How to Succeed in Math: Answer-Getting vs. Problem-Solving

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: “Maths” photo (top) by Robert Couse-Baker. “Math Phobia” photo by Jimmie. Both via Flickr (CC BY 2.0). Phil Daro video by SERP Media (the Strategic Education Research Partnership) via Vimeo.

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


Mindset for Learning Math

Playing with a new image editor, I came across this Winston Churchill quote. What a great description of how it feels to learn math!

If you have a student who struggles with math or is suffering from a loss of enthusiasm, check out Jo Boaler’s free online course on developing a mathematical mindset:

Or explore some of the playful activity ideas for all ages in her Week of Inspirational Math.


howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


Rabbit Trails and Fibonacci Poetry

Homeschooling Memories…

Well, I hadn’t planned on spending my day that way. But one of the great things about homeschooling is the freedom to follow rabbit trails.

While browsing the Carnival of Homeschooling, I found a link to Farm School blog’s article Fib Foolery, which sent me to Gotta Book for his articles The Fib and More Fibbery (read the comments on both threads, but be warned that some are crude) and several other posts, all of which set me off on a morning of poetic fun.

A “Fib” is a Fibonacci poem. It’s based on syllable count, like a haiku, but the lines follow the Fibonacci counting series: 1, 1, 2, 3, 5, 8… Each number is the sum of the previous two numbers.

I knew what I was going to share at our Tuesday Teatime and Poetry Reading that afternoon.

Here’s the best one I’ve come up with so far:

Math:
Word
Problem,
Mental play.
Archimedes shouts,
“Eureka! I figured it out.”

The Kids Join the Fun

While we always enjoyed our tea and poetry times, that day was the only one that inspired the kids to actually write poetry themselves.

My 7yo dd was so proud to be able to count syllables and write:

Cat.
Soft.
Pretty,
But sleeping.

While my 12yo ds really took off, creating more than a dozen Fibs. His first two are still his favorites:

Ducks
Have
No luck,
But they do
Have many feathers.
Hunters like to shoot ducks a lot.

and

Paul
Is
Revered
A lot by
Paul Revere’s Fan Club.
What is Paul’s last name, anyway?

Wouldn’t you like to try it, too? Please share your Fib in the comments below!


Feature photo: “Rabbit” by Save the Bay via Flickr (CC BY 2.0).

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


There Ain’t No Free Candy

Ah, the infinite chocolate bar. If only it could work in real life! But can your children find the mistake? Where does the extra chocolate come from?

Here’s a hint: It’s related to this classic brainteaser. And here’s a video from Christopher Danielson (talkingmathwithkids.com), showing how the chocolate bar dissection really works.

Happy munchings!


CREDITS: Feature photo (top) by Yoori Koo via Unsplash. “Hershey Bar Math” video by Christopher Danielson via YouTube. The infinite chocolate gif went viral long ago, and I have no idea who was the original artist.

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


How You Can Play with Math #106

Do you enjoy math? I hope so! If not, browsing this post just may change your mind.

Welcome to the 106th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle in honor of our 106th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Try This Puzzle

If you slice a pizza with a lightsaber, you’ll make straight cuts all the way across. Slice it once, and you get two pieces.

If you slice it five times, you’ll get a maximum of sixteen pieces. (And if you’re lucky you might get a star!)

  • How many times would you have to slice the pizza to get 106 pieces?

Click here for all the mathy goodness!

Dot Grid Doodling

What can you DO with a page full of dots?

Yesterday, I mentioned my new series of paperback dot grid notebooks, and I promised to share a few ideas for mathematical doodling.

Doodling gives our minds a chance to relax, wander, and come back to our work refreshed. And though it goes against intuition, doodling can help us remember more of what we learn.

Math doodles let us experiment with geometric shapes and symmetries. We can feel our way into math ideas gradually, through informal play. Through doodles, our students will explore a wide range of mathematical structures and relationships.

Our own school experiences can make it hard for us to teach. What we never learned in school was the concept of playing around with math, allowing ideas to “percolate,” so to speak, before mastery occurs, and that process may take time.

—Julie Brennan

I like to doodle on dotty grid paper, like the pages in my math journals, but there’s No Purchase Necessary! You can design your own printable dot page at Incompetech’s PDF generator, or download my free coloring book (which includes several pages of printable dot and graph paper).

Patterns in Shape and Angle

To make a faceted mathematical gemstone, start with any shape you like. Then build other shapes around it. What do you notice? Does your pattern grow outward from its center? Or flow around the corner of your page? How is each layer similar, and how is it different?

