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.
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?
Like this.
6
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
I 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.
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.
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