Summer PD: Productive Struggle in Mathematics- Part 4

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

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.

Stay tuned next week for Chapter One of Mathematical Mindsets! 

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

References: 

• 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.  

Summer PD: Productive Struggle in Mathematics- Part 3

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. 

Because increased student growth and understanding is the goal of productive struggle, there are expectations for the students as well. The list below includes expectations for students during productive struggle: 

• 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. 

The teaching Channel video linked below depicts a class of second graders who are exploring subtraction strategies through the use of number talks. (If you do not see the video embedded below, please click here.)

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? 

Sound Off! What does productive struggle look and sound like in the classroom? 

Read Part 4 here.

Reference: 

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

Summer PD: Productive Struggle in Mathematics- Part 2

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): 

1. Is the mathematics of the task within the students' depth of knowledge?
2. Is the mathematics of the task related to current learning targets?
3. 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 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. 

Sound Off! What kinds of tasks do you use that promote productive struggle? 

Read Part 3 here.

References: 

• 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.  

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: 

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

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. 

Sound Off! What kinds of instructional decisions promote productive struggle? 

Read Part 2 here. 

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

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.

(Quote Reference) 

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. 

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.  

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.  

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. 

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.

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: 

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.