Fostering Mathematical Creativity Through Problem Posing and Three-Act Tasks

Task 1: Encouraging Problem Posing

To introduce the “Whoppers Task” (Fletcher, 2016b), I played a short video (https://gfletchy.com/the-whopper-jar/) of a person dumping packages of whoppers into a jar. While watching the clip, I saw my students’ eyes widen with excitement. Immediately Nick said, “hmm, I think there are 19!” I realized that he had thought of a problem in his head and was answering it. I asked Nick to write the problem he had thought about to find the answer of 19. I hoped to use Nick’s problem to model the process of problem posing. While Nick was writing, I also asked the other students to write down what they wondered about the candies. Anticipating I would see some variety of ideas, all three questions asked the same thing—“How many whoppers were in the jar?” (see Figure 1).

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Figure 1.

Nick writes the estimate of the amount of bags used to fill the jar and the problem he posed. All three students wrote the same question.

While I was excited that my students were able to problem pose, I also realized that they likely needed additional support in thinking about the different kinds of problems they might be able to pose. I decided to play the video again and stopped at another spot to give the students a different view of the candies. I hoped that having the students look at the candy from a different angle would encourage them to pose different kinds of problems. When I paused the video, I asked the students a generative question (English, 1997b) “What else do you know about the items in the video?” The purpose of the generative question was to help my students recognize the important ideas in the video that they could use to aid in posing a problem (English, 1997b). I was excited when Logan asked, “How do we know they all had the same number of whoppers in each package?” This question included one of the other items in the video and extended beyond the procedural question of how many. However, Logan was the only student to ask a question. I wondered to myself, “What else can I do to encourage more problem posing?”

During Act 2 of the task, I decided to model open-ended questions while the students worked for 10 min. The type of modeling I used comes from work in language arts. I chose to model the act of problem posing because “powerful modeling influences can simultaneously change observers’ behavior, thought patterns, emotional reactions and evaluations” (Bandura, 1986, p. 48). I hoped to change my students’ behavior from waiting to solve a given problem to creating original problems or reformulating a problem.

I modeled problems such as, “Do packages of the same size have the same number of whoppers?” and “Does the size of the jar make a difference in the amount of whoppers?” Then I had the students rewatch the video and asked them to pose a problem like the ones I modeled. I also reminded them to think about the items in the video, referencing the generative question I had asked earlier. After watching the video again Nick said, “Now I see there must be more than 100 in the jar.” Logan agreeing to this idea stated, “The packages each have five. So, there must be more than 50.” At first, I was disappointed that the students did not seem to grasp the idea of problem posing. However, Logan paused and asked, “Was one of the packages ripped open?” Building on this idea Cameron asked, “I wonder how many packages there were?” How can we know for sure?” We were excited, the students were beginning to engage in problem posing!

The following day we began Act 3, to give students a chance to share their solutions to the “Whoppers task” as well as share any problems they had posed. I was hopeful that during this act of the task my students would continue to build on the problem posed from the previous two days. However, once I showed the reveal video (https://gfletchy.com/the-whopper-jar/), the students just wanted to know if their answer was correct. I was left wondering; how could I support students in posing deeper mathematical problems across all three acts?

The strategies we used to initiate problem posing, asking a generative question and modeling, seemed to help the students engage in this practice during Act 2 of the task. As seen in Table 1, the problems Logan and Cameron posed during Act 2 were original and showed inventive thinking (see Table 1), two of the characteristics of mathematically creative thinking (Krutetskii, 1976; Torrance & Myers, 1970; Treffinger et al., 1994). The problems showed originality in thinking because Logan and Cameron’s were different from each other and considered various aspects of the task. Logan’s problem was inventive because he considered whether the bag was ripped open, something not shown in the initial task.

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Table 1.

Problems Students Posed During Task 1

Student name	Problem posed by students
Logan	“How many whoppers were in the jar?” (Act 1) “How do we know they all had the same number of whoppers in each package?” (Act 2) “Was one of the packages ripped open?” (Act 2)
Nick	“How many whoppers were in the jar?” (Act 1)
Cameron	“How many whoppers were in the jar?” (Act 1) “I wonder how many packages there were? How can we know for sure?” (Act 2)

The problems posed during Act 2 were an improvement from the one posed during Act 1, “How many whoppers were in the jar?” In the next task, I hoped to encourage students to pose more problems and exhibit different aspects of mathematically creative thinking. I planned to continue modeling and asking generative questions since these strategies seemed to help the students while also incorporating new strategies to encourage problem posing throughout all three acts of the task.

