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.
I’ve been wanting to try my hand at clay portraiture for a while. I made a bust of myself in high school (the chin exploded in the kiln), but that was 30 years ago.
Here are the instructions for a homemade armature I designed and built. It cost me around $40 and uses materials you can get at most box-store hardware suppliers.
Update, April 14, 2019
Here is the sculpture I made with the armature. What do you think?
Please enjoy this Free Valentine Dot to Dot to Dot.
Click the image to download the PDF. Read the directions. Print off the “easy” or “hard” puzzle. Fill in the To: and From: Give it to a special someone for them to solve.
This freebie is a sample of one of my Dot to Dot to Dot skip counting number puzzles. These puzzles have a twist. If you don’t follow the pattern and skip over the extra dots, the picture doesn’t work. Learn or practise skip counting by 2s, starting at 1. Have fun and stay sharp.
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.
Have you been really focused on number sense in your classroom? Are you using number strings, doing number talks, counting around the circle? If so, then you are thinking about numbers and the relationships between numbers. You are thinking about the patterns made by numbers, and the effect of repeated operations on numbers.
Have you ever considered using dot to dots as another way to play with and explore number?
Maybe not. Typically, they are very simplistic, only a few numbers, rarely going beyond the twenties, and only counting by ones; they are a challenge for only the earliest of learners. However, in the last couple of years, I’ve seen a renaissance in dot to dots, going hand in hand with the colouring book trend. There are a couple of very talented designers/artists out there who have created some brilliant dot puzzles, but they still have their limits as to their use in the classroom.
This past year, I’ve been playing with the puzzle design myself, tweaking the structure and mechanics to make something that teachers can use. My puzzles have several essential differences.
First, They skip count by more than just 1s. My puzzles run the gamut from 1s to 10s. This means they help to learn and practise skip counting and growing patterns, and by extension addition and multiplication for almost all the basic facts.
Second, they have different starting points. I have several puzzles that skip by 2s, but some of these start at 1 (1, 3, 5…). Some puzzles skip by 3s, but a few of them start at 1 or 2 (1, 4, 7, 10…; 2, 5, 8, 11…). Solving each number in the sequence in these unfamiliar patterns practices mental addition and number fluency.
Third, there are hard and easy versions of each puzzle, allowing teachers to differentiate for different children’s knowledge and skill levels. The “easy” version is useful for students who are struggling with the concept of skip counting and need some supports to get to each new number. The “hard” version throws distractions and red herrings onto the page in the form of extra numbers. Students doing a “hard” puzzle must really know how to apply the pattern, or risk connecting the wrong dots and creating an incomprehensible scribble.
If you want a more thorough explanation, I detail how these puzzles work and ways to use them in my Teacher Package, available for free here by clicking the link or the images, and on TeachersPayTeachers.com where you can see all the puzzles available. The package explains how to distribute or present the puzzles, tips for differentiating, suggestions for how to tie the puzzles to the number and patterning curriculum, and examples of errors that student might make with the puzzles and what these mistakes might be saying about a child’s skill and knowledge.
Have a look at the Teacher Package or download the free previews for the different Holiday themed collections, also available on this blog. Let me know if you try the puzzles with your students and give me feedback about how it went or suggestions you might have.
Here are two videos I have made that teach how and why to do single stroke printing. Single stroke printing has several advantages over other methods, such as stick and ball.
- Single Stroke allows for more consistent letter formations, because most letters are structured around some basic, well practiced strokes.
- The repeated directionality of each basic stroke helps to eliminate reversals of letters, such as b and d.
- Single stroke printing naturally evolves into handwriting, or at least a hybrid of handwriting and printing.
Or you can watch them below.
Please feel free to share the videos with the educators and parents you know who are concerned about good penmanship.
Is there really no such number as Eleventeen? Well, there isn’t, but bear with me for a minute.
I have been posting about how we math educators can play with language and make our slightly awkward number naming system a little more logical and a lot more transparent. Check out Part 1 to read about playing with number names to clarify our base-ten system. Check out Part 2 to read about how we can make place value and zero clearer. In Part 3, I am exploring how to make borrowing clearer to the new learner.
The number eleventeen was derived by some astute and logical child who figured that since 13, 14, 15, 16, et al. all end in TEEN, then so should 11, and 12 for that matter. Of course, the poor dear didn’t count on the linguistic mashup that is English. There is also every chance that some other child called 11 ‘Oneteen’ and 12 ‘Twoteen,’ bless her heart.
