Saturday, 29 August 2015

OAME 2015: Days 2-3

This is a continuation of the previous Day 1-2 post, regarding the OAME math conference from May 2015.


(2B) Beyond Relevance & Real World: Stronger Strategies.

Friday Keynote Speaker Dan Meyer, basically asking “How do we engage kids in challenging mathematics?” The “When will we use this?” script has never been written down, but we all know it. The question is not ACTUALLY about their future, they’re complaining about their present! They don’t care, and why would they? (To a kid, isn’t unemployment kind of the goal? “Do school for 10-12 years, and we’ll reward you later with work.” Yay?)

There seems to be an attempt to make the covers as interesting as possible: “If math is not real world, it will not be interesting.” But in trying to make math more ‘real’, does math become more real, or more foreign? The real world is unreliable. We don’t think about triangles when looking at whales. “Make math real world” works less often than we’d like, “Make math job world” works even more rarely.

It’s got to be their world, not real world. Actually, not merely their world, it’s got to be RELATABLE. [“Teachers are so eager to get to the answer that we do not devote sufficient time to developing the question” -Daniel Willingham] We often start teaching the way we were taught, it’s hard to free oneself of that. Get creative.

Ideas for jump starting your own creativity: The “math” dial (min -> max). MAX: “Plug it in and get an answer!” How best can we turn the dial backwards, towards MIN: “Ask FOR questions.” Dan showed a graph of animals which had ‘too much’ information on it. Dial this back, so as not to scare people off - subtract the numbers and labels. If you can ask questions about it, it becomes real to you.

NEXT: Ask for wrong answers and best guesses. If you can guess about it, it is in your real world. (The fake world is unguessable.) Numbers brought in only now. NEXT: Start a fight. The effectiveness of constructive controversy - do you agree with what’s been said/shown? If you can fight about it, it is in your real world.

Dan illustrated this by asking for four numbers from 1-25. (17/3/22/8) Which number doesn’t belong, and why? Note there is no “right” answer, and he gave a shoutout to Mary Bourassa’s “@WODBMath (Which One Doesn’t Belong)”. Dan also showed an abstract problem involving circles and squares dynamically changing areas. The big change isn’t stapling a context onto boring math, it’s asking students to do different work with numbers and shapes.

“Barbie Bungee” is a classic problem, we want a fun ride for her but not a fatal one. Dan googled this, and found six different worksheets. The one with an existing table implies rushing to an answer. “There’s lots of dial turning to do before we get to an equation.” And yes, there has to be a Quad 1 grid/graph at some point, but spend time in a less precise land first.

The best move you can make to turn the dial backwards is to take textbook problems and delete things. Not because it’s BAD, but because it’s MIS-TIMED. Cover elements of the problem up, reveal them slowly to ask more interesting questions along the way. “Every new slide is a new dial setting.” Careful, it’s possible to erase too much! “I’m not saying be totally open and ask them where they want to go.” Great formative assessment is possible before revealing more later.

From my session (below): Line Rap
One last piece of advice about engagement: Consider your job, you have lots of them, teachers, explainers, caregivers... how about a salesperson for pain relief? Who is your best customer? Someone who’s feeling a bit of pain - not a migraine. If MATH is the aspirin, then what is the headache? (And how do I create it? If you just give a student a problem, they’re not feeling the headache, so why do they want it?)
[Addendum: Dan got some pushback on his blog here; should we be selling the relief if we’re the ones causing the pain? He counters the pain is there anyway, we’re exposing it.]

Dan likened the pitch to “You gotta take this pill”, so we’ll put some music to it to make it palatable (video shown), but that cheapens what we do. (I take mild offence to this. More on that later.) Instead, create a brief moment of pain, such that the math makes things easier; he did his “choose a point” activity, where labels make that easier. “I’m not here to deny the reality, I’m suggesting that there is other stuff going on, aside from the real world.”

In Summary: Ask yourself, “How can I remove stuff that can always be added back later?”,  “How do I create a constructive controversy?”, in essence, how do I turn the dial down and how do I create a headache? These are harder to answer than changing the context of a question, but often more successful. Also, there are missing places on the dial, because there are missing things we don’t know about student engagement. Find them out, and then come back and talk about them/share them with your students.