Task 2: Concrete Manipulatives to Promote Problem Posing

After reflecting upon my experience implementing the “Whoppers Task,” I decided to also incorporate the use of concrete manipulatives to encourage problem posing during the second act of the task. Manipulatives can help students make meaning of the task (Puchner et al., 2008). I chose to use the manipulatives in the second act of the task because I wanted to gather the students’ initial understanding in the first act and then build on these ideas.

The task we worked on was the “Snowman Task” (Yummy Math, 2019). This task began with an ill-structured problem, one students likely face in everyday life, a picture of a grocery store display of soda boxes in the shape of a snowman. I provided students with manipulatives to recreate the structure anticipating this would help them identify important information from the picture. I then planned to encourage my students to consider how they might modify or extend these ideas (English, 1997b) into problems they could pose.

To begin Act 1 of the task, I showed my second grade students a picture of a snowman (https://www.yummymath.com/wp-content/uploads/soda-snowman.jpg) built using over 100 Coca-Cola boxes. As opposed to the last task, my students did not immediately pose any problems while looking at the picture. So, I asked the same question from the Whoppers Task, “What else do you know about the items in the picture?” However, this still did not elicit any problem posing.

Because of the lack of initial problem posing, I worried that this task may be too complex and the students would not understand the picture well enough to pose problems. As a result, I reframed my generative question and asked: “What are some different mathematics problems do you think Ms. Lewis could ask?” I thought that if I framed the question in this way it would help the students position themselves as the teacher and think about how teachers pose problems. I made this change because for students to pose problems, they need to think about the structure of a problem (English, 1997a).

The generative question I asked helped students begin sharing ideas. Students responded with problems such as, “Are the boxes empty?” and “How many cans of soda are there altogether?” While these initial problems varied in their connection to the mathematics content, I realized that the question, “How much space does the design take up?” could lead to advanced mathematical concepts such as two-digit multiplication or volume. Best of all—each student was able to engage in problem posing.

In Act 2, the students were given a choice of tools such as snap cubes or Cuisenaire rods to help students continue problem posing. Logan used connecting cubes to recreate the real boxes in the snowman tower because he thought seeing arrays in different colors would help him model the exact types of soda used in the problem and assist him in keeping track of decomposing the large structure of boxes. I was excited that Logan was able to explain how the manipulatives would help him make sense of the problem and led to increased problem posing. While building with cubes, I noticed Logan kept posing problems such as, “Do we know if we count those boxes in the corner?”

Instead of answering the problems Logan posed, I chose to respond with questions intended to generate even more problems. I asked him, “How do you know the figure is three dimensional?” and “What question would you be answering if you counted the boxes in the corner?” The purpose of these questions was to push Logan to further investigate these problems himself. From the problems Logan posed, it seemed that manipulatives helped Logan make sense of the picture. I realized he may not have posed such problems with only a flat 2-D picture; the picture may not have helped him think about the shape of each box. I felt encouraging Logan to use manipulatives helped enhance the problems he posed.

After working with Logan, I looked over at Nick’s paper and saw that he represented the array he created using the equation: (5 × 3) – 2. He then began to pose problems. Since Nick wrote an equation and built an array before he started problem posing, I felt that the manipulatives were helping him make meaning of the picture to problem pose. The problem Nick posed to the group was, “I wonder how many cans are in the boxes in the corner part of the design?” I thought this was a unique problem because it demonstrated that Nick was thinking deeply about all areas of the photo. After posing this problem, Nick continued to ask, “Am I correct?” Even though this question will typically elicit a yes or no response, I saw the potential to capitalize on this question as if he was posing a problem. I asked him to explain what he noticed about the picture and how he would solve his question. In response to my questions, he went immediately to his manipulatives and began breaking the snowman into sets of arrays to help explain his thinking.

