The Wonder of Structure

Yesterday I attempted a self guided lesson where students work in groups to learn different concepts. I created a worksheet that had very strategic questions to lead them to specific conclusions but was still open enough that they had to apply critical thinking skills. In my first class, I instructed them to complete the worksheet in 20 minutes and then I would proceed into a lecture to connect the different concepts. After 20 minutes, out of 5 groups, not one was even close to finishing. I am entirely confident that most of the work could have been completed within that time, especially when I was going around helping each group. The problem was that I would get to a group and ask questions which I had prepared before hand but once I would leave, that group refused to do any work. The activity ended up taking 35 minutes up to the end of class. I would spend about 1 minute helping each group before I would move on. Since they only work done was when I was at a particular group, each group would work 1 minute every 5 minutes. Thus, out of a 35 minute activity each group probably only worked for 7 minutes.

However, without giving up I tried to tweak the lesson. They had no motivation to complete the worksheet so for my second class, I gave an incentive. There were 4 questions, each worth 2 bonus points. If I had to help a a group more than once for a particular question they could only receive 1 point for that question. If they are at a standstill in the group, I wanted them to try to figure it out themselves. Not one group asked me more than 1 question on a problem. After 20 minutes, I checked each group's work. They were all completed and the lowest score was a 5 and I could actually finish the lesson.

After talking with my cooperating teacher, he thinks the structure and incentives made a difference but also the quality of the students. I don't know what I think but I wished I would have had the incentives in the first class.

A Bit of Coaching

The following is an exchange of emails between me and a math coach I met from a professional development session.

Hello Joe,
I teach my first lesson today and was wondering what you thought of it regarding the discussion we had.

Thanks

Tom

Tom
I think you have a lot here for one class. There's a lot of information and a lot of thought that goes with it. I think the students will get lost and frustrated. I think the folding activity is good.
Activity I had used were:

Having the class show exponential growth.
Shake one students hand..... Instruct them to shake Two more students hands and so on. Record the # of students who have shaken hands after each event.... That way the students see exponential growth in the class.......

You could do the same with passing along a secret to two students, they pass to two and so on......

If the class creates an activity then you should have an opportunity to dialogue with the students about what is happening. Let the students write what they think is happening and then guess what will happen after 5 events, 10 events etc....

It is when you can get the students voices going and dialogue about the work happening that the students will really understand the concept. If you get caught up in lecturing and giving out factoids you will lose them, they'll sleep or try to do what you ask like good students but they won't get it....

Authentic classes thrive on democratic dialogue. That is what you want to create. Always allow learning to be open to debate. Students need vigorous discussion in the class for education to be active and challenging. Teacher talk depresses students. Teacher monologue limits their speech and development.

Teacher talk will also depress you. Teachers thrive on responses to their labors, and silent classes weaken our moral. Without lively student participation, we risk declining into burnout.

Be bold and challenging. The students have been conditioned into cogs so getting them to participate is difficult.... don't give up though...

1) I sent an e mail to two students. Each student then sent an e mail to two other students

How many e mails were sent after this happened 2xs, 4xs, 10xs,,,, 20xs.....
Try to let the students discover the equations.... I'm sure that what you are working towards with the folded paper..

To make the activity even more authentic tell the students a secret... or pass along a hand shake. These get the class really involved in the process and students can see the process of exponential growth.
Joe

Joe
I would love to do these activities but they don't want it. Wherever I try to do something clever or engaging they look at me like I'm stupid. They wouldn't even fold the paper let alone actually shake hands. Students scoff at me when I give them critical thinking problems. They have confused looks when I relate it to real world situations. A typical educator would just tell me that I must be doing it wrong but I have yet to see it work in a typical inner city high school.

This could be because they aren't used to these types of activities and due to the fact that I am a stranger in the classroom. Also, I am still not convinced that there is enough time in the curriculum. Everything I did today was outlined in the guidelines book. I would love to be able to expand it and go into much greater depth but it seems to be completely unrealistic.

Thanks for all your comments. With your permission I would like to post your email in my online student teaching blog. And I will keep trying different things.

Tom

Tom
These kids are only responding to you in the only language they know. We have conditioned them to do what they are doing. You are now seeing what American educational systems are doing. As a student teacher you are experiencing the normative behavior of the class, teacher talk, student listen. The kids want you to talk while they sit and do little or nothing.

Yes our institutions discourage experimenting and may even punish teachers who rock the boat. Teacher talk can be the path of least resistance. This is not unique to the urban school. It happens everywhere. Most teachers talk and students listen. Dialogue is discouraged.

