Saturday, 25 February 2017

Not Teaching: Week 34

School keeps dogging me. I’m helping to coordinate emails about a conference, I discovered that a “return to system” form I thought was due in April is actually due before March, and my recap post about last Friday’s PD with Peter L has gained a lot of traction on Twitter today. In the meantime, it looks like teachers are now doing our own attendance, so I’ll need to make sure I’m in the system for doing that next September too.

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On the flip side (writing), this week I started posting “Time & Tied” to a new forum: “Royal Road Legends”. I also got approached by someone who runs a time travel blog about perhaps doing a guest post. I caught up in another serial. And, unrelated to any of that, the smoke detectors in the house gave up (they were due to be replaced) leading me to learn more about electrical wiring than I necessarily wanted to know. I can now identify a junction box.

*Item counts run Sunday (February 19) to Saturday (February 25).

Step Count: About 56,000.
Now at 27 straight days of 7500 steps or more - though many days this week weren’t much more than that, and involved a few trips around the kitchen island after 10pm to hit the target.

School Email Count: 78 New (15 sent)
See above for why this is so high; even had an email come in after 11pm today.

Writing/Art Related Items:
 -Inked in comic for last Monday. Wrote and sketched comic for next Monday.
 -Made a FanArt post with links for the math comic.
 -Did Math PD Recaps for 2017 and 2016.
 -Finished Commentary 25 for T&T (stats).
 -Started daily posts of T&T to RRL, and of Comic to tumblr.
 -Wrote some new character descriptions for an old MUX

Non-Writing Items for the past week:
 -Dinner with friends Monday
 -Chipped away at the ice outside in advance of the weekend rain
 -Dealt with smoke alarm issues, including buying new ones

 -Post recap about Anime North (from May)
 -Post recap about CanCon 2016 (from Sept)
 -Post recap about COMA Social (from Sept)
 -Split up “Time & Tied” into short parts for RRL
 -Catch up with web serials I’ve enjoyed
 -Write a TANDQ article on Polling and Bias
 -Write a post about types of praise/encouragement
 -Organize all the paper clutter from school
 -Organize all the electronic clutter from school
 -Weed through/organize emails
 -Do another Parody Math Video
 -French Citizenship project
 -Actually market some of my creative stuff
 -Binging Anime (Magical Index)
 -Binging Anime (Steins Gate)
 -Binging Anime (RWBY borrowed from Scott)
 -Read some of the books sitting at my desk

Still caught up in Timeless (now ended), Bones, and Agents of SHIELD. Supergirl got away from me this week when the site wouldn’t load the new episode. Being up to date in TV, that’s a thing that might be more interesting to track when I’m school-busy.

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Friday, 24 February 2017

Feb PD: 2016

Our board has a day in February devoted to Professional Development delivered by teachers. Here's an account of what happened Friday, February 12th, 2016 (with a 2015 addendum).

The day started with a set of three Carousel sessions. I was actually presenting in slots 1 and 3, the "Probability Project" I use in Data Management (which could be adapted for use in 4C, perhaps). The idea is that students research any topic they find interesting that relates to probability. I have two lists, one for "Events" (struck by lightning) and "Gaming" (boards in Settlers) as past ideas if a student is stuck. The research allows them to narrow down assumptions, either "informed" (I found data on the US, so I'll consider lightning in the US) or "reasonable" (the math gets complicated unless we assume only one person playing 21).

They submit the assumptions, I provide some initial feedback, and then they actually calculate probability and write a final report. The rubric looks at "Assumptions", "Mathematical Conclusions", and "Overall Form + Presentation" (I have them two a one or two minute report of the findings). In the data course, I find it a helpful jumping off before the larger statistical report (culminating investigation). A few people dropped by, one person wondering if I had exemplars - and they were sitting on my desk, but because we were at SirWil (my school) I could run and get them.

In between those two carousels times, I was able to see what others were doing. I lingered longest at the 3D Printing "Maker Mobile", which actually had dates set aside for our school board towards coming and 3D printing items. Didn't take notes, but for more details:


After the carousels, the first session I went to was Writing in Math (by Kevin Cheung, CarletonU).

Kevin started with a poll (, asking which type of writing work do you assign to your students the most? Options: Free writing; (Auto)biography; Blogs/Journals; Summaries; Word Problems; Formal Writing; Hardly Any. See below for more info.

He presented a question: Reduce 16/64 to an equivalent fraction in lowest terms. Note if you cross out the sixes, you get the right answer. Lucky. (A participant notes it also works for 19/95.) You cannot say the answer is wrong - you can only say it is wrong if the context is clear. Other classic examples: Folklore to show 1 = -1. (A certain rule that only applies when numbers are non-negative.) Or “Find x”; “There it is.” The issue isn’t the solution, but the wording, which you wouldn’t see in a math context.

Samuel Johnson: “I have found you an argument; but I am not obliged to find you an understanding.” The Ontario math curriculum lists 7 mathematical processes that support effective learning in math. What are they? (Connecting, problem solving, selecting tools, ...) Of note, COMMUNICATION: The process of expressing math ideas and understandings. There’s key words here which are skills that are transferable - not directly math related. Working through trigonometry you can learn about these things, reflecting and clarifying ideas.

Types of writing in math:
1. Free writing. Whatever comes to mind first. Think journalistic prompts (the Ws: Who, What, Where...).
2. Biography & Autobiography. About people.
3. Learning logs, blogs, journals. More for reflections.
4. Summaries. Concept maps, key words.
5. Word problems. The meaning of words, a more general notion than what we might consider. What does probability study? What’s the meaning of area?
6. Formal writing. Research projects and essays.

BEWARE: “Writing is an art and mathematicians do not cultivate it.” Morris Kline (1973)

Let’s focus on Word Problems. Not “what is the meaning of infinitesimal”, more something requiring a sentence answer. Here's how 50 = 51. A man puts $50 in bank. Then he withdraws 20 (leaving 30), then 15 (leaving 15), then 9 (leaving 6) then 6 (leaving 0). So total starting balance of 30+15+6+0 = 51. That’s “an example of what a con artist would write.” We totalled the leaving, not the withdrawals. With other numbers it makes no sense. Say enough to convince, don’t say more than you need to. If you’re so smart, keep it to yourself.

MAXIM 1: “Bluff and fluff are bad stuff.”

Many times we try to give students the benefit of the doubt. Full marks. But if you want to develop writing, that’s a no-no. If they missed steps, hoping it’s right, being correct is insufficient. We want to be able to convince we are right in a way that is mathematically acceptable. One would be rejected by a journal who can’t understand what you’re saying.

MAXIM 2: Keep the reader informed. (Say what you’ll do, do it, say you’ve done it.)

How to develop writing as a craft? Need opportunities to: Learn, Practice, Improve. To LEARN: Start from scratch. Assume students do not know how to write for mathematics. Ideas for Learning include, 1. Fill in the blanks; 2. Templates; 3. “Reverse Engineering”

Proof of how a negative times itself is positive.
Filling in blanks can help one to think logically. In solving an equation, each statement follows from the previous. If you never allow division, you don’t run into the problem of dividing by zero; always multiply, perhaps with fractions. We don’t want to start with too many options - limit the choices. For a really motivated kid, ask them to logically justify something like (-1)x(-1) = 1. Shows Euclidean geometry proof (it does take time to come up with examples like this).

From individual steps, give a sample template. eg. Rewrite equation into ax^2+bx+c=0. Identify a method for solving that. Conclude by stating solutions. (It’s not efficient, may be a waste of time for some people. We’re teaching the ideas.) For reverse engineering, ask students to come up with a question that best matches a given solution. Let y = tan(2x), then the given equation can be written as y^2 - 2y + 1 = 0. Factoring the left hand side gives (y-1)^2 = 0. It follows that y = 1, so tan(2x)=1, so solutions are x = pi/8 + k/2pi for all integers x.  “For a student who first sees this, it’s incredibly difficult.”

To PRACTICE: Ask students to rewrite a sample solution that is incorrect or badly written. (Don’t give feedback to students on their own work, in that they did something wrong. They tend to get very personal. Emotionally attached, not a good way to practice. Give the sample.) Alternative, ask questions that contain a slight twist. “Determine all INTEGER values x satisfying 2x^2 + x - 1 = 0”. Solutions other than integers may work?

To IMPROVE: Initiate revision/feedback loop for student work. No part marks, must get it by this deadline. Students need sufficient writing skills (and emotional maturity) before they can enter into such a process. As a teacher you have to write a lot and read a lot. It’s tough to pull off, but not many students may get to this level.

