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 texture 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 so 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 and pay attention to the students as they work, making note of successes and struggles, intervening when necessary. Use the language. Apply the procedures. Push the 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.
As an educator, there are few days that go by when I don’t hear some form of teacher bashing or another. If it’s not some angry mother telling everyone who’ll listen she’s going to get some teacher fired, it’s a 6-year-old repeatedly yelling ‘shut your mouth, bitch’ at me because I made her put away a colouring page and join us at the circle to sing.
Those are just the casual day-to-day slams against me and my profession, par for the course, and no different than what nurses, police officers, cab drivers and sandwich artists hear every day in their lives.
However, in these past few months, the slams have been coming hard and fast from parents, community members, Facebook ‘experts’ and even my own provincial government. I say ‘my’ but I didn’t vote for them.
One of the biggest complaints I’ve been reading in comment threads is this.
“I spend hundreds of dollars on tutors for my children every month, because the teachers don’t do their jobs.”
And to this statement, I ask, ‘You’re getting your child a tutor? Why?’
Specifically, Why aren’t you helping your kids yourself? I mean, to read the internet, everyone is an expert in pedagogy. Aren’t you an expert? Of course, you are. An expert who relies on tutors, apparently.
Don’t get me wrong, I think tutors have their place. For some children, they are essential. I was a tutor back when no boards were hiring, thanks to the Harris government cutting teaching jobs.
Truthfully, when I ask ‘why,’ I’m actually being rhetorical. I already know why people who rant and rave about how bad teachers are, end up going to tutors.
This is why.
The material is beyond you
Which is funny, because I hear over and over how ‘In my day, we had to memorize facts and dates and times tables, and we did just fine‘. If you did just fine in the old system, having been taught the ‘basics’ we’re supposed to be getting back to, why can you literally not help your children do their homework or study for a test?
You don’t know that method of subtraction? You don’t get the point of this question?
But you went to school. Maybe you even got good grades. It ‘worked just fine.’
Please, tell me how to teach math and what math to teach. Advise me on which YA romance will finally getting those university-bound kids interested in reading. And then take your 12-year-old to the tutor because you can’t use an open number line.
You always end up in tears
Yeah, you do. You start off with your five-minute warning that homework time is about to start. Then that passes and you now have to get stern. Then angry. Then you’re staring across the table at some grumpy face. This is when the excuses start.
‘I don’t know what to do.’
You read the instructions.
‘I don’t understand.’
You start explaining.
‘The teacher NEVER TAUGHT US that.’
So you teach it your way.
‘THAT’S NOT HOW THE TEACHER DOES IT!’
You’re yelling. Your kid’s yelling. The child storms off to the TV and you scribble out a nastygram telling off the teacher for expecting your child to do work that was never explained or taught.
Out of curiosity, do you know how many times a year a child has told me ‘that’s not the way my dad does it‘ but can’t show me how dad does it? Or how much I hear ‘I know the answer is 6 because my dad told me‘ in response to a math question dad has never seen?
You need a tutor because you can’t handle your child 1 to 1. Yet, you expect your child’s teacher to handle him in a class where she’s 1 to 20, 1 to 25, 1 to 30, even when your child is not the only stubborn one in the class (as if stubbornness is the biggest issue the teacher has to deal with).
You could use your horrible experience with your child to develop empathy and decide to write your government to keep class sizes as low as possible, allowing teachers to maximize the time they have with each student, to deal with the work avoidance and tantrums that go with it.
But no, just hire a tutor and complain about me.
You don’t have time
I’m going to reply to this complaint with a variation of the slam I get regularly.
If you don’t like it, then quit! Give the job to someone who’ll just be happy to have one.
I don’t actually mean that, because I get it. It takes hours for students to learn something new. If they have any learning disabilities or a lack of prerequisite skills or knowledge, it takes even longer. It can take as many as 1000 repetitions for a child to learn something. That’s why you pay the tutor by the hour and not the minute.
Too bad it never occurred to you (while you were writing an angry letter to the superintendent slamming your children’s teachers for failing to meet your kids’ needs), that teachers don’t have hours to work 1 on 1 with your child either. Once that whole-class lesson is delivered, each kid gets a minute, two if they’re lucky or especially struggling before the lesson ends and it’s time to move on. The amount of time they get with the teacher is directly proportional to the number of kids in the class and the amount of time being divided between them. But don’t worry. The 38 minutes they are struggling to learn, alone, while waiting for their turn with the teacher, those minutes build the resilience needed in today’s workforce.
And just as an aside, the kids who do learn quickly are all but ignored by the classroom teacher, because they don’t need the attention. Well, they do, but they aren’t going to get it. Not when EQAO scores and other standardized test results are the driving force behind education policy.
SO MUCH RESILIENCY!
Your child has a learning disability
And you don’t have the expertise, the patience, or the stamina to put in the time, deal with the frustration, and problem-solve or work around the issue.
Now, the 1 to 1 tutor might not be an expert, but they have the time and stamina, because there’s only one kid, with one set of problems to worry about.
Yep, it would be great to spend an hour sitting right next to Johnny with the ADHD so he stops daydreaming long enough to write a sentence. Fantastic if I could spend a whole period repeating the same concept twelve times for the child with audio processing delays and short term memory problems. How wonderful to give my full attention to neuroatypical Mary so that her anxiety stays under control and my desks all stay standing. But I don’t have that luxury.
So you know what. Go ahead and scream at me about how I’m not doing enough to accommodate your child. Throw your tutoring bill in my face. Because you’re right. I’m not doing enough. I can’t provide the support your tutor can. I can’t even match what little you do for your child.
As I said, tutors are great and even essential. If you can’t or won’t support your child personally, then paying someone else to do it still helps. Lord knows I can’t be as effective as a high school kid giving 100% of their time to one child. And lord knows my time with my students is only going to become even more divided. In the end, all I can do is tell you thanks for taking some of the responsibility for your child’s education.
It’s a shame you’re putting out so much of your own money, rather than telling the government to ask taxpayers to properly fund special education supports, instead of cheering on the ‘whiners‘ who are fighting to protect class caps.
But you do you.
This single dot to dot puzzle is a Free Dottoo Dots Thanksgiving Activity. Count by 3s to find the hidden picture. Click the image or this link to see and download the PDF from Teachers Pay Teachers.
Dottoo Dots 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 begin at various start points, increasing the challenge.
Each Holiday collection has over 13 different puzzles, with a “hard” and “easy” version of most of them. 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.