Insights Into Algebra 1: Teaching for Learning
Exponential Functions Teaching Strategies: Instructional Decision Making
This slogan appeared on a basketball T-shirt several years ago: “Just elevate and decide in the air.” The phrase could easily apply to teaching, too.
A basketball player may have to alter his shot because of the reactions of other players. Likewise, a teacher may have to modify a lesson based on the reaction of students. It doesn’t make sense to shoot when a 6’10” defender jumps between you and the basket, and it doesn’t make sense to introduce a second example when students clearly haven’t understood the first one.
The analogy can be taken even further. A successful basketball player keeps her eye on the long term goal – winning the game and perhaps the league championship – when taking a shot. Similarly, an effective teacher understands the goal when delivering a lesson: to ensure that students meet the instructional objectives of the unit.
Students occasionally will hit you with something you never expected. How you react to unanticipated events will dictate what happens next – whether learning occurs or frustration takes hold.
Planning the Lesson
As with basketball, sometimes the best offense is a good defense. A thoroughly planned lesson may help to avoid conceptual misunderstanding and ensure comprehension. Knowing what questions you will ask, what responses indicate that students are ready to move on, and what objectives you hope to reach will help to ensure the delivery of a successful lesson.
Lesson plans are used to structure a lesson and to help with the flow of the class, especially when something in the classroom distracts everyone. In short, lesson plans help to both teacher and students maintain focus and adhere to that old adage: Keep your eye on the prize.
The first step in lesson planning is to answer the question, “What is your goal for the lesson?” To answer that question, of course, you need to know the goals for the unit and for the entire year. The unit and yearly goals are likely determined by state and local standards, so creating lessons that are engaging and coherent and that meet the standards should be the thrust of instruction.
Read what Jane Schielack noticed about the teacher’s long term goals in the video for Workshop 6 Part II:
Transcript from Jane Schielack
In orchestrating this lesson, [Mike Melville] had the big picture of how the lessons fit together into the unit and how the units fit together into the goals of the year. He was able to take advantage of the learning opportunities that came up in the lesson, and he knew that the things he decided not to pursue he would have opportunities to address later.
The choices you make prior to delivering a lesson will dictate whether or not instructional time is used effectively. To ensure success, make conscious and deliberate decisions about the following aspects of your lesson:
What will your students know and be able to do by the end of the lesson? It is helpful to write these objectives in your lesson plans. Having them in writing will be a reminder of what your students ought to accomplish by the end of the lesson, and they will help you keep your eye on the prize.
What learning outcomes do your school, district, or state endorse? As with the objectives, having the standards listed on your lesson plans will keep your class focused.
Consider Jane Schielack’s comments about the importance of instructional objectives:
Transcript from Jane Schielack
Orchestrating student input in a classroom can become almost chaotic if you lose sight of the goal. It’s important to develop some skills for doing that, and teachers have to do that for themselves, but the beginning of that is to go into the lesson knowing what you want the end result to be. Each decision that you make … should be weighed against [the question] “Is this taking us closer to the goal that we have?” There are topics that you might want to record for further discussion in a later lesson. Some teachers choose to actually record those on a chart or a board for the students so that they can let go of them at that moment and know that they’ll come back to that question.
How will you help students accomplish the objectives of the lesson? Will they work on a worthwhile mathematical task in groups? Will they participate in a class discussion? Will you guide them using carefully selected examples? Will they work through an exploration and discover the concepts for themselves? The way you choose to develop the objectives of the lessons varies with each objective and topic. How students come to understand the concept depends greatly on how you develop the lesson.
- Independent Practice
How will you have students practice on their own? Have students engage in activities that move them closer to your instructional goals, and discard the ones that merely provide a distraction. As Jane Schielack says, “Selecting specific homework problems that you know will relate to the goal … can [help focus] the discussion.” Make conscious and deliberate decisions about which exercises you assign and which projects and investigations you use.
Read what Mike Melville says about the selection of homework assignments:
Transcript from Mike Melville
The homework in the curriculum that I’m using right now is a bridge from what we did today to what we’re going to do tomorrow. The homework they did last night was a bridge from what they had done before to what we’re going to do next. The first step on that bridge, the first problem, is an easy step … so [all students] should be able to make that one. The last question on the homework is the big step.
Read Jane Schielack’s thoughts about selecting homework problems to discuss in class:
Transcript from Jane Schielack
It seemed to me that [Mike Melville] had chosen homework problems that would bring out specific discussion about the particular goal [of the lesson]. There were topics that came up, and it was clear that he decided to pursue them at a later date. [He made] some comments to the effect that “We’ll continue with these ideas, the decimals, the discussion of the decimal exponents and what those meant in terms of Alice getting smaller.” They started a discussion and tabled that for a later date, and then went into other types of numbers for exponents as expanding their understanding. So there were many times during the lesson where he made almost visible decisions about what to pursue and what to relegate to another day or another lesson.
In addition to considering what you want students to know by the time they leave your classroom, it is important to consider what they already know when planning your lessons. One way to determine prior knowledge is to refer to past years’ standardized test scores, as well as to assessment results from earlier in the year in your class. Another powerful way to assess student knowledge is during the lesson itself; while interacting with groups and listening to student discussions, it often becomes clear what students understand and what they do not. Based on these informal assessments, you can plan future instruction accordingly.
List three elements that you include in a lesson plan to ensure that you remember to focus on the instructional goals of the unit. How do those three elements help to keep you focused on what students are learning?
