Book: Elementary and middle school mathematics : teaching developmentally

Van de Walle, J. A., Brown, E. T., Karp, K., Wray, J. A., & Bay-Williams, J. M. (2019).
Elementary and middle school mathematics : teaching developmentally (Tenth edition).
Pearson.

Chapter 3: Teaching through problem solving

(pp. 54-80) 

EIGHT MATHEMATICAL TEACHING PRACTICES THAT SUPPORT STUDENT LEARNING.

Teaching Practice	To Enact the Mathematics Teaching Practice, a Teacher:
1. Establish mathematics goals to focus learning	Articulates clear learning goals that identify the mathematics students will learn in a lesson or lessons.
	Identifies how the learning goals relate to a mathematics learning progression.
	Helps students understand how the work they are doing relates to the learning goals.
	Uses the articulated goals to inform instructional decisions involved in planning and implementation.
2. Implement tasks that promote reasoning and problem solving	Selects tasks that:
	Have maximum potential to build and extend students? current mathematical understanding.
	Have multiple entry points.
	Require a high level of cognitive demand.
	Supports students to make sense of and solve tasks using multiple strategies and representations, without doing the thinking for the students.
3. Use and connect mathematical representations	Supports students to use and make connections between various representations.
	Introduces representations when appropriate.
	Expects students to use various representations to support their reasoning and explanations.
	Allows students to choose which representations to use in their work.
	Helps students attend to the essential features of a mathematical idea represented in a variety of ways.
4. Facilitate meaningful mathematical discourse	Facilitates productive discussions among students by focusing on reasoning and justification.
	Strategically selects and sequences students? strategies for whole class discussion.
	Makes explicit connections between students? strategies and ideas.
5. Pose purposeful questions	Asks questions that
	Probe students? thinking and that require explanation and justification.
	Build on students? ideas and avoids funneling (i.e., directing to one right answer or idea).
	Make students? ideas and the mathematics more visible so learners can examine the ideas more closely.
	Provides appropriate amounts of wait time to allow students to organize their thoughts.
6. Build procedural fluency from conceptual understanding	Encourages students to make sense of, use, and explain their own reasoning and strategies to solve tasks
	Makes explicit connections between strategies produced by students and conventional strategies and procedures
7. Support productive strugglein learning mathematics	Helps students see mistakes, misconceptions, na?ve conceptions, and struggles as opportunities for learning.
	Anticipates potential difficulties and prepares questions that will help scaffold and support students? thinking.
	Allows students time to struggle with problems.
	Praises students for their efforts and perseverance in problem solving.
8. Elicit and use evidence of student thinking	Decides what will count as evidence of students? understanding.
	Gathers evidence of students? understanding at key points during lesson.
	Interprets students? thinking to gauge understanding and progress toward learning goals.
	Decides during the lesson how to respond to students to probe, scaffold, and extend their thinking.
	Uses evidence of students? learning to guide subsequent instruction.
Source: Based on Principles to Actions: Ensuring Mathematical Success For All (NCTM), ? 2014.

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Invites creativity Students enjoy the creative process of problem solving, searching for
patterns, and showing how they figured something out. Teachers find it exciting to see
the surprising and inventive ways students think. Teachers know more about their students
and appreciate the diversity within their classrooms when they focus on problem
solving.

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LEVELS OF COGNITIVE DEMAND

Certainly one of the most powerful features of a worthwhile task is that the problem that begins the lesson can get students excited about learning mathematics.

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Classroom A: “Today we are going to use grid paper to show the sub-products when we multiply 2 two-digit numbers.”

Classroom B: “The school is planning a fall festival and class is selling water. The principal said we have 14 cases of bottled water in the storage closet. I looked in the storage closet and could see that a case had 7 rows, with 5 bottles in each row. How can we use the information about one case to figure out how many bottles of water we have?” If we sell each one for $2.00, how much money might we make?

Contexts must reflect the cultures and interests of the students in your classroom

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Using everyday situations can increase student participation, increase student use of different problem strategies, and help students develop a productive disposition (Tomaz & David, 2015).

Picture books, poems, media, and chapter books can be used to create high cognitive demand tasks with multiple entry points. An example of literature lending itself to mathematical problems is the very popular children’s picture book Two of Everything (Hong, 1993). In this magical Chinese folktale, a couple finds a pot that doubles whatever is put into it. (Imagine where the story goes when Mrs. Haktak falls in the pot!). Students can explore the following problem: How many students would be in our class if our whole class fell in the Magic Pot?

