commoncore.aetn.orgcommoncore.aetn.org/training/psmd/PSMD-Day-1-Face2Face-Guid… · Web view(from CCSSM pg 8 “Expectations that begin with the word “understand” are often

Arkansas Common Core Professional DevelopmentProblem Situations for Multiplication and Division and the Nature of “Equals”

(Grades K-6) Face-to-Face Guide

Face-to-Face Workshop Facilitator’s Guide

Session #1: Orientation

IntroductionIn this session, you will:

Meet your course facilitator and fellow students Learn about and practice the Moodle course format Become familiar with course documents and expectations

AccessPreparation:Read the article introducing the instructional model Purposeful Pedagogy and Discourse that will be used as an organizing theme for all of the Arkansas CCSS Mathematics Professional Development Project modules.

Introductions:Introduce yourself to the group. Tell your name, school/district/co-op, job position, and reason you are here.

In small groups, discuss the article you read: Discuss the 3 skills of professional noticing. What do you think is intuitive about them? What

do you think is challenging?

ActivateCommon Core Standards:The course will ask you to refer to the Common Core State Standards for Mathematics, which consists of two parts:

Standards for Mathematical Practice (pages 6 – 8 of the document linked above) Standards for Mathematics Content (pages 9 – 83)

This graphic [CCSSM Circle Diagram] shows an overview of the organization of the mathematics standards.

Personal Notebook: You are strongly encouraged to keep a personal notebook as you work through the class, whether physical or electronic.

Session 1 Page 1



Session #2: Standards for Mathematical Practice Introduction


Briefly examine all 8 of the Standards for Mathematics Practice from the Common Core State Standards for Mathematics (CCSSM)

Study in depth two of the practices – (6) Attend to precision; and (7) Look for and make use of structure.

AccessExamine:The Standards for Mathematical Practice are on pages 6-8 of the Common Core State Standards for Mathematics. These are habits of mind that each mathematics student needs to mature in each year.

The 8 Standards for Mathematical Practice (SMP) are:1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics. 5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

Read:Thoroughly read all of the material on pages 6-8, including the introduction, the descriptions of each of the 8 SMP, and the concluding paragraphs. Reflect on your understanding of each of the practices. Make note of what you think the gist of each practice is and any question(s) you have.

ActivateSmall group/partner discussion:After reading the descriptions of the 8 SMP, discuss their meanings and importance. Consider the following prompts.

Discuss briefly your understanding of each of the Standards for Mathematical Practice by considering these items –

What does this standard mean? What does it look like when students are engaged in this standard? What can a teacher do to model or encourage students to engage in this standard? Why is this standard important?

Several content standards use the verb “understand.” What does the verb “understand” indicate in those standards?

Session 2 Page 2



o (from CCSSM pg 8 “Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut.”)

Did you notice any themes that unite the Standards for Mathematical Practice?o In numerous places the SMP refer to reasonableness or results making sense as being

an important aspect. o It is clear these practices are for STUDENTS (Mathematically proficient students…) to

develop as habits. o Communicating about and communicating with mathematics. o Interaction with others around mathematics.

Analyze:You will meet Mrs. Scott’s 4th grade class in this session and encounter them again in Session 6. The class is engaging with the problem 9 x 7 presented as 9 tables with 7 students at each table. The focus of this exchange is on writing an expression to match how Brian thought about the problem and using the properties of operations. Brian’s original work from the board and the board at the end of the lesson have been recreated here to make it easier for you to follow.

[Note: This class has agreed upon the meaning of a multiplication a x b as being a groups of b items.]

View the video and think carefully about what mathematics is at play, what the teacher does and does not do, and what evidence you see in the students of the habits of mind described by the Standards for Mathematical Practice.

This graphic demonstrates structure in the Standards for Mathematical Practice, showing that standards 1 and 6 are overarching, and the others often occur in pairs depending on the focus of a particular task or problem.

Reflect on the lesson above in light of two of the SMP: (6) Attend to precision; and (7) Look for and make use of structure.

DiscussClass Discussion:After your small group/partner discussion and viewing the video of Mrs. Scott’s 4th graders, discuss how the Standards for Mathematical Practice were involved. Think in particular about these two SMP: (6) Attend to precision and (7) Look for and make use of structure. Consider the following prompts.

In what way(s) are the students dealing with precision?