You are receiving on the job training, burdened with class size, poor recourses, restrictions of required test, texts, and syllabi. You lack any power at all. Only the students have less power than you. It's all run from the top down.

Creating authentic is dangerous, but it is also the greatest thing that has happened to my teaching. It has rejuvenated me and my calling. Today I believe from the bottom of my soul that the only true education happens when authentic dialogue happens in the classroom.

So where to go from here. I would suggest that you tell the students how you feel about the activity that you did. Let them know you are disappointed with the outcomes and that request that they tell you honestly what they felt. You might want them to write it first and share it with one another to get it going. A lot of times this gets them started talking. You must be prepared for honest answers though that may hurt your ego. If you do allow them to speak honestly, without insulting or embarrassing them. They will rise. Your class will rise.

If you can create dialogue in your class you will be able to teach them anything.... but the dialogue must come first.
Joe

Evaluation #1

The following are comments made by my supervisor on an evaluation form during my first lesson. In italics are what I will try to do to improve.

Lesson plan extremely well written. Good objectives, shows knowledge of standards. Informal assessment being used.

Using calculators appropriate for lesson and allowed students to see concept.

You used a lot of critical thinking concepts in your lesson but didn't allow them to be fully developed.

This criticism is hard to talk about. How can I allow them to develop the concepts more? It's is to say it but helping me do it is another story. I guess I could allow more time for my intro activity of folding paper. The concept was that exponential functions grow or decay at increasing or decreasing rates. When folding a piece of paper in half over and over again, the thickness increases slowly at first but since the thickness doubles every time, at some point it explodes. If I put them in groups and let them collect data themselves they may have been able to see the increasing thickness even after you could no longer fold a piece of paper.

Students well behaved, no management problem. It appeared that only a portion of students were actually involved in a lesson.

I agree that I can affect a larger group of students if I create a classroom of active learning but there is no way that I can get to everyone. Education needs to change drastically if we are going to reach all students.

Idea of handing out personal "notes" is a good idea.

This is a response to my idea of making up notes (my way) and then having blanks that they fill in as I lecture. This idea is from Mr. Shilingo, my high school physics teacher.

Make sure you talk to the entire class, you were taking responses mostly from right side of room.

When I am teaching, I am constantly trying to think about where my lesson is going, how am I going to ask some question or how much time do I have. Eventually, I need to sort of go on auto pilot where my focus is on the entire class rather than my lesson. Also, I am going to write names on small slips of paper and ask questions to specific students picked randomly.

You need to give students time to catch up with you. You presented some of the material too quickly. Give them time to respond.

I have been working on this by pausing after I do an example or by asking if I need to slow down.

Try moving around the room to check on their work. See if they are able to follow along.

I love moving around the room while I teach but the classroom I am in is definitely not setup to allow me to do this easily. The room is packed with desks with two teacher desks in the front of the room, severely limiting my mobility. My room would be setup much different. I would like to have my desk in the back or off to the side (so it's not the focus) and I would like to have the desks pointing inward either like a v of a horseshoe. This would give me the mobility I need while creating a different environment where the teacher can move in and out of the spotlight.

You did ask for questions but no one raised any you moved on. It might be good to ask some and call on random students to see if they comprehend.

I will probably use slips of paper with the students' names written on them and randomly pick on students to answer questions. When I try this now, I don't really get any responses.

Classwork question was a great critical thinking question. May have been good to have pairs work together to solve.

Pair sharing is almost impossible to do in these classes. They have no clue how to move into pairs or why it is even necessary. With rigid structure it takes too much time and without the structure they either don't move or they end up discussing with many more people.

A excellent (B)good C satisfactory D fair F poor

Today...

...I rolled my eyes at a student.

I have a tendency to become over bearing and even condescending when talking about a passionate subject but I have always noticed that when I am speaking to an audience from a certain level of authority, I can adopt a comfortable demeanor. It is how I am able to successfully teach people about things I am knowledgeable about. I think the problem arises when I speak to peers and I am under the assumption that they should be as knowledgeable about a particular subject or that their arguments should be as good as mine or better. If this is not the case, sometimes I can be pompous.

However, I have the ability to take on a nonthreatening persona when I stand in front of a class, I think, because I am under a different assumption that these people have a desire to enrich themselves even if they generally don't show it. I know this isn't wrong or even too ideal; Students, under certain circumstances, can accomplish some pretty extraordinary feats. It's this assumption that I am able to have undaunted patience to try different things to spark their interest. If I am successful at this then my job is easy and incredibly rewarding.

