Today was one of those days. Everything went well, and even the things that didn't, well I don't even remember them.

My Geometry classes are diving into proofs, only they don't really seem to know it. Last year proofs went horribly. The kids hated it and fought me every step of the process. "TWO Columns? WHY!" "FLOW CHARTS! Those are stupid. " "PARAGRAPHS! You want me to write, like actual sentences, in math class? No way." This year, we slowly crept into proofs with the points lines, planes project. Then on Monday, we reviewed the best slides for each one. Students who had slides selected were beaming with pride. Students who weren't selected were still trying to argue why their interpretation was correct or why their photo was better. (I got to them! They cared! In math class of all places. And they cared about whether or not their justification and representation of a postulate about undefined terms was best. Whodathunkit).

Yesterday, we did some more Always, Sometimes, Never statements and had the "Best of PLP Project" and the list of postulates as a guide. Explaining that what we are doing today is just like the project, except instead of taking pictures and making diagrams as proof, we have a set of statements that mathematicians have all agreed are true. It went fantastically. The students who were still shaky after the project were rock solid with the postulates to ground them in what was always true.

Today, we kept with the postulates and added in two column proofs and talked about algebraic statements. Not one student even made a peep to complain about the two columns. No one asked why we needed to write a reason for every step. They just bought into the idea of really being precise and explaining everything. I really like coming from a place where they are confident (solving for x) and adding in the new things. I made sure to have a problem set that grew in difficulty. It started with x+2=9, then added in coefficients (6x+3=15), then multiple x terms on one side, then x's on both side, then some challenge problems of solving systems of equations. They haven't formally seen solving systems of equations, but the students who reached those problems were able to figure out what to do in order to solve by using their list of properties of equality. I know they are motivated and engaged learners, but they figured out systems on their own! That was awesome to see. And students who did not have a strong algebra foundation were able to pick up on the pattern quickly. That brought a big smile to my face.

In Stats, a student who I used to have a great connection with has been battling with me all November. I'm part of the problem, it's probably easy for her to tell that I'm frustrated and angry because I am. With all the time we've put in together (including several Saturdays), she seemed to be throwing it in my face, with a lot of foul language and attitude to top it off. Today, magically, it was all better. She worked all class long, asked questions, and was generally pleasant.

The other students in the class finally saw the point of this cardiovascular project we've been working on all week (will post more later). "Ohhh this isn't so bad. The questions are short and help us to figure out what is important in the article. And the math parts are mostly review. The only new part is clicking on a different type of graph in Excel. I guess we have to do a lot of thinking too, but that's not so bad. We have all the pieces, we just need to decide how they go together." I couldn't have said it better myself. Way to go guys!

Last week I was sad about the number of students with missing assignments or who needed to retake quizzes, but this week, they are getting their act together. Yesterday a student stayed after who I keep trying to work with in class, but he brushes me off with, "I'm good. No thanks." We went over a quiz and he said, "Man, if it could just be this fast in class, I'd get everything." When I said, it can be like this, thats what I'm asking to do when I say, "How's it going?" or "Need any help?" He got this confused look, and just said, "Oh. Well then definitely. Let's do it!"

Similarly, in geometry, I have two students who I can see are struggling, one of whom just get angry and talks about how class sucks, the other comes to class late and avoids work by trying to get kicked out. All year I've been trying to engage with them but they weren't ready. Today, both of them called me over and asked for help, and totally got what was going on after some one on one time. I ignored the other kids for a good chunk of class, but the others were ok. Those two were on task the entire work time.

Another geometry student who almost failed last quarter and squeaked by with a D has an A. He turned in one of the best Points, Lines, Planes projects and came today to study for a quiz that isn't until Friday. He didn't realize he had an A and was beaming when I told him. Then it was my turn to blush when he showed me how he was using Notebook and the way I taught him to make 3-D drawings for a history project. Another student who failed last quarter with about 7% has a 97%! He finished his homework on time, caught up on the work from when he was absent, and left with a big, smile.

All in all a great great day.

My attempt to bring math into the 21st century through both content and technology...

