The Common Core maths standards are a topic of hot controversy in the US, especially on the right.

The Daily Signal, a news and commentary site started by the Heritage Foundation, recently featured this picture from Red State editor in chief Erick Erickson’s blog, showing a page from Erickson’s daughter’s third-grade maths textbook:

The above shows a way of doing subtraction that’s probably different than what you’ve learned in school. Rather than the basic “carrying-the-one” approach, students are taught to see subtraction as a way of establishing the distance between two numbers. And the easiest way to do that is to establish the distance between various signpots. So the way to subtract 38 from 325 involves establishing the difference between 38 and 40 (2), 40 and 100 (60), 100 and 300 (200) and 300 and 325 (25). Adding 2 + 60 +200 + 25 gives you the answer: 287.

Anyway, alternative methods like this one, or the “box multiplication” method we’ve looked at before, have raised a stir among some parents and conservative commenters.

The Daily Signal’s Kelsey Harris notes that “students must take about six steps (at minimum, depending how you count) to subtract just two numbers” using this method.

Erick Erickson wrote that this method “makes no freaking sense to either my third grader or my wife.”

There are also complaints that we aren’t teaching children how to perform arithmetic the old-fashioned way. Harris says that the four methods in Erickson’s daughter’s textbook do not include “borrowing and carrying numbers. You know, the old-fashioned-taught-the-same-way-for-decades-granny-method-not-approved-by-bureaucrats subtraction.”

These alternative methods are actually intended to be *supplemental* learning tools, not replacements for the standard methods that have been taught to schoolchildren for decades. The Common Core standards do include learning and using those methods. From the Grade 4 standards for “Number and Operations in Base Ten”:

That “fluently add and subtract multi-digit whole numbers using the standard algorithm” is exactly the borrowing and carrying digits, operating on numbers arranged in columns, that we all know and love.

The “counting up” method (which is what’s depicted in the textbook above) is not intended to replace the standard way. Instead, it captures some of the underlying aspects of subtraction and place value that allow borrowing and carrying to work.

In the standard method, students start at the smallest, rightmost ones digit. If the digit on the bottom is smaller than the digit on the top, they subtract that digit and move on. If the bottom digit is larger than the top digit, they go to the tens column to the left, and “borrow” a ten, adding ten to the ones digit and subtracting one from the tens digit. The student keeps moving to the left until they run out of digits.

The standard method has the virtue of being relatively quick. However, ** why** it works might not be immediately apparent to a second or third grade student who is learning subtraction for the first time.

The standard method is based on place value — the tens digit is representing some number of tens, and it’s possible to move or borrow one of those tens over to the ones place, if we need to.

The “counting up” method illustrated in Erickson’s picture makes the role of place value more explicit. The student starts counting by ones from the smaller number up to the nearest multiple of ten. Then she counts by tens to 100, then by hundreds to the first digit of the larger number, then takes the remaining two digit part of the larger number.

These are all just a different way of subtracting in different place values. Adding these intermediate steps together, the student gets her result.

The point of these alternative methods is to provide a different perspective on a problem, which is often useful in learning maths at any level. Counting up breaks the problem up into smaller pieces, and in the process improves the student’s understanding of subtraction and place value.

Although Erickson didn’t provide pictures of the other three subtraction methods in his daughter’s textbook, they likely also are illustrating other aspects of subtraction and place value.

These alternative ways of looking at arithmetic ideally build up a student’s foundational understanding of these key concepts, making it easier for them to work with the standard algorithms later.

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