Morning Coffee: When Math Makes You Feel Stupid

Posted on by Denise Gaskins
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

This week’s rabbit hole started with a thought-provoking newsletter from Dan Finkel, which led me to his blog…

“Everyone who learns math is familiar with the experience of being stuck on some new idea or problem, banging their head against it, and then, when they finally understand the answer (or having someone tell them), feeling stupid. There’s something fundamental in the nature of mathematics that makes it easy once you get it, and impossible before.

    “These jumps in comprehension can be thrilling, and they’re one reason math is so fun. But they do create a challenge for the student. The evidence that you learned something hard is that you feel like you’re stupid. That stupidity is essential to the process. Students need to know this feeling is the norm when it comes to learning math.”

    —Dan Finkel and Katherine Cook, The centrality of stupidity in mathematics

    Read more about the value of feeling stupid in this second installment of professional development for homeschooling parents.

     
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    “Morning Coffee: When Math Makes You Feel Stupid” copyright © 2025 by Denise Gaskins. Image at the top of post copyright © Kira auf der Heide / Unsplash.

    Puzzle: Henry Dudeney’s Pebble Game

    Posted on by Denise Gaskins
    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.

      Continue reading Puzzle: Henry Dudeney’s Pebble Game

      Charlotte Mason Math: Practical Tips for a Living Math Education

      Posted on by Denise Gaskins
      “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.

      Continue reading Charlotte Mason Math: Practical Tips for a Living Math Education

      Charlotte Mason Math: Wrong Answers and Slovenly Teaching

      Posted on by Denise Gaskins
      "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.

      Continue reading Charlotte Mason Math: Wrong Answers and Slovenly Teaching

      Charlotte Mason Math: The Trouble with Manipulatives

      Posted on by Denise Gaskins
      “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

      Continue reading Charlotte Mason Math: The Trouble with Manipulatives

      Charlotte Mason Math: Finding Time for Big Ideas

      Posted on by Denise Gaskins
      “Woman and Child in the Grass” painting by Renoir

      “Teachers have seldom time to give the inspiring ideas, what Coleridge calls, the ‘Captain’ ideas, which should quicken imagination. How living would Geometry become in the light of the discoveries of Euclid as he made them!”

       — Charlotte Mason, Towards a Philosophy of Education

      The Captain ideas are the great Truths of a subject, the things that make our minds wake up and pay attention, that energize our thoughts and make us yearn for more.

      In math, living ideas are the big principles that tie together many branches of the subject. Things like:

      • Proportion — where two quantities are connected so they scale up or scale down in tandem. For instance, if we double the number of cars in the driveway, that automatically doubles the number of tires.

      • Transformation — how we can change things while keeping important attributes the same. Like, if we shrink a square, its area will change, but the angles stay the same.

      Continue reading Charlotte Mason Math: Finding Time for Big Ideas

      Charlotte Mason Math: How Shall We Teach?

      Posted on by Denise Gaskins
      Woman withchildren painting by Dorothy Kate Richmond, public domain

      Even in Mason’s day, testing drove much of educational policy, but we must not allow ourselves to fall into the trap of teaching for a test. Just as we do not study history in order to win a trivia contest, so we do not study math merely to produce answers on an exam.

      “Arithmetic, Mathematics, are exceedingly easy to examine upon and so long as education is regulated by examinations so long shall we have teaching, directed not to awaken a sense of awe in contemplating a self-existing science, but rather to secure exactness and ingenuity in the treatment of problems.”

       — Charlotte Mason, Towards a Philosophy of Education

      Remember Mason’s twin goals of rightness and reason. Even if you use a math book that focuses on memorizing rules and cranking out answers, you and your child can look for the ideas behind the rules: “Why does this work? How can we know for sure?”

      Not just because the book says so, but because you search out and discover the innate sense of it. That is the essence of mathematics.

      Continue reading Charlotte Mason Math: How Shall We Teach?

      Charlotte Mason Math: Our Educational Tools

      Posted on by Denise Gaskins
      "Woman and Children" painting by Elizabeth Boott Duveneck, public domain

      “Therefore, we are limited to three educational instruments––the atmosphere of environment, the discipline of habit, and the presentation of living ideas. The P.N.E.U.* Motto is: Education is an atmosphere, a discipline, and a life.”

      — Charlotte Mason, Principle 5

      This principle is the key to a Charlotte Mason education. Most of her books consist of drawing out the meaning and implications of this motto.

      When we think about applying Mason’s educational principles to math, we must focus on providing the right atmosphere, developing appropriate habits, and presenting living ideas.

      What is the mathematical atmosphere of our home or classroom? Is math a natural and welcome part of life? Or does it exist only in schoolbooks and in some nebulous “future” for which our children must prepare?

      What about the people in our children’s lives? Do we adults enjoy and use math, or do we dread and avoid it? Is our mathematical worldview positive, eager to learn and grow, or negative, seeing math as a chore to endure?

      Continue reading Charlotte Mason Math: Our Educational Tools

      Charlotte Mason Math: Practice Your Principles

      Posted on by Denise Gaskins
      La Fable, painting by Berthe Morisot

      In our search for a Charlotte Mason math education, we must take into consideration Mason’s approach to all learning, not just the things she said about math. We must be guided by the core principles of her philosophy, even in math

      “We hold that the child’s mind is no mere sac to hold ideas; but is rather, if the figure may be allowed, a spiritual organism, with an appetite for all knowledge. This is its proper diet, with which it is prepared to deal; and which it can digest and assimilate as the body does foodstuffs.”

       — Charlotte Mason, Principle 9

      For instance, we must offer our students living ideas (not mere facts) in math, just as we do in literature and history.

      Masons “20 Principles” outline the essentials of her educational philosophy. If we truly apply these principles to math, it can radically transform how we teach the subject.

      Let’s examine a few of her principles in more detail…

      Continue reading Charlotte Mason Math: Practice Your Principles

      Charlotte Mason Math: Reason and Proof

      Posted on by Denise Gaskins
      “Woman with Child and Two Children,” Léon Augustin Lhermitte, public domain

      The two ideas that Mason considered important in math — rightness and reason — are connected. It is our reasoning that convinces us an answer is right or wrong. How do we know we got a sum correct? We can take the numbers apart and add them another way, to see if we get the same answer. Or we can subtract one of the numbers from the sum and see if we get the other number. Or … well, how would you prove it?

      More than anything else, Mason wanted her students to discover in math a sense of immutable truth, a truth that stands on its own, apart from anything we say or do, a truth we can explore and reason about but can never change.

      This sense of rightness, of solid, unalterable truth, inspires a feeling of wonder and awe — she calls it “Sursum corda,” a call to worship — that delights our minds. It’s that “Aha!” feeling we get when something we’ve been struggling with suddenly fits together and makes sense.

      From the very beginning, children should be doing this sort of informal proof, explaining how they figured things out. Don’t wait until high school geometry to let your children wrestle with ideas.

      Continue reading Charlotte Mason Math: Reason and Proof