Five Things I learned at TMC15: Saturday 7/25/15

1. Vertical non-permanent surfaces (if you want to sound smart) or wall-mounted white boards (if you want to be understood) can increase student engagement and discourse in all the right ways. Peter Liljedahl found that students using non-permanent writing surfaces such as chalk boards or whiteboards tended to start writing faster and were more eager to start, presumably because they could edit their mistakes. Liljedahl also found that vertical surfaces such as chart paper or mounted whiteboards tended to foster participation and mobility, probably because students could more easily see each other’s work.

You can expect to see me taking a trip to home depot to buy some shower board in the next few weeks. Many thanks to Alex Overwijk for the ideas.

2. Visibly random groups can also be good for engagement and discourse. The idea is that students should be in random groups each day, and they should know the groups are random (pull cards out of a hat or something similar). This promotes cross-pollination of ideas, and sends the message that students should be ready to work with anyone. And if it’s a bad grouping, then it’s only for one day. Again, thanks to Peter Liljedahl for the research and Alex Overwijk for the presentation.

UPDATE: As I write this, there is a twitter debate going on about whether purposeful heterogeneous groups can be more useful to distribute students along gender, ethnicity, ability, etc. It seems the consensus is that “balanced” groups can be good sometimes, but prevents similar students from ever working together.

3. Today, Matt Vaudrey introduced the idea of using musical cues to direct student actions. Simply put, the teacher plays a specific song while students are completing a specific action, such as a “getting calculators” song, a “do-now” song, or a “packing up” song. Songs should play for a pre-determined amount of time, and shouldn’t be stopped early or replayed. The songs now serve as a cue for students (instead of teacher nagging) and should hopefully create a sense of internal motivation. It’s such a great idea, and I cannot do it justice with this post. Read about it from Matt here.

4. Are you teaching a math course for the first time, and don’t know the common student misconceptions? Matt Baker has created a google spreadsheet called First Like Third where teachers can note common student misunderstandings as well as what to say and how to correct them. The objective is to make the first year of teaching more like the third year of teaching.

5. Kahoot is a game based learning platform where users answer timed, multiple choice questions against classmates (teacher-made Kahoots) or students from around the world (public Kahoots). At the end of each round, students are shown a leaderboard with the top four players. Students do not need to sign in to play, and any computer or device with internet can be used. I wonder what the pros and cons of Kahoot are over other online quizzing or formative assessment websites and apps. Kahoot looks more like a game, but is there more to it? Thanks to Julie Reulbach for this one.

Five Things I learned at TMC15: Friday 7/24/15

1. John Stevens has created a Math Twitter Blog-o-Sphere Search Engine; a Google search that is restricted to blogs and other sites on the MTBoS. It is great for finding lesson materials on a specific topic, or reading reflections on a specific mathematical idea. Robert Kaplinsky also has a Problem Based Learning Search Engine.

2. Which One Doesn't Belong is a website inspired by Christopher Danielson and created by Mary Bourassa. It features sets of four figures and simply asks: which one doesn't belong. Interestingly, each of the four figures might not belong depending on the context and rational provided. A seemingly simple question can introduce ambiguity and spark rich discussion from kindergarten through calculus.

3. In the past, I have seen very rigid interpretations of standards based grading that either (1) do not allow for the grading of homework, classwork, and larger practice-based tasks, or (2) are seemingly incompatible with a traditional A-B-C-D-F grading scale. Today I found a much more open interpretation that advocates: if it’s important for students to know and do, then make it a standard. That means that there could be a set of “scholarship” standards for good student skills such as completing homework and projects in a timely manner. There could also be a set of “problem solving” standards that highlight the 8 mathematical practices. Of course, these standards would account for a much smaller percentage of a student’s overall grade, but it is possible to incorporate them within a more open view of SBG. Thanks to Anna Hester, David Petersen, and Lisa Soltani.

