Thursday, 2 July 2015

There Are No Dumb Questions

For 15 months, I wrote an education column over at I didn't write a column for June 2015, as things are wrapping up there (instead you got the Ontario Education roundup posts on this blog). However, for the sake of completeness, and in case anyone was unaware of that venture, I now present a quick digest of all those posts - perhaps you'll find something of interest to read?

If there are still questions out there, feel free to fire them my way too - I can't promise I'll get to them, but I'll do what I can.

1. TANDQ: Bias Confirmed (March 2014)
How Can We Ask Good Questions?
In brief: If you can look up your answer with a search engine, that's a problem. Because Confirmation Bias means that the assumptions you have going in will be reinforced.

2. TANDQ: Getting Graphic (April 2014)
How is graphing technology changing in schools?
In brief: Don't market to a system that can't afford to upgrade. Let's see how the Desmos platform is working at replacing TI-83 calculators.

3. TANDQ: Popular Misconception (May 2014)
Why is my network below average, compared to all my friends?
In brief: Your friends experience the same thing. It's mathematics; a few big names skew the "average" for everyone - which is useful if we're talking diseases.

4. TANDQ: Riffing on Khan (June 2014)
How can you tell when criticism is justified?
In brief: Let's look at "Khan Academy" and "Mystery Teacher Theatre 2000". Is it a case of 'THEY don't understand'? Or of 'THEY might have a point'? Somewhere in between?

5. TANDQ: The Education Game (July 2014)
How is teaching like a role-play game?
In brief: "I notice your wizards/students aren't being successful. It must be because they're not casting cantrips/memorizing facts. Let me tell you how you're supposed to do gaming/your job."

6. TANDQ: Around the World: England (August 2014)
What is the education system like in... England?
In brief: A look at the school system in England (not all of the UK), from Standard Assessment Tests (SATs) to A-Levels.

7. TANDQ: Effect of Learning (September 2014)
Why is this activity harder than I anticipated?
In brief: When you don't know much, you tend to focus on what little you do know. As you learn more, your focus shifts, and we see the Dunning-Kruger Effect in action.

8. TANDQ: Average Expectations (October 2014)
Motivation through punishment, or reward?
In brief: Positive reinforcement, negative reinforcement, and punishment are not the same. Meanwhile, regression to the mean will happen regardless of the feedback.

9. TANDQ: Price Points (November 2014)
Why don't more prices end in round numbers?
In brief: If you're planning to buy or sell something, consider the "Left Digit Effect", "Benford's Law" and percentages.

10. TANDQ: Around the World: France (December 2014)
What is the education system like in... France?
In brief: A look at the school system in France, from their 'count down' naming system to 'les grandeécoles'.

11. TANDQ: Rate This Post (January 2015)
Are rating systems skewed?
In brief: Yes; possibly for different reasons. Consider weighted averaging, the voting population, and the scale involved (Roger Ebert hated the four star scale).

12. TANDQ: Text Me Never (February 2015)
When will paper textbooks go away?
In brief: Never, but the industry still needs to adapt. Education is gradually shifting, current prices are unsustainable, and research about reading on electronic screens will be a factor.

13. TANDQ: Pass It On (March 2015)
Why must my password include a capital letter?
In brief: By forcing extra conditions, we actually decrease the total number of options. But humans are not good with random, or comprehending password entropy.

14. TANDQ: Vote of No Confidence (April 2015)
How can I influence a democratic election?
In brief: Vote. No system is perfect, but "First Past The Post" is particularly problematic when compared to both proportional representation, and plurality ranking (used at the Hugos).

15. TANDQ: Animated Discussion (May 2015)
What animation is acceptable at your workplace?
In brief: That's my question to you. I discuss my experience after seeing a convention panel, and remark on the latitude I have as the teacher advisor for my school's anime club.

There's also posts on this very blog too!

Find something of particular interest? Wondering about a follow-up? Let me know in the comments below - or on the MuseHack site using the specific article! (But note I don't have any comment flags set offsite.)

Monday, 29 June 2015

Ontario Teaching Post 2014

September 2015 in Ontario IS NOT like September 2012 - it’s 365 days different. Let’s get that out of the way up front. If you don’t understand what I mean, this post is for you.
I won't force you to read this... but only comment if you do.

Teachers in Ontario have been teaching without a negotiated contract since September 2014. (One could argue it’s been longer, since the last contract was largely legislated, but I’m not here to argue.) And hey, maybe we WILL have a contract this September. (I’m writing this post in June.) But given how Ontario Education Minister Liz Sandals has said “It is typical of school board bargaining that their bargainers have a habit historically of disappearing during July and August” when THAT IS NOT THE CASE HERE... I’m not holding my breath.

The spin is strong. There’s a lot of misinformation out there. A few years ago, I did a media roundup of what’s been going on in the world of Ontario Teacher Contracts. In the interest of informing the public, I think we’re due once more. This post is the SECOND of TWO: It will provide INFORMATION AFTER the Teacher Contracts expired in 2014. Looking back: Here was the first post.

Same disclaimer as before: I’m not an expert, just an Ontario teacher, who thus has a better sense of which search terms to use. Moreover, all I’m doing is pulling information out of the media, though some personal opinions do creep in. Note that most newspaper links do have a limit on the number of complimentary articles you can read.

Final Note: For those out-of-province, Ontario has a Liberal government... though BC also has a Liberal government, and we’ve seen what’s happened there. With the government is OPSBA (Ontario Public School Boards’ Association). On the other side, there are also several different unions in Ontario negotiating simultaneously, including OSSTF (Secondary School Teachers, Secretaries, Custodians, etc), ETFO (Elementary School Teachers), OECTA (Catholic School Teachers) and AEFO (Association des enseignantes et des enseignants franco-ontariens).


What does the future hold for Ontario Teachers?
“Publicly, for the Wynne government, it’s all about optimism, with teacher contracts having expired at the end of last month and negotiations beginning.”