The use of generative questions and manipulatives during the task helped students pose problems during Acts 1 and 2. Overall, the problems students posed were an improvement from “The Whoppers task” (see Table 2). The problems posed during Acts 1 and 2 were all original ideas. Specifically, each student posed a different problem during Act 1. Nick and Logan both posed more problems. Logan demonstrated flexibility in thinking by comparing the boxes to a cube. Cameron was able to pose problems during Act 1 however; he did not pose problems during Act 2. During Act 2, Cameron was working to understand how the tower of soda boxes was constructed. He was so engrossed in this process that when I checked in with him, Cameron explained that he was unable to consider any other problems.

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Table 2.

Comparison of Problems Posed During Task 1 and Task 2

Student	Task 1	Task 2
Logan	“How many whoppers were in the jar?” (Act 1) “How do we know they all had the same number of whoppers in each package?” (Act 2) “Was one of the packages ripped open?” (Act 2)	“Are the boxes empty?” (Act 1) “How many cans of soda are there altogether?” (Act 1) “Do we know if we count those boxes in the corner?” (Act 2) “Do we know if these boxes have three sides like the cubes?” (Act 2)
Nick	“How many whoppers were in the jar?” (Act 1)	“How much space does the design take up?” (Act 1) “I wonder how many cans are in the boxes in the corner part of the design?” (Act 2) “Am I correct?” (Act 2)
Cameron	“How many whoppers were in the jar?” (Act 1) “I wonder how many packages there were?” (Act 2) “How can we know for sure?” (Act 2)	“Is the snowman real?” (Act 1) “Are there layers?” (Act 1)

In Act 3 of the task, the students shared their solutions and I revealed the solution. Then, I asked the students a generative question, “How might you change some of these ideas to make the problem different” (English, 1997b, p. 4) to help encourage problem posing. I made this change because students in the “Whoppers Task” did not have any questions in Act 3. However, the students did not respond to my question with any new problems. Seeing again that the students struggled to pose problems during Act 3, I decided that I would change how I encouraged problem posing during the third act of the next task.

Task 3: Adding “The Sequel” to Promote Problem Posing

I selected the “Gummy Worm Task” (Pearce, 2018) to complete with my students. During this task I planned to continue using generative questions and manipulatives to encourage problem posing. I also decided to have the students create a sequel to the “Gummy Worm Task” to help elicit problem posing during the third act. A sequel follows the third act in a Three-Act Task. Students create sequels based on the solution or by using new information presented in the reveal about the ill-structured problem from Act 1 (Meyer, 2013). “The initial task [e.g., Gummy Worm Task] just serves to set an imaginative hook for the sequels, which is much more demanding and interesting” (Meyer, 2013). I hoped that the increased demand and interest in the sequel would encourage my students to problem pose.

To support my students in exploring the mathematics content within the “Gummy Worm Task,” I brought in a variety of manipulatives: gummy worms, rulers, connecting cubes, and Cuisenaire rods. I noticed that giving students access to manipulatives during the “Snowman Task” (Yummy Math, 2019) helped my students to pose problems connected to the task, and I wanted to provide my students with the same opportunity during this task. I also hoped that by providing students with a wide variety of materials they would be able to make sense of the task and use this information to help them problem pose throughout the task.

I initiated Act 1 by showing a video of worms in a jar. As I had done during the “Snowman Task,” I provided the students with manipulatives. I decided to start the Three-Act Task using manipulatives because it had helped the students to identify a few different problems. Each student was using a different manipulative to help them explore. Logan was using a ruler, Nick was using Cuisenaire rods, and Cameron chose to use the gummy worms. I noticed that using these manipulatives to explore the length encouraged all three students to problem pose during Act 1.

The students posed the following problems: “Are the worms all the same sizes?”; “Is the jar the same size as the one in The Whoppers task?”; “Are the worms the same length as the ones in the picture?”; “How long are the worms if we put them all together?” As I wrote down the problems the students posed for us all to read and see, I realized the problems focused mainly on the shape, size, and quantity of gummy worms. I was excited that the students made comparisons to the “Whoppers” Task (Fletcher, 2016b) jar. I loved the problems the students were posing and was encouraged to see the problems they might pose during Act 2.

In Act 2, I was curious to see if the manipulatives would continue to help the students to problem pose. I noticed Logan held up the worm to the computer screen and asked, “I wonder how long all 25 worms would be?” He started laying the 6-cm Cuisenaire rods on the carpet to model the problem. Then he multiplied 25 worms by 6. Wow, just wow, I thought Logan posed and solved his own problem using fourth grade multiplication! Even though he just engaged in a complex mathematical task, I could see he still had some unanswered mathematics problems.

