Sunday, 30 March 2014

Nov PD3: Region

November 2013 was a good month for Professional Development. I attended four sessions, each with different audiences. Finally, I'm getting to blogging about them.


(This was 3/4 written a month ago. Then I got busy. Seen my new column over at MuseHack?)

The Carleton-Ottawa Mathematics Association has a Conference every year in November - the Ron C. Bender Memorial Conference (named after a professor with the University of Ottawa who passed away in 2007). In 2013, it took place on November 30th. It's attended not only by those in the public and catholic boards of Ottawa, but from other schools around the region. Marian Small (@marian_small) was the keynote speaker.

Marian actually presented during our Subject PD too, but it was at the same time as I was doing my Music Math thing. So I was particularly interested to see what would come out of this event (even though content was different). Coincidentally, she also happens to be back in the news of late: Why the war over math is distracting and futile.


Marian started with a little levity, remarking that the job of a keynote is to "say very little but be funny for a long time". She quickly got down to the idea of subitizing, which essentially means determining small numbers of items without counting them. It was pointed out that this is easier when items are in groups. From there, we went to balancing equations - not how you think.

One of the nice things about the "balance relationship" slides were that the process was valid for all grades. It's NOT about finding a solution, or using algebra (though one supposes it could be), it's about mathematical reasoning. I overheard someone else remark 'Could you time travel back and teach my younger self this? This is where math fell apart for me.'

Sometimes you need a skill before you use an idea. Other times you do not. For instance, do you need to know two digit multiplying before you multiply 52x12? Can't you simply take 520 (52x10), and toss in an extra 104?

Perception plays a large role in this. The next few slides looked at the idea of scales (number lines), and the fact that once any TWO numbers are marked, a number line becomes fixed. So "are 5 and 10 close together numbers, or far apart numbers?" Naturally they could be seen as BOTH. I'm reminded of the problem of placing "1,000" on a number line (with Khan and ViHart). An interesting question might be to place only ONE value on a number line, then ask a student where another one should go.


From there, we went to models - how can you show 2/3, to demonstrate the following: That it's less than 3/4? That it's two 1/3? That it's 2 divided by 3? (Noted that most people do lousy with that last one.) Different representations lead to different perceptions. I mentally extended this to show "square root of 8" is "two times square root of 2"... so again, applicable at all levels.

This can be applied to "proportional reasoning jazz" as well: Is 10% a lot or not? Answer: It depends - but it's NEVER a lot of a WHOLE. Knowing the whole helps define the problem. It was pointed out that measurement itself is always a comparison... whether to a standard "unit" or to another object. (For instance "he is funny"... compared to what?) Marian added that a surveyor had told her that they ONLY measure things in metres. Not millimetres. Not kilometres. Magnitudes of metres. Kind of kills the idea that there's a "right unit" to use for sizes. I've also scribbled here how fractals have not much area but big perimeters. (My web serial has been on a fractal kick.)

Now patterning! There is never only one way to continue a pattern that has started. Something going 2, 4, 6 may cycle, going 2, 4, 6, 2, 4, 6, 2, 4, 6 or even 2, 4, 6, 4, 2, 4, 6, 4, 2... that said, one could ask for which pattern it's EASIEST to determine the 100th term. A class activity involving ratios ended up with the question "Is a 5:3:2 ratio easier to work with than 5:3:3?" Students figured the LAST was easier, as two values were now the same -- yet the total is 11 there, not 10. Hmmm.

It was towards the end that Marian mentioned she had a new rant: "Most of the calculations we do are a waste of time. We should be doing estimating." This rubbed me the wrong way! While I see the value of estimating in terms of daily use, I still feel like Estimation is not Math, as I blogged about here. Estimating "pi" isn't good enough! Math (to me) means PRECISION, unlike, say, science, where our starting point is an observation to some number of digits.

