Monday, 8 October 2018

Math Social: Marian Small

The Carleton-Ottawa Mathematics Association puts on a Social event the last Thursday of September. Sometimes there's a guest speaker (see this recap of 2016), sometimes not (though in 2017 I went to some PD from Peter Taylor). This year, in advance of OAME 2019, we had Marian Small (who is also the chair of that conference - keep an eye open for preregistration).

Her topic for was "Building Thinkers", with examples drawn from both elementary and secondary (and/or a demonstration of how questions could be repurposed). The slides are up on her website ( but here's one of my usual recaps. I've flagged the paragraphs with new questions to consider using bolded stars (*).


In this age of social media, "Followers are nice. Thinkers are even better." Marian hopes that we'll change things into "your version of the idea", don't parrot her. Thinking only happens if you ask the right questions and follow up in the right way. "My goal as an educator is to support teachers, to help kids." Which means help them to think mathematically, not just do math. We want to ask questions to make kids excited, and not all questions do.

What does it take, to influence a student? Being nice, and caring, but it also takes building thinkers, and this talk is about that latter part.

(*)Here's a question for maybe Grade 1: There's three patterns. (Pattern 1: Two squares, circle, two squares, circle... Pattern 2: Square, two circles, square, two circles... Pattern 3: Two triangles, hexagon, two triangles, hexagon...) Which of those do you think are the MOST ALIKE? Why?

Some might pick 1 and 3, for "same same different". Some might pick 1 and 2, for "same shapes used". Will every kid say the same thing? No. That's why Marian likes that question. We develop thinkers by seeing other people having reasons for stuff (which might not be our reason). You might also have you own reason as a teacher. Did anyone pick patterns 2 and 3? One person said Similar Colour. (Patterns were green, orange, red.) Audience member added, or read the last one backwards, it looks like the second.

We want to embrace diversity rather than telling kids, "This is the answer". (That shuts down thinking.)

(*)New Question: Do you think the number 15 is more like 10 or more like 20? Make your case. Our table jumped to the idea of "the 1 in the tens column" so more like 10. Someone else in the audience picked 10 because "same number of prime factors" (3&5 with 2&5, not 2&2&5). Marian added that there are two versions of a hundreds chart that can be posted in a room: one starts at 1 and goes to 100, another starts at 0 and goes to 99. If your version is 0-9, 10-19, you might get "more like 10" because they're in the same line. If you have the other version (11-20) you may get "more like 20". Cool stuff could happen.

An audience member brings up what Marian called "that stupid little rounding thing" as a case for 20. Related, if someone was filling 10 frames, it would be easier to just fill the second one (to 20) versus removing an entire frame. Marian also once had a student say 15 & 20 are more alike because "neither one of them is in any normal counting book" (unlike 10). EXTENSION: Do you think the fraction 4/3 is more like 13/12 or more like 7/3?

(*)New question: Why did Marian pick THOSE NUMBERS for the fractions? Most would go with the thirds being similar, instead of similarity of things being a little more than one. (At our table: When kids think equivalent fractions, they may think 4/3=13/12?) Creating improper fractions also talks to "and a third" in the English language (4/3 is "one and a third"). Someone raised that 4+3=7 so 7 feels like a good choice for the numerator? It's all about perspective, you could argue either way.

We can do this question in place of the question "please turn 13/12 into a mixed number". Marian: "I'm not doing weirdo math here, some might call it basic math." But one of the things to notice is that there is not a divorce between 'thinking' and 'basic stuff'.

For those in even higher grades, is root(20) more like 1.424pi or more like root(25) or more like 4.5? "Why would I say those?" We should see these are not random choices. Figure out why I'm asking this thing, and what would I do. Root 20 makes no sense to a kid, but how is it like 4.5? Between root 16 (4) and root 25 (5). An audience member brings up that 1.424 is like root(2). Marian admits she didn't pick it for that, and (since someone at our table had said the same thing) I HAD to chime in that root(2) is 1.414..., accuracy please.

That said, 1.424 times pi is closer to the true value (4.472...) than 1.414 times pi, and pi is an irrational number, meaning we still have a decimal going on forever. "Everyone in the room could make one up with different numbers if they wanted to."


(*)Maybe in Primary: What sums can you get if you add two next-to-each other numbers? (A soft way of saying consecutive.) What sums can you NOT get? "You can't get a smaller number" than a particular value?