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o They are trying to match an expression with how Brian though about the problem, and in doing so they have to deal with the language of grouping. The class has an agreed upon definition that a multiplication sentence means a groups of b, and they are grappling with what that means.

o They are also needing to deal with the symbolic conventions of representing Brian’s approach and what the different symbolic sentences mean.

In what way(s) is the class trying to look for and make use of structure?o They are using the structure of Brian’s drawing and trying to attach a symbolic

structure that matches.o They are dealing with decomposing one factor or the other and deciding whether that

is possible AND where each factor is visible in the decomposed expression. How are any of the other Standards for Mathematical Practice evident in the students in this

video? For example, do you think there is evidence of constructing viable arguments and critiquing the reasoning of others? Do you see students reasoning abstractly and quantitatively, or just one of those? [Answers will vary.]

o Construct viable argument/critique reasoning: In putting their work on the board, students were trying to create a viable argument that would make sense to their peers and withstand scrutiny. Classmates asked questions and delved into the strategy to both make sense of it and improve on it (by connecting the expressions).

o Reason abstractly and quantitatively: Students and the teacher went back and forth between dealing with the numbers apart from the context and putting the numbers and strategies back into the context. When removed from context and looking at structure, symbols, etc, they were reasoning abstractly.

How did the teacher support the habits of mind students need to exhibit the Standards for Mathematical Practice?

o The teacher asked a lot of questions to keep students thinking about how the expressions they were trying related to the problem. They were “decontextualizing” and she pushed them to “recontextualize” to look for structure.

o She used appropriate notation to model or demonstrate how it could be used to represent a student’s words.

o She got the whole class invested in understanding these different students’ work and the details of their strategies by 1) having everyone work the problem first in a way that made sense to them; and 2) building a classroom culture of seeking to understand strategies and make mathematics connections.

o She restated to make sure she understood and that others heard and understood what was said.

ReflectReflection Journal: While reflecting on the video and your own experiences with students, what you are understanding about these two SMP: (6) Attend to precision and (7) Look for and make use of structure.

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Session # 3: The Nature of Equals


Develop an understanding about children’s conceptions about the equal sign. Analyze and develop ideas about how to facilitate children’s meaning of the equal sign and

understanding equality. Consider the importance of children’s understanding of the equal sign and relational thinking. Consider what this means to them and their instructional practice.

AccessTry It Out:Take a few minutes to think about equality by working through the attached set of equations. Predict how you think students will respond to these problems.

ActivateAnalyze:Part 1 – Equality Video – First Grade [See Instructor Notes Appendix A]Mr. Flud works with a group of eight first grade students. Most of the students struggle with the meaning of equal sign.

Watch the equality video. Attend to the details of children’s thinking and the teacher’s instructional moves and make notes on the note-taking sheet.

How are the students thinking about these problems? What misconceptions do they have? How is their thinking like or unlike your own students’ thinking?

Identify the teacher’s instructional moves. How is the teacher facilitating children’s understanding of the equal sign and equality?

Part 2 – Equality Video – Summer [See Instructor Notes Appendix A]Summer is a middle school student who has just completed 7th grade. [NOTE: The interview sequence comes from this article. The instructor may wish to read it in preparation. The article will be used in a future course. It can be access in Arkansas IDEAS under Resources in the folder “CCSS Math – NCTM Articles” using your IDEAS log in. The article should not be provided to teachers in this course.

Knuth, E.J., Alibali, M.W., Hattikudur, S., McNeil, N.M., & Stephens, A.C. (2008). The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle, 13 (9), 514-519.]

Watch the video of Summer being interviewed about some mathematics equations. The intent was to interview Summer and try to see how she thinks, not to correct her misconceptions. Attend to the details of Summer’s thinking and make notes on the note-taking sheet.

What definition of the equal sign was Summer using?

Session 3 Page 5



What was problematic about her definition? (Include details from the interview that lead you to think that.)

How did it impact how she solved the problems? What decisions did the interviewer make, particularly during the final segment? What impact

did they have on Summer’s understanding of the equal sign and relational thinking?

DiscussClass Discussion:Discuss your impressions of the videos above.

How are the students thinking about these problems and the meaning of equals? What misconceptions do they have? How is their thinking like or unlike your own students? Thinking of the sign as mostly or only separating operation from answer or as telling you to find an answer. Thinking that what is on each side of the sign has to stay separated.