As a conclusion, one could say that problem solving is about cultivating deeper insights. "It's about deciding what you think is important and why, and then focusing on that in your questioning." Marian's website is


There was more, after a break! At this point, the attendees split into junior and senior levels. I went to the latter, which was being spearheaded by Robin McAteer (@robintg) and Anne Holness. They had the room set out upon arrival with papers every third seat through the room, and when two people moved closer together, they were chastised for altering the seating.

The papers featured "Clarence's Quandary", a proportional reasoning question, along with three different student solutions. We were asked to critique them, that "we'll be taking these in to look at them" and "this should be easy" as we do the same thing every day at work. The first invalid solution seemed to be the hardest (for me) to wrap my head around, so when they alerted that time was almost up, I got a bit rattled by the fact I hadn't gotten to the others.

This was, in a word, brilliant.

In less than five minutes, they managed to introduce their topic (that classroom environment is important), doing it by showing instead of telling, and they managed to imprint upon me the fact that:
1) Just because you do something every day, doesn't necessarily make it any easier the next time you're faced with it. (For instance, I have a thing about heights. I couldn't be a window washer. I'm not sure at what point being that high off the ground would EVER feel routine.)
2) Time checks are simultaneously helpful and not helpful. Helpful in that they keep someone from zoning in for too long on any one thing, but not helpful in how they could make you feel if you get anxiety.
3) I suspect that part of the reason I take forever to mark is that for every student, I'm trying to see how they're approaching the problem. Which is a bit silly, in that I rarely have a chance to actually talk to said student later (unless they approach me) - I'd probably make a better tutor.

Once time had elapsed, of course, Robin and Anne encouraged everyone to come down closer to the front, and to talk about the student reasoning in groups, rather than hand the papers in. Notably the one correct answer of the three was unclear, while the incorrect ones had reasonable (if faulty) logic. In fact, it was one of the wrong solutions (the second) that was easiest to mark - it had all the right pieces.

We were then given sticky notes, and the big question: What do you value in your math class? Another reason that drove me to blog about the estimation issue was that I was the only one here to mention Precision. I get upset if students start rounding answers off for no reason. The root of 2 exists for a reason! Yes, I'm a math purist.

Here were the main values, as categorized into themes: Engagement. Attitudes. Inquiry. Communication. Understanding. Collaboration. Now then, how are such values communicated? Not only through body language and written or verbal feedback - but in the classroom environment itself!


"Imagine we had a picture of your classroom." Of course, this can be difficult to personalize if you don't teach in the same room all day, but maybe your ideal classroom. We were shown some pictures and asked what we saw - what we thought the instructor valued more. Were there quotes posted around the room? Manipulatives? What's the desk arrangement? (You can catch a glimpse of part of my room from 2012 in this Day In The Life post.)

We were also asked to discuss our classrooms together; certainly a fresh pair of eyes can bring in a new interpretation. Posting student work was something that arose - often it's only the highest achievements. Perhaps exemplars should include level 1 work as well... though from prior years, without names. Students can then shoot for something "better" rather than feeling they fell short. (If actual student work is a problem, perhaps generate your own exemplar.)

A video was then shown on Activity Based Learning, featuring Alex Overwijk's 2P class (@AlexOverwijk). The activity involved cup stacking; his Twitter avatar is currently a picture of one of the results. (In an aside, it was pointed out that the music in the video helped to set a mood, and I know of some teachers who play music while students are working.) Regarding the activity, it had to be THEIR question to get students on board, and not ALL of the class did the problem with the same model. (For instance, some nested the cups, others didn't.) Again "it's not just the activity, it's the environment" acting like another teacher in the room.

The final aspect of classrooms which was discussed at this session involved feedback. How might we better reflect our values in the feedback we give to students? In particular, students may perceive more FREQUENT feedback as a statement of what's important - whether that's our intent or not. (Noted that teenagers perceive things differently than adults.) Does it have to be written feedback, or can we talk about what the student was thinking? Also, how do we encourage students to use our feedback? Are they able to apply it before the exam? What is the mathematical equivalent of a "rough draft"? Remember, marking and feedback are potentially quite different things.