To encourage Logan to share these problems with the rest of the group, I used revoicing (Chapin et al., 2009) to repeat his question to the group. In response, Cameron asked “Would they [25 gummy worms] all be the same length?” I responded to Cameron by using a generative question: “How could you use the items that we have available to explore this idea?” In response, Cameron and Logan worked together to lay the worms out on the table and took a 5-cm Cuisenaire rod to measure each worm one by one. The students realized that the gummy worms were the same size. I noticed the students were naturally having a mathematical discourse with the tools and the use of generative questions was continuing to lead them toward problem posing.

The problems posed during Act 2 extended the students’ conceptual understanding and helped to foster mathematical creativity. The students were eager to count the worms and seemed genuinely interested in having an original context to measure and create problems about (Lomax et al., 2017). They were original in their thinking to use the rods to compare the lengths of the gummy worms. The problem posing seemed much more natural like sap flowing from a maple tree and was student-led during Act 2 (Pearce, 2018). I felt these changes were due to the choice of tools available to help the students think about the mathematics task. As I concluded this part of the task, I thought to myself, “This is what teachers need for gifted students.” But I wasn’t done yet. I had to push the students to truly create their own task and embrace the act of problem posing in Act 3.

In Act 3, I usually reveal the solution to the problem we agreed upon in Act 1 and had students share their solution strategies. Sometimes, I will also review estimates with the students from Act 1 during Act 3 to help them see how their thinking has changed over the course of the task. Since the focus of this act is on sharing the solution strategy, I realized that I was not creating an environment conducive for problem posing. The development of a sequel during Act 3 does provide a strategy for encouraging problem posing.

Sequels are used during Act 3 of a Three-Act Task to help teachers revisit or reintroduce student questions which were not addressed in Act 2, provide students with an opportunity to create a problem based on the solution, or encourage students to reformulate a problem that was posed (Lomax et al., 2017; Meyer, 2013). The process of reformulating answered questions helps to encourage mathematical creativity (Mann, 2006). I decided if my students created their own sequels using the questions they posed in Act 2, then they would have an opportunity to engage in mathematical creative thinking. Sequels within Three-Act Math tasks work just like books—you get to the ending of the book and you are left hanging—so you need to pick up the next book and continue reading. For Three-Act Tasks, during a sequel you continue to engage in mathematical thinking.

I told my students to write down any problems that were still spinning in their heads after the solutions were shared and the reveal was shown. Then, I told them they would create their own sequel for the group to solve based on the problems they had written or one of the problems they identified in Act 2. I hoped that the sequel would activate their imaginations, encourage original thought, and engage them in problem posing.

Following Meyer’s (2013) advice on his blog for using a sequel I posed these questions to the students: “What did you see in the worm video that we didn’t talk about?”; “Was there some math or information missing that you might want to add?” I felt that these questions would help encourage students to problem pose. I wanted to push my students to think creatively about a problem they could pose and use the sequel to help assess their ability to problem pose (Silver & Cai, 2005).

Logan said he wondered if he could get a few more worms. He then carefully stretched out the middle part of each worm, being careful not to break them. He noticed that as he stretched the worms they were getting very loose in the center. Then he said, “I got it!” He picked up the pencil and started talking aloud and writing, “If you stretched all the worms out as long as you could, will it take less worms to fill up the jar?” He then took his new stretched out worm and the Cuisenaire rods. He said, “The original worm is 5 centimeters using the blocks. If I stretch the worm out it is 7.5 centimeters. So, I predict that it will take less worms to fill the jar.” The problem Logan posed for his sequel was, “How many less worms will it take to fill up a pint size jar with gummy worms if they are stretched out versus still in the package?”

I enjoyed listening to Logan think aloud. He was really problem posing and able to capture all his thoughts into one problem. Logan was so intrigued by the Cuisenaire rods, he said he might need to make a video to teach the class about how to measure with them. So, I encouraged him to continue creating and turned my attention to Cameron and Nick.

