Paarlberg Tomatoes

A little update on the tomato post from a couple of days ago. I looked up the tomato farm and discovered that "This year's [2012] production from Paarlberg Tomatoes would account for over 23 million cans of Red Gold whole, diced, stewed, and specialty tomato products." That's neat!

Here's a pic of another truck at the intersection of 35 and 39. They're harvesting another field now, and still transporting tomatoes.

Tomatoes Galore

Every day on the way to and from work I drive by a tomato crop field. It's a anomaly in the midst of corn and soybean fields. This past week it has been harvested, and sitting near the road are trailers full of tomatoes. I am so curious about these trailers. How many tomatoes do they hold? How many tomatoes did they get off the land? How much money are they worth? How heavy are those trucks when loaded with tomatoes? Where do they get delivered? Today I followed one of the trucks along the highway briefly, and got this picture from the back. The company is called Paarlberg Tomatoes. You can't tell how long the trailer is, so it's hard to get a good sense of how many tomatoes are in the truck, but it's about the size of a semi in length. I wish I could find out more about this operation, and see if I could get my kids involved in an investigation of some of my - and maybe their - questions.

Student-Centered Classroom

Today in one of my geometry classes, I had one of those experiences where I let the students help run the class. It was scary and chaotic and messy, and I hope that the students understand what happened and why I let it happen. I taught a lesson on midpoints yesterday, which was quite easy for them. Today I continued the lesson by having them find an endpoint given the midpoint and the other endpoint. You use the same formula – the midpoint formula – but you have some unknowns in the formula and have to do some algebra to find the endpoint. I didn’t realize how difficult this concept would be for students to grasp. My first hour geometry also struggled with it, so I was somewhat more prepared for this hour, but only for how much they would struggle – I didn’t really come up with a better approach between the two classes. After running through a problem at the board, I had the students work on one while I walked around. I looked at their solutions, and I noticed that one student had done it in her head and had gotten the right answer. She explained it to me, and I asked her if she would be willing to explain it to the class. She agreed, hesitantly. But she got up and explained it, and a couple of students saw it her way. Then my aide took over and started connecting it to the formula. At that point, I think the students felt like I was relinquishing my role as a teacher, and felt a little unsure about what I was doing. So I stopped the aide and told the class that I was letting this happen because I want them to know that I’m not the only one with answers in this class, and sometimes students can explain it better to each other than I can to them. More students started to figure it out conceptually, and then I started directing them to help others who still didn’t get it. I pulled some people up to the board to work with me, and some to work with the aide and the original student who got it. Students were moving all around in the classroom, gravitating toward someone who could explain it. We didn’t have a lot of time left in class to work, but at least a couple more students were getting it, and they were explaining to each other. It felt scary to let go and let them explain because I wonder if the students think I’m not a good enough teacher because I can’t explain it well enough, so I let others do it for me. Honestly, I can’t yet verbalize another way to understand how to use the midpoint formula to find an endpoint. That’s something I’m going to work on this weekend and can hopefully have ready for Monday. I am just surprised that this happened today, and it fits into some of the ideas I learned in my masters in ed program, where the teacher is not supposed to be the only authority. The class today was very much student-centered, and I could feel how different it was. I’m not sure how much I like it because it feels so uncomfortable, but maybe I can foster it and the student will learn to understand it and appreciate it because I think there is some true value in it.

Tangram Tuesday

Somewhere on the MTBoS someone Mr. Stadel mentioned how his students had never done Tangrams, and needed work on manipulating shapes and areas. So he introduced Tangram Tuesday. I thought it would be great for my Geometry class, as would help build their visual and spatial reasoning. Yesterday was our inaugural Tangram Tuesday. Only two or three of the kids had ever played with Tangrams. The rest were clueless. They seemed a little unsure at first, but after about 10 minutes when I said I we were going to move on and I was going to show them the solution page, they all moaned and complained. I was very surprised! Some of the kids seemed frustrated, but by the end of the time, they wanted to keep working. I can't wait for next Tangram Tuesday! Here are some of their creations - some of them were solutions to problems I gave and some of them the students made up on their own.