Arbitrary constraints can lead to mathematically interesting doodles. For instance, create a design out of 45-45-90 triangles by coloring exactly half of every grid square. How many variations can you find?

Symmetry Challenge

Play a symmetry puzzle game. Draw a line of symmetry and fill in part of the design. Then trade with a partner to finish each other’s doodles.

Make more complex symmetry puzzles with additional reflection lines.

Math Doodle Links

  • Who can talk about mathematical doodling without mentioning Vi Hart? If you’ve never seen her “Doodling in Math Class” video series, you’re in for a treat!
  • See if you can draw a rotational-symmetry design, like Don’s “Order 4” graphs.
  • Or experiment with the more flexible rules in John’s “Knot Fun” lesson.
  • And my latest obsession: the “ultimate” tutorial series on Celtic Knotwork, which explores the link between knots and their underlying graphs.
My favorite knot doodle so far.

Inspirations: A Recreational Mathematics Journal
Reflections: A Math Teacher’s Journal
Explorations: A Math Student’s Journal
Contemplations: A Homeschooler’s Journal

Also available through: Barnes-Noble-logo the_book_depository_logo CreateSpace-logo

Before you start doodling: How to Break In Your New Math Journal.


Feature photo (top): Sommermorgen (Alte Holzbrücke in Pretzfeld) by Curt Herrmann, via Wikimedia Commons. [Public domain]

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


Math Inspirations: Why Study Mathematics?

why-study-math

What teacher hasn’t heard a student complain, “When am I ever going to have to use this?” Didn’t most of us ask it ourselves, once upon a time?

And unless we choose a math-intensive career like engineering, the truth is that after we leave school, most of us will never again use most of the math we learned.

But if math beyond arithmetic isn’t all that useful, then what’s the point?

If you or your student is singing the “Higher Math Blues,” here are some quotations that may cheer you up — or at least give you the strength of vision to keep on slogging.

We Study Mathematics…

To Understand Creation

I don’t want to convince you that mathematics is useful. It is, but utility is not the only criterion for value to humanity. Above all, I want to convince you that mathematics is beautiful, surprising, enjoyable, and interesting. In fact, mathematics is the closest that we humans get to true magic. How else to describe the patterns in our heads that — by some mysterious agency — capture patterns of the universe around us? Mathematics connects ideas that otherwise seem totally unrelated, revealing deep similarities that subsequently show up in nature.

— Ian Stewart
The Magical Maze

That vast book which stands forever open before our eyes, the universe, cannot be read until we have learnt the language in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

— Galileo Galilei
quoted by Clifford Pickover, A Passion for Mathematics

To Train Our Minds

The investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings.

— George Washington
quoted by William Dunham, The Mathematical Universe

I told myself, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means.” So I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what “demonstrate” means, and went back to my law studies.

— Abraham Lincoln
quoted by William Dunham, The Mathematical Universe

To Understand History

In most sciences, one generation tears down what another has built, and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure.

— Herman Henkel
quoted by Noah benShea, Great Quotes to Inspire Great Teachers

Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals — the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.

— Martin Gardner
quoted by G. Simmons, Calculus Gems

I will not go so far as to say that constructing a history of thought without profound study of mathematical ideas is like omitting Hamlet from the play named after him. But it is certainly analogous to cutting out the part of Ophelia. For Ophelia is quite essential to the play, she is very charming. . . and a little mad.

— Alfred North Whitehead
quoted in The Viking Book of Aphorisms

To Appreciate the Beauty

The mathematician does not study pure mathematics because it is useful, he studies it because he delights in it, and he delights in it because it is beautiful.

— Henri Poincaré
quoted by Theoni Pappas, More Joy of Mathematics

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful. The ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

— Godfrey H. Hardy
A Mathematician’s Apology

And Most of All, to Play

Mathematics is a world created by the mind of men, and mathematicians are people who devote their lives to what seems to me a wonderful kind of play!

Constance Reid

At age eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined there was anything so delicious in the world.

— Bertrand Russell
The Autobiography of Bertrand Russell

I love mathematics … principally because it is beautiful, because man has breathed his spirit of play into it, and because it has given him his greatest game — the encompassing of the infinite.

Rózsa Péter
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians

Did you enjoy these? You can find plenty more on my Math & Education Quotations page.

  • I would LOVE to hear YOUR favorite mathematics, education, or inspirational quote. Please share in the Comments section below!

 photo exploreMTBoS_zpsf2848a9a.jpgNever Ending Math Problem photo (above) by Danny via Flickr (CC BY 2.0). This post is part of the #MTBoS #MtbosBlogsplosion blogging challenge.


howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.