Elementary Committee Module 2 - Appalachian State NC NAEP Project— Draft Only - Please do not...

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Elementary Committee Module 2

Purpose

The goal of this module is to allow participants to explore forms of single-step and multi-

step addition and subtraction word problems that frequently appear in NAEP. Participants

examine student performance data on such NAEP problems and sets of student work.

Participants familiarize themselves with evidence of student understanding regarding

single-step and multi-step addition and subtraction problems, and investigate various

approaches for teaching these topics.

Overview

The module seeks for participants to view single step and multi-step addition and

subtraction problems from the viewpoint of a student. The module is designed to

encourage participants to examine the difficulties faced by students as they solve addition

and subtraction problems. The difficulties faced may be due to: 1) the number of steps

required; 2) the structure of the problem as determined by the student; or 3) the place

value complexities associated with solving 1-digit or 2-digit problems. The module begins by

the participants’ solving “Sam’s Lunch” problem, a multi-step problem. The participants

then make a list of the different solution strategies employed by the group. The participants

predict the performance of students on this NAEP problem, examine this data, and analyze

a set of student work. In Activity 2, participants examine a set of single step and addition

and subtraction problems. They analyze the structure of the problems and their

mathematical content. Participants learn about instructional methods for teaching this topic

that are based on research (e.g. Cognitively Guided Instruction). They also learn how to

write problems with defined problem structures and predict possible student solution

strategies. Next, in Activity 3, participants examine a set of multiple choice NAEP problems

concerning the topic and analyze the distractors that are included in the multiple choice

answers. They examine the student performance data for these items. This activity

concludes with the participants creating sets of distractor answers for multi-step addition

and subtraction problems. In activity 4, participants view videotapes of students solving

single step and multi-step problems and involving 1 and 2-digits. The participants

summarize the types of student difficulties that they observe in the videos and they record

the types of manipulatives that the children use as problem-solving tools. The participants

are taught how to conduct an assessment/teaching interview. Finally, the participants are

asked to reflect on the previous activities and to write about what they have learned about

problem structures versus problems as interpreted by children.


Background and context notes

Mark Twain’s wisdom includes the following quote of immeasurable importance to

mathematics educators: “It ain’t what you don’t know that gets you in trouble. It’s what you

know for sure that just ain’t so.” For most of the 20th century, American and Canadian

parents, teachers, and curricula have viewed learning single digit addition, subtraction,

multiplication, and division as “learning basic facts.” From this perspective, proper teaching

of these basic facts was achieved through a rote paired-associate—a stimulus-response

approach characterized by memorizing the response (answer) to a stimulus (basic fact

problem). Despite the fact that this approach to teaching and learning was thoroughly

invalidated through research dating back to the 1950’s (Brownell, 1956/1987) and by a

multitude of studies from 1980 forwards (Fuson and Kwon, 1992; Steffe, Cobb, & von

Glasersfeld, 1988; Heibert, 1986; Ginsburg, 1984), this view of teaching basic facts

continues to influence texts and teaching facts in elementary school. In a traditional

approach to teaching facts, a teacher may follow the following instructional trajectory:

1) teach basic addition/subtraction facts using a version of a stimulus-response model to aid

students in memorizing the facts (i.e. use of flash cards); 2) teach application of these basic

facts in single-step word problems; and 3) extend problem-solving involving the facts to

multi-step application problems. This teaching triad may have serious consequences for

student achievement. First, research has established that the learning of basic addition facts

rests on children developing thinking strategies to help them make sense of the facts and to

aid them in remembering the facts. Instruction that does not explicitly include these

strategies lengthens the amount of time required for students to develop proficiency.

Secondly, when basic facts are taught in isolation from problem contexts, it is more difficult

for children to understand the meaning and purpose for the facts. Thirdly, if single step

word problems routinely follow instruction of an operation, then sensible children will come

to recognize the expectation to use the taught operation and will not bother reading the

problems that come at the end of the section. This failure to read and meaningfully

interpret problem situations is often revealed in students’ inability to solve multi-step

addition and subtraction word problems, a difficulty that is bewildering to teachers who

think their children have “mastered” the basic facts. As the learning of basic facts and

problem solving involving the basic facts comprises a large portion of the early elementary

curriculum, it is important to become aware of research-based instructional alternatives

that will support student learning of basic facts and problem solving skills.