The 1 Million Dollar Question: How do you make students do the work? From Dan Meyer: “I teach high school math. I sell a product to a market that doesn’t want it, but is forced by law to buy it.” The student might hate math. They might even hate you. Perhaps reserve for the top 10%, make it optional, to reach a 4+ over a 4? Have this be the “extra 10%” part of the course.

Some resources? He couldn’t find any that are targeted to high school math writing. CEMC past paper solutions are good resources. Some books for university level: “A Primer of Mathematical Writing” (Steven G Krantz); “Handbook of Writing for the Mathematical Sciences” (NJ Higham); “Writing in the Teaching and Learning of Mathematics” (John Meier & Tom Rishel); “Writing Mathematics Well” (Leonard Gillman, out of print). Websites: and

“Do, please, as I say, and not as I do, and you’ll do better.” Some brief discussion after the presentation included notion of discrete math (something is either 0 or 1) and the formal writing in a Data Management statistics or probability culminating investigation. Lewis Carroll also came up (math in fiction vs. non-fiction).


After the snack break, my second session was Data Management with Mike Campbell. This year, he had four sections, diverse group of students. Some students who get it socialize, and detract away those who don’t get it. First question: Who skipped breakfast? Are males more likely than females? (Be aware, gender issues are becoming more of an issue these days. Phrase as “Those who identify as”.) Show the split bar graph. Used to let them play around on graphs, wasn’t sure that was fruitful.

Mike has a set of 1000 cards with data (printed from CensusAtSchool). Ask students, “To know about this population of 1000, how many cards would you like to have?” (eg. How many females are likely in the box?) The first group said 10, so everyone said that. Mike deals out ten cards each to everyone, to check. NOTE: “Don’t divide a box like this into groups, it’s impossible to shuffle 1000 cards randomly.”

How many teach confidence intervals? (I don’t teach any formulas I don’t get myself.) Here it’s done experimentally. We tell Mike how many people out of our 10 cards were female. Placed as dots on a number line. As many cards are below 5 as above 5? What about given a sample of 20? 50? (Mike happens to know “because I’m omniscient” that there’s 510 females so 51% overall.) A confidence statement should capture most of the dots. The person who got 8 females is 2.9 away. We can say that 10 out of 11 times we’re within 2.9 of the true answer. (Often close to what you get with the formula.) Thus, with only 10 cards, your answer may be off by 57%. Larger sample, more accuracy. There’s “Census at School” data, also 2006 Census Microdata.

“German Tank Problem.” During WWII, allies want to retake Europe. How many tanks do the Germans have? Spies are trying to count. Germans knew this, and would move the tanks around and change markings to make it hard to count. Asked the statisticians as well, who used serial numbers on numbers of parts from destroyed tanks. Assuming sequential numbers, what calculation gives a good estimate for number of tanks? This activity didn’t go too well (maybe a Friday) but good to bring history into the class. “Titanic Survival Data” (put together by amateurs) is one people enjoyed.

Session became a bit of a sharing roundtable, as Mike forgot some of his items at home. Mike added he stopped using the textbook entirely this year, it’s old (2003) and we don’t have enough for all students. He’s starting to work the kinks out of sets of practice questions. For the Culminating Project, a peer critique idea: Once project is underway, give whole period to share and comment for peer review on rough versions. For probability, games day, one period to do the whole thing (or story critiquing).

Sources of data? Gapminder. CIAWorldbook (fact book) database, where life expectancy is a requirement. What can be done with data - Ability to use computers, spreadsheets. Programming language of R. (Also called GNU S, for statistical analysis and graphics.) 

There's experimenting with cycling through the topics. CMEF conference and Peter L (Good questions. Visibly random groups. Vertical nonpermanent surfaces.). Doing random groupings can itself be a study. Hand out a playing card (to group), track the number of times you get it. What are the chances you end up “in same group 3 times this week.” Keep things open, look for opportunities.

And that was the OCDSB PD Day for 2016.

2015 Addendum

I seem to have misplaced my notes from that year. I know it was on February 13th, and that I went to a session from uOttawa about looking at things like paradoxes and infinity. For instance, the 0^0 issue, or how we define a function.

My second session was put on by Scott McEwen, who had participated in an exchange to Australia. I'd been writing "TANDQ" columns about other education systems at the time, and thought I might use the information. One of the main things that sticks is how Scott did a house exchange too (his whole family went), and how the low temperatures in winter (our summer) for Australia are problematic. Since even though it doesn't get as cold as it does here, they don't have heating to the same extent.

If I ever find my notes, I may spin that into another post. February PD for 2014 was actually pushed back into November 2013 (because the province was forcing "unpaid days" on us), and I did recap that event in this post. So, thanks for reading all the way, feel free to comment if you have thoughts!

Thursday, 23 February 2017

Feb PD: 2017

Our board has a day in February devoted to Professional Development delivered by teachers. Here's an account of what happened February 17th, 2017. (Less than a week later! Why so fast? Well, I'm not teaching at the moment.)

The day started (after initial announcements) with a number of carousel sessions around the cafeteria at Adult High School. Here's what I dropped by to see:

“First Lesson Idea for 3U Trig”, Ron Watkins.
Give students: Table of Angle, Height. Table of Angle, Base. Angle goes by 15 degree increments. In GSP (Geometer’s Sketchpad), he has a unit circle with a triangle, and measures out the height and base as he rotates it. He stops at 75 degrees and has them continue to 450 degrees. “They struggle, they try to find patterns.” Then in a third column, Ron has them calculate Sine (or for base, Cosine) - it’s not really opposite over hypotenuse. Then they make the graph from the data. This gets the concepts all in one shot.

“Messy Math: Transferring Mathematical Knowledge & Strategies into Doing Science”, Ann Arden.
Ann’s been sharing a prep room with the physics teacher, 4U. Students aren’t recognizing d=vt+0.5at^2 as quadratic in t, or want to change t into x. (Note: actual equation has “delta” on d and t, with vector arrows on d, v, a.) Or equation v=root(Gm/r), to isolate r they don’t square to remove root (they start by multiplying). Yet isolating variables is Grade 9.

Noted that this is the homework question from the math text that we would normally skip. Not suggesting we assign it - but do it on board? Can also ask the question “how is solving for r different than solving for G”. This is an issue post-secondary. Nipissing University has a site:

Other problems: Students don’t know deltas, they don’t know subscripts. Both of these are used in defining slope, again Grade 9. For chemistry, when asked a question and told to “include units” one student write “units”. We should talk units in math, not chuck them. Also in chem, concentration is square brackets [], and ln isn’t “one-n”, it’s base e.

”Using Google Quizzes for Formative Assessment”, Bradley Pinhey
Has used the quizzes in Data Management. For units, and also for review - if you can do the standard deviation question, no need to review that. Limited to multiple choice and checkboxes. Questions can come from Mastery Sets for MDM 4U and NY State Regents exams. Example:

”Puzzles, Proof and Problems”, Michael Campbell
Mike some manipulatives for proving the Pythagorean Theorem, after a Grade 9 student asked “show me why”. Place all red (a square and four triangles) onto the blue square, covering it. Remove the red square (which is c-squared) and rearrange what’s left to show two blue squares (which turn out to be a-squared and b-squared, equivalent to what was removed). An alternative cutout (see image) had a different shape using a and b lengths to rearrange for the proof.

I also wandered by the COMA table and other displays here. Chatted with Ian Winter, who is using Google Docs for Data Management, in terms of creating graphs, etc.

”Balancing Skills and Problem Solving in Academic Classes”, Shawn Godin
There are pros and cons to Problem Based Teaching. Shawn tried implementation in Grade 10. A focus on quadratic algebra would split a skill into stages, moving from concrete to abstract. Example: Factoring with tiles, all positives. Then with negatives. Then factoring using the grid. Then factoring with algebra. Then factoring of any degree (not only quad). There are quizzes of six questions to move between stages. An issue arises when students are at different stages at different times; it can be more work, not less.

Version two was done for Grade 9. Version three is moving towards self-regulation, having a skill list, tracking skill level using green/yellow/red, and with a portfolio summative. Plan going forwards is to not give homework, expect work to be done on topics to improve skill. Good problems can come from CEMC and Crux Mathematica.


The first thing after that sharing session was the keynote, delivered by Peter Liljedahl. I placed that into a separate post, for a quick and easy reference.

Next was an “unconference”. Two timeslots, a board for post-it notes to link topics to rooms, much like an edCamp. Most people seemed to be staring at when I approached, and not feeling like going to one of the three options, I put up “MDM 4U Data Discussion/Share” and went to room 312.

