Of all the myths about mathematics, the one I find most blatantly wrong is the idea that some people are just born knowing the answers. In my experience, when you confront a genuine puzzle, you start out not knowing, no matter who you are.
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Moreover, “knowing” the answers can be a trap; learning mathematics is about looking at what you thought you understood and seeing that there’s deeper mystery there than you realised.
If you’d like to practice learning mathematics by confronting genuine puzzles, Dan’s “A Mathematician at Play” series looks like a wonderful place to start.
Some of these puzzles are classics, others are original. All of them involve some kind of thinking or insight that strikes me as pretty, or surprising, or delightful.
Dan plans to post new puzzles on the Math 4 Love blog every Monday for the next few months. And sharing spoilers on each following Friday, if you want to verify your answers.
“Teach mathematics the way we learn any other subject: Make it visual, make it concrete, not dependent on meaningless, abstract symbols, employ all the senses!
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If math is such an important subject (and it is) why teach it in a way that is dependent on a child’s weakest mental ability: memory, rather than her strongest mental ability: imagination?”
“Earlier we considered the argument, ‘Twice two must be four, because we cannot imagine it otherwise.’ This argument brings out clearly the connexion between reason and imagination: reason is in fact neither more nor less than an experiment carried out in the imagination.
“People often make mistakes when they reason about things they have never seen. Imagination does not always give us the correct answer. We can only argue correctly about things of which we have experience or which are reasonably like the things we know well. If our reasoning leads us to an untrue conclusion, we must revise the picture in our minds, and learn to imagine things as they are.
“When we find ourselves unable to reason (as one often does when presented with, say, a problem in algebra) it is because our imagination is not touched. One can begin to reason only when a clear picture has been formed in the imagination.
“Bad teaching is teaching which presents an endless procession of meaningless signs, words and rules, and fails to arouse the imagination.”
But my favorite way to celebrate any new year is by playing the Year Game. It’s a prime opportunity for players of all ages to fulfill the two most popular New Year’s Resolutions: spending more time with family and friends, and getting more exercise.
So grab a partner, slip into your workout clothes, and pump up those mental muscles!
For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find. This year may prove to be a challenge.
Use the digits in the year 2018 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.
You must use all four digits. You may not use any other numbers.
Solutions that keep the year digits in 2-0-1-8 order are preferred, but not required.
You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
My Special Variations on the Rules
You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
You MAY use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.
This lovely puzzle (for upper-elementary and beyond) is from Nikolay Bogdanov-Belsky’s 1895 painting “Mental Calculation. In Public School of S. A. Rachinsky.” Pat Ballew posted it on his blog On This Day in Math, in honor of the 365th day of the year.
I love the expressions on the boys’ faces. So many different ways to manifest hard thinking!
Here’s the question:
No calculator allowed. But you can talk it over with a friend, as the boys on the right are doing.
You can even use scratch paper, if you like.
Thinking About Square Numbers
And if you’d like a hint, you can figure out square numbers using this trick. Think of a square number made from rows of pennies.
Can you see how to make the next-bigger square?
Any square number, plus one more row and one more column, plus a penny for the corner, makes the next-bigger square.
So if you know that ten squared is one hundred, then:
… and so onward to your answer. If the Russian schoolboys could figure it out, then you can, too!
Update
Simon Gregg (@Simon_Gregg) added this wonderful related puzzle for the new year:
Do you enjoy math? I hope so! If not, the links in this post just may change your mind.
Welcome to the 114th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of articles by bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.
By the way, I found a cool, semi-self-referential trivia tidbit about our carnival number: 27 − 14 = 114. And if you put 114 dots into a 1←7 Exploding Dots machine, you’ll get the code 222. Pretty neat!
As you scroll through the links below, you find several puzzle graphics from the wonderful Visual Patterns website. Use them as conversation-starters with your kids: What do you notice? How does each pattern grow? For older students: Can you write a formula to describe how each pattern? What will it look at stage 43?
Pattern #7, Trees
A BIT OF FUN
Setting the mood: Enjoy this bit of seasonal fidgeting from Vi Hart (@vihartvihart).
If you don’t understand some of the references, that’s normal! Pick a phrase, Google it, and enjoy the fun of learning something new.
TABLE OF CONTENTS
And now, on to the main attraction: the blog posts. Some articles were submitted by their authors; others were drawn from the immense backlog in my rss reader. If you’d like to skip directly to your area of interest, click one of these links.
I’ve been following Sonya’s Arithmophobia No More blog for a couple of years, and I love the work she is doing. But this month, she’s teamed up with Lacy at Play, Discover, Learn (another great blog to follow!) to offer a humongous bundle of playful math.
