xy PLUS 12 Ways to take Your Teaching to the Next Level

Volume 1Issue 1

Aaron StrongEDBE 8P29 Portfolio

MATHWhat CanPERPLEXITYDo For You?Hint: It’s ENGAGING

GettingIMPROPERWith FractionsBreaking the rules withCOMMON DENOMINATORS

When SIZE MATTERSSettling the debate on UNIT RATES

Is STRESSAffecting YourPerformance?ANXIETY in the math classroom

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in this issue 

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old school, old story

hollywood’s math problem

the problem with problem sets

the migration of a flappy bird

get smart with a Smart board

perplexity: making math meaningful

patterns, patios, and parallelograms

it’s hip to be squared

when common denominators are uncommon

quantity or quantity

crash and burn

the three part lesson: your key to success

about the author

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on the cover 

What CanPERPLEXITYDo For You?page 14

GettingIMPROPERWith Fractionspage 22

When SIZE MATTERSpage 24

Is STRESSAffecting YourPerformance?page 10

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Think: Why is the sky blue during the day and black at night?

old school, old story rethinking how we think about mathematics

A word from the editor

Welcome and thank you for picking

up the very first issue of MATH. We

hope you enjoy it.

The focus of this publication is to

communicate opinions and advances in

mathematics education in the 21st

century. With technology developing so

rapidly, it is easy to see where our

education might lag behind. However,

more than the need for devices and

apps for every student is the need to

shift our focus to catch up with the

constructivist era of teaching. Too often

are students forced to learn math in

isolation, memorizing formulae by rote

and regurgitating them on tests, only to

be forgotten as soon as the unit is over

and done with at the end of the month.

Math is more than just plugging

numbers into a formula. Math is a

language. And if we are to consider the

ways in which children develop, with

particular focus on Vygotsky’s social

constructivist theories, we see that

children do not learn best in isolation.

To bring math into the 21st century

we must make it a communicative and

collaborative exercise. We must promote

student inquiry and creativity. And we

must leverage our understanding of the

Universal Design for Learning to play to

the strengths of our students. Teaching

math is so much more than simply

“knowing your stuff,” so please check

back again as we continue to explore the

implications of a new math mindset.

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The prevalent vision of a traditional classroom is one where students are arranged in rows. Here, teaching is facilitated through rote memorization, and students work and learn independently. In this classroom, one student’s illustration of rounding rules was captioned “horrible.” Perhaps it is a reflection on the entire student experience in this isolated environment.

hollywood’s math problem opinion: the implications of a mathematically illiterate society

There was recently an article

published on boston.com regarding

“Hollywood’s math problem.” As one of

the seemingly few who has always loved

math, this article spoke to me; Indeed it

seems all too common that people take

pride in their inability to do even basic

arithmetic; But math plays an important

role in many aspects of our society;

When elections are won on economic

promises, how can one make an

informed decision if they have no

concept of number sense;

Ms. Johnson points out how the

so-called math problem diverges from

what we might see as a language

problem; While being bad at math is a

boastful opportunity, a narrow

vocabulary is not seen in a similar light;

Nor, I imagine, is illiteracy;

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I prefer to compare the math

problem to a grammar problem; I don't

have to know how to properly use a

semicolon in my everyday writing

because no one expects me to; But that

semicolon is calculus, and no one is

asking you to master it;

Having no number sense would be

similar to never constructing a proper

sentence; But we do pretty well when it

comes to using commas, and periods,

and quotation marks; So let's make sure

we give math its commas, and periods,

and quotation marks and give it the

chance it deserves; And you can leave

the calculus to me; And I'll leave the

semicolons to you;

A response to “Hollywood’s Math Problem” by Carolyn

Johnson. Originally written for boston.com (01/30/13)

http://fw.to/YKxMlgM

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What does success in math look like? Though most of us shy away from it, we use math every day. It doesn’t take a bachelor’s degree in number theory to calculate a tip on a bill, or to budget your paycheque to save for a vacation. Success in mathematics goes beyond a mark on homework from the textbook, but our mindset is only just starting to change.

the problem with problem sets why your child’s textbook work probably isn’t being marked

Dear MATH,

Yesterday when acting as a

chaperone for my son’s field trip, I had a

moment to engage his math teacher in

conversation while the kids were eating

lunch. It was at this time that his teacher

made a flippant remark about how she

does not mark the problem sets that she

sends home for practice! I was so upset

I could barely hold my tongue. But when

I got home I immediately called the

school to report on how this excuse of a

teacher is failing our children. How is

anyone supposed to learn if they don’t

get marks back? And what about report

cards? Are the marks simply made up?