Literature ideas are also found in articles in journal articles (e.g., Teaching Children Mathematics) and teacher books (e.g., Math & Literature Series, Using Children’s Literature to Teach Problem Solving in Math (White, 2014)).

Connect to Other Disciplines. Interdisciplinary lessons help students see connections among the courses/topics they are studying, which often feel completely separate to them

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For example, in kindergarten, students can link their study of natural systems in science to mathematics by sorting leaves based on color, smooth or jagged edges, feel of the leaf, and shape. Students learn about rules for sorting and can use Venn diagrams to keep track of their sorts. They can observe and analyze what is common and different in leaves from different trees. Older students can find the perimeter and area of various types of leaves and learn about why these perimeters and areas differ.

The social studies curriculum is rich with opportunities to do mathematics. Timelines of historic events are excellent opportunities for students to work on the relative sizes of numbers and to make better sense of history. Students can explore the areas and populations of various countries, provinces, or states and compare the population densities, while in social studies they can talk about how life differs between regions with 200 people living in a square mile and regions with 5 people per square mile.

Beware of the low-level cognitive demand tasks cloaked in clever artwork—they may look fun, but if the mathematics is not problematic, those graphics are not going to help your students think at a high level. Make sure it is the mathematics itself that is clever and engaging.

Task Evaluation and Selection Guide

Just as students become adept at problem solving strategies, with time and commit-ment you will become adept at evaluating and adapting tasks to better support student learning.

To help students become productive mathematical thinkers, teachers must be comfortable with uncertainty, ask key questions, be able to respond to students, probe student thinking, prompt students to reflect on their thinking, and know the difference between productive and nonproductive struggle (Heaton & Lewis, 2011; Kazemi & Hintz, 2014; Towers, 2010).

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TALK MOVES FOR SUPPORTING CLASSROOM DISCUSSIONS

Developing mathematical communities

• Encourage student—student dialogue rather than student—teacher conversations that exclude the rest of the class. When students have differing solutions, have students work these ideas out as a class. “George, I noticed that you got a different answer than Tomeka. What do you think about her explanation?”
• Encourage students to ask questions. “Pete, did you understand how they did that? Do you want to ask Antonio a question?”
• Ask follow-up questions whether the answer is right or wrong. Your role is to understand student thinking, not to lead students to the correct answer. So follow up with probes to learn more about their answers. Sometimes you will find that what you assumed they were thinking is not accurate. And if you only follow up on wrong answers, students quickly figure this out and get nervous when you ask them to explain their thinking.
• Call on students in such a way that, over time, all students are able to participate. Use time when students are working in small groups to identify interesting solutions that you will highlight during the sharing time. Be intentional about the order in which the solutions are shared; for example, select two that you would like to compare presented back-to-back. All students should be prepared to share their strategies.
• Demonstrate to students that it is okay to be confused and that asking clarifying questions is appropriate. This confusion, or disequilibrium, just means they are engaged in doing real mathematics and is an indication they are learning.
• Move students to more conceptually based explanations when appropriate. For example, if a student says that he knows 4.17 is more than 4.1638, you can ask him (or another student) to explain why this is so. Say, “I see what you did but I think some of us are confused about why you did it that way.”
• Be sure all students are involved in the discussion. English Learners, in particular, need more than vocabulary support; they need support with mathematical discussions (Moschkovich, 1998). For example, you can use sentence starters or examples to help students know what kind of responses you are hoping to hear and to reduce the language demands. Sentence starters can also be helpful for students with disabilities because it adds structure. You can have students practice their explanations with a peer. You can invite students to use illustrations and actual objects to support their explanations. These strategies benefit not just the ELs and other students in the class who struggle with language, but all students.

Warning signs that you are taking over children’s thinking include interrupting a child’s strategy or explanation, manipulating the tools instead of allowing the child to do so, and asking a string of closed questions (Jacobs, Martin, Ambrose, & Philipp, 2014). Taking over children’s thinking sends the message that you do not believe they are capable and can inhibit the discourse you are trying to encourage.

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Questions focused on con-ceptual knowledge and making connections include, “Will this rule always work? (Why?)” “When will this strategy work?” “How does the equation you wrote connect to the pic-ture?” and “Why use common denominators to add fractions?”