Building computation fluency and problem-solving skills requires that basic addition and

subtraction facts be introduced and illustrated through a wide variety of single step word

problem settings. These word problems, which may be introduced through children’s

literature, provide a context to help children make sense of the facts and to encourage


them to use materials as thinking tools and to use pictures and diagrams to help them first

learn the facts as problem-solving situations. This provides the teacher with opportunities

to explicitly teach children how to find the answers to basic facts problems by using

strategies such as counting on, making ten, or using doubles. Teaching basic facts in this

way also provides teachers with additional instructional time during which they may

introduce children to a wider number of addition and subtraction problem structures

(Fuson, 1992; Carpenter, Fennema, Franke, Levi, & Empson, 1999). Developing facility with

a wider range of problem structures aids children in developing meaning for multi-step

problems in two ways: 1) children have learned to carefully read and interpret even single

step problems; 2) children have developed more sophisticated individualized mental

structures for problems and so are better prepared to cope with the strategies and note-

taking required to successfully solve multi-step problems. This module intends to expose

participants to single-step and multi-step NAEP problems and familiarize them with student

performance data for these problems. In addition, the module intends to familiarize

participants with research-based strategies for teaching single-step and multi-step word

problems.

Preparing to teach the module

Teaching notes are included in the margin to assist facilitators in leading the module’s activities and discussions. Before teaching the class sessions in the module, the facilitator should read at least some of the following sources to provide additional background for instructing the module activities:

Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically. Portsmouth:

Heinemann.

Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Chidlren's

Mathematics: Cognitively Guided Instruction. Portsmouth: Heinemann.

Towmey-Fosnot, C., & Dolk, M. (2001). Young Mathematicians at Work. Portsmouth:

Heinemann.

Warfield, J. & Meier, S. (2007). Student performance in whole-number properties and

operations. In P. Kloosterman and F. Lester, Jr. (Eds.). Results and Interpretations of the

2003 Mathematics Assessment of the National Assessment of Educational Progress (pp. 43-

66). Reston, VA: National Council of Teachers of Mathematics.

Wright, R. J., Stanger, G., Stafford, A. K., & Martland, J. (2006). Teaching Number in the

Classroom with 4-8 year-olds. London: Paul Chapman.


Introductory PowerPoint presentation for the module (attached)

Teaching the module

Goals

This module is intended to help participants:

Develop understanding of some of the characteristics of multi-step addition and subtraction word problems involving whole numbers.

Develop understanding of the differences between single-step whole number word problems involving addition and subtraction and multi-step problems.

Based on videotaped examples, develop understanding of common student approaches to single-step and multi-step problems.

Become aware of a variety of instructional approaches for teaching these topics that are research-based.

Practice predicting student thinking on a particular task or item.

Practice analyzing and assessing student work

Practice analyzing and developing distractor answers to multiple choice questions.

Observe children using manipulatives as thinking tools.

Time required

Activity 1: 2 hours

Activity 2: 2 hours

Activity 3: 2 hours

Activity 4: 1 hours

Activity 5: 30 minutes

Mathematics Addressed

Single-step and multi-step word problems involving whole number addition and subtraction

Addition and subtraction of single digit, two-digit, and three-digit whole numbers


Addition and subtraction of decimal numbers

Estimation

Problem solving strategies

NAEP Resources Used “Sam’s Lunch” NAEP Problem M068901

NCTM Principles Teaching, Learning, Assessment

NCTM Process Standards Problem Solving, Connections, Representation

NCTM Content Standards Number and Operation

Grade Band(s): K -5 6-8 9-12

NAEP Content Strand Number Sense, Properties, and Operations


Materials

Activity CD-ROM No. Materials Number Needed

1. Examining Strategies for Solving Multi-Step Problems: “Sam’s Lunch”

1. “Sam’s Lunch” Powerpoint Presentation

2. Copy of “Sam’s Lunch” NAEP Problem

3. “Sam’s Lunch” NAEP Performance Data

4. “Sam’s Lunch” Student Work

5. List of Relevant Sources

6. Chart Paper

1. 1 per instructor

2. 1 per instructor via Powerpoint or individual participant paper copies of the problem