How did some of the teacher decisions facilitate children’s understanding of the equal sign and equality? See notes from earlier in this session.

What does this mean across grade levels? How does what a primary teacher teaches related to equality relate to what the expectations are for older students? Summer did not benefit from a concerted effort to explore the meaning of equals as an older student. In not developing many of the properties of equality as a younger student (such as the idea that if you add the same thing to two equal quantities you maintain equality) she is now restricted in ways she should be able to solve equations. The first graders were dealing with recognizing it as “the same as” and moving up a couple of grades you might want to have a conversation about something like (3 x 4) – 2 = (6 x 2) – 2 and just continue to push the understanding until using properties of equality is natural for students.

Session 3 Page 6



Session #4: Multiplication and Division Problem Situations

IntroductionIn this session you will:

Develop an understanding about the basic types of multiplication and division problem situations.

Identify the number sentences that goes with each problem situation and why it is critical to distinguish between the problems and the number representations.

Consider what this means to you and your instructional practice.

AccessTry It Out:Write a story problem to go with the number sentence below. Draw a picture of the solution to this problem.

3 x 6 = a

Write two story problems that would be solved differently to go with the number sentence below. Draw a picture of the solution to this problem.

18 ÷ 6 = b

ActivateSmall group/partner discussion:Share your different story problems.

What is the meaning of 3 x 6 = a? The most common interpretation is 3 groups of 6 items. Some may say 3 items in each of 6 groups.

Do you all agree about the meaning of the multiplication problem or not? Perhaps not. It is sometimes preferable within a learning group (classroom) to agree on a definition or convention to make communication more clear. For this course we will agree to the multiplication equation as saying “x group of y items.”

Look at the division problems. Are your stories structurally alike or different? How? Here you’ll find that the division may have take on two structures: one where the number of groups is unknown and one where the number per group is unknown.o Discuss how the strategies are alike or how they are different? If the number of groups is

unknown, it is likely that you will make the equal sized groups defined by the problem and then count the number of groups (dole them out). If the number per group is unknown and the number of groups is known, it is more likely you will distribute items one at a time into the groups, a process sometimes called “dealing” – like dealing cards.

o If you were to write multiplication number sentences to go with your story problems, what would they look like? Discuss with the group. If the convention for multiplication is agreed upon that it represents N groups of Y items, then the multiplication equation for

Session 4 Page 7



a situation where the number of groups is unknown would be b x 6 = 18 and the multiplication equation for a situation where the number per group is unknown would be 6 x b = 18.

Analyze:Jordan is a first grader about a third of the way through the school year. He is making sense of three different problems like the ones you just discussed. Watch the three videos below and consider the following prompts:

What did Jordan do? How does his strategy relate to the structure of the problem? What number sentence would go with the way he interpreted the problem?

Video 1 –There are 6 bags of candy. There are 3 pieces of candy in each bag. How many pieces of candy are there? This is a multiplication problem. Jordan begins by making the groups (bags) as loops on his paper. He puts 3 blocks in each and counts the total number of blocks. 6 x 3 = ___

Video 2 – Miguel has 24 brownies. He has 8 plates. He wants to put the same amount of brownie on each plate. How many brownies will he put on each plate? This is a division situation where the number of groups is known but the number of items per group is not known. Jordan at first interpreted this as 8 groups with 8 items per group by drawing 8 “plates” and trying to put a stick of 8 blocks in each. Then when he seems stuck the interviewer reads the story again. He recognizes the need for his total to be 24, which pushes him to rethink his strategy. He then tries groups of 5 blocks in each. When he runs out he takes one from each existing group making them have 4 to see if 4 will work for all of them. He runs out again, and ends us recognizing that 3 will work. 8 x ___ = 24

Video 3 – Kathy has 24 cookies. She wants to put 6 cookies on each plate. How many plates will she need? This is a division situation where the number of items per group is known but the number of groups is not known. Jordan initially draws 6 “plates” which causes the interviewer to read the story. Then Jordan creates groups of 6, but continues to want to fill the 6 plates he already drew. The interviewer reads the opening sentence of the problem again. ___ x 6 = 24

In general Jordan struggles to make sense of the missing factor in both of the division problems, seemingly wanting the given factor to take on both the role of number of groups AND number of items in each group. This causes him to lose sight of the total. When he redirects himself back to the total he is able to figure out both problems.

Read:Read the summary of the Basic Problem Situations for Multiplication and Division.