## Wednesday, November 30, 2011

## Sunday, November 20, 2011

### Process vs. Product

In the last post I wrote about one of my students getting pretty upset about me not telling him if he was right or not. When he told me, "I've never had a math teacher be like this," I wondered if there was an emphasis on math. Are his other teachers like this? Is it just math that he expects to be delivered in neat, compartmentalized chunks of notes, problem sets, and correct answers? This was a student who aces every test and project, so his reluctance wasn't an issue of not comprehending the statements or assignment. The discomfort he felt with not having me tell the correct answer is exactly the reason I wanted to do the project this way. I could have done the same project but hinted, and suggested, and told them if they were right. If done that way, I'd guess that they students would earn better marks on a multiple choice test about points, lines, and planes, but then they would still think that there was one correct answer. They wouldn't have that same feeling as when Ms. A and I debated if there was such a thing as two coinciding planes. If there was, would that count as an intersection?

In forcing them to actually invest, to think, to struggle, the math was getting to them. I had more kids be frustrated, throw fits, and tell me they were quitting. But they didn't. I told them to pick an answer, and move forward with it. If they got more stuck, they needed to back up and try a different answer. Or talk it out with a classmate.

It was also really eye-opening to me to see how different the process focused approach is from being product oriented. In the past I have claimed to be process oriented. My school is art focused so I hear a lot of discussion about being process oriented. Typically it goes hand in hand with a discussion about how students work at different paces and having rigid deadlines or only accepting one format of answer is stymying creative output. As a result I decided to allow students to retake tests, to emphasize that the process of learning is important, not learning by a deadline. In actuality, that is still product oriented. There is a right answer (a product) that I want them to reach, and I don't care how they get there or how long it takes, just that they get there. I also decided to allow for multiple formates on most of my projects. During reviews students can turn in videos or word problems or a review game or a set of textbook problems. Again, I felt like I was being process oriented, but there was still an outcome fixed in my mind. I wanted them to review. I didn't care how they did that, just that they did.

The execution of the project was the first time I feel like I have actually been process focused. I fought every urge to tell them "No! That's not right!" I fought (almost) every urge to hint, "Hmmm. Maybe look at number 2 again," or "Well, do you really think a plane never ends?

Ultimately, with the recap next week, there is a product. I want the students to know that if two planes intersect, they always intersect in a line. In a math class, I can't deny that there is content, some nugget of information I am hoping that they will learn. In showing a "best" answer, I hope that I am not diminishing my initial goal, for the students to learn that it's ok to struggle in math, but that they need to push forward and argue for why they think their answer is correct.

The one time I broke down was when a student had a fantastic photo of why a plane never had an edge (I'll upload it once he turns his project in). He thought the statement "Planes have an edge" was always true. I asked why and he said, "well it's impossible for something to not end." I asked if lines ended or went on forever and ever. He said, "well lines go on forever and ever," so I countered that it was possible for things to not end. He thought for a minute and said, "So if two lines cross in an x and they both go on forever and ever and you fill in the space in between the lines with like paper or more lines or something, then is that a plane? Wait! That's like the graphs we make. OH! that's called the coordinate plane isn't it? Is that why it's called the coordinate plane? Is that a plane?" I initially felt bad that I was nudging him, but after that grand epiphany I was grinning ear to ear. He was visualizing and making connections and it was beautiful. Not bad for last period on a Friday.

In forcing them to actually invest, to think, to struggle, the math was getting to them. I had more kids be frustrated, throw fits, and tell me they were quitting. But they didn't. I told them to pick an answer, and move forward with it. If they got more stuck, they needed to back up and try a different answer. Or talk it out with a classmate.

It was also really eye-opening to me to see how different the process focused approach is from being product oriented. In the past I have claimed to be process oriented. My school is art focused so I hear a lot of discussion about being process oriented. Typically it goes hand in hand with a discussion about how students work at different paces and having rigid deadlines or only accepting one format of answer is stymying creative output. As a result I decided to allow students to retake tests, to emphasize that the process of learning is important, not learning by a deadline. In actuality, that is still product oriented. There is a right answer (a product) that I want them to reach, and I don't care how they get there or how long it takes, just that they get there. I also decided to allow for multiple formates on most of my projects. During reviews students can turn in videos or word problems or a review game or a set of textbook problems. Again, I felt like I was being process oriented, but there was still an outcome fixed in my mind. I wanted them to review. I didn't care how they did that, just that they did.

The execution of the project was the first time I feel like I have actually been process focused. I fought every urge to tell them "No! That's not right!" I fought (almost) every urge to hint, "Hmmm. Maybe look at number 2 again," or "Well, do you really think a plane never ends?

