Premier Doug Ford blames elementary school teachers for the ‘failing’ grade six math scores in the province. He asserts that the consistently large percentage of 11-year-olds getting Cs and Ds on EQAO tests is the result of teachers’ lack of math knowledge and poor teaching methods.
He is essentially stating that if a student doesn’t learn something, it is because their teacher doesn’t know what they’re doing or how to do it. If this is true–and Mr. Ford certainly thinks it is–then he must believe that today’s teachers don’t know math because their own teachers didn’t know math or what they were doing.
Here’s why the Premier might be right, in spite of himself.
The Ontario College of Teachers website shows the vast majority of educators in this province are over 30, with more than half over 40. This means most of Ontario’s teachers were educated in the 90s or earlier. In the 70s, 80s, and 90s, my colleagues and I were taught from textbooks and mimeographed pages of fill-in-the-blank equations. We drilled adding and multiplication tables. We were shown formulas, told when and how to use them, then given lots of practice plugging numbers into those algorithms to get answers. These methods were required for rote knowledge and mechanical application. According to Mr. Ford, they also failed to teach any of us teachers how to do math.
So I ask, if these methods failed today’s educators, why on Earth would anyone want to go back to them now after we’ve come so far?
If you’re over 30 and you didn’t study it in university or use it in your profession, then you probably don’t know math like you should. Not thoroughly. Not deeply. Your children are coming home with work in their backpacks, and you’re frustrated and confused when you try to help. You’re struggling with the open-ended tasks and numberless word problems. You’re sending nastygrams to your child’s teacher because you have no idea what a stem and leaf plot is. You’re yelling “What do you mean there are 11 kinds of adding questions?” and “Just carry the one and go to bed!”
You’re not alone. When I was ten, I memorized the value of Pi and used it to find the area of many circles. I was 39 years old when I found out that Pi is equal to the number of times a circle’s diameter fits in its circumference. 39. I have two university degrees. I’m not stupid. Neither are my colleagues. Neither are you. We can all do grade six math, but we have gaps in our understanding, and we are unpracticed in our thinking.
There is nothing wrong with memorizing adding facts. That’s why it’s been in the Ontario curriculum since at least its revision 15 years ago. There’s nothing wrong with learning formulas and how to apply them. That’s why the curriculum says to use student-generated AND standard algorithms. But the curriculum goes well beyond these expectations.
If today’s teachers are struggling with some of the math and how to teach it, it’s because the focus of our own math education came up short. It’s because we’re emulating pedagogy we saw when we ourselves were taught. We are working very hard to do better than was done before. Which means, if today’s students are struggling, it’s because we expect more from them than was ever expected of you or me. We’re doing more than just the ‘basics.’
I am on the third edition of my dot to dot book, and it is already more successful than both the previous versions combined. The only thing I’ve changed is the cover.
I am an art school grad. I’ve worked as a designer. I took the Ryerson Publishing programme’s book design course. Dripping with the skills I thought I possessed, boosted by an undeserved sense of marketing savvy, that describes me when I started on this project.
I mean, my first attempts at cover design weren’t Edsel or mullet-bad, but I’m no Chip Kidd.
The first covers I created were for the PDF versions of my puzzles, which I publish on Teachers Pay Teachers. I tried to be mindful of a number of factors.
Teachers Pay Teachers is like Amazon. You see the thumbnail in the search first. This cover was supposed to catch your attention and tell you what the book was about with a glance. I thought I did that, but I did not.
Upon reflection, I decided it would be a good idea to actually show what picture you’d be drawing. I released these puzzles in a collection and as singles. Each individual cover featured a boring little outline of the finished image. Each one also had a giant graphic of a pumpkin outline, just to confuse matters.
I realized that the most unique feature of these dot to dots was missing. I got rid of the giant counting dots and focussed on the puzzle images and all the extra dots.
When I decided it was time to release the paperback to the world, I wanted to showcase all the different holiday themes in the book. I hadn’t learned the lesson about the bad title and I ended up making the dots too small. You can’t read most of the text, but the drop shadow is nice.