Wouldn’t it be nice if our number names made more sense? Well, what’s stopping us?
Read this list of numbers to yourself. Then tell yourself what comes next.
10, 20, 30, 40, 50, 60, 70, 80, 90
Did you follow Ninety with One Hundred? Of course you did.
Now, how about this string? Read it to yourself and say what comes next.
100, 200, 300, 400, 500, 600, 700, 800, 900
Did you say One Thousand? Or did you say Ten Hundred? Which one is right? What’s the difference?
Most would say that One Thousand is technically the correct response, but in common vernacular, Ten Hundred is acceptable too. We usually said Nineteen Hundred Ninety-Something a couple of decades ago, and we all knew what we meant.
My idea is to take this already common and, quite frankly, logical exception to naming a number, and apply it to clarifying the act of borrowing.
Look at this equation.
If we subtract these numbers in columns, without any deriving, manipulating, or expanding of the numbers (let’s go ‘old school’ for a moment), we immediately end up in a situation where we have to borrow. And how do we do that?
We can’t do 4-7, so we have to borrow 1 TEN from the Tens column, effectively breaking that decade unit into 10 ONEs. We add those to the 4 ONEs already in the Ones column. Now we have 14-7. But look at how we’ve notated that borrowing on paper. A new learner would look at that top number and lose it. That number now makes no sense. It was 234, now it’s 2214, which looks to me like Two Thousand, Two Hundred Fourteen.
But of course it isn’t actually. It’s 2 HUNDREDS + 2 TENS + 14 ONES.
So here’s my thinking. Let’s actually call it that. The original number was Two Hundred Thirty-four. Now it’s Two Hundred Twenty-fourteen.
You know, as in Twenty-eight, Twenty-nine, Twenty-ten, Twenty-eleven, etc.
If we can say Ten Hundred, why can’t we say Twenty-fourteen?
Typically we abandon what the Minuend (the number we are subtracting from) is called (thereby losing the value) the moment we break it up to borrow. We hash it up, cross things out, write things in. It’s no longer an identifiable number. However, while we broke the rule about how many digits are allowed in a column, and put more than 9 ONEs into the Ones column, we can try to keep order and at least follow the rest of the rules by naming the new number logically.
Rather than jumping in to solve 14-7, I’ve read that it’s a good idea to do all the borrowing before any subtraction is actually done. So, let’s keep borrowing and move on to the Tens.
We can’t take 5 TENs from 2 TENs. We must borrow 1 HUNDRED, break it up into 10 TENs, add them to the 2 TENs already in the column, and get 12 TENs altogether.
Now, we’ve done all the borrowing we have to do, and the minuend is completely mangled. Any thoughts on what we could call this new number? What do we name 12 TENs sitting in the Tens column?
Seventy, Eighty, Ninety, Tenty, Eleventy, Twelvety!
Therefore, I propose the new number we are subtracting from is One Hundred Twelvety-Fourteen.
One Hundred minus One Hundred is Zero Hundreds (or No Hundred, if you read Part 2). Twelvety minus Fifty is Seventy. Fourteen minus Seven is Seven. So the ‘answer’ is Seventy Seven.
Without naming the new number and making the borrowing obvious, there is so much potential for confusion. Children don’t understand how the carried over numbers apply. They think that the number has suddenly increased in magnitude. They forget to cross things out…
But, do you see how, by naming borrowed numbers this way, we are following through with the properties of our number naming system, staying true to the Hundreds, Tens, and Ones structure of base-ten, using the name system to impose order and make what we did with the borrowing process obvious?
Would I exclusively name numbers this way. Absolutely not. That does no one any good. However, by renaming the minuend, we are playing with language. Kids love playing with language. By playing with the language, we are making our number system clearer, exposing rules that have been hidden due to linguistic laziness/pragmatism over time. A clearer sense of number in this gaming context makes the learning stick in a meaningful, more flexible and fluid way. And, when we learn any language, our goal is fluency. The language of math is no exception.
If you want to read Part 1 where I discuss the number words we have for the TEENS and TENS, click here.
For Part 2 of tackling number naming conventions, another issue that stands in the way of many students truly understanding the base 10 system is the role of the zero, and what we call it.
Zero is a quantity, a benchmark, and placeholder. All students learn what zero is as a quantity. They connect 0 to the idea of having nothing very easily. They also catch on to the 0+n=n equation pretty fast; not just the idea of adding to ZERO but the formula itself, even at 8 years old. ZERO as nothing is easy.
