Category Archives: Pedagogy

How to do Single Stroke Printing

printingHere 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.

Here are the links.          Part 1        Part 2

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.

Grammithmatic Part 2

If you want to read Part 1 where I discuss the number words we have for the TEENS and TENS, click here.

SLOBSFor 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 as a quantity. They connect 0 to the idea of having nothing very easily. They also catch on to the 0+n=n 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 HUNDRED 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.infinite

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 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 202nice 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 HUNDRED and ZERO TENs and ZERO ONEs. Over here 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 that that number is there, it has a purpose and a value even if its value is nothing.

If you keep this going, we have Twenty-NoneTwenty-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?

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.

Grammithmatic Part 1

HELLOI have come across a number of factors that stand in the way of children understanding our number system. One issue is our language (English, if you haven’t noticed yet), and the way we name numbers. Number names aren’t always logical, and can confuse many of our students. Here are some ideas to mitigate language confusion for those struggling to gain number sense and understand our base 10 system, and to enrich understanding for those who “get it,” but can go deeper.

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.

blackboardAfter 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.nineteen

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.

Click here for Part 2, naming the ZERO

Fluid Hundred Charts-100 Tiles

This next week, I will be giving two presentations at the Ontario Association of Mathematics Educators conference in Toronto. One will be on using fluid hundred charts, which is essentially using 100 numbered tiles to teach about number sense concepts.

I’ve made a video that describes how to make a set of 100 tiles.

My plan is to post about the different ways to use the tiles, and to link those posts with other videos.

Supporting Students with Autism Spectrum Disorder (Math)


A spectrum of numbers

As part of a Math AQ course, I put together this Dos and Don’ts list for things to consider when working with students with autism. Many of the suggestions can be applied to more than math instruction.

However, there is a saying amongst those who work with these children. “If you’ve worked with one child with autism, you’ve worked with one child with autism.” Generalizations about this disorder, thanks to the spectrum nature of it, are impossible to make. Therefore, when referring to this resource, keep in mind that, while these are pretty sound suggestions, the degree of application of any particular suggestion must be tailored to each child individually.

In this list, the first Do corresponds with the first Don’t, the second Do with the second Don’t, etc. The downloadable PDF pairs them in columns.

I hope you find this useful.

  • refer to general resources that support your math planning for students with ASD.
  • get to know your student’s interests, because those will be the entry points to the math.
  • make the math stimulating, fun, and interesting.
  • use the same kind of differentiated methods of instruction as with other students.
  • use visual aids and concrete materials to make the abstract real and accessible.
  • break learning up into manageable parts.
  • help the student to generalize math concepts and procedures.
  • incorporate rote learning into your program: Many children with ASD are strong in this kind of learning.
  • assessments that allow the student to be most successful in sharing thinking, playing to strengths.
  • use precise praise for effort and success.
But Don’t
  • assume the resource is definitive, e.g. most recommend visual aids, but some students with ASD are stronger auditory and/or kinaesthetic learners.
  • be afraid to broaden the student’s horizons by introducing new things.
  • over-stimulate, distract, or agitate the student: know the triggers and warning signs that tell you if things are about to take a turn.
  • forget the communication skills such as vocabulary, and being able to explain the math, be it with pictures, numbers, or words.
  • forget the hierarchy of visual aids: from most effective to least are real objects or situations, facsimiles or models, colour photographs, colour pictures, black and white pictures, line drawings, graphic symbols and written language.
  • do the lesson without relating it to the other small parts and the whole.
  • let manipulatives take on one fixed meaning or purpose: They are variable models that can represent many ideas.
  • limit your lessons to rote learning: It’s important that this knowledge be connected to, and is in service of understanding the big ideas and concepts of math.
  • forget to take the student’s weaknesses into account: be sure to consider stamina, mood, and external factors to decide when, how, and how much you assess at any one time.
  • be stingy with the non math-based positive reinforcement that will foster a positive attitude and ideal brain state.

Here’s the link to download the document.

Number Lines – Made to measure (patterning)

measuring tapeI recently posted instructions for a hands-on activity wherein the students make their own meter measuring tapes. I promised I’d explain how I’ve used them so far. Below is a description of ways to use the tapes to teach patterning. This will be followed in another post with ideas for using them with number sense and then, of course, measurement.


growingPatterning is the act of analyzing the relationships between elements in a string of repeated occurrances of those elements. 😛 In this case, we are looking at the relationship between one number and the next after something has been done to that number. We can focus on the individual numbers, or on the change happening to those numbers.