4. A boring lesson in person is still a boring lesson on video. While creating flipped video lessons for students introduces the ability to predict, rewind, and re-watch, we must be careful not to push mounds of procedural blather onto students. Andrew Stadel suggests creating videos that are quick and to the point with off screen preparation. He also advocates for videos with errors; complete a problem incorrectly and ask your students to find the error. Princess Choi suggests that students create their own videos as a review strategy. Whatever the method, don’t make your videos boring!

5. Pump up your class. Give everyone a high five when they come in the room. Give neuron stickers when they help each other. Let students hit the gong in your class. Create varsity math t-shirts. Build rapport. Find something you love and do more of it. Be fast, fair, friendly, firm, and funny. And remember, you can’t learn from someone you don’t like.

6. Five miles seems shorter when running with other math teachers.

Five Things I learned at TMC15: Thursday 7/23/15

1. Don’t give your students mathematical definitions. Instead, have groups of three or four students write a definition for a well-known mathematical idea or geometric figure on their own. Then, have each group pass their definition to the next group. Groups should now try to “break” their definitions by providing a counter example. Once they find a flaw in the definition, they refine the definition and continue passing the paper.

2. Here’s an idea for exit tickets. Make a grid with 3 by 3 inch boxes and write each of your student’s names in a different box. Place your grid on the wall near the door. Have your students complete their exit tickets on a sticky note, and stick them in their box on the grid as they leave class. This system makes it easy for students to hand in their exit slip and even easier for the teacher to grade. It is best used for reflective exit slips, as students could cheat on a skills-based or assessment-style exit slip.

3. Mathmistakes.org is a site is about compiling, analyzing and discussing the mathematical errors that students make. It is edited by Michael Pershan, a middle school and high school math teacher from NYC.

4. Manifold is an origami style game by The Incredible Company. It challenges players to take a small sheet of paper with scattered black and white boxes and fold it such that all the white boxes appear on one side and all the black boxes appear on the other. While it isn’t tied to any topic or standard, this is a fun activity in spatial reasoning that will either frustrate or leave you addicted.

5. Another game about spatial reasoning that will either frustrate or leave you addicted. Okay? is a simple iPad and iPhone app that challenges users to bounce a ball off a series of blocks in one shot. Best of all, you pay what you want for this app.

6. (Extra Credit) Dueling Piano bars are the best.

Five Things I Learned at PCMI: Friday 7/17/15

1. When you set goals, set them in stone. The RoP staff asked us to set goals for the upcoming school year as a result of our experiences at PCMI. They asked us to consider possible solutions to challenges we might face along the way. And then we made it official; we made and submitted a video of us explaining our goals to be reviewed later in the year. Although it is a bit awkward to create a video of yourself, I doubt I will forget my goals any time soon.

2. All of the afternoon working groups presented today. Our group presented Geometry Transformed, a webinar style professional development to help teachers transition to Common Core geometry, which relies heavily on transformations. There were also a number of other professional development resources presented, and I will be exploring them all very soon. Most notable were the presentations about growth mindset, the social dynamics of math discussion, writing in math, and a problem set from Ben.

3. Bowen Kerns demonstrated the math games at http://solveme.edc.org/. There is a Mobile puzzle that develops intuition for solving equations, a “who am I” game for number sense, and Ken Ken style puzzle that imposes higher order thinking on the basic operations. Each game also gives students the ability to make their own puzzle.

4. I learned how to transform an algebraic function if the rule is in this form: (x, y) -> (f(x), g(y)). I also learned about how permutation groups and symmetric groups relate to reflections and rotations of regular polygons. There are so many connections between geometry and algebra; you would think it was the focus of the conference. Wait a second…

5. The MAA held a "debate" over the relative merits of the transcendental numbers, pi and e, at the Family Weekend for Williams College in 2005. The debate is largely humorous, but does hide some important mathematical ideas. Thomas Garrity, who served as e's champion, was one of the faculty members at PCMI.

