing off-the-shelf customization; research-based content in core math concepts and pedagogy; interactives that offer parallel teacher and student activities; insights into student thinking; and learning communities of facilitated discussion forums.

Each three-week course module provides opportunities for teachers to reflect on their own learning and to identify obstacles to learning and misunderstandings through dynamic discussions with peers that support the continuous and sustained learning experience. In the first week of each module, the core algebra topic is introduced – often with an RTT Interactive (a small manipulable Java applet that is cross-platform and runs over the Internet) – to ensure that the participating teacher has a solid conceptual understanding of the math. In the second week, the teacher has an opportunity to explore student understanding of the math concept by analyzing video vignettes of classroom students and expert commentary on how students learn. Finally, the third week links teacher and student understanding to the local curriculum and presents suggestions that support the teachers in integrating course elements into their instruction.

The modular online course structure allows schools, districts, states, and even colleges and universities to offer customized sequences of the modules to best meet local needs. The sequence can be adjusted easily for schedule and length. Licensees have the option of using their own facilitators or PBS TeacherLine’s professionally trained mathematics facilitators to lead the courses. Upon request, PBS TeacherLine can provide specialized facilitator training. Ready to Teach Algebra can also be used in pre-service teacher education programs as a component of middle or high school mathematics methods courses.

If you’re ready to participate or are interested in more information about Ready to Teach Algebra professional development opportunities, please contact Raymond Rose (ray@concord.org) or Beryl Jackson (bjackson@pbs.org) to discuss your specific needs.

Participants from the Pilot Study say:

The activities are great... they really make you re-examine the way you teach the topics in your own classes. Things like the Qualitative Grapher interactive have exciting implications for use in the classroom. The opportunity to interact with so many colleagues on a regular basis is invaluable.
I am a better mathematics teacher because of the two modules we did.
The course presented some basic core concepts in a very hands-on, productive way. I love the stuff.
The course really helped me to be more aware of potential areas of student misunderstanding, and helped my teaching as I was taking the course.

Raymond Rose (ray@concord.org) is Vice President of the Concord Consortium, Beryl Jackson (bjackson@pbs.org) is Director of Mathematics Content for PBS TeacherLine.

Article Links & Notes

Seeing Math – http://seeingmath.concord.org
PBS TeacherLine – http://teacherline.pbs.org
Ready to Teach – http://rtt.concord.org
RTT Interactives – http://rtt.concord.org/interactives

Got a Plan?

The cell phone plans can be represented as piecewise linear functions. The Piecewise Linear Grapher connects symbolic and graphic representations of piecewise linear functions, so that changes in one representation cause changes in the other representation.

In the RTT Algebra course, the goal of a problem is not only to arrive at an answer, but also to learn from the involved, sometimes messy steps in between. As you do this activity, observe your process as a teacher, an expert learner, and a problem solver. The complete “Got a Plan?” activity is available online at http://rtt.pbs.org/rtt/courses.cfm.

The Problem

Three cellular phone companies are offering promotions for new customers that are represented in the graph at right.

Three users with different profiles – an attorney calling her assistant and clients from the courthouse; a retired pediatrician who makes long-distance calls to his grandchildren on weekdays; a high school math teacher who keeps in touch with her daughter’s nanny during free periods and makes occasional calls to schedule appointments – each want to purchase a cell phone. They are each trying to decide which plan suits their individual needs.

Your Role in the Story

You are a customer service representative for Cell Zone, a company that produces reports on cellular phone companies and helps customers choose plans that fit their needs and lifestyles.

Your Task

As the customer service representative, you will make preliminary recommendations to your three customers. Use as many tools as you need to reach each decision and make your recommendations clear to each of your customers. Include any decisions you reach as to what plan or plans might be best suited to each of them and the conditions under which each customer might find the plan or plans most useful.

RTT Algebra Modules

Overview

Participants learn to navigate the online course environment as they meet fellow teachers through personal web pages and online discussions. They examine obstacles to interpreting time vs. distance graphs and try out the first interactive, the RTT Qualitative Grapher. Through these activities, they examine the nature of algebra, assumptions underlying their own understanding, and approaches used in their students’ curriculum.

Professional Development Goals

Build an effective online learning community for studying algebra
Approach learning and teaching algebra in more profound, varied, and flexible ways
Help students interpret time vs. distance graphs correctly

Ratio, Proportion, and Scale

This module prepares the transition from a primary focus on arithmetic and skills with algorithms (typical of elementary and middle school) to a focus on algebra, where students use multiplicative, as well as additive, thinking. The common terms – ratio, proportion, and scale – are placed within the broader mathematical themes needed in algebra.