Have you ever noticed that your students can skip count, no problem, rambling on and on until they reach the biggest numbers they know? Have you noticed that they don’t actually know what 10+2 is, despite being able to count by 2s, or what 15 + 5 is, despite knowing how to count by 5s; they don’t actually know how these patterns are constructed or how they work?

This is because they have memorized these patterns the way they memorize the lyrics to the latest Katy Perry song; they know the words, but have no idea what they are actually saying!

To avoid this meaningless regurgitation, it is important to teach the students how the pattern works at the same time as teaching them the pattern. Better still, is teaching them about how changes happen in patterns and letting them figure out the different patterns that can happen because of those changes. Specifically, we can continually add on, or take off a fixed quantity, and say the new number we get, and eventually, we notice the same numbers happening over and over until we see a pattern.

To accomplish this learning, the skip counting must be accompanied by a concrete understanding of the quantity we start with, the quantity being added, and the quantity that we end up with, and the students must understand that it is a recursive process wherein the quantity we end with becomes the quantity we start with.

thermometer-mediumThese measuring tapes (and any other number line example, such as thermometers, hundred charts, etc.) provide both a concrete example of quantity, and a way to figure out, check, and keep track of the pattern until it is memorized with meaning.

Circle Counting

With the students sitting in a circle, with their personal measuring tapes in hand, start at zero, and go around the circle, adding the number you’re skip counting by. As you go around, the children mark the place on the number line with their fingers, and they move to each new number in sequence as they hear it called out.

If you followed my advice in the first post, and numbered only the 5s and 10s, then you should start the students practicing this activity with 5s. This will help those who struggle with counting, as the numbers are clearly marked.

Then you can do 10s, noting that you skip the 5s.

Then you can do 1s, easy because they should know the sequence orally, difficult because the numbers aren’t marked on the strip.

The students will lose their places on the tapes. That’s fine. Once they hear the number that came before, they will be able to catch up. Either, they will be forced to go right back to the beginning and count out (because they haven’t yet realized that the number on the tape always represents the same fixed quantity), which is valuable practice, or they will, hopefully, eventually, realize it’s easier to go to the closest anchor of 5 or 10, and work out from there where the next number is.

For example, “The previous number was 34, I can find 34 because it is one down from 35, and now add 2 more, that makes 36.”

Over and above developing a true understanding of the skip counting process, this activity builds estimating skills, builds fluency with number and the distributive property, and sets the ground work for operations with ‘friendly numbers.’

Follow the same activity to count by 2s.

In Ontario, grades 1 and 2 have to count by 1s, 2s, 5s, and 10s, starting from a multiple of those numbers. Grades 3 and higher have to be able to count by those numbers from any starting point, and be able to count by other factors as well. It’s easy to adapt the circle counting activity by changing the factor being added, and/or changing the starting point.

Using the tapes forces the students to figure out what comes next, by adding/subtracting the right amount to/from the previous number. This is better than just hearing the pattern and parroting it back. Also, as patterning becomes algebraic study in the later grades, the students become familiar with the recursive process, or the algorithm of skip counting (n=n+1), so the formula will have meaning when they eventually learn it.

What concrete thing are we counting exactly? Make sure that the students understand we are counting centimeters, “the spaces between the lines.” This might require some groundwork on what centimeters are, in measurement terms.

If centimeters are too abstract, use counters of some kind, small enough to fit between the lines on the tape. Place the correct number of counters, directly on the tape, with each new iteration of the pattern.


In and Out 

There is a game played in the Junior grades where the players give a number to put in to the ‘machine’ and the leader gives the number that comes out (input and output). For example, a student says 5, the leader writes 10. A player says 10, the leader writes 15. You can use these tapes to figure out the algorithm for what happens to the input to get the output. By marking on the number line where the input is, and the output is, commonalities will reveal themselves, and a rule will be developed.

The patterning in this case lays in the repeated operation. What is always happening to the number given to get the new number. Incidentally, this is an A pattern. +1, +1, +1, etc. is A, A, A, etc..

hundredchartEssentially, any number pattern game you can think of can be done with these measuring tapes. If you have games you use to explore the numbers on 100 charts, you might be able to adapt them to these number lines, taking advantage of the linear nature as compared to the array layout of the 100 chart.

If you have any other ideas for ways to use the measuring tapes to build number pattern knowledge, please share them in the comments below.

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