Part 1: https://youtu.be/whpAX30vjoE
Part 2: https://youtu.be/i1hYL0ccqm0
Part 3: https://youtu.be/DytqTMmi_os
Part 4: https://youtu.be/9CCBqinb744
Part 5: https://youtu.be/Xp_znceoeWo

6. (Extra Credit) The cheese and ice cream from Heber Valley Milk & Heber Valley Artisan Cheese is delicious. Two vans full of teachers drove down from the Zermatt to eat some ice cream and see the cows. I split the south west cheese curds with matt, and most of us walked back to the resort.

7. (Extra Credit) The people I have met at PCMI are some of the most amazing, thoughtful, dedicated, and human educators that I have ever met. Darryl and Bowen helped me remember the highs and lows of being a student again. The entire Reflecting on Practice staff facilitated sharing, discussion, and reflection about pedagogy. In my afternoon working session, Gabe and Jim gave me more insight into transformational geometry than I ever had hoped to have. Every single one of the participants at PCMI 15 was willing and eager to share their ideas both in and out of sessions. The entire three weeks served as an open forum for refining old ideas, gaining new ideas, and plenty of time in the pool.

On the last day, most of the TLP got all of the leftovers together for one last meal, one last volleyball match, one last hot tub, and one last late night card game. I hope to see you all again soon.

Five Things I Learned at PCMI: Thursday 7/16/15

1. I learned a great deal from Dylan Williams. See the full post here: http://mrjanesmath.blogspot.com/2015/07/dylan-williams-at-pcmi.html

2. Chris introduced me to a question structure for inquiry. After students have become familiar with a problem, scenario, or task, they should be asked to generate at least one or two questions in each of the following categories. Level 1 questions are questions that they know the answer to, level 2 questions are ones that they don’t know the answer to but have an idea of how to approach it, and level 3 questions are ones that they don’t know how to approach. All three levels are important. It might seem trivial to write down questions that students already to know the answer to, but it is important to parse out what information is known and what is unknown. Besides, it also helps validate student thinking. Chris also noted that throughout the activity, questions should be moved from one level to another as they are answered, or more insight has been achieved.

3. We discussed “My Favorite No” today. I’ve seen this before, but I want to incorporate this in my class. Rather than boring you, let’s go straight to the source: https://www.teachingchannel.org/videos/class-warm-up-routine

4. We have also discussed 5-Practice Routines for the past two weeks, but now is a good time to bring them up. Again, the source a la Christopher Danielson: https://christopherdanielson.wordpress.com/2011/08/26/five-practices/

5. Marty talked to me about The SimCalc Project at UMass Dartmouth. The objective of the project was to create software that allows students to collect data on computer simulations relevant to algebra and calculus classes. The software and curricula are now available online for purchase: http://www.kaputcenter.umassd.edu/products/. However, Marty cautioned that other programs that used more hands-on simulations and data collection had more success. I’m not teaching calculus next year, but I’ll pass this one on to my colleague who is.

6. (Extra Credit) Beth A Herbel Eisenmann is a mathematics education researcher at Michigan State University who focuses on promoting student discourse. Her book Promoting Purposeful Discourse was recommended to be by a colleague at PCMI.

Dylan Williams at PCMI

Dylan Williams spoke to the PCMI Teacher Leadership Program today via webinar. He answered questions, one of which was mine. Here are some major take aways and quotes:

- There are only two recommendations that educational psychologists can agree on. First, mass practice is less effective than distributed practice. That is to say, we should integrate our topics such that students have many opportunities to practice a particular skill. This notion is closely aligned with Ebbinghaus’s Forgetting Curve. Distributed practice also helps with truant students because no topic is addressed in only one day, making it easier for the absent student to catch back up.

- Second, frequent testing without giving grades helps students because it helps them practice retrieval. Students often practice memory storage when they hear a lecture, experience an activity, or study for a test, but they do not regularly practice memory retrieval. It seems that testing, when used the right way, can significantly benefit learning.