--- My Angry Disbelief

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Dear MAD,

We understand your concern, but

there are a few things you should know!

First is that problem sets from a textbook

are not a good, reliable source for

teacher assessment. The answers to all

of the questions are often listed in the

back cover, so it would be meaningless

to mark this work, not to mention that

there aren’t enough hours in the day to

do so! There is a difference between

marks and assessment. The teacher

likely forgoes marks on these problems,

instead using them as a means by which

your son may practice and reflect on his

understanding. This is an important

component of formative assessment, so

be sure to ask her about the rest of her

assessment plan, as this is a great

strategy to help students learn!

--- MATH

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Problem sets can be a good way to practice learned material, but are not necessarily the best way to introduce concepts to students. Textbooks often have solutions inside the back cover, thereby giving students an opportunity to check and reflect on their own work.

the migration of a flappy bird how it’s better to be frustrated than frustrated while doing math

Ask any student with a smartphone

about the game Flappy Bird and you will

probably get an enthusiastic response.

They will likely tell you just how much

they hate the game, how it is “evil,” or

how frustrating it is to play. But they

continue to play it anyway.

The key to Flappy Bird’s success

is how impossible it is to play, and how

that inspires a sense of camaraderie in

our failure. It’s too hard, and we all fail at

it. Math however is different. We all learn

math at different rates and levels, so it is

really unfair to provide students with

rapid-fire type video games that assess

their ability to work through standard

algorithms. Afterall, where in the

curriculum does it say that students have

to be able to answer math problems in a

split second?

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The ability of a student to answer a

question quickly and accurately is

certainly an indication of their mastery of

the topic, but what does it do for other

students? How do we differentiate?

Games like Math Castle (1991)

and Puppy Canoe (2015) were designed

to incorporate student interests (video

games) with curriculum content, but the

time constraints of these activities make

them stressful. Where does anxiety fall

into process expectations? Math video

games are often no more than a tool to

assess student ability to keep calm when

under pressure. The attempt at

differentiation to student interests is

admirable, but the overall experience is

intimidating. Perhaps there are better

ways we can engage students with

technology, hopefully, without tears.

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Video games seem like a great way of combining curriculum material with student interests, however they often induce anxiety due to time constraints and visual distractions. Math Castle (1991) and Puppy Canoe (2015) illustrate how little has changed in our failed attempts to engage students.

get smart with a Smart board embracing technology and empowering your students

What is the role of technology in

the math classroom? If video games

aren’t the answer for making math more

engaging to students, what other forms

of technology might we use? It is a

question that has been asked for years,

but despite the rapid change in

technology outside, few ideas have

translated well into the classroom.

When math and technology were

combined with platforms such as

Mathville VIP (1998), the greatest

limitation was the lack of technology

available to students at the time. Not

every computer had a CD-ROM to install

such software, and the number of

computers and their availability was

fairly limited once spread across the

entire school. But computing power has

come a long way since then, and it isn’t

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uncommon for most students to have a

smartphone in their pocket, or a tablet at

home. Many schools even have class

sets of iPads available to the students

on rotation.

While we often think of math

software as being complex, sometimes

simple is best. Smart boards and simple

iPad apps such as Noteshelf and

ShowMe empower students with

something they are already familiar with:

handwriting. There is virtually no

learning curve with these technologies,

and students can now create graphs,

diagrams, tutorials, or a number of other

products in a matter of minutes. The

incorporation of tablets and projectors

makes student work easy to share, alter,

and build upon, and is a great

opportunity for collaboration.

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Smart boards and apps (Noteshelf pictured) are a powerful and easy way to differentiate the learning process. These tools are very effective for modeled exercises, and can incorporate student interaction for a release of responsibility. Apps such as Educreations and ShowMe allow students to leverage technology to create videos and sound recordings that can demonstrate a process or product.

perplexity: making math meaningful supporting student learning by being less helpful

A great man by the name of Dan

Meyer once said to “be less helpful” to

your students. While this might sound

absurd at first---how are students

supposed to learn without your help?---it

is a strategic move on the teacher’s part

to promote student independence and

inquiry. Student independence is

bolstered by the refusal to address

questions such as “I don’t get it!”

