Today we examine a time-tested method to help kids reason about math: Leave out the numbers.
First up, there’s Brian Bushart’s numberless problem bank for young students. Then we’ll look at Farrar Williams’s modern revision of a math teaching classic with problems for upper-elementary and middle school students.
Have fun thinking math with your kids!
Word Problem Bank
Word problems are commonplace in mathematics classrooms, and yet they regularly confound students and lead to frustrated teachers saying things like:
“They just add all the numbers! It doesn’t matter what the problem says.”
“They don’t stop to think! They just start computing as soon as they’re done reading the problem.”
Brian Bushart offers a collection of ready-to-go slide presentations that walk through the steps of making a word problem make sense.
Discover Farrar Williams’s book Numberless Math Problems: A Modern Update of S.Y. Gillian’s Classic Problems Without Figures, available in ebook or paperback.
Williams writes: “In order to answer the question, they’ll have to explain it, because the problem doesn’t give you anything to calculate with. The only way to answer is by explaining your process. See how sneaky a numberless problem is? It makes students really think about the process of solving the problem.”
“When students face a word problem, they often revert to pulling all the numbers out and “doing something” to them. They want to add, subtract, multiply, or divide them, without really considering which operation is the right one to perform or why.
“When you don’t have numbers, it sidesteps that problem.
“For students who freeze up when they see the numbers, this can be a really good way to get them to think about their process with math.”
—Farrar Williams, Math With No Numbers
CREDITS: Feature photo (top) by saeed karimi via Unsplash.com.
If you teach children in the primary grades, you’ll enjoy this new series from the wonderful Steve Wyborney. Every day for the rest of the school year, Steve will post a new estimation or number sense resource for grades K–8 (or any age!) at his blog:
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!
“What is the best curriculum for my children? They are four and six years old, and I’m afraid of letting them fall behind.”
I remember being a young parent, eager to start homeschooling. I used to get mad (without letting it show, like a true introvert) when people told me, “They are young. Just let them play.”
Now I see the wisdom in it.
The most important thing for your children right now, by far, is for them to enjoy learning. The joy of learning is a child’s natural state. As a parent, your primary job is to keep yourself from stomping it out.
But our parental fears can push us into joy-trampling before we realize it.
And our own experience of school makes it hard for us to see how much of our children’s play really is learning. We expect education to look like schoolwork, but natural learning looks nothing like that.
You’ll need a 6-sided die, a hundred chart (printables here), and a small token to mark each player’s square. A crumpled bit of colored construction paper works well as a token.
Take turns rolling the die. If you roll:
1: Move either 1 or 10 squares, your choice.
2: Move either 2 or 20 squares.
3–6: Move that number of squares.
The first player to reach the final square by exact count wins the game.
Variation #1: For a shorter game, the first player to move off the board wins. You don’t have to hit the final square by exact count.
Variation #2: For a longer game, if you cannot move your full roll forward, you must move backward. Rolling 6 is a “wild card” — you can move any number from one to ten.
Variation #3: Count down. Start at the highest number on your chart and subtract each roll, moving toward zero. If you have a chart like the original shown above, a player whose move goes past zero into negatives will add the number on their next roll.
More Ways to Play on a Hundred Chart
A hundred chart can provide mathematical play from preschool to high school. The list on my blog began many years ago with seven activities, games, and logic puzzles.
Problem-solving is a habit of mind that you and your children can learn and grow in. Help your kids practice slowing down and taking the time to fully understand a problem situation.
Puzzles Are Math Experiments
Almost anything your child notices or wonders can lead to a math experiment.
For example, one day my daughter played an online math game…
A math journal can be like a science lab book. Not the pre-digested, fill-in-the-blank lab books that some curricula provide. But the real lab books that scientists write to keep track of their data, and what they’ve tried so far, and what went wrong, and what finally worked.
Here are a few open-ended math experiments you might try:
Pick out a 3×3 set of dots. How many different shapes can you make by connecting those dots? Which shapes have symmetry? Which ones do you like the best?
What if you make shapes on isometric grid paper? How many different ways can you connect those dots?
Limit your investigation to a specific type of shape. How many different triangles can you make on a 3×3 set of dots? How many different quadrilaterals? What if you used a bigger set of dots?