One common pattern goes like this: teacher asks a question, student answers the question, teacher confirms or challenges answer. This “initiation-response-feedback” or “IRF” pat-tern does not lead to classroom discussions that encourage all students to think. Another pattern is “funneling,” when a teacher continues to probe students in order to get them to a particular answer. This is different than a “focusing” pattern, which uses probing ques-tions to help students understand the mathematics. The talk moves described previously are intended for a focusing pattern of questioning.

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Who is doing the thinking. You must be sure your questions engage all students! When you ask a great question, and only one student responds, then students will quickly figure out they don’t need to think about the answer and all your effort to ask a great question is wasted. Instead, use strategies to be sure everyone is accountable to think about the ques-tion you posed. Ask students to first write their ideas on a notecard or individual white board, then talk to a partner about the question, and finally ask for contributions as a whole class. This think-pair-share strategy maximizes student participation.

When you confirm a correct solution rather than use one of the talk moves, you lose an opportunity to engage students in meaningful discussions about mathematics and thereby limit the learning opportunities. Use student answers to find out if other students think the conclusions made are correct, whether they can justify why, and if there are other strategies or solutions to the problem.

Three things that teachers need to tell students are (Hiebert et al., 1997):

• Mathematical conventions. Symbols such as + and =, terminology, and labels are conventions. As a rule of thumb, symbolism and terminology should be introduced after concepts have been developed as a means of expressing or labeling ideas.
• Alternative methods. When an important strategy does not emerge naturally from stu-dents, then the teacher needs to introduce the strategy, being careful to introduce it as “another” way, not the only or the best way.
• Clarification or formalization of students’ methods. It is appropriate and necessary to help students clarify or interpret their ideas and point out related ideas. A student may add 38 and 5 by noting that 38 and 2 more is 40 with 3 more making 43. This strategy can be related to the Make 10 strategy used to add 8+5. Drawing everyone’s attention to this connection can help other students see the strategy, while also building the confidence of the student(s) who originally proposed the strategy.

Writing improves student learning and understanding (Pugalee, 2005; Steele, 2007). The act of writing is a reflective process and involves students in metacognition, which is connected to learning (Bransford, Brown, & Cocking, 2000). Metacognition refers to conscious monitoring (being aware of how and why you are doing something) and regulation (choosing to do something or deciding to make changes) of your own thought process.

Writing plays a critical role in classroom dis-course. Writing can serve as a rehearsal for a class-room discussion. It can be difficult for students to remember how they solved a problem.

(Alison’s Note – could this be used during discussion time? Would it be helpful for akonga with short term memory issues (e.g. ADHD) to write down questions or thoughts, or would this provide too much distraction? Could this be useful for akonga with autism to practice what they want to say?)

The I-THINK framework supports the problem solving process and metacognitive skills (Lynch, Lynch, & Bolyard, 2013; Thomas, 2006) pg77:

Individually think about the task

Talk about the problem.

How can it be solved?

Identify a strategy to solve the problem.

Notice how your strategy helped you solve the problem.

Keep thinking about the problem. Does it make sense? Is there another way to solve it?

Using organizers like I-THINK can help students be more aware of their mathematical thinking and better at communicating that thinking to you, which provides you with better formative assessment data.

Chapter: 4

After reading this chapter and engaging in the embedded activities and reflections, you should be
able to:
4.1 Explain the features of a three-phase lesson plan format for problem-based lessons.
4.2 Design lessons using a planning process focused on mathematical inquiry.
4.3 Describe specific lesson design ideas, including ways to differentiate instruction.
4.4 Explain strategies for working with families, including effective homework practices.

…a problem-solving lesson

• approaches learning as a complex process,
• builds on each student’s prior knowledge,
• prioritizes making connections among mathematical ideas,
• incorporates the mathematical practices.

Preparing a lesson shifts from preparing an agenda of what will happen to creating a “thought experiment” to consider what might happen (Davis, Sumara, & Luce-Kapler, 2008).

A Three-Phase Lesson Format

1. what happens to set up that inquiry (before phase)
2. what happens as they explore (during phase)
3. what happens after the task is solved (after phase)

The Before Lesson Phase

The before phase is the first 5 to 10 minutes of a lesson and is intended to get students ready for a
focus task or exploration that engages them in higher-level thinking and using the mathematical
practices.

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Activate Prior Knowledge

• remind students what they have previously learned
• connect to their personal experiences

If the problem does not begin with a context you can add one. This helps students see mathematics as relevant and helps them make sense of the mathematics in the problem.
Sometimes activating prior knowledge involves vocabulary and possible tools that might be used for the focus task.

