Insights Into Algebra 1: Teaching for Learning
Direct and Inverse Variation Teaching Strategies: Concepts First, Skills Later
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
Audio from 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. Part-way through my teaching career, I got involved with a project that was developing a new set of curriculum materials that were based on context with the idea that the traditional method worked for some students, but not everybody. And this project was looking for a better way to reach more students. And then, I got excited about it. My students, I get a variety of responses from them. Some love it, because they have to think; some hate it because they have to think. But I really believe that I would be doing them a disservice if I didn’t challenge them to think and apply and connect. the richness that I get from teaching math in a context I would not change.
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.
Transcript from 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.” What I find is that students have struggled much less when they have something to hook the concept to, such as the idea of an oil spill spreading or a scale factor when you’re increasing the size of a geometric shape, if you relate it to a person’s body shape, their shoe size and their height, or something that they can get a handle on. And now, when I have these same students two years later, and we come back to this topic for some reason, I can mention oil spills and they remember what the concept was that we were studying. But if I just said “direct proportion,” then those words don’t have any meaning for them necessarily.
A typical traditional lesson typically has four components:
The teacher gives (or the students read) a brief overview of what material will be covered that day.
- Direct Instruction
The teacher explains a concept and presents an example to illustrate the idea.
- Guided Practice
The teacher and class work together on some examples.
- 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.
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.
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.
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Audio from Bea 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.” And we will provide several examples and then give students problems, and the expectation is they will crank out more equations for similar type problems. But in this particular situation, the students had no idea of what the equation was that they were looking for. It was just through a series of events they came to a mathematical equation, which I thought was really nice.
Peggy ended the lesson where many teachers may begin the lesson. The summary was the natural place to talk about the mathematical vocabulary. She was connecting the appropriate mathematical terms to the things that the students said, where they were using their own natural language and examples, and modeling of what happened during the lesson. I thought that was very important, because by attaching the proper math terminology to what the students already knew, understood and believed in, that they are going to make more connections with the proper terminology to the concept and most likely they are going to retain this information over a longer period of time.
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
Audio from Bea 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. But the use of the proper math terminology will come at a time when it’s appropriate, once the concept has been developed thoroughly. One of the most powerful elements of the lesson was the use of summarization at the end of the lesson. It allowed students to reflect on their experiences for the day, the applications that they had done, to really look at what does an inverse variation mean, what does it consist of, and compare that to a direct variation. And do a compare-contrast of the direct to the inverse. If students can do that, that really shows that they have really developed the concept to a point where they can describe what it is, what it isn’t, and that really means that they understand.
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?
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
Audio from 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. And I was reminded of that again today when one group of students had not worked through the homework problems and I asked them to refer back to the previous day’s activity that we had done, and one student couldn’t remember. He had to be prompted and helped with that. And I just think it’s critical for students to spend some time doing independent practice, whether it is more skill-building, or more application.
My students can expect an assignment every class period, just about. I mean, there are exceptions to that, but I think it’s important for them to have a homework assignment and some independent practice time during class for them to clear up any misconceptions they may still have, but also to spend some time outside of class working on the types of problems that we dealt with that day.
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?
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.