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The Routty Math Teacher is a web-based company created to support K-8 mathematics instruction through blogging, collaboration, and instructional materials.

## Wednesday, August 10, 2016

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The Routty Math Teacher Has Moved!

## Thursday, July 28, 2016

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Mathematical Mindsets: Chapter 4

**Chapter 4 Summary**

**My Big Takeaway**

**Ways to Use this New Knowledge to Support Our Students in the Classroom**

Developing strong number sense is an important way to help students become more successful math students; however, in order for students to be truly successful, it must be accompanied by a mathematical mindset.

** Sound Off!** How do you develop number sense in the classroom?

**References:**

## Thursday, July 21, 2016

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Mathematical Mindsets: Chapter 3

**Chapter 3 Summary**

**My Big Takeaway**

**Ways to Use this New Knowledge to Support Our Students in the Classroom**

Changing how we view the role of and purpose for mathematics in the classroom will help our students see it as more than a series of rules, procedures, and calculations. By showing them what real mathematics is and what mathematicians do, we may encourage more positive feelings about and increased success in the subject.

** Sound Off!** How do you describe mathematics?

**References:**

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## Thursday, July 14, 2016

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Mathematical Mindsets: Chapter 2

**Chapter 2 Summary**

**My Big Takeaway**

**Ways to Use this New Knowledge to Support Our Students in the Classroom**

As we prepare to return to school for the fall semester, it's important to consider the role mistakes will play in our classrooms. Will our students hide and shyly respond to questions of which they are unsure of the answer or will they be proud of their thinking, even when it is wrong? While this way of thinking about the role of mistakes may be foreign to many of us, it's imperative to helping our students find success in mathematics.

** Sound Off!** How will you celebrate mistakes this year?

**References:**

## Thursday, July 7, 2016

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Mathematical Mindsets: Chapter 1

**Chapter 1 Summary**

**My Big Takeaway**

**Ways to Use this New Knowledge to Support Our Students in the Classroom**

** Sound Off!** How will this idea of a fixed/growth mindset change your classroom for this next school year?
**References:**

## Wednesday, June 29, 2016

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Summer PD: Productive Struggle in Mathematics- Part 4

**Want to learn more about mindsets?** Join The Routty Math Teacher and 12 other bloggers for a collaborative book study focusing on Jo Boaler's __Mathematical Mindsets__. Grab a copy of the book and read along with us or join us online for chapter summaries, biggest takeaways, and tips for applying the new learning in the classroom. The study goes live on Thursday, July 7th.

**Sound Off! **What are some other ways we can promote a growth mindset in our students?

**References:**

## Wednesday, June 22, 2016

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Summer PD: Productive Struggle in Mathematics- Part 3

**Reference:**

## Wednesday, June 15, 2016

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Summer PD: Productive Struggle in Mathematics- Part 2

## Wednesday, June 8, 2016

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Summer PD: Productive Struggle in Mathematics- Part 1

Thank you for joining me for week two of Summer PD! For the next three weeks, I will be talking about productive struggle and how we can use it to promote a growth mindset for our students. Part I of this mini-series will define productive struggle, advocate for its purpose and usefulness in the classroom, and illustrate how it is reflected in a teacher's instructional decisions.

Imagine two sixth grade classrooms with two teachers, Mrs. K and Ms. S, who are presenting the following problem solving task:

Ms. S is very prescriptive in her response. She tells the students to draw a strip diagram and use pictures to represent the number of animals when compared to the cows. She then instructs them to label the entire rectangle as 96 to represent the total number of animals. The picture below shows an example of the strip diagram created by Ms. S. Finally, she instructs the students to use the diagram to determine the number of each kind of animal.

Mrs. K approaches the situation in a different way. She begins by asking for the class's attention and then encourages the class to analyze the situation. Together, they begin to complete a KWC chart to identify what they know, what they want to know, and the special conditions for the problem. Mrs. K then begins a discussion to probe the students about what should be done next. After several suggestions have been offered, Mrs. K encourages the students to consider the ideas that were shared and choose a path to explore.

As a result of the teachers' instructional decisions, the students have had very different learning experiences. While Ms. S used a prescriptive approach to direct the students' thinking and lead them to the correct solution path, Mrs. K helped the students analyze the problem and encouraged them to choose a starting place at which to begin. Ms. S's students have learned that if they are stuck and can't move forward, Ms. S will rescue them. Mrs. K's students have learned that when they are stuck and need help to move forward, Mrs. K will give them some support to help get them back on track.

