Download Draft #1 - 10/01/13

Draft #1 - 10/01/13
George Gadanidis 2013
I don't like math ... I love it! (v.1)
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About this book
If you ask a child about a movie they like, they will eagerly share with you the exciting moments,
the plot twists, and how they felt about the experience.
Ask the same child “What did you do in math today?” and the typical responses will be:
“Nothing,” “I don’t know,” or “Fractions.”
Where did all the good math stories go? Why are children not coming home eagerly saying, "Let
me tell you what we did in math today - it was so cool!"
Why is this the case and what can we do about it?
This book is a work in progress, and it is freely available as pdf from www.joyofx.com and as an
iBook from iTunes.
Constructive feedback is appreciated - please email George Gadanidis at [email protected].
Updated versions will be posted as more chapters are completed.
Please follow George Gadanidis' Twitter feed @joyofx for update announcements.
I don't like math ... I love it!
ISBN 978-1-926699-72-1
(c) 2013 George Gadanidis
All rights reserved.
George Gadanidis 2013
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About George Gadanidis
George Gadanidis a professor and researcher of mathematics education at Western University. He
spends about 50-60 days each year in elementary school classrooms collaborating with teachers
to design better ways of engaging children with mathematics.
With his daughter Molly Gadanidis, he has co-authored several children's stories that engage
young children with big ideas of mathematics and science, available at www.BrainyDay.ca.
Previous reviews of George Gadanidis' work
I could not put it down and have referred to it several times since.
The author's love of mathematics rings throughout the text.
Teaching Children Mathematics
National Council of Teachers of Mathematics
The book makes sense. It should be required reading before all mathematics curriculum meetings
or classroom teaching.
Mathematics Teaching in the Middle School
National Council of Teachers of Mathematics
It's amazing that they're learning this math in grade 3. I thought she couldn't do it but she really
did. I hope you give more homework like this.
It's great to see my son excited about school and about math.
Grade 3 parents
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1. Where did all the good math stories go?
If you ask a child about a movie they like, they will eagerly share with you the exciting moments,
the plot twists, and how they felt about the experience.
Ask the same child “What did you do in math today?” and the typical responses will be:
“Nothing,” “I don’t know,” or “Fractions.”
Where did all the good math stories go? Why are children not coming home eagerly saying, "Let
me tell you what we did in math today - it was so cool!"
You might be thinking, "What do stories have to do with math?"
The fact of the matter is that stories are a big part of what makes us human.
We think in terms of stories, we understand the world in terms of stories that we have already
understood, we learn by living and accommodating new stories and we define ourselves through
the stories we tell ourselves1,2,3. Our lives make sense when shaped into narrative form4.
Story is not simply entertainment. Story is a biological necessity, an evolutionary adaptation5.
Story makes us human and adds humanity to mathematics.
What person really wants to spend 10 or more years studying a subject that they can't talk about
by sharing a good story?
You are probably thinking, "How can you tell a story about mathematics? Math is not fiction. It's
logic and fact, it's right or it's wrong. How do you make a story out of that?"
Let me share an example ...
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2. Escape from Flatland
If I asked you to tell me everything you know about parallel lines, what would you say?
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2.1 Parallel lines never meet
In 2008 I started the band Joy of X (www.researchideas.ca/jx) and since then we have been
performing math concerts for K-8 schools, with funding from the Fields Institute.
The songs we sing come from project classrooms, where I collaborate with teachers to develop
better ways of engaging children with mathematics. The concerts are a way of sharing with other
schools what we do in the project schools.
Before we sing the Parallel Lines song (based on the work of second grade students), we invite a
couple of volunteers to the stage and ask, “Please tell me everything you know about parallel
lines.”
The typical responses are, “parallel lines are straight” and “parallel lines never meet.”
“Are you sure?” we ask. “Is it mathematically wrong to say that parallel lines might meet?”
A chorus of “Yes!” is the reply.
I’m always surprised that students have this rigid view of parallel lines. Two lines are parallel or
they are not. It’s always right to say “parallel lines never meet” and it’s always wrong to say
“parallel lines can meet.”
You are probably thinking: “Well, that’s because parallel lines never meet.”
You might also be thinking: "Math is right or it's wrong - it's a subject where there is always
certainty."
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Well, keep an open mind because I'm going to try to convince you that you are wrong - on both
counts!