As a benchmark, it’s iffy. It applies mostly to negative numbers which baffle teenagers sometimes. However, talks about rulers, scales, and centigrade thermometers can help students to at least see that we generally start counting from zero.
What is really tough is when it comes to place value, especially when borrowing and carrying in columnar adding, and regrouping when applying the distributive property. It’s overwhelming sometimes. And yet, without a ZERO, our number system is impossible. As such, it is imperative that students gain a strong understanding of how the ZERO works and what it means. Here’s a distracting link to Schoolhouse Rock you can show your class-but preview it first because you never know how people muck around with stuff.
So, how do we overcome some of this confusion?
Be aware the ZERO is there, and make sure students understand that it means something. When we have a number like 20, or 304, or even 2, the ZEROs go unmentioned, like they aren’t there; but they are, they have to be.
It’s not actually 20. It’s not even Two TENs. It’s Two TENs and ZERO ONEs.
It’s not 304. It’s Three HUNDREDs and ZERO TENs and Four ONEs.
The simple act of explicitly bringing the ZERO to a student’s attention, and giving a name to the position of the ZERO makes place value much more obvious.
Think of that ZERO as an empty box. Whenever we see a ZERO, there is a place we can put another number. We don’t think about it because we don’t say it (it takes too long to do such a thing practically, but for educational purposes…). If you think about it, even the number 2 has ZEROs. There are infinite ZEROs both before and after that 2, but for brevity and clarity’s sake we don’t mention them.
What’s important to understand is that empty boxes can be filled. You have experienced children who write 198, 199, 200, followed by 2001, 2002. This is because they have misunderstood that the ZEROs are placeholders for numbers that can be added to the HUNDREDs. They think that the ’00’ works like a plural S or some kind of suffix. It must be made clear that the first 0 after the HUNDRED is for any TENs that might come along, and the next ZERO is for any ONEs. What’s nice is the 0 looks like a cozy little container you can put stuff in.Again, labelling the ZEROs with their place value will support this understanding.
Here we have 200…Two HUNDREDs and ZERO TENs and ZERO ONEs. As well, we have Two ONEs. I’m going to add those ONEs in the empty ONEs container of 200 to make Two HUNDREDs and ZERO TENs and Two ONES.
I have Given ZERO a name. I know, ZERO is its name. So is nil, and cipher, and nought. But that’s not what I mean. If you think about 3, it can be THREE, it can be THIRteen, it can be THIRty, or THREE HUNDRED. And each name for 3 implies it’s location in the place value system (again, follow the link to Part 1 to see how to make this explicit). Well, I think we should do the same for ZERO.
So, we have the number 0 alone, and that is called ZERO.
What about for 10? First, we break 10 down and say it is One TEN and Zero ONEs. Then, when that is understood, or maybe in order to make sure it’s understood, we take things a step further, and name it One TEN None.
Did you see what I did there? To clarify, 21 is Two TENs ONE. Therefore, 10 is One TEN NONE. The TENS are no longer seen solely as large collections of ONES. By naming the empty ONEs column, we make the students aware a number is there, it has a purpose and a value even if its value is nothing.
If you keep this going, we have Twenty-None, Twenty-One, Twenty-Two….
We have Fifty-None, Fifty-One, Fifty-Two….
Next, 300 becomes Three HUNDRED, Empty-None. Get it? Fifty=Five TENS, Thirty=Three TENS, Empty=No TENS! (I’ve also toyed with Nonety or Nunty, but that might get confused too easily with Ninety. I’m still experimenting.) The point is, by naming the ZEROs, we emphasize the value of each part of the number as well as the ZERO’s placeholder role. This should prevent numbers like 60024 when writing about how many kids go to our school
Let’s do 2045? Two THOUSAND, NoHUNDRED, Forty-Five. Or perhaps you prefer None HUNDRED?
This can go on, but once we’re in the thousands, the irregularities of naming just repeat themselves. We, as a culture, haven’t used large numbers enough for the language to evolve shortened, lazier forms of big number names. How would you say 3 000 000?
Three Million, NoHundred Empty-None Thousand, NoHundred Empty-None.
Finally, I’ll give you a challenge that one of my students gave me. How would you write the number Empty-Six in numerals?
I’m pretty impressed that he understands that whole infinite ZERO thing, and I see this insight of his as proof that this idea of naming the ZEROs has some real legs. Let me know how it works for you if you try it, and tell me about any confusions or issues that popped up that I haven’t experienced yet.