On your grid paper, let one dot “hold hands” with two others. How many different angles can you make? Can you figure out their degree without measuring?
Are there any angles you can’t make on your dot grid? If your paper extended forever, would there be any angles you couldn’t make?
Does it make a difference whether you try the angle experiments on square or isometric grid paper?
How many different squares can you draw on your grid paper? (Don’t forget the squares that sit on a slant!) How can you be sure that they are perfectly square?
Number the rows and columns of dots. Can you find a pattern in the corner positions for your squares? If someone drew a secret square, what’s the minimum information you would need to duplicate it?
Does it make a difference whether you try the square experiments on square or isometric grid paper?
I’d love to hear your favorite math explorations or journaling tips!
Please share in the comments section below.
P.S.: Do you have a blog? If you’d like to feature a math journal review and giveaway, I’ll provide the prize. Send a message through my contact form or leave a comment below, and we’ll work out the details.
Today we have a guest post — an exclusive tale by Sasha Fradkin and Allison Bishop, authors of the new math storybook Funville Adventures. Enjoy!
Funville Adventures is a math-inspired fantasy that introduces children to the concept of functions, which are personified as magical beings with powers.
Each power corresponds to a transformation such as doubling in size, rotating, copying, or changing color. Some Funvillians have siblings with opposite powers that can reverse the effects and return an object to its original state, but other powers cannot be reversed.
In this way, kids are introduced to the mathematical concepts of invertible and non-invertible functions, domains, ranges, and even functionals, all without mathematical terminology.
We know about Funville because two siblings, Emmy and Leo, were magically transported there after they went down an abandoned slide.
When they came back, Emmy and Leo shared their adventures with their friends and also brought back the following manuscript written by their new friend Blake.
It’s a short book with plenty of great stories, advice, and conversation-starters. While Danielson writes directly to parents, the book will also interest grandparents, aunts & uncles, teachers, and anyone else who wants to help children notice and think about math in daily life.
“You don’t need special skills to do this. If you can read with your kids, then you can talk math with them. You can support and encourage their developing mathematical minds.
“You don’t need to love math. You don’t need to have been particularly successful in school mathematics. You just need to notice when your children are being curious about math, and you need some ideas for turning that curiosity into a conversation.
“In nearly all circumstances, our conversations grow organically out of our everyday activity. We have not scheduled “talking math time” in our household. Instead, we talk about these things when it seems natural to do so, when the things we are doing (reading books, making lunch, riding in the car, etc) bump up against important mathematical ideas.
“The dialogues in this book are intended to open your eyes to these opportunities in your own family’s life.”
My youngest daughter wanted to do Singapore math. Miquon Red was her main math text at the time, but we added a bit of Singapore Primary Math 1B whenever she was in the mood.
We turned to the lesson on subtracting with numbers in the 30-somethings.
The first problem was pretty easy for her:
30 − 7 = _____
I reminded her that she already knew 10 − 7.
She agreed, “Ten take away seven is three.”
Then her eyes lit up. “So it’s 23! Because there are two tens left.”
Wow, I thought. She’s catching on quickly.
Mom Always Talks Too Much
We went to the next problem:
34 − 8 = _____
“Now, this one is harder,” I said. “But you know what ten minus eight is, right? So we could take one of these tens and—”
She waved at me to be quiet.
I was just getting started on my standard speech about how to turn a tough subtraction like 34 − 8 into the easy addition of “2 + 4 + two tens left.” But her mind was still on the last problem, specifically on the two tens and the seven.
“If you have 27,” she said, “and you add three more, you get 30. And four more is 34.”
“Um, yes, but…” I interrupted.
She shushed me again.
“And then you can take away the four. And then you can take away the three. And then you can take away one more…It’s 26!”
Mom Learns a Lesson
She continued through the next page that way. For every problem, she started with whatever number struck her fancy, usually containing at least one digit from the problem before. She added enough to get up to the 30-something number in the book.
Only then would she deign to subtract the number in question.
I don’t think she ever saw the point of the mental math technique the book and I were trying to teach, but she did have a lot of fun playing around with the numbers.