During the Lesson
Of course, even the best laid plans can go awry. It’s easy to follow a script when things proceed exactly as you hope, but what do you do when a discussion veers off in a different direction? How do you proceed when students give an indication that they understand the material before you get through all of your examples? And what do you do when you’ve done your best but some students still are struggling to understand?
A simple example of on the fly decision making is paying attention to students while you make your way around the room. Which students are having interesting discussions about important concepts? Which student has discovered something that she should share with the entire class? This is sometimes called “the purposeful walk.” Teachers not only listen to student conversations, but they decide the order in which they will ask students to present their thinking. Often, teachers choose to elicit student comments so that the comments progress in complexity. Although these choices are made during class, predicting these types of decisions in your lesson plan will make instructional decision making much easier.
Read about one way Mike Melville chooses students to share solutions with the class:
Transcript from Mike Melville
For today’s homework, I already had decided that I was going to have a particular person from each group present. As I walk around [while the students are discussing the homework in groups] I’m looking for volunteers, I’m looking for different approaches to a problem.
|Listen to audio clip of teacher educator
Transcript from David Webb
In this lesson, you’ll see Orlando visiting student groups and gaining additional insight into where students thoughts are on this particular pattern. And all along the way, Orlando has the end goal in mind. He knows students will eventually get there, and he knows the techniques that he needs to use to guide that discussion. But he’s going to let these ideas emerge from the students first, and that way, students will be able to make sense of the formal notation that will come later in future lessons.
Because teaching is dynamic, you have to be flexible. If a student brings up an interesting observation or makes a mistake, you have a choice: pursue the topic and deviate from the lesson, or table the discussion and proceed with the lesson. You need to decide if the deviation will likely lead to the learning of significant mathematics or if it is simply a diversion that will keep the class from reaching the instructional objectives.
During Mike Melville’s lesson on exponents, the first discussion focused on the following problem, which he had assigned the previous night for homework:
Which is more money?
- One billion rallods.
- The amount obtained by putting one rallod on one square of a chessboard, two rallods on the next square, four on the next, and so on, until all 64 squares are filled.
In the course of discussing this problem, students offered the following attempts at a solution:
- Kelsey: “What our group came up with was, if it keeps doubling and there are 64 squares on the chess board, then you need to do 263 … and 63 because the first one, it’s … he starts with putting one on the chessboard, and 20 is 1. So, 63 instead of 64. That’s a lot bigger than a billion; it’s a quintillion.”
- Irving: “Basically, what we did is, since 20 equals 1, this would be the first square. And then this is pretty much this, just in a different sort of form. [On the board, he indicates 20 + 21 + 22 + … + 263 = 22016.] And after using the additive law of exponents, which is when we add the exponents, and this just means it goes all the way up to 63 … And that [he points to 22016] is just all of these exponents being added together.”
Notice that both of these attempts only hint at a correct solution to the problem. However, Melville does not focus on the solution. Instead, keeping in mind the goal of the lesson, he chooses to focus on students’ representation of powers, their observation of patterns and discovery of rules, and their understanding of exponents. Ultimately, he will get to the solution and to a solution method, but not quite yet. When he does think it appropriate to pursue a solution, he might approach the problem by having students look at a table of values like the one shown below:
Students should notice that the number in the “Running Total of Rallods on All Squares” column is always one less than the next power of 2.
Read Jane Schielack has to say about Mike Melville’s decisions during the Rallods discussion:
Transcript from Jane Schielack
There are many examples in Mike’s instruction [of] making decisions about what to pursue and what not to pursue because of the goal that he has in mind. For example, there are some representations of the first homework problem that don’t particularly connect to the question of what’s the sum on the chessboard. But because the focus of the lesson is on the use of exponents and the laws of exponents, his questions really focus on that part of the content, and he clearly decides not to worry about whether they are talking about sums or just the representation of each square, because that isn’t really an important part of the goal for his lesson that day.
While delivering a lesson, it is important to ask yourself, “Will asking a question that corrects an error in a student’s answer, points out a connection to another area of mathematics, or encourages an investigation that moves us closer to the goal?”
Hear what Jane Schielack has to say about opting not to correct student errors:
Transcript from Jane Schielack
There are some interesting examples in this lesson of places where our everyday language interferes with the mathematical understandings. For example, it’s very common and very well understood in the regular world that when we divide something in half, we are separating it into two equal parts; in mathematics, when we use the words “divide in half,” we think of the division symbol and we think the number 1/2. But those two things don’t go together symbolically in that situation … Mike chose not to follow up on some of the imprecise language about the exponents and about the bases and about division and multiplication because he wanted to focus on the patterns and generalizations for that particular lesson. Knowing that they would be using exponents for quite a bit after that, and he would have opportunities to refine that language as they went on.
It may also be necessary at times to engage in activities that may not seem relevant to students at the moment, but are necessary for you to attain your instructional goals. In Melville’s class, one student had represented the fraction 1/4 with the decimal 0.25. Melville asked the student to rewrite the decimal in fraction form. Although he didn’t explain to the class why he would like it to be written as a fraction, he knows that students will be more likely to see a pattern if the numbers are represented in fractional form. This instructional decision, which required the use of approximately 10 seconds of class time, set the stage for later activities.
Consider a time when a student’s action or response caused you to modify your instruction. Explain what the student said or did that caused you to modify your lesson plan. What modification did you make, and why did you think this would improve student progress toward the objectives?
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.