There was a lunch after, but I had to get to a fundraiser. I loved how a lot of Marion Small's activities could be generalized out to any grade level. I'm still not sold on estimation though - and here's another argument against it. If people cannot estimate, they may feel they cannot do math, which is WRONG. At a family New Years' gathering, we did trivia, and one question was to estimate the average salary of a basketball player. I had NO clue and was WAY off... but that's not because I cannot do mathematics. It's because I cannot handle the real world situations that math activities so often throw at us. (Again, estimation is important, but it shouldn't be all consuming.)

As far as the environment aspect goes, I can see how it really is a factor, and one that's hard to quantify - as are most of the important things in teaching. My first PD post remarked that "watching better teachers doesn't make you any better if you don't know what they're doing"... but could it be that seeing their classrooms might be a step in the right direction? With the semester turnover in February, I shifted my setup from sets of desks together facing front to entire rows together facing front. I'm still not sure about it.

Then again, as soon as you become sure of something, you might stop questioning. As I said in my MuseHack column, we should take steps to avoid that.

Monday, 10 March 2014

AMV Friday Roundup II

Time to recap again. The first roundup was here, which also provided the backstory, and details on my arbitrary qualifications:
A) Posted by Creator
B) Few Subtitles
C) Low View Count
D) Single Song OR Single Anime
E) Not a Slideshow

I also arbitrarily declare that a 'Zed' not a 'Zee'

Not much has changed in three months. In fact I'm not sure who (if anyone) is even aware I'm doing this - but then, my web serial was equally as obscure when it started. It's also occurred to me I should probably post an actual comment saying that I liked the anime music video, but I'm having the "no insight" problem I blogged about, or maybe paralysis of another sort. Still, we'll see.

The bits in boldface below are what I was actually searching on when I found the AMV; if there's no bold, it was usually a side suggestion from a different search. Ideas and requests (specific or general) still accepted (@mathtans), subject to above qualifications. In particular, I'm less likely to watch a video if it's an anime I don't know, and there's a TON of those. To track the ones that follow after this, follow #AMVFriday. To track the ones that came before, it's...


13) Lucky Star. Song: "Do Re Mi" (from Sound of Music)
Channel: Tonymcy
Views when found: 11,870 (Dec 20/13)

14) Itsudatte my Santa. Song: My Only Wish (Brittany Spears)
Channel: NoAnimeNoLife360
Views when found: 607 (Dec 27/13)

15) Soul Eater. Song: High School Never Ends (Bowling for Soup)
Channel: pokeEUReKA
Views when found: 940 (Jan 3/14)

16) Melancholy of Haruhi Suzumiya: Endless Eight. Song: Here It Goes Again (OK Go)
Channel: drakeblazenheart
Views when found: 94 (Jan 10/14)

17) Banner of the Stars. Song: Ballad of Serenity (Sonny Rhodes)
Channel: caliztar
Views when found: 103 (Jan 17/14)

18) Various! Song: Fireflies (Owl City)
Channel: Ikaros39tails327
Views when found: 67 (Jan 24/14)

19) He is My Master. Song: Bitch (Meredith Brooks)
Channel: DilaaaxAiomi
Views when found: 776 (Jan 31/14)

20) Mai Hime. Song: Addicted (Kelly Clarkson)
Channel: shiznat4eva
Views when found: 34,736 (Feb 7/14)

21) Various! Song: Only Hope (Mandy Moore)
Channel: Ariettychan
Views when found: 1,088 (Feb 14/14)

22) Various! Song: Payphone (Megan Nicole/Dave Days cover)
Channel: minisaku25
Views on Feb 21/14: 9,858

23) Master Keaton. Song: MacGyver Theme (Randy Edelman)
Channel: psygyl
Views on Feb 28/14: 3,429

24) Revolutionary Girl Utena. Song: All About Us (t.A.T.u.)
Channel: Angel Aito
Views on Mar 7/14: 1,118

If you want to read the little blurbs I posted with each AMV, search #AMVFriday. If you want to comment, do so below!