(I’ll give the government this - MUCH better handled than in 2012. Of note, the article contrasts Ontario and the Strike in BC, which was still ongoing at that time. It’s worth reading in full.)

Ontario teacher talks hit quick snag in new system
“After just four days of talks at the new, more formal provincial negotiating table ... OSSTF said parties were unable to agree which issues they can hammer out centrally, and which to leave to local bargaining.”

(That went downhill fast. This relates to Bill 122 in my prior post: what issues are local? Because local conditions in Toronto are NOT the same as those in Ottawa. A reminder here that teacher representatives were ready to talk over the summer - could we not have reached this point sooner? We don’t disappear, at any rate.)

OCTOBER 1, 2014
OSSTF remains ‘optimistic’
“Public high school teachers, who have been without a contract since August 31, will vote next week on whether to strike. ... Thomas said the union is still hopeful for a positive outcome from its talks with the province.”

(The vote ended up in favour of a strike, should it be necessary. OSSTF teachers actually had to vote twice - once on whether we would strike provincially, and a second time on whether we would strike locally. Given the two tables under the new Bill 122. That’s important, we’ll come back to it later.)

DECEMBER 3, 2014
Elementary teachers vote 95% to give union strong strike mandate
“Elementary teachers president Sam Hammond said yesterday the strike vote is a normal part of the process to provide leverage at the table for negotiators to ‘bargain seriously’.”

(Elementary teachers have now joined secondary school public teachers in sending a message to the provincial government.)


MARCH 14, 2015
High school teachers’ strike possible for Peel, Durham, Halton: Union
“Elliot said the talks between the province and the union, which also represents school support staff, have been fruitless, with parties yet to gather for the ‘central bargaining date.’”

(Other regions which would possibly be affected: Rainbow (incl. Sudbury), Thunder Bay, Waterloo and Ottawa-Carleton. Here’s where things can get murky. Remember how there are now two tables, Provincial and Local, both running in parallel - recall, a lot of the early issues involved deciding which issues were CENTRAL and which were NOT. So, the seven boards listed above were identified as local problem areas, even as complaints were listed against the province.)

APRIL 7, 2015
Possible Strike Looming for Ontario School Boards
“The Ontario Secondary School Teachers’ Federation is taking the final legal step to clear the way for strikes in the seven hand-picked regions, with two boards already a week into a mandatory 17-day countdown”

(That’s three weeks later. I'm pointing out that there are a lot of steps you have to take before you can legally strike. In case you were wondering.)

APRIL 15, 2015
High School Teachers’ Union OSSTF walks away from negotiations
1-“‘We’ve said all along that this is a net zero round of bargaining and we remain committed to that’ ... [Liz] Sandals added.”
2-“Dave Barrowclough, president of the Durham branch, District 13, does not feel the breakdown of provincial talks affects the local situation.”

(First, I believe that going into a negotiation saying “These are our expectations! We’ve told them to the media!” is the wrong philosophy. You should go in saying “These are our proposals. What are yours?” Of course, OSSTF also broadcast possible local strikes in March, so read into things as you will. Incidentally, OSSTF would later return to negotiations. Then leave again.)

(Second, we again see the split here between local and provincial - and at this point there was a clock ticking down locally in Durham (just outside Toronto) even as central talks broke down. Could their local deal be reached before a provincial one, to avert a strike? Well... it would have been a first.)

APRIL 24, 2015
Catholic teachers give union strong strike mandate
“The Ontario English Catholic teachers (OECTA) voted 94.2 per cent in favour of striking.”

(My most recent articles are on OSSTF, but other unions are attempting to negotiate too.)

APRIL 29, 2015
As teachers’ strikes loom over Ontario, the fight isn’t about salaries, though it is about the bottom line
“It’s been over a week since public high school teachers in Durham went on strike, closing schools for 21,000 students, and no one seems to be able to explain why.”

(Durham and Sudbury are now on strike, Peel possibly joining them next week. That article mentions how “elementary teachers are ready to strike as early as May 10th, and Catholic teachers are taking steps towards a walkout”. A key part of the issue - in this other article - notes how “the province wants to eliminate class caps on class sizes”.)

MAY 8, 2015
Ontario Elementary Teachers Crank Up Pressure, Release Job Action Details
“Teachers will withdraw administrative functions until the government and the Ontario Public School Boards’ Association (OPSBA) take some of their demands off the bargaining table.”

(Those administrative functions included administering standardized tests and writing report card comments. I haven’t talked much about OPSBA yet - the local people who sit at the provincial table alongside the government. Again, personal opinion here, I think that’s part of the trouble, as it makes things look like this editorial cartoon by Steve Nease. But that’s me.)
Visit neasecartoons for more of his work

MAY 15, 2015
Ottawa High School Teachers to Begin Job Action Thursday
“[OSSTF] gave the school district its five day notice of strike action on Friday.”

(There’s specifics in that article if you like. Basically, Ottawa-Carleton - and Halton - wouldn’t be doing a full walkout, but would be withdrawing administrative services. Also mentioned in that article, Liz Sandals decides to seek advice from the Education Relations Commission of Ontario.)

MAY 26, 2015
Ontario High School Teachers’ Strikes Illegal, Labour Board Rules
“The labour board determined that the local strikes at the three school boards are in contravention of the School Boards Collective Bargaining Act.”

(It likely doesn’t mean what you think. It’s not so much that the strike was illegal, it’s that the content of the strike was illegal. Remember, the Act they refer to was only passed in April of 2014 - and the OLRB took days to consider this, as it would set a precedent. There’s an interesting Primer on the Teacher’s Strike Decision at that link. I’ll summarize (TL;DR) for you:

Again, two tables running in parallel: Provincial and Local. The argument by the school boards was that the teachers weren’t striking over local issues, but rather provincial ones. As evidence, there was at least one sign about ‘Class Size’, which is a provincial table issue. In the end, the OLRB agreed, and told OSSTF they had two weeks to remove all the signs that included provincial matters, after which they could go back out on strike - exclusively locally. And during those two weeks, teaching had to resume. Make sense?)

