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.

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.

]]>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.**

**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.

**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.

**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!

**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.**

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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?

]]>Click the image to download the PDF. Read the directions. Print off the “easy” or “hard” puzzle. Fill in the To: and From: sections. Give it to a special someone for them to solve.

This freebie is a sample of one of my Dottoo Dots 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 practice skip counting by 2s, starting at 1. Have fun and stay sharp.

A complete set of Valentine-themed Dottoo Dots is available at TeachersPayTeachers.Com, Etsy, and in print form at Amazon.com and Amazon.ca

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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.

Roy

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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 practice 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 on TeachersPayTeachers.com. It 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. Let me know if you try the puzzles with your students and give me feedback about how it went or suggestions you might have.

]]>- 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.

]]>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.

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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.

**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,

Let’s do 2045? T*wo 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?

Like this.

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.

]]>You will likely have come across the fact that in some language families, such as Chinese and Algonquin, number names actually describe the number structure. Take 24 for example. In English it’s “twenty-four,” whereas these other languages say the equivalent of two-tens-and-four. The unitizing of TENS and the addition of ONES are plainly obvious. The very act of learning number words in these languages teaches the number system.

So, in my class, I make the language we use equally explicit. Here is how.

**What is 10?** In our system, TEN can be a series of or a pile of single ONES. It can also be a single unit that contains 10 ONES. I make the comparison between 10 pennies and a dime, or eating 10 cookies one at a time versus buying 1 bag of 10 cookies. I talk about having 10 birthdays and being 10 years old. This comes up when we are practicing skip counting, growing patterns, and working with money. Students must have a strong sense of TENness before they can understand base 10 and place value.

**After 10, we have 20.** While not as explicit as in Chinese, the word TWENTY has clues to how many it stands for. TW- comes from TWO. -TY implies ten. So TWENTY is Two TEN. We build on that. As a class, we look at THIRTY to see if there are any clues to how many it is. Then we look at FORTY, FIFTY, etc. Don’t forget to look at the words in relation to the numerals. To make it even clearer, I will list all the 10s that make the number in a column, and add them together in a sum below, counting the number of TENS, One TEN, Two TENS, Three TENS, that makes THIREE-TENS, or 30.

**Now do 10s with 1s**. Twenty-One means Two TENS and One ONE. Fifty-Four means Five-TENS and Four ONES. This ties nicely into columnar adding without carrying. Count the TENS, count the ONES, relate how many of each you counted to the final sum and the structure of that number’s name and numerals.

**Now we look at the teens.** There are several issues with the teens. One issue is that they use a different suffix to imply a group of ten.The language of the teens has very little in the way of a pattern to latch on to. You might have noticed children who can count by ones in the 20s, 30s, and higher, but still struggle to say what comes after 11. So, it is important to make TEEN explicitly understood as One TEN, and when we count by tens we are really counting One TEN, Two TENs, Three TENs, and so on.

**What’s worse**, ELEVEN gives no clue to how many it is. Neither does TWELVE. So we have to look at the numerals 11 and 12, and question why it is that we don’t say TEEN for these numbers, even though they have the same numerical structure. The numbers really should be said as ONEteen and TWOteen.

**Notice for 20 we say the TENS first**, then we say the ONES. However, when we use the TEENS, we say the ONES first, then the TEN. Many children reverse their numbers because of this. If the language of our number system was a little more logical, we might say Teen-eight, Teen-nine, Twenty. Or better yet, we could say Onety-Four, Twenty-Four, Thirty-Four. So, in my class, I actually do use this language. Not always, and not exclusively, but I do use it.

*What’s Ten plus Nine? It’s Ten and Nine more. It’s Teen-Nine. But because we have crazy English, it’s Nine-Teen.*

This discussion of the language makes students meta-aware of the structure of our numbers and how they express those numbers.

**I use conventional language and the invented naming** interchangeably. You might be worried that this would cause students problems in the future, with accountants who can’t be understood because they learned how to count from crazy Mister Reed. But this isn’t the case. The predominant convention will dominate. Making the exceptions of our naming system explicit, we create a stronger understanding of the rules, and create connections between the names, numerals, and quantities which creates deeper understanding of number.