What Is Multiplication, Anyway?

At some point during the process of teaching multiplication to our children, we really need to come to terms with this question:

What IS multiplication?

Did your device hide the video? Find it on YouTube here.

“What’s my answer? It’s not one that society’s going to like. Because society expects — demands, even — that mathematics be concrete, real-world, absolute, having definitive answers.

    I can’t give a definitive answer.

      Multiplication manifests itself in different ways. So maybe the word ‘is’ there is just too absolute. And it’s actually at odds with what mathematicians do.

        Mathematicians do attend to real-world, practical scenarios — by stepping away from them, looking at a bigger picture.”

        —James Tanton, What is Multiplication?

        For Further Study

        You may also enjoy these posts from my blog archive:

        Memorizing the Times Table: A Life Skills Approach

        Continuing on my theme of times table facts, here’s the inimitable James Tanton:

        Did your device hide the video? Find it on YouTube here.

        “If our task is to memorize this table, please make it about mathematics — about thinking your way through a challenge, and what can I do to make my life easier.”

        —James Tanton, Making Memorising Multiplication Facts (if one really must) a meaningful Life Skill Lesson

        For Further Study

        You may also enjoy my blog post series about working through the times tables, paying attention to mathematical relationships (and a bit of prealgebra) along the way.

        Times Tables Series

        Click the button to see the whole series. Scroll down to the first post to go through it in order.

        What Are Mixed Numbers?

        I just discovered a fascinating fact: In some places in the world, mixed numbers apparently don’t exist.

        So that made me curious about my blog readers:

        • Did you learn about mixed numbers in school?
        • Do you ever use mixed numbers in daily life?
        • Are your children learning to work with them?

        And if you DO know mixed numbers, can you simplify this mess:

        [If you enjoy dry math humor, the answer is worth the work.]

        Continue reading What Are Mixed Numbers?

        Mathematical Days of Christmas

        Enjoy this bit of seasonal fidgeting from Vi Hart.

        If you don’t understand some of the references, that’s normal! Pick a phrase, Google it, and relish the fun of learning something new.

        Did your device hide the video? Find it on YouTube here.

        For More Holiday Math

        CREDITS: Lamppost photo (top) by Aaron Burden via Unsplash.com.

        Have a Mathematical Thanksgiving Dinner

        With the pandemic still raging, most of us will have to adapt our normal holiday traditions to fit the new reality. We may not be able to have a big family gathering (except over Zoom), but we can still enjoy great food.

        So for those of you who are planning ahead, here is a mathematician’s menu for next week’s Thanksgiving dinner.

        Optimal Potatoes

        Green Bean Matherole

        Borromean Onion Rings

        Thanksgiving Turduckenen-duckenen

        And for Dessert

        May I suggest some of Don Cohen’s Infinite Cake?

        Click here for cake

        CREDITS: “Thankful” photo (top) by Pro Church Media via Unsplash.com. Food videos by mathemusician/doodler Vi Hart.

        Playing Math with Michael and Nash

        Michael and Nash have been creating and posting new math games with astonishing regularity throughout the pandemic. Their YouTube channel is a great resource for parents who want to play math with elementary-age children.

        Today’s entry: Closest to Ten, a quick game for addition and subtraction fluency with a tiny bit of multiplication potential.

        And here’s one of my favorites for older players: Factor Triangles, a card game for 2-digit multiplication.

        Check out their channel, and have fun playing math with your kids!

        Visit Michael and Nash on YouTube

        That’s Mathematics

        Here’s a bit of fun I found on YouTube. Happy Friday!

        Mathematicians and maths educators in order of appearance:

        Eddie Woo @misterwootube
        Hannah Fry @FryRSquared
        James Tanton @jamestanton
        Chris Smith @aap03102
        Bobby Seagull @Bobby_Seagull
        Jo Morgan @mathsjem
        David Wees @DavidWees
        Matt Parker @standupmaths
        Michael Stevens @tweetsauce
        Lieven Schiere @lievenscheire
        Ben Sparks @SparksMaths
        Rob Eastaway @robeastaway
        Nira Chamberlain @ch_nira
        Ed Southall @edsouthall
        Steven Strogatz @stevenstrogatz
        Simon Pampena @mathemaniac
        Rachel Riley @RachelRileyRR
        Alex Bellos @alexbellos
        Simon Singh @SLSingh
        Katie Steckles @stecks
        Craig Barton @mrbartonmaths
        Kyle Evans @kyledevans

        Math That Is Beautiful

        One of the sections in my book Let’s Play Math: How Families Can Learn Math Together — and Enjoy It encourages parents to make beautiful math with their children.

        Do you have trouble believing that math can be beautiful?

        In “Inspirations,” artist Cristóbal Vila creates a wonderful, imaginary work studio for the amazing M.C. Escher. You’ll want to view it in full-screen mode.