Thursday, 6 March 2014

The Rational Divide

If you know me, you know I'm fond of "Day in the Life" initiatives. This week, there was a proposal to do a "Single Class Edition". (You can read about it on Tina's blog here.) That's new! In most of my previous ventures, I said "then I had this class" with minimal elaboration.

Quick background info: The course is 3U (Ontario Gr 11 Functions), the unit we're in is "Equivalent Equations" and we're currently simplifying "Rational Expressions". Introduced the topic Monday, concept of holes Tuesday, adding and subtracting Wednesday. Which brings us to Division/Multiplication.

There's 28 students in my classes. The desks are in three rows of 10, smushed together so students can talk with each other. I'll be breaking down a 75 minute period, 8:40-9:55.


8:40 - Period begins. Morning announcements.
8:45 - Period actually begins. I have up the last slide from the previous day, where I rushed a bit to have ten minutes to hand back tests. Also hand back tests for a couple who were away. Start of class is time for individual questions based on homework. One student asks for the second last slide from previous day; I oblige, someone else wanted it too. Not getting much in the way of questions. Take attendance; students help when I realize that (once again) there's an absence where I'm not cluing in who, due to not seeing the face.

First Slide
8:55 - New topic: Put up the multiplying/dividing slide, as well as homework for tonight (to start, for those getting it fast). Correct spelling of multiplication, thanks to a student. Give them some time to look at it and copy down, any other possible questions individually. One such question is student asking if answer is 1.5/8, I note answer is just 1.5

9:00 - Ask class how we divide those fractions. Student I was talking to answers: we find a common denominator, then divide the top. This throws me off because, well, I'm not used to getting that answer. I write up effectively that method, ask if anyone did other things. Another student says they flipped the second fraction and multiplied. I said yes, that works too, show that, and point out that the REASON it works is because, by dividing with the common denominator, things would become "unity" anyway.

9:05 - A student asks "so why did you even show us that?" Glad you asked! Because now we have to apply this, not to whole numbers, but functions. I walk through the first example of that, pointing out how the common denominator generates a domain restriction on x that wouldn't be visible otherwise. Also show how it can be done using the reciprocal of the second, but you STILL need to track the original asymptote. Check for questions. Someone says "back up!", I back up and reiterate a few things; again, homework's on the board for those already getting this.

9:15 - On to second slide with second example. Give them time to try that. Wander around to help individually. Couple people canceling around the additions, I remind we can't do that. Some having issues factoring the common denominator part. I keep wandering, students also talking with neighbours for help.

Second Slide (after writing)
9:25 - I show how it factors, encourage keep going through the rest. One student notes there's a lot of "holes" for this graph. I agree, saying something like "Holey Question, Batman", get a few chuckles. Some are asking me if answer is correct - I honestly don't recall (made up these examples last year), so follow their logic, looks good. Now that I know, makes checking later ones easier.

9:35 - I put up the solution in full. I use the "common denominator" method, emphasizing that the reciprocal method works (careful with restrictions) and that canceling the (x-3) right off the bat works too "but I'll pretend I'm so zoned on Common Denominators that I don't notice - hey, I'll get the same answer". Check for questions. Guy is wondering what the result (x/(x+1)) actually looks like. I pull up internet and plug it into Desmos. Shape reminds me that, yeah, it's basically 1/x, moved one left, then vertically stretched by "x", so I say that. Someone who wasn't there the first day we did holes (when I zoomed in on the Desmos graph to show how it breaks) wants to see it zoomed. I start doing that, before remembering I didn't enter an equivalent equation - there's no restrictions here, as in the original equation. Derp.

Third Slide. Too cruel?
9:41 - I put up my multiplication example, saying I'll see if I can backtrack to Desmos while they're working on this one. End up having issues because I'm already so zoomed in; a student next to the curious one has his laptop, says he'll just show her the Desmos whackyness.

