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.