Cameron and Nick wanted to build a super worm. They smashed their worms together to create the Super worm. Cameron then started writing on his index card, “If a Superworm is made of four worms, is it really four times as big?” Nick wrote, “Will it take one-fourth of superworms the worms to fill up the jar?” To solve the problems they posed, Nick and Cameron took out a ruler to measure the length of the worm. Nick observed that, “The worm really isn’t much longer, just fatter.” Then Nick asked, “I don’t know how to measure how fat he is.” I suggested to Nick that this would be a great sequel.

While the students worked on their sequels, I began jotting notes to help me reflect on their work. Logan demonstrated creativity with and the originality of his idea. Once I wrote this, I thought, its creativity in a nutshell. Gifted students can be creative through problem posing and the sequel is a great way to extend the idea of problem posing further. My students could be imaginative, original, and independent in creating their own sequels to Three-Act Tasks (Polya, 1945; Torrance & Myers, 1970). I realized I was able to make “important insights into children’s understanding of mathematical concepts and processes, as well as their perceptions of, and attitudes towards, problem solving and mathematics in general” (Brown & Walter, 1993; English, 1997a, p. 1; Meyer, 2013).

The addition of the sequel assisted the students with continuing to problem pose throughout the entire task. I noticed that the problems students posed were an improvement from Task 2 (see Table 3). Logan was able to pose more problems during Task 3. The ability to pose several problems showed his fluency in thinking. He was original in his ideas and demonstrated flexibility by considering the idea of stretching out of the worms. Cameron was able to pose problems during different parts of the task, something he struggled with during Task 2. He also showed originality and flexibility in his thinking. Although Nick posed the same number of problems during Task 3, the problems he posed demonstrated flexibility and originality by considering the use of a superworm. In Task 2 the problems Nick posed were original but did not consider a perspective that was not presented in the picture. Overall, through problem posing, the three students were able to engage in mathematically creative thinking during the Three-Act Task.

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Table 3.

Comparison of Problems Posed During Task 2 and 3

Student	Task 2	Task 3	Assessment of questions
Logan	“Are the boxes empty?” (Act 1) “How many cans of soda are there altogether?” (Act 1) “Do we know if we count those boxes in the corner?” (Act 2) “Do we know if these boxes have three sides like the cubes?” (Act 2)	“I wonder if all the worms are the same sizes?” (Act 1) “Is the jar the same size as the one in “The Whoppers task?” “(Act 1) “Are the worms the same length as the ones in the picture?” (Act 1) “I wonder if I can use the rods to determine the length of one worm?” (Act 2) “I wonder how long all 25 worms would be?” (Act 2) “If you stretched all the worms out as long as you could, will it take less worms to fill up the jar?” (Sequel) “How many less worms will it take to fill up a pint size jar with gummy worms if they are stretched out versus still in the package?” (Sequel)	Areas of Improvement from Task 2 to 3: 1. Increased the number of questions 2. Compared objects from one task to another task 3. Posed problems during the Sequel (Act 3) 4. Posed problems that went beyond solving the initial task Area of Continued Growth: Increase the complexity of the problems posed in Act 1
Nick	“How much space does the design take up?” (Act 1) “I wonder how many cans are in the boxes in the corner part of the design?” (Act 2) “Am I correct?” (Act 2)	“I wonder if they are all the same color?”(Act 1) “How long are the worms if we put them all together?”(Act 1) “Will it take one-fourth of the worms to fill up the jar?” (Sequel) “If a Superworm is made of four worms, is it really four times as big?” (Sequel)	Areas of Improvement from Task 2 to 3: 1. Posed problems that went beyond solving the initial task 2. Posed problems during the Sequel (Act 3) 3. Drawing connections between mathematical concepts 4. Considers mathematical ideas beyond those shown in the picture Area of Continued Growth: Increase the number of problems posed specifically in Act 2
Cameron	“Is the snowman real?” (Act 1) “Are there layers?” (Act 1)	“Would they [25 gummy worms] all be the same length?” (Act 2) “If a Superworm is made of four worms, is it really four times as big?” (Sequel)	Area of Improvement from Task 2 to 3: Drawing connections between mathematical concepts Areas of Continued Growth: 1. Increase the number of problems posed across all three acts 2. Pose problems that go beyond solving the initial task Consider various perspectives when viewing the materials presented in Act 1