MAY 27, 2015
Teachers’ plan to resume strike unacceptable: Ontario education minister
“Minister Sandals said that back-to-work legislation -- known as Bill 103, or the Protecting the School Year Act -- will make the resumption of strikes in those three boards illegal for the rest of the school year, because the Education Relations Commission had said the school year is in jeopardy.”

(This article also notes that NDP leader Andrea Horwath does not support the legislation - which is worth a mention, because if she had supported it, the legislation could have passed two days earlier. Before the OLRB ruling even came down. As things stood, it passed the very next day, May 28th. So even after the two weeks were up, the three boards that had been striking would have to keep teaching, due to Bill 103.)

JUNE 5, 2015
High School Teachers file for Conciliation
1-“If parties are in negotiation, they must seek conciliation to try to resolve issues before engaging in a strike. ... The next steps for the secondary school teachers union will be the appointment of a conciliator and a meeting between the parties and the conciliator.”
2-“Francophone teachers voted 93 per cent in favour of a strike this week.”

(This article gets a bit cluttered between references to both OSSTF and AEFO - though that does give me a two-for-one. At this point we see most teachers - or at least, their unions - agreeing that the province and OPSBA aren’t doing a very good job of negotiating. It’s worth noting that we’ve reached this point despite a third party mediator - Kevin Burkett, one of Canada’s most respected labour arbitrators - being called in, back in May.)

JUNE 9, 2015
Instead of a report card,
have this X-Ray.
Don’t Expect Report Cards, Boards Warn Parents
“Some boards had hoped to hire extra help for principals to input marks so that report cards could be completed, given members of the Elementary Teachers’ Federation of Ontario, now in the midst of work-to-rule, won’t be doing that or providing any report card comments. However, the Ministry of Education is expected to announce no money will be provided to do so”

(This is during the second phase of ETFO job action, which now includes not holding grade promotion meetings. The report card announcement resulted in some backlash, given how teachers were providing marks to school boards, merely not entering them - and how long does that take anyway? This particular article goes on to point out how the Ministry of Education saved $40.4 million in salaries when OSSTF teachers went on strike, wondering why they didn’t use that money.)

JUNE 17, 2015
TDSB about-face on report cards prompts other boards to promise marks
“A ‘tidal wave’ of schools across Ontario will now provide students’ final marks or a bare-bones report card after a surprise announcement by the Toronto board that it will send out the information to families by mid-July.”

(Basically, with an extended deadline and volunteers, marks are possible. Even the president of ETFO, Sam Hammond, tweeted out a thanks after this. Of course... the cynical side of me believes they merely solved a problem that they themselves created, in order to look like heroes. But that's my opinion.)

JUNE 22, 2015
Teacher lockout ‘last resort’ says Liz Sandals
“Education Minister Liz Sandals says teachers frustrated with the state of contract negotiations could be locked out, but only as a last resort.”

JUNE 25, 2015
School’s out, but will it be back?
“Every teacher in the province will likely be in a legal strike position by September, opening up the possibility September will usher in a new level of nastiness to the standoff between unions and the province.”

There’s not much more I can say after those two headlines. This is where we’re at. Hopefully, the province/OPSBA stops being obstinate about controlling every moment of a teacher’s day (including class sizes), and the unions can compromise given the constraints the government is apparently under.

Either way, I ask you, the reader: please don’t say that September 2015 is like September 2012. Not when it’s so much more than that.

Sunday, 28 June 2015

Ontario Teaching Pre 2015

September 2015 in Ontario IS NOT like September 2012 - it’s 365 days different. Let’s get that out of the way up front. If you don’t understand what I mean, this post is for you.

I won't force you to read this... but only comment if you do.

Teachers in Ontario have been teaching without a negotiated contract since September 2014. (One could argue it’s been longer, since the last contract was largely legislated, but I’m not here to argue.) And hey, maybe we WILL have a contract this September. (I’m writing this post in June.) But given how Ontario Education Minister Liz Sandals has said “It is typical of school board bargaining that their bargainers have a habit historically of disappearing during July and August” when THAT IS NOT THE CASE HERE... I’m not holding my breath.

The spin is strong. There’s a lot of misinformation out there. A few years ago, I did a media roundup of what’s been going on in the world of Ontario Teacher Contracts. In the interest of informing the public, I think we’re due once more. This post is the FIRST of TWO: It will provide THE BACKSTORY leading up to when Teacher Contracts expired in 2014. Looking ahead: Here is the second post.

Same disclaimer as before: I’m not an expert, just an Ontario teacher, who thus has a better sense of which search terms to use. Moreover, all I’m doing is pulling information out of the media, though some personal opinions do creep in. Note that most newspaper links do have a limit on the number of complimentary articles you can read. I’ve also pulled some articles from my previous post (post 2012) for context, and you can look there if you want more specifics.

Final Note: For those out-of-province, Ontario has a Liberal government... though BC also has a Liberal government, and we’ve seen what’s happened there. On the other side, there are several different unions in Ontario negotiating simultaneously, including OSSTF (Secondary School Teachers, Secretaries, Custodians, etc), ETFO (Elementary School Teachers), OECTA (Catholic School Teachers) and AEFO (Association des enseignantes et des enseignants franco-ontariens).


APRIL 11, 2012
Ontario education minister warns teachers, school boards about local bargaining.
"Ontario Education Minister Laurel Broten has launched a pre-emptive strike against the province's teachers aimed at discouraging their unions from engaging in talks with individual school boards."

(Part of the problem in 2012 was that there was a ‘provincial table’ and a ‘local table’. Here, the Minister was trying to curtail local bargaining - the province had a framework in mind. Some said her actions interfered in negotiations; we’ll come back to this later. The province would reach a deal with OECTA on July 5.)