9:46 - Conscious of time, I show the factoring for the multiplication example. Solve it. Ask if there's questions for the last five minutes, either about this or the homework - and yeah, look at those questions if you haven't yet. Student is wondering about the restrictions on last example, what with the fractions. I add the explanation to the slide, she asks "will that be on the website", I assure her it will be. Also show how monomials divide (without mentioning power laws) since I didn't get to my last example of such.

9:52 - Guy with the laptop shows me how when he moves the window a bit, the insanely zoomed in Desmos graph tries to redraw itself differently. Hadn't realized that before. Interesting.

9:54 - In the last minute, I show a couple of crazy graphs "like you might end up seeing in the Advanced Functions" course. Then, before the bell, remind them that there will end up being a quiz on this week... after the March Break. To see what's been retained.
Student: "I'll study." His neighbour: "You liar." Student: "I'm not lying!"
9:55 - Bell rings, class switchover.


Second verse, same as the first? Well, same material, different students, so not quite the same as the first period. But I'm thrilled to have the same course twice for the first time in a year and a half... so here we go. Spot the changes.

10:00 - Bell rings, I presume, I'm still talking to a couple students after having handed back yesterday's tests from those absent. Head to front to pull up same first division slide from before. (Forget to fix the spelling.) Remind again, first ten minutes or so for questions, if none, have a look at this. Back to the students; one offers the help the other, other says he doesn't like having to ask for help. I say something like "yeah, it's tough, but once you understand you can always help the next person, like the circle of life or something". First guy says it's the "circle of math", I say I like that better.

10:05 - Attendance. I keep circulating in the room. One question about previous day. One question about TODAY'S homework from a person who's already started it. Field those.

Seen the Canceling Song Parody yet?
10:10 - Back to the front. How do we divide fractions?? This time I get the answer "you take the reciprocal of the second fraction". (Yes, student used the term reciprocal.) I respond with yes, but WHY? No one offering up the common denominator this time, so I show why doing that works, and why that means taking the reciprocal is valid. Some nods.

10:15 - I say apply one of those methods to first function example; don't walk through it immediately. Circulate. Some ask me if they're getting it; most missing the common factor of 4 but have the idea. I then put it up on the board, writing something more or less like in first period. I don't stroke through the denominator this time, more underline it; I like that better.

10:26 - Second slide/example goes up to try. More circulating just like earlier, and prior period. Some questions about this - some are about the factoring, others are getting it, but then I have to remind them to go back and deal with restrictions. The one girl from the start of the period has another question related to the new homework (it's a common factor issue).

10:40 - I've now put up the factored version of the question, as I think most people made it at least there. Mention the answer (sans restrictions), encourage keep trying it.

10:45 - Full solution now on board, I offhandedly make a "Holey Batman" comment again. (It worked better first period.) Mention how the simplified graph relates to 1/x, say that I showed it this morning, do people want to see it? Some nods, so pull up Desmos again, there we go. Someone wonders how that came from the original division. I say I'll pull that up, but first, multiplication question to try.

Why you gotta be So Mean?
10:48 - Multiplication question up, I note how the operation is generally easier, as factoring makes everything multiplications anyway... so the factoring for my question is quite difficult, to keep things interesting. Student: "That's not nice." Me: "No. Sometimes life is cruel."

10:55 - I show in Desmos how the two graphs collide to make the division from previous example. Question about the nature of holes (and why it's hard to show them on a graph), I note how a value works for 3.00000001 and 2.99999999 but not 3 - how to show that on a zooming scale? Keep circulating. More factoring issues. One student in particular seems to be tuned out, but I see he's made some progress today, so I'm not going to push it. I remind everyone that I do pick tricky examples to show at the front, do have a look at the homework, might be more straightforward.

11:05 - I put the solution on the board, making sure to show tile chart for the hardest one, as I did in first period. Student asks to see chart for the cubic. I do, but point out that as it's just a common factor, there's just the one row. Pause. Me: "Did that help?" Student: "No." Honesty is good. I head around to talk with her and her neighbour.

