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.
PART 1: SCHOOL PD
PART 2: SUBJECT PD
PART 3: REGION PD
PART 4: HEADS PD
(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 www.onetwoinfinity.ca
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.