You get math journaling pages, games, creative task cards, thought-provoking worksheets, and video training resources to help you build your child’s understanding of math from arithmetic to early algebra. Wow!
These activities are perfect for homeschooling families or anyone looking to supplement their child’s current math curriculum with effective discovery-based activities. If you’ve ever wondered what to do with those Cuisenaire rods you picked up on sale way back when, this bundle is for you.
I’m so looking forward to using some of these ideas with my elementary homeschool co-op kids next year!
If you’ve been reading my blog for very long, you’ve probably seen how much I love the blog, books, and classes available from the Natural Math folks.
Their newest book is just off the presses — Funville Adventures, a math adventure chapter book.
And until December 20, they’re having a holiday sale. Make your own bundle of any Natural Math books. Playful algebra, calculus for 5-year-olds, inquiry problems and more: Great deal!
(US customers only: We’re sorry we can’t offer bulk discounts for our international readers, but the complexities of international duties and tax laws are too much for this very small family business.)
Do You Know of Any Math Deals?
Apollonian greetings from my homeschool co-op kids, and best wishes for a grace-filled holiday season.
If you’ve seen a great deal or holiday price on a math resource you love, please share!
Add your deal to the comment section below, so we can all take advantage of the math joy this season.
Each carnival brings you a great new collection of puzzles, math conversations, crafts, teaching tips, and all sorts of mathy fun.
This month we have whole-body math, notice-and-wonder puzzles, a game to build math vocabulary, intransitive dice, making sense of trig identities, and playing Go on a hundred chart. And plenty more!
Do you write an education or family blog? Classroom teacher, math coach, homeschooler, parent, college professor, unschooler — anyone interested in helping kids play around with math?
Please consider volunteering to host the MTaP blog carnival for one month.
We still need a home for the last carnival of 2017.
Or plan ahead: 2018 is wide open.
You choose the month that fits your schedule and decide how much effort you want to put in. Writing the carnival can take a couple of hours for a simple post — or you can spend several days searching out and polishing playful math gems to share.
Check out the new playful math blog carnival at Find the Factors blog. Iva put together a great collection of math games, activities, and teaching tips:
The carnival features comics, literature, talking with kids, favorite numbers, classroom management, a bulletin board that actually gets read, and plenty of math art. Along with several fantastic math puzzles to explore.
You want your child to succeed in math because it opens so many doors in the future.
But kids have a short-term perspective. They don’t really care about the future. They care about getting through tonight’s homework and moving on to something more interesting.
So how can you help your child learn math?
When kids face a difficult math problem, their attitude can make all the difference. Not so much their “I hate homework!” attitude, but their mathematical worldview.
Does your child see math as answer-getting? Or as problem-solving?
Answer-getting asks “What is the answer?”, decides whether it is right, and then goes on to the next question.
Problem-solving asks “Why do you say that?” and listens for the explanation.
Problem-solving is not really interested in “right” or “wrong”—it cares more about “makes sense” or “needs justification.”
Homeschool Memories
In our quarter-century-plus of homeschooling, my children and I worked our way through a lot of math problems. But often, we didn’t bother to take the calculation all the way to the end.
Why didn’t I care whether my kids found the answer?
Because the thing that intrigued me about math was the web of interrelated ideas we discovered along the way:
How can we recognize this type of problem?
What other problems are related to it, and how can they help us understand this one? Or can this problem help us figure out those others?
What could we do if we had never seen a problem like this one before? How would we reason it out?
Why does the formula work? Where did it come from, and how is it related to basic principles?
What is the easiest or most efficient way to manipulative the numbers? Does this help us see more of the patterns and connections within our number system?
Is there another way to approach the problem? How many different ways can we think of? Which way do we like best, and why?
What Do You think?
How did you learn math? Did your school experience focus on answer-getting or problem-solving?
How can we help our children learn to think their way through math problems?
I’d love to hear from you! Please share your opinions in the Comments section below.
CREDITS: “Math Phobia” photo by Jimmie (blog post title added) via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.
Do you write an education or family blog? Classroom teacher, math coach, homeschooler, parent, college professor, unschooler — anyone interested in helping kids play around with math?
Please consider volunteering to host the MTaP blog carnival for one month.
We still need volunteer hosts for fall semester 2017.
Or plan ahead: 2018 is wide open.
You choose the month that fits your schedule and decide how much effort you want to put in. Writing the carnival can take a couple of hours for a simple post — or you can spend several days searching out and polishing playful math gems to share.
Denise Gaskins' Let's Play Math 