RESEARCH, REFLECTION, PRACTICE

B a r b a r a O ’ D o n n e l l a n d A n n Ta y l o r

A Lesson Plan as Professional Development? You’ve Got to Be Kidding! “I … thought I didn’t need to plan anything or even think about it because I had the lesson/problem right there. I can see the difference when we did take the time to really think about each aspect of the lesson.”—Gwen, after using a four-column lesson plan in a lesson study

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esson planning is central to good teaching, and yet it is also the butt of many jokes by teachers: Its purpose is seen as pleasing others, such as university supervisors or principals. Teachers often find the process tedious, time consuming, or just another hoop to jump through. So why would we suggest it as a way to improve teaching? We believe the lesson plan format proposed in this article requires teachers to grapple with content, examining it from many angles; use strategic knowledge to negotiate troublesome teaching situations and analyze student actions; and, above all, reflect on the teaching-learning process. According to Shulman (1987), teaching is a process of “pedagogical reasoning and action” that involves the need for teachers to grasp, probe, and comprehend an idea, to “turn it about in his or her mind, seeing many sides of it. Then the idea is shaped or tailored until it can in turn be grasped by students” (p. 13), but still this is not enough. Teachers also need to develop strategic knowledge (Shulman 1986) to confront troublesome, ambiguous teaching situations and build a “wisdom of practice” (p. 13). A multicolumn lesson-plan format is part of most lesson studies (Fernandez and Yoshida 2004; Barbara O’Donnell, [email protected], and Ann Taylor, [email protected], teach undergraduate and graduate courses that focus on mathematics methods and problem solving at Southern Illinois University, Edwardsville, IL 62026-1122. O’Donnell’s research interests include problem solving and action research as a way to improve teaching. Taylor is interested in all aspects of teacher development, especially classroom dialogue. Edited by Alfinio Flores, alfi[email protected], who teaches mathematics methods courses to future and current teachers in the Department of Curriculum and Instruction, Arizona State University, Tempe, AZ 85287-0911. “Research, Reflection, Practice” articles describe research and demonstrate its importance to practicing classroom teachers. Readers are encouraged to send manuscripts appropriate for this department by accessing tcm.msubmit.net.

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Lewis 2002) and has been used by many other educators, notably the Middle Grades Math Project, which evolved into the Connected Mathematics series (Lappan et al. 2004). The multicolumn lesson plan format helps teachers, professional developers, and mathematics educators raise important questions about pedagogy, mathematical content knowledge, and students’ thinking about and learning of mathematics.

Teachers’ Responses to the Four-Column Lesson Plan

For the past two years, as part of our research on using a four-column lesson plan (see fig. 1), we have collected data from thirty-three teachers and sixty preservice teachers. Teachers’ responses to this lesson-planning process indicate that they find it to be thought provoking and worthwhile, although they do not believe the process flows easily. They recommend using this format for problematic lessons when “you need to think it out from different angles”; it helps “keep us [teachers] from straying away from higher level thinking in our lessons.” In addition, beginning teachers also benefit from using it to “work out all the details” in a lesson. Figure 2 shows a complete listing of benefits and concerns raised by research. Here we summarize the teachers’ comments about the benefits: 1. Understanding the mathematics from students’ point of view and anticipating their responses. One teacher commented, “It [the four-column lesson plan] requires you to think in the mind of the student.” Another teacher noted that “the switch in roles allowed me to think as my students … to have a more personal relationship with my students.” These statements correspond directly with Shulman’s (1987) transformation subcategory of adaptation and tailoring to student characteristics in that this lesson plan asks teachers to “fit the represented material to specific students in one’s classroom rather than students in general” (p. 17).

Teaching Children Mathematics / December 2006/January 2007 Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Figure 1 Components of the four-column lesson format with suggested planning questions for teacher reflection

Description of Task Portions with Allotted Time • Who is the principal participant of each segment? • Do students have ample segments in which they are principal investigators? • Am I, as the teacher, dominating the lesson by directing most of the lesson? • Am I allotting enough time for students to understand and engage in the activity within each segment?

Teacher Activity • Have I worked through the mathematics of this problem myself? • Do the questions I ask and the statements I make help students understand the concepts? • Am I using proper terminology and phrasing to help students make the connections needed? • Am I giving students too much information or not enough to engage with this activity?

Anticipated Student Activity and Thinking • Have I anticipated students’ misconceptions as well as their understandings? • Have I anticipated at least three different ways students may understand the mathematical ideas in response to my directions?