Meanwhile, inquiry is promoted once you

give your newly independent students

engaging, meaningful tasks.

Dan Meyer would refer to these

tasks as ones that cause perplexity.

Problems that will not cause students to

become confused, but curious. So how

do we go about designing lessons

centered around perplexing moments?

What will be perplexing to students?

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We took a timeout from our

publication here at MATH to consider

how one might design perplexing

problems for students, and after

watching possibly every Dan Meyer talk

on YouTube we must admit to being

even more perplexed, or possibly

confused, than when we started.

If we had to describe our view on

what perplexity is, we would say that it is

a cross between inquiry, engagement,

and creativity---not all things we

generally associate with math. It is a

change in mindset on both the teacher’s

and students’ part, and something we

are actively pursuing right now. So to

begin our transformation in mindset, we

started by creating a “digital math”

problem from an image we had taken,

with hopes of perplexing students.

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Is architecture perplexing? Our original Digital Math problem considered how tall the tree could grow, but this is a fairly closed-ended question.

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Would this mess at Ollivander’s wand shop combine the inquiry, engagement, and creativity we felt were required for students to feel “perplexed?”

“There should be a wand-finding app on his phone so he doesn’t have to do all that work.”

Our first attempt at a perplexing

digital math problem consisted of an

image of the Simon Fraser University

rotunda. Designed by the same people

that built Alcatraz, SFU has striking

brutalist construction, with mesmerizing

stairways and skylights throughout.

Students often wondered about the tree

at the center of campus (that is, before

someone knocked it down) and whether

it was able to grow any taller where it

was located. This is certainly curiosity,

but is figuring out the overhead

clearance an engaging problem? Is it

creative for students?

While we liked and appreciated the

architectural display, we are a bit

subjective. How then do we get to the

core of this transient idea called

perplexity? The key is to be less helpful.

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We took the opportunity to design

a problem set in the world of Harry

Potter as this would be a great way to

engage many junior/intermediate

students. Using a photograph taken at

Universal Studios Orlando of Ollivander’

s Wand Shop, we decided to ask

students how long it might take Harry to

find his wand if he lost it in the shop.

Students really seemed to like the

context of the problem, and did not seem

to have any issues with the math, but it

was their comments that were most

helpful. Perplexing? You decide!

“Harry has magic and would use that to

find his wand.”

“He should lift the boxes and open only

the heavy ones.”

“He just needs to check at least half of

the boxes, and chances are he’ll find it.”

patterns, patios, and parallelograms the story of an artistic student hidden in a math classroom

Kasim was in grade five when he

started to get bored at school. He slowly

stopped participating in his classes, and

spent more and more time doodling in

the back of his rarely used math

notebook. His favourite things to draw

were dragons.

By grade seven Kasim was

frustrated with school. Teachers kept

pestering him for missing homework that

he had no intention of ever doing, and

even his dragons were getting boring

because he only ever had one colour to

draw them with: grey. Teachers thought

Kasim was a lost cause---he never did

anything in class, and he had no idea

what was going on in the algebra unit.

So when the class was moving on to

geometry, everyone just assumed Kasim

would float by as usual.

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Kasim wasn’t very excited about

geometry. He liked all of the shapes, and

that the teacher was no longer bugging

him for algebra homework, but he was

so far behind in math that he was used

to just being shrugged off by his

teachers and peers.

But a funny thing happened during

a lesson on geometric patterns. For the

first time ever, relationships were being

expressed with shapes. Kasim’s interest

was piqued, so when it came time to use

trapezoids to tile a rectangular patio,

Kasim quickly got to work, alternating

each tile to form rows, which were then

added to to make the patio. Meanwhile,

all of the other students were literally

tiling in circles! Kasim thought he might

not be such a “lost cause” after all!

