**MODULE 6: REFLECTION**

Reflection on Implementation of 3 Acts Math Lesson Using Technology

For more info on the 3 Acts Math Curriculum Design scroll down to module 4 and watch the Video OR visit Dan Meyer’s Website. If you are interested on why mathematical storytelling is so important to me, read my blog post: This Girl is on Fire!

Check out my reflection below to read how I used to metacognitive thought processes to inform my instruction and ensure I effectively implemented technology as a tool, not a distraction.

**MODULE 5: LESSON PLAN WITH TECHNOLOGY**

3 Acts of a Mathematical Task: Surface Area and Volume of a Pop Can

Below is my lesson plan with technology implementation. The last part of the lesson plan document is a detailed account of each mathematical act with supports and questions identified for possible misconceptions. Below the lesson plan document, the specifics regarding content, pedagogy, and technology are addressed.

**Content:** The lesson content focuses on volume of cylinders and is part of 3D Geometry unit for an 8^{th} grade math course. The students will use iPads and the 3-Acts math lesson I created to identify which pop can has more pop.

**Connection to Common Core: **8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Mathematical Process Standards (reasoning, proof, problem solving, communication, connections)

Students typically struggle with abstract problems, so by presenting the problem using the 3-Acts Math lesson, I am able to provide a concrete scenario that allows the learner to make sense of the problem through mathematical abstraction. The overarching goal is that students are able to reflect on and consider how volume and surface area affect consumer and producer choices. In order to reach that goal, students will need to decide what information matters in a situation or problem scenario and what information they need in order to solve the problem. They need to know how to find the volume of each cylindrical can, how to write proportions for unit conversions, and how to interpret their findings. Students will use their mathematical computations to develop a real world answer and go on to consider the surface area of each can as well.

**Pedagogy:** The lesson is set up to be a problem solving and problem finding task. It was carefully designed to ensure learners are able to identify the problem, accurately communicate their thinking, apply reasoning skills, make connections to prior knowledge, and understand complexities in various forms. Each “act” in the lesson provides supports that are grounded in constructivist principles and that fit within a metacognitive framework. By implementing this pedagogical design, I am able to focus on giving each learner the opportunity to explore and create his or her own understanding through differentiated instruction at a level that makes the content meaningful.

**Content & Pedagogy:** The ideas behind three-act math lessons were the building blocks, or driving forces if you will, for my lesson plan on surface area and volume. Each act maps onto a mathematical task. Act one captures the students’ attention and naturally allows for them to wonder about the problem at hand. They are not handed over the problem; they identify what the problem is. They become innovative thinkers and problem finders. Act two requires learners to be aware of what they know and how they may use it. It also scaffolds learning in such a way that they know when more information is needed and how to find it. The best part about three-act math is that each task is designed so learners are able to naturally construct new knowledge and build on prior knowledge. Moreover, by implementing tasks that stimulate collaboration, students interact with others and question new ideas. This process helps them move from the known to unknown by building on ideas they are already familiar with. Discovering the connections and regularities within knowledge they already have is empowering. It is within this process that learners identify similarities amongst contexts and learn to apply their thinking to new situations. Reflective periods allow students to learn from their mistakes and experiences by accommodating what they thought to be true with what they have found to be true.

**Technology:** I am implementing iPads and digital media resources to present this lesson. The main reason I am using the iPad is so that students can learn surface area and volume through a series of photos and videos. By designing experiences digitally I can show rather than tell, scaffold mathematical abstraction, crowd source patterns, and prove the math works! The technology allows for the content to be presented in a completely different way- allowing the learners are able to move beyond computational skills and achieve mathematical application skills. Also, the data collected on the iPad allows the teacher to see trends in misconceptions and responses.

**Technology & Pedagogy:** The iPad and the 3 Acts Math curriculum design mesh together to establish a foundation for mathematical tasks that is both prescriptive enough to be *useful* and flexible enough to be *usable. *The digital tools allow the problem to be presented in a completely different way, a way that allows the learner to build their understanding as they work through the problem. The mathematical tasks embedded in the design force students to consider what they already know and how they may use that information. By interacting with the math task and each other, the students will assimilate their experiences by relating them to old experiences and existing frameworks, or they will accommodate for the new information by reframing their mental representations to fit the new experience. In essence, each “act” is designed to build on what learners already know and establish new or more extensive relationships within their mental frameworks. By involving learners in the formulation of the problem solving process, the learners are able to see that problem solving is an ongoing process and is not about finding the answer or being right.

**Technology & Content:** Using multimedia- pictures and videos- to present the lesson does much more than a textbook problem can. Even the greatest textbook problems have to provide or pose a question at some point. By using a video, I can provide a concrete scenario that forces students to inquire about something or establish a problem. Textbooks, at best, could provide a problem about volume and pop cans where the problem has already been abstracted, which robs students of the skills needed to formulate and solve problems. The 3-Acts Math lesson not only involves students in the learning process, it also stimulates worthwhile discussions. The learners develop skills that help them identify what information matters in a problem scenario and what additional information they need. By learning this way, the learners are able to consider much more complex concepts. They are able to address real world problems and present real world solutions. They are able to see how their math computation skills can be applied to real world processes and consider mathematical concepts from different perspectives, such as a product consumer or producer. Also, the learners are able to record their answers and reflect on their new understanding right on their iPad as they work.

