Friday, February 5, 2016

The MTBoS, Week 4 (Day 98)

Before we get to the final lecture, let's get to the last week of the 2016 MTBoS Blogger Initiative. But I'm having a problem with this week's topic. This comes from Tina C., a Massachusetts teacher:

[T]his week you’ll be sharing about a lesson. Perhaps you’ve already chosen it and compiled all your resources to share. If not, think to the week ahead and consider what you’re teaching (or observing) so that you can be sure to take a photo, gather samples of student work or record part of the conversation (#phonespockets anyone?) to be able to include direct quotes. None of these are required but all require planning ahead so think now about what you might want in your post!

In describing this topic, Tina C. quotes a more well-known MTBoS blogger, Kate Nowak, a New York curriculum developer.. Dan Meyer, the King of the MTBoS, once gave Nowak her own title -- "high priestess of the high council." Here's a link to the High Priestess's blog:

Nowak doesn't want a spectacular lesson here -- she urges the MTBoS to "please be more boring." To fulfill this week's topic, we can teach an ordinary lesson -- the important part is to emphasize the interaction between the teacher and the students.

I am only a substitute teacher, which means that what I teach is based on the luck of the draw. Last week, the MTBoS Week 3 topic fell right into my lap -- I subbed for a math teacher who strongly believed in having the students question each other, when the MTBoS topic was about questions.

But this week, I'm covering a biology teacher who's out on paternity leave -- meaning that I taught very little math this week. Last week, many of the other participants struggled with their Week 3 posts while I cruised on by, and this week I'm the one who's having problems. This is why I won't consider myself a full member of MTBoS until I become a real math teacher, not just a sub. The real teachers with real classrooms and real lessons, of course, are having no trouble with Week 4.

Still, I started this Blogging Initiative, and I want to finish what I started. And so without further ado:

Date: About a year ago
The Lesson: Today I am posting a Pythagorean Theorem Activity from last year.
My Teaching: I've never taught this lesson in a classroom, but I had back when I was once a math tutor myself. I enjoy letting the students visualize exactly why the Pythagorean Theorem holds.
Student Responses: Whenever I have students try this out, it's always easier for them to complete the puzzle with the a^2 and b^2 squares than with the c^2, since it must be placed at an angle.

The worksheets that I'm posting refer to this Pythagorean Theorem activity. I'm hoping that I can teach this in the classroom soon.

Here are some math lessons that I was able to teach during the past week, despite spending most of it in a biology lab:

Date: Friday, January 29th
The Lesson: Just before being assigned to the biology class, I was placed in a middle school special ed class, but covered sixth grade math during the conference period. The students had a worksheet where they reflected points over either of the axes given their coordinates.
My Teaching: The class was already divided into six groups of six students each, so I played the Conjectures/"Who Am I?" game (MTBoS Week 2) with the students.
Student Responses: This game didn't go well at all. Group 6 was working way ahead of the game and answering most of the questions before I asked them. But Group 4 struggled with almost every question I gave them. I tried to help Group 4 out by letting them know which coordinates they had to keep and which ones they had to change the sign to. But the other groups grew restless, and I had to abide by the rules of the game by declaring Group 4's answers wrong and not awarding a point. This led to Group 4 getting upset, and one girl even started to cry.

Date: Wednesday, February 3rd
The Lesson: On the first day in the biology class, the students actually had an assignment that involves mathematical modeling. This is the classic predator-prey problem, where students had to model the populations of sea lions (the predators) and sardines (the prey), and see how the population of one affected that of the other. Each group of four students had a square foot of blue construction paper to represent the bay, 50 small squares of paper to represent sardines, and several small squares of paper to represent sea lions. For the first generation, the students would toss two sea lions into the bay, and however many sardines each sea lion was touching it would eat. Any sea lion that didn't eat three sardines would die, and for every pair that survived, a new sea lion was born. Then this would start the next generation, and students would repeat the process for 25 generations.
My Teaching: The students actually began this activity on Tuesday, the day before I arrived. I tried to understand the rules of the activity but I was confused. Based on my reading of the instructions, the sardines that were eaten were removed from the bay. But then I couldn't see how there could ever be any sea lions after the 17th generation -- during the first generation, one sea lion eats its quota of three fish while the other sea lion fails to catch any fish and dies. Then for generations 2 through 16, the one remaining sea lion eats three sardines per generation, for a total of 48 fish eaten. This leaves only two sardines available for the sea lion at Generation 17, and so it dies.
Reflection: Simulating a predator-prey model is a great idea, provided that the students are given much clearer instructions. Of course, then students would have to keep track of both the sea lions and the sardines after each generation unless they're given a simple rule, such as "Replace all 50 sardines after each generation."