Intervention: Anticipated Actions and Questions to Keep Task at High Level of Cognitive Demand • Are my interventions helping students get at the mathematical concepts and connections in the problem? • How will my interventions maintain or increase the lesson’s high cognitive demand on some or all children?

Follow-Up Reflection Questions • Was I able to anticipate what my students would do and say? • Were my interventions effective? What ideas did my students have that I did not anticipate? • In what ways did the lesson plan help my teaching of the lesson? • Did the quality of my interactions with students improve? • What changes do I need to make to maintain the mathematical challenge for all my students? • Why were these students off task? • What possible mathematical misconceptions might they have to make this happen? • What didn’t these students understand? • What was preventing them from completing the task?

As a result, teachers felt that their lessons were “better developed” and that they connected more fully with the “deeper meaning in the lessons” and student learning. Many also stated that the plan helped them think about “maintaining higher level thinking.” Because they were able to see the dialogue that would unfold in the lesson, they stated overwhelmingly that they needed to improve their questioning techniques by devising more “openended questions,” “allowing more think time,” and “not telling or leading their students.” This process of anticipating student responses was both the greatest benefit and the greatest challenge, because anticipating what students would say or do—in particular, determining what questions or problems students might have—is not a familiar step for many teachers. We expected this response from preservice teacher candidates, because of their limited exposure to teaching and planning, but we also found that new and veteran teachers had the same reaction. As one teacher pointed out, “Only

having three years of teaching experience, I didn’t feel comfortable enough [in] knowing exactly how to question.” Even teachers with as many as twenty-two years of experience felt that anticipating student questions and actions was new to them, “something unfamiliar” to their everyday work. We suggest some possible reasons for this difficulty in anticipating student responses. First, teachers who use direct instruction methods and rely on the scope and sequence of their text typically are focused on the lesson and their actions and do not necessarily think about how students will make sense of the lesson. Second, many teachers are grappling with content. They may not understand the connections within the discipline and are unable to dissect the content to anticipate students’ misconceptions or questions. Place value is one such area; although teachers instruct students in how to use algorithms, they do not always understand what they represent and why they work (Ma 1999).

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Figure 2 Facilitators’ perceived benefits and concerns about the four-column lesson plan Benefits

Concerns or Problems

• Requires focus on students and their learning • Anticipates and considers student thinking for a wide range of abilities • Raises questions about different ways of solving a problem • Specifies exact language; imagines sequences of dialogue and how to develop dialogue • Links teacher pedagogical moves to student content ideas • Requires anticipating short cycles of interaction in which mathematics connections as well as main ideas are taught • Requires thinking through the content and highlighting content uncertainties • Dissects each teacher action, examines its effectiveness for students, and provides interventions • Shifts lesson planning away from thinking of teaching as a sequence of activities to thinking of teaching as the interaction between teaching and learning • Breaks out of the false procedural-conceptual dichotomy; switches the conversation from the teacher’s actions to what the teacher is doing, saying, and thinking in relation to what the student is doing, saying, and thinking • Can be used for any topic • Can be used individually or in a collaborative group

If teachers have so much difficulty anticipating student questions and actions, one might say that this plan is too difficult, not appropriate for preservice teachers. According to data collected, preservice teachers appreciated the level of detail involved, literally “everything that happens during a lesson … not just how to teach it.” They felt their chances of teaching a successful lesson were improved because they were prepared with specific questions and had analyzed the content to anticipate students’ responses. As one preservice teacher stated, “It [the four-column lesson plan] forces you to expect the unexpected.” Another acknowleged that, as a result of the thorough planning involved, “it has helped build my confidence in math.” 2. Planning for misconceptions and problems, which leads to extending mathematical thinking. Another benefit that many teachers identified was that they were able to extend the lesson into more complex 274

• May be perceived as irrelevant by preservice or novice teachers • Not a natural action for new teachers in “the fantasy stage” of teaching • Challenging to do, requiring thought and persistence • Not a regular part of experienced teachers’ habits; they are not used to thinking about teaching in such detail (teacher culture militates against such thought) • May not know content well enough to anticipate how the task will unfold • Lack of experience in asking critical questions and formulating ones that are other than procedural • Overuse of textbooks, indicating that teaching has become a process of managing and distributing worksheets • May require word processing skills to achieve layout

mathematical ideas. Specifically, their prediction of students’ misconceptions and their planning for how to respond led to modification and adaptations. One teacher stated, “I began to brainstorm ideas of how to extend the lesson…. I realized many other things I could be doing to extend the activity.” This lesson plan requirement supports Shulman’s (1987) statement that effective teachers comprehend “how a given idea relates to other ideas within the same subject area and to ideas in other subjects” (p. 14). In this same vein, other teachers remarked that the detailed planning involved in this lesson helped address classroom management problems, because the teacher anticipated problem areas in the lesson.

