Mental Math: Three Basic Principles

Doing mental math on the couch

“We know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.”

— Pam Harris, Math Is Figure-Out-Able Podcast

Whether you work with a math curriculum or take a less-traditional route to learning, do not be satisfied with mere pencil-and-paper competence. Instead, work on building your children’s mental math skills, because mental calculation forces a child to understand arithmetic at a much deeper level than is required by traditional pencil-and-paper methods.

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Charlotte Mason Math: Living Books

“The Reading Lesson” painting by Jonathan Pratt, public domain

[An addendum to my earlier Charlotte Mason Math series.]

“Our business is to give [children] mind-stuff, and both quality and quantity are essential. Naturally, each of us possesses this mind-stuff only in limited measure, but we know where to procure it; for the best thought the world possesses is stored in books; we must open books to children, the best books; our own concern is abundant provision and orderly serving.”

— Charlotte Mason, Toward A Philosophy of Education

Most homeschool teachers, whatever our curriculum or schooling approach, understand the importance of teaching with living books. We read aloud biographies, historical fiction, or the classics of literature. We scour library shelves for the most creative presentations of scientific topics that interest our children, and encourage our high school students to go back to the original documents whenever possible.

And we teach math with a textbook.

Not that textbooks are inherently bad, because math is an abstract science. We need to meet the ideas  — the “mind-stuff” — of math on their own terms, and textbooks can help with that.

But it’s not enough.

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Morning Coffee: When Math Makes You Feel Stupid

Morning Coffee Lifelong Learning for Parents

One of the best ways we can help our children learn mathematics (or anything else) is to be lifelong learners ourselves.

Here are a few stories to read as you sip your morning brew. . .

Download your printable Morning Coffee journal

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Puzzle: Henry Dudeney’s Pebble Game

photo of girl playing with pebbles on the beach

English mathematician and puzzle-meister Henry Ernest Dudeney once wrote:

“It may be said generally that a game is a contest of skill for two or more persons, into which we enter either for amusement or to win a prize. A puzzle is something to be done or solved by the individual.

    “The example that I give here is apparently a game, but, as in every case one player may win if he only play correctly, it is in reality a puzzle. The interest, therefore, lies in attempting to discover the leading method of play.”

    Below is the puzzle game as Dudeney explained it.

    Play it for fun at first, then see if you can solve the puzzle.

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    Charlotte Mason Math: Practical Tips for a Living Math Education

    “Young italian woman with two sleeping children on coast’ painting by August Riedel, public domain

    Focus on the logic of reasoning.

    Correct answers are important, of course, but as children explain their thinking, they will often catch and fix mistakes on their own.

    “Two and two make four and cannot by any possibility that the universe affords be made to make five or three. From this point of view, of immutable law, children should approach Mathematics; they should see how impressive is Euclid’s ‘Which is absurd,’ just as absurd as would be the statements of a man who said that his apples always fell upwards, and for the same reason.”

     — Charlotte Mason, Towards a Philosophy of Education

    “Most remarks made by children consist of correct ideas badly expressed. A good teacher will be wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.”

     — W. W. Sawyer, Vision in Elementary Mathematics

    • Tip: If you’re not sure how to draw out your child’s reasoning, read Christopher Danielson’s wonderful examples and advice on talking math with your kids: Talking Math with Your Kids.

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    Charlotte Mason Math: Wrong Answers and Slovenly Teaching

    "Playing with the kittens" painting by Emile Munier, public domain

    The second place where a surface-level reading of Charlotte Mason’s books can lead to misunderstanding involves the treatment of wrong answers. Mason wrote:

    “… quite as bad as these is the habit of allowing that a sum is nearly right, two figures wrong, and so on, and letting the child work it over again. Pronounce a sum wrong, or right — it cannot be something between the two. That which is wrong must remain wrong: the child must not be let run away with the notion that wrong can be mended into right.”

     — Charlotte Mason, Home Education

    Does this call to mind images of your own childhood schoolwork? It does for me: laboring over a worksheet or quiz and then taking it to my teacher to be graded. Right was right, and wrong could not be mended. In such a performance-oriented setting, mistakes can take on the flavor of moral failure.

    Is this authoritarian approach the way Mason wants us to teach math to our children? Where is the summa corda — the joyful praise — in that?

    No. Please, no. Very definitely no.

    Mason wanted us to avoid slovenliness in our teaching. In this passage, she warned against several forms this might take.

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    Charlotte Mason Math: The Trouble with Manipulatives

    “Mother Playing with Child” painting by Mary Cassatt, public domain

    Two passages in Charlotte Mason’s writing about math are in my opinion widely misunderstood. The first relates to the proper use of manipulatives.

    Mason believed strongly in the importance of physical objects and oral work (mental math) in early math education. In her priorities, the use of written calculation fell in distant third place.

    “A bag of beans, counters, or buttons should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to ‘do sums’ on his slate.”

     — Charlotte Mason, Home Education

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