“Cultivating thinking skills is the main reason for teaching math. It is the mind’s perfect playground for shaping up.
To begin developing thinking, you must first have a child who is curious. For without curiosity, there is only forced thinking.
The problem with traditional math is it jumps to the punchline.
Absolutely no mystery or suspense is developed in traditional math books. Why? Apparently, someone thought math was without mystery. That math is a definitive subject of rules and algorithms that all have been discovered.
We must persuade children that math is a worthy pursuit through interesting stories, examining quirky math properties, and asking good questions.”
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.
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.
“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!
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!
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?
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.
You don’t have to celebrate Christmas to enjoy many of these activities — but really, I couldn’t find much for the other winter holidays. A few calculation worksheets with clip art, which is not my idea of playful math.
Do you know of any great math-related seasonal games, crafts, or activities I missed? Please add them to the comments section below!
Clarissa (@c0mplexnumber) demonstrates how to make beautiful, challenging origami snowflakes. She recommends beginners try the first few folds — which create a pretty cool design on their own. Let it Snow… You may also enjoy her other Christmas projects.
If you’ve followed my blog for long, you know I like to play with dot grid paper. So of course, I was delighted to find Spatial Learning’s Isometric Dot Paper Activities, and the follow-up Cube Stack Activity. What a great way to build geometric intuition!
My daughter is struggling with online homework in her calculus class — not because the math is too hard, but because the interface is anti-intuitive. So David’s (@davidwees) post resonates with me: Online Practice is Terrible Practice. And I love his challenge to find and teach to the Big Ideas of math.
I’d like to wrap up the carnival with an article you may have seen before. If you haven’t read it, you’re in for a treat. And if you have, well, it’s very much worth re-reading. Annually. As we wrap up the old year and prepare for the new … Francis’s (@mathyawp) Mathematics for Human Flourishing.
“Shalom and salaam, my friends. Grace and peace to you. May you and all your students flourish.”
And that rounds up this edition of the Math Teachers at Play carnival.
I hope you enjoyed the ride.
The next installment of our carnival will open sometime during the week of January 22–26, 2018, at … well, we don’t know!
We need more volunteers. Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn hosting the Math Teachers at Play blog carnival, please speak up.
To share your favorite blog post with the carnival, please use this handy submission form. Posts must be relevant to students or teachers of preK-12 mathematics. Older-but-still-relevant posts are welcome, as long as they haven’t been published in past editions of this carnival (at least, not in recent memory).
When you get to the Nrich website, click a number to go to that day’s math.
For Teens and Adults
“This year we’ve decided to bring you some of our favourite Plus videos. There’s nothing more soothing that a bit of fascinating maths, explained by a fascinating mathematician, that doesn’t even require you to read stuff. Happy watching!”
When you get to the +Plus Magazine website, you can tell which links are live because they jump to a larger size when you tap or mouse over the picture.