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Most students have little difficulty with geometric patterns involving familiar quadrilaterals such as squares and rectangles. When posed with the task of designing a square patio using trapezoidal paving stones, students would repeat the pattern without flipping or rotating tiles, thereby resulting in a circle.

it’s hip to be squared teacher candidate blown away by derivation of area

“We never did anything like this in

school when I was a kid,” said the Brock

University teacher candidate specifically

interviewed for the purpose of writing an

article in MATH on this very topic. “It

just kind of blew me away how it linked

grade eight curriculum content to

concepts like limits in high school

calculus.”

The teacher candidate was, of

course, talking about finding the area of

a circle. The age-old method of just

telling your students it’s “pi-r-squared”

has gone out the window, in favour of a

discovery method involving wedges. By

having students divide a circle into more

and more wedges, they are able to see a

familiar shape take form: the rectangle.

By grade eight, students are well-versed

in finding properties of

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rectangles such as area and perimeter,

so it comes as no surprise that this

‘wedge’ method came easily to them. “It’

s just a rectangle,” said one student.

“Circles aren’t so scary after all.”

This revelation had come to the

teacher candidate after years of studying

math in what we can only assume was a

traditional classroom setting---a place

where perimeter and area are found by

systematically plugging values into

equations from the blackboard and

hoping for the best, or put bluntly, the

kind of place math enthusiasm goes to

die. After the activity, a poll was

conducted on the grade eight students

that indicated a renewed joie de vivre

from their newfound understanding of

the origins of one of math’s most

mysterious expressions.

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Introducing circles in grade eight can be very uncomfortable and confusing to some students. This is an example of where the formula for area is derived from. Though it was not often taught in traditional math classrooms, students are being allowed to explore its implications more.http://ccssimath.blogspot.ca/

when common denominators are uncommon editor’s review of three generations of math texts in ontario

MathQuest, Math Power, Math

Makes Sense. These three texts make

up nearly thirty years of Ontario math

resources, but while the curriculum has

certainly been revised in that time, little

has changed about our textbooks. This

is particularly true if you consider the

chapters on fractions, and in particular,

dividing two fractions.

The proposed means of dividing

two fractions is invariably that a student

should follow the trusty method of invert-

and-multiply to find a quotient (or is it a

product now?) While there is a method

in the madness, it is certainly not

immediately apparent why this is the

case. I studied math all the way to the

end of university, and it was long after

grade eight that I realized why this

algorithm for division even works.

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But the idea behind our trusty

algorithm is simpler yet than taking

calculus in university. The rationale can

be found in the most basic of

manipulatives for our unit: the fraction

strip. Consider what happens when you

divide 1/1 into groups of 1/3. You can

easily count that there are now three of

them. What about 1/2 divided by 1/4?

Now there are two.

It is not much of a stretch at this

point to teach division using common

denominators. You might rewrite 1/2 as

2/4, and then you could do this problem

by dividing the numerators and

denominators. And the the beauty is that

this works for any two fractions.

Students have long been deprived of this

simple understanding, so ignore your

text and teach it in a logical way.

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Fraction strips are a powerful way to compare fractions with some of the most common denominators. We can use them as manipulatives to add, subtract, multiply, and divide.

quantity or quantity making unit rates accessible to students

Recently in an Ontario classroom a

problem was posed to a class of grade

eight students dealing with the purchase

of pizza. Two options were offered: the

first that three medium pizzas of 30 cm

diameter could be purchased for $25,

and the second that two large pizzas of

35 cm diameter could be purchased for

$20. It was thought that the aspect of

pizza might interest and engage

students, however the instructor soon

noticed that students were stalled by the

familiar concept.

Students had no issues deducing

that they should find the area of the

pizzas in each option in order to

compare which was a better value. They

began by finding the areas of the

medium and large pizzas individually

using the familiar πr2 algorithm they had

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derived the class before, and then

multiplied the areas by the number of

pizzas in each case. All of the students

easily found that buying the medium

pizzas resulted in more food, but with

that came a $5 increase in cost.

Students intuitively knew that the

additional cost was not justified by the

small difference in area, but were unable

to communicate mathematically why this

would be so. At the teacher’s suggestion

of trying unit rates, students only

became more confused as they simply

could not relate the dollar value to the

area of pizza in square centimeters.

Students insisted they must find a way to

compare the number of slices of each

pizza, however the slices also differ in

size, making this a fruitless effort, unless

of course we’re talking Hawaiian pizza.