JULY 30, 2012
Start of Ontario's School Year in Jeopardy if Labour Talks Stall
"Education Minister Laurel Broten said at a news conference on Monday that the government will introduce legislation if school board trustees are 'unwilling or unable' to negotiate agreements with teachers before the existing contracts expire on Aug. 31."

(Sorry, I must rant here. I am still twelve kinds of bitter about this. Yes, contracts would run out on August 31, 2012. They also ran out out in 2008 - and 2014. Teachers still returned to work. Liberals didn’t need to legislate a damn thing, just negotiate! Contracts are retroactive! The one issue I will grant them was the automatic wage increases had to be addressed - but the government made it sound like teachers weren’t going back to work in September! AND PEOPLE BELIEVED THEM! See my 2012 post for more detail.)

SEPTEMBER 11, 2012
Ontario Teachers Vow to Curb Extracurricular Activities as Bill Passes Freezing Wages
"Up to 136,000 public school teachers will drop extracurricular activities like running clubs and coaching teams on Wednesday to protest Ontario's new law."

(This was the notorious Bill 115, the ‘Putting Students First Act’, which gave the government the power to impose a contract. Conservative Tim Hudak loved it, partly why it passed despite a Liberal Minority government. Incidentally, the Bill included the following language, one reason it’s STILL an ongoing court case:
 "14. (1) The Ontario Labour Relations Board shall not inquire into or make a decision on whether a provision of this Act, a regulation or an order made under subsection 9 (2) is constitutionally valid or is in conflict with the Human Rights Code." )


JANUARY 21, 2013
Liberals repealing controversial Bill 115, Jan 23
“Education Minister Laurel Broten, who promised to kill the legislation after imposing the contracts on Jan. 3, said repealing the law will remove what became a lightening [sic] rod in the province's labour battle with teachers.”

(The government ultimately used their Bill to impose contracts - then repealed it. To me, it’s sort of like shooting someone, and then saying the gun is gone, so what’s all the fuss about? Perhaps coincidentally, that repeal was right before the provincial Liberal leadership convention.)

(Kathleen Wynne won the Liberal Leadership on January 26, 2013. Liz Sandals became the Education Minister on February 11, 2013. A prorogued legislature was recalled on February 19, 2013 with the Throne Speech.)

I'm new here, I can make this all better...

FEBRUARY 27, 2013
Extracurriculars make slow comeback as Ontario teachers, government resume bargaining
“Negotiations are expected to focus on coming contracts that will take effect in the fall of 2014, and on non-monetary issues relating to existing contract terms.”

(Yes, even after the imposed conditions, details still needed to be worked out. Locally, in particular. In the end, secondary teachers would resume extra curriculars before elementary teachers.)

APRIL 19, 2013
Ontario’s public high school teachers approve deal
“The Ontario Ministry of Education, which went back to the OSSTF and the ETFO to negotiate a collective agreement, says that no new money went into the deal.”

(The deal was ratified 84% by OSSTF. A lot of that news article is the Conservatives complaining that extra money actually did go in, hence the quote. Perhaps it’s true, in the form of working conditions - I don't know enough to say. But Bill 115 did completely strip out the sick leave plan that had previously been in place, things like that had to be addressed.)

JUNE 5, 2013
Ontario Universities bracing for teachers college funding cuts
“Ontario’s government is following through on plans to double the length of teacher education programs, but has knocked universities off balance by demanding they make the switch with 20 percent less funding for each student.”

(Okay, that article is only tangentially related, but honestly, the government seems to have it in for education? It could be my imagination. And totally unrelated to that: Dalton McGuilty Resigned as an MPP later that month.)

JUNE 18, 2013
Premier Wynne defends pay hike for public elementary teachers
“Wynne’s government has reached a memorandum of understanding (MOU) with ETFO that agrees to a 2% raise starting in September 2014, giving their members parity with French and Catholic school elementary teachers.”

(Meaning maybe it is my imagination. Wynne did point out there that the teachers wouldn’t get the pay raise for two years; Conservative Tim Hudak countered that he would freeze all broader public sector pay. The fact is, ETFO took a pay hit back in 2008, this closes that wage gap. Their members ratified this deal on June 23, 2013.)

(This means three key Ontario unions now had a deal: OECTA (July 2012), OSSTF (April 2013) and ETFO (June 2013). I’m honestly not sure what happened with AEFO, someone feel free to enlighten me.)


APRIL 8, 2014
New Ontario law will guide teacher negotiations from now on
“The School Boards Collective Bargaining Act ... formalizes what was a fuzzy, often voluntary system ... after Queen’s Park took over education funding 15 years ago”

To decide what's local,
we'll flip a coin...
(This Act, also known as Bill 122, was first proposed back in October 2013. It replaces the old Education Act, and tries to fix the negotiation process so that we don’t have the 2012 situation again. Recall: When Laurel Broten’s “pre emptive strike” interfered in local bargaining - my first article, above. Now, issues will be separated between the tables.)

JUNE 13, 2014
Four more years: Ontario awakes to a Wynne-led Liberal majority
“The Liberals defied almost all predictions, with most pre-election polls predicting either a Liberal or PC minority.”

(So we now have a Liberal majority. OSSTF had also elected a new president by this point. Then, summer. And when summer ended, in September 2014, teachers went back to work. Despite not having a new contract in place. Seriously, still twelve kinds of bitter about 2012.)

Incidentally, you might have noticed that, in 2012, negotiations were underway well BEFORE the contracts expired in August. Not so much in 2014. It is my understanding that OSSTF filed to negotiate at one of the earliest possible dates, back in June 2014. One of the main delays, of course, was the provincial election: Who would teachers even be negotiating with? There’s also the fact that Teacher Contracts were previously 4 years, but Bill 115 legislated for only 2, putting everyone back at the table a lot faster - but I don’t know if that’s even relevant.

That said, since the Liberals were re-elected, I don't know why the government wasn't taking steps to negotiate last summer. Feel free to enlighten me.