Two people dropped by - the person currently teaching Data at my school, and Bradley (from the carousels above). “How to lie with charts” has good deceptive graphs, and I need to do a better job of organizing my directory. It was also mentioned that Pisa scores are interesting when correlated, but SATs are more iffy, some states have low participation rates to factor in.

For the second slot, I went to the room on “Activity based learning”; some were still talking “Useful apps”. Like EZSolver (works like TVM) and talk of Desmos and turning off grids. The “Activity Based” seemed to be Alex O talking with JP about his card tossing, but I knew them and knew about that already so wandered back to the board. Met up with Nour and we went to “Assessing math practices”.

Pierre Trancemontagne had proposed the topic. (From French board office, CEPEO.) Someone else was there too. Talked of how teaching to the test, never a good thing, yet can show increase of performance. Teaching from the textbook, you’re not focussed on EQAO, but that doesn’t mean the textbook is richer.

Pierre mentioned that the newer practices is easier to implement when there’s fewer numbers than in a larger board. He works with 8 French schools, whereas on the English side there’s 50 schools (and Catholic board which has funds). For EQAO in Grade 6, number sense was higher in French board vs English; thoughts were that on French side the standard algorithm is held back until Gr 5/6. More “personal algorithms” used. In fact, in discussing with Pierre later, there’s a few other differences, like “Proportional Reasoning” in Grade 4. Apparently the 2005 curriculum revamp was done in parallel for languages, not integrated.

Discussion shifted to how evaluating EQAO is different from what’s in classrooms. I mentioned the 10/20/30/40 from an OAME session last year, including how a right answer with no work is a 20. Pierre’s high school colleague (also CEPEO French board) noted how French doesn’t need to have the two summatives. So can treat the EQAO as 30% exam, to not overload Grade 9s coming into high school. (Some did 15% multi choice and 15% own exam, there was a directive, if you’re going to mark it, mark all of it.) Mentioned how field questions on EQAO may not be the same across the province, though official questions are standard. (I didn’t know that.) Things wrapped up, we were already off topic.


After lunch was the session I had specifically signed in for - it's something that I don't think I do a great job of teaching.

“What are Career Opportunities if you are good in Math?”. Rafal Kulik, from OttawaU.

Rafal’s goal is to “convince you, so that you can convince your students, that knowing math or even having a degree in math pays off” - brings lots of career opportunities. He quipped that he’s not sure if we’ll want to change careers after this, but “we need you in schools”. (He has two teenage daughters, can’t imagine teaching a whole class.)

He started with many things we may already know (then moved to small exercises we could develop more later). Some key words related to math are: identifying/solving problems, analyzing data, identifying patterns, simulating and computing, writing and presenting what you learned. The point is, mathematics is everywhere. Yet if you ask what kind of career will need math, you tend to get: Teacher, Researcher in Academia. (Nothing wrong with this, but there’s more.)

Statistician/Data Scientist: One thing you hear nowadays is “big data”. There’s 150 openings at this moment at very different levels - starting at sports teams.
Actuary: Things like calculates your pension (how much you will get) and predicts your future life. Must be updated every year.
Data Analyst: For computers, economics, operations. To students who say “I want to do computers but not using much mathematics”, writing a program is using mathematics, just in another language.
Researcher in Industry: Applied sciences, like economics. StatsCan (in government) must predict based on research. HealthCanada needs to manage vaccinations, to target particular parts of the population. Also analysis of finance, risk. Or modelling in medicine (epidemiology, physiology).
Computer Programmer: For data that needs to be analyzed very quickly. Transportation, optimization of routing of planes, or how cars are built (using a big wind tunnel, optimizing the flow of air to minimize drag, have good fuel consumption). They must hire people who know differential equations (airflow equations). Also Communications/Information Security: Cryptography, electronic passwords, etc. “Mathematicians are specialists in problem solving.”

The “Wall Street Journal” reported on in 2015: Top jobs in US (rankings considering environment, salary, job satisfaction): 1-Actuary. 3-Mathematician. 4-Statistician. 6-Data Scientist. (There’s no “Teacher” because one drops lower in ranking if there’s high stress...) And, College Salary report 2015-16 gave the top 40 bachelor’s degrees (ranked by salary expectations), many are based on mathematics. (Engineering, computer science, economics, mathematics, statistics...) The message again is, here is the evidence, if you do good in math you can have a good job. (Not the same pay as a lawyer, but the stress is less, very often more flexible hours.)

Where do our students work? (Rafal had a sample of his students.) Government agencies (Statistics Canada, Health Canada, Environment Canada, NAV Canada) - consider image recognition for Canadian Border and Security - as you age, you change. IT companies (Google, Microsoft, BlackBerry). Financial industry (TD Canada, Bank of Montreal, Export and Development Canada). Research hospitals (CHEO has a research unit, Heart Research Institute) - need (Bio)statistics, to see how drugs work, variability in people, plus applied mathematics. CSIS (pure math). DND (Department of National Defense).

Here are some related classroom activities.

Activity 1: How does GPS Work? A bit of history: In 1957, USSR launched satellite Sputnik. US military launched first GPS in 1978. Major development started in 1983, Ronald Reagan declassified the GPS after the tragedy of Korean Air 007 (shot down in Russian airspace due to navigation issue). Full operational capacity with 24 satellites was announced in 1995. Satellites orbit the Earth twice a day at altitude of 20,000 km. Could use clip of Liam Nieson. In “Taken 2”, he calls his daughter to help him where he is.

How GPS works in a nutshell: You are lost at OttawaU campus. Ask someone, where am I? “You are 200 m from the University Centre”, gives a circle of radius 200 m. “You are 150 m from Math department.” Next circle narrows the intersection down to two points. “About 50 m from the main building.” That circle must intersect with the other two, and hits your unique point. This exercise should be accessible to grade 7 students, “I’m not familiar with curriculum but this is what my gut tells me” - uses radius, perhaps circumference.

Is this it? No, this is just the beginning. The earth is a sphere (could talk equation of sphere) so need spherical co-ordinates (which requires cosine/sine function). A GPS receiver (your smart phone) actually sends a signal (speed of light) to a satellite. GPS measures the time that the signal travels and converts it to the distance. (d = c*t, c is speed of light.) Then heavy math is involved, related to signal processing. “This is not something you can do in 15 minutes, but three circles is the basic idea that each kid can understand.” (Note: GPS communicates with 4 satellites; in theory 3 is enough but 4th to check.)

Can also ask, “why do you have mistakes when you use GPS?” If the time is not recorded property (in reality tiny fractions of a second) then resulted mistake is 300 m off. Satellites have atomic clocks to be as good as possible. So kids, “If you do a mistake in your calculation”, you may end up with very bad results. What careers does it lead to? Telecommunication companies (wireless phone, radio software). DND (Department of National Defense). See also book source: “The mathematics that power our world: How is it Made?” if you’re looking for some inspirations (“I’m not getting any royalties”).

Activity 2: Cryptography. Bob sends a secret message to Alice. Both know the key. With only digits 0-9, we have 10*9 possibilities; using a computer, anyone can quickly break the code. Relates to Combinatorics in Data Management. Modern encryption keys are based on prime number factorization. “Really needs an hour to explain how it works” but RSA encryption in brief: Bob sends a number N, a product of two prime numbers. The public can only see N. Factorization is difficult, and breaking the code means finding these prime numbers. (Link to factoring. Also, if number is 2178, we see it is divisible by 3; this is why it’s useful to know divisibility properties. Don’t test numbers you know won’t work.)

Is this it? No, next step is modular addition, number theory, exponential functions and the Euler function. Have computers and algorithms that work better and better - also, Quantum Computing. What careers does it lead to? CSIS, IT companies. (Need B. Science in Mathematics)

Activity 3: Financial mathematics. Motivation? Rafal says he was once a day late with a credit card statement, had to pay $100 interest. Was told “There’s a formula in your credit card contract.” Or mortgage advisors, they may want a good deal from their perspective, not yours. If you have experience in finance, you can pick up when someone is trying to do that.

Need a loan of $10,000. At the end of each year, you pay $2000. Annual interest rate is 10%. How long am I going to pay my loan? (Not 5 years! After year 1, the balance left is $9000.) Ask kids to come up with a general formula using symbols before giving exponential equation. Is this it? Can add complications, payments every month (interest rate division by 12). Random interest rate (if coin flips heads, then 10% rate, if tails, drops to 5%) which leads to different scenarios, adds combinatorics and expected value.

What careers does it lead to? Employment opportunities in financial industry (Banks, Export and Development Canada). His former student (Sabrina) is a Market Risk Analyst at BMO after working on “some financial things” with her over the summer. What I try to tell students: This is when you learn math and then do the applications. It’s better than learning applications, and then trying to understand the math, that’s more difficult.

