THURSDAY MAY 7
The day started very meta, as I dreamt about being sleep deprived at a math conference. They were trying to sell bibles at registration? I got nothing.
(1A) Pythagoras, the Man and his Theorem
C. Chan started with some of the background on Pythagoras (lived 570-490 BCE), and his school “All is Number”. In particular, some of the links made between numbers and characteristics; 1: Source of All; 2: Female/Opinion; 3: Male/Harmony; 4: Justice; 5: Marriage. 10 was the Divine, the sum of the four spatial dimensions (1+2+3+4). (Granted, we now see ‘1’ as the ‘0’ dimension.) Johannes Kepler once said “Geometry has two great treasures” (one of those being a^2 + b^2 = c^2).
Looking at the Theorem itself, the Berlin Papyrus 6610 (1300 BCE) was related to Pythagorean Triples much BEFORE Pythagoras. Babylonians (YBC 7289) also tried to calculate the diagonal of a square, and found proportions close to root(2) in their system of Base 60. (The special isosceles right triangle case, approximated: 1 + 24/60 + 51/60^2 + 10/60^3 ) And China had the “Gougu Theorem” (3-4-5) with 1 chih at about 23 cm.
The presentation then looked at the numerous proofs of the theorem over the years, and included a handout illustrating many. Euclid’s proof appears in his “Elements” (~250 BCE), Book 1, proposition 47. Extended in Book 6, Proposition 31 to any figure (not merely squared). A proof from the 3rd century AD is here: http://donwagner.dk/Pythagoras/Pythagoras.html Chan also showed how you can also create a fractal tree from the Theorem.
The big question: “Did Pythagoras prove the theorem?” Ancient sources do not say this, and Euclid’s “Elements” did not mention him. There’s a legend saying “Pythagoras sacrificed 100 oxen” upon the discovery, but the guy was a vegetarian so... unlikely? Either way, his name is linked now; I note it is spelt with “A”s there is no “U”.
(2A) There’s Student Mindset, what about Teacher Mindset?
Thursday Keynote Speaker Marian Small: “You’re used to me being all math all the time. ... This will be a bit more philosophical.”
As a teacher, what do you think about most in terms of your teaching?
(1) Covering the curriculum (good, that’s what they pay you for) and keeping kids engaged (given different mindsets: busy/interested/other);
(2) Higher EQAO (provincial test) scores (if you’re practicing they must matter somehow - is it boosting the higher students or bringing up the lower ones?);
(3) Instilling love (or ‘not hate’) for math (which secretly means being successful);
(4) Improvement of your own practice;
(5) Convincing other teachers to try what you do;
(6) Fitting in with current ideology (all teachers like belonging, just like kids).
|How do you settle this?|
As teachers, do we not think enough about what math is, or is not?
-VIEW 1: Do nice algebra stuff. That’s all it says to do. VIEW 2: The kid should understand that any relationships can be described in different ways. View 1 teachers bring out different highlights; there’s different mindsets here. That’s not to say one is always better.
As teachers, we have different assessment beliefs too. How do we measure success? A View 1 teacher is happy when a student can add fractions. A View 2 teacher is happy if the student can provide an addition using different denominators, such that the answer is out of 15 parts. Both of you are right, it’s just different perspectives. Don’t fixate on “Growth mindset is good, fixed mindset is bad”, think about your perspective!
Is math about getting answers to problems, or is it about having a deeper understanding of mathematical relationships? Is fun critical? Did you mean fun, or engagement, and is that enough? Marian then provided a number of statements, asking which one people agreed with more. For instance: You are a better math teacher if: (A) You know the math you’re teaching deeply OR (B) If you’ve struggled with math yourself so you better understand potential student struggles. (This isn’t “fixed vs growth”; it seems to be something else.)
Other statements: (A) If you work at your teaching, you improve and student learning improves. OR (B) You can only control some of what kids learn; most of it depends on the attitudes of your students. (I feel like that depends a bit on the day!) When you struggle with a new strategy: (A) You keep trying until you get it OR (B) You try a few times and abandon ship to look for others. (I’m definitely ‘B’, there’s too much out there, move on. It was noted that both are valid as long as you DID try the strategy!)
