Five Things I Learned About International Geometry at ICME-13

Throughout the Thirteenth International Congress on Mathematical Education (ICME-13), I had the opportunity to listen and reflect on a number of competing geometry education paradigms. Despite the wide range of philosophies, curricula, and pedagogical strategies, I found that almost all perspectives can be placed on a continuum between a traditional “axiomatic-style” and a more reformed “discovery-style”, with some room for outliers. During a survey of current geometry curricula, Nathalie Sinclair (Canada) noted that, “Proof, and more generally geometry curricula, are bound to socio-cultural norms.” Interestingly, the style of geometry that each country favored tended to be a reflection of their culture. In this overview, I will explore where various countries fall on the continuum, and how their views are ultimately intertwined with their culture.

1. Eastern Europe: Eastern European countries (e.g. Russia, Hungary, and Romania) typically have a very axiomatic view of geometry. In these countries, most geometry curricula start with an emphasis on Euclidian axioms and proceeds to develop other notions from this standpoint. This style may have its roots in university education. During the ICME survey of geometry curricula, multiple panelists reported that secondary teachers in Russia often view themselves as content experts first and teachers second. A few also commented that teachers often use the pedagogical techniques of a more traditional university professor, techniques such as lecture and practice. Since Euclidian geometry as written in The Elements, it is reasonable to assume that there is a relationship between

2. Western Europe: Other European countries also employ a more axiomatic style, but for different reasons. Halfway through the conference, I had a conversation with Bernie O’Donoguhe (Ireland) about the role of Geometry in Ireland. She reported that secondary education in Ireland is largely motivated by two exams: the Junior Certificate and the Leaving Certificate. Both exams have mandatory sections in mathematics with an emphasis on axiomatic geometry. While many other western European countries have changed or done away with the geometry content in their end of course exams, Ireland’s history of moderate isolationism from mainland Europe may help explain why they have not followed this trend. It has only been very recently that the Irish government has discussed deemphasizing the role of axiomatic geometry on the Junior Certificate.

Most Western European countries (e.g. France, Spain, Italy) have recently reformed their curricula to either move away from the axiomatic method or decrease the role of geometry altogether. In a panel on the teaching and learning of geometry, Maria Bartolini (Italy) explained that the concept of proof in Italy is no longer tied exclusively to axioms. Rather, students often believe that accurate measurement and repeated trials can lead to a rigorous proof, and not just a mere conjecture. Nathalie Sinclair quickly added that Dynamic Geometry Environments (DGEs) such as Geogebra can be used to bridge the gap between empirical conjectures and more theoretical approaches to proof. By using check boxes, sliders, and drag features, students can both collect data to make conjectures, and get a visual-spatial understanding of how a particular theorem works. Bartolini agreed and noted that many Western European Countries have decreased the breath of geometry content on their exams, allowing for more classroom time to explore geometric theorems using DGEs.

3. Northern Europe: During the thematic afternoon, I learned that while many of the reforms in Western and Central Europe are new, the Netherlands has used Realistic Math Education (RME) for some time. Although the didactics of RME are not uniform across the country, the basic tenets of RME are the same. According to Marja Van den Heuvel-Panhuizen (Netherlands), RME should both start with and end with a context that is imaginable and real to the student. Problems must be novel, but can include an intra-mathematical or extra-mathematical context.

In a similar session, Guenter Krauthausen (Germany) explained that RME lessons in the Netherlands are often task-based with room for multiple approaches and natural differentiation. In natural differentiation, the teacher presents all students are presented with the same complex task. Then, students informally self-select the solution method, manipulatives, notation, and group members that they will interact with. This provides a very open and social learning environment, where geometric axioms are only taught if apply to the task, and only if the student requires it. As a result, students in the Netherlands are exposed to less content, but more depth.

