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: http://www.engineering.uottawa.ca/makermobile
WRITING IN MATH
After the carousels, the first session I went to was Writing in Math (by Kevin Cheung, CarletonU).
Kevin started with a poll (https://pollev.com/kevincheung), 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: http://files.eric.ed.gov/fulltext/ED544239.pdf
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.|
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: http://www.artofproblemsolving.com/articles/how-to-write-solution and http://web.cs.du.edu/~mkinyon/mathwrite.html
“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.
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!