DiscussClass Discussion:

Session 4 Page 8



What are the three basic multiplication and division problem situations? How are they alike and how are they different? [See summary above]

How did the different problem structures influence Jordan’s strategies for solving the problems? What details did you see?

Why does problem structure/situation make a difference instructionally? What did your students do? How do their strategies relate to Jordan’s?

[Teachers may bring up concern that they don’t have time to interview all of their students like this. First of all it is important to realize that you can find out a lot about how your students are thinking by listening as they talk to each other or by stopping to conduct shorter interviews as everyone in the class is working on a problem. Obviously you can’t pay equal amount of attention to every student every day. As you gain understanding of how each student thinks, it becomes easier to interpret his/her work. Also, conducting interview periodically will help you hone your professional noticing skills and make you more adept at interpreting with a bit less conversation or by looking at written work. These interviews were done for the purpose of helping teachers develop those skills.]

Session 4 Page 9



ASSIGNMENT – Between Day 1 and Day 2:

1. Read the article Fostering Relational Thinking While Negotiating the Meaning of the Equals Sign by Marta Molina and Rebecca C. Ambrose.

1. Pose some true-false equations to your class to assess your students’ understanding of equality. Consider their thinking.

What surprised you about the students’ thinking? How are the students thinking about these problems? What misconceptions do they have?

2. Reflect: What did you learn about equality and children’s understanding? How does your thinking relate to the article Fostering Relational Thinking While

Negotiating the Meaning of the Equals Sign? How does this relate to your students? What are the instructional implications?

3. Pose the problems on the next page to your class. Use problems that are appropriate for your students. Select problems that are beyond what you typically give them to encourage them to solve the problem in ways that make sense to them.

Assignment Between Day 1 and Day 2 Page 10



Pose to Students: Multiplication and Division Problem Situations

Pose three problems to your students (multiplication, measurement division and partitive division). Select one problem from each problem situation type. The goal is to get your students to solve the problems in a way that makes sense to them like Jordan did. Select problems that you have not taught yet or you think students would be forced to think about in their own way as opposed to using an algorithm you have already taught them. Collect the work from your three focus students for later discussion. Be prepared to look at how the problem structure influenced or didn’t influence them and how they thought about the problems.

Multiplication:Marie has _____ bags of cookies. She puts ______ cookies in each bag. How many cookies does she have?(4, 2) (6, 5) (8, 9) (18, 10) (18, 25)

Martina is feeding ____ pet turtles. She gives each a _______ of box of turtle food. How many boxes of turtle food does she need?(12, ¼) (27, ¼) (36, 1 ¼) (36, ¾)

Measurement DivisionJacob has _____ toy trucks. He puts _______ trucks in each box. How many boxes will he need to put all of his trucks away?(12, 3) (25, 5) (36, 4) (63, 7) (256, 10) (336, 56)

Lisa was making cupcakes. She has ______ cups of frosting. She put _____ of a cup of frosting on each cupcake. How many cupcakes can she frost?(5, ¼) (13, ¼) (37, ¾) (37, 1 ¼)

Partitive DivisionMarcus has ______ cookies. He puts them onto _______ plates. If he puts the same number of cookies on each plate, how many cookies will be on each plate?(12, 4) (24, 6) (137, 10) (228, 3)

There are ____ friends sharing _______ giant cookies. If they share equally, how much cookie will each student get?(8 sharing 14) (5 sharing 13) (6 sharing 10)

Assignment Between Day 1 and Day 2 Page 11



Instructor Notes for Part 1: First Grade Equality Video – Shane Flud

Shane Flud works with a small group of first grade students. Most of the students struggle with the meaning of equal sign. Watch the equality video. Take notes on the details of children’s thinking and the teacher moves.

How are the students thinking about these problems? What misconceptions do they have? How is their thinking like or unlike your own students?

Identify the teacher moves. How is the teacher facilitating children’s understanding of the equal sign and equality?

How did his moves influence children’s thinking? How did Shane use his understanding of children’s thinking to facilitate understanding about

the equal sign and equality?

Shane starts by asking the students what goes in the box to make this number sentence true.4 + 3 = ☐ + 2The students respond the following 9 (3 students), 7 (3 students) and 5 (2 students). The students struggle because they do not see meaning equal and have a rigid definition of equal related to operations (it means to do something). His goal is to pose problems to students to get them to see that the equal sign is about being equal on both sides.