I dropped by E303 briefly between the keynote and my next session.

(3B) Growth Mindset in MCR3U

Gordana Rakonjac and Alison Pridham have looked at problem-based lessons in MCR 3U (Grade 11 Functions course). This from a ministry funded but teacher directed Leadership Program; colleagues worked on the Advanced Functions/Calculus course. They basically walked us through their course; it involves a lot of activities and group work.

They start with a hotdog contest commercial, and an apple tree problem, which prompts self-directed linear/quadratic review. The “lesson” is student consolidation, as they tell a scribe what to write. Next a function sort video and classification. Exponential functions in the context of hamburger toppings, paper folding and spreading diseases. (Make sure students mingle for this last one; infected trading with infected also creates realism.) TIPS documents can be used as homework.

From that into Trigonometry, a sorting terms activity, and spaghetti use to model the graphs. (I’ll be honest, I’m not an activity guy, but trying to break spaghetti in a precise manner had my interest.) This is put in the front of the book to refer back to later; then there are trig carousel activities. Of note, the Discrete Functions (sequences and series) is done as an Independent Study in order to free up two weeks, since this instruction method can take longer.

It was noted that you should still take at least 5 days for the Finance component towards the end of the course. There is student interest and it is relevant! There was a handout of the powerpoint, and they had us doing most of the activities. I can see this being good for someone with a different headspace from where I am now.

At this point it was my lunch, but having seen people yesterday walking into session having having picked theirs up, I decided to do the same. I went to OAME Ignite, which I did blog about in May. Because of the double format session, they actually started their second round at the beginning of break, so I stuck around... meaning 20 minutes into the next session... that made me a bit late for:

(5B) MathsJam!

Envelope folding...
This was with Dan Allen and Chad Richard. Because I was late, I missed the introduction, though I was already somewhat aware of it through Twitter. “MathsJam!” meetings started in the UK, and there are now two gatherings in Ontario... effectively people coming together to play math games, do puzzles, etc. That was in full swing when I arrived.

I helped to create a couple Sierpinski triangles from size 8 letter envelopes. I also looked at a few puzzles that had been offered - and created a much more involved solution for the nested circles question than was necessary. (Multiple methods...) Someone else there also proposed one involving nested radicals equaling 6. And while I’ve never been into games, there was one with funny cards that people were playing.


I went to the OAME AGM, because it’s not long and gives me an idea of the inner workings. Finances are still an issue, in that the books close long before the report is presented. Also moving the AGM needs a motion AT the AGM (it’s in the Constitution) which will likely happen next year (to implement for 2017). OAME 2016 announced at Georgian College in Barrie. I chatted briefly with William Lundy and Tim Sibbald; everything was adjourned by 4:18pm.

I’d opted not to go to the Banquet this year; it hadn’t really thrilled me the previous year, and moreover, I felt like I might want time away from people. Which was good, as I’d actually had a complete mental breakdown Wednesday night, upon arrival, when my keycard wouldn’t work. Crying and everything. I needed quiet time here.

I ended up taking a long walk, all the way to the nearest grocery store, where I bought some supper, brought it back, and wrote the post “You’re a Good Teacher”. It relates to people I heard discussing whether it was right that Marian Small was posting up tweets, and my feelings about Dan Meyer slamming music - it begins "There’s nothing quite so simultaneously invigorating and demoralizing as going to a math conference..."

(1C) Problem Solving without Algebra

To start Saturday was Serguei Ianine’s session focussed on ratios and proportions. (He had another for percents, but it was opposite my session below.) He’s a private school teacher teaching 7-9 and 10-12, and his session included a booklet of problems.

Students are introduced to the idea of using “x” really early, and while equations are good, that can be wrongly timed. Technical aspects should go in parallel with problems, we shouldn’t introduce operations, and then the applications, 80% of kids get lost in the transition... for the simple reason they lack experience. The very abstract isn’t tangible, and creating the equation is difficult.