Ultimately, with the recap next week, there is a product. I want the students to know that if two planes intersect, they always intersect in a line. In a math class, I can't deny that there is content, some nugget of information I am hoping that they will learn. In showing a "best" answer, I hope that I am not diminishing my initial goal, for the students to learn that it's ok to struggle in math, but that they need to push forward and argue for why they think their answer is correct.

The one time I broke down was when a student had a fantastic photo of why a plane never had an edge (I'll upload it once he turns his project in). He thought the statement "Planes have an edge" was always true. I asked why and he said, "well it's impossible for something to not end." I asked if lines ended or went on forever and ever. He said, "well lines go on forever and ever," so I countered that it was possible for things to not end. He thought for a minute and said, "So if two lines cross in an x and they both go on forever and ever and you fill in the space in between the lines with like paper or more lines or something, then is that a plane? Wait! That's like the graphs we make. OH! that's called the coordinate plane isn't it? Is that why it's called the coordinate plane? Is that a plane?" I initially felt bad that I was nudging him, but after that grand epiphany I was grinning ear to ear. He was visualizing and making connections and it was beautiful. Not bad for last period on a Friday.

### Teaching Points, Lines, and Planes

The time has finally come! I get to teach the lesson I spent weeks developing! For a bit of background: see my previous post about the PCMI project. Also: I teach 3 sections of geometry. My co-worker, Ms. A, teaches 2 additional sections. All of our 9th graders are enrolled in geometry except for a few students who took it online last year. There are a few sophomores, juniors, and seniors in the classes as well, typically students from other districts who still have Algebra in the 9th grade. We both taught the project at the same time.

When Ms. A and I decided to teach this lesson we wanted it to be as an introduction to proofs. As a result, we wanted to emphasize them arguing for whatever they felt was correct. The photos, diagrams, and sentences would be their supporting statements.

I started the activity by going through the definitions worksheet (their givens). I changed the third column to be "real-world approximations." I have a number of students who would need the scaffolding of having a list of what points, lines, and planes could be represented with. I was really glad we did that because when students were struggling I just sent them back to the definitions sheet. I was originally going to have the students use geogebra to make the diagrams, but I didn't ever really teach them how. I left it open as an option, but encourages them to do it by hand or use the shapes/lines tools in SMART notebook (which they all have installed on their computers). I really wish I had spent more pre-activity time teaching them how to make the drawings and explaining why they were important to know how to do. Ms. A spent more time on explaining the drawings, so we shall see if she gets better results.

When I was explaining the project I spelled it out really clearly that part of my goal was to introduce them to a new way of thinking about mathematics. During this unit, and going forward, there might not be one right answer. I don't really care whether they pick always, sometimes, or never, as long as their photos, diagrams, and sentences back up their argument. I would not be telling students if they were right or wrong at any point. In fact, I wouldn't really be offering much help at all. They have teammates, classmates, a definitions sheet, a textbook, and everything online.

I explained that there were a few statements that were purposefully unclear. They needed to choose an interpretation and run with it. I talked about how my math teacher friend and I argued for hours at math camp and that Ms. A and I got into a fight about the answer to one of the statements the night before. One smart alek asked if we threw punches or just swore at each other. When I said, "neither. It was pretty civilized," he turned to the class and said, "So it was both. They totally cussed each other out. Ms A took a swing, but Ms. W knocked her out just after." At least he was loyal and picked me to be the winner.

Throughout the project they kept trying to get me to tell them, but I stayed strong. One student was having a hard time with "Two lines that don't intersect are always parallel." He recognized that with the lines being on separate planes they could never intersect and not be parallel and wanted to know if that was right. I kept evading his demands to tell him if he was right and he got quite upset. Eventually he told me, "I've never had a math teacher be like this. Math has an answer. You know the answer. JUST TELL ME! Isn't it your job to teach? You're not doing anything" I just reiterated that in mathematics, people don't know if what they are researching is correct or not, they just have to make assertions and proceed. After he calmed down and said, "Well, we haven't really talked about two planes before, so I'm going to assume it means one plane and go with Always True." At the time, I was a bit sad that he went that route, but if I'm saying it's about their thought process and argument, I have to be really pleased with that moment. He got over his frustration, chose a path, came up with an argument, and worked it out, which is what I wanted.

In being more hands-off, I know that a lot of the groups have some major misconceptions. But I'm ok with that. Points, Lines, Planes, and those crazy drawings of them intersecting etc are always a confusing topic for my students. I'd rather them learn that math can have that fuzzy I don't know if I'm right or not, but I'm going to go with it feeling too. After we come back from break, I will use Kate Nowak's wrap-up activity of presenting the "best" 10 statements and have them correct 4 slides with misconceptions.