Professional Development Goals

Understand the relationships among ratio, proportion, and scale
Identify equivalence by comparing symbolic expressions used to express ratios and proportions
Articulate how the concepts of ratio, proportion, and scale, although closely related, serve different purposes in expressing quantitative relationships of linear, two-dimensional, and three-dimensional measures

Linear Functions

The Linear Functions module introduces algebra through the mathematically cohesive concept of functions and grounds it by modeling real-life situations.

Professional Development Goals

Distinguish between functions and equations
Use piecewise functions to interpret the meaning and characteristics of linear functions in the context of real-world situations
Identify ways that multiple representations express and enrich mathematical concepts
Develop student activities based on module concepts

Transformations of Linear Functions

Participants observe relationships between graphic and symbolic forms of a function. They explore how changes to the graphic representation of a function alter its symbolic representation, and vice versa.

Professional Development Goals

Observe graphic and symbolic transformations of linear functions, using interactives
Represent, categorize, and use families of linear functions in multiple formats
Interpret the concept of slope in different contexts

Linear Equations

Most algebra curricula introduce linear equations before linear functions. In this module functions are discussed first, showing an equation as a particular instance of a function.

Professional Development Goals

Understand the rationale for manipulating symbols when solving problems with equalities or inequalities
Deepen the distinction between equivalence of function and equality of value
Gain facility in moving between symbolic and graphic techniques when solving equations, whether presented in symbolic or story (text) form

Quadratic Functions

Using models and problem solving, participants examine how the general nature of quadratic functions informs the particular instances described by quadratic equations. Participants also use multiple representations – tables, graphs, and equations – as powerful tools to describe physical situations.

Professional Development Goals

Learn to shift from recognizing physical patterns to describing them in symbolic language, and vice versa
Develop additional strategies for working with symbolic manipulations and syntax
Distinguish between local and global representations of a function
Recognize the different roles of functions and equations
Model a variety of situations using quadratic functions

Transformations of Quadratic Functions

Using interactives, participants observe how changing symbolic expressions alters their graphic representation, and vice versa. By working with families of quadratic functions, they deepen their understanding of the role of each symbolic form in gleaning information about a function.

Professional Development Goals

Represent, categorize, and use families of quadratic functions in multiple formats
Understand how each of the three major symbolic forms – polynomial, product, and vertex – serve a different purpose and how each informs graphic transformations
Execute translations, reflections, and dilations in graphic and symbolic forms
Understand how the function notation of transformations such as f(x), af(x), f(ax), f(x+a) and f(x) +b deepens knowledge about various classes of quadratic functions

Quadratic Equations

This module makes explicit the relationship between quadratic functions and quadratic equations. Because textbooks and tests devote a great deal of time to the skills of factoring and finding roots, participants also use graphical means, as well as successive approximations in tabular form, to reach the same goal.

Professional Development Goals

Identify the three symbolic forms for quadratic equations – polynomial, product, and vertex – and clarify how each form’s role informs not only its graphic representation, but also how to find the root of its function
Delineate techniques for solving quadratic equations by comparing quadratic functions, both graphically and symbolically
Solve quadratic inequalities, in both graphic and symbolic forms, as a natural extension of comparing functions

Descriptive Statistics

This module discusses data sets, measures of center (mean, median, mode), range and outliers, and linear fits to data sets. Research suggests that when students calculate a measure of center, they use algorithms without associating the measurement with a characterization of the entire aggregate.

Professional Development Goals

Model and describe real-world data sets
Connect general measures of data with the set they describe
Understand options for characterizing large data sets
Use measures of center and linear fit to describe particular data sets
Use interactives to view and understand data sets in different ways

Project

This module provides a framework for project design and a set of rubrics to assess those who are taking the course for credit. It consolidates Ready to Teach concepts, participants’ understanding of algebra, and their philosophies about teaching practice. Participants say farewell to their coursemates and facilitator in a brief graduation “speech.” They also compose a concrete plan for using new insights in their own thinking and teaching, and learn from the plans of colleagues.

Article Links & Notes

Seeing Math – http://seeingmath.concord.org
PBS TeacherLine – http://teacherline.pbs.org
Ready to Teach – http://rtt.concord.org
RTT Interactives – http://rtt.concord.org/interactives

Spring 2004 @Concord