Ohio Council of Teachers of Mathematics
Educate, Engage, Inspire ... with the Power of Mathematics!
One day in a College Prep Precalculus class (I call it CPPC, which is a palindrome), we worked on a problem involving volumes of three different objects. Find a relationship for a sphere, a cylinder, and a cone that all have the same radius and volume. The students’ objection was that there were “all variables, no numbers!” It looked like this:
The students discovered that the smallest surface area would be the sphere.
Rumor has it that is known already by everybody but me. But it’s cool that the
kids figured it out. Mr. McMonigle encouraged me to utilize his students’ skills
again if I could….I actually did last year, too.
Here is a mini story about that: I was getting ready to do a presentation about
some of my favorite real world problems, including one about the safe angle at
which a ladder should be leaned against a wall. I really enjoy this lesson in class
because I send out a teaser to the kids the night before: a video from OSHA
about ladders in the workplace. The kids have NO idea what is going on!
The next day, I send them a TI-nspire document with an interactive graphic of a
sliding ladder, and the angles at which it is leaning. I ask them to slide it back
and forth until they come up with a “range” of safe angles.
After that, I wish I could have the students interact with a real ladder, but I think
it’s against a rule somewhere. I wasn’t allowed to do it at the presentation in a
hotel/conference center, either. I guess nobody wants a lawsuit. So, I bought a
mini ladder online that comes from a set of WWF Wrestling dolls and
accessories. But then I wanted another. So I asked a student to make one for
me on the 3-d printer. I guess it was a stretch for this kid to do it, and he tried
very hard. When he brought me his finished product the day before I left for the
conference, he was a little embarrassed, but I was so proud! I showed it
around during the presentation, and gave it away to a teacher to take home.
The question I posed was do folding ladders come at a pre-determined safe angle?
Extension ladders must be placed against a wall or surface at a safe
angle by the human who wished to climb it. I have several pictures from my
summer Mission Trips where kids are up on ladders at a worksite, and I am
responsible for their safety. We “measure” the slope and angles using the
TI-nspire calculators to determine if the ladders are safe. I will continue to use my
mini ladder and the technology to engage students (or teachers) in this “real world”
I never thought I would get to GO BACK to China, but I did! So, I experienced the same surprise this summer when students did not know what "HL" stood for when proving right triangles congruent...but at least I was prepared for it. When I explained, I got to hear the cool "Ohhhhhhhh" sound from about 28 students all at once. Two periods in a row! This year, there were 2 classes of about 29 kids each, which my colleague and I taught back to back periods. Some days, we taught from 8 to 9:20 and 9:30 to 10:50, and other days it was afternoon sessions. We seem to always stumble into our classes a few hours after we arrive in the country, usually on a weekend day. This year, we taught from Saturday to Saturday without a day off. Phew! I explain that our American students probably couldn't handle that. (maybe American teachers couldn't handle it, either?)
I decided to make the lessons a little more interactive this year, to display the "American style" of teaching and learning. I felt that I had to prepare some materials ahead and take them with me, in addition to packing supplies that could be used for activities "on the fly".
My first activity was the Box Volume problem from Algebra 2. Every pair of students was given a piece of centimeter grid paper, and asked to cut out a specific square from each corner. (of course, I left the pretty neon paper at home and had to settle for local, recycled paper!) Each pair then folded up the sides to create the open box, and calculated the dimensions and the volume. We put all of the class data on the board, and then I also entered it into my Ti-nspire calculator on the projected touch screen. (sort of a Smartboard) We all agreed which box seemed to have the greatest volume, but they didn't know how to answer more specifically what the dimensions would be of the absolute MAXimum volume. Using scientific calculators, I imagine they would do some guess and check for a while until their teacher was satisfied that they had gotten close enough. I used the technology to show them how the graph can be analyzed to find a very accurate maximum ordered pair. We had fun discussing the quite different shapes of the boxes, and how certain boxes would only be good for storage of certain items such as pencils or erasers. The very flattest box was called a tray. Students were working on their English conversation skills at this point which was very engaging for them all.
One other activity we did involved the Parent Functions, as I call them, and Even and Odd Symmetry. In each case, I had graphs glued onto sticky notes, and had students come to the board to arrange the graphs properly according to the names or categories in which they belonged. Each time we did this activity, there were not enough graphs to go around, so I asked some students to be the teachers and find mistakes that were made by their classmates. Mistakes would be anonymous, of course, because many stickies were up on the board
at that time. No one was embarrassed by putting a linear graph with the word quadratic by mistake. After one iteration of students correcting the work, both classes were at 100% accuracy. The students learned many new vocabulary words through these lessons, including the names of some basic functions (quadratic, cubic and so on) and also using Even and Odd to speak of rotational symmetry or reflective symmetry. Over the brief 8 day series of my lessons, many words were repeatedly used, and so students began to use the words more rapidly when I posed questions to them. (You see, repetition does work!)
I never thought I would be teaching math in China! Last June, I had the opportunity to go teach in China for a couple weeks, and I took it. In the heat of July, I was there. My colleague (an American History teacher) and I met together, and planned some activities for teaching English, then we each did our own planning in our subject areas. I had to plan enough material for teaching 10-12 two hour classes, without knowing whether I would have internet access, a SMARTBoard, or access to copies! I knew that it could mean being "stripped" of the many luxuries of teaching in my little bubble of the world in Room 221 in Northeast Ohio, USA! What have I gotten myself into?