        How many mathematical objects could you identify?

        Vila offers a brief explanation of the history and significance of each item on his page Inspirations: A short movie inspired on Escher’s works.

        Read about the inspirations, and then try making some math of your own.

        “I looked into that enormous and inexhaustible source of inspiration that is Escher and tried to imagine how it could be his workplace, what things would surround an artist like him, so deeply interested in science in general and mathematics in particular. I imagined that these things could be his travel souvenirs, gifts from friends, sources of inspiration…”

        —Cristóbal Vila
        Inspirations: A short movie inspired on Escher’s works

        Math Makes Sense — Let’s Teach It That Way

        I had forgotten this video, and then rediscovered it yesterday and loved it just as much as ever. Perhaps you’ll enjoy it, too — especially if you think of yourself as “not a math person.”

        Annie Fetter is talking to classroom teachers, but her message is just as important for homeschoolers. Math is all about making sense. Let’s help our kids see it that way.

        “Sense-making is the first mathematical practice for a reason. If we don’t do this one, the rest of them don’t matter. If we’re not doing this, our children are not going to learn mathematics.”

        —Annie Fetter
        Sense Making: It isn’t Just for Literacy Anymore

        You can download the notes for Fetter’s updated session on sense-making and find several links to wonderful, thought-provoking posts on her blog:

        How Can We Encourage Sense-Making?

        Here are some ideas from Fetter’s updated notes, which expand on her comments in the video above:

        • Get rid of the question. Literally.
        • Ask students “What could the question be?”
        • Get rid of the question and the numbers.
        • Give the answer.
        • Or give several answers.
        • Ask about ideas, not answers.
        • Ask “Why?” or “How did you know?” or “How did you decide that?” or “Tell me more about that.”
        • Use active reading strategies.

        Get this free downloadable poster from Teacher Trap via Teachers Pay Teachers.

        A Few Resources to Practice Sense-Making

        In no particular order…

        “I implore you, stop ‘cracking the math code.’ Make sense-making the focus of every single thing you do in your math classroom.”

        —Annie Fetter
        Sense Making: It isn’t Just for Literacy Anymore

        And if you haven’t seen it before, don’t miss Annie Fetter’s classic video “Ever Wonder What They’d Notice?”

        CREDITS: “Building a rocket ship” photo by Kelly Sikkema via Unsplash. “Reading is thinking” poster by Teacher Trap via Teachers Pay Teachers.

        Mathematics Is Worthy

        “When I began my college education, I still had many doubts about whether I was good enough for mathematics. Then a colleague said the decisive words to me: it is not that I am worthy to occupy myself with mathematics, but rather that mathematics is worthy for one to occupy oneself with.”

        Rózsa Péter
        Mathematics is beautiful
        essay in The Mathematical Intelligencer

        Rózsa Péter and the Curious Students

        I would like to win over those who consider mathematics useful, but colourless and dry — a necessary evil…
        No other field can offer, to such an extent as mathematics, the joy of discovery, which is perhaps the greatest human joy.
        The schoolchildren that I have taught in the past were always attuned to this, and so I have also learned much from them.
        It never would have occurred to me, for instance, to talk about the Euclidean Algorithm in a class with twelve-year-old girls, but my students led me to do it.
        I would like to recount this lesson.
        What we were busy with was that I would name two numbers, and the students would figure out their greatest common divisor. For small numbers this went quickly. Gradually, I named larger and larger numbers so that the students would experience difficulty and would want to have a procedure.
        I thought that the procedure would be factorization into primes.
        They had still easily figured out the greatest common divisor of 60 and 48: “Twelve!”
        But a girl remarked: “Well, that’s just the same as the difference of 60 and 48.”

        “That’s a coincidence,” I said and wanted to go on.
        But they would not let me go on: “Please name us numbers where it isn’t like that.”
        “Fine. 60 and 36 also have 12 as their greatest common divisor, and their difference is 24.”

        Another interruption: “Here the difference is twice as big as the greatest common divisor.”
        “All right, if this will satisfy all of you, it is in fact no coincidence: the difference of two numbers is always divisible by all their common divisors. And so is their sum.”
        Certainly that needed to be stated in full, but having done so, I really did want to move on.
        However, I still could not do that.
        A girl asked: “Couldn’t they discover a procedure to find the greatest common divisor just from that?”

        They certainly could! But that is precisely the basic idea behind the Euclidean Algorithm!
        So I abandoned my plan and went the way that my students led me.

        — Rózsa Péter
        quoted at the MacTutor History of Mathematics Archive

        For Further Exploration

        Note: When the video narrator says “Greatest Common Denominator,” he really means “Greatest Common Divisor.”

        CREDITS: “Pink toned thoughts on a hike” photo courtesy of Simon Matzinger on Unsplash.