Today the bio students took a test on this unit. There were two multiple-choice questions directly related to this question. The first asked, what curve shows hows population increase when there are unlimited resources? Nearly every student correctly answered "exponential" -- only a few students answered "logistics curve." The other question presents a logistics curve and asks the students to identify which of five labeled points corresponds to the carrying capacity. The correct answer is E, the rightmost point, but some students chose C, where the curve is increasing the fastest. Notice that many Common Core Math questions in modeling are similar to this one.

Date: Thursday, February 4th
The Lesson: The biology teacher has an AVID class during third period. Twice a week, on Tuesdays and Thursdays, college tutors come in to help students with various lessons in any subject. I predicted that many of the students would have questions about math -- and I was absolutely right. The class consisted of all freshmen. Most were in Integrated Math I, but some were in the Honors version of the same class. Regular students were working on systems of equations, while Honors kids were working in a chapter on radicals. I found this surprising as Integrated Math I students typically don't even work with quadratics, much less simplifying radicals. Then again, I'm not completely sure what the Honors Integrated Math I curriculum is like.
My Teaching: I first helped one of the Honors students with simplifying sqrt(144) + cbrt(8) -- where "sqrt" and "cbrt" are ASCII for square root and cube root, respectively. Of course, the question wasn't written in ASCII, but in standard radical notation with an index of 3 to indicate cube root. The student thought that maybe the 3 was a coefficient, like sqrt(144) + 3sqrt(8), until I told her that it stood for cube root. Then I moved on to a regular student who had to solve this system: {x + 5y = 28, -x - 2y = -13}, and she decided to use substitution. I told her that it was much better to use elimination for this system, since the x-terms were x and -x.
Student Responses: The girl then started to multiply one of the equations by -1. When I told her not to do so, she tried to multiply the other by -1 instead. Then one of the college tutors intervened and told me that it was better for me to use an inquiry approach (shades of the MTBoS Week 3 topic), rather than just tell her not to multiply by -1. But then the tutor added, "Don't worry. Almost every sub has tried to just tell them the answers."
Reflection: Perhaps a good question for me to have asked the girl is, why would we need to multiply an equation by -1? Then perhaps she would have learned that it's the coefficients of the variables (including their sign) that determine which equation(s) need to be multiplied.

Notice what one student wrote on her Tutorial Request Form:

Prompt: I gained a new/greater understanding of my point of confusion by/when ...
Student: When our substitute [yours truly] explained to me that it's an exponent and not a multiple.

Oh, and that reminds me! In David Kung's Mind-Bending Math, Lecture 24, his final lecture, he discusses "The Paradox of Paradoxes" -- if paradoxes so difficult to figure out, why are they fun? He explains that people naturally enjoy the mental challenge of solving a puzzle. He mentions the famous 4 T's puzzle, which has a similar answer to the c^2 puzzle -- fit the 4 T's in the box at an angle! He also summarizes all the paradoxes described throughout the course. Most real numbers are not rational, most functions are continuous nowhere, and most functions that are continuous everywhere are differentiable nowhere. (Thanks, Sam Shah!) This idea that what's most familiar is least common permeates science -- most life forms are bacteria, most animals are insects, most matter is dark matter, most ordinary matter is plasma, and so on.

Here are the three links for this week, two right above mine and then the next Blogspot. All of these appear to be Geometry lessons, so that's great:

Geometry -- oh, that's right. Kung's course and the MTBoS have dominated my blog this past month, but after Lincoln's Birthday weekend, this will return to a Common Core Geometry blog. So below are images of the Pythagorean Theorem Activity and two Core-inspired activities.

For those who are reading this post as part of the MTBoS, thanks for participating in the 2016 Blogging Initiative. Hopefully there'll be a 2017 Blogging Initiative -- see you next year! Otherwise, my next Geometry post will be on Tuesday, February 9th.

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