Consider the following open-ended task exploring perimeter (based on Lappan & Even, 1989).

Instead of beginning your lesson with this problem, you might consider activating prior knowledge
in one of the following ways:

• Draw a 3-by-5 rectangle of squares on the board and ask students what they know about the shape. (It’s a rectangle. It has squares. There are 15 squares. There are three rows of five.) If no one mentions the words area and perimeter, you could write them on the board and ask if those words can be used in talking about this figure.
• Provide students with some square tiles or grid paper and say, “I want everyone to make a shape that has a perimeter of 12 units. After you make your shape, find out what its area is.” After a short time, have several students share their shapes. Students can also use a virtual geoboard, like the one found at the Math Playground.

Each of these warm-ups uses the vocabulary needed for the lesson. The second activity suggests the tiles as a model students may elect to use and introduces the idea that there are different figures with the same perimeter.

Be Sure the Task Is Understood

analyze the problem and anticipate student approaches and possible misinterpretations or misconceptions (Wallace, 2007). Time spent building understanding of the task is critical to the rest of the lesson.

Ask questions such as,

• What are we trying to figure out?
• Do we have enough information?” and
• What do you already know that can help you get started?”

The more questions raised and addressed prior to the task, the more engaged students will be in the during phase.

The most difficult facts can each be connected or related to a fact already learned, called a foundational fact or known fact

Ask, “When you learned addition facts, how could knowing 6 + 6 help you find the answer to 6 + 7” and follow up with asking, “What might a known fact be in multiplication?”

In the case of story problems it is important to help students understand the meaning of the sentences without giving away how to solve the problem. This is particularly important for ELs and students who are struggling readers.

Questions to ensure understanding include:
What is the problem asking?
How does the candy store buy candy?
What is in a carton?
What is in a box?
What does that mean when it says ‘each box’?
The last question here is to identify vocabulary that may be misunderstood. Asking students to reread a problem does little good, but asking students to restate the problem or tell what question is being asked helps students be better readers and problem solvers.

Establish Clear Expectations.

There are two components to establishing expectations:
(1) how students are to work and
(2) what products they are to prepare for the discussion.

it is essential to have students be individually accountable and also work together.

One way to address both individual accountability and sharing with other students is a think-write-pair-share approach (Buschman, 2003b). The first two steps are done individually, and then students are paired for continued work on the problem. With independent written work to share, students have something to talk about.

writing supports student learning in mathematics, and having multiple ways to demonstrate knowledge is important for providing access to all learners (multiple exit points).

One effective strategy is to have each student write and illustrate their solution independently, then present the team’s solution as a group, with each person sharing a part of the presentation.

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These before phase goals may occur in any order.

The During Lesson Phase

Once curiosity has been piqued, students engage in mathematical activity alone, with partners, or in small groups to explore, gather and record information, make and test conjectures, and solve the mathematical task.

In this during phase, students need to have access to tools, such as models, images, diagrams, and notation. In making instructional decisions in the during phase you must ask yourself,
Does my action lead to deeper thinking or is it taking away the thinking?
These decisions are based on carefully listening to students and knowing the content goals of the lesson.

The four goals of the “during phase”:

• Let Go!
• Notice Students’ Mathematical Thinking
• Provide Appropriate Support
• Provide Worthwhile Extensions

Once students understand what the problem is asking, it is time to let go. Encourage
students to embrace the struggle. Doing mathematics takes time, and solutions are not
always obvious. It is important to communicate to students that spending time on a task,
trying different approaches, and consulting each other are important to learning and understanding
mathematics. Although it is tempting to rescue students who are feeling frustrated
and uncertain in the during phase, they will learn more if you provide support without just
showing/telling them how. Avoid these three teaching moves, which often result in you (the
teacher) taking over a student’s thinking: (1) interrupting a student’s strategy, (2) manipulating
the tools, and (3) asking a series of closed questions ( Jacobs, Martin, Ambrose, & Philipp,
2014). Instead try to understand the student and focus on helping them navigate the problem.
Ask questions like:
“What is this problem asking you to do?”
“How have you organized the information?”
“What about this problem is difficult?” and
“Is there a different strategy (or manipulative) that you might try?”
These questions support student thinking, yet do not tell them how to solve the problem.