Whiteboarding in Geometry: Points, Lines, and Planes

We did whiteboarding in Geometry today, and it was amazing! On Friday, we did a mini lesson on points, lines, and planes, basically just definitions. Their homework involved answering questions about a diagram that showed two parallel planes intersected by a line. I felt like they might not have gotten enough practice with these concepts, and we hadn’t done any drawing of points, lines, and planes, so today I planned to have them do more practice. Out come the white boards! It was their first time using them. Following is the list of statements I used. They had to draw what each statement described:

• Points X and Y lie on line CD.
• Points A and B are collinear.
• Two planes do not intersect.
• Two planes intersect.
• Line LM and line NP are coplanar but do not intersect.
• Line m intersects plane R at a single point.
• Lines s and t intersect, and line v does not intersect either one.

What I liked about this experience of whiteboarding is that the problems weren’t drill problems, like I had done in the past. These required some real thinking. The diagrams of points, lines, and planes that students drew were entirely new to them – they hadn’t seen anything like it. And many of them didn't think to represent some of the diagrams in 3-dimensional space as opposed to 2D. So for example, when students were asked to draw a line m that intersects plane R at exactly one point, they came up with things like:

and

 when they really needed something like:

We got to have some good conversations about why some of these diagrams didn't work. For example, in the second picture, you can see the student was thinking about the line intersecting at only one point on the plane. But planes go on forever. Even though the diagram makes it look like the plane is contained, it really goes on infinitely in all directions (in 2D). So if you extended it, it would look like the first picture, where the line is on the plane, rather than intersecting it at a single point. 

It was fun to watch students think about how to draw these things. The ones who struggle to pay attention and take notes were into it, and so were the ones who get bored easily by class. So I'd say it was a success!



Students also had to work on an Always, Sometimes, Never activity for points, lines, and planes. I like the level of challenge of some of these questions. I stole these from someone, but I can't remember who, so thanks to that anonymous person! It came from here.

Always True, Sometimes True, or Never True?

Points and Lines

1. Given one point, only one distinct line can be drawn through it.

2. Given two points, only one distinct line can be drawn through both of them.

3. Given three points, only one distinct line can be drawn through all of them.

4. Given four points, only one distinct line can be drawn through all of them.

Points and Planes

5. Given one point, only one distinct plane can be drawn through it.

6. Given two points, only one distinct plane can be drawn through both of them.

7. Given three points, only one distinct plane can be drawn through all of them.

8. Given four points, only one distinct plane can be drawn through all of them.

Lines and Planes

9. A given line is contained in a plane.

10. A given line is contained in two planes.

11. A given line is contained in three planes.

12. Two lines are contained in the same plane.

13. Three lines are contained in the same plane.

 

First Week

I've only taught three days at my new (and first!) school, and it feels like I've been there at least a month already. The first week went well enough, and I enjoyed the math I did with the students in those first few days. Though I probably didn't implement them to their full potential, these problems really got students thinking and gave me some insight into their abilities in math.