3. 1 via Powerpoint

4. 1 set per participant pairs

5. 1 per participant

6. ½ piece per individual; 2 pieces per group of 4-6 participants

2. Who’s Problem Structure is It? Examining Problems from the Perspective of the Expert and of the Student

1. Who’s Problem Structure is It? Powerpoint

2. Copy of Children’s Solution Strategies chart

3. Copy of Strategies for Solving Problems

4. Copy of Finding a Problem for the Strategy

5. Sticky Notes

6. Chart paper

1. 1 per instructor


3. 1 per every 2-3 participants

4. 1 per every 2-3 participants

5. 5 or 6 per participant (depends on the number of groups)

6. 1 piece per 2-3 participants

3. Multi-step NAEP Items: Exploring Distractor Answers

1. Multi-step NAEP Items: Exploring Distractor Answers Powerpoint

2. Set of Multi-Step NAEP Items

3. Performance Data for the NAEP Items

4. Chart paper

1. 1 per instructor

2. 1 set per participant

3. 1 set per pair of participants

4. 1 piece per group of 4 participants

4. Conducting Interviews and Analyzing Problem-Solving Videos

1. Interviews and Videos Powerpoint

2. Problem-Solving Videoclips

1. 1 per instructor

2. 1 per instructor


4. 1 bagged set per


3. Interview Template

4. Manipulatives (list)

pair of participants

5. Wrap Up 1. Reflection Prompts PowerPoint

2. Index cards or journals

1. 1 per instructor


Activity 1: Examining Strategies for Solving Multi-Step Problems: “Sam’s Lunch”

The module begins with participants’ solving “Sam’s Lunch” problem, a multi-step problem. The participants will make a list of the different solution strategies employed by the group. The participants will predict the performance of students on this NAEP problem and will examine this data and analyze a set of student work.

Solving “Sam’s Lunch” NAEP Problem

Sam can purchase his lunch at school. Each day he wants to have juice that costs 50¢, a sandwich that costs 90¢, and fruit that costs 35¢. His mother has only $1.00 bills. What is the least number of $1.00 bills that his mother should give him so that he will have enough money to buy lunch for 5 days?

Give each individual one half of a sheet of chart paper. First, ask participants to solve the problem individually on their chart paper. Ask them to make brief notes about how they thought about the problem and the steps they used to solve it. Next, ask participants to form into small groups (4-6). Provide the groups with a sheet of chart paper and ask them to make a list of the strategies that were used by group members to solve the problem. Bringing the entire group together, ask each group to briefly share the strategies that were applied by the group members, illustrating their answers with their charted solutions. Draw the participants’ attention to the fact that a seemingly “routine” problem, nonetheless was solved using a variety of strategies.

Predicting Fourth Graders’ Success with “Sam’s Lunch” NAEP Problem

Reform the small groups of participants, and ask them to make a prediction as to the percentage of fourth graders who solved the NAEP problem correctly and the percentage who did so incorrectly. Have them include this prediction on their chart paper. Furthermore, ask the participants to make predictions as to the types of student errors that they would expect to see on this problem. Require the small groups to provide responses to the following questions:


1. How many steps are needed to solve the problem? (Describe a range of reasonable approaches)

2. Examine the language used to express the question. What parts of this language might be difficult for students to interpret? Would any of the language be especially challenging for English Language Learners?

3. What type of mathematical errors would you expect to see fourth graders make on this problem?

Draw the small groups’ attention together briefly and show the fourth grade NAEP performance data concerning “Sam’s Lunch” problem to the participants (see Powerpoint). Ask the participants to discuss the question: How did your predictions of the fourth graders’ achievement on the problem relate to the actual performance data?

Examining the Fourth Graders’ Work on “Sam’s Lunch”

Provide each small group of participants with a set of fourth grade students’ work on “Sam’s Lunch.” Ask the group to examine the student work and make the following comparisons:

1. For students who completed the problem correctly: How did their solution methods compare to the solution methods applied by the members of your group?

2. For students who completed the problem incorrectly:

Did the student seem to be aware that “Sam’s Lunch” was a multi-step problem or did they solve it as if it were a single step problem?

Did you see any evidence that indicated that the student did not understand the language used to express the problem?

How did the actual errors made by the fourth graders compare to the error predictions made by your group?