Mrs. K's instructional decisions displayed in the vignette supports students "struggling productively as they learn mathematics" (p. 48). The National Council of Teachers of Mathematics (NCTM) states that Mrs. K's classroom instruction "embraces a view of students' struggles as opportunities for delving more deeply into understanding the mathematical structure of problems and relationships among mathematical ideas, instead of simply seeking correct solutions" (p. 48). NCTM also suggests that using productive struggle in the classroom has long-term benefits that will allow students to apply their learning in a variety of new situations and contexts.

**Reference:** National Council of Teachers of Mathematics. (2014). *Principles to actions: Ensuring mathematical success for all*. Reston, VA: National Council of Teachers of Mathematics.

## Wednesday, June 1, 2016

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Summer PD: No More Keywords!

**Show a child some tricks and he will survive this week’s
math lesson. Teach a child to think critically and his mind will thrive for a
lifetime.**

Welcome to Summer PD! Because many of us devote time during the summer months to look for opportunities for professional growth, I will be presenting an 8-week summer PD blog series this June and July. Join me each Wednesday for a new topic! Happy Reading!

The focus of this week's Summer PD is the dangers of using keywords to solve math word problems. This article presents arguments against the use of keywords and offers a new strategy to refocus students' learning on critical thinking and sense-making.

**The Problem**

Van de Walle and Lovin (2006) and Van de Walle, Karp, and Bay-Williams (2012) provide four arguments against the use of key words:

1. Keywords can be dangerous! In fact, they can be used in ways that differ from the way students expect them to be used and lead students to an incorrect solution strategy path. Consider the problem in the illustration below. If students misunderstand the phrase "6 more" to mean that Caty has six more baseball hats than Derek, they will incorrectly respond with an answer of 16 rather than 4.

2. The use of keywords focuses on looking at the words in isolation and not in the context of the problem. "Mathematics is about reasoning and making sense of situations" (Van de Walle & Lovin, 2006, p. 70). Students should analyze the structure of problems in context not just dissect them for keywords.

3. Many problems, especially as students begin to advance to more sophisticated work, have no key words. Consider the problem in the illustration below. Because the problem does not contain key words, students who rely on this approach will not have a strategy on which to rely.

4. The use of key words does not work with more advanced problems or those with more than one step. Therefore, students who do not attend to the meaning of a problem while solving it will be unsuccessful in completing the problem.

Tina Cardone, author of "Nix the Tricks," a guide to avoiding non-conceptually developmental short-cuts, suggests having students think about the words of the problem as a whole and focus on what is happening in the problem in context. Additionally, she suggests that the use of student-drawn illustrations will help students understand the problem and make sense of the words before completing computations. (Grab a copy of Tina's book here.)

**Math Makes Sense**

Instead of using keywords, I would like to encourage the use of the operation situations.

The illustration above shows the multiplication situation and the two division situations. In the multiplication situation, equal groups are counted until a total is found. Problems illustrating a multiplication situation involve finding the total amount in a certain number of equal groups. In the division situations, a total amount is divided into a specific-size group or a specific number of groups. Problems illustrating a division situation will involve finding how many groups or how many in each group.

**Give it a try!**

Here are some ideas to move your students from keywords to the operation situations:

1. Use basic word problems from a grade-level resource or textbook as a sorting activity to allow students to practice visualizing the situations and matching them to an operation. Be sure to have students identify the situation when they provide the operation.

**Looking for more?** You can find a complete version of my Operation Situation pack with the
full-size illustrations of the operation situations in my Teachers Pay Teachers
Store. Click here to see it now!

**Sound Off!** How do you teach your students to analyze word problems?

**References: **

## Monday, May 30, 2016

## Friday, May 27, 2016

###
Solve It Friday!- Task #35

**Here's how Solve It! Friday works:**

1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task.**Note: **In some cases, I may post more than one version of the task to reach a wider variety of grades.

2. Before the next Friday, use the task with your students.

3. Have students solve the problems individually or with a group.

4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions.**Note:** The blank backs of old book covers make great poster paper!

5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students.

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results.

I look forward to seeing your students' work! Thanks for sharing!

**Solution: **The real beauty of this task is in the process. Please emphasize that with your students. It may take some time to solve this problem. Validate their efforts and ask questions to move them in a different direction if needed. For your convenience, I have provided the solution below:

**#RMTSolveItWeek35: **There were 20 red fish, 10 green fish, and 6 yellow fish.

## Contributors

Hello! The Routty Math Teacher has moved!