4. East Asia: While there are stark cultural and pedagogical differences between the Scandinavian countries and the East Asian countries, their current approach to geometry is remarkably similar. Both regions teach through tasks and only discuss content that is relevant to the current task. Interestingly, most of the tasks in Japan’s national geometry curriculum contain some aspect of geometry. The Japan Society of Mathematical Education (JSME) led a session that featured six tasks from their curriculum. Although each task had multiple facets, every task required a non-trivial level of geometry content to complete. For example, one task involved folding a paper pentagon to make new shapes. Another task asked students to chop a cylindrical log into rectangular prisms of lumber. Unfortunately, I was not able to determine the prevalence of geometry in The Netherlands curricula, and thus I can only assume that Japan may emphasize it slightly more.

Another point of similarity between the Netherlands and Japan is the amount of mathematics content each country reportedly focuses on. According to Jinfa Cai, both countries cover less than 70% of the content on the Programme for International Student Assessment (PISA) and Trends in International Mathematics and Science Study (TIMSS), and yet they have some of the highest scores on these tests. In fact, Japan claims to explicitly cover less than 60% of the content on the most recent TIMMS. Towards the end of the conference, the JSME facetiously demonstrated how small and skinny their textbooks were compared to their larger and bulkier American counterparts. The display was a bit tongue-and-cheek, but the point is well taken. In Japan, less explicit curriculum can translate to higher test scores.

5. Afterthoughts: As the conference concluded, I was left with three lingering questions:

• How can two very different cultures produce such a similar geometry curricula? 
• Why do these countries perform so well on international assessments? 
• Can this success be replicated in other cultures, such as ours?

I suppose the answer to these lie in additional research about these countries, their curricula, and their cultures. In an effort to reach out to other teachers, I plan on disseminating the information I have gathered here through regional conferences and publications in New England. Hopefully, this will bring myself, and the math education community as a whole, one step closer to answering these questions.

Five Things I Learned at TMC16: Monday 7/18/16 & Tuesday 7/19/16

1. The Variable Analysis Game: Joel Bezaire presented The Variable Analysis Game: a simple game that encourages pattern seeking and algebraic thinking. The class is presented with three (or more) rows of numbers, with column headings labeled a, b, and c. Students must find the relationship between the numbers in the same row and develop an equation that uses all of the numbers in each column, and works for every row. I'll leave it to Joel to explain more and give some examples, found at variableanalysis.info.

I plan on using this game in my classroom because it allows students to work at different speeds, communicate their reasoning without "giving it away", and has a natural extension.

2. More Coloring Books: Edmund Harris (Current Prof at the University of Arkansas and my roommate for this TMC16) has released Visions of the Universe, a sequel to Patterns of the Universe. Both are adult coloring books steeped in mathematics. This iteration includes map projections, fractals, a representation of all the ways you can add to 11, and a number of lesson plans for bringing a bit of art into math class.

3. Nominations: Kathryn Belmonte encourages students to put more thought and effort into open ended assignments by nominating each other to share their work in front of the class. See the details here.

4. Birthday Functions: Hannah Mesick shares one of her favorite ways to celebrate her student's birthdays: by using them as an analogy to teach functions. Find more in her Prezi here.

5. The MTBoS is not a Plug and Play: Dylan Kane delivered a great keynote about moving beyond resources and into solid petagogy. His slides are here and the video is here (thanks to Glenn Waddell).

6. (Extra Credit) Using Photos: Sarah VanDerWerf uses images to remind herself of the little things she tends to forget while teaching. I'll try to find examples of these, but the internet is a big place.

Five Things I Learned at TMC16: Sunday 7/17/16

1. The difference between further and farther: Farther is used to refer to actual distance, but further refers to a metaphorical, or figurative, distance. Often, further can be used to discuss depth in a curriculum. For example, "we can take the similarity of congruence further by extending it to non-polygons, such as circles"

2. The Price is Right and Probability: The mathematics used in the game show The Price is Right varies greatly, and allows for a natural way to differentiate a probability or statistics course. Flip Flop gives a nice introduction to the main concepts of probability (and a great ending in this clip). Games like Bonkers let us think about random guessing versus strategy. Others, such as Punch a Bunch, have multiple levels of probability built in which can be great for a low-floor high-ceiling task. Better yet, they have the individual clips online. Thanks to Denis Sheeran for pointing this out. Even better, this Slate article has strategies for every game, along with how viable that strategy is. This would have been great to use at last year's TMC trip to The Price is Right... if only we had made it onto the stage.

