Insights Into Algebra 1: Teaching for Learning
Linear Functions and Inequalities Teaching Strategies: Appropriate Use of Technology
“Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning.”
The above quote comes from the National Council of Teachers of Mathematics’ (NCTM) discussion of the Technology Principle in the Principles and Standards for School Mathematics (PSSM). NCTM also states:
Electronic technologies – calculators and computers – are essential tools for teaching, learning, and doing mathematics. They furnish visual images of mathematical ideas … and they compute efficiently and accurately … Students can learn more mathematics more deeply with the appropriate use of technology (Dunham and Dick 1994; Sheets 1993; Boers-van Oosterum 1990; Rojano 1996; Groves 1994). Technology should not be used as a replacement for basic understandings and intuitions; rather, it can and should be used to foster those understandings and intuitions … In short, technology can help students learn mathematics.
(Professional Standards for Teaching Mathematics, 1991, p. 24-25)
Both Janel Green and Tom Reardon teach lessons in which technology is essential in helping to foster students’ understanding and use of intuition, and to focus on decision making, reflection, and reasoning. When used appropriately, technology can enhance students’ learning opportunities and connect the development of skills and procedures to the more general development of mathematical understanding.
Listen to what Diane Briars has to say about Tom Reardon’s use of technology in the video for Workshop 2, Part I:
|Listen to audio clip of teacher educator
Transcript from Diane Briars
One of the most impressive things about Tom’s class was his use of technology. Having that SMART board added so much to the lesson itself. The students could not only see the results of what was graphed on the graphing calculator, but they could actually watch the steps he was going through as he was entering information. When he was writing things on the board, you could just see there was a structure there, and the color highlighted different pieces that he wanted to emphasize. It added a whole new dimension to the class. I think [Tom] really illustrated how new technologies can be used as effective tools in the mathematics classroom.
Fostering Understanding and Use of Intuition
Teaching is not telling, and an excellent way to help students understand mathematical concepts is to use technology appropriately so that they can look at relationships, draw conclusions, and reach an understanding on their own that is deep and meaningful. “With calculators and computers students can examine more examples or representational forms than are feasible by hand, so they can make and explore conjectures easily,” according to NCTM. “The graphic power of technological tools affords access to visual models that are powerful but that many students are unable to generate independently.” (PSSM, 2000)
In Tom Reardon’s class, the students were able to use technology to see the connections between the parameters in their function and the resulting graph and table. The slope and y-intercept weren’t just numbers in an equation but had a meaning, and this meaning became apparent in the table of values and the graph.
Read what Janel Green says about the importance of understanding different methods for solving equations:
Transcript from Jenel Green
The key concept was for students to relate making a table and solving algebraically and making a graph to solve any problem. Often people never connect that idea, but today I believe we accomplished that; we were able to make the connection between all three methods.
Think about a lesson you teach that uses technology to foster understanding and intuition in your students. Describe several ways that the activities in this lesson demonstrate an appropriate use of technology.
Focusing on Decision Making, Reflection, and Reasoning
“Using technological tools, students can reason about more general issues, such as parameter changes, and they can model and solve complex problems that were heretofore inaccessible to them,” according to NCTM (PSSM, 2000). Tom Reardon encouraged decision making in his class when he led a discussion of the different ways students chose to solve the problems. Some of the students were substituting values into the equation, some were looking for values in the table, and some used a graph. He then asked them to verify that all the methods produced the same solution. Janel Green encouraged her students not only to learn all three methods for solving equations and inequalities but also to reflect on which method might be most efficient in a given situation.
In Janel’s class, technology was helpful because students were learning how to solve equations and inequalities using tables, graphs and algebra for the first time. Later on in their mathematical studies, however, it might be considered inappropriate to use a calculator to solve these problems; students should be expected to find solutions mentally. Therefore, teachers must make sure students improve their mental math skills when using technology. Students need to be able to examine an electronic graph or table, for example, and determine that it makes sense and is reasonable, and they should be able to explain thoughtfully why a solution generated by a calculator is correct or incorrect.
Read what Fran Curcio has to say about how calculator use can help students reason about the important aspects of a problem:
Transcript from Fran Curcio
Exploring the window of the calculator and what the window will help to reveal with respect to the input is very important in being able to use the calculator effectively. So spending the time in this particular lesson talking about the window, the ymax, the xmax, the ymin, the xmin, will help students be attentive to the frame that poses the scene for the particular problem as it will appear in the display of the calculator.
Tom Reardon talks about the benefit of using different colors with the SMART Board to help students improve reasoning skills:
Transcript from Tom Reardon
I would try to highlight what we did in numbers and in the words that went with it, so that they would see a subtle, subliminal suggestion. When we subtracted the same number from both sides, I usually did that in red. When we divided both sides, I usually did it in green or blue so that kids would see that, and they would start to say, “Oh yeah.” Those types of messages are being sent to kids without them really realizing it.
Reflect on a lesson you teach and describe how you can use technological tools to help students reason about the lesson’s general issues and concepts.
Enhancing Students' Learning Opportunities
According to NCTM, “Teachers should use technology to enhance their students’ learning opportunities by selecting or creating mathematical tasks that take advantage of what technology can do efficiently and well – graphing, visualizing, and computing.” (PSSM, 2000) Janel Green used a familiar problem context and graphing calculators to support her students in reaching an important and difficult mathematical goal: the ability to solve equations and inequalities using tables, graphs, and algebra. Tom Reardon used a variety of technologies and a seemingly more complex problem to help his students deepen their understanding of linear functions and how slopes and y-intercepts relate to real-world situations.