Implementing the Four-Column Lesson Plan

If possible, find a colleague or two who are interested in improving their teaching and invite them to

Teaching Children Mathematics / December 2006/January 2007

read this article. Decide how much time you want to use to try this process. Teachers we work with soon find that the process is so meaningful that they want to spend three to four hours planning just one lesson. Remember: You are completely in charge, so you can spend as much or as little time as you want on the process. Start with a one-hour block or arrange several shorter sessions. Check to see if your administrator will support this as professional development time. In preparation, identify one lesson that was particularly difficult for you or your students. Do not be tempted to think of this as a “rewriting” of a block of curriculum. Remember that the purpose is intensive planning of one lesson. Now go through the following four-step process. Do not worry if you do not complete each step; often you will want to move on to another segment of planning and then cycle back through an earlier step in light of new insights.

Step 1: Description of task portions with allotted time Divide the lesson into its natural segments and add the expected time commitment for each segment of the lesson. For example, the lesson might begin with a segment in which the teacher introduces a task or problem and invites the students individually to construct their own ideas about the problem. A new segment could begin with the students talking in groups, sharing the ideas they just formulated about the problem. As activities change, add new rows to the lesson plan along with approximate times for each segment. Add these new rows under column 1 with the approximate time. Questions for reflection would include these: • Who is the principal participant of each segment? • Do students have ample segments in which they are principal investigators? • Am I, as the teacher, dominating the lesson by directing most of the lesson? • Am I allotting enough time for students to understand and engage in the activity within each segment?

Step 2: Teacher activity Start writing out exactly what you will say to students at each stage of the lesson. Record the exact wording in column 2: “Teacher activity.” To do this, you will need to understand the deeper concepts embedded in the lesson. Formulate probing, thought-provoking questions that promote higherorder thinking, and be prepared with follow-up

questions you can ask to help students unpack the mathematics in the lesson. Enjoy dissecting and investigating the lesson concepts and sequence and content yourself with gaining a deeper understanding of the concepts embedded in the lesson. Example: One teacher was attempting to use the four-column lesson plan for a problem-solving activity that involved a task commonly known as the Locker problem. In the problem, students open and close 100 lockers on the basis of a sequence of multiples. This teacher realized that to complete column 2 in the lesson plan, she had to thoroughly solve the problem and work through it step by step. By planning exactly what she intended to say, she was able to see the reasoning behind why some lockers remained open and others were closed. Do not be surprised if you need to think and research the mathematical ideas—this is part of the process. You may find yourself reading past issues of Teaching Children Mathematics or sections of books such as Chapin and Johnson’s Math Matters: Understanding the Math You Teach, Grades K–6 (2000) or doing research on the Internet. Questions for reflection would include these: • Have I worked through the mathematics of this problem myself? • Do the questions I ask and the statements I make help students understand the concepts? • Am I using proper terminology and phrasing to help students make the connections needed? • Am I giving students too much information or not enough to engage with this activity?

Step 3: Anticipated student activity and thinking So far, so good. Now comes the challenging part: Try to anticipate what students will think and do in response to your actions and statements. Draw on your past experience in teaching this topic. Just as Shulman (1986, 1987) suggests, you are delving into your “wisdom of practice.” Ask colleagues what they believe students’ mathematical responses will be. You may do more research here if you like. Completing this column is difficult, so just do your best and make a note to pay particular attention during the lesson to see if you can identify new and different ways students understand the mathematics. Add these ideas to column 3. (After teaching the lesson, you may find it helpful to revisit this section and incorporate questions that the students raised during the lesson. Doing so will help you begin to build your own collection of valuable classroomtested resources.)