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2.2 Where’s the proof?
Over 2,000 years ago, the Greek mathematician Euclid tried to prove his Parallel Postulate. He
was surprised that he was not able to. Many mathematicians after him tried as well, and they all
failed.
How could this be?
The properties of parallel lines seem so obvious. What’s wrong with these mathematicians? One
young student remarked, “Maybe they didn’t try hard enough.”
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What colour was the bear?
Back at the Joy of X concert, we thank the volunteers who shared what they knew about parallel
lines and they return to their seats.
We then pose this riddle:
Molly steps out of her tent.
She walks South 1 km.
She then walks West 1 km.
She sees a bear and gets scared.
She runs North 1 km, arriving back at her tent.
How is this possible?
And what colour was the bear?
On occasion, some students read ahead in their program and find the solution to the riddle in the
lyrics of the Parallel Lines song.
But typically, although we get a lot of interesting answers (for example, “The tent is gigantic and
in her 3 km walk Molly never leaves it.”), it’s rare that someone solves the riddle by challenging
the idea that parallel lines (the North and South paths travelled by Molly) never meet.
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2.2 Parallel lines in second grade
I love working with young children. Everything is beautiful to them. Everything is interesting.
And they have amazing imaginations.
In one second grade classroom, the teacher asked students what they knew about parallel lines.
She recorded their ideas, and then students searched for parallel lines around them.
They noticed parallel lines in “tiles on the ceiling,” “lines on the cupboards,” and “wires on the
guinea pig cage.”
Then students read the math fairy tale, Do Parallel Lines Meet? which I have written with my
daughter Molly, who is now in sixth grade (available at www.BrainyDay.ca).
In this story, Wolf and the Second Little Piggy are running along parallel paths through the
woods. Piggy is puzzled when Wolf, huffing and puffing because he’s out of shape, suggests:
“Let’s meet where our paths come together.”
Piggy asks, “How can our paths come together? They are straight and they are parallel.”
The teacher then posed the riddle about Molly and the tent, and students shared and discussed
their ideas.
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As a hint, the teacher gave students mini globes of the Earth and asked them to use them to
explore the parallel paths of Wolf and Piggy. We purchased mini-globe sharpeners at a local
discount store for $1 each. Students worked in small groups, shared ideas, and used words and
diagrams to record their thinking.
You can see a classroom documentary at our project website, www.researchideas.ca, with videos
of students working on these problems and singing their song, as well as teacher interviews.
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2.4 Singing about parallel lines
The second grade students’ ideas were compiled to create the Parallel Lines song, which students
performed for another class to the tune "We will rock you" by Queen.
Parallel Lines
Paaaraaalleeell lines
Paaaraaalleeell lines
Tiles on the ceiling
Lines on the cupboard
Wires on the guinea pig cage
Paaaraaalleeell lines
Parallel lines
Never meet
But they meet
at the north pole
Paaaraaalleeell lines
The world is a sphere
A 3D solid
The world is not flat
Like a circle
Paaaraaallell lines
Molly in her tent
How did she get back
She saw a bear
What colour was it?
Paaaraaalleeell lines
Molly went south
Then went west
Then went north
How did she get back?
Paaaraaalleeell lines
Parallel lines
In a triangle
At the north pole
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Is how she got back
Paaaraaalleeell lines
Paaaraaalleeell lines
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2.5 Life on planet Earth
Isn’t it interesting (and quite sad!) that we have such a narrow view of parallel lines yet we live
our lives on a sphere (or an approximate sphere)? A conspiracy theorist might say this is a
devious plot perpetrated by the Flat Earth Society.
Imagine a bug walking on a globe. If its walk is balanced, the bug will walk a “straight” path.
Is this path really “straight”? Well the answer is not a simple Yes or No. The answer depends on
your perspective.
From the view of someone not on the globe, the bug’s path will look curved, since they can see
that the bug is walking on a sphere. However, from the bug’s perspective, the path is straight.
I interviewed mathematician Megumi Harada of McMaster University and she explained this in
terms of a straight path being the shortest distance between two points.
For example, if we try to find the shortest path between the North Pole and the South Pole (on the
surface of the Earth), then that path is going to be one of the longitude lines. Any path between
the poles that is not a line of longitude would be longer.
