Math Intuition

In the classroom where I am placed at Chief Sealth, I am noticing that many students do not have good mathematical intuition or do not tap into it to help them solve problems. Frequently I find students stuck on problems that are conceptually quite simple because they “don’t remember how to do this [type of problem].” Today, students were having trouble converting from inches to centimeters. Some were stuck because they didn’t know how to set up the proportion (which is just one way of solving this problem) or how to solve for the unknown variable once the proportion was set up (many don't seem to understand what cross-multiplying is or why it works) or if they should multiply or divide by 2.54 (the conversion factor). Students did not seem to exhibit an awareness that there were multiple ways to solve this problem and that they can use a different approach if one doesn’t work for them, or that they could use reasoning to help answer their own questions. (For example, a student could reason: A centimeter is smaller than an inch, so a measurement in inches is going to be a smaller number than the measurement in centimeters. So to convert 20 centimeters to inches I would have to divide by 2.54 instead of multiply.) When I approach a situation like the ones I'm describing above at SGS, I'm used to either 1) asking other students in the group to explain how they did it and why, 2) asking students to explain their reasoning and poking at the parts that don't make sense until a student understands the problem for themselves, or 3) trying a simpler problem with a student to unveil the concept that is at work so that the student can arrive at a solution themselves. At Chief Sealth, I feel like these same techniques are not as useful. Students seem frustrated that I don't go right to "Here's how you do it..." or confused when I begin to throw different problems and solutions at them. So right now I'm wondering about the best ways to help students at Chief Sealth tap into their own intuition so that they can become more flexible problem solvers...

Week of 10/5 - Post #1 on Student Experience

On Wednesday my CT did an activity to help kick off recursive functions. Students were given a graph of data: column 1 was the number of bounces and column 2 was the height of each bounce. Students were asked to graph the data, find the ratio between each bounce and model the sequence of numbers with a recursive formula. There was not much of an intro to the lesson. Mostly it was "now we're doing recursive functions" and "here's the activity for the day." I've been thinking a lot about how student experiences can be brought more into the classroom and what kind of experience is most meaningfully to the students and mathematically. Ideally this lesson could be done with an actual ball and perhaps with motion detectors to collect the height of the bounces first hand. But even if motion detectors were not available, there are a few ways this lesson could involve more student experience. First off, kids could see a demo of a ball bouncing repeatedly (or each group could get a ball to bounce if there are enough balls). Rather than going straight to max. bounce height (the graph of which may not be intuitive to some students), they could be asked to approximate a graph of height vs. time which would consist of a series of parabolas smooshed side by side. Then from their first graph they could be asked to sketch a graph of just max height of bounce vs. # of bounces. This graph could be compared with actual data, and students can then talk about whether or not their observations/intuition were correct. These are just some preliminary thoughts about how to bring in student experience and mathematical intuition. My questions are: would bringing in this kind of demo/hook be meaningful to students? From personal experience, my answer is that it would be to some. Some really like physics-oriented demos and experiments, and when I taught physics most students could be drawn in. But then what about the students who prefer connections that are more social or emotional in nature? I will keep pondering this issue...

Classroom Culture

I wasn't really sure how to define classroom culture, so I Googled "classroom culture." This is what I got from http://eltnotebook.blogspot.com/2006/09/first-lessons-establishing-classroom.html:

"Perhaps the best definition of culture which I’ve ever heard is “the way things are around here”. It sums it up nicely. Classroom culture means the often unspoken and frequently unconscious assumptions about how people (both the teacher and the students) will behave during the lessons – Where will people sit, or stand? Who will speak, when, and what about? What types of behaviour are appreciated, tolerated or frowned upon?"

So with that definition in mind, I will launch in to a ramble of what I perceive to be my CT's classroom culture.

• My CT is the one who does most of the talking during class while writing down information on a sheet of paper under a doc cam. Students are expected to sit and take notes (in a specific format) on what my cooperating teacher says. When he's done showing/explaining something, often students will be given an assignment to work on. If they have questions, they are expected to ask their group questions and then, if their questions are still unanswered to ask the teacher. Sometimes the CT is actively helping students, but sometimes he is at his desk working. The messages that I infer from this format is that the teacher is the authority on mathematical knowledge. He also knows the best way for students to learn, which is by taking notes using an inflexible format. Given the lack of two-way dialogue, I also get the sense that the teacher is not interested in understanding how the students think about problems.
• When the CT assigns classwork to students, he usually says something about how if they don't finish, it will become homework, and that they need to do the assignment to get their five points. Sometimes he offers an extra point if a student finishes the assignment in class that day. Rarely is an explanation or demonstration given as to why the math the students are doing is useful or important. What these behaviors indicate about the classroom culture is that math should be done quickly to just get it done and out of the way. It should be done for extrinsic rewards, not because it is interesting, useful, or beautiful.
• In order to motivate students to finish their classwork, the CT will often say something like, if you don't get it done in class, it will become homework. Several students do not in any real way attempt the classwork. Some have their math book out but are talking or doing something other than math. Some don't even make any pretense of working and have nothing out on the desk. When asked, these students generally say that they don't have to do the classwork because they can do it for homework instead. Though unintentional, the classroom culture permits students to not work or learn during class as long as they say they are going to do it at home (many of these students don't actually do their work at home either).
  

There is probably more I could say about the classroom culture, but this is all I'm going to say for now.

Biases

How are your own biases affecting your observations of the classroom?