Come visit my new website at www.therouttymathteacher.com!

Welcome back! For the past four weeks, I have been reading and participating in a collaborative book study focused on the book __Mathematical Mindsets__ by Jo Boaler. Chapter 3 described how the natural creativity and beauty of mathematics connects to the real world, but a disconnect exists between it and what is typically taught as part of the school mathematics program. (Read Chapter 3's post here.) Chapter 4 discusses ways to promote a mathematical mindset by helping students develop flexibility with numbers.

Jo Boaler states, "when students see mathematics as a set of ideas and relationships and their role as one of thinking about the ideas, and making sense of them, they have a mathematical mindset" (p. 34). Boaler also describes a study conducted by two British researchers which discovered that when compared to a group of low-performing counterparts, who had been exposed to the same amount of content, the high-performing students were able to employ number sense, a flexible and conceptual interaction with numbers, to solve problems successfully. The low-performing students, however, employed a standard algorithm which is often difficult to recall and execute successfully. The chapter also includes a robust discussion about purposeful math fact review and homework assignments.

Jo Boaler states, "number sense and mindsets develop together, and learning about ways to develop one helps the development of the other" (p. 36). I've always believed in the importance of helping students develop number sense at all grade levels, but this chapter really helped me solidify this belief. However, the piece that has not been as strong a belief for me is how mindsets support this goal. I now know that they both work together.

Helping students develop number sense must be a regular classroom routine that includes conceptual activities focused on numbers and number facts. I describe my two favorite routines below.

**Number Talks:**Jo Boaler describes them as "appreciating the connected, conceptual nature of mathematics" Number talks require students to describe how they solved an abstract problem mentally. After collecting multiple students' methods, the class then explores how and why the strategies work. I wrote an entire series about using number talks in the classroom when I participated in a book study focused on the book,__Making Number Talks Matter__. Read the series here.**Greg Tang:**His collection of books is fantastic for helping students build numeracy skills. Through his picture books, students learn how to group numbers to flexibly count the number of items in his colorful pictures. My favorite book is__The Best of Times__. It helps students discover strategies for mastering basic multiplication facts. Explore the online version here. I also wrote a post about how I used his books to help my students develop computational fluency skills. Read the post here.

Developing strong number sense is an important way to help students become more successful math students; however, in order for students to be truly successful, it must be accompanied by a mathematical mindset.

- Boaler, J. (2016).
*Mathematical Mindsets*. San Francisco, CA: Jossey-Bass

Welcome back! For the past two weeks, I have been reading and participating in a collaborative book study focused on the book __Mathematical Mindsets__ by Jo Boaler. Chapter 2 described the importance of mistakes in the learning of mathematics. (Read Chapter 2's post here.) Chapter 3 takes an in-depth look at how people view the world of mathematics and how school mathematics differs from real-mathematics.

In this chapter, Jo Boaler discusses how our view of mathematics affects how effectively we are able to learn and understand the subject. She states that people see mathematics as different from other subjects, a series of rules and procedures, because they do not understand the complex nature and beauty of mathematics. Because it is often viewed as a series of right or wrong calculations, Boaler describes school mathematics as a disconnect between "the mathematics that mathematicians use and the mathematics of life"-- "a creative, visual, connected, and living subject" (p. 27).

Mathematics is mathematics. There is no difference between school mathematics and the mathematics that mathematicians study every day. Jo Boaler states, "when we teach mathematics-- real mathematics, a subject of depth and connections-- the opportunities for a growth mindset increase, the opportunities for learning increase, and classrooms become filled with happy, excited, and engaged students" (p. 32). In addition, being good at math is not about being fast or first, it's about being a powerful thinker.

1. Investigate the role of mathematicians. Post a chart entitled, "How to Think Like a Mathematician" with verbs that describe what mathematicians do, such as wonder, guess and check, ask questions, make conjectures, make connections, solve problems, etc. Continue adding to the chart throughout the year as students find other verbs to describe the work of mathematicians. When solving problems, refer to the chart and ask students to reflect on something that they did that made them "Think Like a Mathematician."

2. Ask students to wonder! I love to use Dan Meyer's 3-Act Mathematical Stories for this. For example, show the students the video portion of the Girl Scout Cookies 3-Act with no leading information. Just show the video. After the video is over, go back to the 17-second mark and pose the question, "What are you wondering now?" Allow students to contribute their ideas during a class discussion before focusing on what question to tackle. This 3-Act is set-up to answer the question how many boxes fit in the trunk; however, there are many other questions to explore. Once a question is determined, discuss how students will investigate the question and find a solution. The beauty of this task-- it's tough to find an exact answer, so the focus is on the process, not the final answer. Give it a try!