Read what Tom Reardon says about how the SMART Board has helped enhance his students’ learning opportunities, even when they’ve been absent from class:
Transcript from Tom Reardon
Whatever I do on that SMART Board, I can save it as an HTML file and I can put it on my Web site with the date. And so if a student is absent today, he or she can click on class notes, Math 1, today’s date. The notes are right there, in color. [The student] can view them or print them out in color or in black and white, and also get the assignments and get caught up. Or if [students are] absent for an extended period of time, that chore of having to send assignments home is eliminated, because the kids can go online and do that. And I’ve had many students with success stories [since I’ve been doing this.] For example, I spoke with one girl recently and I said, “I’ve noticed since you [returned after an extended absence] you are doing better.” She said, “Well, every day in class there’s always one thing I miss, but then I go online and check your notes, and so I don’t miss that one thing anymore.”
What types of lessons do you use that incorporate technology as a means to enhance student learning opportunities? How would those learning opportunities differ if technology wasn’t used?
Connecting Skills and Procedures to Mathematical Understanding
“Technology can help teachers connect the development of skills and procedures to the more general development of mathematical understanding,” according to NCTM. “As some skills that were once considered essential are rendered less necessary by technological tools, students can be asked to work at higher levels of generalization or abstraction.” (PSSM, 2000) Because Tom Reardon used a contextual problem and incorporated technology, he was able to help his students understand the general idea that the cost of the phone bill was a function of the length of the phone call. He was able to move naturally between different representations of the linear function that the class used to model the problem. They were able to understand that y = 0.24x + 0.85, C = 0.24t + 0.85, and C(t) = 0.24t + 0.85 were all equivalent representations for the problem.
Read what Janel Green says about how her students began to see the connections between the numeric, graphical, and algebraic solutions:
Transcript from Jenel Green
I thought it was great when the students did see the three methods in the beginning and they saw the power of algebra actually, because a couple of students said, “Well you know what? I would rather solve that break-even equation algebraically than to do it graphically or use a table” – because it was so much faster, which is great. They got to see that algebra actually is powerful. But then, of course, they had to realize, too, that later on they will find functions which can only be graphed and cannot be expressed as an equation – they have to also be able to interpret graphs and look at graphs and tables.
Fran Curcio discusses how technology can help expedite the process of generalizing concepts:
Transcript fromFran Curcio
How frequently do you need to go back and actually connect up to the real-world setting? In [Tom Reardon’s] lesson, there was an initial real-world launch, then there was work with the mathematics. At the beginning, it was in pure mathematics, and then about 15 minutes into the lesson, there was another connection back to the real-world setting … [But] every time we talk about the slope of 24/100ths, should we have said that was 24 cents? Should we have taken the y-intercept of 85/ths and reminded students that it was 85 cents? How often do you need to make those connections? Should students be working in the mathematics for a while and then come back and interpret in the real-world setting?
In the Hot Dog Sales lesson featured in Workshop 2, Janel Green works with her students to find the break-even point, where revenue and costs are equal. She shows them how to solve the equation they generated, 0.50x – 450 = 0, using tables, graphs, and algebra. Then she has her students work in groups and apply and extend these concepts. Even though this was the first time these students had ever been exposed to these ideas, many of them were able to figure out how to use what they had learned about equations and to transfer that information to finding the solution of inequalities. Having the ability to solve inequalities numerically and graphically helped the students in their struggle to figure out the algebraic solution.
Below is an example of how one group of students found the solution to one of the inequality questions in the Hot Dog Sales lesson. The students were asked to show the number of hot dogs that need to be sold to produce a profit of at least $250.
This group of students needed Janel’s help to figure out the correct algebraic solution strategy.
A very important concept for students to understand is that both the numerical method (using tables) and the graphical method are general methods of solutions and can be used to solve equations and inequalities of any function. In contrast, the algebraic method of solution only works for linear equations and inequalities. Students must learn the particular algebraic manipulation needed to solve each type of function. In fact, solving inequalities algebraically is extremely difficult, and sometimes impossible, for any function other than a linear function.
Write about a lesson you teach that helps students develop particular skills and procedures as well as a more general mathematical understanding.
Some schools and teachers believe that the cost of technology prohibits them from using it effectively. However, there are several ways to make sure that your classroom has the appropriate technology.
Listen to what Tom Reardon says about finding ways to purchase appropriate technology:
|Listen to audio clip of teacher
Transcript from Tom Reardon
The SMART Board and the computer projector that we have [were] purchased through a Martha Holden Jennings Foundation grant. Once I got the SMART Board here and I showed it to our technology coordinator, I gave him about a five-minute demo. He left the room and went and ordered five of them. He had some funds from the federal government, and he said “That’s definitely the way we have to go.” And they have definitely changed my teaching style. I’ve also had to learn how to write grants. I’m not the best writer, but we’ve had some English teachers – one in particular – who has helped me quite a bit. She knows the writing and the English and the proper grammar, but I think I’ve found that if you’ve got that enthusiasm behind it and you can get that in your writing, that that’s what helps you get the grants.
What do you consider adequate technology in an algebra classroom? In what ways can you ensure that your classroom provides adequate technology for your students?
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.