Wednesday, September 18, 2013

Conceptual Understanding and Procedural Fluency

I know I am supposed to share my first few years of PBL. I have it all in my head. I just got distracted with school. Things are starting to be "normal" for me so I am going to try and write weekly. Key word is try :). I want to share something that happened in class this week. It is not PBL but a key aspect of helping all kids learn math deeply which is the heart of PBL.

In my Algebra II classes, we started the systems of equations unit on Monday. I always start a unit with a small pre-assessment. It is never more than four multiple-choice or written response questions and it is basically skills students need to be successful during the unit. See this link if you want to know the pre-assessment questions. Well, the pre-assessment let me know that most kids didn't have the necessary background and I would have to do some reteaching as we learn how to solve a system of equations. 

We started with the simplest way to solve--graphing. We reviewed how to convert an equation into slope-intercept form so that they could graph the equations. The example I used was 2x + 4y = 36 and 10y - 5x = 0. I knew from the pre-test that some students knew how to convert from standard form to slope-intercept form so I just asked for volunteers on how to convert the two equations. This gave me the opportunity to see how students approached converting the equation. For the first equation, some students wanted to divide by four for each term. Although this is a perfectly acceptable first step, I asked for another way just to make sure I didn't throw off students by putting fractions in early. 

In each class, various students helped me follow the typical process of subtracting/adding the term with the x variable, then dividing by the number with the y variable at the end. With each step, I reminded students of the desired result (isolating the variable of y) and why we are doing each step. We then completed the last step of placing the equations in the graphing calculator to determine the solution. I send them off to work time like normal and this is where the class becomes interesting. 

I completely expected to help students with using the graphing calculator. We have only had them a week. However, I was shocked by the number of students who were still lost with converting the equation. At the end of the day, I discovered the root of the problem was students lacked the fundamental understanding of equations. They had practiced such a specific algorithm they didn't know what to do if it came in a different form. For instance, the first problem in the book was 2y - 3x = 7 and 5x = 4y - 12. A few didn't know where to begin while others were doing the typical process. As I helped student after student, everyone wanted to subtract 5x as for he second equation.   

Just like other teachers, I thought about what to do the next day. The plan calls for me to move on to substitution. They have a test on Thursday that includes them solving by graphing and substitution. The students don't have a conceptual or even a solid procedural fluency of equations. 

What would you do? How do you help students who have misinformation while staying on track with learning new information?

Thursday, July 18, 2013

My PBL Journey: Year 1

Yesterday, I was tweeting with Harry (@hblyleven) and Al (@alfredbie) about challenges with implementing PBL with math educators. This conversation led to me agreeing to blog about my own PBL journey. I think I will not only blog about my past but I will try to share my journey this year with implementing PBL and the new Common Core State Standards.

My journey began in 2005 when I switched from corporate to education. I was hired at a local charter school. During our orientation, the school CEO said they are a project-based learning school. She showed a video of a project where a student was studying marine life. I was intrigued while watching the video. I thought, "schools really have changed. I like this new way of teaching." After the video ended, the CEO said a few more housekeeping measures and that was it. My only training at that point was a video. Since I was wanting to implement, I asked veteran teachers once school started. I soon discovered that many of the teachers did not teach through projects but in the traditional way. Those who did complete projects where more about cutting and pasting then deeper learning like I saw in the video.

Despite this hit of reality, I was not swayed from my interest in how to do project-based learning. I put my journalism background to use and started researching. It was slightly difficult to find material in 2005 compared to today. However, I quickly found out that PBL had a few names. The most common name was project-based learning however there was also problem-based learning, inquiry-based learning, authentic learning, etc. After reading a few articles, I discovered they were basically the same thing just with different names and styles of delivery. So I moved on to trying to find books that could help me implement the framework in my classroom. The first two books I found were How to Use Problem-Based Learning in the Classroom and a book where students build a city of the future (sorry, I don't remember the name). I remember how excited I was to have these books. The Problem-Based Learning book was great but lacked a step by step process that I was needing. The second book (authored by a teacher who completed it with his students) was a complete step by step. I was so happy to have directions given this was my first year teaching. I was still scared to implement so I tried it with my honors Geometry only.