- Having students predict an answer before learning how to complete a problem, even when they have no obvious entry point, has been shown to increase student learning. The thought is that by predicting an answer, they cognitively struggling. This, in turn, leads to greater learning (more on that later). This is connected to test corrections. Even if students don’t perform well on an assessment, the act of trying and correcting their work aids learning.

- Higher levels of cognitive struggle lead to higher levels of learning. This can manifest itself simply: a student trying to solve a novel problem unaided will learn more than a student completing a procedural problem whose solution is already known. However, other types of cognitive struggle also impact learning. In a psychological study, students who were given a smudged print out of a story were better able to recall the story than students who were given a clean copy because they struggled to read the smudged copy.

- There is no psychological evidence that catering to individual learning styles helps students. In fact, it may lead to lower learning since it would lower the cognitive demand and struggle. Yet, this is not permission to ignore learning styles altogether. Working in a learning style that is not your own is tiring, and so teachers should frequently vary the style they use to ensure equitable learning for all.

- In both Japan and the United States, lessons often start with the teacher demonstrating how to solve a certain problem. In the United States, this is often followed by procedural practice. However in Japan students are challenged to find other solution methods. This leads to natural differentiation because higher achieving students can be prompted to find more complex solutions while the teacher scaffolds more basic solutions for students who are confused.

- The advantages of formative over summative, or comments over grades, is not absolute. The only thing that matters about feedback is what students do with it. If a student in AP Calculus would learn the most from seeing a number grade out of five on their practice test, then they should receive a numerical grade. If a student would benefit the most from teacher comments, then that is what should be done. Moreover, any assessment can be used as a formative or a summative depending on how you use it. Groups of four students could individually take a standardized test, then all pool their ideas as a group to create a fifth “best version” of the test. After finishing, the teacher would find any common mistakes and have groups present their solutions to each other. In this way, a traditionally summative assessment has become formative.

- The goal of feedback is either to give students information about where they are at currently, or how they can improve. However, it is disadvantageous to give both at the same time because students tend to focus on one type and ignore the other.

This talk gave me a great deal to think about, but I’ve tried to come up with a few succinct action steps to improve my teaching. These probably sound familiar:

Integrate the topics in my curriculum more, and don’t hesitate to expect students to remember essential topics from previous units.

Let students predict answers to novel problems before they start them, and allow them to struggle within a well-thought scaffold. 

Use more quick formative assessments. Even if I don’t grade them myself, the act of taking the assessment is important, and it can give students the opportunity to reflect on their peers and on themselves.

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

1. G(Math) is an add-on for Google Docs that allows you to write equations using an equation editor or LaTEX. Although Google Docs has an equation editor, it is rudimentary. For example, G(Math) also allows you to create graphs from a specified function, plot points, and has a “speech to equation” function. Here is the link: https://chrome.google.com/webstore/detail/gmath/hhaencnpmaacoojogjkobikbmkhikjmm?utm_source=permalink

2. The “Rational Tangle Dance” is a simple, yet deep, activity in knot theory that is a favorite amongst math circles and can be adapted down to high, middle, or even elementary school. In the activity, four people hold two ropes that connect them as pairs. The group moves in a series of twists and rotations, eventually creating a complex knot between them. It is the objective of the group to better understand the dynamics of the system, and eventually undo any knot that they have created. It was first introduced by John Conway, but explored by many mathematicians. Here is a link to a video: http://www.mathcircles.org/node/798, instructions: http://www.geometer.org/mathcircles/tangle.pdf, and a paper: http://www.mathteacherscircle.org/assets/session-materials/JTantonRationalTangles.pdf

3. Chris introduced us to The Art of Problem Solving by Richard Rusczyk. The text is designed to prepare students for mathematical competitions, but is highly applicable to the general high school setting. Its goal is to teaching concepts and problem solving methods not traditionally taught in school. One of the main concepts is to take an old problem and list all of its attributes. Then, change or remove an attribute to see what effect it has on the solution. Rusczyk also publishes a set of middle and high school textbooks modeled after his philosophy. http://www.artofproblemsolving.com/