Conclude the activity by drawing the participants’ attention to the increased difficulties faced by students when they are asked to solve a multi-step problem as opposed to a single-step problem. Beginning teachers typically have some appreciation for the difficulties that children have with multi-step problems, based on their own remembered experiences with solving such problems. However, often they are not able to empathize with children in earlier grades who experience difficulty in solving single-step addition and subtraction word problems. They are not aware of the different problem structures framed in such problems and how children can have difficulty in finding the meaning within such problems. Activity 2 is intended to help participants gain some experience in examining the various problem structures that have been defined through research.

Activity 2: Whose Problem Structure is It? Examining Problems from the Perspective of the Expert and of the Student

This module begins with participants examining a set of single step addition and subtraction problems. They analyze the structure of the problems and their mathematical content using a framework from Cognitively Guided Instruction (CGI). Participants also learn about

Comment [S1]: Teaching note: It is possible that participants have made errors while working the Sam’s Lunch item. As much as they are willing to do so, have participants share their errors with the group and how these errors may provide insight into those of students.


instructional methods for teaching this topic that are based on research (e.g. CGI), learn how to write problems with defined problem structures and predict possible solution strategies.

Getting Started

Ask the participants to consider the following problems:

Connor has 10 cars. He gives 3 cars to Tristan. How many cars does Connor have left?

Connor has 3 cars. How many more cars does he need to have 10 cars?

Connor has 3 cars. Tristan has 10 cars. How many more cars does Tristan have than

Connor?

Have participants complete the problems and then discuss the following questions in small

groups:

How would you solve these problems?

How might a first grader solve these problems? What strategies might a first grader use

to arrive at the answer?

After the participants discuss the problems in small groups, have each small group report

out about their discussion.

Examine Problem Types

Provide participants with Chapters 1, 2, 3 in Children’s Mathematics: Cognitively Guided

Instruction and ask them to read this material prior to class. Introduce the Powerpoint

presentation that describes the problem types to the participants. After the presentation,

divide the participants into pairs and ask them to create a set of problems that illustrate the

problem types and to record their set of problems on chart paper.

Join- Result Unknown, Change Unknown, Start Unknown

Separate-Result Unknown, Change Unknown, Start Unknown

Part-Part Whole- Whole Unknown, Part Unknown

Compare- Difference Unknown, Compared Set Unknown, Referent Unknown

Have groups share the problems that were created on chart paper. Discuss why the

problems are or are not the problem type and then discuss how students might solve the

problems. See facilitator note above.

Comment [TGE2]: Teaching note: You are hoping that participants share that they might model, count on their fingers, know some sort of fact, etc.

Comment [TGE3]: Teaching note: Cognitively Guided Instruction helps teachers examine basic word problems and identify different problem structures. Some of these problem structures are harder for children to learn than others.

Comment [TGE4]: Teaching note: Be sure to highlight that typically students see problems with the start unknown as more difficult. Also, make sure participants notice that some of these problems can be solved with addition strategies and others with subtraction strategies. However, it is important that the problems are not separated out.


Examine Solution Strategies

The book, Children’s Mathematics: Cognitively Guided Instruction, includes a CD with videos

of childrens’ strategies for solving the problem types. View the Addition/Subtraction set of

videos with the participants. After each video discuss with participants the strategy that was

used. After viewing all videos, pass out Children’s Solution Strategies Chart (see Supporting

Documents). With the entire group, discuss each problem, label the problem type and

discuss each strategy. Then ask the participants to work in pairs to complete the “How

Would Children Solve These Problems?” worksheet (see Supporting Documents). Have pairs

come to the front of the room and fill out the spaces in the chart displayed on an overhead

projector or a whiteboard. Give participants time to review the completed chart and to

compare their answers to the ones displayed. Do the answers make sense? Is this how a

first/second grader might solve this problem? Are there other possible solutions?

After the discussion, pass out “Finding a Problem for a Strategy” worksheet (see Supporting

Documents). Divide participants into small groups (about 3 participants each). Have each

group complete the exercise and post its answers on the wall. Hand out sticky notes to each

participant. Have participants engage in a gallery walk and use the sticky notes to post

questions or comments on the other groups’ papers. After every group has rotated through

the papers, let each group retrieve its paper and read their comments. Engage in a large

group discussion about the problems. Are there any questions? What did you notice? Is

there something that you are unsure about?

Activity 3: Multi-step NAEP Items: Exploring Distractor Answers

In Activity 3, participants examine a set of multiple choice NAEP problems concerning multi-

step NAEP items and analyze the distractors that are included in the multiple choice

answers. Participants also examine the student performance data for these items. This

activity concludes with the participants creating sets of distractor answers for multi-step

addition and subtraction problems.