Class has been a bit messy, but I'm pleased with the progress and the project. Of course, the day that the kids are laying on the floor (a line on a plane), rearranging photos on the wall, and wandering the high school with cameras, is the day that the principal wanders by, the vice-principal stops in for an observation, and the tour group of potential families parks itself in front of the math rooms rooms. It was a controlled chaos. Ms. A was pretty calm about it and reminded me that with projects, things look a bit messier, and that's part of the point. Messy is ok.

When Ms. A and I decided to teach this lesson we wanted it to be as an introduction to proofs. As a result, we wanted to emphasize them arguing for whatever they felt was correct. The photos, diagrams, and sentences would be their supporting statements.

I started the activity by going through the definitions worksheet (their givens). I changed the third column to be "real-world approximations." I have a number of students who would need the scaffolding of having a list of what points, lines, and planes could be represented with. I was really glad we did that because when students were struggling I just sent them back to the definitions sheet. I was originally going to have the students use geogebra to make the diagrams, but I didn't ever really teach them how. I left it open as an option, but encourages them to do it by hand or use the shapes/lines tools in SMART notebook (which they all have installed on their computers). I really wish I had spent more pre-activity time teaching them how to make the drawings and explaining why they were important to know how to do. Ms. A spent more time on explaining the drawings, so we shall see if she gets better results.

When I was explaining the project I spelled it out really clearly that part of my goal was to introduce them to a new way of thinking about mathematics. During this unit, and going forward, there might not be one right answer. I don't really care whether they pick always, sometimes, or never, as long as their photos, diagrams, and sentences back up their argument. I would not be telling students if they were right or wrong at any point. In fact, I wouldn't really be offering much help at all. They have teammates, classmates, a definitions sheet, a textbook, and everything online.

I explained that there were a few statements that were purposefully unclear. They needed to choose an interpretation and run with it. I talked about how my math teacher friend and I argued for hours at math camp and that Ms. A and I got into a fight about the answer to one of the statements the night before. One smart alek asked if we threw punches or just swore at each other. When I said, "neither. It was pretty civilized," he turned to the class and said, "So it was both. They totally cussed each other out. Ms A took a swing, but Ms. W knocked her out just after." At least he was loyal and picked me to be the winner.

Throughout the project they kept trying to get me to tell them, but I stayed strong. One student was having a hard time with "Two lines that don't intersect are always parallel." He recognized that with the lines being on separate planes they could never intersect and not be parallel and wanted to know if that was right. I kept evading his demands to tell him if he was right and he got quite upset. Eventually he told me, "I've never had a math teacher be like this. Math has an answer. You know the answer. JUST TELL ME! Isn't it your job to teach? You're not doing anything" I just reiterated that in mathematics, people don't know if what they are researching is correct or not, they just have to make assertions and proceed. After he calmed down and said, "Well, we haven't really talked about two planes before, so I'm going to assume it means one plane and go with Always True." At the time, I was a bit sad that he went that route, but if I'm saying it's about their thought process and argument, I have to be really pleased with that moment. He got over his frustration, chose a path, came up with an argument, and worked it out, which is what I wanted.

In being more hands-off, I know that a lot of the groups have some major misconceptions. But I'm ok with that. Points, Lines, Planes, and those crazy drawings of them intersecting etc are always a confusing topic for my students. I'd rather them learn that math can have that fuzzy I don't know if I'm right or not, but I'm going to go with it feeling too. After we come back from break, I will use Kate Nowak's wrap-up activity of presenting the "best" 10 statements and have them correct 4 slides with misconceptions.

Class has been a bit messy, but I'm pleased with the progress and the project. Of course, the day that the kids are laying on the floor (a line on a plane), rearranging photos on the wall, and wandering the high school with cameras, is the day that the principal wanders by, the vice-principal stops in for an observation, and the tour group of potential families parks itself in front of the math rooms rooms. It was a controlled chaos. Ms. A was pretty calm about it and reminded me that with projects, things look a bit messier, and that's part of the point. Messy is ok.

*Neither my vice principal or principal said a word to me about it. Didn't ask why math class needed 10 digital cameras or why the students were all over the school. So my worries were unfounded and I love that they were unfounded.*
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