The problem, that I noticed today, is that it might not be an unrealistic assumption that causes me to be condescending. I think it has something to do with openness or effort. Openness is essential for people to embark on mutual discovery of different perspectives. Passion can be involved but only when there is a sense of reciprocity where I want to hear what you have to say, not so I can disagree but so I can compare your perspective to mine and others in order to investigate logical implications. This is in the hope of discovering new ways of thinking about a subject that might make more sense than it did before. I like to think of it in terms of the literal meaning of the word philosophy, 'the love or pursuit of truth.' All too often people close up because they feel their ideas were attacked just because of some disagreement or strong counter arguments.

Effort, on the other hand, is important for me to recognize in a person because it represents their internal motivation or will to improve themselves and the world around them. If I encounter a situation where I perceive that another person just doesn't seem to care, try to see other perspectives, gives up when faced with positive confrontation or gives up when faced with confusion, I first feel a sense of deep frustration because I see it as an attack on my passion for meaningful conversation and then judge them because I feel they are wrong to give up. Sometimes, my reaction to feeling hurt is to take a negative demeanor where I patronize them.

Today...I had the opportunity to create a lesson that I feel very strongly about. They lesson was on one problem but it was a problem that was about finding a well reasoned strategy and then justifying it. I introduced this new kind of math problem to them by saying that too often math is left in a vacuum where relevancy is rarely discussed or poorly discussed. Problems arise everyday and some have solutions and some seem to not have any solution or many solutions. I started the lesson with a simple question: "What is a problem?"

During my third lesson two female students, that often express their negativity during class, made a few comments that struck at the heart of my passion and enthusiasm. When they moved into a group activity one expressed that she "didn't know what the hell I was talking about" and that she didn't understand why I was talking about "social sciences." I tried to explain to her that I was trying to broaden the class to encompass many different ideas and to bring in other subjects. She did not hide her disapproval. While presenting a possible solution to the problem I asked a question which was more of a rhetorical question to highlight an important part of the problem. They both made comments about how confused they were and how much they thought what I was doing was stupid. I rolled my eyes. Being frustrated, I asked if they would have rather I started the new chapter and went to a lecture style class. Expecting that they would choose this type of lesson I was understandably surprised when they voiced their opinion to start the new chapter. Not able to continue with the solution I addressed the whole class positively about why I believed that what I was doing was important. I needed to take my frustration and vent it in a way where students were not attacked. It worked (at least for me) and I continued the lesson.

Why do I believe they were so unhappy with the lesson? I think they don't want to be challenged. They have gone their whole lives never being challenged in school. High school is merely a necessary step to get past in order to go to college or start working. They don't want to exert that type of effort. They want me to give them answers so they can regurgitate it on the test. Too bad they can't even do that very well. I want to get them in a room and shout, "What do you want from me?" They were not open to a different kind of lesson and they didn't want to expert much mental effort to figure out a real math problem. This got to me. But after reflecting I have come to this conclusion: We failed these students. I believe that if I show them something different, if I show them what math is really about, then they will eventually come around and start being engaged in their learning experience.

Observer Effect

The observer effect shouldn’t be confused with the Heisenberg Uncertainty Principle. It is the idea that the act of observation of a given event fundamentally changes that event. The observer, in a way, becomes part of the event, therefore making it impossible to observe the desired event. I thought about this recently while preparing a lesson for a review of the PSSA (state assessment). This test is called a benchmark which is used by the federal government (No Child Left Behind) to determine if the school is making adequate yearly progress (AYP). Students essentially need to constantly improve their scores on state assessment exams in order for their school to continue operating without any federal interventions or regulations.

The key word I used was benchmark. I would define a benchmark as a test to assess some predetermined level of quality or it may be used to compare the results of one test to others who have taken the same test. There are benchmark programs for computers where you run it and get an arbitrary score which is compared to other computer users who run the same benchmark. The significance of the test is seen through the comparison. You can see how different hardware configurations can result in higher or lower scores which may affect which pieces of hardware consumers want to purchase. A benchmark within education is a state test which is compared to previous tests by the same school to determine if that school achieved a level of predetermined progress. The idea is that each school needs to be constantly improving their test scores.

The problem with benchmarks is that people tend to construct unrealistic scenarios where benchmarks produce a higher score without providing an accurate measure of performance. Nvidia, a popular graphics processor manufacturer, came under fire a few years ago because it programmed specific instructions into the processor code which allowed certain Nvidia graphics cards to score higher on a popular benchmarking program. This is called benchmark inflation. Unfortunately, this now occurs in education. Nvidia inflated its scores to sell more graphics cards which make them more money; Schools try to inflate scores because they don’t want any federal regulation which, in many cases, results in them losing money. If tests scores of a school do not meet AYP (annual yearly progress) then that school goes under review by the federal government. This review process may result in any number of interventions.