Multiple-choice items provide students with a few options from which to choose the best

response. To be effective, multiple-choice items must have some choices that are

reasonably close to the correct response. These choices, called distractors, may be based

on common misconceptions or student errors. In this activity, participants explore how

distractors are chosen, how to create effective distractors, and how to evaluate their

effectiveness.

Comment [TGE5]: In a Gallery Walk, all participants will walk around the room and examine each group’s poster. They will respond to with questions and comments written on sticky notes. The purpose of the Gallery Walk is to allow participants to examine one another’s ideas and serve as a catalyst for discussion.


Investigating Distractors

Form the participants into small groups. Provide each group with the item “Becky’s

Popcorn” (See Supporting Documents).

“Becky’s Popcorn Balls” NAEP Problem

On Thursday Becky made some popcorn balls.

On Friday she made 6 popcorn balls.

On Saturday she made 12 popcorn balls.

Becky took all 23 popcorn balls she made to a party.

How many popcorn balls did she make on Thursday?

A) 5

B) 11

C) 17

D) 41

Have each group work the item and select the correct response. Then, have participants

explore how a student might have arrived at each of the incorrect responses. Remind them

to consider the problem in steps as they did in Activity 1. This should more clearly reveal

how students could arrive at each of the incorrect responses.

Have participants in groups discuss how they determined that a student might have

selected the incorrect options. For the “Becky’s Popcorn” item, these are choices B, C, or D.

Participants observe that for each of these distractors, students may have used the

numbers in the item to complete a calculation that was either incomplete or inappropriate.

Option D, 41, results from finding the sum of all three numbers provided in the item. It is an

incorrect approach, but the calculation itself is correct. Options B and C result from

subtracting one the number of popcorn balls from day one from the total. It is a correct

initial step in finding the answer, but it is incomplete. Only response A, 5, results from

correctly completing the two steps of this problem. Participants see that Options B and C

are selected by students who approach the Becky’s Popcorn item as a single step rather

than a multi-step problem. This misunderstanding is a common error on which distractors

are written.

A different item that can be discussed is “Estela’s Notebooks.” In this item, students must

estimate the number of dollar bills Estela needs to buy two notebooks.

“Estela’s Notebooks” NAEP Problem


Estela wants to buy 2 notebooks that cost $2.79 each, including tax. If she has one-dollar

bills and no coins, how many one-dollar bills does she need?

A) 3

B) 4

C) 5

D) 6

The problem is a multi-step problem just as “Becky’s Popcorn” is, but the concepts now

include estimation as well as number and operation. Because of this, the distractors also

differ. For example, common errors in estimation, such as rounding up or down

inappropriately for the problem context, inform the writing of the distractors. Provide

groups with the item and have them work the item and select the correct response. Then,

have participants explore how a student might have arrived at each of the distractors. You

may wish to again remind them that the steps themselves can provide the origin for an

incorrect response. Have participants explore what errors or misconceptions were involved

in a student’s choosing an incorrect response.

Student Choices – Exploring Distractor Selection

After exploring the items and distractors, ask participants to predict how many students

would have correctly responded to the two items. Have groups predict which distractors

might be selected more often and why.

Provide each group with a handout of student performance information for the items that

were discussed Ask participants to consider how the actual data compares to their

predictions. Engage the whole group in a conversation about the student data. They

should observe that on the Estela problem, there was one distractor that was much more

commonly selected than the others, while the incorrect selections on the Becky problem

were more uniformly chosen. Ask the participants to discuss reasons for these results.

Writing Good Distractors

Returning to the “Sam’s Lunch” item, an open response item, participants can develop their

own group of responses to recreate it as a multiple-choice item.

Ask the groups to look back at the item and their calculations for it. Encourage participants

to create four options for the item, one correct and three incorrect ones. Then have

participants look back at the student work samples to see how their suggested incorrect

responses compare to the student work. Have students discuss the distractors.

Comment [TGE6]: Teacher Note: The second step in this problem is that of estimating. A discussion of estimation techniques is appropriate.


Were your suggested distractors representative of incorrect student responses?

Were your suggested distractors all procedurally-oriented?

Did they represent correct concepts but incorrect calculations or vice versa?