11:10 - Back up at the front, quickly show the same monomial division as I did in first period, in case people need it. Also show the few crazy advanced functions style graphs, and remind of quiz. I mistime things - pretty much done talking with 30 seconds left. (Lost my watch yesterday, actually. Found it when I got home today.) So students start shuffling to the door. Blah. Pet peeve, but I quip about how people seem eager to get to lunch.

11:15 - Bell goes, lunch starts. One student had talked about getting help at lunch, but either something I said individually answered the problem, or he forgot. There's another student who'd asked to talk, slipped my mind to ask if lunch was possible until too late, hopefully we connect up again.


No time for a lunch break for me at this point! Anime Club meeting in my room; ended up watching Lucky Star. Third period after lunch was my prep, had to deal with quizzes and math competitions (though more made notes for this post). Fourth period - data management (4U statistics)... the second unit of which I've been messing up, after doing a decent first unit. Sang "It's Probability", Carly Rae Jepsen song to close off the period, which I'm waffling on adding to my published material.

After school, I'm spearheading the musical rehearsal until after 5pm, so not leaving school until 6pm at the earliest. Coincidentally, 6pm is around when I made it to the washroom for the first time - probably helps that I never got around to drinking anything since leaving the house in the morning. Despite the fact that I still have marking to do, I wrote this thing up and came home.

I feel like I do more "direct instruction" teaching than not. Random activities isn't my thing. So I try to balance "me slapping down examples" with letting the students talk more. Some days I succeed at that better than others. Questions? Advice? Let me know in the comments. Hope you enjoyed reading.

Monday, 3 March 2014

Nov PD2: Subject

November 2013 was a good month for Professional Development. I attended four sessions, each with different audiences. Finally, I'm getting to blogging about them.


The Ottawa-Carleton Public Board has one PD day during the year designated for subject-specific PD. All math teachers meet at one school, all science teachers at another, et cetera... or all history teachers visit the War Museum, whatever the subject council has pulled together. Normally that day is in February, after the insanity of semester turnaround. With the province forcing "unpaid PD days" on teachers this year, the day was shifted to Friday November 15th.

Given the change, there was an earlier than normal call for presenters. I offered to repeat my "Musical Mathematics" 75 minute workshop from last year. I checked off both sessions 1 and 2, noting I'd be fine to present in either... resulting in me presenting for both. Which is fine - just didn't think there would be that much interest, and you won't see as much in this post about other sessions.


We were at Glebe high school, where things started with Bruce McLaurin making welcoming remarks. Then my first session. 6 people were signed up; 2 showed. A 100% increase from when I did this last time, at TMC! Then for the second session, 5 were signed up and all 5 came. Neat.

Pic of Bruce from later that day
If you want the music links and things I went into, just send me a message. What follows is a few random thoughts. (Which actually held up this post for two months. I'm not good at random thoughts.)

1) Creation time. I gave them time to try to create a few couplets or verses of their own. This went better in the first session, as the two people were more interactive than the five. There was some resistance to trying it, but in the end one remarked "This is ridiculously entertaining. I may have to get them to do it." So groups and pairs is probably a good thing to integrate.

2) Staying topical. I had a few videos showing rhyming (from Whose Line) or misheard lyrics (cdza) connected to songwriting. Someone remarked that they weren't necessary, seemed like I was "filling time". I see the point. Part of the reason I didn't show as many math videos this time was that I had created a 10 minute clip show version. That may have been an error in judgement.

3) Student elements. Last time (before TMC) that I did this, there'd been the valid question of what students got from it. I brought along one student rap this time, and (if memory serves) mentioned that the June feedback hadn't included anyone who found it detrimental - though that could be due to how I phrased the question. I need to jot down more of the anecdotal things, I tend to forget them.

Anyway, people seemed to enjoy it and found me enthusiastic. Was also the first time I taught the same thing twice in one day since... I don't remember when. Incidentally, someone mentioned "Calculus: The Musical", anyone heard of that?

Between the sessions, I also learned that, at the Data Management session, there was an attempt to get a network going of the teachers - made a note to sign in with that. Some of the other sessions were also posted to the Math folder in our Board Mail program (which is where I got one of John Katic's handouts "More Gems and Insights").


