Five Things I Learned at PCMI: Wednesday 7/8/15

1. There is a deep connection between transformation T(a+bi)=(a+bi)^2 and all sorts of algebraic geometry such as transformations, polar coordinates, Pythagorean triples, and hyperbolas.

2. There is an online companion to the text Principles to Actions (2014), called The Principles to Actions Professional Learning Toolkit. It consists of a modules containing videos, tasks, and student work designed to demonstrate the effective teaching practices. Fred Dillon, a math teacher and Teacher2Teacher Associate for the Math Forum, introduced it this morning during the "5 minute shorts". Take a look: http://www.nctm.org/Conferences-and-Professional-Development/Principles-to-Actions-Professional-Learning-Toolkit/

3. The "convince yourself, convince a friend, convince a skeptic" technique from Jo Boaler and Cathy Humphreys' book Connecting Mathematical Ideas (2005) is an excellent way to get students thinking about the reasonableness of an answer and practice delivering and analyzing arguments.

4. When asking a hinge question (aka quick check), it is important that the task be quick for students to complete (under 2 mins), easily gradable by the teacher (30 seconds), and crafted in such a way that students can't arrive at the correct answer accidentally. While multiple choice questions are a poor choice, "select all that apply" can be very useful.

5. The best people to be around are those who take their losses in stride. Today, we went tubing down the Provo River and what was advertised as a "lazy river" was full of rapids. We lost one pair of shoes, three water bottles (found one), two beers (found both), and one tube (found 15 minutes later). And yet, it was one of the most exciting activities that I have done all week - all thanks to the people I was with.

6. (Extra Credit) The best place to picnic is across from a grocery store. When faced with no dinner plans and no kitchen to cook in, we had to come up with an idea quickly. Fortunately, there is a small park complete with charcoal grills across the parking lot from the local grocery store.

Five Things I Learned at PCMI: Tuesday 7/7/15

1. I can construct an ellipse using only compass and straightedge tools in Geogebra. It took much longer to figure it out than I originally anticipated.

2. There is an app called "Plickers" that can be used as a low-cost quick response system that is similar to clickers. Each student gets a paper with a QR-like code on it and holds it in the air a certain way to submit their answer. The teacher then uses their phone or ipad to scan the room and collect the data. I haven't used it yet, but it sounds promising. Check it out: https://plickers.com/

3. There are many ways to check for student understanding, but the most important part is that it is planned and purposeful. One method is a variation whiteboard speed dating where students each receive their own problem, become an expert in that problem, then exchange problems with a partner. In case there is an issue, they can just ask their partner, the expert, for help. Other methods include two truths and a lie, or simple problems that target one conceptual understanding. Here's the original white board speed dating: https://kellyoshea.wordpress.com/2012/01/22/whiteboard-speed-dating/

4. If you're a farmer that has a scale with two sides, you can maximize the amount of corn you can weigh by using only weights that follow the powers of three: 1, 3, 9, 27, etc. This came from a solution to a problem posed during the "Pizza and Problem Solving" session. Oh, and I have the tendency to eat too much pizza.

5. Line dancing can be fun.

Five Things I Learned at PCMI: Monday 7/6/15

1. There are many ways to explain why the slopes of perpendicular lines are opposite reciprocals. One way is to recognize that perpendicular lines are just a 90 degree rotation away from each other. Since the algebraic function for a 90 degree rotation is R(x,y) = (-y,x), the slopes of the two perpendicular lines will be y/x and -x/y. Of course this is only a rotation about the origin, but a rotation around another point would just add a constant to our function, and would only effect the y-intercepts of our lines.

2. The Dekker & Querelle method (2002) can help students analyze worthwhile tasks. After students have completed a short task, they should analyze a number of sample solutions (more than just a few). Then, either ask them to pick one incorrect and one correct to discuss in depth, or use some sort of critiquing rubric (I have a 4 point one). This technique works well after working on open middle problems. Speaking of open middle: http://www.openmiddle.com/

3. It is difficult to caramelize a large amount of onions in a high-walled stock pot. If there is too much moisture and not enough surface area, then the onions just end up steaming themselves.

4. There is a high level of variance in standards based grading across schools. Some schools have standards set for teachers by administrators, others ask teachers to come up with their own. There is also a debate as to how many standards you should have, if they should follow through the entire year or just one unit, and if practice standards should also be included. The most interesting one was a hybrid SBG system where 50% of the final grade is given to content standards, 30% is given to practice standards, and 20% is given to scholarship standards (homework, participation, etc). I also learned about Easy Grade Pro, a program for standards based grading. It’s pretty neat, although I’m not sure how it would fit into our existing grading structure. Take a look here.

5. I can now create an origami augmented dodecahedron.

6. (Extra Credit) I learned some math chants:

Rah rah sis-boom-bah!

2, 4, 6, 8, Even numbers are really great!

1, 2, 4, 8, We can exponentiate!

Ski slope: lots of fun! Math slope: rise over run!

Our love for America is IRRATIONAL!

Count off PCMI! 1! 1! 2! 3! 5! 8! 13!

M-A-T-H M-A-T-H! MATH!

Pictures below are from the 4th of July weekend: the Park City parade, lunch at The Bridge, dinner at the Wasatch Brew Pub.

Five Things I Learned at PCMI: Friday 7/3/15

1. Geometric transformations and notions of symmetry can be used to both visualize and prove properties of many figures including: traversals, isosceles triangles, various types of quadrilaterals, and more.

2. Validate others always. Today I had the experience of being in a working group whose ideas were invalidated shortly after they were presented to the class, and suddenly all the math anxiety of my high school career came flooding back to me.