The timeline continues in PART 2: Ontario Teaching Post 2014.

Tuesday, 23 June 2015

Multiplication is Ridiculous

Interpret the following: 6X-3

Do you see “6 times negative three”? Do you see “6 times x minus 3”? After all, it could be either. Because multiplication is ridiculous. Yes, I said it. Multiplying: the operation that can be expressed using nothing, or practically any other math symbol already in use elsewhere. Why do we do this to students? Why do we do this to ourselves?

X marks the spot... of no return.


The “X” symbol was first used for multiplication by William Oughtred, which was published in his book “Clavis Mathematicae” (1631). However, around that same time, Thomas Harriot was using the elevated dot “ ” symbol. Gottfried Leibniz preferred that notation... supposedly with the complaint that the “X” resembled an unknown “x” too much. (Even THEN they saw this problem coming!) Incidentally, Leibniz also used the inverted U (“cap”) symbol for multiplication, which now denotes intersection.

Owing to general disagreement, it wasn’t until the 1800s that “X” became popular in arithmetic for denoting multiplication. But the elevated dot is still used too - except where it denotes a decimal point in British textbooks (as Ben Orlin found out). But wait, in senior math for North America, the elevated dot is also used to represent the DOT PRODUCT (or scalar product) of two vectors, not to be confused with the CROSS PRODUCT (or vector product) - that’s the one which uses the X. Though we can write dot product as |a|x|b|x(cosC), where those x’s are (of course) multiplications, not cross products.

Confused yet? No? Okay, then let’s make it worse.


Oh dear...
In elementary school, the convention is to teach X means multiplication (why??). Later on, with order of operations, we can put math into parentheses/brackets. What’s inside them gets done first, as in (3-1)X(2+4). Yet once you get to high school, we don’t use that X *AT ALL* (provisionally, until cross product), we simply use the parentheses for multiplying. Except now you must do these special "multiply" parentheses AFTER you do any exponents, despite PEMDAS/BEDMAS! It only makes sense - exponents are a form of repeated multiplication, and you need to multiply before you can multiply. Right?

Take a step back now to evaluate: (5)(4)2

If you’re like a LOT of students, you’d write (20)2, not realizing that the expression is really (5)(4)(4). Of course, by the distributive law of multiplication, it still means (20)(20), because the 5 distributes to both 4’s, yeah? (NO! WRONG! Distribute on addition only.) What also doesn’t help is the HORRIBLY inconsistent way brackets are then used in conjunction with exponents in textbooks: (5)4 is actually the same as (54), and used interchangeably, but they’re NOT the same as (-54) because now the (-) is NOT on the base. Granted, I admit I’m guilty of being inconsistent with parentheses that way myself, unless it’s pertinent to the question at hand.

Which brings me to the huge problem: Whether something involving brackets is multiplied or not tends to be inferred from CONTEXT. Which is terrible when we come to f(x), the notation for a function. There it’s NOT multiplication, those parentheses aren’t the same parentheses as before! Perhaps rightly, every year this BAFFLES some students, who invariably see f(x)=5x as some weird multiplying ritual. To find out when it equals 20, they will write 20(x)=5x. Then divide by 20. (Or worse, subtract 20. x = -15, right?) Similarly, ask the student to find “f(3)” and their last step is invariably a division by that 3, unless they learned the context clues.

Okay, so X is confusing, but parentheses are JUST HORRIBLE by comparison. What’s left? Well, context for multiplying can’t be THAT hard, right? How about we express multiplying without any symbol at all! What could go wrong?


Perhaps this “no symbol” idea could work - if we weren’t in a place value system!! Combine that issue with early use of “x” as an unknown, and we’re screwed. After all, why can’t 4x mean 4 in the tens place, and an unknown ones place? (Heck, I suppose it can, if we let x=10, so that the ones place is zero.) Then there’s a typical high school expression like “5-2x”. Is it any wonder students combine unlike terms? There’s a subtraction RIGHT THERE, and no other operations! So 3x, right? (Wrong again!)

That's a deadly Sin...
Except now there’s also the problem of OTHER times math uses no symbol, when we DON’T mean multiplication! Exponents being the more obvious one, what with " 32 " not meaning (3)(2). (Except didn’t exponents mean multiplication after all?) As another example, if I write “sin x”, that doesn’t mean “sin” multiplied by x. “sin” by itself is meaningless! Yet despite that, there’s always that one student who will try to solve “sin 3x=90” by dividing out “sin 3”. (And I cry a little bit. Though here’s a follow-up question: Should I cry more or less if they instead divide the 3 before doing inverse sine?)

Just to round things out, multiplication can also mean division. Because if you’re dividing by 2, that’s the same thing as multiplying by 0.5 - and we generally want to do this if we end up with a rational or trigonometric expression that has fractions on top of divisions. (Such as [(x+1)/(x+2)]/2 ) Oh, and what do we use to show multiplication of two rational expressions? Often the X symbol.

I mean, students don’t know cross product with vector notation yet, so they can’t get confused, right? And once we reach matrices, they’ll know 3x4 is a 3 by 4 matrix, not a calculation, yeah?

What. The. Hell.


The thing that bugs me the most about all confusion this is that there’s an obvious solution to it. Computer scientists have been using it for decades. It’s the asterisk/star (*) symbol! Which you have probably used yourself online, perhaps even in writing up a mathematical blog post!

And before you argue that * is a recent addition, the choice wasn’t completely arbitrary. The asterisk was (supposedly) used in Germany back in the 1600s. Possibly even by Johann Rahn, the same guy who popularized the obelus symbol for division. (Though for division, Leibniz preferred using the colon. Bringing this article full circle.)

Enough! We thought WE were your X girlfriends!

In conclusion, I claim the X isn’t working. So why haven’t we switched to * in this day and age??