2a. If you want students to play the games themselves, check out online games, such as this Deal or No Deal simulation. This provides a great discussion about expected value.

3. Instilling racial competency in teachers doesn't have to be painful: I attended a professional development session not too long ago that aimed to make teachers aware of their own privilege. Instead of meeting this objective, it inadvertently made examples of teachers with less socioeconomic privilege. Some participants left crying, others confused.

In contrast, Jose Vilson and Wendy Menard led a very comfortable, yet deep session on Racially Relevant Pedagogy. They started by placing the words Race, Ethnicity, Gender, Class, and Religion on different walls of the room. Then, Jose would pose a question and ask teachers to respond by walking towards one of the words on the wall. Questions started with, "Which one do you most identify with?" but quickly became deeper, "Which one do you talk about least with your parents?" Along the way, teachers were prompted to volunteer reasons for their answers, but rationals were never forced, and responses were always appreciated. The setting was so comfortable, I didn't realize how many "difficult" topics we had addressed until it was over. I hope I can take the spirit of this session back to my school in the fall.

4. The Dean's Feedback Meetings: Feedback meetings can be an especially strong way to create a culture of learning, both as a math teacher and as a grade level team leader (dean). Anna has graciously put her resources up here.

5. Primary and secondary teachers have a lot we can learn from each other: Tracy Zager began her keynote with the graph below from Math With Bad Drawings. She went on to discuss how both content and pedagogy are important for primary and secondary teachers, and how we should use both physical and electronic interactions to connect and ask for feedback from teachers outside of our own grade level. The keynote discussed a number of rich anecdotes that probably deserve a separate post. However, Tracy's two final calls to action were concise and clear:

- Look at who you're following on Twitter and diversify those voices to include different grade levels.

- What are the current obstacles to cross-grade level collaboration? What can I do to tear them down?

6. (Extra Credit) Explore Math: Sam Shah created a site called Explore Math. It serves as a launching point for students to browse topics not traditionally covered in a secondary mathematics curriculum. He asks students to create about four of these open-ended explorations a year, and finds that it works best as a low stakes, high reward activity. More information can be found here.

7. (Extra Credit) Voroni Maps: Dave Sobol gave an amazing presentation that focused around finding perpendicular bisectors between two points on a map. They're called Voronoi Maps, and you can use them to divide up a the USA into sports regions, find the closest airport to you, and more. The best part is, you can then compare the Voronoi Maps to actual maps (e.g. media markets) and discuss the differences.

8. (Extra Credit) Nice Ride: You can rent a Nice Ride bike in Minneapolis for three days for only $10. What a deal!

Five Things I Learned at TMC16: Saturday 7/16/16

1. Since the start of my career, I have believed that math lessons should be relevant and engaging to students. For some time now, I have struggled to deliver a level of relevance on a consistent basis. My ideas often seem either contrived, or a carbon copy of someone else's lesson I have graciously borrowed from online. However, Dennis Sheeran presented a framework for increased relevance. The acronym is below, but more information can be found at his site. For now, just an overview; these five thing posts are supposed to be short, after all.

Infusing your life

Natural flow

Sudden changes

Television and pop culture

Awareness of your surroundings

National events

Two or more disciplines 

2. GeoGuessr is an online game that uses Google StreetView images. Players are dropped in a random location and are challenged to estimate where they are in the world. Players receive points for each guess, and the more accurate the guess, the more points are received. Besides being moderately addictive, this game makes an excellent data collection tool and jumping off point for a number of statistical discussions. We explored the relationship between accuracy of guess and points awarded. It turns out the model is not linear (as might be expected), but follows an exponential decay as the estimation moves farther from the actual location. More interestingly, the coefficnets of the model will change depending on what version of the game you're playing (America only, whole world, just cities, etc), but the general exponential trend remains stadic.

