Talk to any kindergarten teacher, and she’ll tell you about her students who transpose numbers. Two-digit numbers puzzle many younger students. It’s not unusual to hear students confusing the numbers “13” and “31” or writing the number “14” as “41.”
This common mistake is sometimes called a transposition. When students transpose numbers, they write down all of the correct numbers, but they don’t put the numbers into the right sequence (place-value order).
Transposition errors often occur in two-digit numbers. For my students, the most commonly transposed numbers are the numbers 12-19. These mistakes with the teen numbers actually reveal the child has a good understanding of the spelling patterns for numbers and words. Mistakes with numbers greater than twenty may indicate that the child needs more place-value practice.
Today, I want to empower you with effective tools for addressing transposition errors. First, let’s figure out why students are confused.
Are Your Students Reversing Numbers?
Another type of mistake is writing the mirror image of a numeral, and we usually call those mistakes reversals. Click here to find my free article on reversals and my paid program for stopping number reversals. Some kids get tripped up both both transpositions and reversals.
Why Do Students Transpose Numbers?
As adults, we’ve been working with numerals for decades, so it’s an effortless task. Because we’re so fluent with numbers, it’s sometimes hard to understand why kids transpose them. Dig a little deeper, and you’ll discover that the teen numbers are more complex than they seem.
To get to the bottom of this conundrum, we need to investigate English spelling!
Understanding the Language of Numbers
Let’s look at how the words for most two-digit numbers are spelled.
In two-digit numbers, you can hear and write two word parts or morphemes. For example, say the number twenty-three:
First you hear the ten’s place — twenty. I’ve shown this in red. Then you hear the one’s place — three. I’ve shown this in green. Since we write from left to right, this is easy to record:
Examine any number between 20-99 and you’ll see that every number is spelled first with the ten’s digit then the one’s digit. We hear – and write – the ten’s place followed by the one’s place.
The Teen Numbers Follow Another Pattern
It’s no surprise that kids almost always transpose the teen numbers. These numbers do not follow the same spelling pattern as other double-digit numbers, so children need to understand a completely new pattern.
Let’s investigate the sequence of morphemes inside thirteen. Say thirteen aloud:
What you don’t hear is the pattern we just described above: the ten’s place followed by the one’s place. Instead you first hear the one’s place – thir. Then you hear the ten’s place – teen. We can’t just listen to the morphemes and record them from left to right. See what I mean?
The teen numbers don’t follow the same pattern of morphemes as all of the other two-digit numbers. If they did, we would pronounce 13 as “teenthir!”
The teen numbers actually make a lot of sense if we look at the words’ origins in Old English. By visiting the Etymology Dictionary we discover:
- The morpheme “teen” means ten more than. “Thirteen” then means three and ten more. This pattern explains: thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, and nineteen.
- Eleven also comes from Old English and means one left (over ten).
- Similarly, twelve means two left (over ten).
The morphology of number words actually helps build mathematical understanding!
Morphology: How to Use Word Parts to Help Your Students
Some students need explicit instruction. I recommend:
- Explain that number words are built out of word parts. Explore some of the double-digit numbers between 20 and 99. Can your students listen carefully to the words? Do they hear the individual word parts? Can they hear the ten’s place followed by the one’s place?
- Now explore some of the teen numbers between 13 and 19. Have your students listen carefully to the words. Which word part do they hear first? That’s right! It’s the one’s place!
- With your students, investigate the origins of the teen numbers in an etymology dictionary.
How to Use Manipulatives to Clear Up Confusion
For some students, a quick explanation may be enough to solve transpositions. Other students will benefit from using math manipulatives. I recommend using base-ten blocks to clear up confusion.
Younger students, or older children with dyslexia, dyscalculia, or language disabilities, may need to be shown that words, blocks, and numerals can all be used to represent the same number. I use these steps:
STEP 1: Start by building a standard two-digit number (between 21 and 99) with manipulatives. First, create a graphic organizer that clearly shows the ten’s place and one’s place. Download my free copy here. Show the student how to build this number using base-ten blocks.
STEP 2: Now, show how to record this number with numerals. Point to the ten’s place and say: “I see four tens in the ten’s place, so I’m going to write the numeral 4. I see two ones in the one’s place, so I’m going to write the numeral 2.”
STEP 3: Explicitly link the concrete blocks with the symbolic numerals. Give kids a chance to verbalize the connection between the blocks they’ve placed and the numerals they’ve written. I might model: “I see the same number with the blocks and the written numerals. I see four ten-rods and I also see the numeral four. I see two one-blocks and I also see the numeral two. Both the blocks and the numerals show the number 42.” Practice with standard two-digit numbers (21-99) and teen numbers (13-19).
STEP 4: The student practices.
Extension: Start with symbols. Once the student can identify the number from the base-ten blocks, work the other direction. Write out two-digit numerals and ask the child to build the number with blocks.
BONUS PRACTICE GAMES
To give your students extra practice, here’s my set of free number cards. You’ll find over 40 cards with four different representations for the numbers 11-19. Each number is represented with base-ten blocks, ten frames, the word, and the numeral.