3. Look for math in nature and have students describe how what they see relates to mathematics. Collecting and posting pictures of math in nature is a good reminder to students that math stretches beyond the walls of the classroom.

- Boaler, J. (2016).
*Mathematical Mindsets*. San Francisco, CA: Jossey-Bass

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Last week, I began participating in a collaborative book study with 12 other bloggers focusing on the book __Mathematical Mindsets__ by Jo Boaler. Chapter 1 described what happens in the brain when we learn new things and illustrated the difference between a fixed mindset and a growth mindset. (Read Chapter 1's post here.) Chapter 2 discusses the power of mistakes in mathematics.

Making a mistake is good! It lights up our brains causing it to "spark and grow" (p. 12) even when we don't know a mistake was made. Coupled with a growth mindset, when an individual believes he or she has the ability to learn and that mistakes are just part of the learning process, mistakes will cause our brains to grow, especially when we participate in challenging tasks.

Mistakes should be valued, celebrated even. When I read this chapter, I immediately began thinking about my years as a classroom teacher and how many times I may have given the impression that making a mistake at a certain point in the year with curriculum that we had learned months earlier was wrong. I know that my intention was to build proficiency, but I'm not sure that is the message my students received.

After reading this chapter, I understand the power that was in the mistakes my students made with curriculum we had learned sometime earlier and how taking the time to analyze them carefully may have led us down a new path. A path that may have led to a deep misunderstanding that the students still had-- an opportunity I missed to help them become more successful learners.

1. When we are reviewing assignments, move from analyzing assignments for correctness and begin to analyze them for mistakes. We should ask ourselves, "What do the mistakes reveal about where my students are in the learning process?" Then create an appropriate plan of action to address the mistakes.

2. Use Jean Piaget's concept of "disequilibrium" to promote a growth mindset. Helping students understand this concept will send two important messages: 1.) It's okay not to know the answer right away and 2.) It's okay to struggle (Carter, 2008). Providing tasks that put students in a state of disequilibrium will help them develop and understand the role of productive struggle in the successful learning of mathematics.

3. Celebrate and highlight mistakes! "My Favorite No" is a teaching strategy that does just that. Click the video link below to see the strategy in action in a real math classroom. Don't see the video? Click here.

As we prepare to return to school for the fall semester, it's important to consider the role mistakes will play in our classrooms. Will our students hide and shyly respond to questions of which they are unsure of the answer or will they be proud of their thinking, even when it is wrong? While this way of thinking about the role of mistakes may be foreign to many of us, it's imperative to helping our students find success in mathematics.

- Boaler, J. (2016).
*Mathematical Mindsets*. San Francisco, CA: Jossey-Bass - Carter, S. (2008). Disequilibrium and questioning in the primary classroom: Establishing routines that help students learn.
*Teaching Children Mathematics*, 15(3), pp. 134-37.

Today marks the start of a collaborative book study I am participating in with 12 other bloggers focusing on the book __Mathematical Mindsets__ by Jo Boaler. This idea of mindset has sparked a lot of discussion in the world of mathematics education since the book's publication earlier this year.

This work began with Carol Dweck's research detailed in __Mindset: The New Psychology of Success__ published in 2006. Dweck's work revealed that we all have a mindset-- a core belief about how we learn. (Boaler, 2016). People with a growth mindset believe that smartness increases with hard work, whereas those with a fixed mindset believe that you can learn things but you can't change your basic level of intelligence" (Boaler, 2016, p. x). This idea of mindsets, then, becomes incredibly important for educators because the research conducted by Dweck and Boaler reveal that different mindsets "lead to different learning behaviors" and, in turn, "different learning outcomes for students" (p. x).

The driving force behind this book pushes us to understand the power of mindsets and how we, as educators, can use them to change our students' learning pathways and allow them the opportunity to achieve higher levels of success. Join me each week as I, and my fellow bloggers, explore each new chapter of the book. And, if you're really interested, grab a copy and read along with us! Happy Reading!

As we encounter new ideas, electric currents begin to fire in our brains making connections between the various areas and regions-- the more complex and intense the new learning, the more lasting the connections will be. This lays the foundation for our work with students; if we provide them with the tools they need to successfully accomplish more complex tasks, they will foster a growth mindset and believe that they can be successful mathematicians.