You can see the interview with Megumi Harada on the theme of parallel lines at
www.fields.utoronto.ca/mathwindows.
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2.6 Great circles
When a bug walks a straight path on the globe, it walks a great circle.
A great circle is the biggest circle that you can draw on a globe. Or, as a sixth grade student
described it, “It’s the smallest circle through which the globe will fit.”
Lines of longitude and the equator are great circles. They are the paths of shortest distance
between points. They are straight paths.
Lines of latitude, on the other hand, are not great circles, and they are not straight paths. If a bug
had to walk along a line of latitude it would have to walk an unbalanced path – it would have to
limp or lean to one side.
Lines of latitude are not the shortest paths between points. This is why an airplane traveling from
Tokyo to New York does not fly along a line of latitude. Rather, it travels along a great circle that
passes over Anchorage, Alaska.
If two identical planes race from Tokyo to New York, the plane that travels along the path over
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Anchorage would get there before the plane that travels along a line of latitude.
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2.7 Are lines of longitude parallel?
Megumi Harada explains that one way to describe that two lines are parallel is to say that both
are pointing in the same direction. All lines of longitude point North-South, so they are parallel
from this perspective.
We also know that lines of longitude are the straight lines of a sphere. They are the shortest paths
between the North Pole and the South Pole.
So, from the bug’s point of view, lines of longitude are straight and parallel. But unlike the
parallel lines of a flat surface, lines of longitude do meet.
Cool!
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2.8 Euclid’s problem
Euclid tried to prove his Parallel Postulate as a theorem. The problem is that his Postulate is not a
theorem but rather an assumption.
If we make different assumptions about parallel lines, we get different geometries. Each of these
geometries makes sense mathematically, but parallel lines behave differently.
Here is one way of looking at this.
We have a straight line and a point near the line. How many straight lines can we draw through
the point that will not cross the first line?
If we assume that the answer is 1, then we get a flat surface geometry, also known as Euclidean
geometry. This is the geometry that the Flat Earth Society wants us to buy into, where parallel
lines of longitude never meet.
If we assume that the answer is 0, that there are no lines through the point that don't cross the first
line, then we get elliptical of spherical geometry. This is the geometry that students walk on
every day. Isn't it about time they started learning about it?
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If we assume that the answer is infinity, that there is an infinite number of lines through the point
that don't cross the first line, then we get hyperbolic geometry.
Isn’t this so much more interesting than “parallel lines never meet”?
So doing math is more like writing fiction rather than following a recipe. Different mathematical
assumptions create different mathematical worlds, and new problems to explore. This is what
mathematicians do.
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2.9 How many degrees in a triangle?
We’ve all learned in school that there is only one answer to “How many degrees are there in a
triangle?” and that answer is 180o.
One way to demonstrate that the sum of the angles in a triangle is 180o is to cut a triangle from a
piece of paper (any triangle will do), tear off each of the three corners, and then put the vertices
together. You’ll notice that they fit on a straight line, or 180o.
But is this always true? Is the sum of the angles in a triangle always 180o? Well, it depends on the
geometry we’re working in.
In a flat surface geometry, the answer is yes. In a spherical geometry, the answer is much more
interesting.
Let’s create a triangle on a globe using three straight lines: two lines of longitude and the equator,
as shown in the diagram.
The two lines of longitude meet the equator at an angle of 90o. So the sum of these two angles is
180o. But we still have to add the angle at the pole. However, the size of the angle at the pole
depends on which two lines of longitude we use.
So on a sphere, the sum of the angles in a triangle can have many different answers, depending
on which triangle we use.
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What are the smallest and largest sums of the angles in a triangle drawn on a sphere?
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2.10 How could it be we never knew?
I teach a variety of online math-for-teachers courses for teachers. Some of these are freely
available at www.researchideas.ca/domath and you are welcome to take one.
In one of the course activities we explore parallel lines. Teachers are always surprised to discover
how narrow this concept was presented to them in school. One teacher wrote:
I feel like I was misled, misguided, told the half-truth about parallel lines. It is the first
time that I have realised/felt that math isn't just BLACK & WHITE and can cause quite
creative outcomes/discussions.
Another teacher added:
For me, the discussion about latitude and longitude in relation to parallel lines opened my
eyes to the possibility of questioning math’s rigidity.