Activity 4: Statistics. Say we divide a 30 person class into 3 groups, and ask individuals “Do you like the school?”. In the whole class (the population) 50% do, but in a group of 1-9, 10% do and in a group 9-1, 90% do. (Last group is 5-5.) Statistics is about sampling and variability, the table illustrates sampling variability — some groups are not representative. To generalize, if you ask 1000 Cdns, “do you like Donald Trump” and 500 say YES - does it mean that 50% of all Canadians like him, or is there a sampling bias. Leads to the science of how to infer results from a sample to a population. Enormous employment opportunities now: Government agencies (Stats, Health, Environment); research hospitals and everywhere. The job market for graduates is so huge, there is difficulty getting Statistics students into Masters programs.

Activity 5: How does Netflix recommend its movies? Data Science. Assume 3 are available (Pirates, Diary, iCarly). A 15 year old girl logs in, which one to suggest? To answer, we use a Decision Tree based on Historical Data: Gender, Age, influencing Movie Chosen. What is more important (for top of tree), gender or age?

Given this data, we see all males watch Pirates, regardless of age; women watch others. The age is not a determining factor - if a man logs in, we know what to recommend. If a woman logs in, now we go to age. If she is 15 years old, there is a 66% (2/3 historically) she will like iCarly. Current decision tree is gender at the top, then age. Whenever you provide your data, it targets.

What careers does it lead to? Different Machine Learning algorithms. Jobs at Google, Amazon, Netflix, etc. Any institution that needs prediction, classification (e.g. credit card fraud detection, self-driving cars). TD Canada Bank is opening a Data science group in Ottawa (need joint honours in Math and computer science).

There is outreach at uOttawa. Math Horizons and Math Camp (Joseph Khoury) in June. Holiday lecture series. NEW: Video competition for high schools on “What is math”. (What do you think math is? Cash/Scholarship prizes.) NEW: Regular visits at uOttawa or at your school. Working on next: Math fair, present a project on a particular topic like the ones you’ve seen. Teach kids that math is good!

There was a question at the end about the joint Math and Economics program, Half and Half. High end requirements into program, it’s very difficult. Very little room for exploring other subjects (three depts: school of business, economics and math).

And that was it - after a return to the auditorium to draw for door prizes. Thanks for reading! Feel free to comment with anything that jumped out at you. As a reward(?) for getting through, here is a link to a music video a student produced for me last year. It's "Vertex" to the tune of "Go West" by Pet Shop Boys. Enjoy.

Wednesday, 22 February 2017

Feb PD: Peter Liljedahl

Our board has a day in February devoted to Professional Development, delivered by teachers. For 2017, our subject council brought in Dr. Peter Liljedahl from Simon Frasier University in British Columbia, to deliver the keynote. I present that here, along with some follow-up at the end; the rest of 2017's PD Day is in that separate post.


Peter started “with my last slide”, so that you could find him and/or see this presentation:

He’s based his findings on a lot of literature, and he flashed up papers noting “some of these are not so boring”. Our story starts about 13-14 years ago. Peter was asked to come in and help a middle school implement problem solving. He was a former high school teacher and PhD student at the time.

This was great. Though the first thing the middle school teacher said was she didn’t “need your glee and enthusiasm” - she wanted tasks, not a co-teacher. Peter said okay, but if he gives a task, he gets to see how it’s used. Which brings us to Jane’s Class (2003), and the first problem, one of Lewis Carroll’s: “If 6 cats can kill 6 rats in 6 minutes, how many cats are required to kill 100 rats in 50 minutes?” Peter had used it in many settings, it had gone well, he thought “this is going to be awesome”. Note “Jane” had never done problem solving before this.

Peter asked us, “What do you think happened?” (The audience is silent until someone pipes up “Like what’s happening now.”) Disaster! Peter explained it as, arms went up, and the teacher ran between students for 40 minutes, trying to push them to do it. As soon as the teacher left, they sat there, and hands went back in the air. After, Jane came to see him in the back of the room. I figured “one and done, I’m out of here” but she asked for another task. This woman’s got tenacity! Second time, disaster again. Gave her another one. After the third, she said, “I think that’s enough.”

It had been painful for everyone. Students were in pain, Jane was exhausted, and Peter was bleeding internally. He asked, “Can I just stay in class the rest of the day to watch what you normally do.” Then stayed there for 3 days. He had an epiphany.

At no point during the regular teaching had the students been expected to think. Sure, they were doing stuff, solving questions, but all the thinking had been sucked out, either through direct instruction or modelling or the like. Peter went to another math classroom and saw the same thing. He went all over the lower mainland of Vancouver and kept seeing same thing. It doesn’t matter how good texts are, or how smart a Smartboard is, or whether desks have seats attached to them versus sitting at tables. Nothing is going to matter until we get students thinking.

The person doing the thinking is the person doing the learning. Right now, the best one can hope for is good behaviour. Peter started on a quest.

He spent lots of time in classrooms. Good ones and bad ones. Came to a realization: Classrooms all over the world are more alike than they are different. It doesn’t matter if it’s Grade 2 or Grade 12. It doesn’t matter if you’re in a low socio-economic setting or a privileged private school. It doesn’t matter if we’re in Canada or Taiwan or Chile. Classrooms for the most part look the same.

Students sit in desks, usually oriented towards a teacher, who provide knowledge to those students. It’s been the same in every classroom for 100 years. Yeah, desks now have wheels and the board is white, but fundamentally, nothing has changed. Maybe it’s time to pay attention to some of the things that are taken as “non negotiable” in classroom settings. Peter started on a project. (Aside: “I have no idea what my next slide is.”)

He recruited more than 400 teachers to work with (casting about), and started with a contrarian approach. Figuring the best way to start is to just do the opposite, and see what the effect is. Students are used to sitting? Stand them up. They’re used to us answering questions? Let’s stop. They’re used to furniture in the room? Take it out. (They got calls on that one.) They’re used to us talking? Let’s not talk at all. (Aside: “That doesn’t work.”) The project always started with the contrarian approach; sometimes, it was the best option. Always, it brought them closer to what was the best way.

Peter also experimented with problems. It’s a myth that if we want students to learn through problem solving, we just find a magical book that will wash knowledge all over the student. “I don’t have that book.” He spent a year and a half on problems. Also this:
-How we give the problem. We could write/project on a board, or use a textbook, or handout. Or other ways.
-How we answer questions. We’re implicit with “learned helplessness”; being helpless is a good way to move forward.
-Room organization. Desks in rows, in pairs, in clusters, desks like trapezoids, round tables. Arranged in horseshoes, or the Ikea model, where you walk around the whole room to get to the door.
-How groups are formed. It’s February 17th; in 12 days, elementary teachers all over Canada will create new groups. “You two can’t sit next to each other.” Sometimes that’s about ability, sometimes about management... does it matter?
-Student work space. Could be desks, computers, tablets.
-Automony. Some students have none; some have more, and play in the back.
-How we give notes. Essentially, how to get them from us to students. On a board & copied, upload them online, fill in blank sheets?
-Hints and extensions. What’s the best way to provide them? Or is it everyone try this for 3 min 42 seconds, then reveal the solution and continue. Related: How we level.
-Assessment. Still playing with this, we’re probably doing it for our whole lives.


The teacher who had no furniture in their room, they learned from that. They learned “don’t do that”. Sometimes, they learned things that didn’t work, but that was because they needed to tweak it. Or have someone else try it. The project zeroed in on the things having a powerful effect. On what answers: “Does this contribute to student thinking?” Will the students think more, will they think better? Here are the results from those above points.

-Problems. With a good category of problems for the first week, afterwards, problems can come out of the textbook. But still have to think about it, ixnay on pre-teaching.

-How we give problems: Use oral vs written. We’re used to the written form, but what worked better than anything is oral. Every demographic, didn’t matter, giving problems verbally worked better every time. “To be fair, I can’t give you the problem 3x to the one half plus..." etc. etc. Some problems we can’t give verbally. Some we have to show them text or graph. But all instructions have to be verbal, it has a huge effect.

Because when instructions are textual, the first things students do is decode and read. And in a group, they read separately. But verbally, the first students do is talk to each other, even if they don’t like each other. What’s the absolute worst source for students to do problems from? The textbook. Orally was good, this ‘other method’ wasn’t bad, down here (Peter motions low) is the textbook. “Sorry to the publishers here.” It’s because textbooks have embedded an institutional norm. The assumption is you’ve been shown/read about how to do the textbook problem in advance.