There are two types of mindsets: fixed and growth. Students who shut down when they reach their frustration point because they foster a belief that being good at math is an intelligence which you obtain at birth have a "fixed mindset." On the other hand, students with a "growth mindset" believe that the intelligence needed to be successful in mathematics can be gained through effort.

1. Provide short, prolonged opportunities to interact with new skills so that students' brains have the opportunity to make connections and change structurally.

2. Foster a belief that no one is born "good at math." Rather, emphasize the idea that making mistakes is part of the learning process and how we growth as mathematical thinkers.

3. Provide feedback that is related to actions and not student characteristics. For example, say, "Your work shows that you thought a great deal about how to approach this problem" instead of, "You are so good at this!"

As teachers, understanding mindsets is a powerful way to begin a new year with our students. Not only will it help us understand our own mindset about mathematics, it will help us understand how the way we think about and teach mathematics can play a deeper role in what our students take away with them at the end of our time together.

- Boaler, J. (2016).
*Mathematical Mindsets*. San Francisco, CA: Jossey-Bass - Dweck, C. (2006).
*Mindset: The new psychology of success.*New York: Random House.

Welcome back! Last week, I presented a list of expectations for both students and teachers during productive struggle and provided an opportunity to see productive struggle in action via a Teaching Channel video. (Read Part 3 here.) Today, I want to connect productive struggle with growth mindsets and share a great resource with you.

Susan Carter (2008) introduced her first grade class to the concept of disequilibrium to help them better understand and identify the discomfort they felt when they solved math problems. This allowed her students to do two things. First, it let them know that it was okay not to know the answer to a problem right away. Second, it "gave them permission to struggle" (p. 136). NCTM (2014) advocates that "teachers must accept that struggle is important to students' learning of mathematics, convey this message to students, and provide time for them to try to work through their uncertainties" (p. 50).

This idea of disequilibrium is so important for students to understand as many people believe that math is either right or wrong. They believe that there is only one answer and there is a process, or series of steps, to take to get to the solution. Students, and teachers, who view mathematics in this way cripple themselves because they miss a major component in the successful learning of mathematics-- growth is the result of struggling through a process. Remember Lev Vygotsky's "zone of proximal development" that was discussed in Part I? (Read about it here.) Growth occurs between what we can do alone and what we cannot do.

So where then do growth mindsets fit in? Despite the willingness of a teacher to allow students to work through disequilibrium at their own pace, some students will still shut down when they reach their frustration point because they foster a belief that being good at math is an intelligence which you have at birth-- you either have it or you don't. Research conducted by Carol Dweck (2006) has termed this type of thinking as having a "fixed mindset." On the other hand, those students with a "growth mindset" (Dweck, 2006) believe that the intelligence needed to be successful in mathematics can be gained through effort. These students are more likely to tackle a challenge because they view it as "an opportunity to learn and grow" (NCTM, 2014, p. 50).

How do we promote a growth mindset in our students?

- acknowledge and show value for the effort that students make
- ask questions that allow students to move forward without giving away the answer or solution strategy
- provide specific feedback on students' progress
- reinforce the idea that struggle is part of the process

Stay tuned next week for Chapter One of __Mathematical Mindsets__!

- Carter, S. (2008). Disequilibrium and questioning in the primary classroom: Establishing routines that help students learn.
*Teaching Children Mathematics*, 15(3), pp. 134-37. - Dweck, C. (2006).
*Mindset: The new psychology of success.*New York: Random House. - National Council of Teachers of Mathematics. (2014).
*Principles to actions: Ensuring mathematical success for all*. Reston, VA: National Council of Teachers of Mathematics.

Welcome back! Last week, I offered a more definitive illustration of productive struggle and how it occurrs in the classroom. (Read Part 2 here.) Today, I want to offer a list of expectations for both students and teachers during productive struggle (NCTM, 2014) and provide an opportunity to see productive struggle in action via a Teaching Channel video.

In order to create an environment where productive struggle can occur naturally, pre-planning on the part of the teacher is required. The list below includes actions that teachers can take to plan for and support students' struggle during instruction:

- Choose tasks that require students to use their critical thinking and reasoning skills in order to be successful.
- Encourage students to continue working through a challenging task, even when it feels insurmountable.
- Provide support without removing the challenge and opportunity for student growth from the task.
- Convey the message that the journey is just as important as the destination and encourage students to explain and justify their solutions.
- Give students the opportunity to evaluate and validate a variety of strategies and solutions.
- Provide access to tools that may support students throughout the process, such as manipulatives like number lines, counters, measuring tools, calculators, etc.
- Ask questions that are reflective of the students' thinking rather than that of the teacher.