Avoid being the source of right and wrong

The correctness of an answer lies in the justification, not in the teacher’s brain or answer key.

Letting go also means allowing students to make mistakes. When students make mistakes (and when they are correct), ask them to explain their process or approach to you.

Notice Students’ Mathematical Thinking

• what are different students are thinking
• what ideas they are using
• how they are approaching the problem

This is a time for observing, listening, and interacting with students. “Professional noticing” means that you are trying to understand a student’s approach to a problem and decide an appropriate response to extend that student’s thinking in the moment (Jacobs, Lamb, & Philipp, 2010).

This is very different from listening for or leading students toward an answer or approach that you have pre-selected or assumed.

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The during phase is a great opportunity to find out what your students know, how they think, and how they are approaching the task you have given them. As students are working, any of the following prompts can help you notice what they know and are thinking:

• • Tell me what you are doing.
• • I see you have started to [multiply/subtract/etc.] these numbers. Can you tell me why?
• • Can you tell me more about . . . ?
• • Why did you . . . ?
• • How did you solve it?
• • How does your picture connect to your equation?
• • I am not clear on what you have done here. Will you explain it so I can understand?

Don’t be afraid to say you don’t understand their strategy. When you are open to learning, you help students feel more comfortable with engaging in the mathematics.

Be aware that your actions can inadvertently shut down student thinking and damage self-esteem. “It’s easy” and “Let me help you” are two such statements.

communicate to students the real messages you want to send: “Doing mathematics takes time and thinking. You can do it—let’s see what you know and go from there.”

Provide Appropriate Support

Consider ways to support student thinking without taking away their responsibility of designing a solution strategy and solving the problem in a way that makes sense to them.

• “What have you tried so far?”
• “Where did you get stuck?”
• “Have you thought about drawing a picture?”
• “What if you used cubes to act out this problem?”

• “Try drawing a picture or a diagram that shows what 10 percent off and 20 percent more means.”
• “Have you tried picking a price and seeing what happens when you increase the price by 20 percent and then reduce the price by 10 percent?”

Notice that these suggestions are not directive, but rather serve as starters. After offering a tip or suggestion, walk away—this keeps you from helping too much and the students from relying on you too much.

Provide Worthwhile Extensions.

Students solve problems at different rates. Anticipate how you might extend a task in an interesting way for those that finish early without it seeming like extra work.

If a student finds one solution quickly, say, “I see you found one way to do this. Do you think there other solutions?” continue to push their thinking by asking questions like, “Are any of the solutions different or more interesting than others?

Questions that begin “What if . . . ?” or “Would that same idea work for . . . ?” are ways to extend student thinking in a motivating way.

To extend student thinking, ask, “How would you do that on a calculator?” and “Can you write two equations that represent this situation?” These are ways of encouraging children to connect 9 + ? = 12 with 12 – 9 = ?.

The After Lesson Phase

In the after phase of the lesson, you facilitate a classroom discussion. Your students will share, justify, challenge, and compare various solutions to the task they have solved. Ideas generated in the during phase must have a chance to “bump against each other” so that mathematical ideas can emerge (Davis & Simmt, 2006, p. 312).

The after phase is an important time to make drawings, notations, and writing visible to others; to make connections between the ideas that have emerged; and to create spaces for students to take up, try on, and expand on the ideas of others.

It is in the after phase where much of the learning will occur, as students reflect individually and collectively on the ideas they have explored. The expectations for the after phase require careful consideration of possible student responses, recognition of the responses generated in the during phase, and a willingness to be open to unanticipated responses.

Twenty minutes is not at all unreasonable for a class discussion. It is not necessary for every student to have finished, but all students need to have something to share. This is not a time to check answers, but time for the class to share and compare ideas.

Promote a Mathematical Community of Learners

Engage the class in productive discussion, helping students work together as a community of learners. In a community of learners, students feel comfortable taking risks and sharing ideas, students respect one another’s ideas even when they disagree, and ideas are defended and challenged respectfully, and logical or mathematical reasoning is valued. You must teach your students about your expectations for this time!

Listen Actively without Evaluation

As in the during phase, the goal here is noticing students’ mathematical thinking and, in addition, making that thinking visible to other students.

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Resist the temptation to judge the correctness of an answer. When you say, “That’s correct, Dewain,” there is no longer a reason for students to think about and evaluate the response.

You can support student thinking without evaluation: “What do others think about what Dewain just said?”