4. Inside Mathematics is a website that houses a number of model lessons, videos of teachers, performance assessment tasks, and classroom resources that grew out of the Noyce Foundation’s Silicon Valley Mathematics Initiative. Cathy Humphries (from “convince yourself, convince a friend, convince a skeptic” fame) is one of their teacher mentors. Take a look: http://www.insidemathematics.org/

6. (Extra Credit) A dilation by a complex number is equivalent to a rotation. Huh…

7. (Extra Credit) Olympic Park, near Salt Lake City, is an active training facility for ski jump and freestyle aerial as well as an outdoor adventure park. When there is no snow, the ski jumpers land on wet turf and the freestyle aerial skiers land in a pool with jets pushing them back up. Besides watching the athletes practice, we completed couple of ropes courses, raced down the alpine slide twice, and were first in line for the extreme zip line. The perfect day finished with a trip to a High West Saloon in Park City.

Five Things I Learned at PCMI: Tuesday 7/14/15

1. James Tanton has a video and series of videos and lessons called “exploding dots”. They start with a very intuitive look at number systems and trace through arithmetic, polynomials, decimals, sequences & series, irrational numbers, negative bases, and more. As another teacher at PCMI remarked, “Every time I watch, I am amazed at how far these concepts reach. If our kids could see this from when they were little, think of what they would understand.” This one was found by Matt. Watch Tanton here: http://gdaymath.com/courses/exploding-dots/

2. While you’re working with Tanton’s exploding dots and number systems, you might want to get a better sense of scale for what 10^9 or 10^-5 really mean. Check out The Scale of the Universe, and interactive applet that lets you zoom in and out and see objects at different scales. Thanks to Chris for this one: http://scaleofuniverse.com/

3. Ilana Horn is a professor at Vanderbilt University who wrote the book Strength in Numbers: Collaborative Learning in Secondary Mathematics. In it, she outlines how to effectively use group work to create a learning environment in the secondary mathematics classroom. She goes on to explain how students experience learning mathematics in collaborative settings, and how teachers can develop tasks, concepts, strategies, and tools that create successful group work. She has some interesting views on group work, some I agree with, and some I would modify. Here is a link to her blog: https://teachingmathculture.wordpress.com/about/

4. We had a discussion about group quizzes today. In her book Strength in Numbers, Ilana Horn suggests that group quizzes may be used as a review for an upcoming test. Groups of students are asked to complete two to four big problems without the teachers help or any help from outside the group. Upon completion, the teacher collects and grades just one paper from the group (pg 59). While this practice has its merits, some teachers at PCMI felt that students would tend to let the highest achieving student do most of the work, and then copy off of his or her paper before they were collected. Some solutions to this issue include having students work alone first, grading every students paper, having a higher level of supervision during the work, and asking students to grade each other participation. However, the best solution is to give the group 15 minutes to work together right from the start. They can use this time to ask questions, formulate ideas, or attempt the problem. After 15 minutes, they receive a small grade for their collaboration, and then must finish the task individually. I like this solution because it promotes collaboration and questioning, yet still places the final burden on the individual student.

5. Joe introduced the group to an intuitive way of measuring irrational numbers. I believe it was presented at an NCTM conference, and has been reposted in a few different places. Yet, it’s a good activity that deserves to be passed around. Here is one incarnation: https://thescamdog.wordpress.com/2011/05/31/radical-ruler/

6. (Extra Credit) The International Congress on Mathematical Education (ICME) is being held in Germany next year, and they’re looking for representatives from PCMI to go. Hmm…

7. (Extra Credit) I presented both a 10-minute share and a 5-minute short today at PCMI. The 10 minute share focused on mathematics and music. Although there were some technical issues, the presentation went pretty well. The 5 minute short was with Matt, and we explained how to have students create a survey, and then use those results to create game-show style questions. We gave a survey to the PCMI teachers and created some game show questions – the results (without the statistics) are below.