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Pizza is very common to math problems. It can be used for circles, fractions, and sometimes even unit rates, however we should take caution with the latter. Comparing aspects such as area and price may not be as intuitive to students as using another criterion such as slices.

crash and burn how one aspiring teacher decided to wing it and what followed

It was an unusually warm fall

morning when we sat down with a

teacher candidate from Brock University

(as per his request we have withheld his

name from this publication). We had

been tipped off by rumours of disaster

from a group of grade seven students,

and decided to follow up with the matter

immediately.

Interviews with students and staff

have revealed that the teacher candidate

had requested the opportunity to lead an

activity with a group of students, but that

this responsibility had soon evolved into

a full lesson on variables and collecting

like terms. The associate offered to

model the lesson in first period, before

allowing the teacher candidate to

attempt the same content again after

lunch.

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“I took very detailed notes that

morning,” said the candidate. “I really felt

like I had a plan this time.” The

candidate spoke at length of leading an

activity in his own math class at Brock,

and how he wasn’t pleased with the

result. “I need to tone it down,” he said,

before completely zoning out and

watching leaves fall from the trees

outside.

The candidate eventually returned

from his daydream to tell us how he is

now much more comfortable in front of a

class and that he is only a little

embarrassed about what transpired. “All

I had were a couple of notes and quotes.

It was a little rough, but I did my best to

follow a three part outline, and the

students were really responsive to it. I

can’t wait to use a full plan in practice!”

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“I did my best to follow a three part outline, and the students were really responsive to it. I can’t wait to use a full lesson plan in practice!”

What is x? Many students struggle with the concept that x is just “a number.” That is, x could be any number, not a number in particular. In addition, students are uncomfortable with simplifying expressions as it does not result in a single numerical value like they are accustomed to.

the three part lesson: your key to success “if you don’t know where you’re going, you’ll end up someplace else”

While Yogi Berra’s statement on

preparedness might seem a little

nonsensical, it rings true particularly

when teaching a lesson. Showing up to

teach a lesson when you’re unprepared

will make it difficult to address curriculum

expectations. Surely the students will

learn something, but how will it fit in with

subsequent lessons? Having a clear

lesson plan is the best way to map out

how you will teach each component of

the curriculum, and how students will

demonstrate what they have learned.

The three part lesson plan is an

organizational tool to help teachers

access the full potential of their students.

We begin the lesson with a “minds on”

activity, where we find out what students

know about a topic by asking them

exploratory questions related to their

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prior learning or understandings. This is

not an introduction to a new concept, but

a diagnostic assessment that acts as a

primer for the material that is to come.

This is followed by the action, or how the

new material will be taught, and finally a

consolidation exercise where students

reflect on and share what they have

learned.

While three part lessons are

becoming common in all subject areas,

they are particularly important in math.

The shift to incorporate reflection and

discussion takes students beyond the

confines of a traditional education based

on rote memorization of relationships

and procedures as defined by Euclid

more than 2000 years ago, and

harnesses the power of collaboration

and metacognition to enhance learning.

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Teaching is not an all or nothing thing. Some strategies may work for some students, while others will not. Use three part lessons to assess where your students are and where they’re headed. Effective lesson planning will enhance student learning, even if your students sometimes have to sit in rows like a “traditional” classroom.

about the author

Born and raised in a sleepy part of Hamilton, Aaron was identified as gifted in grade three and spent the next seven years in what we might identify as being a constructivist classroom. Aaron attended Westmount Secondary School’s self-paced program, cramming in all of the math, science, music, and Latin credits he could before progressing on to a degree in chemistry at McMaster. He then moved to Vancouver for three years while working towards a graduate degree as a research assistant for the National Research Council's Institute for Fuel Cell Innovation. Out west Aaron continued to pursue teaching opportunities and ran the photographic darkroom club and workshops on campus.

Aaron recently moved back to Hamilton, married, bought an old house, and spent a little time working as a mailman before finally deciding to follow up on the plans he had had to enroll in teacher's college after graduating from Mac. In his free time you might find Aaron running a marathon, playing with his pets, doing yoga, fixing up the house, drinking coffee endlessly, or developing film in the basement.

twitter @strongam_instagram @strongampinterest @strongam

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