From my session (below): Fraction rap
Do simple solving of equations and simple applications, then something slightly more difficult for both - don’t use equations in your applications. Eventually you reach problems that can’t be solved by inspection or other methods, and can merge then. Also watch for a tendency of students to work from left to right, instead of using order of operations.

Ratio problems: Can be a ratio of three. You’re a lawyer, problem of a man’s last will, his wife is pregnant, “If it is a son, he should get twice as much as his mother. If it’s a girl, the mother should get twice as much as the daughter.” (Sexist challenge? If you reverse it, there are no problems. Interesting.) The wife delivers TWINS, one boy, one girl, so how much should each person get?

There’s always a couple kids saying ‘let’s call this x and this y’, but we don’t need the abstraction - go visual. Son has more than mother, who has more than daughter. Partition out a line. Always start with the smaller part, end up with ratio 4:2:1 and can now solve based on initial amount. Don’t require the kids to remember words, establish concepts, the linking can come later (even at Grade 10 level).

Another problem was presented involving a timeline (not to be confused with a number line), where a difference between two ages stays the same, but the ratio of the numbers CHANGES (always decreasing, for mother and son) - and how old are they? Shift after this into rate problems, the “per one” (e.g. km/h) is not easy for students, and even the word “rate” is an abstraction.

Proportion problems: This is more than setting two fractions equal; there is both Directional Proportionality and Inverse (reciprocal) Proportionality. Again, don’t run to an equation. Classic example: “20 birds eat 20 kg of seed in 20 days. 30 birds eat how much in 30 days?” What does your intuition tell you?

If we kept days the same, and doubled birds, the kg would double. If we kept kg the same and doubled birds, days would cut in half. The first two are directly proportioned, while the outer two are inversely proportioned. But “I don’t need any variables here, only how to multiply fractions”... keep days the same, 1 kg/bird. Now divide down to one day, then pump back up to 30, all using direct reasoning. Now have 3/2, pump that up to 30 birds, and finished.

That problem should not appear too early, but somewhere in the middle of the process. Don’t expect students to go through these questions at your speed either - it WILL take longer, half a school year for sure. But if you try to set an equation for a problem like that, you will never win. Proportions also has applications in Grade 9 applied, more so than Academic. Do things over and over and over again, and remember there are alternatives to using ‘x’ as some unknown number.

(2C) Teaching & Learning Math Using Spatial Thinking

Saturday’s Keynote Speaker was Nora Newcombe. She approached the math from a more psychological viewpoint, noting “I might be right or wrong. I’m eager to get your reactions.”

There’s a starting point back when we’re born: A generalized magnitude system, for both spatial and mathematical thinking. Development occurs even before getting to a formal school system. Then there’s differentiation of space from number, and number for little kids becomes discrete integer values. Which is a different sense of quantity from continuous! Our intuitions about continuous quantities get superseded, even though we need them later, for fractions.

Piaget said metric coding of space appears late - “Children cannot measure, cannot compare heights that aren’t on the same base.” So there’s a confusion between number, length and density. Nora argues it’s not that this observation is WRONG, but what it actually TELLS us about a child’s understanding might have been incorrectly interpreted. Babies do look longer at places where things are “surprising”, like things popping out of a sandbox where they were not originally hidden.

Containers make the difference.
There are TWO TYPES of Quantity. Intensive (Proportional) vs Extensive (Amount). Given two half full containers, remove the context, and one actually has more in amount. It’s that proportional quantity that babies DO come with. They look more when 1/2 isn’t 1/2. Good news! Proportional/Intensive quantities are what we need for scaling, like reading maps. (Three year olds aren’t accurate in such activities, but neither are they random.) The implication is we can build scaling early... but teachers tend to resist this, and cite that Piaget study.

There are more connections. Spatial level ability at age 3 predicts spatial at age 5 (makes sense) but ALSO math at age 5. And spatial at age 5 predicts math (or “approximate symbolic calculation”) again at age 8. The caveat is, spatial is not the only thing that predicts mathematical functioning, there’s other variables (like “jumping around too much”). A number line (age 6) is seen as both spatial and numerical.