3. Julie Wright gave a quick but rich presentation about feedback quizzes. About twice a quarter, students in her class take a quiz in which the left third of the paper is left blank. Upon completing the assessment, student work is alphabetized and scanned. Instead of placing a numerical grade on each tests, Julie uses an annotation software to type comments into the blank column in each student's paper. Besides shifting the focus from grades to feedback, this system also allows the teacher to copy and paste comments, allowing for more efficient grading. After printing and passing back the papers, students fix their mistakes.

Upon reflection, I'm not sure that I am comfortable with only giving feedback in this manner twice a quarter. Yet, I do see and understand the massive time commitment required to give specific thoughtful feedback on every question for every student. Perhaps a compromise is to provide feedback on a single exit slip question every day and provide students the opportunity to reflect during the next class. This would allow for feedback but decrease the grading load on the teacher. However, the logistics involving scanning the papers requires a bit more thought.

4. Sara VanDerWerf has a backwards bike, and no, it's not a bike that runs in reverse. A traditional bike is steered by moving the handle bars and pointing the front wheel in the direction you want to go, but the backwards bike has a set of gears that cause the front wheel to turn in the opposite direction of the handle bars. The bike was first created and ridden by Destin from the YouTube channel Smarter Every Day. In his video, Destin explains that it took eight months for him to un-learn how to ride a traditional bike and learn the backwards bike. Surprisingly, it only took his young son two weeks. He attributes this to a higher level of neuroplasticity in kids than in adults.

Back at TMC16, Sara explained that riding the bike can teach us lessons in persistence and struggle. Just as with the bike, our students may sometimes know how perform a task, but are still unable to complete it. As both Destin and Sara have said: knowledge is not understanding. Just because you may be able to verbalize the mechanics of the backwards bicycle, it doesn't mean you can ride it. This rings true for both teacher and students. First year teachers may know what good teaching looks like, but may find it difficult to meet their own expectations in the heat of the moment. Similarly, students may understand a set of mathematical procedures, but freeze when these procedures are applied in a multi step context. It is crucial that we understand this struggle and this frustration, and provide safe spaces for students to fall, dust themselves off, and get back on the bike.

5. Minneapolis is a great place to run.

6. (Extra Credit) I learned how to wobble... poorly.

Five Things I Learned at Descon16: Friday 7/15/16

1. Desmos is now accessible to students who are visually impaired. Pressing command + F5 activates a text to speech function that will not only read the functions as you type, but will also identify if your cursor is in the denominator, subscripts, superscripts. Pressing option + T will activate an audio tracer that will audibly read the coordinates of points on your graph as you press the left and right arrow keys. Pressing option + H will play an audible tone whose pitch will change as the function increases or decreases from left to right across the graph. I hear that Chris L was trying to get a graph to play "Mary had a Little Lamb". Unfortunately, I can't figure out how to make these work on a Windows OS. Hmm...

2. teacher.desmos.com now includes bundles of activities. Bundles are a collection of activities in a suggested order around a specific topic, such as functions or quadratics. Each bundle has key understandings for that topic, as well as commentary on how each activity fits into the larger sequence. Of course, it's up to the teacher to fill in the gaps, but I'm interested to see how well the sequencing, pacing, and scaffolding provided by these bundles holds up in the classroom.

3. Access to Desmos at home can be a big step towards equity in the classroom. Sara VanDerWerf, besides being one of the most passionate and kind people I have met, gave a great keynote that touched on technology and equity. She noted that a large barrier to students is often home access to technology, which is why she implored the group to have all students

4. Alice Hsiao introduced me to the Chicken McNugget Problem (Also called the coin problem), which is stated below:

McDonalds sells Chicken McNuggets in boxes of 6, 9, or 20. Obviously one could purchase exactly 15 McNuggets by buying a box of 6 and a box of 9.