In Geometry, we did the Square-ness problem from Balanced Assessments. Students are given 9 rectangles and asked to put them in order of "square-ness." Many had no problem with this, but struggled with the follow-up questions asking why they arranged them like that, whether they could find a rectangle with an intermediate value of square-ness if given two other rectangles, and if they could come up with a formula for measuring square-ness. I didn't expect students to have all of the answers, just to be challenged and to start thinking geometrically. I let them work for most of one day on it, and we went over it two days later. I didn't spend very much time coming up with an accurate formula for measuring square-ness, but we came up with something that was on its way to being accurate. Many students realized you could subtract the side lengths to determine how square something is. Then in my last hour of geometry, when we were talking about finding the difference between the side lengths, one student piped up that it wouldn't quite work. He asked what would happen if one rectangle was much larger than another -- then our measure of square-ness wouldn't work. He came up with the idea of dividing the side lengths. Others thought maybe we could scale one rectangle to have an equivalent side length to the other rectangle, and then subtract the sides. That's much harder to put into a formula, but the idea works. Many students conflated the concept of area with square-ness while working on this activity, and now I think I see why. First of all, when asked to come up with a formula, many came up with l * w because that's a formula they already know related to rectangles. They didn't think about whether it works or not. But I think there's a deeper intuition here, as well, that some students had. While area doesn't affect square-ness -- a square is a square no matter how large it is -- you have to think about area when coming up with a measure of square-ness to avoid accidentally allowing area to mess up your measure. That's what the student was getting at in my last hour of geometry. I was surprised and delighted with his insight. I thought most students were barely holding on to the argument, and here he was ahead of me. One of the signs of a good task is a low entry point and a high ceiling, and I think this task hits the mark on both of those.

In Math 8, we did the Handshake (or Supreme Court Welcome) problem from Illuminations. I also showed this video of Sandra Day O'Connor talking about shaking hands with the other justices during meetings.

 I had students make guesses at the beginning of class to get them invested. Most came up with 81 (9*9), which was more of a quick calculation than a guess. I saw some good representations on paper as I watched them work. In my smaller class of 13, we had time to let students present their solutions to the class. I wanted them to see that there are different ways of representing math using diagrams. No one came up with the network solutions, so I showed them that, and they were excited by the pattern it makes. In the larger class (35 students), we didn't have time to go over it, which I regret. But I hope it still got them thinking and gave them a good first experience with 8th grade math.

Now that the first few days are over and we are heading into the curriculum, I feel like there's less room to do activities like this. My goal is to find ways of integrating problem solving tasks like these into my curriculum, but I'm not even near that yet. One step at a time.

First Day

There are many reflections I could make about my first day of school and teaching, but there is one that I want to write about here. I teach 8th grade math and high school geometry, and I have my room set up in groups of four. One of the other math teachers was talking to me about how that might (or might not) work out. She has taught 8th grade before and from experience has found that the kids get distracted and talk too much during lectures if they do not face front. I just read someone's blog about why she became a believer in groups (will have to find who), and one of the reasons is that in groups, students only have about three people to distract them; whereas in rows, they  have people to every side, so more like 8 students to distract them. However, when looking at it in my classroom, I see that in rows, students are all facing the same direction, so it's harder to get distracted and start conversations. In groups, students are facing each other, so the temptation is stronger. So I'm not really sure where I'm at with that yet. I want to play around with groups for a while longer and see what my experience is.

During that conversation, I said I'm not sure how much lecturing I will be doing vs. group work. I might not have to worry much about students getting distracted during lectures if we're doing mostly group work. (Although from today I can see how easily they get off topic while in groups.) In talking about 8th grade, she said that she has found there's hardly any time to do group work and fun things because there are so many standards to cover. I want to believe that I can teach in a way that prepares students for standardized tests without pushing them through procedures and concepts. I need reassurance that it's possible. I haven't finished all of Jo Boaler's MOOC on How to Learn Math, but she cites research that shows that students who took math classes where they did tons of problem solving and didn't see standardized tests for three years (or something like that) did better than students in traditional classrooms. While I want to believe that's true, I really don't know. I don't personally know any teachers who do it. Sometimes I think it takes a veteran teacher to make that happen, and that's certainly not me. I want to grow into the type of teacher that can teach problem solving and prepare students for the world - and standardized tests - without lecturing all the time. Yet traditional teaching methods are mostly what I see in schools, and sometimes it looks like it has better results. And I feel pressure to teach that way. I think that if I try something different, and students don't do well, it's because the traditional way of teaching is better. I'm scared of that being true.

I am really hoping that in the near future, Indiana switches to PARCC assessments where problem solving is more important than procedural work. That would make me more comfortable teaching in a non-traditional way with an emphasis on math thought and process rather than procedure.