## ImagineIT - Phase Three

**I. Identify Desired Results**

The focus of my ImagineIT project is perseverance in problem solving in mathematics. Students should be able to, once presented with a problem, put forth a reasonable effort to solve it. Note that my goal is not for students to solve perfectly every problem they are given; rather, they should (a) determine the best way to approach the problem, including deciding where to begin, given their background and knowledge, (b) following through on their initial strategy, whether it is fruitful or not, and (c) not being afraid to either pursue an avenue that seems risky or to have the confidence in their abilities to start over when things go wrong.

These three results are key to the discipline of math (and life in general) because being able to apply what one has learned in a real-world way is the only reason why we learn math. We don’t just learn it to solve for x, as many students believe; rather, we learn it so we can build on what we have learned in a general sense to apply it to specific problems and issues. These problems can be solved by someone who understands this, the essence of why we learn math and why we must learn to fully follow through when solving problems, and can apply different approaches and persevere in their work.

This application of math skills to other disciplines is also key, because a job in which students are given an equation and asked to solve for x is non-existent. Students need to be able to apply math to all other subjects, even if it is a subject that they find beneath them or boring (which is not as uncommon as one might think). In addition, they need to be able to communicate their ideas and work together to persevere in their work, but also recognize when someone else’s work is well-thought-out or poorly conceived.

After unpacking all of the processes of persevering, students should not give up on problems. They should pick them apart and dissect them. They should be able to play around with ideas, strategies and solutions without being afraid of being wrong. They should also become more confident in their problem-solving abilities and become better thinkers.

**II. Determine Acceptable Evidence**

There are many different types of performances of understanding I will be using as I go forward with my project.

The first performance of understanding will be one low-stakes pre-test and various post-tests that will follow. The pre-test will occur on the first day of school, where I will give the students an assignment that is impossible to finish or solve. I know that sounds mean, but I need to determine where a student’s baseline perseverance level is - in other words, how long will they stick with a problem before giving up? I’d like to see where they began and where they ended up, but I’d also like them to write a paragraph or two explaining their rationale(s) for their process(es) and have them justify their answer. Once they learn my dirty trick and have some chances to practice persevering, I will assign them various weekly post-test problems, which will be the same format, but will be solvable. I will grade a few of them, but I will also like their peers to grade them. Once they are familiar with the facets of persevering, they should be able to determine if a peer possesses this ability or not. This will allow me to see the growth they’ve shown in working through difficult problems. Plus, since these problems will all deal with real-world and cross-curricular problems, it will be a more accurate picture of the skill level of each student.

Another performance of understanding I would like to use is interviews various times of the year to help determine specifically which aspects of perseverance are the problem areas for students. Once students are able to articulate to me where their deficiencies in problem solving are, we can start a dialogue and I can help them overcome them. In addition, if enough students share the same issues, I could either create a lesson addressing them, or I could form a study group or after school group for them to meet with me in groups and further elaborate on their problem solving and perseverance issues.

A final performance of understanding that I would like to use is simply classroom observation. When students are working on a problem, I would wander through the classroom looking and listening to see how each student is using the principles of perseverance to work on a problem, to see how diligent they are. This would give me a quick snapshot of where they are, and could clue me in on things they maybe didn’t want to say in interviews or in post-test questions.

**III. Plan Learning Experience and Instruction**

One of the main ways I plan on addressing this issue is to implement a flipped classroom. Now, I have never used this method before, and I hope to jump in and do it nearly every day. When I think about my students, in terms of their backgrounds and struggles, I think this would work well. Typically, in my Calculus classroom, there are between 24 and 31 students, depending on scheduling. Nearly every student has had me as a teacher for the previous class (Precalculus), which makes transition easier, and allows me to better assist them in their learning, since I can recall where their strengths and weaknesses lie. I know from taking Precalculus with me that they are, on average, “good” at math. They can manipulate formulas and use them to solve problems, though some students have issues with factoring, dealing with fractions and expanding polynomials. Each student is required to have a TI-83/84/Inspire calculator, whether that means that they buy one on their own or they borrow one from me. This is reflective of the technological support I have at my school: within reason, most technologies that I would like to use are either at school or can be purchased.

How would a flipped classroom look in my room? I think it would work in a fairly simple way: I would post a video of a general concept that students would watch for homework. They would also be required to post a comment about the video, be it a discovery, a question, a comment on someone else’s comment, etc. The following day, we would be begin to unpack the concept together, through the use of word problems that would not only serve to help students better understand the concept, but would also allow them to practice persevering in those problems.

Now, practicing perseverance is a difficult idea. That is why I have broken it down to three general ideas, as stated earlier: (a) determining the best way to approach the problem, including deciding where to begin, (b) following through on their initial strategy, whether it is fruitful or not, and (c) not being afraid to either pursue an avenue that seems risky or to have the confidence in their abilities to start over when things go wrong. Here are the ways I would implement learning on each of these ideas:

- The best way to start a problem: For students to practice doing this, at first, we would work on a series of exercises that would present a problem. Students would write down their initial thoughts of what the problem was asking, what they feel the answer could look like (units, magnitude, etc.) and what they feel their first few steps should be. This could be done in groups or individually, but would always involve a share out and final group discussion which would include solving the problem.
- Following through on initial strategy: This would follow from the previous idea, where students would write down their initial ideas on a strategy and follow it to its logical conclusion, regardless of whether it would be correct. They would work with another student and one of the two would come up with the strategy, and they would work together to attempt to solve the problem using only that strategy, even if the other student didn’t believe it was the best strategy. They would then write a reflection on their strategy, commenting on whether it worked or not, and what they might have changed. This would also be shared with the group.
- Not being afraid to change course: This also builds off of the other two ideas. One of the bigger issues I’ve seen is when students are so paralyzed with fear of making a wrong decision that they don’t even know how to start solving the problem. I would hope that once we’ve gone through the previous two aspects of perseverance, this would no longer be an issue. If not, conferences and interview would help me further address those needs if they come up. The second part of fear in perseverance is for a student to have the self-reflection to change what they are doing if they feel it is leading them to an incorrect solution. This is tricky because students don’t want to redo work or do additional work. So in order for students to get into this mindset, they need to start looking at others’ work. In groups, students would look at examples of problems either fully or partially solved, or successful or unsuccessful case studies and determine if the strategy used is sound, or if it needs to be reworked, and if it does need to be reworked, they would need to find the correct steps. They would then share out with the group. This would lead to individual work on such problems, which would lead to students working in pairs, each starting a problem, exchanging it with their partner and deciding whether their partner needs to rework anything. Finally, this would culminate in the complete solving of a problem, including a reflection on the entire process, discussing where they started, where they ended up, how they feel about their answer and any pitfalls they encountered while solving.

For the flipped classroom, I would be using either ScreenCast-O-Matic or Camtasia to produce the videos, which I would upload to my YouTube account for students to view. I would then leverage the YouTube comments to assess the number and quality of comments and questions provided by students after they watch the video.

In terms of presentation, I would be using the iZiggi wireless document camera (in conjunction with a ceiling mounted-projector) to allow students to share their ideas with the class. This would allow them to project their ideas on to the screen without needing to get up from their seats. More than one student could comment on the work without all of them needing to get up and interrupt class.

Finally, technology that I would use daily in class would be my TI-84 on-screen calculator. I would also be using Wolfram’s Mathematica at various times throughout the year to assist in learning, especially in visualization of more difficult concepts.