-How we answer questions: There’s 3 types of questions. 1) Proximity. You’re close, the most student-ly thing I can do is ask you a question. 2) Stop thinking questions; asked so the student can stop thinking. Most popular, “is this right?”, second most popular, “is this on the test?”. 3) Keep thinking questions, clarification questions asked to get back to work. Either of first two feed into the “learned helplessness” - answer only the third type. Caveat: A hilarious thing happens in a kindergarten class if you don’t answer. They follow you, and ask the question again. One teacher had 7 kids following after her, like a conga line. So, what to do instead of answering the question? They tried all sorts of things (like turning it back, “What do you think?”). The best thing? Is to just smile at them, and then walk away. The walking away says I’m deliberately not answering.

-Room organization: The best rooms for thinking were chaotic, in disarray. De-fronting the room means that, with no discernible front, the students are more apt to engage. They don’t know where to point at. Stop anchoring their desk as being their place to think. All of a sudden, defronted, the whole space becomes something to use.

-How groups are formed: Use visibly random groups (VRG). It worked better than anything else.
-Student work space: Vertical non-permanent surfaces (VNPS). More on those in a minute.
-Autonomy: To think, we have to create autonomy. Create a space AND push them into it. We have to be ambiguous, we have to be elusive, and give them choices about what to do about this ambiguity. But also had to push the students to use this ambiguity.

-How we give notes: Don’t, at least not how we normally do it. When students were copying down what was on the board, in every class Peter studied it, more than half the class was at least one example behind. And never more than 5 students went back to their notes. (Usual number was 3 students.) Fill-in-blanks notes turned out to be WORSE, the students were paying attention, but only listening for key pieces (to fill in). They’re still working on this. Best, students write their own notes (as “Notes to my future, dumber self”).

-Hints, etc is about “Managing flow”. Level to the bottom. Four purposes of assessment.

What do all these things have to do with thinking? All are not equally powerful. If you go to school Monday and the only thing you do is not answer questions, you’re going to have a bad day - that’s a finishing item. What’s the best way to start transforming? Begin with sledghammers and chainsaws, those being: Good problems, VRG, VNPS. Framing hammers are next, then finishing tools. When we’re ready, we slide into the next set of tools. Not answering questions, that’s in the middle (along with defronting, and oral instruction).

The project found very quickly that students were most engaged when standing up and working on a whiteboard. The cynic in Peter wanted to test this, so did a controlled experiment in five different classrooms, Grades 10-11-12. He let the teacher decide what the groups would be. (Strategic grouping.) Then Peter decided which group would work on which surface. Some had a whiteboard standing, some had it on desks, some stood and worked on flip chart paper, some had that paper on desks. A control group worked in notebooks the way they normally do. Did in five different classrooms in five different schools, randomly assigning students to a surface.

Their proxies for engagement: Time to task, time to first math notation (versus deciding on a group name). Also on a score of 0 to 3 (0 if none) they measured: Amount of dissuasion, eagerness to start, participation, persistence, knowledge mobility, and non-linearity of work (was it messy; shows real thinking, not copying). “The thinking process is messy.” They averaged scores for groups of same type.

Peter put the results up. Discuss. “Someone tell me something you notice.” (Responses included “Nonpermanent seems to be performing better”; “Notebook performed well to start”). The better non-permanent scores was determined to be since “they can make mistakes”, it’s easy to erase, freeing them to risk more. The irony is, they don’t erase. But because they can, they’re quicker to start.

Mobility of knowledge sucked when they were sitting down. Hard to work between groups that way. Standing nonpermanent is better than sitting nonpermanent - but why is standing good? They’re still unpacking that. (Audience suggestions: “Because you’re on display” “Everyone can see everything equally”; Peter remarked “If your bum is numb, your brain is dumb”.) So they stopped saying “whiteboards”, because vertical mattered, and nonpermanent mattered, hence hashtag #VNPS (Vertical Non-Permanent Surfaces).

What’s it look like? Looks like this. (Peter showed a number of rooms. One was Alex Overwijk’s. One a careers class. They brought in bar tables instead of desks; the principal made the teacher call them cafe tables.)

On to VRG (Visibly Random Groups). Random groupings started to show more success over other methods. Peter went into the harshest environment he could think of, a high school in Vancouver that was racially bifurcated, 50% caucasian versus 50% asian. No wars in the parking lot, but they create their own class system. There was social stratification, but also split in two. They moved through each other, they didn’t integrate.

Peter spent 6 weeks in the room of a teacher. After 3 weeks, this is what he saw: Students became agreeable to work in any group. (Became. To start there were trades, subversions - “How is there a group of 5, really?”) Ultimately they’d decided “I can work with anybody for an hour”. He presented a story, where individuals were learning about individuals in the other racial group.

VRG means elimination of social barriers. The mobility of knowledge between students increases. A reliance on co-constructed intra- and inter- group answers increases, while a reliance on the teacher for answers decreases. Engagement in classroom tasks also increases, Peter admitting, “I didn’t expect that one”. At parent-teacher interviews, eight parents said “my son/daughter loves coming to your class”. Students became more enthusiastic about math class.

Put those together, here’s what it looks like. Peter points out, “This guy belongs to that group [over there].” While a guy in front is part of the group where “this guy” is. These are highly collaborative, active, engaging spaces. BUT “I want to emphasize something here”. What was learned goes way beyond vertical surfaces around the room. That’s just the opening act. The rest of the tools are what get at the thinking, what drive the performance and engagement.

The work continues. We’re still learning. “I’ve added two tools this year.” You can clap, we’re back to his last slide.


There were breakout sessions with Dr. Lilijedahl through the day. I didn't sign up for these, so I don't know exactly what went on - but part of the reason I didn't sign up was because I had the opportunity to attend one of Peter's sessions back in 2014 at the Canadian Mathematics Education Forum (CMEF). See this link for a full recap. Here's a TL;DR (Too long; didn't read) version, since that was a long conference:

Picture a graph, with a horizontal axis for "Skill" and a vertical one for "Challenge". You need to vary challenge as skill increases. Little skill with too great a challenge creates Anxiety. Great skill with too little a challenge creates Boredom. There is a channel between these (picture a direct variation from where they join) which creates FLOW. From research by Csikszentmihalyi (1990) there are nine items that help people experience flow.

Three of these are external, which teachers can effect: There are clear goals every step of the way. There is immediate feedback on one's actions. There is a balance between challenges and skills. The other six are internal: Attention is focussed on one's actions. Distractions are excluded from consciousness. There is no worry of failure. Self-consciousness disappears. The sense of time becomes distorted. The activity becomes satisfying in it's own right.

Managing flow is nigh impossible in a class of 30, but is achievable using smaller groups (a reason why VRG is part of the first steps). You cannot create an engaging task that hits "an expectation", because it would remove the possibility of solving the task using other options. You can merely create a problem that goes in the right direction, and see what different groups do with it.

If you were at one of the breakouts, feel free to comment with whether this relates to what was covered, and whatever else might have been said. Or simply remark on something you found interesting. You can also check out Alex Overwijk (referenced above), who has blogged about these topics as well.

As for me, I thrive on order and structure, so bombshells like this take me time to process. We'll see where it goes. I also do unconventional in a different way - for instance, my personified math comic is currently doing a derivative arc. Okay, done now, thanks for reading.

Saturday, 18 February 2017

Not Teaching: Week 33

This was another school-ish week. Had a math meeting Monday, and Friday was the subject PD (Professional Development), which I attended. Got to chat with math teachers not only from my school, but around the board. Add to that how, in the middle of the week, I went to a board meeting about the closure of the high school in my neighbourhood. Also mid-week-ish, Valentine’s day, where I successfully cooked a small ham with cloves.

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Finally, the “Vertex” music video, done by a student last year, is now online. In less than 2 days, it’s matched the view count for the 7 weeks my holiday special has been online (19 views). I need to not look at stats. Better that I look at fan art; someone drew Hyper for me today. O.o :D

* Item counts run Sunday (February 12) to Saturday (February 18).

Step Count: About 65,100.
I’m at 20 straight days of 7500 steps or more, though only Monday had a high count.

School Email Count: 58 New (6 sent)
Busy-ish week.