I did get wise about the font size.
Eventually, I figured out that the title needed to go. I also mention the extra dots this time. I thought the specially shaped dots used in the book would be a nice touch. I thought wrong. They look like ink blotches or moles.
Giving myself credit where it’s actually due, I have sold many copies of the dot to dot puzzles on TPT, despite the graphic horrors. However, I have to admit that the competition on the site is not made up of award-winning artists and marketing experts. Clip-art reigns supreme over there, meaning I could potentially sell even more with the right cover.
When I decided to release a third, expanded edition of the paperback book using Ingram instead of Create Space (Another blog topic), I took advantage of the opportunity to change the design almost completely.
This is what I came up with.
First, I simplified the title. Playing with the text orientation makes it stand out from all the other titles on the search page. Next, I further emphasized the puzzle image and the extra dots by using the image at a 1:1 scale. You don’t see the dots as numbers in the thumbnail, but the mass they create contrasts very well against the big, bold line drawing of the cat. You are curious to see what they might be. Finally, I kept the text simple, big, and informative. Even shrunken down, most of it’s legible.
Since the upload of the redesigned book, two weeks ago, I’ve sold 22 copies online without even trying. I had 13 pre-orders before the title was officially available because a few die-hard searchers noticed it among the thumbnails on page 17 of their Amazon search “extreme dot to dot.”
Needless to say, I will be revising the layouts of all 88 puzzles and collections on TPT.
The process of redesigning these covers over the past three years has taught me a lot. Even if I hire someone else to do my next cover, here is what I must remember.
- Don’t go with the first design. Too bad if you’re anxious to launch and you want it done right now. Instead, just do it right.
- Keep It Simple, Stupid. Leave the filters, tricks, and design-school cleverness to the experts who can afford to experiment because they’ve got a million-dollar ad campaign forcing the book on people, bad cover or not. I still have some drop shadows on the back cover (wink).
- Even if you’ve hired an expert (especially if), get feedback from people you don’t have to share a dinner table with. You need honest opinions. Ask artists what they hate about it. Ask philistines what they like.
Keep this in mind when designing your book because, in the world of online shopping and search page scrolling, you will absolutely be judged by your cover.
This May, I presented at the OAME (Ontario Association of Mathematics Educators). My session was on combing visual arts and mathematics to create rich cross-curricular learning experiences.
A lot of art has direct connections to math. You don’t have to look long to find examples of geometry and patterning and, in some ways, number. The art elements shape, form, line, and space are all found in the geometry curriculum.
The design principle of balance is connected to the concept of equality and every painting, symmetrical or asymmetrical, can be looked at in terms of equations. The design principle pattern/repetition is ubiquitous in graphic and product design, using colour, shape, orientation and texture to create harmony and movement.
Behind the scenes, all strands of math abound, especially measurement. An artist cannot make art without knowing and applying math concepts and skills. From preparing a canvas stretcher (perimeter and area) to mixing plaster (capacity, proportion, time) to calculating the shrinkage rate of clay (ratio, percentage), there is math that the artists must do.
Teaching Math with Art
When teaching math with art, the goal is to notice the obvious and hidden math, name it, and apply it. Watch this video to see my first project, Mandalas with Geometry and Pattern. The video gives a captioned explanation of each step while the math connections pop up as the video plays. You can pause at certain moments to think about and discuss the math as you watch. Download this PDF for a written lesson plan that goes with the video.
You can use such an art project at the beginning of your unit, letting the hands-on experience be the concrete modelling your students explore before they move to paper and pencil tasks.
Or, you can do these projects to apply the math they’ve already explored in other contexts, spiralling back to previous learning, turning the knowledge into understanding and application.