Wrapping up, is the point of school: to teach expectations? to build problem solvers? to build curiosity? to make learning attractive? More to the point, do you think the “purpose” affects the way you teach, and what do you expect of yourself and your colleagues? Marian says that neither she, nor school boards have the right to tell you what you think about being a teacher. The right to expect you to try something (not reject OR accept out of hand) but not the right to your thinking.
Teachers don’t have time to stop and think. That’s a social-political issue that she has no power over. Still, try to think about what your mindset it, and it may influence you.
(3A) Be More Dog: Enthusiastic and Curious Classrooms
Dogs are playful and curious, they like to explore (a dog was first into space). They also have better senses than we do (smell & sound). Useful qualities for investigating mathematics; don’t be scared of adventure. What you didn’t learn from the books on mindsets: Changing mindsets is really hard. “Becoming is better than being.”
Have an element of surprise driving learning, and piquing curiosity. Dogs also live in the moment. They like to make messes - and math can be pretty messy. Big Idea: The same object can be described by people using different measurements and/or levels of precision. (Can we deliberately choose units so that the measurement is more valuable? A puddle is “10 steps wide” vs “12 steps wide” - smaller steps mean you’re right?)
Multiple Methods: “If 24 items are to be handed out to Ann and Ben in a ratio of 3:5, how many would each person get?” Answer via: Intuitive (guess). Additive (keep giving 3 & 5). Equivalent Ratios (9:15 same as 3:5). Finding the Unit: 24 whole, 3 subunits of 8. Try opening questions; instead of “there are 30 children on a bus, five more boys than girls”, there are more than 30, you decide the exact number.
Dogs are visual learners... okay, maybe that’s a lie, but let’s talk about visualization! Follow up questions can get students thinking this way. (“Where do you see 12? Why do you think a subtraction sentence was used?” “Why could you always write a division sentence if you have a multiplication?”) The same algebraic expression can be used to describe two different situations: “5p” could be total amount of money at $5/hr after p hours... or five fingers per hand means work out number of people.
More dog links: “Be dogged” means persistence, not finishing until you’re done. If you want what is buried, dig deep to find it. Dogs are also very social. A final performance task could be questions surrounding a picture; can week out non-mathematical ones by topic sorting, ranking what is a good question. “We are all in this together.”
(At this point, 4A, I had lunch and browsed the displays.)
(5A) They Can Do it! Supporting All Students’ Thinking.
This was Featured Speaker Chris Suurtamm. “All students are able to engage in mathematics. And to extend their mathematical thinking.”
She presented five ways to encourage students’ mathematical thinking:
1) Value prior learning and experience
2) Provide space for students’ own solutions
3) Pay attention to students’ thinking
4) Focus on connections rather than hierarchies of solutions
5) Encourage opportunities for success
#1: Consider “What math looks like at home” vs “What it looks like at school”. We need to connect things to prior learning, as a ‘blank slate’ leads to memorization. What models to students use, what understandings do they have? We were told to partner off (I was near to Jennifer M from St. Catherines) and discuss models for integers. “It doesn’t make sense for me to begin a lesson with a number line if prior knowledge is different.” (Use temperature? Algebra tiles? Money?)
#2: “Research has shown that allowing students to develop their own strategies deepens their own understanding of the problem and processes involved.” Because their method is seen to be valued. Turn off timers, no “mad minutes” use RPM (Reasoning strategies, Practice and Monitoring). We considered a ‘children on a school bus’ example (not unlike in Amy’s session above).
#3: The most important thing! It’s so easy for us to keep talking. She showed a student solving b+b+b-30=12. Of note: Listening to student thinking helps me with what they know, but it DOES NOT tell me what they DON’T know. “Just because someone doesn’t do something, doesn’t mean they don’t know it.” You can solve with one method, but still know others. You can drive through a stop sign, but still know you’re supposed to stop.