Well, I wish I can say it was the most beautiful experience ever. Although it was not terrible, it did not have the outcome like the video. The students excitement about the project made me want to continue the journey. Here are my lessons learned from that first year:

  • A step-by-step process is more harmful than helpful. I realized that this is a learning experience and following directions is not a learning experience. If I did have steps, I needed to look at those steps as guidance only. What worked for that teacher and his students may not be exactly how I need to do it for my style and my students.
  • Work completed at home has varied results. I announced to the students that they were going to build a city of the future. I gave them a sheet that included the parameters and told them it would be due in a couple of weeks. For the next two weeks, class was just like normal with them working at home on the project. I didn't even teach anything that was related to the project. According to the book, this was a way to "check" to see if they got the information previously taught. When two weeks were up, I had an array of "cities". All of them were elaborate but definitely showed signs of only them working on it. I wish I could say this moment made me realize that I should have them work in class. It didn't. I just thought this is the result of projects and some will be good and some not so good. However, it did make me realize the next point.
  • Rubrics are a way of helping students know quality and expectations. I realized that I only had parameters not details of expectations. This made grading really hard. Out of guilt, I just gave everyone an "A" for effort. I realized that I probably should use a rubric in the future. 
  • Limited resources does not mean limited teaching/PBL. My first year of teaching was really interesting. In addition to finding resources to learn PBL, I was trying to figure out how to teach with limited resources. I had one class that did not have textbooks, a 100 copy a month limit, class sets of copies had to be submitted for approval (3 day minimum wait), a whiteboard that didn't wipe off, a teacher desktop computer and an overhead projector (yes, overhead not projector). Although I had almost nothing, I realized that I could teach with little or no resources. I found creative ways to help students learn. This experience made me want to learn more about inquiry and assessment. I wanted to learn if there is a way to incorporate these strategies into PBL. 
My first year was a huge learning experience. It gave me a lot of goals to make for my second year. So, please share your first year lessons. 

Friday, January 4, 2013

Calculus: Derivative Project Ideas

AP Calculus AB is typically the highest course for many high school students. Coming up with a possible Calculus project has been challenging. Not because the course is hard. It has been challenging because my desire to keep the project authentic and connected to the major concepts of the subject. Every time I create a project, I want it to be real-world work. I also want to have students think deeply about the "power standards" of the subject.

Identifying the power standards for Calculus was simple. I agree with Steve Strogatz's post in The NY Times,  "Calculus is the mathematics of change. The subject is gargantuan - and so are its textbooks. But within that bulk you'll find two ideas shining through. Those two ideas are the 'derivative' and the 'integral'."  Although knowing that a project on derivates or integrals was easy to identify, the real world application of it has been much harder. Every time I run into a person whose profession is mathematics based such as an engineer, I would ask them how do they use calculus in their work. They would often laugh and say they don't. 

Despite these roadblocks, I am determined to find an authentic use of calculus in the workplace. After searching, I think I have some ideas for a derivative project. The main criteria I am using in thinking of ideas is that there are multiple solutions. Please let me know what you think of them and please add other possible projects in the comments below. 
  • Drug Flow: Healthcare is huge topic today. It is a constant discussion at dinner tables. One aspect of healthcare is drugs. Many people do not understand the affects of these on the body. The Center for Policy Analysis wants to educate the public on how various drugs affects the body so that people will be more empowered about their health care. They want to explain their website to include this information. It must include how to calculate the effects in layperson's terms (idea is adapted from extended applications in Calculs with Applications by Lial, Greenwell and Ritchey).
  • Classic Box: Kellogg is in the process of repackaging their cereal boxes to respond to market demands. Increase in gas over the last decade has caused the company to increase the cost of cereal. However, recent changes in tax code is lowering Americans ability to pay increases in cereal. The company wants to redesign the boxes to achieve maximum volume but cost less to produce. The company's director of product development is requesting a proposal to meet the company's needs. It must include calculations of the cost to produce and the volumes of the new products (idea is adapted from the classic box problem in most calculus textbooks. There is also an activity by TI-Instruments with their TI-Nspire calculator).
  • Start-up Company: Terri Miller has left her position as a manger of a local company to start her own business. She makes lady handbags. The demand has really taken off and she is in full production. She needs to get more capital to expand but that requires creating a proposal that details various financial aspects of the business. Investors are wanting a lot of information including her marginal cost, revenue and profit for the organization. She is small and don't have the staffing to help with the creation of this document. She is seeking assistance to help grow her business (idea is adapted from Calculs with Applications by Lial, Greenwell and Ritchey).