- Persevere through challenging tasks even when they are frustrated and want to quit.
- Communicate thinking coherently with the use of appropriate mathematical vocabulary and terms.
- Ask questions to help clarify the explanations of others or when an explanation is not fully understood.
- Use drawings and math tools to make sense of the tasks.
- Communicate verbally with others while working through a task to help move the solution strategy forward.

Questions to consider while watching:

1. How do the teachers' actions promote productive struggle?

2. How do the students persevere through the task?

3. What tools do the students use to help them understand and respond to the task?

4. How do the students interact with each other?

5. How does the teacher support the students during the lesson?

- National Council of Teachers of Mathematics. (2014).
*Principles to actions: Ensuring mathematical success for all*. Reston, VA: National Council of Teachers of Mathematics.

Welcome back! Last week, I begin discussing the topic of productive struggle and how it can be used to encourage critical thinking and promote a growth mindset in our math classrooms. (Read Part 1 here.) This week I want to focus on better defining productive struggle and how to achieve it in the classroom.

This idea of productive struggle reminds me of an idea Lev Vygotsky first began to theorize called the zone of proximal development in the 1930s, The "zone of proximal development" describes an area of learning between what a student can do alone and that which he or she cannot do (see the illustration below). This "zone" is where productive struggle happens. It is the place where students work to accomplish a more challenging task with the support of their teacher. Simple right? Here's the tricky part, as teachers, we want our students to be successful, so we give them a task that may be slightly too easy or we give them a task that is too hard and then drag them through the solution strategy process. In either case, the students' understanding and level of thinking will not advance.

With that said, what makes a struggle productive? Here are three key ideas (Peterson, 2016):

- Is the mathematics of the task within the students' depth of knowledge?
- Is the mathematics of the task related to current learning targets?
- Does the task require sense-making?

Selection of the mathematical task is one of the most important aspects of the productive struggle process. For this reason, the task at hand is central to whether or not the students' struggle will be productive.

First, in order for the students to be willing to tackle the task, it must be be in the zone of proximal development discussed above. If the task is too hard, students will be easily frustrated and want to shut down. If the task is too easy, students will gain nothing from the experience.

Second, if the task does not involve mathematics with which students are familiar and are able to do, they will not be successful with the task. However, if the mathematics involves something with which the students have been working, then they are much more likely to continue to work at the task until they have completed it successfully.

Third, in order for students to gain a deeper understanding of mathematics, they must be able to make sense of what they are doing. The selected task is central to this idea. If students cannot make sense of the mathematics, no knowledge will be gained. Growth occurs when students begin to make sense of something they did not initially understand.

Let's return to the example with Mrs. K from last week's post and examine her instructional decisions. (Read it again here.)

Mrs. K understood this need for the students to struggle with the task and proceeded to help them analyze the problem and determine a place to begin and direction to follow. Unlike Ms. S, she did not walk the students through each aspect of the solution strategy process. Instead she offered the support students need to stay in their zone of proximal development so that they could gain the most from the experience.

**References:**

First, in order for the students to be willing to tackle the task, it must be be in the zone of proximal development discussed above. If the task is too hard, students will be easily frustrated and want to shut down. If the task is too easy, students will gain nothing from the experience.

Second, if the task does not involve mathematics with which students are familiar and are able to do, they will not be successful with the task. However, if the mathematics involves something with which the students have been working, then they are much more likely to continue to work at the task until they have completed it successfully.

Third, in order for students to gain a deeper understanding of mathematics, they must be able to make sense of what they are doing. The selected task is central to this idea. If students cannot make sense of the mathematics, no knowledge will be gained. Growth occurs when students begin to make sense of something they did not initially understand.

Let's return to the example with Mrs. K from last week's post and examine her instructional decisions. (Read it again here.)

Mrs. K presented her fifth grade class with the following problem:

Farmer Brown’s niece Angie is in charge of her uncle’s farm while he is on vacation. He gave her strict instructions to make sure none of the animals ran away. When Angie counted the cows, chickens, and sheep, she counted 96 animals. There were three times as many chickens as cows and twice as many sheep as cows. How many sheep did she count?