Writing/Art Related Items:
 -Finished other half of OAME recap, and posted
 -Finished T&T part-split edits through Book 1
 -Drew comic for next Monday

Non-Writing Items for the past week:
 -Yoga Monday
 -Math Minutes Meeting Monday
 -Attended Board Meeting about school closure Wednesday
 -PD Day Friday

 -Post recap about Math PD (from Feb, Times Two)
 -Post recap about Anime North (from May)
 -Post recap about CanCon 2016 (from Sept)
 -Post recap about COMA Social (from Sept)
 -Split up “Time & Tied” into short parts for RRL
 -Catch up with web serials I’ve enjoyed
 -Write a TANDQ article on Polling and Bias
 -Write a post about types of praise/encouragement
 -Organize all the paper clutter from school
 -Organize all the electronic clutter from school
 -Weed through/organize emails
 -Do another Parody Math Video
 -Replace smoke alarms with Christmas ones
 -French Citizenship project
 -Actually market some of my creative stuff
 -Binging Anime (Magical Index)
 -Binging Anime (Steins Gate)
 -Binging Anime (RWBY borrowed from Scott)
 -Read some of the books sitting at my desk

There was an excessive amount of snow this past week. I’ve been amusing myself today digging trenches for the thaw.
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Thursday, 16 February 2017

OAME 2016 Ignites

This is a post I should have put up over 8 months ago. (These recaps were typed up on Friday, May 6th, the day the presentations occurred.) I nearly posted it here to end May 2016, but ended up musing on Supergirl instead. Since then, I’ve felt like maybe I should do more with it. At this point, it’s about getting it out there. I may have still beat the videos. You can also see prior OAME Ignite 2015 and OAME Ignite 2014 on this blog.

For those unaware of the format, speakers have 5 minutes to deliver a talk, using slides that advance automatically every 15 seconds. There were 15 speakers (down one due to a family emergency) and presenters played “two truths and a lie” where they gave 3 “facts”, revealing after which one was incorrect. I didn’t track that. For more context about the conference where this occurred, you’ll want to read this prior post.

Enjoy these rundowns, typed in real-time, then slightly edited months later for readability.

1) Matthew Oldridge - from Peel

-Math is...
-It is what parents and teachers make it. Which is what this story is about. Amazing, terrible, awe inspiring beautiful...
-There’s innumeracy in the world, like the infamous double decimal (gas station).
-Andrew Wiles’ proof of Fermat’s Theorem, “This is Mathematics” is not school math, so what is school math?
-Shows quote from 1990, we’re already one generation within the reform math movement.

-We’re very good at perceiving patterns, and at teaching this. The exertions of common sense by other means is doing mathematics. We want this judgement by students; deal is a rip off.
-What do kids say, a way of describing our world is math. Malke Rosenfeld from Iz, Age 10. Also DFJH_Nitchell, Stella, less happy. On Twitter. Early Math is Serious Math. But we have to get playful too.
-“I have infinity money, I have more than I can count.” (like Donald Trump) What is the biggest number you know? We don’t even know place value yet.
-Infinity wondering is as dividing down towards zero. Things we can play with.
-Don’t be like the goat saying “I was just never any good at math”, we have the power to shape students attitudes towards math.

-Strategies and activities play a part. Classes should be talking, thinking, conjecturing, wondering. Yes, basic skills and concepts, but also thinking, always yes.
-Grade 6 student: #MathIS the most powerful force in the universe... use it wisely, use it well.

2) Ron Lancaster - from Toronto

-What to be “passionate about”.
-Ron considered a career in Computer Science after being a finalist at the 1970 Int’l science fair. The idea died when almost failing CS at McMaster University; an Assembly Language grade of 57.
-Passion rekindled in 1985 when a student brought a Casio graphing calculator to class. July 1990 saw the TI-81, then in July 1993, the TI-82 REALLY changed everything, had a port for sharing programs. Len invited him and Bob Hart to do more, T^3. (Teachers Teaching with Technology.)

-On Dec 31 1994, students ran a program to have the TI screen take 10 seconds to drop the New Year’s ball. More programs with putting two calculators together, creating pictures.
-Using it to make a graph is nice, but to do these kinds of things (seen on slide) is at a much higher level. Look up these programs, there’s many for the last 25 years. The trend is to make the programs interesting. There’s a Stereograms program! We can get kids interested in programming this way.
-We need to be more open minded, not linear in how we view mathematics, or we’ll say there’s no time. We need to get with “the program”. :)

-Final thoughts: Ron is grateful to Len Catleugh (retired after 43 years) for inviting him to be part of the T^3 family. 60 outstanding Cdn educators.
-In 1996 the OAME game Len an award for contributions. He’s retiring in June. Talk is to thank Len as much as this is about coding.
-Ron left as a MYSTERY which of his three intro statements is the lie.

3) Cathy Bruce - from TrentU

-Spatial reasoning. @drcathybruce
-With young children and teachers, read a UNICEF study, 12% achievement gap between high and low resource children. Didn’t think it was so high.
-Math is the best predictor of school success. All students are capable. M4YC (Math For Young Children), fundraisers of the research over the years.
-It’s polyhedra, but spatial reasoning is also: non-verbal reasoning, orienting, scaling, etc. (many words on slide)

-Imagine you are a bird flying directly over this flower (imaginary perspective taking), which you would you see?
-Consider symmetry, not only horizontal and vertical, but it’s all over our world and 3D figures. (Symmetry versus mirrors of each other.)
-Grids, orienting, mapping, are foundations to coding; a form of pre-coding. Spatial reasoning is dynamic; this tweet shows visual proof with a mirror for circle thinking.
-Playful Pedagogy, exploring this, asking children to play with math ideas. View things in different ways, then verbalize, verify through exploration. Open tasks with no floor or ceiling, use bodies to explore equality ideas.

-Working with under resourced children, they’re surpassing their peers, and weirdly and wonderfully they’re also showing gap closing in number (magnitude) also.
-Mental rotations versus crossword puzzles. The former did better with addition problems.
-Spatial Reasoning is a great dance party, it spans across geometry into the other strands of mathematics and our daily lives. “Taking Shape” (published a book), activities to develop in Grades K-2.
-Look at those girls, they deserve to do spatial math and complex math.

4) David Costello - from PEI
“I’ve never used a microphone before, I’m pretty loud as it is.”

-Units of Study. Building blocks or thinking blocks. How do we plan our math?
-How it started: @dr_costello tweets “why is math broken into units”. Learning goal, how does the yearly plan impact student understanding.
-Connection to the classroom. What does your yearly plan look like? “We’ve always done it that way”, it was originally designed as a method of separating concepts. (Teach - test - move on.) What is taught is not necessarily what is learned.

-Similar to “Speller” program: Short, discrete units. Words are separated based on prefixes, sounds, etc. Spellerization. Such short term compartments, encourages the idea it’s something you have or you don’t have, without focussing on the understanding.
-Fractions quiz. 21/20, solved all of it, but don’t know how to get it. Comprehension vs Decoding. In regards to math, completing procedures. Need a Multilayered approach to instruction. Connections made between concepts to help synthesize learning.
-Place Value. 4 Math Operations. Why separate them? Wouldn’t it make sense to integrate? Name one math concept that does not relate to another. “Laughter is good”
-Spiral vs Linear, we still treat as not. Moving short term memory to long term memory, for a spiral. Processes that we focus on, yet when reporting, we structure it back through discrete stoics and strands.

-Flexibility is a necessity not an option. How I got here from PEI.
-Instructional Goal: Not to cover curriculum. It’s to get students to be multidirectional problem solvers. If we teach in silos, we promote compartmentalized thinking.
-If I can ace a test but not use it, what’s the point? It’s not about covering curriculum, it’s about students understanding.
-“I’m counting down the last few seconds, what a great looking crowd you are.”

5) Marian Small - from Ottawa

-Years ago, I (Marian) was the mom of school aged kids. I was involved. “I was a good mom”. Supporting the school and teacher.
-I was also an elected school trustee in NB and I fought for what I thought was right. Fighting to save music program, for better teachers/admin
-One battle was fierce, I sided with parents against the board. Eventually I was no longer a trustee and I told my kids I just couldn’t care anymore.
-Wise daughter: “Mom, some day I’ll have kids and you’ll get involved again.” So here we are. She was right.

-Being a citizen, here’s what we need in a school system. Teachers who are kind, supportive, positive with kids. Teachers who “know” their students. At interviews, I don’t want to know the mark, I want that you know her.
-I expect teachers to work hard at improving their craft - and you are those teachers, or you wouldn’t be here today. I want teachers who “personalize” their teaching. Given “This is what I do” vs “We’re supposed to do this”, I preferred one, guess who.
-I expect a system that makes it easier for teachers to collaborate. One that doesn’t turn a blind eye to teachers who are not kind to children and/or who don’t teach well. Should ensure leaders, in schools or central, are both expected to be, and given time to be, instructional leaders (not managers).
-Curriculum is more about instilling a love of learning than covering expectations. Do help students see math as a way of thinking, not as something you do for 60 (or 75) minutes a day. We shouldn’t make teachers choose between meeting students needs and delivering expectations. Your curriculum document doesn’t give enough information, in my opinion.