If a student selected one of your incorrect options, what might that reveal about that student’s understanding?

Collect distractors from all the groups and provide an overall list to the group. Have the

participants select the “best” three distractors from among all of those suggested by

groups. Explore why these are considered the best.

Distractor Analysis – an Extension

The extension activity is most effective if completed in conjunction with a field-based experience, such

as service learning, student teaching, or a practicum. Participants will evaluate the effectiveness of the

distractors they created for the Sam’s Lunch problem by administering the item to a group of

elementary students.

Prepare participants for administering the item by discussing the purpose of multiple choice items and

distractors. Remind participants that the distractors are meant to differentiate between students who

genuinely understand a concept and/or can apply a skill and those who cannot. Distractors may do a

very good job of differentiating between those students, but if they are not well-chosen, they may not.

Consider the example that follows:

Sara has fifteen marbles in a jar. Five are blue. The rest are green. Suppose she reaches into the jar

without looking and randomly draws out a marble. Which of the following colors is she likely to have

drawn?

a) blue b) green c) red d) yellow

In this item, the options of c and d are impossible to have been selected; the colors are not even

mentioned in the problem. Test-takers can use cues from the item, such as the lack of either of these

two options, to quickly eliminate two choices. Therefore the probability of a student guessing the

correct option is much greater than if the choices are reasonable.

Prior to administering the items, have each group prepare a handout that includes the Sam’s Lunch item

with multiple-choice options. The handout should have well-written directions, such as circling the

letter of choice or placing a check in the blank next to the choice. Have participants discuss how

effective they believe their distractors are. Then, depending on the structure of your field-experience,

allow participants to administer the item and collect the work.

When participants reconvene to continue the activity, have each group complete the Student Response

Sheet (See Supplemental Materials) by recording the number of students who selected each item. Have


participants calculate the percentage of students who selected each item as well. Post group results at

the front of the room.

Guide participants through a discussion of the effectiveness of their items. Questions to explore

include:

What percentage of students selected the correct response? What could this indicate?

Were the distractors evenly selected? If not, which distractors were selected most often?

What might the selection of distractors indicate?

How could the distractors be improved to more accurately gauge student understanding?

Activity 4: Conducting Interviews and Analyzing Problem-Solving Videos

In activity 4, participants view videos of students solving problems. The participants are asked to summarize the types of student thinking that they observe in the tapes and they will record the types of manipulatives that the children use as problem-solving tools. The participants are also taught how to conduct an assessment/teaching interview.

Examining Multi-digit Problems

Ask the participants to begin by solving the following problems in pairs. Ask them to discuss ways that students might approach solving these problems. Ask them to carefully document the thinking strategies that students might apply to solve the problems.

1. Samuel had 37 baseball cards. He collected 28 more baseball cards this summer. How many cards does Samuel have in his baseball card collection now?

2. Kelly has 24 tomato plants for her garden. She gave 16 plants to her mother. How many tomato plants does Kelly have left to plant in her garden?

Continue the session by viewing the multi-digit videos on the CD included with Children’s Mathematics: Cognitively Guided Instruction. Ask the participants to view the clips that illustrate students applying the following thinking strategies to solve the problems. Ask them to choose the videos that best fit the strategy they applied to solve problems 1 & 2.

Direct Modeling with Tens

Invented Algorithm- Incremental

Invented Algorithm- Combining Tens and Ones

Invented Algorithm- Compensating

Problem Solving: Conceptual Understanding and Algorithms


During this activity two videos are shown to the participants. The participants are asked to describe the similarities and differences that they observe in the children’s problem solving strategies.

Begin by showing the video clip entitled: Multi-digit, Invented Algorithm Incremental -Separate (Result Unknown) from the Children’s Mathematics: Cognitively Guided Instruction CD to the participants.

Problem: Kevin had 84 gumdrops. During a week he ate 29 gumdrops. How many gumdrops does he have left?

Next, show the participants the video clip of Gretchen, Video clip #3, from IMAP.

Gretchen’s problem

After viewing both videos, ask the participants to form groups of 2-4 to discuss the following questions. Generally, it is necessary to play the clips more than once so that the groups can make hypotheses about the students’ thinking and then confirm or refute their hypotheses by watching the videos a second or third time.

1) What written algorithm or artifact did the boy use to solve Kevin’s Gumdrop problem?

2) What thinking strategies did he use?