You can even use these projects to evaluate your students’ math knowledge by observing them and discussing the math as they work. Can they use a protractor properly to make a mandala with nine segments? Can they tell when their art isn’t symmetrical? Note: when doing culminating assessments, you can’t rely on the finished product alone because undeveloped artistic skill might get in the way of showing the math properly–if your student can tell you their folding wasn’t congruent or their rotations weren’t quite equal, then they are demonstrating they know the math, even if they aren’t that precise with the art-making.
Have a clear list of success criteria that covers knowledge, understanding, thinking and application. Pay attention to the students as they work, making note of successes and struggles, intervening when necessary. Use the math language. Apply the procedures. Push understanding and thinking by doing more and more challenging work.
Links to my Art Math videos and PDFs:
Mandalas with Geometry and Pattern – Video – PDF
Tessellations with Geometry and Pattern – Video – PDF (in the works)
Animal Collage with Geometry – Video – PDF (in the works)
Collage with Number Patterns – Video – PDF (in the works)
Please share with anyone you know who loves doing art and math.
This single puzzle is a Free Dot to Dot to Dot Thanksgiving Activity. Count by 3s to find the hidden picture. Click the image to download a copy of the puzzle.
These are not your mother’s connect the dot puzzles. They skip count by 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, and 10s, and they have extra dots. Follow the pattern, skip the extra dots, and reveal the picture.
Each Holiday collection has over 13 different puzzles, with a “hard” and “easy” version of most of them. There is Halloween, Christmas, Valentine’s Day, and Easter, as well as a Spring & Summer and a Snowflake collection. Incorporate these puzzles into your number sense, operations, and patterning instruction and assessment. Use them in whole-class instruction, as part of your math centres, or for a fun but educational holiday activity to do with your class.
Get the Free Teacher Package that shows you how to manage the puzzles in the classroom, describes ways to include them in your math lessons, and how to analyze your students’ errors to know where they are on the Number Sense continuum.
Please leave your feedback to help me make these collections the best they can be.
So, I was driving to work, listening to the CBC (Canadian Broadcasting Corporation – for U.S. followers), and they were talking about the Heartbleed bug.
Well, they were actually talking about changing passwords, and how important it is to make them fairly complex, differentiated, and not in any way predictable or guessable.
I am SHER locked.
Of course, this level of complication makes passwords extremely forgettable.
Wait. I want to digress a bit. Let’s play a game. This is how you figure out your Movie Star name. Your first name is the the name of the first pet you ever had. Your last name is your mother’s maiden name. My Movie Star name is Snoopy Masters. What’s yours?
And while you’re at it, why don’t you just save me some time and tell me your father’s middle name, the street you grew up on, and your favourite fictional character, because clearly you want to be the victim of identity theft.
Most webpages let you reset your password, or at least get a hint, if you can accurately answer a security question that you set up when you registered. You know what I mean. “What is the name of the first street you grew up on?” “What was the name of your first pet?” Imagine my shock when I heard the CBC hosts talking about how con artists set up these posts and threads, collecting people’s information with innocent little internet games.
Not as advanced as the scene in Now You See Me, but just as effective.
So, NO MORE “What’s your Rocker Name” games for me. Or you, I hope.
The first theme from Brockman’s collection of intelligence-fostering essays, This Will Make You Smarter, is that we would all be better off if we saw the world the way a scientist does. There are three essays in particular that extol the objective processes of the scientific method and experimentation as the procedures for doing everything, from allaying irrational fears to overthrowing fascist governments.
Full disclosure, so that you’re not changing the entire way you think based on my interpretation of a small data set. These essays are all very short, and are not to be considered exhaustive. But, they do encourage a lot of deep thinking and reflection, in me anyway.
The Controlled Experiment – Timo Hannay
This essay is about following the scientific method to make decisions in our lives. Hannay asserts that we are too quick to judge, and those judgments are based on prejudice, preconceptions, and a lack of imagination. According to him, we should be striving to be more objective and data driven, rather than going by some introspective gut feeling, or asking around to see what others think. I believe that his assertions are based on data collected in controlled experiments, but I only have his word for it.