**Assessment:** I am assessing my student’s ability to work through a real world math application problem using what they learned previously regarding surface area and volume of cylinders. I will be looking for them to identify which components matter and then using those ideas to work through the task. Since I am more interested in my learner’s problem solving process, not whether or not they can memorize formulas, I am giving them the conversions and formulas they need once they identify that they need them. I will be assessing their ability to work through a problem using math computation skills and vocabulary they learned previously. I will also be checking to see if they are learning from their mistakes. They may not have reached the correct solution, but I am more interested in how they recover once they watch Act 3. I am also assessing their ability to make sense of their answer in terms of products and packaging. Most of the assessment will be formative. Through walk around assessment, effective questioning, individual conversations, class discussion, students posted work/answers, and individual reflections I will assess what the students know and what gaps in understanding they may still have.

**MODULE 4: DIGITAL STORY**

A Problem Worth Solving

A visual look at two different methods for teaching volume of cylinders and how each method is associated with what the learner learns. I used the constructivist approach with metacognitive problem solving as a foundational framework throughout.

Rationale:

My digital story focuses on how changing the way something is taught changes the way something is learned. Specifically, textbook problems typically teach math computation rather than problem solving and reasoning. The textbook problem gives the key pieces of information, which turn out to be components that fit nicely into a formula found, say, two pages back. So, the question really isn’t asking you to solve a problem at all. In fact, there isn’t even a problem to be solved; all of the information is given, it just needs to be plugged into a formula and computed.

The trouble with this approach is it doesn’t require learners to think critically. Really, can you think back to a time where you solved a meaningful, worthwhile problem where you knew all of the necessary information beforehand? Probably not, because in most meaningful problems we encounter, identifying what the actual problem is is half of the battle.

One of the biggest pitfalls of current education practices is that educators give out a problem, develop a commonsensical path leading to the solution, and congratulate learners for following the approach, yet they fail to involve the learners in the process of formulating the problem. Our job as educators is to provide diverse learning experiences and investigations that engage learners’ minds and imaginations. Further, instead of giving out problems, we need to involve learners in the formulation of the process. Albert Einstein describes this process to a tee*: “*The mere formulation of a problem is far more often essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances” (*Good Reads,* 2011). So, in my digital story I take the fairly compelling elements from a traditional textbook problem and rebuild it in a way that supports reasoning and metacognitive problem solving.

The textbook sample question in the digital story gives the learners the components they need to plug into a formula and figure out how much wax each candle uses; however, Dan Meyer (Father of 3 Acts Math and math blogger) suggests that the sub-steps be removed from the problem, which leaves it up to the learner to formulate them. The ability to decide what matters in a situation or problem has become such an **underdeveloped concept in education**, but it is a necessary skill for 21^{st} century learners and innovative problem solvers. So, the sample problem is left at which candle is bigger and the learners are left to decide whether components such as height, size, etc matter. Then, instead of asking that question, you provide a picture or video clip that makes the learners wonder about the size and shape of the candle. At the end of the clip, typically without even asking what students are thinking, a student says, “So which can is bigger?” This is how you know you’ve baited the hook.

Through mathematical story telling the learners will discover the problem on their own, build on previous knowledge, and construct new knowledge rather than having the information handed to them. This method provides the learners with a framework that they can build and expand on to create an individual approach rather than basing their understanding on someone else’s.The main goal is to help the learner see problem solving is an ongoing process and is not about finding the answer or being right.

Moreover, this learning approach gives students the opportunity to interact with others and question new ideas. This process helps them move from the known to unknown by building on ideas they are already familiar with. Discovering the connections and regularities within knowledge they already have is empowering. It is within this process that learners identify similarities amongst contexts and learn to apply their thinking to new situations. Perhaps they will make mistakes within this process, but by accommodating what they thought to be true with what they have found to be true they are learning from their mistakes and experiences. By using the constructivist approach with metacognitive problem solving as a foundational framework, teachers can focus on giving each learner the opportunity to explore and create their own understanding through differentiated instruction at a level that makes the content meaningful.

Changing the model of pedagogy to meet the demands of the 21^{st} century is crucial. Learners need the opportunity to learn how to learn, discover and formulate problems, build on other peoples’ insights, and adapt their abilities to various situations. As Polya (1957) says, “**A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.**” Today’s learners need to rediscover the values of education and how they address real life needs. They need to be reflective in their thinking and discover their talents and abilities on their own.

**MODULE 2: AUDIO INTERVIEW**

For my audio interview project, I asked two college graduates and one high school student what geometry is.

First I want to clarify the organization and tone of this podcast. I used a “news” theme because it captures how the interviews felt. While all three candidates are comfortable with me, they tensed up the minute I said “math” and “record.” The first candidate told me they were sweating the entire interview, the second candidate kept mouthing, “I don’t know what to say” when I would ask her to elaborate, and the third candidate said he felt like he was in the hot seat. All three of the candidates asked me if they gave the right answer when we finished recording, which I think says something about how people feel about math in general: it’s either right or wrong. While the conversation was very casual, I think there is something nerve racking about anything being permanent, even if it is just a recording about geometry. Once I turned the recording device off, I was able to get the participants to elaborate on things like the mini golf problem in the textbook. The whole thing reminded me of fox news, so that’s the theme I went with.