Well, aside from the fact that traditions are really hard to break (hey, I’ll admit I’m not using it yet), there’s the problem of needing to reprint all the textbooks. Textbooks that schools can’t afford to buy. And online, we might also lose out on Khan Academy videos like “Why aren’t we using the multiplication sign?”. So I guess we’re stuck. Until the computer uprising.

For more on Mathematical Symbol Origins go to this link! Alternatively, read my prior rants about y=a(x-h)2+k or cross multiplication. Or merely comment below.

Sunday, 31 May 2015

OAME 2015 Ignite

I posted about OAME Ignite last year, in 2014. In brief, it’s a double session at the OAME (Ontario Association of Mathematics Educators) conference, whereby presenters have precisely 5 minutes to deliver a talk. Their slides advance automatically. The topic can be anything. Each speaker introduces the next one.

It still tests my shorthand, except this time my shorthand was typing directly into my laptop. (Is there a term for that? Honestly, the main problem was autocorrect.) For this post, I’m keeping the point form I used, while cleaning up the sentences for you. Doing otherwise may cause it to linger on my hard drive even longer - not unlike my recap plans for the rest of the conference. I’m not sure if this format means I end up capturing the heart of their talks - what do you think?


1) Sunil Singh: Mathematics Is...

-1971: An invitation by Willie Wonka, nestled in the movie scene was a song.
-“Creativity is intelligence having fun.” -Albert Einstein
-Show having fun with numbers. Start with pi, both irrational and transcendental. Then see numbers 1-49 in factors.
-“Prime Climb” is a game on Amazon, primes are colour coded! A Seattle based company “math 4 love” made Prime Climb.
-Also see James Tanton’s yellow book, “Arithmetic = Gateway to Love”
-How do teenagers see math? Bounded (textbooks) or boundless?
-BOOKS: “The Math Olympian”; “The Crest of the Peacock”; “Mathematician’s Lament”
-Mathematics is Imagination. But there’s more, let’s bring in Keats, Mathematics is also Beauty and Truth.
-“What was our philosophy again?” Math has a right to... beauty.

2) Matthew Oldridge (@MatthewOldridge): Mathematical Surprise 

-On Mathematical Mess: Are we prepared to be surprised by student thinking and our own thinking?
-Given time and space, your students will ask questions. Try a “wonder wall”, an open space for thinking, with questions written on chart paper.
-Teaching Rates: “We knew the math and we still didn’t predict all the strategies.”
-Using Minecraft to construct bar graphs? (The game is literally an infinite space, what could happen there?)
-“I didn’t think of that.” There are so many methods and strategies. The thinking is there, we have to let surprise emerge.
-The start of a proof for area of a trapezoid? Critical insights.
-“What’s a fair price?” (Go to price per kilogram? Conclusion: Food in vending machines is obviously overpriced.)
-One student generated a model for points scored by a hockey player.
-“I left this slide blank if I was running out of time.” But it’s space to think, let’s call this the metaphor.

3) Chris Suurtaam: What’s Important?

-In math teaching and learning, RESPECT everyone: Teachers, students, parents, administrators, colleagues.
-All students are able to engage in mathematics and to extend their mathematical thinking. It’s a right.
-We need to think about equity issues. “There is no time limit in terms of exploring mathematical ideas.” Everyone has the right to feel value learning and to feel capable as a learner.
-Students learn in different ways and it’s that diversity that should be valued. They won’t all be hitting the same target at the same time. Don’t measure a lesson like that.
-A lesson is successful if by the end of it, the students’ thinking has moved from where they were before.
-Value the math that students bring, and the various ways they work. All solutions involve mathematical thinking, not just those that are symbolic.
-We learn mathematics by connecting mathematical ideas. Mathematical thinking is at the centre of learning.
-Assessing What Matters: What IS the important mathematics? I’m not talking about an assessment event (e.g. test) but the ongoing listening and thinking.
-Assessment shows students what you value and what mathematics is important to know and be able to do. If we say we value problem solving, then assessment should value those things.
-Attentively question, listen and respond to students’ thinking and engage them in important mathematical tasks. 

4) Ron Lancaster: The Rubik’s Cube

-On the Rubiks Cube, Contests, TV commercials, and a great teaching and learning tool for all.
-Books and puzzles: A Rubik’s clock is just as good as the cube.
-There was a 1981 Rubiks cube contest at Ron’s high school, led to teaching how to solve. It’s the math in the cube that is amazing.
-Sequence of turns and order. Doing movements over and over again, the cube will come back to original position. Can provide a big picture view of mathematics.
-1/7 loops back around. Related: f(x) = (x-3)/(x+1), the x comes back in sequence, f(f(f(x))). 8 perfect shuffles will restore a deck of 52 cards. Inverse function can be taught through this.
-Number of positions for a 3x3x3 cube: only by visiting the 12 worlds. If you unstick a colour and swap it elsewhere, the cube becomes insolvable. You must divide by 12 to get the overall Rubik’s formula.
-Becel margarine commercial, a move that twists the centre of the U; centres of the cube also.
-Doug Henning trick on NBC special: Tosses cube into the air, it’s solved. LIVE DEMO

5) Kyle Pearce (@MathletePearce): Engagement

-Hi. Web:
-Engagement is a buzzword. There’s many types of student engagement but no clear answers.
-It does NOT look like making good notes. “A simple question like the following will appear on your test.”
-Behavioural engagement might improve with technology, but not their engagement of the MIND. What measure is being used?
-Next tech tool: Cards and Smartboards. That saw improvements to success rates, had a working formula for standardized tests - but are students just being compliant?
-OAME2012 shift, and joining the MTBoS, hundreds of math teachers.
-Saw that innovative uses of technology can improve things, but it’s merely a bandaid solution.
-Traditional lesson plan worked due to familiarity rather than engagement. Promoted memorization rather than connections.
-Do not give a crossword puzzle with all the solutions listed on the page. Algorithms aren’t engagement or understanding. Task based assessment format allows interconnectedness.
-Never stop trying to find new ways to leverage the natural curiosity that we all have.
(You can view Kyle's Ignite on YouTube: thanks @jgibson314 !)

