Every year, the Ontario Association for Mathematics Education (OAME) holds a conference somewhere in the province. I "live blogged" about it last year. This year, not so much... but after an extended wait, here's a summary of what I saw!
1A) Musical Intervals & Mathematical Means
Mike Reiners is co-author of a new book "Fostering Mathematical Thinking through Music", which is sponsored by Casio. (There was a rep there from the company.) The book acts as a series of musical investigations, and we looked at some of these as they pertained to means, or "averages".
To start, numbers can be attached to note values, like how a MIDI datafile uses values from 0-127 assigned to "pitch", "key velocity" and "duration" to produce what sounds like a musical note. Noted that as long as you are consistent within the system you choose, you can get interesting results (and what system might result if "dividing by 1" was undefined?). The arithmetic mean (add values, divide by total) for the 12 tone chromatic scale lands on... the note that makes equal musical spacing. An augmented 4th and diminished 5th between octave notes. Sometimes the 4th is first, sometimes the 5th, depending on the initial note.
That having been explored, we moved to frequencies, with A=440 Hz (a "pure pitch" tuning fork note). It was found that the arithmetic mean between notes creates a PERFECT 5th (diminished 6th), which is more pleasing to the ear. The geometric mean (multiply values and take the root)... well, first, this IS a mean, producing a "middle value". Picture a rectangle of some length and width. Now what is the side length of a square having an equal area? Square root of length*width: The geometric mean. The side has a value between the length and width. Also relates to proportions in the altitude of a right triangle, creating 3 similar triangles. News to me, at least.
Now, the geometric mean, when applied to those frequencies, produces the Augmented 4th (diminished 5th) pattern mentioned earlier. Not only is this interesting, it indicates that the arithmetic mean applied to 'note values' was a 'false positive'... there is a geometric relationship. (Makes sense, and consider that in the equal tempered music system we use, note frequencies progress via the twelfth root of 2.)
|Is this a classroom or a circus?|
We didn't have time to explore that mean in the session, but it was pointed out that an entire keyboard can be mathematically created from only a single pitch. Also, this sort of music activity can help someone more artistically inclined get involved in a mathematics classroom... they can refer to "note E" while others use "note 8". (ASIDE: I learned later there's also a logarithmic mean. Also here is a post talking about uses for the various means.)
2A) Keynote - Championing Educational Change
Sugana Mitra started the "Hole in the Wall" computer system in India back in 1999. Basically, he put a computer behind a wall, with the monitor and keyboard visible. The kids asked "What is it?", he said "I don't know" and walked away. Notably he found he HAD to leave for kids to explore - they wouldn't while he was there. THE OBSERVER AFFECTS THE OBSERVED, leading to unsupervised learning. Also leading to a problem in that there was no way for researchers to know HOW kids were learning, only WHAT they were learning.
Over 9 months, there was a high correlation between time and literacy. (Of note, "You gave us a computer that only understands English, so we had to teach ourselves English.") Of course, the internet doesn't know what's monitoring the child, and plagiarism did result, but here's an interesting question: How did they know WHAT to copy? Are they learning how to learn? There's also the question of framing assignments to motivate learning. Instead of saying "This program can teach you algebra", offer it as "This program is supposed to teach algebra, let me know if it's any good".
His next experiment: Can children teach themselves anything? Even advanced genetics? Sugana gave kids a test (0%) then came back later (30%). The response a kid gave was "Apart from <complex genetic phrase> we understood nothing". He decided that to go further, they needed encouragement - the 'Grandmother Method'. That is, someone to stand behind them and say "Wow, that's fantastic, I never did anything like that at your age". Such encouragement brought scores up to 50%.
|UK advantage: Already know English.|
With respect to "the importance of writing" (like taking notes), Sugana questioned whether writing itself wasn't merely an inefficient tape recorder. To engage with any creative writing, there needs to be some understanding between the reader and the writer. The whole thing rests on "The Edge of Chaos". (An example: taking something we may understand, like a pendulum, and banding it to another pendulum of different length, resulting in a more unpredictable swing system - something chaotic.) Pointed out how some "gaming" (off topic work) on computers may result, but it's "occasional chaos" and will go away. The main thing is, you can't timetable things.
Sugana got a standing ovation, then took a few questions. One person asked about going beyond the 50% score mentioned earlier. Another person asked about how such methods would apply to adolescents (experiments weren't done on students over ~13). Sugana responded that part of the issue is breaking the mentality of "I'm worried people will think I'm a fool". He pointed out that more context may be helpful for older people, and methods must always be adapted - keep experimenting.
3A) Reading & Writing in Math
David Pugalee discussed how learning math is tied into other skills. For instance, how are we using our textbook? There are whole passages of explanations there, and usually we just assign practice problems - do students know how to read a text? Or are we in an age where they would prefer to watch an online video anyway? It's not just about course CONTENT it's about process SKILLS for finding things out on their own. "Reading is more than reading what's there. It's an opportunity to activate what they already know about the topic."
There can be other language issues too (particularly for ELLs - English Language Learners) in that some words have multiple meanings. A math "solution" isn't the same as a chemical "solution", and even within math itself, when you "round" something, are you making it circular, or approximating pi? When you have a "function", do you have a purpose, or do you have an expression with a clearly defined output? (Or is that the same - opportunity for some wordplay there.) The importance of schemata, activating what students already know (even in other contexts?) was mentioned.
At this point, the session became a list of possible strategies (with examples shown) for incorporating reading and writing. Examples ranged from Anticipation Guides to Cloze Exercises, Frayer Models to Word Walls based on a student centred problem. Everything was subsequently provided to participants electronically in a pdf. David closed with a story written by a student: "An Equals Sign in Function Land". People had already started filtering out; chatting briefly with the presenter afterwards, he said he probably should have had more interactive examples.
|z-score excitement is Normal|
Also stopped by the McGraw-Hill booth because they have a new Data Management book coming out. We have no money for a class set at this point, but you never know. Besides, the current text we're using is over 10 years old, so the example about a histogram from "CD playlists" catches some students off guard. I had a look at the other booths, but didn't really linger.