3) What written algorithm or artifact did Gretchen use to solve her problem?

4) What strategies did Gretchen apply?

5) Which student was more confident about their answer? Why?

6) What implications do these videos suggest for instruction?

Interviewing Students

It is essential for students to be able to articulate their reasoning while learning addition and subtraction facts and basic problem solving. It is useful for students to explain their problem solving in order to help them clarify their thinking and to spur conversations between students about alternative ways to solve the same problem. Inviting students to participate in such mathematical conversations also provides teachers with opportunities to assess students’ thinking. Learning to lead mathematical discussions in class can be daunting to beginning teachers. One step in that direction is to learn how to conduct an assessment interview with an individual student. Learning to interview individual students helps teachers learn how to scaffold mathematical tasks and how to ask open-ended questions to solicit student thinking. As teachers plan their interviews, encourage them to solve the problems themselves first, and then practice the interview with a peer before interviewing a student.


Ask participants to complete the attached Interviewing Students Assignment in a field placement.

Activity 5: Wrap-Up

Discuss the Events of the Module

Lead participants to discuss the key features of this module: recognizing complexity of

single step and multi-step addition and subtraction problems, using items to investigate

students’ understanding of such problems, and creating multiple choice distractor answers

to explore and assess students’ thinking.

You may wish to allow some initial time for individuals to compose their thoughts before

encouraging small groups to discuss. After adequate time for small group discussion,

facilitate a whole group discussion in which participants summarize the activities.

Questions included in the Introductory PowerPoint may be helpful in guiding participants to

think on these ideas.

What have you learned about single step and multi-step addition and subtraction problems?

What have you learned about teaching single step and multi-step addition and subtraction problems?

What have you discovered about student thinking about single step and multi-step addition and subtraction problems?

Reflect on the module

Provide each participant an opportunity to reflect in writing about the activities they have

just completed. Writing prompts can be found in the Introductory PowerPoint, and you

may select all or only a few to include. These prompts are suitable for journal writings or

index card responses and include:

What are the key concepts that students need to understand about single step and multi-step addition and subtraction problems?

What pedagogical strategies can be applied to teach these concepts?

What have you learned about students’ understanding of single step and multi-step addition and subtraction problems?

How can we assess students’ learning of single step and multi-step addition and subtraction problems?

What are some ways that a problem’s distractor answers can be written to determine what students know?


Finally, the participants are asked to reflect on the previous activities and to write about what they have learned about problem structures versus problems as interpreted by children.


References

Bay-Williams. J. (2009). Field Experience Guide: Resources for Elementary and Middle School Mathematics – Teaching Developmentally. New York: Allyn & Bacon.

Brown, C. & Clark, L. (2006). Learning from NAEP. Reston, VA: National Council of Teachers of

Mathematics.

Brownell, W. A. (1987). AT Classic: Meaning and skill—Maintaining the balance. Arithmetic Teacher, 34(8), 18-25. (Original work published 1956).

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Mathematics: Cognitively Guided Instruction. Portsmouth: Heinemann.

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Supporting Documents


Activity 2:

Find a Problem for the Strategy

1. Jacob counts out 14 cubes. He counts, “1, 2, 3, 4, 5” pulling one cube away from the

group for each count. He then counts the remaining cubes, “1, 2, 3, … 9.” The answer is

9.

2. Paula says, “18, 19, 20, 21” and holds up a finger for each count after 18. She says, “the

answer is 21.”

3. Tristan says, “10, 9, 8, 7” and holds up a finger for each count after 10. He looks at his

fingers and says, “the answer is 3.”

4. Connor draws 3 lines as he counts, “1, 2, 3.” He then skips some space and draws five

more lines as he counts, “4, 5, 6, 7, 8.” He looks at his drawing and counts the second

set of lines and. “1, 2, 3, 4, 5, the answer is 5.”


Strategies for Solving Problems

Problem Problem

Type

Direct

Modeling

Strategy

Counting

Strategy

Derived

Facts

Strategy

Other

Connor had a

bucket of cars.

For his birthday,

he received 9

more. Connor

counted his cars

and he now has

32. How many

cars did Connor

have originally?

On a scavenger

hunt, John

collected 14

prizes and Julie

collected 8

prizes. How

many more

prizes did John

have than Julie?

Anasofia has ten

dolls. She

decided to give

three dolls to

Cora. How many

dolls does

Anasofia have

now?


