Rose Curves, Tattoos, and Weekly Journals

Last year and the year before, I stared every Thursday’s class with a journal entry. The students wrote to me in response to the questions “how was your week?,” and “what did we do?,” as well as some third thing. This third thing was almost always a writing prompt about the math that we had done that week, or that we were about to do. Then I wrote back to them on little post-it notes. It created this nice sense of ownership of their learning, offered them a chance to reflect on and write about their week of mathematics, and provided a space to give me feedback about the class. I would always dread reading these on weeks I had deemed bad, but nobody ever wrote anything terribly negative.

Towards the end of last year I stopped doing this. I felt that it was taking too much time out of the day Thursday, which at our school is only scheduled to last 30 minutes (impossibly short!). I also felt that writing only one day out of the week wasn’t organic since the most prompt-worthy topics might have been covered on Monday. I told myself I would have them stop and write every week whenever it seemed like a good time to put math into sentences, but since the schedule wasn’t forcing me to do it, I hadn’t been since.

NEW SOLUTION: weekly journals will still be due each week, but there is no specific class time carved out to do them. They are expected by class on Monday, and students can do them after school, in the morning, over the weekend, and whenever they get done early in class. I will see if it works and let you know. I already got a “good luck with that” from one fellow teacher.

I polled the math 4 students about journals (yes or no about bringing them back) on an assignment from Monday involving plenty of purposeful writing. I will likely do a whole separate post about that because my polar unit got really excellent this year, but basically they had to discuss the visual and numeric patterns of the number of petals on rose curves with fractional coefficients. This is the best visual that I have found for that. I summarized the results on excel:

tats roses

In addition to the more mathematical questions, I asked them about the journal (the aye’s have it! good thing I was going to re-institute it anyway!) and which rose curve I should get as a tattoo. Yet another thing to stay tuned for – new math tattoo!

My major conclusion right now is that I should do more polls. When I was in high school we would do this thing we called “carpool car poll” where anyone could just declare a poll and we’d all have to say our opinions on a topic. I want a catchy phrase like that and more chances to give opinions.

Launch – Explore – Discuss

Happy Saturday, blogosphere! I’m currently located at school, about to do my second round of ACT/SAT prep of the day, and furthermore drink my third cup of coffee of the day. In between, my colleagues and I were tabling at a conference for twice-exceptional students. Tabling typically involves me excitedly explaining our school to anyone willing to listen, and ramping up imaginary rivalries with our fellow small schools. I did try to tone that down this time, though, because I suddenly want to do some classroom observations (aka spying) at other small schools.

The congregation of the church we rent our space to is slowly arriving, and that is the only thing weirder than being at school by myself. There are also two cats in the office building instead of just one like usual,

None of this is what I had intended to discuss! I have been implementing and blogging about collaborative learning for years, explaining that we had learned all of these techniques and ideas in college and I had wanted to try it. All this time I had been ignoring what I have recently discovered is an important element: the discussion. What you’re supposed to do is launch the task by explaining the directions and process, have the kids explore the material on their own, and then have a discussion¬†as a full class to bring together their ideas. Since I was so focused on having the kids prove to me and themselves that they knew the material as an individual, I’ve been skipping the last step.

Such a poor choice! Wednesday we spend most of class on a discussion of the Rectangle Pattern Task I discovered last year. I love this task so very much. It’s the physical embodiment of “look for and make use of structure.” As I had learned to do in college, I structured the discussion around which groups had discovered what in a nicely-phrased way, making sure that everyone would have a chance to speak. Here are my notes:

notes

The groups shared the patterns they had recognized. They predicted step 0 as just 2 blue squares, then determined that each time you add 4. To find the pattern, I wrote that out like this:

chart

This made it easier for them to see that the nth iteration would have 2 + 4n blue squares. Love it!

To discover the green and red patterns, which are quadratic, we went more visually. That’s the squares you can make out in my notebook. The students described that the pattern of green squares is that on each side, you add another row, and that row will include 2 more than the previous. I drew this out as they described it:

structure A

Then I asked them to find the largest square in the first one. They said 1. I asked for the largest square in the second. They said 4 at first, then clarified that that was 2 by 2. For the last one, someone pointed out that there are many different 2 by 2 squares in there, which is true. I asked them if they could re-arrange the boxes to create a larger square than that (should I have waited to pose this?). One poetically described that process as “you take the wings off of the bird, and you put them on the bottom.” I illustrated it slightly differently, and then gave everyone paper to cut in order to create squares. This was the end result here:

structure B

The pattern for one side is then n squared plus n!

The next day the room was entirely silent and productive as brilliant math learners completed their independent reflections and then determined the patterns on yet another set of box designs. Next week we’ll discuss those, then I want to give them an assessment on this [which I have yet to write], and then we’ll work on factoring in a hopefully visual / non-terrible way.

Math Words Task

I have always been fascinated by languages. Being able to switch seamlessly between languages has always seemed like the coolest, most useful skill to me. Earlier this year I was thinking about the idea of counting in other languages, specifically counting in languages that are not base-10. Counting 1, 2, 3, … 8, 9, 10 and then starting again 11, 12, is something that we take for granted but several languages do this really differently. Super cool!

The task I assigned the students comes directly after some time spent on logarithms and logarithmic scale. I got the idea to do a linguistics, number-word task because they both have to do with bases. I also thought this would be a good, less-algebraic precursor to modular arithmetic.

I first gave the students a passage from The Crest of the Peacock, Non-European Roots of Mathematics that outlines four different types of number systems. This includes our base-10 system, and the illustrated body numbers I took a picture of out of the book:

Image

As one of the students pointed out, if you wanted to say you were 17, you’d say “I’m neck years old,” which is totally how that worked.

Then they were asked to fill out a blank Pascal’s Triangle, where all the outside numbers are 1’s, and each number is the result of adding the two numbers above it:

Image

So all the boxes on the outside are 1’s, and the box below the 1 and 2 would be 3, and so would the other one like that. What numbers come next?! ARE YOU HAVING FUN YET?!

Sidenote, I wrote a whole Pascal’s Triangle task for one of my other classes, and that was awesome too.

They compared their Pascal’s Triangle to an Ancient Chinese Pascal’s Triangle I found in order to understand Chinese numerals. There is a whole part of the book that’s about Ancient China that has the exact same graphic of the Pascal’s Triangle. I’m giving it to my one student who is way ahead and completely done to read tomorrow. The number 9 is nowhere on the Chinese triangle, but it does go all the way to 8, and 10’s on there, so I asked them to figure it out. These are the important pattern recognition skills that humans need in life! This is what the Common Core means by “look for and make use of structure”!

Then here’s the next part of the task. It’s a partially filled-in chart of number words in Ventureno, part of the now-extinct family of Native Chumashan languages spoken in present-day California. Students once again had to “look for and make use of structure”:

Image

Can you figure it out? They had a pretty hard time with this because it’s not as strong a pattern as some of them.

I’m so excited about this task, and if there were an award for “activity you would have most wanted to do in high school,” I would totally nominate myself. I’ll be back with more dispatches soon, I’m trying to blog the most out the rest of the academic school year.