Spread the cards out on the ground and ask students to match up numbers. You can also use the cards to play the game Memory or Go Fish.
Let’s Do This!
With some insights into English spelling and base-ten blocks, you and your student should be set.
Great math skills begin with self-confidence. Many students who feel challenged by math feel uncomfortable with numbers. It doesn’t have to be this way! From the very start, we can show students how to understand and express themselves with numbers. Reading and recording two-digit numerals is a wonderful way to build this self-assurance.
I love connecting with readers! I respond to every e-mail I receive, and it brightens my day when I hear from a parent or teacher who finds something helpful. Let’s connect in the comments. I’d love to hear from you.
Free Downloads
So excited over the fact that you have addressed number switching. Up until now I hadn’t found any help targeting this problem. Thank you so much!
I’m so glad you found the article helpful. Number switching isn’t a problem that I see all that often, but I have had it come up a few times over the years. When we see students making this mistake, it becomes a valuable teaching opportunity. Let me know how the techniques work for you.
Thank you, Anne Marie! Best article I have ever seen on this!
Della Blanton, LDT, CALT
Detroit ISD Dyslexia Therapist
Thanks, Della! That’s so nice of you.
I have done this all my life not just teen numbers but in addresses or someone gives me 4 numbers I get them correct however almost always switch the sequence around. I’m 72 now. It never goes away
Absolutely agree with Della … fantastic article, and so nice to see in print the sorts of things many ed. therapists do every day! An underrated profession but I love it 🙂
Thanks, Kristina!
I’m 65 and I still do this. I’ve known about it fore decades. There are times I look at numbers and I *see* them flip on the page. It doesn’t happen with letters. Is there anything to do? It causes me constant problems.
Number switching can also be an indication of dyslexia. My daughter did this and we were informed that it was normal. It didn’t become abnormal enough until the middle of 2nd grade when she has other signs as well. Parents and educators should be in tune to these signs so early intervention can take place.
I’m glad you brought this up, Laura. I’ve certainly seen students with dyslexia both reverse and transpose numbers. Generally speaking, we do give students until the beginning of second grade before number reversals become a red flag for a language-based learning difference. I’d recommend reading over an article I wrote on the connections between number reversals and dyslexia.
If parents are seeing number switching in isolation, this is generally not a cause for concern. However, if your young student displays persistent and deep challenges with learning their letter sounds, sounding out words, and breaking words into sounds to spell AND number switching, it’d be a good idea to talk to the classroom teacher, the school learning specialist, or an educational therapist. I can also recommend parents visit the International Dyslexia Association’s website. They have a very brief Dyslexia Screener for School-Age Children.
Laura, I’m so glad you brought this up! If parents have concerns about dyslexia, I can’t tell you how important it is to seek out early intervention. Great comment 🙂
I just found this site. I heard of a child I will be working with who reverses his numbers when writing them. Two things I like about this are: it is so visual, with easy directions, and it meshes perfectly with my trusty Math Their Way and Math a Way of Thinking activities. I have the manipulatives, too! In my little ‘grant’ program I’ve always used the ‘old’ MTW to teach ones, tens, hundreds, but the actual way to ‘word’ the numbers, especially the teens, but also those above, is new and will be helpful.. (I’ve gone so far as to call numbers like 37 ‘three-ty seven’ or ‘five-ty-four, etc. This is my first time to your blog, and I can see there’s lots of great stuff on it. Thank you!
Judy, I’ll have to research Math their Way and Math a Way of Thinking. Those are two new programs for me. I love your example of “three-ty-seven.”
Thank you, thank you, thank you!!! I can’t wait to try the math sheets for transposition of numbers. I will let you know how it goes!!
Thank you, Judy! I hope the math sheets are helpful.
My son number switches but only the teen numbers. He is in the 1st grade. I also notice when he writes it correctly he writes the tens place number first. I can’t get him to write the 1 first. It’s a habit he needs to break so he doesn’t make mistakes but I don’t know how.
Do you know if their is help for adults who transpose numbers, and have dyslexia? I am not aware of having problems with letters. But when working with calculating a lot of numbers, I often mixed them up. Also, I have difficulty remembering peoples names.
So far, I am impressed with what I’ve read on your site. I’ve been teaching SEN for about 15 years and am always looking for something else to try. With the reversals for numbers, you could possibly add numicon and money illustrations also.
When working with students who transpose numbers 21-99 where do you start. Do you work on just building numbers in the 20’s then move on to the 30’s etc… or do you work on building 23 and 32, the 24 and 42 etc…
Hi Kathleen, great question! You can experiment and see what works best for you students. Personally, I’d move through the decades sequentially, one at a time, until they student has reached mastery. E.g. start with 20-29, and then move on the thirties. How does the student do with lots of building with the base ten blocks?
So useful! Am currently teaching a child with dyslexia who has some of these maths problems and your article explains it beautifully. It will help me to explain to him useful ways of thinking about this.