My cooperating teacher (CT) has a very different approach to teaching math than I do, an approach which I perceive to be ineffective. As a result, I tend to view all that he does through a pretty negative lens. I am quick to see how things aren't working and slower to see the successes that some kids are experiencing.

At the same time, I am open to questioning the way in which I do things. My pedagogy is still not completely formed, and I see flaws in what I do as well. I am also realizing that one way of doing things will not reach all kids. So I am open to seeing that the way in which my CT designs his curriculum and works with kids may be more effective for some kids than the way in which I would choose to do things. I also recognize that he has more experience teaching math in a public high school than I do, and I believe I have a lot to learn about how to teach effectively in that context.

I know that one of our class objectives is to "observe and gather information without interpretation or judgment." I think I am still capable of doing this, but I would have to be intentional and give myself a concrete task like recording different types of statements teacher and student are saying or tallying behavior of various kinds. My automatic reaction is to judge what my CT is doing as right or wrong.

Coming from a private middle school to a public high school, I expect that the student body will be different from what I am used to. But I think for the most part, I am remaining open to my students and don't have many preconceived notions of them. I am finding that in general they are smart, kind, and willing to work. Many of them come in with preparation to be working at a higher level than the class is pitching to them right now.

I've had a couple of instances where I think some students are trying to test whether or not I am prejudging or stereotyping them. For example, I was talking to one student about the fact that her arm was sore from getting an HPV vaccination shot. An African-American male student at her table, with whom I've had a number of positive interactions, then said that his arm too was sore from being shot...with a (some type of gun/caliber--I can't remember). I'm not sure what kind of reaction I showed, though I'm sure my face revealed something. Inside I felt surprise that he said this, because I was nearly certain that he was joking and that his joking was about more than just being playful and because we have had so many positive interactions before this. He quickly said he was just kidding and then changed the subject. I found the situation to be kind of confusing, because I had the feeling that he was testing me to see what my reaction was. Was he wondering what kinds of preconceptions I, as a white teacher, might have about a male, African-American adolescent? I have no idea if I passed his test.

How does technology change the way we teach and learn mathematics?

I don't think technology necessarily has an impact on the way we teach and learn math. There are poor ways to use technology and excellent ways.

But I can do some things more or better with technology than I can without it:

• More play (constructivism): Can play with ideas like the slope of a line, the parameter "a" in a parabola. Pencil and paper are slow. In a class period, a 7th grade student can only reasonably make about 5 graphs (and they would most likely complain about doing that many). With technology, they could make dozens and be able to see how changing parameters affects a line.
  
• More conceptual: Like bullet point #1. Students develop better intuition through play.
  
• More hands-on/real-world application: Students can easily measure real life objects and/or events through capturing video or photos and work with these real-life measurements. Data can be analyzed faster and more transparently. Example: Students can film someone on a swing (pumping at a constant rate), overlay a meter stick on say iMovie, share the movie with others, then at stations watch the movie and create a graph of height versus time. Hopefully they will see a beautiful sinusoidal curve!
  
• More organic sharing of products (of work): Kids can make final products like a voice thread or digital story and post online for kids to view and comment. There's no need for everyone to sit and listen to everyone's poster or Powerpoint presentation and then get bored after the third one.
  
• Less paper (but more energy use and toxic ewaste)

Day 4

What did I learn?
I love the example of students turning in a photo of a normal distribution. I've heard of doing this with parabolas. I teach algebra 1a and I'm wondering how this assignment can be meaningful with linear patterns. Lines abound, of course, but many are not that interesting from a math stand point.

What do I want to know more about?
I still want to know how to use photos and video to measure and graph and model motion. I'm emailing Dan Meyer (of the blog dy/dan) to see if he will share how he did it.

How do you use Sketchpad with students in way that is more open ended rather than cookbook style?

How can I connect this stuff to the classroom?
My biggest take away here was using digital story or movie software for kids to create math stories. This is something I want to think more about (for example: which unit, when, requirements, etc.) My first though was to do graph stories.

What did I learn?
I didn't know how box and whisker plots worked until today. Fathom was completely new and completely fabulous.

What do I want to know more about?
Same as yesterday. I'm interested in visual and audio media. Plus I'm interested in how technology can be used for more creative assessments (Fathom looks rich for opportunities).

How can I connect this stuff to the classroom?

Algebalance
This could be a useful addition to my algebra class. We use algebra tiles to learn how to solve equations. I talk about how it's important to keep the sides balanced, and some kids get this idea, but the idea is lost on a few. This activity would be useful to illustrate the idea that you must do the same thing to both sides to keep it balanced (or take away or add a zero). It would be great if you could create pages with set equations for kids to practice with. You could put kids in pairs. Each creates a balanced pan and then switch to solve their friend's equation.

More Slope Games
My two favorites here are matching and archery. Archery is a lot like guess the slope but way more fun. Also, it's not competitive (between students) so it's less likely to turn off students who shy away from competition or failing.

Parabolas in Factored Form
Don't like this ap. Confusing layout, don't like the fact that r1 and r2 are in parentheses (the unnecessary notation will confuse and distract some students from the key points here), and too hard to use.

Box and Whisker
The dynamic nature of this activity allows kids to play around with the concept and come up with their own understanding of what a box-and-whisker plot represents.

Fathom
The connections to statistics are obvious. i teach algebra and so i'm thinking about how to connect fathom to algebra. i would have students do something like what we did today: find data with two continuous, quantitative variables, scatter-plot the data and find the best fit line and then tell a story about the data. i would probably do this in pairs.

okay my arm feels like it's going to fall off, so i'm signing off.