Could you purchase exactly 17 McNuggets? How would you purchase exactly 53 McNuggets?

What is the largest number for which it is impossible to purchase exactly that number of McNuggets?

What if the McNuggets were available in different sized boxes?

I intentionally haven't added any links as not to spoil a solution. After all, I'm still working on one now...

5. Drinks in Minneapolis are annoyingly cheap during happy hour, especially when there is a minimum charge. Thanks to everyone at the Desmos team for their time and generosity!

6. (Extra Credit) Make a list in Desmos with L = [1, 2, ... 10]. Check back in with Sara's Desmos Dictionary. Make your own marble slides, card sort. Hide some folders. Talk about Desmos, Geometry, naming points, and back end vs. front end. And lastly, read The Art of Evangelicalism.

Mathematics and Music in the Classroom

Typically, we differentiate our instruction by adding more visual or kinesthetic activities, but it is rare that we allow students the opportunity to actually hear the math they're doing. Below are a few suggestions on how I have used music to demonstrate mathematical concepts, along with resources.

Resources from the 2016 Twitter Math Camp Presentation

Slides from the presentation

Resources from the 2015 Global Math Department Presentation

Slides from the presentation

The book that I took my musical excerpts from

The recording of the presentation

Rhythm and Counting

1. Use rhythm to divide note values into fractional parts.

Then, add and subtract the note values. Let students hear the rhythms.

2. Play varying tempos (beats per minute) to demonstrate different speeds, ratios, slopes, etc. Here is the metronome I typically use, but Google has an Online Metronome too!

3. Play two or more meters, note values, or tempos at the same time (called a polyrhythm). You will be able to hear the conflicting ratio of the two meters, and you can use the idea of a least common multiple to determine when the two meters will line up. Here is an African Trance Polyrhythm Beat Generator.

Sound Waves and Sine Waves

4. Ask students to transform a sine function by changing the amplitude or period. Then, play the tones that correspond to those functions and let students notice the impact amplitude has on volume and period has on pitch and frequency. Here is an online tone generator.

5. Ask students to add two sine functions together, then let them hear the tones that correspond to the functions, first separately, then as a composite function. For example, noise canceling headphones work by taking the sine waves produced by the environment and adding the corresponding opposite function to produce a zero function. Complex tones, such as that of a guitar, are built from the addition of many sine functions, all with different frequencies and amplitudes.

6. Use a Fast Fornier Transform (FFT) to decompose complex periodic functions into their component sine waves. Let students experiment to see what sounds produce different component functions. Here is a free online FFT and instrument tuner program, however it's partially in Japanese. If you dig I bet you can find a free app in English...

Scales, Rational, and Irrational

7. Ask students to order various rational numbers on the number line. Then, play them the corresponding intervals and ask them to order them from the smallest interval to the largest interval. Allow students to compare their results.

8. Have students build a musical scale by multiplying various intervals together. Let them listen to the scale they created by playing the corresponding tones.

9. Have students physically build pipe instruments by taking the ratios used to build the scale and cut pipes to length. Here is one option that uses hard PVC pipe. I suggest getting plastic golf club holders and cutting them to length, similar to these BoomWhackers.

10. Have students investigate the reasons why the first pianos were tuned using a rational intervals (also called Just Intonation), and why we currently tune pianos using a irrational intervals (also called Equal Temperament). More information here.

Geometry, Transformations, and Composition

11. Introduce three musical transformations that composers use to create melodies: transposition, inversion, and retrograde. Play examples of each one of these transformations. Then, have them experiment with compositions to develop "rules" or "theorems" about how they work.

12. Allow students to create their own compositions using the transformations.

13. Push students to investigate these musical transformations further by verifying the existence of inverse operations, the commutative property, closure, and more.