What is the distribution of number of years teachers have been in education? Normal with a positive skew.

Who likes the food at lunch more: tea drinkers, coffee drinkers, or both the same? Coffee drinkers.

What percentage of PCMI teachers are from an east coast state? 56% ± 15% teachers.

What do teachers enjoy more: Darryl and Bowen's comments or the comics before RoP? Darryl and Bowen's comments.

Which had a lower standard deviation: how much teachers enjoy afternoon cookies, or how much teachers enjoy goats? The cookies (although both have high standard deviations, and are very divisive topics).

Who likes goats more, teachers with dietary restrictions (vegetarian, vegan, gluten intolerance, etc) or those without? Everyone loves goats the same (difference was not statically significant).

Is there a correlation between teacher’s enjoyment of breakfast food and lunch food? Not as much as you’d think, r = 0.45.

Five Things I Learned at PCMI: Monday 7/13/15

1. I currently display question stems (i.e. talk moves) on a poster at the front of the room, but as Jennifer’s morning presentation showed, putting them on small laminated index cards is even better. In her class, each student gets a stack of question stems bound together with a ring. Green cards are for probing questions or new ideas, red cards are for questions or disagreements. Jennifer noted that her students liked using them so much that they started to raise their hand while holding the question stem they were about to use.

2. Jo Boaler has a document on youcubed that outlines seven classroom norms for growth mindset and how to effectively implement them. In our Reflecting On Practice session, we tried to recreate some of her norms with a bit more, um... color. Read the original here: http://www.youcubed.org/wp-content/uploads/Positive-Classroom-Norms2.pdf

3. Alice told me about a recent episode of This American titled Episode 550: Three Miles. The show is about a program that brings together kids from two schools. One school is public and in the country’s poorest congressional district. The other is private and costs $43,000/year. They are three miles apart. The hope is that kids connect, but some of the public school kids just can’t get over the divide. The episode focuses on what happens when you get to see the other side and it looks a lot better. Listen to it here: http://www.thisamericanlife.org/radio-archives/episode/550/three-miles

4. The TI-Nspire has a number of interactive applets that provide great for discovery lessons throughout the middle and high school math curricula. I got the chance to use them today, and I was impressed by the number and versatility of the calculators. However, I’m not convinced that these applets don’t exist for free online. Since all of my students have laptops, it might not be worth the investment, especially considering that the cost of a cheap Chromebook is just about the same as a TI-Nspire plus a textbook. The jury is still out for me, but you can form your own opinions here: https://education.ti.com/en/us/activities-home

5. PCMI will be hosting a mini PCMI weekend session somewhere in the Boston area. Details still need to be worked out, but it sounds like a good time!

Five Things I Learned at PCMI: Friday 7/10/15

1. When sand, snow, or any other loose material is piled on top of a polygon, the pile will have ridges equidistant from the nearest sides (also known as angle bisectors), and peaks at the center of an inscribed circle (also known as the incenter).

2. Written feedback is often difficult to write well and not read by students. Carol Dweck’s famous study on growth mindset suggests that we should not score or evaluate student’s work, but instead give specific feedback that relates to their effort or methodology on a particular problem. Such feedback should also move a student to think or reconsider their error rather than giving hints or being evaluative. It may even be useful to give feedback on correct answers to extend student thinking. Research has shown that these techniques increase desire to attempt difficult tasks, allow students o recover more quickly after setbacks, and increase academic achievement in the long run. Of course, the source: http://mindsetonline.com/

3. James Tanton has a series of eight books called Thinking Mathematics that run from basic arithmetic through AP Calculus and AP Statistics. My roommate Matt describes it as, “A bit long, but very easily accessible. He provides many examples and develops concepts intuitively before putting a name and a formal definition on them.” He also publishes curriculum letters monthly. If your interested, take a look at his philosophy: http://www.jamestanton.com/

