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Insights Into Algebra 1 - Teaching For Learning
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Workshop 6 Exponential Functions Teaching Strategies
Teaching Strategies:

Affective Domain

Instructional Decision Making
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Instructional Decision Making

Planning the Lesson
During the Lesson

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:

Read transcript from teacher educator 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... Read More

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:
  • Objective(s)
    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.

  • Standard(s)
    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:

Read transcript from teacher educator Jane Schielack
Orchestrating student input in a classroom can become almost chaotic if you lose sight of the goal... Read More

  • Development
    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:

Read transcript from teacher 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... Read More

Read Jane Schielack's thoughts about selecting homework problems to discuss in class:

Read transcript from teacher educator 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]... Read More

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.



Reflection:
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?

record your thoughts in your journal


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:

Read transcript from teacher Mike Melville
For today's homework, I already had decided that I was going to have a particular person from each group present... Read More

Hear what David Webb has to say about keeping the goal in mind:

Listen to audio clip of teacher educator
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... Read More

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?
  1. One billion rallods.
  2. 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:

Square Number of Rallods on Square Running Total of Rallods on All Squares
1 1= 20 1
2 2= 21 3
3 4= 22 7
4 8= 23 15
5 16= 24 31
6 32= 25 63
64 263 264 - 1

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:

Read transcript from teacher educator 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... Read More

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:

Listen to audio clip of teacher educator
Jane Schielack
There are some interesting examples in this lesson of places where our everyday language interferes with the mathematical understandings... Read More

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.

Reflection:
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?

record your thoughts in your journal

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