Wednesday, June 22, 2016

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 II 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? 

Stay tuned next week for part four and the conclusion of this mini-series! 

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

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 15, 2016

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 I 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. 

Stay tuned next week for part three of this mini-series! 

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

References: 

Wednesday, June 8, 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. 


Stay tuned next week for part two of this mini-series! 

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

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

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. 
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.  

Monday, May 30, 2016

Summer PD


Check out my new 8-week Summer PD series beginning Wednesday, June 1st! 

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.  

Thursday, May 26, 2016

Thursday Tool School: Critical Thinking Tools- Games

Thank you for reading my "Thursday Tool School" series. Today's post marks the last "Thursday Tool School" post until September. Until that time, I will be featuring a Summer PD series from June through July. Be sure to check back each week for some new summer learning, in between relaxing at the pool and vacationing of course! :o) 


Today's post features one of my favorite critical thinking games. It is sure to keep your students engaged in these last few days as it is open-ended and can meet the individual needs of your students. "Make 24" is a game that promotes problem solving skills when students use four numbers to add, subtract, multiply, and divide their way to the number 24. 


Freebie Alert! Grab a free copy of this game here

Sound Off! What are your favorite critical thinking games?  

Tuesday, May 24, 2016

Transformation Tuesday: Engaging Critical Thinking Activities- Web Games

Thank you for reading my "Transformation Tuesday" series. Today's post marks the last "Transformation Tuesday" post until September. Until that time, I will be featuring a Summer PD series from June through July. Be sure to check back each week for some new summer learning, in between relaxing at the pool and vacationing of course! :o) 


Today's post features my favorite critical thinking web games. They are sure to keep your students engaged in these last few days. There are two Coolmath.com games that I would like to share with you-- Bridge Crossing and Water Jars. Both games require students to use their strategic thinking skills to successfully complete the activities. They originally appeared in my 2014 "Problem Solving Palooza" series. Check it out here.

The first challenge is one of my favorite web games. It's called "Bridge Crossing." See the screenshot and game details below. 



Grade Level: This website is more suited to the reasoning skills of upper elementary/ middle school students. (See variations below for lower elementary challenges.)

Objective: Help all of the characters cross the bridge.

Task: Only two characters can cross the bridge at the same time. The lantern must be used when the characters cross the bridge. Each character shows how many minutes it takes him/her to cross the bridge. The lantern will lasts for 30 minutes; that's how long the characters have to cross the bridge because they cannot cross the bridge in the dark. 

Note: This is a challenging task, but it is possible. I've had students master this task in the past! 

Ways to Utilize the Activity: This website is a great way to get your student thinking about how to solve problems such as these. Consider displaying the website to the class and review the directions. Then, as a class, discuss ways to approach the task. Ask questions such as: 
  • What is the task asking us to do?
  • What special conditions do we need to consider?
  • Which characters take the shortest time to cross the bridge?
  • Which characters take the longest time to cross the bridge?
  • How can we pair the characters together so that we use the shortest time possible? 
  • Which character can be easily used to travel back and forth across the bridge and hold the lantern?
You may also want to consider allowing students to work in pairs to approach the task and discuss ways to meet the goal together. 

This applet makes a great fast finisher activity. It can also be included on a math menu or used on a menu of problem solving station options. 

Variations: The two websites below are more suited for younger elementary students. They are similar to the challenge above but include special conditions, such as a small penguin cannot be left alone on a side with an unrelated adult penguin. The special conditions add an additional element of challenge, but these tasks are easier than the "Bridge Crossing" task. 

The second challenge is another one of my favorite web games. It's called "Water Jars." See the screenshot and game details below. 


Grade Level: This website is more suited to the reasoning skills of upper elementary/ middle school students.

Objective: Measure 6 liters of water from a 5-liter jug and a 7-liter jug

Task: Fill-up the jars and use them to measure 6 liters of water. The jugs can be emptied, refilled, and transferred to the other container multiple times until the goal is achieved. 

For example, if you fill up the 5-liter jug and then pour it into the 7-liter jug, you can refill the 5-liter jug and pour it into the 7-liter jug leaving 3 liters in the 5-liter jug. You've just measured 3 liters. 

Students will need to make moves like the one above to measure 6 liters. 

