Introduction

AMTE’s (2017) Standard C.2 emphasizes that pre-service teachers (PSTs) need opportunities to learn to select, adapt, and design rich mathematical tasks that promote reasoning, problem solving, and multiple solution strategies. To address this standard, I incorporated a project into my middle-grades mathematics methods course related to designing three-act tasks. The students in this course are in their first semester of a year-long teaching residency and will be certified to teach mathematics to grades 4-8. In the course, we discuss the National Council of Teachers of Mathematics’ (NCTM) Mathematics Teaching Practices (NCTM, 2014). Specifically, we discuss how the practice of implementing tasks that promote reasoning and problem-solving can open doors for the utilization of other practices.

Project Overview

For this project, PSTs worked in groups to create a Three-Act Math Task for in-service teachers. Three-Act Math Tasks utilize storytelling to engage students in mathematics (Champagne & Suh, 2021; Meyer, 2011). Act 1 begins with a short, intriguing video or picture to introduce the “conflict” of the task by prompting students to ask questions. In Act 2, students determine what information they need to answer their questions and solve the task. Then, the “conflict” is resolved in Act 3 by revealing the answer and connecting student responses. The PSTs had experience with Three-Act Tasks before this project was assigned. They participated in tasks during class, watched video examples, and analyzed pre-made tasks. The instructions for the class project stated:

For this project, you will work with your group to design and create a Three-Act Task for the specified standards. A teacher will facilitate the task with their class. Afterwards, you will analyze the student work from the task. This project provides experience in creating rich mathematics tasks, and it helps teachers by offering a ready-to-use task for their lessons. 

Four in-service teachers for grades 4-8 participated in this project. At the beginning of the semester, the teachers sent the standards they would be covering in a two-week period towards the end of the semester. The PSTs worked on their tasks throughout the semester. Once completed, I emailed the task materials to the in-service teachers to use in their classes. After implementation, the in-service teachers sent back pictures of students’ work for the PSTs to analyze in class. 

Creating the Task

The PSTs were grouped based on their preferences for the grade level and standards given by the in-service teachers. To brainstorm ideas for their task, the groups followed three steps: 

1. Brainstorm ideas
2. Use AI to generate ideas
3. Select final idea

Step 1: Brainstorm Ideas

For the first step, I provided each group with a document containing their standards and a designated brainstorming space. The PSTs analyzed the standard for which concepts it included. Then, they made a list of ideas for their task related to that concept. Figure 1 shows an example of a group’s brainstorming document based on their given standards. 

Figure 1. Student Brainstorming Example

Step 2: Use AI to Generate Ideas

After brainstorming possible ideas for a task, the groups utilized AI to generate more ideas. The PSTs were instructed to begin with the following prompt: “What are some Three-Act Task ideas for these [insert grade level] math standards: [copy and paste standards here]”. Then, they copied the AI response into the document. After the initial prompt, the groups were given the option to ask further questions to refine the AI ideas. The document also included space for the PSTs to review the AI ideas to practice their AI literacy skills. They answered questions regarding the alignment of the task idea to the standards and the three-act structure. Figure 2 shows an example of the AI responses for one group. The response to the initial prompt included ideas for a backyard patio puzzle, sports field makeover, and bakery counter display. The group refined the idea by asking AI for a scenario like the bakery counter display that involved organizing makeup. 

Figure 2. Generating Ideas with AI Document Example

Step 3: Select Final Idea

Lastly, the groups discussed which idea they wanted to select for their task. They recorded their final idea and reflection of the process in a table in the document. One group decided to use the original idea they brainstormed instead of an AI-generated idea. Two groups decided to utilize an original idea of theirs but noted that AI helped spark ideas for enhancing it. An example of this is shown in Figure 2. One group stated that they used an idea generated by AI but adjusted them to fit the standards better. Overall, the PSTs found AI helpful to generate ideas, but they ultimately made adjustments to make the ideas their own. 