As soon as Mrs. K presents the problem, the students begin to show signs of struggle and she overhears several students say that they do not know what to do. In order to respond to the students' cry for help, Mrs. K asks for the class's attention and then encourages the class to analyze the situation. Together, they begin to complete a KWC chart to identify what they know, what they want to know, and the special conditions for the problem. Mrs. K then begins a discussion to probe the students about what should be done next. After several suggestions have been offered, Mrs. K encourages the students to consider the ideas that were shared and choose a path to explore.If we analyze the scenario, we can see that the problem is within the students' depth of knowledge since it only requires knowledge of basic operations in order to obtain the solution. Understanding how to compute with basic operations is central to the mathematics of fifth grade. The algebraic foundation of this problem definitely requires students to make sense of it before beginning to tackle it; so, this becomes the place where students struggle productively. They have all of the tools necessary to solve the problem, but must create a solution strategy that will address the specifics of this task.

Mrs. K understood this need for the students to struggle with the task and proceeded to help them analyze the problem and determine a place to begin and direction to follow. Unlike Ms. S, she did not walk the students through each aspect of the solution strategy process. Instead she offered the support students need to stay in their zone of proximal development so that they could gain the most from the experience.

- National Council of Teachers of Mathematics. (2014).
*Principles to actions: Ensuring mathematical success for all*. Reston, VA: National Council of Teachers of Mathematics. - Peterson, B. E. (2016). How and why to let students struggle: Thoughts from research. Retrieved from: https://mathed.byu.edu/~peterson/NCTM%202016%20-%20Productive%20Struggle%20-%20San%20Fran%20handout.pdf on June 14, 2016.

Thank you for joining me for week two of Summer PD! For the next three weeks, I will be talking about productive struggle and how we can use it to promote a growth mindset for our students. Part I of this mini-series will define productive struggle, advocate for its purpose and usefulness in the classroom, and illustrate how it is reflected in a teacher's instructional decisions.

Imagine two sixth grade classrooms with two teachers, Mrs. K and Ms. S, who are presenting the following problem solving task:

Farmer Brown’s niece Angie is in charge of her uncle’s farm while he is on vacation. He gave her strict instructions to make sure none of the animals ran away. When Angie counted the cows, chickens, and sheep, she counted 96 animals. There were three times as many chickens as cows and twice as many sheep as cows. How many sheep did she count?As soon as each teacher presents the problem, the students begin to show signs of struggle and the teachers overhear several students say that they do not know what to do.

Ms. S is very prescriptive in her response. She tells the students to draw a strip diagram and use pictures to represent the number of animals when compared to the cows. She then instructs them to label the entire rectangle as 96 to represent the total number of animals. The picture below shows an example of the strip diagram created by Ms. S. Finally, she instructs the students to use the diagram to determine the number of each kind of animal.

Clipart by Pink Cat Studio |

As a result of the teachers' instructional decisions, the students have had very different learning experiences. While Ms. S used a prescriptive approach to direct the students' thinking and lead them to the correct solution path, Mrs. K helped the students analyze the problem and encouraged them to choose a starting place at which to begin. Ms. S's students have learned that if they are stuck and can't move forward, Ms. S will rescue them. Mrs. K's students have learned that when they are stuck and need help to move forward, Mrs. K will give them some support to help get them back on track.

Mrs. K's instructional decisions displayed in the vignette supports students "struggling productively as they learn mathematics" (p. 48). The National Council of Teachers of Mathematics (NCTM) states that Mrs. K's classroom instruction "embraces a view of students' struggles as opportunities for delving more deeply into understanding the mathematical structure of problems and relationships among mathematical ideas, instead of simply seeking correct solutions" (p. 48). NCTM also suggests that using productive struggle in the classroom has long-term benefits that will allow students to apply their learning in a variety of new situations and contexts.

Welcome to Summer PD! Because many of us devote time during the summer months to look for opportunities for professional growth, I will be presenting an 8-week summer PD blog series this June and July. Join me each Wednesday for a new topic! Happy Reading!

The focus of this week's Summer PD is the dangers of using keywords to solve math word problems. This article presents arguments against the use of keywords and offers a new strategy to refocus students' learning on critical thinking and sense-making.

Van de Walle and Lovin (2006) and Van de Walle, Karp, and Bay-Williams (2012) provide four arguments against the use of key words:

1. Keywords can be dangerous! In fact, they can be used in ways that differ from the way students expect them to be used and lead students to an incorrect solution strategy path. Consider the problem in the illustration below. If students misunderstand the phrase "6 more" to mean that Caty has six more baseball hats than Derek, they will incorrectly respond with an answer of 16 rather than 4.