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Figure 3 A portion of a four-column lesson plan developed by an in-service teacher This in-service teacher introduced the problem this way: “How many of you have ever had a lemonade stand at a garage sale or ball game? Today, we are going to see what Maria and Chuck did with their lemonade stand. Let’s read the problem together.” The teacher and students read the following problem together. Maria and Chuck have set up a lemonade stand at the local Little League Park. Their parents have given them all the ingredients and supplies. They are going to sell each glass of lemonade for 25¢. How much money will Maria and Chuck make if they sell 100 glasses of lemonade? 250 glasses? How many glasses must they sell to make $100? What pattern can you discover in the sales data? After answering any questions, the in-service teacher sets the students to work individually, first to think through the problem and then to try different strategies. Here is the next segment of her lesson. Description of Task Portions with Allotted Time Individual work time (5 minutes)

Teacher Activity

Intervention: Anticipated Actions and Questions to Keep Task at High Level of Cognitive Demand

Anticipated Student Activity and Thinking

I begin with having the students work independently to formulate ideas about how to start the task with these directions:

Some students may make a table to develop the pattern.

“Please record your attempts in your math notebook, and when something occurs to you, write down what you are thinking.”

Some students will begin multiplying 100 glasses of lemonade and the cost of 25¢ per glass (100 × 0.25).

They may fill in every line of the table, showing sales for 1 glass all the way to 100 glasses.

Other students may look at the table and see the pattern that for every ten glasses of lemonade, the sales amount increases by $2.50. Other students may try the last question first. They would divide $100 by the cost per glass to get the answer. This division problem may be a stumbling block because of the decimal point.

I will look at the table and how the students are calculating the data. I may prompt, “I see that you are completing the table line by line. Is there a way you might fill the table faster with larger increments?” I will look at their algorithm and ask them, “Why did you choose to multiply?” or “Why did you choose to use those numbers?” I will ask them to explain by using the following questions: “What pattern do you see?” “What number would come next in the table?” I may prompt, “Is there another way to solve this problem without using division?”

Note: Other sections—strategy sharing with the whole group, group work time, discussion, and conclusion—followed the portion of the lesson depicted in this sample.

Example: In the Fencing task, students are asked to build a rectangular rabbit pen that allows the most space when using 24 feet of fencing (Stein et al. 2000). Many students’ thinking while solving this problem is faulty. One error in student thinking that should be listed in column 3 is that students may confuse perimeter with area and construct a fence surrounding an area of 24 square feet. Questions for reflection would include these: • Have I anticipated students’ misconceptions as well as their understandings? • Have I anticipated at least three different ways students may understand the mathematical ideas in response to my directions? 276

Step 4: Intervention: Anticipated actions and questions to keep task at high level of cognitive demand Misconceptions are part of student learning, and if the lesson is going well, these misconceptions will begin to surface. Effective teachers think about student errors but seldom record these in detail in a lesson plan. Think about how you will respond to each of the anticipated student responses recorded in step 3. Start first with the most significant ones, those that relate to the key mathematical ideas, and prepare your specific questions and activities to engage the students

Teaching Children Mathematics / December 2006/January 2007

in rethinking their misconceptions. Add your responses to column 4. Example: Let’s return to the Locker problem. As students work through the problem by using concrete objects and by constructing charts, they often jump to conclusions about the patterns they are recording. They see a pattern of open lockers that increases by 2 between each closed locker. They can then solve for the number of closed lockers fewer than 100, but they still may not realize why the lockers are closed. The teacher needs to redirect their reasoning by asking such questions as, “Why are these particular lockers closed?” “How many people touched the open lockers?” “How many people touched the closed lockers?” “What makes the closed lockers different from the open ones?” Such interventions are easily omitted yet are a crucial part of planning. Research presented by Stein et al. (2000) illustrates that preparing a challenging task or lesson is not enough. How the teacher and the students allow the task to be implemented determines the amount and type of student learning. Questions for reflection would include these: • Are my interventions helping students get at the mathematical concepts and connections in the problem? • How will my interventions maintain or increase the lesson’s high cognitive demand on some or all children? See figure 3 for an example of one portion of a four-column lesson plan created by an in-service teacher.

Step 5: Go teach it! Now that your planning is complete, it is time to go teach your four-column lesson plan! Remember that the purpose of this planning process is to help you dig deeper into the complex activities you engage in daily and to support your deeper understanding of how your students learn mathematics. Questions you can ask yourself would include these: • Was I able to anticipate what my students would do and say? • Were my interventions effective? What ideas did my students have that I did not anticipate? • In what ways did the lesson plan help my teaching of the lesson? • Did the quality of my interactions with students improve?

• What changes do I need to make to maintain the mathematical challenge for all my students? If you find yourself commenting on classroom management issues, such as “Groups were tempted to get out of control and off task by talking to students in other groups,” try asking the following questions to examine how the mathematical ideas may be linked to behavior issues: • Why were these students off task? • What possible mathematical misconceptions might they have that caused this to happen? • What did these students not understand? • What was preventing them from completing the task? In short, reflect on what happened, revise your plan, add what you did not anticipate, change what did not work, and use your plan again.

