Tuesday, October 23, 2018

The Importance of the Unit: Part I

An area of struggle for students is manipulating numbers once they get beyond basic counting numbers. A big reason for this struggle is that when manipulating multiple digits, students get intimidated by the larger numbers. They are also confused by how various digits work in algorithms. This shows up with operations on multiple-digit numbers, fractions, and decimals.

One early source of the challenges may be the way we count. In many Latin-based languages, the teen numbers follow their own naming rules and don’t obey anything resembling logic. For example, what on earth does “eleven” mean? And what about “sixteen” -- why is the “teen” or tens named after the ones?

Counting returns to normal once you get to the twenties, with twenty-one, twenty-two, etc. following place value order and the way we write them. But those teen numbers confuse young brains and can be responsible for many of the errors young children make, such as writing “seventeen” as 71 and not 17. They are being logical, generalizing the rules for writing numbers based on language; it’s our language that is illogical.

Contrast this to languages such as those spoken in China, Japan, and South Korea, in which counting proceeds according to place value. They count 1-10 with individual words, like in English, but subsequent numbers are represented by place value: ten-one, ten-two, ...., ten-nine, two-ten, two-ten-one, two-ten-two, etc.

This language advantage leads to a mathematical advantage. Language researchers believe this is the reason children from those countries can, by the end of preschool, count to 100 consistently, and solve almost three times as many simple arithmetic problems as US children. This gap can be closed by teaching the “say ten” way of counting, mirroring the counting used in these Southeast Asian countries. For example, children would learn to count the usual way, but then to rename the counting sequence using Say Ten counting.

In conjunction with working with manipulatives to show that every number is a sum of ones, and that a ten is a shorter way of naming ten ones (renaming), students can develop a strong sense of place value that makes numbers easier to work with. They learn that digits work similarly, but it’s the place that gives the digit its value. Using the Matholia place value disks tool and place value strips tool can help the student see these dynamics in action.

Students can also use the 10-row Rekenrek to track the tens and ones as they count. This will reinforce the concept of the “make-ten” strategy for addition and subtraction, which then can be generalized to multi-digit numbers. For example, adding 12 to 29, a student can decompose either number to make a ten and then add the remainder. A visual example of the mental math, using number bonds, is shown below. These strategies are practiced extensively in Primary 2.

Then if the student encounters 290 + 120, the same strategies can be used, just renaming it as “29 tens plus 12 tens,” understanding that the sum will be in tens, so 41 tens or 410.
The ways this understanding extends to fractions and decimals will be addressed in future articles.
Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.

The Matholia Team

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