6) Marian Small (@marian_small): Pushing Deeper
*Quick happy birthday for this weekend.

-No news! It’s all about high expectations. I am just “one more push”.
-You can settle for correct answers or you can push for more insight, more connections, more sense-making. You can settle for “That’s right” or suggest “That’s great, now what about this?”
-Practice your “one more pushes”. Such as (moving through grades):
>‘What sums can you get if you add two next to each other numbers? What sums can’t you get?’
>‘If two sides of a triangle are 4 and 6, what perimeters are impossible?’
>‘Draw the greatest obtuse angle possible. Are you sure it’s the greatest?’
>‘Can 25% of one number be 75% of another number?’
>‘Remove one data value so that the median goes down more than the mean.’
>‘When might a line with a slope of 3/2 look steep? Look not so steep?’
>‘If an angle doubles, does the cosine for it double, more than double, less than double? Does it depend?’
>‘Create a spinner with unequal sections where the expected value is -0.5’
-I push me all the time to refine and improve, to differentiate, to share substance as well as meaning. You can push students for deeper insight and understanding, as well as more “generalizing” (things bigger than one little problem).
-Make thinking about your practice and questioning your normal game plan.
-We all started with “one more push” (pun on birth).

7) Dan Meyer (@ddmeyer): The Express Lane

-Which line in a grocery store is fastest? “Think about that for a second. Now five more seconds.”
-“The express lane has great PR unless it’s a scam. ... Yes it is.”
-What information matters/doesn’t matter to find out? (asked audience)
-Number of items someone has is a particular idea. As items increase, what happens to total time? Get data, model with equation. Total time = 3 * items + 35 seconds.
-There’s a flat rate of time to pay and chat with checkout person!
-Situation on slide: Need 173 seconds for 4 people with single digit num of items, versus only 155 seconds for the single 40 item guy.
-Featured on national TV, “In the Fast Lane”, art of zigging and zagging. (Dan lost. Models don’t always describe things perfectly. “I got nailed by produce.”)
-“All models are wrong, but some are useful.” - George E. P. Box
-A slide to reiterate the 6 step modelling with Math: Shouldn’t only be calculation. There’s the humiliation of the Validation step.
-Remember: Each person in line is a standing version of 10 items (due to flat rate).

8) Amy Lin (@amylin1962): Creative Spaces

-Our math classrooms are usually more row based. We’ve been trying to improve those classrooms, like a game it comes with instructions.
-In the game of mathematics, it’s about addition, we’re adding more games and technology, to increase scores and grades... because that’s winning the game.
-Is it? What is the goal? What is winning?
-Turn class back into a playground, play is valuable. Play is now apps on a phone and video games.
-We can create successful classrooms by being creative and giving students choices. Maybe the goal shouldn’t be adding, but empowering. We’ll get feelings of confidence. A better chance of deeper understanding.
-Problem of math anxiety? Can add a fun game, they get high scores and rewards, or can have tutors and video lessons and worksheets to help prepare for exams.
-“I’m going to launch a campaign about creativity and to abolish grading.”
-Promote student thinking and conversations. Want to get a space designed for students to WANT to engage. Maybe the new solution is more about what you cannot measure.
-I want to start “The Math Movement”, a community of educators who sense that something is wrong with the old game. I think we will win because we will see students light up and be driven to share what they learn with us.
-Open minds and hearts play in creative spaces. That’s a win for us.

(Insert Session Break)


9) Jonathan So (@MrSoclassroom): Doers v Doing

-We want students to become “Doers of Math” instead of just doing the math
-“The purpose of teaching is to help students learn.” (Fosnot and Doik.) Could see them as two separate things. Or “If learning doesn’t happen there is no teaching.”
-Common complaints: “Didn’t they learn this last year?” “Why am I doing review over and over again?” If I am the common denominator each year, then I need to change. We need to build mathematicians.
-A close minded way of learning fractions - only looking for answers? Use same thinking, but now explaining solutions and being forced to talk about the math they’re seeing.
-Students need to present solutions in a generalized way. They need to feel comfortable trying out new ideas. Where we can take risks.
-A Working Framework: Role of the Teacher, Environment of Learning, Accountable Kids.
-What really impacts our students? Gr 4/5 class: talk died in asking rote questions but with big idea questions, then kids talked.
-Three Types of Questions: Interrogation, Going Beyond, and Comparing. With all three there’s a big idea to every one of them.
-Two Important Talk Moves: Wait Time and Revoicing. Allow time to think.
-Don’t be passive observers of the math. Are kids just doing the math or being doers?

10) Mary Bourassa (@MaryBourassa): Big & Small

-Making big shifts and small steps to grow as a teacher.
-Change your words, change your mindset. Set the culture in your classroom, and then can connect with all students.
-Big changes requires collaboration. Join the MTBoS; Twitter is her daily PD.
-Investing time in social media has paid dividends. Read blogs every day, find out what other teachers are doing. (Mary has shoutout of 5 websites.)
-Next, start your own blog. Share your practice, it helps you reflect, you get feedback. “If I can blog, you can blog.” “I didn’t think anyone would ever read my blog.”
-If you’re ready for a bigger shift, spiral an entire course with activities.
-If that’s overwhelming, smaller changes: Warm ups. Can be big impact in small time. Gets students into math mode. They can argue about math, start collaborating.
-Warm Ups Sites: Estimation180, Visual Patterns, Which One Doesn’t Belong (a book of shapes, goes to graphs, then Mary runs further with it). 
-More small changes: No hands up. Hinge questions. Random groups. @dylanwilliams review stations. 3 strikes (N Kraft). Teamwork. activities.
-“Change can be had, but it all starts with a big or small step from you.”
-“Be willing to take risks, students need to see us do that.”