Figure 3. Selecting Final Idea Example

A group’s final Three-Act Task is shown below. They ultimately changed their idea of filling a drawer to filling containers with half gumballs. Figure 4 shows a picture of the Act 1 video. The video portrayed half gumballs being placed quickly into the containers, leading students to the main question: How many half gumballs can fit in the bottom of the containers?

Figure 4. Act 1 Example

The information given to students in Act 2 is shown in Figure 5. The dimensions of the containers are depicted as a composite figure, and the diameter of the half gumball is given. 

Figure 5. Act 2 Example

The Act 3 video revealed the solution to the task. A picture is shown in Figure 6.  The Act 3 video was the completed version of the Act 1 video with the total number of half gumballs that fit in the containers. The group also included a Sequel question to extend the task. Their sequel question asked: How can you figure out how much area of the container is left uncovered by the half gumballs? 

Figure 6. Act 3 Example

Analyzing Student Work

After the teachers implemented the Three-Act Tasks, they emailed their feedback and scanned copies of student work for the PSTs to analyze. Each group answered questions based on the student work. The questions aligned with Jacobs and colleagues’ (2010) framework for professional noticing of children’s mathematical thinking, which involves attending to students’ strategies, interpreting their understanding, and deciding how to respond based on their understanding. The first set of questions prompted PSTs to describe what they observed in students’ work, including their strategies, representations, and solutions. The second set focused on interpreting student thinking and understanding based on their responses. The third set asked the PSTs to consider instructional responses by developing assessing and advancing questions (NCTM, 2014) for at least two students and by planning how to select and sequence student work in accordance with Smith and Stein’s (2018) Five Practices for orchestrating productive mathematics discussions. The final set of questions invited PSTs to reflect on this process and to consider revisions they would make to the task based on student responses and teacher feedback. 

Considerations for Teacher Educators

Each semester, I refine this project based on challenges that arise. First, I found that providing a template for the PSTs to use made it easier to send the materials to the teachers. During the first semester of this project, I did not specify how to format the task, so they were all formatted differently. Adding a template provided guidelines for what to include and consolidated everything into one file to send to the teachers, rather than multiple files. While it offered guidelines, the template was basic, allowing PSTs to customize it as they saw fit. Second, a challenge was the PSTs’ understanding of the structure of a Three-Act Task. Some groups wanted to tell students the key question, rather than using the Act 1 video to lead them to the question. In the future, I will include a requirement for the PSTs to practice their task with someone else first. Third, some of the standards were more difficult for the PSTs to create tasks for than others. For example, the 7th grade TEKS 9.C “determine the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles” (Texas Education Agency, 2012b) was easier for the PSTs to create ideas for than the 5th grade TEKS 4F “simplify numerical expressions that do not involve exponents, including up to two levels of grouping” (Texas Education Agency, 2012a). It is crucial to consider the standards and help the PSTs understand how to apply them.

Conclusion

Overall, the PSTs and in-service teachers enjoyed this collaboration. The PSTs appreciated their task being used in a classroom and seeing how real students solved it. The in-service teachers commented that their students were engaged in the tasks and had fun solving them. This collaboration supported the development of the PSTs by giving an authentic opportunity to design, implement, and reflect on rich tasks. Additionally, incorporating AI as a tool for idea generation encouraged PSTs to engage critically with technology by refining AI-suggested ideas to ensure alignment with content and student thinking.

Link to materials.

References

Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. https://amte.net/standards

Champagne, Z., & Suh, J. M. (2021). 3-act math tasks: Authentic engagement with mathematical ideas.

Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202. JSTOR.

Meyer, D. (2011, May 11). The three acts of a mathematical story. dy/dan (blog). https://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/

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

Smith, M. S., & Stein, M. K. (2018). 5 practices for orchestrating productive mathematics discussions (2nd ed.). National Council of Teachers of Mathematics.

Texas Education Agency. (2012a). Texas Essential Knowledge and Skills for mathematics, §111.26. Grade 5, adopted 2012. https://tea.texas.gov/sites/default/files/ch111b.pdf

Texas Education Agency. (2012b). Texas Essential Knowledge and Skills for mathematics, §111.28. Grade 7, adopted 2012. https://tea.texas.gov/