Relatedly, use praise cautiously. Praise offered for correct solutions or excitement over interesting ideas suggests that the students did something unusual or unexpected. This can be negative feedback for those who do not get praise. Comments such as “Good job!” and “Super work!” roll off the tongue easily, but do nothing to help the student or those listening to know what was good about the work, statement, and idea. In place of praise that is judgmental, Schwartz (1996) suggests comments of interest and extension: “I wonder what would happen if you tried . . . ” or “Please tell me how you figured that out.” Notice that these phrases express interest and value the student’s thinking.

There will be times when a student will get stuck in the middle of an explanation. Be sensitive about calling on someone else to “help out.” You may be communicating that the student is not capable on his or her own. Allow think time. You can offer to give the student time and come back to them after hearing another strategy. Remember that the after phase is your window into student thinking (i.e., formative assessment); listening actively will provide insights for planning tomorrow’s lesson and beyond.

Summarize Main Ideas and Identify Future Tasks

A major goal of the after phase is to formalize the main ideas of the lesson, making connections between strategies or different mathematical ideas. It is also a time to reinforce appropriate terminology, definitions, or symbols.

If a task involves multiple methods of computing, list the different strategies on the board.

There are numerous ways to share verbally, such as a partner exchange, where one partner tells one key idea and the other partner gives an example.

Following oral summaries with individual written summaries is important to ensure that you know what each child has learned from the lesson. For example, exit slips (handouts with one or two prompts that ask students to explain the main ideas of the lesson) can be used as an “exit” from the math instruction. Or be creative—ask students to write a newspaper headline to describe the day’s activity and a brief column to summarize it. Many engaging templates and writing starters are available online.

Process for Preparing a Lesson

The decision-making that goes into designing a three phase lesson…

Step 1: Determine the Learning Goals

Working of the curriculum, ask:

• What should my students be able to do when this lesson is over?
• What content (conceptual and procedural) is important?
• What mathematical practices/processes will be developed?

Content Goals: Fixed Areas.

the development of observable and measurable objectives. There are numerous formats for lesson objectives, but the key is that they tell the things you want your students to do or say to demonstrate what they know.

In our perimeter example, the objectives might be:

• Students will be able to draw a variety of rectangles with a given area and accurately determine the perimeter of each.
• Students will be able to explain relationships between area and perimeter.
• Students will describe a process (their own algorithm) for finding perimeter of a rectangle.

As a counterexample, “Students will understand that the perimeter can change and the area can stay the same” is not a well-designed objective because “understanding” is not observable or measurable.

Step 2: Consider Your Students’ Needs

What do your students already know or understand about the selected mathematics concepts? Perhaps they already have some prior knowledge of the content you have been working on, which this lesson is aimed at expanding or refining. Examine the relevant learning expectations from previous grades and for the next grade.

Be sure that the mathematics you identified in step 1 includes something new or at least slightly unfamiliar to your students. At the same time, be certain that your objectives are not out of reach.

Questions to consider include:

• What context might be engaging to this range of learners?
• What might students already know about this topic that can serve as a launching point?
• What misconceptions might need to be addressed?
• What visuals or models might support student understanding?
• What vocabulary support might be needed?

Step 3: Select, Design, or Adapt a Worthwhile Task

With your goals and students in mind, you are ready to consider what task, activity, or exercise
you might use. Chapter 3 provided extensive discussion on what to consider for this step (see
Tasks that Promote Problem Solving), so here we simply offer ref lective questions:

• Does the task I am considering address the content goals (step 1) and the needs of my students (step 2)?
• Does the task have potential to engage my students in the Mathematical Practices?
• Will the task require students to apply problem solving strategies?

If the answers to these questions are yes, the task can become the basis for your lesson. You may still decide on minor adaptations, like adding in a children’s book or changing the context, to better connect to your students. If you find the task does not fit your content and student needs, then you will need to either make substantial modifications or find a new task.

It is absolutely essential that you do the task yourself. It is only in exploring the task that you can identify possible challenges, anticipate student approaches, and determine the strategies you want to highlight.
Teachers who consider ways students might solve the task are better able to facilitate the lesson in ways that support student learning (Matthews, Hlas, & Finken, 2009; Stein, Remillard, & Smith, 2007).

Step 4: Design Lesson Assessments

Thinking about what you want students to experience and how they are going to demonstrate their understanding of the content goals is an important consideration that occurs early in the planning process, not at the end. It is important to assess in a variety of ways (see Chapter 5).