(No Paul Alves - not sure what happened)

11) David Petro (@davidpetro314): Pi

-There’s a link at the bottom for all references:
-This particular March 14 was Epic Pi Day. 9 decimals deep into pi. Einstein was born on Pi day. Coincidence?
-Random nature of Pi guarantees that your birthday is in Pi. (website can find it)
-Feynman Point (repeated digit) that looks not random is the definition of random.
-Exists unofficial record for digits of Pi: not acknowledged by Guinness Records.
-There’s a poem that goes 31 decimal places deep into pi.
-Pi in the Bible, Kings 7:23? Pi in Congress, in Alabama wanted to make it 3? NOT TRUE. BUT Indiana did try to make pi “3.2”.
-Could the average sinuosity of all rivers in the world be pi?
-Buffon’s Needle, dropping of toothpicks: what is probability of toothpicks lie on line? (Asks low/high estimate?) It’s 2 / pi, yet no circles are there.
-Two transcendental numbers and an imaginary number walk into a bar. There’s a proof from first principles on an Ignite session out there.
-“Reflect” on PIE/314.

12) Nora Newcombe (@kittydundana): On Quantity

-As a Cognitive Psychologist, knows Piaget: Space and Number take years to develop. Children confuse number, length and density. This is both true and untrue!
-It’s true in some ways. Toddlers can remember where we hid something in a sandbox.
-Babies are able to notice PROPORTIONAL quantities (compared with another) more than ABSOLUTE quantities. Proportional reasoning is what supports scaling.
-Demonstrations of babies isn’t about number, it’s about magnitude and general quantity.
-As they grow through preschool into elementary, they’re connecting things. Spatial is linked to number lines. The number line is both spatial and numerical. It joins integers with values in between.
-Yet students eventually LOSE some of the between, because we focus on discrete values so much.
-They don’t remember that a unit is a distance. They think a unit is the slash marks on the number line. Perhaps don’t line up everything with zero? Kids in early elementary school lose the sense of measurements.
-Also fractions are difficult (but important). How to get around problems in that reasoning? Proportional reasoning and scaling are deeply related initially, yet proportional reasoning also is related to fractioning.
-Think about number and space together. Use CONTINUOUS as well as discrete representations.
-Putting lines in between integers shouldn’t be of a smaller size. Number line can then be used for fractions and negatives.
-Push the number line so that it’s important, not merely discrete integers.

13) Don Fraser (@DonFraser9): Take Numb out of Numbers

-How to use the new technology? The best motivator is success, but also important is the hope of success. “We can’t spell success without U.”
-Look at the numbers: 1,2,3,4 and in your mind, circle one. Mostly people pick 3. Why? I have no idea. (Do you?)
-Shopping is an excellent source of real world math. (Image in store: “Was $3.99 now $3.99 save $0.”)
-A growth mindset in math means?
-New book: Becoming Steve Jobs. At $25/bag, air travel is becoming very expensive. Measurement. Peanuts comic.
-Linking math and writing. On average, Americans open their refrigerators 22 times per day: There’s no way this is true? Add all values together, averaging.
-Canadian license plates show there’s at least two aspects to math. “Yours to Discover” (Ontario) and “Je me Souviens” (Quebec).
-Do more rich people put toilet paper over the top? 2/3 of Canadians are right kissers, tilting right to kiss. Regional thing? Pose questions.
-Thank you for making a difference with the students that you ignite.

14) Al “The Big O” Overwijk (@AlexOverwijk): Reinvention

-If you told me years ago I’d be standing here, I’d have told you you were crazy.
-You know that I love to tell stories: My courses were unit based, moving easy to difficult, review, test. Do it six or seven times, call it a course. Then exam.
-They’d feel good about themselves, I’d feel good about myself, but it didn’t work for all students.
-What I valued was the content in the course.
-I told students I was the world freehand circle drawer, and despite the show, I still had disengaged students. Bruce said that I was the problem, I needed to change.
-We focussed on process. Focus on uncovering curriculum rather than covering curriculum.
-Here’s a prompt: (image of shirts). What do you notice, what do you wonder? How long to fold them? Why do you have that many shirts? How much surface area? (image of shirts in hallway)
-The sky’s the limit. Create something with low floor, high ceiling. Like “Beardo Weirdo” data when growing beard (on blog).
-Using “Visible Random Groups” & “Vertical NonPermanent Surfaces”.
-Get more teachers: Collaborative lesson studies. We learned about student learning. Testimonials from students.
-Some groups go the “wrong” way, we learn from their mistakes. Some take an inefficient way, some go right to an expectation.
-“It is better to make a story than to tell a story. It is better to have students make mathematics than to tell them mathematics.”
-I look forward to hearing your stories. If not now, then when, if not you, then who.

15) George Hart: Printing Manipulatives

-3D printing for the Mathematics Classroom (
-Want a Sierpinski Tetrahedron? His start was 3D Printed sculpture.
-You can create shapes and colours. Complexity is FREE. You can make complex things the same way you would make a cube or a sphere. Easy to program a fractal in two or three lines.
-How to take these things into the classroom? If you want a “MakerBot Cupcake”, “Strut Connector” or “Twelve-Stick puzzle”: can print the pieces. Mathematical models of all sorts of things.
-(10,3) a Lattice. Stellated Rhombic Dodecahedron. If you make a half dozen of them it’s so easy to see them fit together. Hollow and light.
-Screw puzzles. He has a bunch, come see him later. Tricky to assemble.
-“I’m not a classroom teacher, so I’m leaving it to you to see how you can use it in the classroom.” Have students create things and/or argue over how to work with things.
-You can write an equation for anything and make a 3D print. (Venn Diagram Candy dish shown - or anything else they want.) There’s some way you can use this to get students to be really excited!
-Classroom modelling. It’s cool! It’s fun! Shows the creative side, and teaches the importance of exact details. Gets students/teachers excited about math.

That was it! I left pretty quickly to get to another session.  Feel free to comment on whatever stood out the most for you!