2. The use of keywords focuses on looking at the words in isolation and not in the context of the problem. "Mathematics is about reasoning and making sense of situations" (Van de Walle & Lovin, 2006, p. 70). Students should analyze the structure of problems in context not just dissect them for keywords.

3. Many problems, especially as students begin to advance to more sophisticated work, have no key words. Consider the problem in the illustration below. Because the problem does not contain key words, students who rely on this approach will not have a strategy on which to rely.

4. The use of key words does not work with more advanced problems or those with more than one step. Therefore, students who do not attend to the meaning of a problem while solving it will be unsuccessful in completing the problem.

Tina Cardone, author of "Nix the Tricks," a guide to avoiding non-conceptually developmental short-cuts, suggests having students think about the words of the problem as a whole and focus on what is happening in the problem in context. Additionally, she suggests that the use of student-drawn illustrations will help students understand the problem and make sense of the words before completing computations. (Grab a copy of Tina's book here.)

Instead of using keywords, I would like to encourage the use of the operation situations.

In order for students to become successful at solving word
problems, they must become proficient at identifying what’s happening in the
problem situation. Students can do this by visualizing the situation and
creating a mental picture of the actions that are taking place. Once they
understand the actions, students can then connect them to symbols. This is the
power of using the operation situations. They allow students to see the problem
as a whole, like a scene from a movie, and match an operation to the picture.

While it looks like there are a lot of situations, there are
really only six-- two addition, two subtraction situations, one multiplication, and two division situations.

The illustration above shows the two addition situations. In a joining situation, sets are being joined together. Problems illustrating a joining situation involve looking for the total or one of the addends. Similarly, problems illustrating a part-part-whole situation involve looking for the whole or one of the parts.

The illustration above shows the two subtraction situations. In a separation situation, a group is separated and something is left behind. Problems illustrating a separation situation involve finding what's left or what changed after separation and the initial amount before the change. Problems illustrating a comparison situation involve comparing quantities and looking for the larger amount, the smaller amount, or the difference.

The illustration above shows the two addition situations. In a joining situation, sets are being joined together. Problems illustrating a joining situation involve looking for the total or one of the addends. Similarly, problems illustrating a part-part-whole situation involve looking for the whole or one of the parts.

The illustration above shows the two subtraction situations. In a separation situation, a group is separated and something is left behind. Problems illustrating a separation situation involve finding what's left or what changed after separation and the initial amount before the change. Problems illustrating a comparison situation involve comparing quantities and looking for the larger amount, the smaller amount, or the difference.

You'll notice that some of the situations have a missing component within the operation side of the number sentence, such as with What's the Change (Joining), What's the Start, and What's the Part. Sometimes, students must use an inverse operation to
obtain an answer. For example, joining situation #2 is an addition situation,
i.e. two groups of spiders are joined together. However, because the second
group is an unknown, the problem sets up as 5 + __ = 11. Students then have to
subtract to find the answer. Yes! This is an early algebra concept. However,
when students are presented with problems like these, they will determine the
correct path and eventually see subtraction as the most efficient way to find the missing information.

The illustration above shows the multiplication situation and the two division situations. In the multiplication situation, equal groups are counted until a total is found. Problems illustrating a multiplication situation involve finding the total amount in a certain number of equal groups. In the division situations, a total amount is divided into a specific-size group or a specific number of groups. Problems illustrating a division situation will involve finding how many groups or how many in each group.

Here are some ideas to move your students from keywords to the operation situations:

1. Use basic word problems from a grade-level resource or textbook as a sorting activity to allow students to practice visualizing the situations and matching them to an operation. Be sure to have students identify the situation when they provide the operation.

2. Need an anchor chart idea? As the students encounter
different problem types, record the word problem and the problem type on a
labeled operation poster, like "Addition Problems". Keep adding to
the anchor chart throughout the school year.

Van de Wall, J. A., Karp, K. S., & Bay-Williams, J. M. (2012). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson.

Van de Wall, J. A. and Lovin, L. H. (2006). Teaching student-centered mathematics: Grades 3 - 5. Boston, MA: Pearson.

1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task.

2. Before the next Friday, use the task with your students.

3. Have students solve the problems individually or with a group.

4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions.

5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students.

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results.

I look forward to seeing your students' work! Thanks for sharing!

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