The natural thinking is can't get EVENS when you add that way. Marian turns that around, if you believe you can't get a hundred (which is even), did you TRY? Convince me. That's purposeful. Marian believes said "this is what I think teachers don't do right a lot of the time". They'd agree with the assertion about evens, and then move to the next kid, instead of asking for a test of the theory. THAT'S where we develop thoughtful kids (with the proof).

"Notice I'm not just asking for the yeses, I'm also asking for the nos." (What numbers can you NOT get.) The pragmatic reason for that, is when kids try stuff and get a wrong answer, they feel like they have an answer they can't use after putting in all that effort. Here you still get your money's worth.

It's not for all kids, but she's met enough where understanding what does and what doesn't work means you have a better conceptual understanding. 'What if I pick next to each other decimal numbers' was raised (by Jimmy?). For instance, 1.5 and 1.6. "That was my reason for not telling you what consecutive means." Mom or Dad may have taught some other stuff at home, we're not saying there's a way this has to sound.

(*)EXTEND: Can you add five consecutive whole numbers, and get a number whose digits add to 5? (There was some discussion at our table. I was thinking, should be possible, either side of "5" and the tens place will fold together? Oh wait, add to 5, we don't want a 5 in the tens place. Also want a small number in ones - zero? 8+9+10+11+12? Oh.) We want kids to figure stuff out, what kinds of numbers can you get. Then don't ask for a right answer.

"Who had a number that was five consecutive but it didn't work"? "Did anyone get 14 as their answer? Or 32?" We play that game. My list: 50, 140, 230, etc, they all end in zero. Would they have to end in zero? It doesn't promote the same level of thinking if I give those to start. I'm not telling you any answers.

(*)Next was an image, bar diagrams (maybe for primary?). The total in the blue (a full length) is the sum of the two blocks above (yellow and grey, latter a small length). "The number 50 is in one section, what numbers make sense for the other sections?" (At our table, 'damn we just did ratios - in primary?'. 'Can we cut it up and fold it?')

"I got you to do more of the work, deciding where is the 50." Some kids will be better at seeing proportions than others. Then, having let you play around with it, here's what we do now. "I could say 'what are your numbers' but I'm going to say 'I saw someone's paper (I'm lying) and they put 70 as one of the other numbers, so where do you think they put the 50?'" We could also say, did anybody have 30 and 20 as the other two numbers? (Maybe someone did, if they're uncomfortable, you can bring it up and still have those discussions.)

Key action: Don't talk about what the answer is, ask another question.

Again we extend: The number 8pi is in one section. What irrational numbers make sense for the others? "I did, like, nothing." There are children in the room who don't know that pi is an irrational number. They'll go to square root guys, "but notice all I'm asking is 8, and you stick pi on the end". But not all kids KNOW that can just have pi everywhere (and use integers). Looking only at stuff for our grade level is not a good plan, stuff from other grade levels is super easy to adapt - if it's more about THINKING and not about the mechanics of how to do stuff. The mechanics IS more tied to grade level.


(*)Back to primary. Subtract two numbers, also add them. Those answers are 50 apart. What could the numbers be? Marian actually came around to our table this time during the discussion. My knee jerk reaction had been 25 and 25, to use zero as the subtraction. Someone else at our table said 25 and 75. Marian asked about more answers? Did we notice anything?

Within the audience, someone had 25 and 26. Marian generalized this one, you had 25 and anything. There's a number line (she draws in the air), do you see it. I'm going to put my finger there, and show +25 and -25. My finger was on the anything. Lots of good stuff happening in the question.

(*)Adaptation: One answer is 2/3rds of the other. (Note: Not 2/3rds more than, if it was, you could do exactly the same thing.) Some algebra kids could do it, but also can be done numerically, or there's other things you can do. Adapting from primary to a bigger grade. (Notice x and 5x will work. 4x and 6x.) "I didn't tell the kids to write equations, just gave them a question." Good things emerged.

(*)Maybe a Grade 3-6 question: If two sides of a triangle are 4 units and 6 units, what perimeters are possible? What are impossible? (This can lead to the Triangle Inequality. Sides 10-1-2, no triangle, can't just put 4 and 6 with 100.) As well, 4 and 6 are non-threatening numbers. "I don't need to use decimals or weirdo stuff, and it makes them think about what's possible and impossible." (Me, I'm thinking cosine law given three sides. Not always possible.)