Formative assessment allows you to gather information that can be used for adjusting the direction of the lesson midstream or making changes for the next day, as well as informing the questions you pose in the discussion of the task for the after phase of the lesson.

Summative assessment captures whether students have learned the objectives you have listed for the lesson.

Assessment Considerations: Fixed Areas.

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Step 5: Plan the Before Phase

As discussed earlier, the before phase should elicit students’ prior knowledge, provide context, and establish expectations. Questions to guide your thinking include:

• Would a simpler version of the task activate prior knowledge, introduce vocabulary, and/ or clarify expectations?
• In what ways can you connect the task with previous mathematical experiences, other
  disciplines, or an interesting current event?
• What presentation strategies and questions will minimize misinterpretations and clarify
  expectations?

Options available for presenting tasks include:

• having it written on paper,
• using their texts,
• using the document camera on a projection device;
• or posting on an interactive whiteboard, chalkboard, or chart paper.

Students need to know:

1. the resources or tools they might use;
2. whether they will work independently or in groups;
3. if in groups, how groups will be organized, including assigned roles; and
4. how their work will be presented (e.g., completing a handout, writing in a journal, preparing a team poster) (Smith, Bill, & Hughes, 2008).

Step 6: Plan the During Phase

The during phase is an opportunity for students to fully engage in the task and for you to notice
what students are thinking, and provide support and challenges when needed. Carefully prepare
prompts that can help students who may be stuck or who may need accommodations that will
give them a start without taking away the challenge of the task. Have options of other materials
such as geoboards or grid paper. Prepare extensions or challenges you can pose to students.
This phase is also a time for you to think about which groups might share their work, and in
what order, in the after phase of the lesson.

Step 7: Plan the After Phase

The after phase is when you connect the task to the learning goals. Even if the mathematics is obvious to you, students may complete the activity without making the intended connections. That means careful planning of the after phase is critical. The following questions should be taken into consideration (Smith, Bill, & Hughes, 2008):

1. How you will organize the discussion to accomplish the mathematical goals (e.g., which solutions will be shared and in what order)?
2. What questions you will ask to help students make sense of the mathematics, make connections to other mathematics, see patterns, and make generalizations?
3. How will you involve all students (over time, not in every lesson)?
4. What evidence are you seeking that will tell you the students understand?

Importantly, the point of the after phase is not just to hear student solutions, but to compare and analyze those tasks, making connections among the strategies and generalizations about the mathematics.

Plan an adequate amount of time for your discussion. A worthwhile problem can take 15 to 20 minutes to discuss.

Step 8: Reflect and Refine

A well-prepared lesson that maximizes the opportunity for students to learn must be focused and aligned. Steps 5, 6, and 7 result in a tentative lesson plan. The final step is to review this tentative plan in light of the lesson considerations determined in steps 1–4, making changes or additions as needed. There is often a temptation to do a series of “fun” activities that seem to relate to a topic but that have different learning goals. Look to see that objectives, assessments, and questions are aligned. If the questions all target only one objective, add questions to address each objective or remove the objective that is not addressed.

Questioning is critically important to the potential learning (see discussion in Chapter 3). Using your objectives as the focus, review the lesson to see that in the before phase you are posing questions that capture students’ attention and raise curiosity about how to solve the problem. In the during and after phases, you are using questions based on the objectives to focus students’ thinking on the critical features of the task and what you want them to learn.

Research on questioning indicates that teachers rarely ask high-level questions—this is your chance to review and be sure that you have included some challenging questions that ask students to extend, analyze, compare, generalize, and synthesize. And, be sure you have a plan to make sure each student is thinking about and responding to each of these high-level questions.

Chapter 10: Developing whole- number place-value concepts

(pp. 246- 274) 

Learner outcomes

10.1 Identify the pre-base-ten understandings based on a count-by-ones approach to quantity.
10.2 Recognize the foundational ideas of place value as an integration of three components: base-ten concepts through groupings and counting, numbers written in place-value notation, and numbers that are spoken aloud.
10.3 Demonstrate how to develop students’ skills in place value through the use of base-ten models.
10.4 Explain how students can use grouping activities to deepen their understanding of place-value concepts.
10.5 Explain strategies to support students’ ability to write and read numbers.
10.6 Recognize that there are patterns in our number system that provide the foundation for computational strategies.
10.7 Describe how the place-value system extends to large numbers.