#4: Every bus solution (from #2) involved mathematical thinking. Avoid the idea of “I did it but I didn’t do it the math way.” They’re ALL math ways (merely some are more symbolic). What strengthens understanding is not that they can see one representation over the others but that they can make the connections between the different forms. Demonstration here from youcubed.org
#5: Problem: It’s easier for a student to not try and not hand it in and then fail, versus to ACTUALLY try and then fail. Chris mentioned a student who had produced work in class, but when work was due, didn’t have it and said they were willing to ‘take a zero’. We need to be rethinking ability-oriented language. Moving mathematical thinking forward is the goal, and a successful lesson is one that helps to do that. (We shouldn’t throw out success criteria, but don’t narrowly define them, we’re not aiming every kid at the same target.)
THURS EVENING INTO FRIDAY
After the Thursday sessions, there was the usual Wine and Cheese. I ran into a couple people, but my memory fails me. Then a number of Ottawa teachers got together. I saw Balazs, my former math head, now vice-principal (actually returning to our school this Sept). He proposed oral explanations, for anxious students (about tests or generally). It doesn’t need to be time consuming, you can record one student, then they interview the next as you’re back in the class, and so on. Listen to the recording later, you’ll know if they were prompting each other.
Then we went to “Lone Star”, as we had last year, for a “Tweet Up”. (I got a lift with Tania A and some other teachers.) I remember hearing a bit from Al Overwijk about some class struggles, and there was a fascinating conversation with Ann Arden about her sessions. She signed up for a “Themed Thread” (I couldn’t thread through, I was presenting during one) where they had been talking about so-called basics like “definition of multiplication”.
Here’s a problem: As an expert, you cannot see how two identical things could be perceived as different. But as a novice, you may not see how they are the same. Particularly when done with gestures and representations. (This may have pushed me into my recent popular post “Multiplication is Ridiculous”.) A rubber band model was referenced as one representation. Onwards to Friday:
(1B) Supporting Students with Learning Disabilities in Math
This was Connie Quadrini and Connie Gray (their Featured Session, not part of the “Thread” set). I was impressed that someone was translating into sign language in real time over on the right.
|Matchstick task is on the screen.|
a-Build figure 10 and “1 by 1 Count”. (“Creating a model” or “Concrete Representation”)
b-Extend the pattern with repeated addition (table; Additive Thinking)
c-Extend the pattern with a rate triangle (graph; Extrapolating)
d-Create an Explicit Pattern Rule (algebra; Multiplicative Thinking)
There was mention of the YCDSB (York District School Board) Middle Years Collaborative Inquiry Project. It was started with an eye to proportional reasoning and extended to patterning/algebra. Involved teachers Grades 4-9, special education teachers, consultants, administrators, 3 families of schools, 6 days per teacher. Idea of unpacking IEPs through a mathematics lens.
There were pre/post assessments. Mult choice, and open response scoring was assessed not only on completion but also math strategies used. Also clinical interviews. Some results:
-Students who arranged using a linear format with non-visible groups tended to use Additive reasoning, as opposed to Multiplicative (see above; multiplicative can also be linked to determining a unit rate).
-Students who interchanged multiplier and constant had perceptual reasoning listed as an area of need in IEP.
-70% of students indicated that using tiles supported them in learning patterning/algebra, both for those who had perceptual reasoning as a strength and as a need.
-Someone who had “memory” as a need liked the hands-on because ‘you can remember it’.
We also looked at a Caterpillar task and one related to “Doing 6 good deeds every 4 days” extended to 14 days. Despite materials being available, a student working on the latter did not use them. Yet once the learning tool was presented with the model (tracking both parts simultaneously) the student made links easily. Remember: Non-verbal mathematical actions can reveal important mathematical thinking, and learning tools can enable student perseverance through a mathematical task.
That’s all for Part 1, this will be concluded in Part 2 on Saturday (including mention of my own session).