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: http://algebra.nipissingu.ca/mistakes.html
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: http://pinheymath.pbworks.com
”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.
AFTER THE CAROUSELS
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.
MATH CAREERS
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 Careercast.com 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 Payscale.com, 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.
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