School math is taught as if answers are always right or wrong.
But real mathematics is a complex, interesting and imaginative human creation. As a teacher in
one of my math for teachers courses once commented: “I learned that math can be discussed with
your family and friends just like you would a favourite book or good movie.”
Another of my students, Victoria Smith, summarized her learning about parallel lines by
rewriting the lyrics to Neil Diamond's "Sweet Caroline".
Sweet Parallel Lines
When I was young
The lines in my life were lonely
Like parallel lines
along a city street
But now that I’ve grown
You’ve taken a chance
and shown me
A beautiful way that
they can meet
Lines, touching lines
At the poles
Touching North
Touching South
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(Chorus)
Sweet parallel lines
(BUM, BUM, BUM!!)
Who’d have ever thought
they would?
I’d been resigned
(BUM, BUM, BUM!!)
To believe they never could
But – now - I
Look at the globe
And all of the spheres around me
How could it be I never knew?
The world isn’t flat
Lines that connect surround me
And parallel lines can do it too
Lines! Touching lines!
Reaching out!
To the North!
And the South!
(Chorus)
Sweet parallel lines
(BUM, BUM, BUM!!)
Who’d have ever thought
they would?
I’d been resigned
(BUM, BUM, BUM!!)
To believe they never could
(Repeat & Fade Out…)
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3. Two plus two is always 4?
You might be thinking, "OK, that's cool! I get that there's much more to parallel lines than I
learned in school. But what about basic math skills, where kids need lots of drill and practice?
How do you make that into a better math story?"
My short answer is that practice is good and necessary, but children need to practice skills that
help them develop mathematically - and such skills do make for good math stories. Let me give
you an example.
I need your help with this one.
Please take out your wallet.
Take all your money and spread it out in front of you – every bill and every coin.
Now please count it.
All right, now how did you count your money?
If you are a typical person, chances are you first counted the $20 bills, then the $10 bills, then the
$5 bills, and so forth, finally counting the coins, with the pennies last.
If you counted your money this way, what I want to know is this: How did you ever graduate
from elementary school?
Let’s take two numbers, say 87 and 29.
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The way you were taught to add numbers in school was by first adding the ones column and then
the tens column. But when you counted your money, you added the tens column first! You added
from the left, not from the right.
What is interesting about addition is that if you don’t teach children which way to add, they
naturally add from the left. Why don’t we let them?
Maybe it’s because of the recent rash of right-wing governments. Teaching our kids to add from
the left may be politically incorrect.
... more to come
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4. Math as a good story
A big problem with mathematics education is that our teaching philosophy is based on "make
math easy to learn."
You're probably thinking: "Of course!"
The problem is that to develop mathematically children don't need easy-to-learn math. Rather,
they need experiences where they have to think hard.
School math is hard precisely because it is designed to be easy-to-learn.
Let me explain this to you in terms of a movie.
You sit in the theatre and as the scenes unfold, you guess ahead. Your mind naturally tries to
figure out what might happen next. If the movie is easy-to-learn, then your guesses are correct.
You might take some pleasure from this the first or second time it happens, but eventually the
movie becomes predictable and boring and you lose interest. Easy-to-learn movies are hard to
watch. Imagine watching easy-to-learn math movies, day in, day out, for at least 10 years of
school.
We take pleasure from movies, and stories in general, when our guesses are wrong, when we are
surprised, when there is cognitive conflict, when we have to think hard and flex our imaginations
to make sense of an unexpected situation. We are storytelling beings and we are naturally
attracted to experience and share good stories.
Easy-to-learn math is not a good story to experience, its is not a good story to share when asked
"What did you do in math today?", and it does not motivate us to engage with it or to want more
of it.
... more to come
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Endnotes
1. Bruner, J.S. (1990). Acts of Meaning. Cambridge, MA: Harvard University Press.
2. Bruner, J.S. (1996). The Culture of Education. Cambridge, MA: Harvard University Press.
3. Schank, R. (1990). Tell me a story – A new look at real and artificial memory. N.Y.:
MacMillan Publishing Company.
4. McIntyre (1984). After virtue: A study in moral theory. Notre Dame, Indiana: University of
Notre Dame Press.
5. Boyd, B. (2001). The Origin of Stories: Horton Hears a Who. Philosophy and Literature,
25(2), 197-214.
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