4. Collaborative Mathematics is a website by Jason Ermer that posts mathematical video challenges and encourages users to discuss their successes and failures both in chat form and in video response form. These are great problems for teachers and students alike. As Ermer puts it, "one of the goals of Collaborative Mathematics is to help cultivate a productive attitude toward challenging problems: one of creativity, resourcefulness, self-confidence, and perseverance." Take a look here: http://www.collaborativemathematics.org/

5. Just three sets of one minute planks before lunch will get you ripped abs in no time! Or at least, that’s what Samantha and Alice told me.

6. (Extra Credit) The Uinta Mountains in North East Utah are beautiful. I went backpacking for two nights with Matt; pictures are below.

Five Things I Learned in the High Uinta Mountain Range

1. Ground mats are essential. You might not think that a thin strip of foam between your sleeping bag and the ground would be a big deal, but without it the ground saps the heat out of you.

2. Starting a fire against a rock is a blessing and a curse. It often provides a good wind break and reflects the heat back at you. Unfortunately, it forces the smoke to blow right at you if the wind is uncooperative.

3. You can drink out of mountain springs without boiling or purification.

4. There is still snow on the ground in July, but only if you're at 11,000 ft.

5. The small clearing on the far side of Amethyst Lake in the High Uinta Mountains is one of the most beautiful camp sites I have ever seen.

Five Things I Learned at PCMI: Thursday 7/9/15

1. I discovered how to use patty paper and reflections to draw the conic sections, and then presented my patty paper at the end of today’s Morning Math session.

2. Vision is important. For the past few weeks, we have been working on our online common core geometry course with varying degrees of structure. Some days, we would know exactly what to do, other days it was a struggle to stay together. However, after a bit of a discussion today, I think we are on the right track for the remainder of PCMI.

3. Wendy gave a presentation on her end of year assignment for her geometry class. She created a grid with all of the units they had covered for the year on the top of the grid and various levels of Bloom's Taxonomy on the side. The center was filled with various activities, and students had to pick one from each column and each row. In this way, students were exposed to a wide variety of topics and modalities.

4. I learned strategies to make students into teachers. Ben gave a presentation on his initiative to turn his class into cooperative group of student-teachers. He created videos to model what effective tutoring looks like and discussed common pitfalls such as steamrolling. He explained how he sorted the class into student-teachers and not-yet-student-teachers via exit slips from the day before. Ben also shared some of his tactics for when students struggle with being the teacher, such as only allowing student-teachers to ask questions or collaborating as a class to come up with question stems.

5. I’m going to try using giant 2 ft by 3 ft whiteboards in class. Earlier in the week, I had a discussion about how writing either horizontally (as part of a group) or vertically (teaching a group) can affect learning. In a separate talk, I thought about how important it is for students to reflect and revise their work - especially when working in groups. Lastly, I realized how expensive sticky-note poster paper is. Taken together, it makes sense to have students write and present with over sized whiteboards rather than disposable poster paper. They’re reusable, revisable, can be placed on the chalk sill or window sill, and make great student work displays.

6. (Extra Credit) In projective space, parallel lines intersect at infinity. This is because projective space can be represented by all complex numbers (x,y) such that (hx, hy) is equivalent to (x,y) for any non-zero value of h. In other words, all pairs of numbers that are proportional to each other are equivalent. We could also say that the pair (x,y) is equivalent to another pair when the ratio y/x is equivalent to another ratio. But, what happens when x=0? Normally we can’t divide by zero, but in this case it simply stands for the pair (y, 0) which we label as the point at infinity. Projective space can also be represented as the set of all lines through the origin (where the vertical line represents infinity) or as a circle (where the north pole represents infinity). Thanks to Dagan Karp from Harvey Mudd for his lecture, both in the pool and at the chalk board.