Teacher resources and professional development across the curriculum

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Insights Into Algebra 1 - Teaching For Learning
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Topic Overview Lesson Plans Student Work Teaching Strategies Resources
Workshop 7 Direct and Inverse Variation Teaching Strategies
Teaching Strategies:

Questioning Techniques

Concepts First, Skills Later
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NCTM Standards


Peggy with group
Concepts First, Skills Later

A Traditional Lesson
An Upside-Down Lesson
Independent Practice

Teacher Peggy Lynn's mathematics classroom probably looks different from many classrooms you have seen. That's because Peggy teaches math in context, using special activities to bring out math concepts.

Listen to audio clip of teacher
Peggy Lynn
I've been teaching for 17 years, so when I first started teaching, I taught much the way I was taught -- which was very traditional. Read More

A Traditional Lesson

The term "traditional lesson" is loaded with somewhat negative connotations. It is generally associated with a boring, monotone lecturer at the front of the class and students dutifully taking notes in their seats. But the traditional classroom was not entirely bad. After all, didn't some of us learn math that way? Still, says Peggy Lynn, the traditional method of solving a few sample problems and then letting students try some on their own is not the most effective way to reach all students.


Read transcript from teacher Peggy Lynn
I think students struggle with anything if it's only given to them in a cookbook form. Such as, "This is an equation, this is a graph, and you go on." Read More

A typical traditional lesson typically has four components:
  1. Introduction
    The teacher gives (or the students read) a brief overview of what material will be covered that day.
  2. Direct Instruction
    The teacher explains a concept and presents an example to illustrate the idea.
  3. Guided Practice
    The teacher and class work together on some examples.
  4. Independent Practice
    The students work on some problems, individually or in small groups, and the teacher only helps when necessary.
A quick search on the Internet found the following traditional lesson plan. (The file has been modified slightly, and names have been removed.) This lesson contains these four components typically found in traditional lesson plans, and, for all intents and purposes, provides all the information students need to solve standard direct variation and inverse variation problems.

Likewise, a Web-based learning site provided another traditional lesson plan on direct variation. This lesson presents a list of objectives, a definition, and a one-paragraph discussion of direct variation. This is followed by three sample problems (none of which have applications to the real world). The lesson ends with this statement: "Now, you should understand the concept of a direct variation and be able to solve problems involving direct variations."

What's missing from these lessons, however, are the subtle nuances of direct variation that students in Peggy Lynn's class found. In reading the traditional lesson plan, you have to wonder if any of the subtleties will ever become evident to students. The discovery of those details is what solidifies deep conceptual understanding. In contrast to the traditional lesson plan and the lesson on the Web-based learning site, Peggy Lynn's lesson ensures that students will be exposed to those details.

Reflection:
Is your classroom a "traditional" math classroom? Do you teach by modeling solutions for students and then letting them practice on their own or in groups? If so, explain why you feel a traditional lesson is the most effective means for teaching students about direct and inverse variation. If not, explain why you would use another method to teach these concepts.

record your thoughts in your journal


An Upside-Down Lesson

Beatrice Moore-Harris, an educational consultant from Houston, TX, describes how Peggy Lynn's lessons on variation differ from traditional lesson plans.

Listen to audio clip of teacher educator
Beatrice Moore-Harris
Many times lessons start off with, "Here's the equation that we're going to use today," or, "This is how we will represent this situation mathematically."
Read More

Peggy's lesson on direct variation began with a brief introduction to the context of oil spills. Then, she described a situation in which students were going to investigate oil spills using toilet paper sheets (representing pieces of land) and drops of colored vegetable oil (representing oil). From there, students explored the situation, made scatterplots and tables from the data, and drew their own conclusions. Peggy then led a brief discussion in which the key ideas about direct variation were summarized, and finally, students were asked to practice some applied problems on their own.

For her inverse variation lesson, the mathematics and the experiment were slightly different, but the flow of the lesson was the same: introduction, exploration, summary, and independent practice.

Beatrice Moore-Harris explains how Peggy Lynn's lesson is effective in developing conceptual understanding and increasing retention. By presenting vocabulary words at the end of this lesson rather than at the beginning, students learn the terms in relation to a concept they already understand. "The proper math terminology will come at a time when it's appropriate, once the concept has been developed thoroughly," Beatrice says. In addition, the use of an end-of-lesson summary allows students to reflect on their own learning.

Listen to audio clip of teacher educator
Beatrice Moore-Harris
They had been working with direct variation, so it was just natural that they will say, "Well, this seems to be the opposite, so it's indirect variation," which I thought was really great because the students are really spending more time developing the concept, not getting so bogged down with the terminology. Read More

Reflection:
The pieces of a traditional lesson - introduction, direct instruction, guided practice, and independent practice - are evident in Peggy Lynn's lessons on variation. Yet her lessons look very different from those we might think of as traditional. What advantages, if any, are there to teaching mathematics using hands-on investigations and discussing vocabulary at the end of the lesson instead of the beginning?

record your thoughts in your journal


Independent Practice

Peggy's lesson develops conceptual understanding by allowing students to first learn a concept by experience, and then introducing the terms they need to know. This differs from the traditional lesson in which terms are presented at the beginning. Consequently, you might think that Peggy's lesson has no relation to a traditional classroom, but that's not the case. Building and practicing skills is an essential part of Peggy's philosophy, and independent practice - especially in small groups - is fundamental to her students' success.

Listen to audio clip of teacher
Peggy Lynn
I think it's very important for students to take what they've experienced in class and practice it, because once again, they will remember it better. Read More

Reflection:
Students in Peggy Lynn's class appeared to understand the concepts of direct variation and inverse variation from the investigation alone. Why, then, did Peggy assign additional problems to students if they already understood? What role do practice problems and homework assignments serve in your mathematics class?

record your thoughts in your journal


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