-I expect a system that values intellectual debate about it’s workings. You’re allowed to have constructive criticism with those who are in charge. Have an honest look at the dilemma of marks.
-I hope this is not just my dream. I am not a teacher or education official. But I hope those of you who are, share my dreams, and help make them a reality.

6) Jill Gough - from Atlanta, Georgia

-What do you see when you look at this face? Honours student? Someone curious? High achiever? Low?
-Leadership is not about being in charge. It’s about taking care of those in your charge. Simon Sinek reference. This girl is working hard. How are we helping her attend to precision, maybe in more than one way?

-“I know that before, I am good at math”. We want that for every learner in our care. Unlocking the Hidden Potential of the Dyslexic Brain. The battle to develop confidence, resiliency and positive self image is large won or lost in school.
-“The Myth of Average”, Todd Rose TED talk. Average doesn’t really exist. We need to be aware, even if we don’t say them aloud: Words Matter. Can we let go of expectations?
-Do people who learn by making things have the opportunity to show it? How do we offer opportunities for those students? Or those who need to think deeply before talking? Those who are dreamers, who make statements that don’t seem to be connected? We don’t want them to be shut down.
-Are our environments communal? We want them to become confident mathematicians. Do they know how to make sense of a task and persevere? Do we show our thinking, so that they can begin to learn and grow?
-Teacher says to a child that “this does not make sense”. Encourage them to make that thinking visible, then we can see their success in many ways. Flexible, artistic, creative students. Then when they understand, we help them to #ShowTheirWork

-Do we teach our students to use colour and imagery to help things make sense to the reader, and to have a community where feedback is sought instead of feared? Has a picture “Feedback needed”.
-When we hear the question WHY, why show things in more than one way? Because our children need us to, to take care of those in our charge.

7) Mary Bourassa - from Ottawa (OCDSB)

-150,000 Questions. That’s a rough estimate of the number of questions asked of students. Of course, I had a concussion when I came up with that number.
-But how many were good questions, helping students to move forward? Reality is, probably not that many. Wonder how big your inner circle would be if you asked that question.

-Enter the MTBoS (Math Twitter Blog O’ Sphere). A number of them have helped in considering better questions for deeper thinking. Like Dylan Kane. “Change one digit to make the equation easier to solve.”
-Uglier, hard questions don’t necessarily develop understanding. More thoughtful questions can’t be solved by doing a sequence of steps. Want to change questioning away from algorithms.
-Sam Shah with Better Qs blog, has many meaningful posts. Tab called “On Questioning” with a great list of questions broken up by what you are trying to accomplish. Not just good questions, but asking questions for the right purpose.
-Good questions take planning, we have to be intentional when asking them. Don’t make it a leading question. Tailor the question to student or class. Wait time is really hard, but we need to give them time to think.
-Want student to work at a deeper knowledge then a level of 1 or 2. Thinking about mathematics, not simply stating facts or following steps. Get students talking to hear their points of view. Helps to expand their repertoire of tools. has many pathways of getting to an answer. And (which is my favourite one). Slides: “You have about 8 seconds to figure out how many tiles are in the next.”
-Showing patterns are equivalent, which is so powerful. For algebra. And (which one doesn’t belong).
-Open questions with Marian Small and Amy Lin. Here’s an example. “Which of these are most alike and why?” (Slide: lines with varied slopes and intercepts)

-In summary, need to: Orchestrate conversations to not give away thinking, note wait (processing) time, make every interaction count.
-For the next 150,000 Qs - ask better questions does not have to be complicated, but purposeful. Focus on sense making, not only right answer, find questions to spark curiosity.

8) Amy Lin - from Peel

-“Grade Expectations.” How to transform a culture from grading into assessment.
-It’s not really about that, it’s about motivation, and nature vs nurture in the classroom. I didn’t realize I wasn’t supposed to do what I did, I simply did it?
-Math should allow an exploration of educational avenues. The push towards good grades has minimized the ability to innovate. Losing that learning is messy.
-We have to let leave that grades measure the knowledge of a certain math topic. What do “Average Marks” really tell you?

-If test comes back, first thing is students look at their mark, even if they have written comments. Calvin & Hobbes comic. Second thing they do is look at someone else’s mark. “Schoolies” comic. Students may start cutting corners or compromising ethics.
-Only gave descriptive feedback for 1D course, mastery form. Then have students self-assess, and involved in grading process when comes time to give a mark for report card.
-So instead of testing assignments, they’re all over the table and arguing and discussing what they did and what they knew and what’s the important concepts. They’re usually more honest than you think, not inflating their marks.
-This is an example of a high school math class when we need to relinquish control. Students aren’t naturally accepting of no grade. (“Sit, Say”, never “Think, Innovate”.) Will I lose marks if I do this or don’t? Then 3 weeks in, the innovation to learn math starts to shift.
-Final summary from a student: I feel good about math when I am learning more than yesterday. (Versus having good marks.)

-Three Wishes: I wish to never have curiosity crushed by conformity. To still have passion for learning. That only type of learning isn’t grading.
-It’s not about facts being stored in your head, it’s about creating a person that appreciates math intellectually and socially. We’re creating an illusion that worthiness is a result of calculations. It’s time we’re being trusting of a different story. “Follow your North Star.”


9) Alex Overwijk - from Ottawa (OCDSB)

-Peter Liljedahl: I spent a weekend with him in 2014 and two days this last school year. He’s had a massive influence, I’m drunk on his koolaid. #ThingsPeterSays
-Replace instruction with activity based teaching, go in more depth as you go through, spiral the curriculum.
-You need good tasks, open ended with low floor and high ceiling, multiple entry points, multiple solutions, hands on activities, and the list could go on.
-VNPS (Vertical Non-Permanent Surfaces), students stand and solve, the feedback is timely.
-VRGs (Visibly Random Groups), a deck of cards, figure out size of groups, boards are labelled 1-10 and cards are too. Get up and get at it.

-Stage 1: Very blunt and in your face, but easy to implement. About room and students. Give oral instructions. If 20% of kids understand, then that understanding will make it’s way around the room. DeFront the room to make it about what students do, instead of what teacher does.
-What questions do we answer? There are 3 Types. 1) Proximity Qs. Don’t answer simply because you’re near. 2) Stop thinking Qs, eg “is this right”? Don’t answer those either. 3) Keep thinking Qs - you can engage in those.
-That’s Stage 2, less blunt, more difficult to implement. About students and teacher and room. “Middle size gear.”
-Communicate with students to tell them where they are, to report grades, and to be clear about what we value. Evaluate what we value; if we want tests, it’s a problem, because students are smart, they’ll figure out what we REALLY value.
-This is how we level knowledge: Not bringing everyone up to where we want them to be, then the ball falls back down. Level to the bottom instead, watch no one fall down. Totally absorbed and in the moment.
-Create flow, a balance between skills and challenge. Avoid Boredom and Anxiety. Keeping students in flow is good teaching.
-To manage flow, increase or decrease students in a group. Take best member, put somewhere else. Teacher could join a group. Then could play dumb. Give a hint, or give a group an easier task.

-How we assess and how we level and manage flow is Stage 3. Very subtle, very hard to implement. Small size gear. All teacher driven.
-If not now, when, when if not you, then who. Make the massive change. If not, students and other things will pull you back down.

10) Jon Orr - from Lambton-Kent District

-The Hero’s Journey: A mathematics tale. A philosophy of teaching. Idea on story, an English teacher’s idea.
-Begins with our favourite characters from TV or movies. Time versus tension, put it on a graph for math. Time passes (horizontal), tension is speed (vertical), what the character feels.

-Beginning. All stories start the same. Battle forces of evil. Crisis. Learn things about themselves, strengths and weaknesses. Needed to make the ending awesome.
-As story ends, the tension drops back to normal, but it’s a new normal. That’s learning student curve as well. This is a process we go through when we learn something of value.
-Let’s talk about math class. It was always taught the same way. I taught it this way for ten years. Job 1 is to take up homework, it’s in the manual. You put it up, or students put it up. Then Job 2 is to have definitions, procedures. Then do examples, then homework. Then you rinsed and you repeated.
-Put this on the same tension-time graph. Time of a lesson versus tension the student is feeling. Begins the same, but the tension then falls.
-Reveal, discovery of math rules and strategies. Students should feel the NEED for math. The procedures mean nothing, we’re robbing them of a chance to bring value to what they’re learning.
-Let’s take this old model of math learning and flip it to the new model, discovery of math rules. This is the flipped class. Want them to be begging for a better way to solve problems. Productive struggle with inquiry. The time is right for this.