The Double-Blind Control Experiment – Richard Dawkins
Dawkins is of a like mind, but twice so. I will infer he has been reading the same journals as Hannay. He wants us to be critical. He also thinks we need to be more objective in the face of personal opinion, because of the bias inherent in it. He wants us to stop generalizing from a handful of anecdotes, in favour of coming to conclusions based on evidence. For him, it all comes down to improving the population’s understanding of how controlled experiments work (specifically, I believe, how they filter out as much bias and external influence as possible, ensuring that true causality is established), so that we develop a healthy, habitual skepticism that will keep us from being tricked or lied to.
Promoting a Scientific Lifestyle – Max Tegmark
In this entry, Tegmark laments the “fact” that scientists are pretty much useless at getting their message out to the public, and that, when the message is disseminated, it has often been watered down, misinterpreted, or bastardized (I’m generalizing here, but HELLOOO, every last initiative in education I’ve seen in the past 15 years).
He laments this because so many of the decisions we make in our lives are based on inaccurate and incomplete information. He believes we are too busy, too biased and set in our ways, and too overwhelmed by the volume of information available. I’m going to add too skeptical of things we think will make our lives complicated, boring, or inconvenient.
These restrictions of our intellectual development, he believes, result in us being easily manipulated into voting for stubborn politicians who turn their backs against evidence that undermines their biases. We allow ourselves to be duped by sound bites and propaganda that play into our fears and prejudices. And we end up blindly following charlatans and buying books about wish fulfillment, looking for the simple, quick fix to all our problems. If he had his way, scientists the world over would be using the same tactics as politicians and corporations to get their messages out, and that message would be “Look, smarten up.”
My purpose behind this blog entry is to explore how these ideas might influence me and my pedagogy. What would be the result of teaching students about controlled experimentation and critical thinking? Shall I question the authority of these experts before I make any decisions?
Visions of riot shields and fire hoses come to mind when I think about challenging authority. Is that what these folks mean? Do they mean rebel against authority? Don’t trust anyone over 30? I don’t want a child to touch the stove because he’s been taught to be skeptical of me when I say it’s hot. I don’t want him to find out on his own. Or do they actually mean to question authority? Well heck, I do that all the time.
Questioning is not necessarily the act of open rebellion some politicians and other community leaders make it out to be. It is just the act of asking questions. “Why is that the case? How do you know? Based on what evidence? Have you considered other data? Which ones?” I don’t really have a problem with this, as long as I have substantiated responses to provide. And as long as the student isn’t just questioning for the sake of being a disturbance, and can provide her own evidence to the contrary, if necessary. Anyone who does question can certainly be given the opportunity to find out the answers for himself, through research and experimentation. I’m comfortable with this, because I’m fine with finding out I was wrong.
Which brings me to an idea these writers have not touched upon, but I believe must be pointed out. If we are going to question others, then we must also question ourselves. Riot police are called out when two factions of self-righteous people clash. But, those who scream that they know better than the authorities inevitably end up authorities themselves, with their own violent detractors to repel. Humility and openness to others’ skepticism seem much more productive than arrogance and close-minded certainty. When we experiment, we suppose. We are bound to find some of our suppositions were wrong.
In the end, I agree with these three arguments, and yes, to acknowledge my bias, this is mostly because they match my experiences. But, when I say experiences, I mean from both perspectives. I have been on the receiving end of ignorant dictates (from several education ministers and their leaders since ’95). And, I have been guilty of an unbending, near-sighted hubris of my own (Dear Editor). I’ve made myself the authority and then been challenged because of it (Literacy Coach). I’ve challenged and been shut out. I’ve lead the charge, only to find myself the first to bare the brunt of a counter attack (Facebook Page).
If I choose to see those experiences as experiments, then I can look at them objectively. I can question myself and decide how to conduct the next experiment better. I can assess for bias. I can restrict parameters. I can gather data. I can learn.
I can use that learning to question.
I can use that questioning to teach.