Teaching the Standard Algorithms

[Feature photo above by Samuel Mann, Analytical Engine photo below by Roͬͬ͠͠͡͠͠͠͠͠͠͠͠sͬͬ͠͠͠͠͠͠͠͠͠aͬͬ͠͠͠͠͠͠͠ Menkman, both (CC BY 2.0) via Flickr.]

Babbage's Analytical Engine

An algorithm is a set of steps to follow that produce a certain result. Follow the rules carefully, and you will automatically get the correct answer. No thinking required — even a machine can do it.

This photo shows one section of the first true computer, Charles Babbage’s Analytical Engine. Using a clever arrangement of gears, levers, and switches, the machine could crank out the answer to almost any arithmetic problem. Rather, it would have been able to do so, if Babbage had ever finished building the monster.

One of the biggest arguments surrounding the Common Core State Standards in math is when and how to teach the standard algorithms. But this argument is not new. It goes back at least to the late 19th century.

Here is a passage from a book that helped shape my teaching style, way back when I began homeschooling in the 1980s…

Ruth Beechick on Teaching Abstract Notation

“Understanding this item is the key to choosing your strategy for the early years of arithmetic teaching. The question is: Should you teach abstract notation as early as the child can learn it, or should you use the time, instead, to teach in greater depth in the mental image mode?

Beechick-EasyStartArithmetic

“Abstract notation includes writing out a column of numbers to add, and writing one number under another before subtracting it. The digits and signs used are symbols. The position of the numbers is an arbitrary decision of society. They are conventions that adult, abstract thinkers use as a kind of shorthand to speed up our thinking.

“When we teach these to children, we must realize that we simply are introducing them to our abstract tools. We are not suddenly turning children into abstract thinkers. And the danger of starting too early and pushing this kind of work is that we will spend an inordinate amount of time with it. We will be teaching the importance of making straight columns, writing numbers in certain places, and other trivial matters. By calling them trivial, we don’t mean that they are unnecessary. But they are small matters compared to real arithmetic thinking.

“If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find that she likes arithmetic more. And when she does get to abstractions, she will understand them better. She will not need two or three years of work in primary grades to learn how to write out something like a subtraction problem with two-digit numbers. She can learn that in a few moments of time, if you just wait.”

— Ruth Beechick
An Easy Start in Arithmetic (Grades K-3)
(emphasis mine)


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5 thoughts on “Teaching the Standard Algorithms

  1. This booklet, short as it is, was one of our math mainstays when our three girls were young, along with Miquon Math. I referred back to it whenever I needed reminding to keep things simple and commonsensical. (The oldest one graduated with a BSc in math last year, so I guess that says something?)

  2. Okay. I have this book. I read it a long time ago right before I started homeschool. I completely agree with this book. I love the hundred chart.
    So far, all of my 4 kids like math. We do lots of mental math, some arbitrary, some real life calculations.

      1. Thanks, Denise,
        Your kids are fortunate to have a mom who wanted to play games with them.

        Have you seen the books by Joy Hakim, called “The Story of Science, Aristotle Leads the Way”?( there are 3 science ones).
        On pg. 144 there’s hundred chart made by mathematician Rich Schwartz.
        I found it on his website:

        He has children’s books, too. I think you might like him.🙂

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