-Let’s use Amy Lin’s and Marion Small’s open problems and Dan Meyer’s 3 acts, and WODB math and Desmos (my crush), to have this journey. Give the #RealFlippedClass a try.
-Allow students to transform their own story, put them through the hero’s journey.

11) Lisa Lunney Borden - from St. Francis Xavier U

-Show me Your Math: Education for Reconciliation.
-Quarter life crisis, quit job to get PhD. More indigenous knowledge to mathematics.
-Quill boxes made from birchbark, measure three times across, and a thumb width, works out every time. “That’s pi”; they say “it’s just common sense”.

-Invite the kids to talk with the elders, not us with the elders. Let the students be the researchers. Mi’kmaw kina’matnewey community. “Show Me Your Math” wanted a help desk contest; instead, for last 10 years it’s been a math fair.
-Exploring the math in community practices. The math in making a wigwam. This is bringing back stories that may have been lost. Our Elders were mathematicians too. Learning together, coming together, restoring cultural knowledge.
-We’ve shifted to Inquiry Projects in schools. Eels Kataq Project. Created scale models of eel habitats at low tide. Maple Syrup making. Blindfolded to see where their gathering place is. “Show me your language” to look at language laws and traditional language and Gaelic community too. Shared history.
-Making a drum, from steaming and bending wood to stretching leather, intergenerational learning happens. “I remember Auntie Caroline doing that at the basket shop”.
-Making an 8 pointed star, birch bark bitings. Innovating. Stencils. Making Canoe paddles with fractions and symmetry, using traditional methods.
-Grandfather’s Lunch, sharing our finished products and hearing the stories. Only as they became adults they can do it.
-Wholeness resists fragmentation, cultural synthesis. Building their own personal self.

-I invite you to think about how you might find your allies and community content, and invite them to show and you, to show me, their math.

12) Graham Fletcher - from Atlanta, Georgia

-1,025,109.8 is the number of words that make up our english language. Don’t ask me about the .8th, probably from southern states where we use “fixit” and “y’all”.
-Words are put together as Command-Exclamation-Question-Statement. Which of these allow students to talk more, and us to talk less? 16,000 words in a day. A man once spoke 48,000 words in a day. Probably a math teacher.

-Where do we use all our words? 17% (asking), 83% (showing, modelling, telling, giving). This perplexed me, how can I take this idea and flip it? Here’s 6 lessons I learned on the road to becoming an 83 percenter.
-It took time before the lesson came into play. To collaborate with colleagues and figure out, “I can’t have time to not have the time”. Plan up front, ask questions that can’t simply be answered with a yes or no. Questions beginning with “Should have” or “Can you do” become my Franks RedHot sauce - put it on every student.
-Max Ray: Listening TO student responses is greater than listening FOR student responses (2 < 4); go to his Ignite Talk.
-Voice in my head is saying stop talking, you don’t need to clarify or justify, just listen. Yo, zip it, keep mouth closed... and it’s hard to do.
-Fourth lesson: Wait time is hard. Silence is powerful and awkward. (Graham didn’t speak for the time that slide was up.) What that silence does, when we embrace it, is shows what students have to say is more important than anything we’d have to say to them.
-No standing zone. Where are you positioned in a classroom? Questions from the front are far less effective. Walk amongst the tables, with the students, does a better job of talking less and listening more.
-Lesson six: As teachers we are in a position where we can talk a lot, but we need to choose our words and our sentences carefully. The less I talk, the more they actually listen to me (when I do), because I’m not rambling on.

-As you go back to your classes, choose your words and sentences effectively, or you’ll sound like Mrs. Donovan. If you don’t know who that is, just ask Charlie Brown.

13) Chris Suurtamm - from UOttawa

-Everyone is a Mathematical Thinker. I don’t just mean everyone in this room, I mean everyone outside of the room too. Struggling learners, mothers, siblings, students.
-We see math in the fractals of broccoli, in the spiral of a shell, in a pinecone, in the structures we build all around us. Look at this building, in this building at the Faculty of Arts.

-But some feel math people are special. In this drawing from a young student, “math guy” has glasses and mismatched socks. But that’s not what a math thinker looks like. We’re all math people.
-We often hear administrators or parents or friends at cocktail parties who say they were “never a math person”. But even as a young chid, we’ve all: sorted, organized, grouped, looked for patterns, and tried to make sense of the world around us + objects we interact with.
-As we get older, we have to use objects to help think of what makes sense. Math tools are needed for everyone, not only struggling learners. We use technology to generate graphs, and we consider periodic functions, how do we change the amplitude or period? What if I dip a cube into a bucket of paint?
-Use this model. We are all curious. Not just George (monkey picture). As teachers, we need to value the curiosity that students bring to us. We need to listen to student thinking. I was going to use Charlie Brown and that teacher (see prior speaker); we can’t listen in our teacher voice.
-School math vs Math at home. Using cookies versus adding numbers. “Their mind is not a blank slate.” We need to use their intuition and thinking.
-I’m thinking of a number, if I do these operations, what is the number? They can solve an equation, all we do is formalize the way they already think when writing and solving it in the classroom.

-Our most important job as mathematics teachers is to question, listen, and respond to students’ mathematical thinking. And to respond in ways that move that thinking forward.
-Value what students bring with them, connect new ideas to their ideas. Everything is connected. All students have the right to many things, including being valued and most importantly being heard.

14) Doug Duff - from Thames Valley (TVDSB)

-As an elementary school principal, maybe I look at it differently. Can a math focus improve a school? Nine points of change. (Top “10 minus 1” change)

-In working with two large schools, what you focus on changes! Level 4s in both. (EQAO Gr 6) Belief that kids can learn.
-What happens when you focus on changes? Everything changes.
1. Formalize the Focus. Number flexibility, connections among strands.
2. Planning. What’s your path through curriculum. Front loaded notion of number.
3. Broadcast the Goal. Interactive Bulletin Boards, wish we had thought of it earlier. Display all the kids work. Representing numbers, string moving through and connecting decomposition, etc. Also “Fraction of the day”, reasoning and proving that fraction, what’s halfway.
4. Assessment Data. Schoolwide common tasks. Moderate with “Criteria for Analysis”, based on really deep number understanding (Kindergarden task, matching worms). Kids are doing things teachers didn’t know they could do, can now remediate. Develop progressions, early number understanding; FDK Task shown. Let’s look at complex cardinality, move kids through progression.
5. Timetables. Numeracy prep teacher. Best math teacher goes around and does inquiry in classes. Number progressions, move from additive to multiplicative.

-Second fastest improving school in the province, Princess Elizabeth Public school in London. “Don’t just stand there, do some math.”
-To learn the rest of the list, come to our school.

15) Paul Alves - from Peel

-Be More Cat: A rebuttal to Amy’s “Be Dog” talk from last year.
-Sit back and relax. A talk would be stealing most of that other stuff anyway. So listen to a story, The Hero’s Journey. Don’t be that dramatic, because teaching is hard.

-“A Teacher’s Story”. Grade 10. Reality of different students in class: Distracted one, the cool kid, the keen people. (pictures of cats for these)
-Recollecting. A whisper “Your fly is down”. I’m literally giving them everything I’ve got, then it’s April Fool’s. Ha ha.
-I felt more prepared but maybe not for what I was teaching. Some students just want 3 math credits and to get out of there, others want to go further, how to meet the needs of all.
-One of them says “When will I ever need this”. Did my best to pull the wool over his eyes, and everyone else in class at that time. But if I’m bored, I can’t imagine how the kids feel. Things will change. Talked to another teacher.
-“Why not”, what’s the worst that can happen. Rich Task: “Game of Frogs”. VP came in during the game, using the kids as live frogs. “We’re doing math, just relax”. Using slides and jumps to get across.
-One of my students came up and said had solved 10 by 10 frogs, did you find a model? “It’s just S’s and J’s man, all 120 of them”. Posted his solution.
-Jane’s Story, one student outside my door with a crumpled assessment. She was just tired, could see the pain in her face. Let’s just do it again, why not - do you know it? If not, let’s find out. When you do, let me know, and then we’ll try it again.

-Without deviation form the norm, progress is not possible (Frank Zappa?)
-“That wasn’t the slide I asked Amy to include at the end”. Final shoutout to Amy Lin who put all this together and got all the speakers, a lot like herding cats. Thanks.

And that’s the whole set. Hope you enjoyed these short recaps of similarly short presentations. Feel free to let me know if anything stood out for you in the comments below, and/or check out the rest of the OAME 2016 conference.