Please Join Me

I have taken a semester off from school to work on creating projects for all grade levels. I ask that you join me in helping shape this work. Your help is really simple, you can do any of the following:

  • Share likes
  • Give a suggestion of how to improve
  • Try it out
Any and all comments are welcome. This is the biggest endeavor of my life. I hope you will join me in the adventure.

Thursday, September 17, 2009

Smith Project-Day 1

The class began with a review of the format of my class, rules and expectations. The day I started the project was the second day of getting all new students. Originally, I was teaching Geometry and Algebra II. Now, I am only teaching Algebra II. I had to catch up the new students but also review some information regarding systems with the old group. The class ended with the introduction of their first project.

To understand how I introduced the project, you must understand the set up of my classroom. Some of my readers know about my classroom. However, for those of you who are not familiar with my classroom, it is a fictional corporation called Logic Inc. Students act as consultants and managers that solve real world problems of individuals or businesses. The mission of the company is to help people see math in all areas. In addition to helping clients, the students promote the company through a "company website" where they provide tutorials and information on math concepts.

Since my class is a company, the students get all their information from the company's intranet site ( It is here the class gets information on the company's first "fictional" client. Students understand the clients are fictional right now so that they can gain experience solving problems. Students read the clients situation and their driving question. Students then created a concept map in mywebspiriation of what they know and didn't know about the clients problem.

To see the actual results of day one, check out the information below:
  • This is the powerpoint presented in class-

Wednesday, September 16, 2009

Project in Full Swing

I am discovering quickly that once you start a project it is hard to sometimes stop and write about the flow of it. It is also difficult when all of your class dynamics change before you introduce the first project. One week ago today, the students were introduced to the Smith Project. Let me explain the situation regarding the Smith Project and the make-believe family in it.

Kate Smith is a single parent who lives in the area. Kate lost her job at the GM plant. Since losing her job, Kate has had three other jobs. She is now in a stable job at a local manufacturing business. However, Kate is not making as much money. While working at GM, Kate used to take her and her three daughters to the state fair. Since losing her job, they have not been to the state fair. She really wants to take the girls this year. The girls are teenagers and Kate wants to do as much with them as possible before they graduate. She heard we may be able to help her go to the fair on her limited budget. She has $150 that she can spend. The driving question(focus of the project): How can Smith make the most of her $150 budget for the state fair?

This is a very small almost seemingly easy answer at first. However, once students investigate the cost associated with the fair, it has some complexities to it. The goal of the project is to teach systems of equations and inequalities while also helping to get the students prepared to work in teams, think critically and talk mathematically.

I have several things I did to complete the project. The project ends Friday. I will discuss what occurred each day over the next few days. I can't wait to talk to you. Stay tuned for details and links to student work.

Saturday, September 5, 2009

Presenting a PBL

This is the second portion to my video posting. Below is a video of me practicing presenting a PBL to a group of educators. The educators are pretending to be like my high school students.

Video on Getting Money for Education

Finally, I was able to find the time to make the videos from my filming at the IMSA institute. It is two different videos. Below is my information on getting funding for your professional development and classroom.

Sunday, August 30, 2009

Contemplating Projects

There is absolutely to many things on my plate. I am a member of the alumni association of Northwest Classen High School. As a member, I agreed to produce the newsletter. I am five months behind producing the newsletter. I was supposed to have it out in April. I got completely caught up with understanding student council responsibilities and then a summer full of activities. As I sit trying to pull the articles together, my alarm sounds and I realize it is time to BLOG!

The funny part about this blog is along with completing the newsletter, I am trying to formulate all the projects that my students will complete this semester. I hope to have at least some ideas jotted down that match the standards I have to cover. I can't wait to get started but I have to get this newsletter done before our meeting tomorrow at 4:30 p.m.

Please make sure you check back into my blog to give me your advice on the projects I come up with tonight.

Wednesday, August 26, 2009

Great Meeting With Team

Yesterday, I discovered why it is great to work in teams. As I said in the earlier post, I was confused on the order to complete items. All of my colleagues are going to follow the book, chapter by chapter. This will help me with following an order even though I will still operate on concepts rather than specific chapters. I also found out that matrices are not as necessary of a skill for calculus preparation so I will be able to concentrate only on the other three methods to solving a system-graphing, substitution and elimination. I would rather teach a few ways that are strong then have students confused on a concept.

I also had a great lead for a project-circuits. I asked the team what is the emphasis for complex numbers. This lead to a possible project about electrical circuits. I have a colleague who is going to share with me what she does and I may be discussing soon how I will make it into a project. I am EXCITED!