My math lessons included topics such as equations, inequalities, systems, trigonometry, congruent triangles, sequences, and polynomials. The students seemed to be the equivalent of my sophomores back in the USA. They were excited to hear me teaching in English, and to assimilate what they already knew about each topic. Of course, there were things that were simply translation issues. (including my jokes!) We used translators to help, but discovered that often an English word and a math word don't translate very well! Another thing I knew I couldn't use was "American shortcuts" such as FOIL, that only work if you think/speak in English. I'm not sure the students knew the word "Distribute", but they knew how to do it.
One day, I decided to do triangle congruency with them. No problem, right? The students knew each of SAS, AAS and ASA very well. All of a sudden, we came to a few right triangles and they used HL before we even discussed it....they didn't know what H or L stood for, but they knew that the logic of the proof was correct. When I introduced the words hypotenuse and leg, there was a unanimous "Ohhhhhhhhhh" to express their understanding of the method of proof. Enlightening.
After a day with the Unit Circle and some basic Trig (again, skipping the SOHCAHTOA mnemonic), I moved on to using the reciprocal functions of cotangent, secant, and secant. The students had never heard of these! What? I guess they are truly not necessary, since you can always change the problem into its reciprocal and solve it from there. But what about the graphs? I thought it was worth playing around with, so we did graph them one day. I liked to think that I actually "taught" some math to this group of students! And it didn't really matter, because it wasn't on any test for them, and I wasn't giving them a grade.
Let us not forget, that one of the best parts of teaching is the different groups of students we get to work with every year or every semester. I had a group of 25 kids from China, who had dreams of coming to the U.S. for education in the next few years. I enjoyed asking them about their favorite movies, foods, and more. They were thrilled to ask me things about life in America. Unfortunately, sometimes those conversations turned to talk about drugs and gun violence. They wondered if I was afraid in my home, and if I would buy a gun. Discussions that would not likely take place in my classroom at home.
After two weeks, my colleague and I figured out that we were actually teaching the kids through their last week and even their last day of school! (Who wants to be teaching someone else's classes on the LAST day of school?) The day was July 28th, and it was probably 90+ degrees outside. Our last act as their teachers was to attempt to buy "ice cream" from the local guy just outside the campus gates...rather challenging to communicate how many and how much! We think it cost us about $20 to buy each kid an ice cream bar of some sort. :)
Stay tuned for the next installment...
I'm blogging today to let you know about a newly published article from the Ohio Journal of School Mathematics. Catherine Lane from Baldwin Wallace University has written a remarkable article, "The Joy of Following Students Down Unexpected Paths." The article documents Catherine's experience when she decided to let her students think and reason mathematically FOR THEMSELVES (imagine that! An all-too-rare occurrence in many classrooms, right?).
As you read the article, you'll be struck how students will take unexpected paths, and how letting them do so---even when working with the most familiar of tasks---can lead to surprising connections. In this article, Dr. Lane describes new paths her class took to generalize the sum of the interior angles of convex polygons.
Check out the article here. Share it with your friends and fellow teachers! Spread the word about the excellent ideas of our Ohio teaching colleagues!
The Ohio Journal of School Mathematics has just posted a newly published article, What Are The Next Three Terms In This Sequence?, by Janet Walker and Matthew McBurney of Indiana University of Pennsylvania.
Here's a link: https://library.osu.edu/ojs/index.php/OJSM/article/view/5816. The authors explore a task that---on its surface---appears unremarkable. However, upon closer inspection (and with the help of some algebra and technology) yields some cool mathematics. Arguably more importantly, they illustrate the danger of assuming that routine tasks have only one solution.
I love their use of TI-Nspire and the Online Encyclopedia of Integer Sequences (OEIS). You really must check out their paper! :)
Coming in October… Global Math Week!
The Global Math Project aims to engage students and teachers around the world in thinking and talking about the same appealing piece of mathematics during a series of annual Global Math Weeks. Inspired by the work of code.org, which makes coding accessible for millions of students across the globe, the GMP team will share the inherent joy, wonder, relevance, and meaning of mathematics with students everywhere and create a forum for the global celebration of creative mathematical thinking.
What Happens During Global Math Week?
The very first Global Math Week takes place this fall. Beginning October 10, 2017, one million students will experience Exploding Dots, a popular topic developed by Global Math Project founding team member James Tanton. During Global Math Week itself, teachers and other math leaders are asked to commit to spending the equivalent of one class period on Exploding Dots and to share their students’ experience with the Global Math Project community through social media. Teachers can choose a low-technology presentation format by using downloadable pdf lesson plans as a guide. Alternatively, they can opt for a technology-intensive experience, developed by the Canadian education technology company Scolab, which will consist of a collection of visually appealing “islands” representing Exploding Dots topics. For those who wish to delve deeper, additional materials will be freely available on the Global Math Project website to support further exploration of place value, arithmetic algorithms, negative numbers, alternative bases, polynomials, formal infinite series, and more.
How Can I Get Involved?
Consider becoming involved in some or all of the following ways:
See mathematics like you've never seen it before and take part in a global conversation. Get started at http://gmw.globalmathproject.org.
In the Spring 2017 issue of the Ohio Journal of School Mathematics, Anne Berger discusses an exciting way to engage our students in conversation and collaboration while developing their procedural fluency and mental math skills. In the article, "Using Number Talks to Build Procedural Fluency through Conceptual Understanding," Berger describes her use of number talks to provide brief, daily opportunities for students to discuss, connect and develop their strategies for solving problems. Here's a video of children engaged in a number talk with a guru of the technique, Jo Boaler. Here's a link to the Spring 2017 Issue of the Ohio Journal of School Mathematics.
OCTM is an affiliate of the National Council of Teachers of Mathematics.