Activity Three:

Becky’s Popcorn Balls

On Thursday Becky made some popcorn balls.

On Friday she made 6 popcorn balls.

On Saturday she made 12 popcorn balls.

Becky took all 23 popcorn balls she made to a party.

How many popcorn balls did she make on Thursday?

E) 5

F) 11

G) 17

H) 41

Estela’s Notebooks

Estela wants to buy 2 notebooks that cost $2.79 each, including tax. If she has one-dollar

bills and no coins, how many one-dollar bills does she need?

E) 3

F) 4

G) 5

H) 6


STUDENT RESPONSE SHEET

Look at the student data you collected using your Sam’s Lunch item. Count the number of

students choosing each option. Record those in the spaces below by writing the complete

options you provided in column one and the number and percentage of students selecting each

in the appropriate column. List your correct option and data in the first row.

Options

Number of

Students Selecting

Percentage of

Students

Selecting


Activity Four Interviewing Students

Interviewing your student— As a teacher, you will need to learn to pay attention to your

students’ understandings of the concepts studied in your class and be able to adjust your

instruction accordingly. This assignment assesses your ability to follow a child’s thinking as he

or she works through mathematics activities. After observing the student in the whole

classroom setting, you will interview the student, after the lesson or as soon as possible

thereafter. You will develop appropriate interview tasks for your individual student interview

based on the mathematics concepts that were taught by the classroom teacher during the

lesson when you observed your student. You may use the following Interview example and

video cases suggested in class to help you develop your interview tasks. Use the attached

template to help you plan your student interview.

Interview Plan Example:

Addition with Regrouping Grade 3

Materials Needed: Problems, base-ten manipulatives, Unifix cubes, paper and pencil.

Problems (Tasks) Selected for the Student Interview and the Rationale for the Problems Chosen: The following problems were selected to help us learn about the child’s skill with whole number addition involving regrouping. The first two problems are included so that we may determine if the child is able to demonstrate knowledge of basic facts and to determine if he/she can easily add two-digit numbers that do not require regrouping.

35 42 56 39 47

+ 23 +16 + 28 + 45 + 54

Interview Process: In the beginning of the interview, your objective is to record what the student currently knows about the concept. Show the child one problem at a time. Ask the child, “Can you solve this problem?” Make brief notes about what the child does and says. If the child asks a question or makes a comment, such as, “I don’t remember what 9 + 5 is,” do not provide the answer at this point. Instead, ask the child, “How could you figure that out?” Do not tell the child how to solve the problem. If the child is not successful with the initial tasks, modify the tasks until you find the point at which the child begins to experience some success. Throughout the interview ask questions such as, “How did you


do that?” Let the student do the work and do not teach. Follow-up on the student’s work or comments with questions, such as, “What were you thinking?” or “Why did you do it that way?” Only after you have recorded where you think the child is conceptually, should you offer direct help. Also keep account of what instructional assistance you offer and whether it is helpful to the child. Focus on asking the child helpful questions.

List of questions that you should answer following the student’s activities during the interview:

How did the child solve each problem? Did he/she use any manipulatives or pictures? What algorithms, if any, were used? What errors, if any, did the child make? What might this tell you about his or her

understanding of addition with two-digit numbers? (Modified based on an interview pattern from Field Experience Guide: Resources for Elementary and Middle School Mathematics – Teaching Developmentally by Jennifer Bay-Williams, 2009).


Student Interview Template

Grade level: _____________

What mathematics concept was taught in the whole class setting?_________________

________________________________________________________________________

Materials Needed: ________________________________________________________

Problems (Tasks) Selected for the Student Interview and the Rationale for the Problems Chosen:

Interview Process:

List of questions that you should answer following the student’s activities during the interview:


Teachers’ Comments (Interview summary and episodes from the interview transcript to be

added here.)


ELL Issues Specific to this Material

Preservice teachers may be surprised to realize that students from different cultural and language backgrounds may have different ways of thinking about early number ideas, such as subtraction. Both the algorithmic approaches, notation, and language used may be quite different from those typical of American classrooms. Recognizing such differences before a lesson is respectful of the diverse backgrounds of students. Comment [TGE7]: Other material is to be added

to this section.

Elementary Committee Module 2 - Appalachian State NC NAEP Project— Draft Only - Please do not...

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