3. While post-it note poster paper is great for groups to write on and present with, it’s expensive and often goes to waste if a student makes a mistake. Instead, I am going to make giant whiteboards, 2 ft by 3 ft. I envision putting them on the chalk sill or window sills and letting students use them whenever students ordinarily need poster paper. Students will be able to erase mistakes easily, and the boards will serve as a good student work decoration when not in use.

4. In order to visualize 4D objects, it is useful to first think about how can represent 3D with a 2D projection. Thus, we can represent a 4D object with a 3D projection, and then print off the projection with a 3D printer/ Of course, some projections are better representations than others and it’s important to study those projections. Also, 3D printing is just generally pretty cool.

5. Miegakure is a platform game where you navigate a four-dimensional world to perform miraculous feats and solve puzzles. It's not out yet, but it looks interesting. See it here: http://miegakure.com/

6. (Friday Bonus) Give It Away by the Red Hot Chili Peppers is not a good karaoke song. Sugar We're Goin Down by Fall Out Boy is also not a good karaoke song. But, I Want It That Way by the Backstreet Boys is an excellent karaoke song - especially when it comes with an open invite to all male math teachers.

Five Things I Learned at PCMI: Thursday 7/2/15

1. Jo Boaler at Stanford University a website called youcubed.com that provides resources and online courses for teachers, parents, and students with a focus on growth mindset in the mathematics classroom. The site focuses on practices that promote depth over speed, equitable group work, and math efficacy. There is also a free set of videos and lessons for the first week of school that are designed to promote growth mindset, introduce the 8 mathematical practices, and set the tone for the year. Take a look here: http://youcubed.org/

2. There are multiple ways to complete congruence proofs with transformations. You could use transformations of the entire plane to map one figure onto another, starting with one point. Next, assume that the other points do not line up, and use properties of your figure and of transformations to explain why they do. Of course, this proof can take many forms, but there is another method too. You can express the properties of a figure by using circles to preserve distance and rays to preserve angles to create a locus of points where the vertices of the figure could exist. If there is only one set of vertices, then the figures are congruent.

3. Although there is some debate, the term "constant of proportionally" should generally be reserved for relationships between a domain and a range, and has the form y = k x. On the other hand, "scale factor" refers to a relationship between two geometric figures. It should normally be presented as one number, not a ratio. However, it can be useful to use ratios to compare lengths within one geometric figure. For example, we could say that two triangles with sides 4:6:8 and 6:9:12 have a 3/2 or 1.5 scale factor. You can thank Bill McCullen for that.

4. I can construct the incircle of a triangle using Euclid: The Game, level 15. And yes, I already knew the properties of an incircle and incenters, but the construction takes an extra trick.

5. The mountains in Utah are amazing. We took a 6 mile hike that turned into an 8 mile hike to Middle Mountain in Wasatch Mountain State Park. The trail was heavily washed out and it was a bit steeper than anticipated, but the view from the top and the experience was worth it.

Five Things I Learned at PCMI: Wednesday 7/1/15

1. Besides being a champion pinball player, Bowen Kerins is a project developer for the CME Project (part of EDC’s Center for Mathematics Education) and has had a hand in writing their high school textbook series. The most striking feature of these texts is the way that the subtle scaffolding from the morning problem sets is mirrored in each section. While I’m not endorsing Bowen’s book wholesale, I know that these scaffolds can create strong habits of mind in students. Check it out here: http://cmeproject.edc.org/

2. Even if you don’t know where you’re going, you’ll probably get somewhere if you just start. While this may apply to many of the morning math sessions, today this lesson is about hiking. We hiked up Park City Mountain, but instead of finding the correct trailhead, we just started hiking up the steepest ski trail. Eventually we came across a park ranger who directed us to the trail, but we never would have gotten to the top if we didn’t start at the bottom.

3. Utah has the most restrictive yet nuanced alcohol laws of any state.

4. I can find the center of a circle using Euclid: The Game, level 13 in 4 moves. However, the game says that the minimum possible number is 3 moves. Hmm...

5. I have learned a new vocabulary of math puns. PCMI participates in the Park City Fourth of July Parade, and we had a brainstorming session. When you get a bunch of math teachers, grad students, and professors in room and ask them to think of math puns that relate mathematics to American independence, you know you’re in for a good time.

Five Things I Learned at PCMI: Tuesday 6/30/15

1. There is a paired down version of Geogebra called "Euclid the Game" in which players construct a given figure with limited tools (you start with points, segments, rays, and circles). Each figure you construct can be used as a tool in later levels. For example, once you make an equilateral triangle, a “draw equilateral triangle” button appears. It’s a great way to learn Geogebra and understand the Euclidian notion of constructions. Try it here: http://euclidthegame.com/

2. To quote Bill McCallum, “The high school standards were written with the assumption of four years. Last time I checked, there were four years in high school, however apparently there are many who think there are not”

3. Again Bill McCallum, “I don’t think that aggressive testing and high standards can exist at the same time, at least not this way. Something has to give, and I think Common Core has unintentionally started a conversation about assessment… However, I think that PARCC and SBAC are one of our greatest hopes for changing the testing landscape. These people are trying hard, and although the test bank is small, we’re moving in the right direction… Of course the standards can be seen as difficult, but isn’t the idea to set high standards? Many of the issues we face are unfortunately policy issues… But, who owns the Standards anyway?”

4. Although not immensely publicised, there is a Common Core progressions document for Geometry by Hung-Hsi Wu of University of California, Berkeley. Here it is: https://math.berkeley.edu/~wu/Progressions_Geometry.pdf

5. I have met the local wildlife, including: four goats (one pregnant), two horses, a family of badgers, a few magpies, and five pocket gophers.