He begins with the following number sentences.2 = 22 = 2 + 0Some students see these as begin true but some of the students struggle to see how these make sense.

This problem begins to shift thinking. Some of the students struggle with the equal sign being in the beginning of the equation but that idea is temporarily forgotten since 4 is not equal to 5. They are able to suspend the backwards number sentence. 4 = 4 + 1As students talk, they bring up the idea that if the equation started with 4 + 1 then it would be equal to 4 + 1. Shane uses that idea to press further on the students’ thinking and write the number sentence to match their suggestions. This builds on the idea that 4 + 1 is equal to 4 + 1 while pushing the students to accept two expressions and away from the equal sign to do something or an operation comes next. 4 + 1 = 4 + 1

Shane pushes further by using the commutative property to see if the students will still accept that it is the same on both sides. Students accept it.1 + 4 = 4 + 1

5 = 5

Appendix A



0 + 2 = 2 + 0

Again he pushed combining two expressions this time using an inequality. The students quickly see that 2 + 2 is 4 and that is not equal to 2. Some students still struggle to understand this.2 + 2 = 0 + 2

2 = 2 + 0

2 = 0 + 2

He presses on a number that is close to the original open number sentence. The students again recognize that these two are not equal (easier to deal with a false and suspend their issues related to the structure of the problem). 4 + 2 = 4 + 36 ≠ 7

This problem is based on the previous conversation. One student said that if the number sentence was 4 + 2 = 4 + 2 or 4 + 3 = 4 + 3. So Shane asks what would it take to make the following a true number sentence.4 + 2 + ___ = 4 + 3

The students come up with one going in the blank and then justify their thinking in three different ways. The first student thinks about it relationally. If you add 2 + 1, that would be three, so 4 + 3 = 4 + 3. Another student, student added 4 + 2 to make 6 and said plus one would be 7 and 4 + 3 is also 7. The other student adds both sides and gets 7 on each side.

Shane follows up with a similar problem to the first one. The students respond 4 because 6 + 2 was 8 and 4 + 4 was 8.6 + 2 = ___ + 4

The students are beginning to see that the equal sign means equal or the same as. This is learning though that takes place over time.

Appendix A



Instructor Notes for Part 2: Equality Video – Summer

Summer is a middle school student who has just completed 7th grade.

Watch the set of videos of Summer being interviewed about some mathematics equations. Take detailed notes about Summer’s thinking.

What definition of the equal sign was Summer using? Summer thinks of the equals sign as indicating the need to perform an operation to get an answer.

What was problematic about her definition? (Include details from the interview that lead you to think that.) It kept her from seeing that she could have an answer on both sides. It limited her to thinking about performing arithmetic only on one side of the equal sign to get a result. It keeps her from identifying helpful structure by comparing the two sides.

How did it impact how she solved the problems? On the problems where she was finding the value of m it limited her to guess and check. She did not see that if she subtracted (10 or 7) from both sides she would still have equal expressions. In part (c) she still did not see this. She also seemed to struggle with the box and the “x” times sign in this problem (might she have struggled less if it had used a letter variable as with the earlier problems (2m + 15 = 31; 2m + 15 – 9 = 31 – 9)

What decisions did the interviewer make, particularly during the final segment? What impact did they have on Summer’s understanding of the equal sign and relational thinking? The interview deliberately helped Summer realize she was thinking about the equals sign differently and led her back to the earlier problem to see if she could use her new understanding to think about that problem again.

3 + 4 = 7

What is the symbol called the arrow is point to?What is the definition of this symbol?------------------------------------------------------------------------------------------------------------What value of m will make the following equations true? ()(a) 4m + 10 = 70 (b) 3m + 7 = 25

(c) Is the number that goes in the ☐ the same number in the following two equations? Explain your reasoning.2 x ☐ + 15 = 312 x ☐ + 15 – 9 = 31 – 9

(d) In the equation ☐ + 18 = 35, the number that goes in the ☐ is 17. Can you use this fact to help you figure out what number goes in the ☐ in the equation ☐ + 18 + 27 = 35 + 27? Explain your reasoning.

Appendix A



Appendix A

Documents

commoncore.aetn.orgcommoncore.aetn.org/training/psmd/PSMD-Day-1-Face2Face-Guid… · Web view(from CCSSM pg 8 “Expectations that begin with the word “understand” are often