(*)Maybe in Junior or Intermediate: About how many times do you turn on a light in one month? Immediately we think it's probably different on a school day versus a weekend, and then seasons - dark days in winter not summer. Tons of thinking to give any reasonable value. And no one in the universe knows the answer to this question, but we do have the ability to hear a kid make sense or not make sense. "I think being a thinker means you make sense." The idea is to ask those kinds of questions, Fermi questions. (see link)

(*)Related questions: If all the fruit on this cake were blueberries (currently blueberries, strawberries and peaches), how many of them would there be? Some kids would be freaked out by the white spaces (even on the original image), others wouldn't be. (Is it one-to-one replacement or space covering?) Or how long for a million new apps, if about 40 new ones are available every hour? Student reaction, "I don't have a million on my phone..." Even on EQAO we can have thinking.

(*)Can 25% of one number be 75% of a different number? (At our table again: "3 is 75% of 4, but 1/4 of 12." I immediately wondered about generalizing.) Marian notes the interesting thing is the relationship between numbers, and "I did start with nice percentages". Again, maybe don't ask what numbers are, but rather, which was the bigger number, the 25% or the 75%. Because most will say their bigger number is the 75% number, as it's a larger percent.

How about 60% of another number? "We're not going to do it." Marian says she could have put any number in the universe in that second position. "I'm not trying to teach them to push a button on the calculator, but rather to teach them that every number is a percent of a lot of different numbers."


(*)Create a set of data. Remove one value so that the median goes up more than the mean does. That's a much better question than "here is data, what is the median and the mean". (I like this for Data management. At our table, the educator next to me said we should start with a 1 and a 2, as removing the 1 later will change the mean very little.)

(*)Without calculating, list 10 fractions for a/b so that 2/3 divided by a/b HAS to be more than 1. We want kids to understand the answer is how many of the a/b fit into thirds. Unless you have a guy that's littler than a/b, you're good, and "I ask this question to force that".

(*)In a higher grade, adapt to "create an algebraic expression that is sometimes more than -2m, but not always". Start with something always greater than that, they'll say +m, and they're wrong, because m could be negative. (Not always greater. Answers the original question.) If you're an intermediate teacher, we need kids to learn that -2m+10 is always more. The boundaries of "m" make this trickier.

(*)For Grade 9 or 10? Why might a line with slope 2/3 look steep? Why might it not? Marian assures, "I swear to you I can make it look steep." If you mess with the scale. This is huge for kids to understand, many grade nine kids look at a picture and don't read the scale. "I can mess with them. And I think our job IS to mess with them." Not to be mean, but that's why to ask a question like that. Stretch the thinking. "Okay, you're looking very tired."

(*)For Grade 11: Can every function be written as a composite function? (That means 'is a function of the function'.) Or an exponential function is a LOT like y=2^x, so what might it be and how are the functions similar/different? Or you evaluate an algebraic expression when x=2/3 and get 1 and a quarter, what could the expression have been. (Rapid fire slides.) "Some of the high school teachers I meet are very enamoured with algebra."

(*)We pause here: One thing they do in higher grades is solve a linear system, or set of equations. There's something called elimination. Question: After elimination, we were left with 13y = 22, so what could the original two equations have been? She doesn't offer suggestions - "I'll just leave it with you, but it's a neat question, particularly for academic students." Makes them think about what happens with this elimination 'thing'. We're not even talking about the other variable, could be x or z or q.

In conclusion: "I believe: The game is not about what the curriculum requires or does not require. It is about what we want our kids to be. Doing math is part of it but clearly, to me, not enough." It's on us. Some children in our system get other things in their homes, and some don't, we're it.

It's a moral obligation to create thinkers, while still doing what we have to do.

Marian then put on her different hat, saying please help make OAME 2019 a big success! Lots of interesting featured speakers, sessions in English and French, Desmos is helping us (sending Eli as a featured speaker and helping with other speakers on Saturday). It's a long weekend in May, come with families, there's Ottawa's tulip festival, we hope it works like it did 10 years ago.

Check out, there will be an ad in the Gazette December issue. We would LOVE for you to offer to speak, proposals are being accepted now. We would LOVE for you to offer volunteer to help. We would LOVE for you to register to come. (Marian also has a third hat, someone asked her to run for Canadian Director for NCTM, to represent Canada. Voting starts tomorrow, if you're an NCTM member.)

There were then some drawings for registrations and Wipebook flipcharts, and I was able to talk to some fellow teachers about what they were teaching and my recap writings. ^.^

Hope you found something of use here, or at least of interest! Thanks for reading, and consider coming to OAME, registration opens on February 15th, 2019. I also personify mathematics over at, if you're wondering about the graphics. Keep on thinking!

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