5A) Assessment & Evaluation in Data Management
Data Management is a course in Ontario (MDM 4U). This session was by Wayne Erdman - he won lifetime membership this year too - and it was sponsored by McGraw-Hill's new textbook. But as in the first session by Casio, this didn't feel like a product advertisement.
After some partner discussion, Wayne began by pointing out how the MDM course curriculum states students will "make connections, through investigation USING DYNAMIC STATISTICAL SOFTWARE"... not with/without technology, like in many other Ontario math courses. Here it should be integrated. Question: "Can you describe your assessment practices?" Instruction has tended to have a focus on theory first, then word problems, but right now we're trending towards a more integrated model.
The new text has "Minds On" questions to engage thinking, as well as an "Achievement Check Problem" - which is NOT a "Chapter Problem" as before, but a scaffolded question where hopefully everyone can get part (a) and progress to (d) as their mastery allows. Wayne also encouraged looking at such problems in small groups, as the pendulum in instruction swings in that direction. Assessment For/While/Of Learning was highlighted. More questions will target concept, versus calculation.
The biggest thing in the MDM course is Strand 5: The Summative Project. The probability project story was mentioned (and is in the McGraw-Hill text), apparently having it's origins in the Japanese version of the "3 Bears". What are good topics and bad topics? One universally "bad" topic in feedback from teachers was "Global Warming". It's too big, and a point of dispute depending on where you obtain your data. Also, if a student finds mistakes and reflects, this should be full marks; ignoring errors is what costs marks.
Other tidbits: Do the probability first, not because it's "more difficult", but in order to better understand the VARIABILITY when you reach statistics. Standard deviation was said to be a variation on the "distance" formula, and all Grade 9 activities can be repeated in this course - with the added component of regression. StatsCan's "The Daily" can provide articles, to which one can ask: What can you ADD to their study? Parallel questioning is a possibility (answer this on hockey or this on figure skating). Finally, giving space to answer gives an indication to the student of "how long an answer should be" - is that GOOD or BAD?
6A) Musical Mathematics
|One sing-ular sensation|
One of the first things we did was to get participants to try and come up with a rhyming couplet about a hard to recall topic, in groups of 2 or 3. This seemed to go well (one group had an entire verse!). After sharing, I played a few clips. Mike spoke about inspiration ("a times a is a-squared", which was put to a beat), and how there is online software that can allow you to pitch shift an mp3 into a better key (transposr.com). There's also a site that lets you create karaoke versions of songs.
Towards the end, I performed "Mean", Mike (and Kermit) performed "Forms of Quadratics", things were well received and we called it a day. I also got a lead on a new YouTube channel.
Following the sessions there was a "Wine and Cheese" social back with the publishers (apparently it was in BOTH sections where there were publishers - I didn't realize) and I got a chance to catch up with some other teachers. Including Steve Pritchard who had been in Dan Meyer's workshop, and Kate Mackrell who was an instructor at Queen's when I was in Teacher's College.
After that I went out to a "Tweet Up" at the Lone Star down the road, which included Mary Bourassa, a couple guys marketing the "netmaths" application out of Quebec, Dan Meyer asking our server how many US states had borders that touched Canada (and mentioning the time he was given a Calgary Flames hat to wear in Edmonton), a couple guys from Windsor talking about needing to get more people out from their district... and more. Sorry to those I missed, and that my ability to remember names is lousy. I headed off before 9pm.
|I am in this picture. Look close.|
1B) Mathematical Paradoxes
Session by Douglas Henrich; I realized I recognized this presenter from "Social Justice" last year. As then, he asked us to sit near a quote that spoke to us. I selected: <<Beethoven could easily write an advertising jingle, but his motivation for learning music was to create something beautiful. Why does math have to be useful?>> We started with introductions around the room.
The first paradoxes he presented were "standard" - Escher's Hands (notice has a left and right hand), Ames Room (it's a trap...ezoid) and Russell's Paradox (sets that contain themselves lead to contradictions). "Change our perspective, change what you think is true." He also showed the video of Numberphile's infinity paradoxes. Someone said that 'paradox' comes from 'para' (Beyond) and 'doxa' (Belief). To solve a paradox, we must show the contradiction was only in appearances, or that it rests on unreasonable grounds in the first place.
Douglas said that we SHOULD resolve such paradoxes for students, or it can cause anxiety. (30% don't learn from paradoxes.) This in part because the conflict affects our "schemata", our system of beliefs. A few other numerical paradoxes were also shown which come from a "lack of math sophistication". For instance:
|Is a puzzling quadratic a Para Paradox?|
2. log(X^2) = 2log(X), yet the domain of the left side graphs for all real numbers, while the right side is restricted to x > 0. Really it should be log(X^2) = 2log|X|. (absolute value X)
3. Derivative Paradox: x^2 = x+x+x+x+...+x ; now take the derivative of both sides. You get 2x on the left and 1 "x" times on the right, so 2x=x? In fact, on the right you must invoke the product rule, you have f(x) "x" times leading to "1x + x1" = 2x. Paradox resolved.
The session concluded by pointing out that there are still a number of paradoxes out there which are still points of contention, like 0^0. (And the idea that adding 1+2+3+4+... can result in -1/12.)
At this point, I went to the keynote by Jo Boaler... but as this recap is getting FAR too long, I'm going to cut off here and pick up the rest in my subsequent post. Sorry to be such a tease. Feel free to complain or otherwise comment below.