But the mapping of abstract numbers onto discrete objects is problematic. There’s interference, it’s harder to measure (3.4 mL) because they don’t know what to COUNT, it’s harder to deal with proportions (3:4) because they look for the COUNT, it’s harder for fractions (3/4) because those exist between the COUNTING values. Piaget’s argument that children do not understand number until the age of 6 or 7 is based on that confounding of density, weight, etc - continuous measures, of which there WAS awareness.

Many paradigms led to the conclusion that infants (and nonhuman animals) know number. It’s probable that small numbers (1-3, maybe 4) are subitized, perceived by a separate system. Though experiments in 2010 showed a sense of “moreness” or “lessness" is present (generalized magnitude system). Children then get hooked on integers, and count numbers on rulers - looking at a length from 2cm to 5cm, there are 4 numbers (2-3-4-5), so it’s 4cm long. (A “hash mark error”.) We must highlight the COUNTABLE units. Can use plastic/unit chips, or show with fingers a unit apart.

On Proportional Reasoning: Unfortunately, many children and adults struggle with this, even though infants and younger children are sensitive to such relationships. We have to get ourselves AWAY from the impulse to count. Proportional reasoning is better with continuous quantities. Don’t draw a picture breaking the drink down into discrete unit chunks, remove that thing that’s deceiving (“seducing”) them.

On Fractions: They’re difficult but important. Strong predictive relations exist between early fraction understanding and later math achievement (controlling for other factors). Kids in preschool (age 3) can add 3/4 and 1/2, with portions of a circle. When they do make errors, it’s based on “count words”... two thirds is really TWO OF THREE, but they fixate on “two something”. Different languages are a factor here! English: “One fourth” VS Korean: “Of four parts, one”. American children did better with fractions when done in the Korean style of terms, and we CAN do that without sounding like idiots. OF is a great word. 

Nora closed off by reminding us that she is not a teacher, but that the research suggests early math teaching should use continuous as well as discrete representations. Use number lines in which whole numbers do not dominate the fractions - and they can help teach negative numbers later too. “Taking away a negative number” means you should be adding to the positive side, not “you give someone five cents, so negative five cents”... five cents is five cents is a DISCRETE object. We need more ways to get across what negative numbers mean, like zero as a fulcrum.

(3C) Musical Mathematics

In the last slot was my session with Michael Lieff. It was kind of a blur; we’d also pitched a double, but only got a single. After some general introduction and pop music word frequency charts, I got into the idea of rap vs. parody, and encouraged people to try writing their own song about a difficult concept, possibly with a partner. I and Mike circulated, and got some people to share afterwards. I took photos, and asked permission to post; they're throughout this post, including here on the right.
From my session (left): Trig Rap

Towards the end, I did my live version of “Polar Plot”, as promised in the blurb. We also took questions. One person asked me about how I marked my Data Management song assignment (it was an option instead of doing a probability project), and I said that from the outset they’d have to have a minimum number of terms included - it wasn’t that different from “Counting Stories”, if you’ve heard of it.

Someone asked about students recording while performing; Mike pointed out that he’d asked for his act to be recorded. Me, I started with “don’t” and have since backed off to “if you like, but don’t post it anywhere”... mainly you need the students’ respect, I guess. One teacher had a horror story of posting something online, then someone (not a student) took her audio rap and made their OWN video, which included profanity! Be sure to click “No Video Response” when you YouTube post so others can’t do that without permission.

I got a couple other sites, updated my Music document (I now have over 60 links) and sent it out after the conference because one person had wanted it and I couldn’t remember who. Someone else responded too, saying thanks because they’d wanted to come, but had to leave early. (There were about 20 people who attended, much better than any previous time it’s been run.) And so that’s basically that!


As I typed/edited this, one theme I’ve noticed running through the conference is the whole “multiple methods” angle. Both in terms of student solutions, and teaching strategies. Diversity is definitely seeing a lot of attention these days (on a number of fronts), I think in part because humans have difficulty with it. I know once I’ve found something that “generally works”, I tend to come back to it. Related to that there’s also the fact that what might be typical for me might be new for someone else, and it’s not always easy to recognize that.

Will I be back again next year? Well, I rarely make decisions on such things so far in advance; I guess we’ll see. Thanks for reading, I hope you found something in this post to interest you.

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