Schools try to inflate scores by both aligning their curriculum according to the test (i.e. changing the year in which algebra II is taken – Alg I, Geometry, Alg II, PreCalc) and by specifically allocating lessons used to prepare students for the tests. Every Monday a class period is used to review problems that may appear on the PSSA. Every Tuesday a 15 minute practice PSSA quiz is given to the students. This is over 1/5 of the curriculum spent on reviewing for a benchmark. How is this not a disservice to the students? And how is this supposed to accurately measure students’ ability?

The goal of the federal government with NCLB is to assess how effective schools are and then hold educators accountable for any failures that may occur. Their mandates for state assessment exams are an attempt to observe the effectiveness of the public education system. This observation is fundamentally changing the system. High accountability is forcing schools to change strategies and build new curricula around these state assessments. We are no longer observing or assessing the original education system, hence, the Observer Effect. Could this be their goal? Maybe but how then are we to determine how effective the assessment exams are? Who determines the quality of the assessments? How are we to assess the assessments which assess the assessments…?

Prompt #3

What is your greatest teaching challenge?
What are effective teacher behaviors?
What does effective teaching mean to you?

Well from a student's point of view I think effective teaching would very unique to each student. Given that, effective teaching is meeting all student's needs. I'm not sure to what extent I agree with this statement but teaching is one of the more altruistic positions where teachers need to be dedicated to the students. My teaching aspirations define what I think effective teaching is. I want to be able to give students an understanding of mathematics that reaches much further than content. If a teacher is able to use content to teach and model ethics, leadership and engagement while being understanding and entertaining then students are given an experience rather than a boring place to go during the day.

Effective teaching behaviors are those in which teachers maintain a structured class while also creating an environment where students have the reasonable opportunity to be engaged. I say reasonable because a traditional classroom does not provide an atmosphere where students can be inspired. Some students are able to succeed but their motivation has nothing to do with the class. These students are naturally curious and engaged and need little instruction. Other students need to be shown that engagement in the classroom can occur. It is almost like they need to be taught curiosity. This is what I mean by provide a reasonable opportunity. You could also think about it as providing a reasonable choice. Students choose to be engaged in the classroom. If you were a student, would you enjoy your lessons or would they be boring?

This will be my greatest challenge, trying to provide these engaging lessons. These students haven't experienced this type of instruction. They don't understand what it means to think of mathematics outside of the textbook. To them, its a subject not an idea. To them, its necessary instead of interesting. To them, it can't be fun. With everything teachers need to do, how can we be expected to constantly develop creative lessons. These lessons require time to do research and effort to adapt personal style. In a typical day, where the curriculum is rigid and content focused and with only one prep period, how is this to be done?

First Full Day

Last Thursday, I taught 3 algebra II lessons which my cooperating teacher planned. It was basically a review of a section on complex numbers and the quadratic equation. It was nothing special. Half the lesson was showing examples on the board and the other half was a worksheet. The only criticism I received was that I talk to the board too much, meaning, I should be talking to the class while looking for confusion or disruptions. It’s an interesting criticism considering that while I teach I consciously think about looking at the class. It is interesting when you think about the things you do subconsciously that undermine the things you try to do consciously. This makes me wonder about my voice inflection. I try not to have a pattern or a habit with my voice. I always try to be loud and sometimes a little crazy. But I may have an inflection pattern or, dare I say, a repetition word.
On Monday, I developed a lesson and taught 4 algebra II classes. It was a review for the practice PSSA quiz on Tuesday. I tried to set it up in three parts. I used the first 15 minutes to model how to complete the problems. Then I gave them a practice quiz which they were to complete by themselves in 15 minutes. For the remainder of the period, I had them pair up and check their answers while completing on quiz with both names of the partners. There were many problems, the main being that a large number of students have absolutely no motivation to do quality work. Even though I model how to solve each problem on the board, they refuse to write very many steps down. Many asked questions when not understanding but an alarming number of them refuse to even ask for help.
Another issue is a procedural one. I said they could work in pairs but I knew that it wouldn’t work like I wanted it to. I say it was a procedural problem because the cooperating teacher did not establish it at the beginning. If I were to have assigned partners there would have been a problem. If I were to have said, “turn to your left…” there would have been a problem. They ended up working with many people and most people turned in their own quiz. I could have either concentrated on making sure people were working correctly or help with questions. I don’t know how to make this any better for upcoming lessons.