Note: This is a challenging task, but it is possible. I've had students master this task in the past! 



Ways to Utilize the Activity: This website is a great way to get your student thinking about how to solve problems such as these. Consider displaying the website to the class and review the directions. Then, as a class, discuss ways to approach the task. Ask questions such as: 
  • What is the task asking us to do?
  • What special conditions do we need to consider?
  • How can we measure different amounts, other than 5 liters or 7 liters, using these two jugs? Let's try to measure 3 liters. 
  • What's the purpose of being able to fill the jug and transfer it to the other jug?
You may also want to consider allowing students to work in pairs to approach the task and discuss ways to meet the goal together. 

This applet makes a great fast finisher activity. It can also be included on a math menu or used on a menu of problem solving station options. 

Variations: This activity can be simulated in the classroom with actual jugs of water. Set the situation up by saying that you need to measure ____ liters of water but you only have a ____-liter jug and a ____-liter jug. Try using this model to support the students' understanding of the task before using the website for struggling or younger students. 

Sound Off! What websites do you like to use to challenge your students' critical thinking? 

Friday, May 20, 2016

Solve It! Friday- Task #34


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:

#RMTSolveItWeek34: There are 7 ways for the girls to spend exactly $2 on candy and chips: 20 candies and 0 chips, 17 candies and 2 chips, 14 candies and 4 chips, 11 candies and 6 chips, 8 candies and 8 chips, 5 candies and 10 chips, or 2 candies and 12 chips.  

Thursday, May 19, 2016

Transformation Tuesday: Critical Thinking Tools- Pattern Blocks

In September 2015, my Thursday Tool School series titled, "Critical Thinking with Pattern Blocks" introduced using pattern blocks as a critical thinking tool. (Read the original post here.) Today, I would like to rewind to that series and take another look at how pattern block activities can be used to increase your students' critical thinking skills. 


Click the image to read the post!
Here's a short summary of the purpose of pattern blocks. A set of pattern blocks contains six basic shapes: a yellow hexagon, a red trapezoid, a blue rhombus, a green triangle, an orange square, and a beige rhombus. The pieces are proportional to each other which extends the number of ways in which they can be used. 


My favorite activity from my "Critical Thinking with Pattern Blocks" series is "What's the Common Attribute?" This activity can be used year-round and makes a great starter activity, especially during that your geometry unit. The only prerequisite skills needed are some basic vocabulary terms related to shapes, i.e. sides, angles, congruent, equal, etc. I created a freebie pack of tasks to accompany this activity. You can find a copy of the freebie pack with the original post!


Click the image to read the post!

The next activity was inspired by an article in the August 2015 edition of Teaching Children Mathematics. It involves having students determine the cost of each element of a pattern block design given the total cost. This task provides the foundation for essential algebraic thinking skills and offers a high-level problem solving task with multiple solutions. You can find a copy of the activity page with the original post. 





Click the image to read the post.


The next activity is similar to the common attribute task above. For "Odd One Out," you display four of the pattern blocks. Students study each figure and determine which shape is the "Odd One Out." The caveat-- there isn't necessarily one right answer. In fact, the name of this game is critical thinking. The goal is for students to scrutinize each shape and use discrimination to isolate one shape 
from the group.





Click the image to read the post.
The next activity emphasizes logical reasoning to determine in what order to place a collection of shapes. For "Pattern Block Line-Ups," students try to place pattern blocks in the correct position using a set of clues to find the correct placements. This activity is a great way to integrate vocabulary and build critical thinking skills at the same time! You can find a free set of frames with the original post!



Click the image to read the post.

Pattern blocks. Fractions. No way! Yes, way! My favorite way to use pattern blocks in the classroom is to teach fractions. The red trapezoid, blue rhombus, and green triangle all fit proportionally inside of the yellow hexagon, which makes them great tools to use to model fractions.


The last activity is called, "What's the Whole?" and supports students' understanding of the 'whole' when compared to the 'part'. Specifically, this activity requires students to view a whole as more than the area inside of a single space, such as the hexagon; it helps them understand a whole to be a unit, as defined by the size of each part. You can grab a free copy of this activity sheet with the original post!

Freebie Alert! Be sure to check out the series link here for freebies and printables to accompany this series! 



Sound Off! How do you use pattern blocks in the classroom?