How You Can Use This Process

1. On your own—This lesson-planning process benefits both preservice teacher candidates and in-service teachers. Although it cannot be used for every lesson you teach, it is a good analytical process for problematic or important lessons. One teacher used the plan for her yearly lesson evaluation and received rave reviews from her principal for her questioning techniques. She attributed her success to the detailed questions she prepared and her anticipation of student responses. 2. With colleagues—Working with a small group of teachers who can help analyze the lesson and suggest interventions is very effective. Many teachers around the country are forming lesson-study groups to analyze and improve their own teaching. This lesson plan provides a tool for discussion of content, student thinking, student misconceptions, possible interventions, and critical questioning. 3. At a professional development workshop— This process works well in a half-day format. If a number of groups are involved, follow-up sessions can be scheduled to discuss the results of the lessons. 4. In graduate or undergraduate courses—We try to plant the seed of always considering three things: the content, how students think about the content, and how to respond to misconceptions about the key ideas. The process helps preser-

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vice teachers develop their critical questioning techniques and focus squarely on how students learn. Why does producing a detailed lesson plan feel so satisfying to teachers? Teachers often report that they are not allowed time for such in-depth planning. They are often on their own in their classrooms and responsible for their own encouragement and professional development. We believe this lesson plan provides those teachers who want to change their practice the capability to do so, alone or in groups. This planning process requires that teachers focus on teacher-student interaction, not just lesson activities; grapple with the mathematical content of the lesson and how students think and act on it; formulate questions that facilitate conceptual learning over procedural understanding; develop critical thinking skills, such as questioning; and reflect on the entire process. Thus, the time invested in the four-column lesson plan results in real benefits to teachers and their students. Put simply, teachers feel that they are developing as professionals and doing what they think teaching

is all about. As Shulman (1987) has articulated, they are engaging in “pedagogical reasoning and action,” and the ensuing “new comprehensions” are very satisfying (p. 13).

Final Remarks The format proposed in this article is consistent with the work of Shulman (1987) as presented in his model of pedagogical reasoning and action. The lesson plan causes teachers to think about the interactions taking place during the act of teaching instead of merely focusing on directing a sequence of activities. As Shulman’s model proposes, teachers work through the cycles of comprehension, transformation, and instruction when preparing this lesson plan. But while teachers such as Gwen are experiencing the benefits of working with others in lesson studies (Lewis 2002; Takahashi and Yoshida 2004), not all teachers can find the time or the colleagues to engage in a full lesson study. We have found that teachers working alone find great benefit from the thinking that is required from planning a single lesson using the four-column lesson plan.

References Chapin, Susan H., and Art Johnson. Math Matters: Understanding the Math You Teach, Grades K–6. Sausalito, CA: Math Solutions Publications, 2000. Fernandez, Clea, and Makoto Yoshida. Lesson Study: A Japanese Approach to Improving Mathematics Teaching and Learning. Mahwah, NJ: Lawrence Erlbaum, 2004. Lappan, Glenda, James T. Fey, W. M. Fitzgerald, Susan N. Friel, and Elizabeth D. Phillips. Connected Mathematics. Needham, MA.: Pearson Prentice Hall, 2004. Lewis, Catherine C. Lesson Study: A Handbook of Teacher-Led Instructional Change. Philadelphia, PA: Research for Better Schools, 2002. Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum, 1999. Shulman, Lee S. “Those Who Understand: Knowledge Growth in Teaching.” Educational Researcher 15, no. 1 (1986): 4–14. ——. “Knowledge and Teaching: Foundations of the New Reform.” Harvard Educational Review 57 no. 1 (1987): 1–22. Stein, Mary Kay, Margaret Schwan Smith, Marjorie A. Henningsen, and Edward A. Silver. Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development. Reston, VA: National Council of Teachers of Mathematics, 2000. Takahashi, Akihiko, and Makoto Yoshida. “Ideas for Establishing Lesson-Study Communities.” Teaching Children Mathematics 10 (May 2004): 436–43. s 278

Teaching Children Mathematics / December 2006/January 2007

A Lesson Plan as Professional Development?

In what ways did the lesson plan help my teaching of the lesson? • Did the quality of my interactions ... the teacher anticipated problem areas in the lesson. Implementing the .... (2000) or doing research on the Internet. Questions for reflection ...

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