Lunch was a chance to connect up with math teachers from around the board. I remember chatting with Mary Bourassa (@MaryBourassa) about Twitter Math Camp (and my uncertainty about going, which I later blogged about). There were a few others (like Sofya?), but of course I didn't think to write this stuff down at the time, so, oops, apologies.

Then at 12:50 there were then three "rapid-fire" sessions... 15 minute sessions run by various teachers who were just showing an interesting aspect of what they were doing. (For instance, @JPBrichta talked about Desmos.) I went first to Bruce McLaurin's session (@BDMcLaurin) on "Letting Go of Units". I had seen some of this idea back at OAME (and last month posted about my first attempts at it in the 3U course).

Bruce questioned: "Why do we stick with units?" Structure. Routine. To test related topics. New question: "What's the advantages of letting go?" Using current events. Connecting across strands. More opportunities for problem solving. (Versus "I will use an exponential model, because this is the exponential unit.") Also apparently can create more time; less stalling for everyone to be on board, lends itself to differentiated learning.

Evaluation becomes more activity based - which, I admit, is where I have difficulty, because activities are well outside my comfort zone. The cycles aren't planned in advance either, it's more "I'm getting nervous that it's been almost four weeks with no trig". That said, I was trying to mess around with the underlying principle in 3U, as I said.

Alex highlights some discoveries
Second session I went to was "Questioning" with Al Overwijk (@AlexOverwijk). Central to the presentation was an activity whereby a number of objects were presented to the class. Groups were made, given their own coloured pens, and they came up with questions related to the various items. (For instance, empty timbit boxes from Tim Hortons, where a question might be "Why are there no timbits?") Then they placed checkmarks next to what they thought were the BEST three questions, and WHY.

A few things came out of this.
1) There was difficulty articulating the WHY. What MAKES a good question - what are the criteria? For instance, having it relate to the subject matter is only one aspect. The idea that "we don't know what to do right away" connects, though does a good question need to be something that can definitively be solved?
2) The fact that students ask great questions - but don't necessarily rank them highly. For instance, with a starfish toy that grows in water, a great question was seen to be "What's the volume of the starfish?" as opposed to whether it's actually 600% larger.

Students also struggled with some of the collaboration, and listening aspects. A suggestion was to get the next student to paraphrase what someone else said before talking. In the end, better questions were asked later in the course, and there was definitely the idea that questioning is a skill that takes work.

Andrew discusses equality
The third and last session I went to was "Mathematics and Science can coexist!" with Andrew Cumberland. He presented us with a handout containing a set of questions as "a starting point for future and enduring questions" about the nature and applications of numbers. It was framed as whether you saw certain equalities as True or False. For instance: "a = A"; "1 = 1.00000"; "1.0 m = 1.00000 m"; "-1.0 m < 1.0 m".

We were given time to come up with our own thoughts, then share with our neighbours. Needless to say, we didn't always have the same answers. Obviously the idea of significant digits comes in, but if -100 is less than 1, physics is a problem... the negative merely describes direction! One could also argue that 'm' is a variable, as opposed to being a unit of measurement - or at least I brought that up, being contrary.

Andrew addressed us at the end, asking us what is the nature of measurement, and abstraction. He said that one teaching method is: Come in with a prompt. Let students talk it out. It evolves into discussions. Things are never just true or false. Even "equal" may not mean what we think in some contexts. Things here could have become psychological or philosophical, but that was the end of the fifteen minutes.


That was also the end of the PD Day, evaluations were filled out and we went on our way. So, given it's just over three months later, what still resonates?

What I mostly remembered (before actually checking my scribbled notes) was that cycling is good, good questioning is hard, and decimals (to me) remain a more scientific measure. I should also possibly find a new thing; while I'm enthusiastic about math and music, I'm not sure it's a draw beyond my enthusiasm. That said, I'll be presenting a variant of it with Michael Lieff at OAME 2014 in May.