Writing to Learn Math: Problem-solving cares less about whether an answer is right and more about whether a solution makes sense.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Problem-Solving Prompt

When children face a tough 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, then forgets it and goes on to the next question. Problem-solving cares less about whether an answer is right and more about whether a solution makes sense.

Students who care about problem-solving want to explore the web of interrelated ideas they discovered along the way: How can they recognize this type of problem? Can this one help them figure out others?

What could they do if they had never seen a problem like this one before? How would they 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 manipulate the numbers? Does this help the problem-solver see more of the patterns and connections within our number system?

Is there another way to approach the problem? How many ways can they think of? Which do they like best, and why?

Journaling Prompt 91: Sci-Fi Puzzle 2

Aliens from the planet Crimbat double their population every year. They send out lots of colony spaceships to keep from overpopulating their home.

A Crimbatti settlement on Earth contained 6,400 aliens in its 6th year.

In which year was the population half that size?

How many Crimbatti settlers were there in the first year? How long until the aliens outnumber the humans on Earth?

This week, we have the first mathematical adventure of Alexandria Jones, which takes place in the early days of her father’s archaeological expedition mentioned in the introduction post.

Herein we meet the one major character who is not named after a mathematician from history: Alex’s nemesis, Simon Skulk.

While my fiction doesn’t rise close to the level of “art,” I hope it offers a glimpse into the playful nature of mathematics for you and your children.

The Secret of the Pharaoh’s Treasure

Alexandria Jones stood outside her father’s tent. The glare of the sun on the rocky desert hurt her eyes. Holding up a hand to shield her gaze, she spotted her dad (the world-famous archaeologist) arguing with the foreman.

Poor Dad, she thought. He was sure this was the right site, but so far he’s found nothing.

She looked down at her feet, where her faithful dog Ramus waited, panting. “Well, Rammy, it looks like Dad will be busy for while. What do you say? Shall we go exploring?”

Alexandria ducked into the tent for her backpack and canteen.

Thump! Something bounced against the side of the tent. Ramus barked.

Alex stepped outside and looked quickly around. No one was in sight. She saw a fist-sized rock beside the tent, with a note tied to it. She picked it up and read:

Ha! The real Pharaoh’s Treasure lies under a pyramid of stones, and it’s mine. You can’t stop me this time!
—Simon Skulk

“Find him, Rammy. Which way did he go?”

Growling, the dog ran down the trail to the valley. Alex followed him, but pulled up short when Ramus started into a dark cave. She swung her backpack off her shoulder and drew out a flashlight.

“The treasure must have been buried in a cave, as some of the later pharaohs were,” she said. “Okay, Rammy. Lead on.”

The cave ended at the foot of a head-high pyramid of small, rough-cut stones. Her father’s enemy was just setting his lantern on the ground.

“You creep!” Alex shouted. “That treasure belongs in a museum.”

The man laughed and reached for a stone. “Not if I find it first. The Pharaoh’s Treasure should bring me a pretty profit!”

Alexandria Jones ran to grab a stone from the pyramid. If she could just uncover the treasure before he did…

Can You Find the Pharaoh’s Treasure?

The Pharaoh’s Pyramid is a strategy game for two players. Print out the PDF gameboard or draw your own pyramid of stones.

• Each player will need a pencil or colored marker. Youngest player gets the choice of going first or second.

• On your turn, draw an “X” on one or two stones, to remove them from the pyramid. You do not have to work from the top down; you may mark any stones you wish. But if you want to mark two stones on the same turn, they must be touching each other.

• Take turns marking stones until they are all gone. Whoever marks the last stone uncovers the Pharaoh’s Treasure and wins the game.

If you draw your own game, experiment with different sized pyramids. How does changing the number of stones affect the strategy of the game?

The Pharaoh’s Treasure, Continued

“I can’t believe it!” Simon Skulk threw down the last stone in disgust and walked away. At the mouth of the cave, he turned back and shook his fist. “You haven’t seen the last of me, Alexandria Jones.”

Her muscles aching, Alex sank to the ground and hugged her dog. The she gave him a little push toward the front of the cave. “Rammy, go get Dad.”

Ramus barked once and took off running.

Alex turned back to look at the Pharaoh’s Treasure. Where the last stone had stood was a hole. In the hole lay three wooden sticks, like tent pegs, and a long loop of rope with twelve evenly-spaced knots.

What could it be?

It seemed as though Alex sat there forever, shivering as the desert sun went down, her muscles growing stiff.

Finally, Ramus came bounding into the cave and licked her face. Her father ran in behind the dog, giving her a hug that chased the stiffness away.

Then, slipping on a pair of vinyl gloves from his pocket, Fibonacci Jones reached into the hole and gently touched the pegs and rope.

“You found it,” he said. “The Pharaoh’s Treasure! Isn’t it beautiful?”

Alexandria frowned. “But what is it?”

The ancient Egyptians were careful builders, Fibonacci Jones explained. To keep their pyramids, temples, and other buildings squared up properly, Egyptian surveyors (called “rope stretchers”) used a powerful design tool: the right triangle.”

How to Use the Rope Triangle

You can make your own rope stretcher’s triangle with a piece of string and three straight pins.

Tie eleven evenly-spaced knots, about 2 inches (5 cm) apart, and then tie the ends of the string together to make the twelfth knot. Getting the knots right is the hardest part. Check your spacing with a ruler or by folding the string several times, to make sure the knots all line up.

Then use the straight pins as your pegs, and use the cardboard back of a spiral notebook or the side of a cardboard box as your ground:

• Pound one peg into the ground where you want the square corner. Loop the rope around the peg with one knot exactly at the peg.

• Move along the line of your wall a distance of three knots on the rope. Pound in the second peg, with the rope looped around it and with one knot exactly at the peg.

• Slip the last peg into the rope loop and pull it tight, four knots from the first peg at the corner. Pound this last peg into the ground, making sure it is exactly at its knot.

This makes a right triangle with sides of 3, 4, and 5 lengths on the rope. The corner is perfectly square.

A Puzzle for Older Students

Play around with your Pharaoh’s Treasure model.

• Can you explain why the instructions above form a right angle?

• There are other ways to form a right angle using the rope and pegs. Can you find an alternate method?

You need to find some way to make two lines that are perpendicular to each other. Feel free to use extra ropes or pegs, if needed. The Egyptian rope stretchers could use as many or as few ropes as they liked.

If you want a hint, consider Euclid’s Book 1, Proposition 11.

And if you get stumped, I will put one possible answer in a future post. But don’t give up too quickly. Remember the math adventurer’s rule.

Math Adventurer’s Rule: Figure it out for yourself!

When I tell you the answer, you miss out on the fun of solving the puzzle. Figure it out for yourself—and then check the answer just to prove that you got it right.

To be continued…

This game requires students to make logical deductions about number relationships.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Digit Disguises

Math Concepts: addition, subtraction, multiplication, division, algebraic reasoning.

Players: two players or two teams. Best with teams.

Equipment: printed gameboards (optional), pencils and paper, or whiteboard and markers.

Set-Up

The FREE 68-page printable (pdf) Prealgebra & Geometry Printables file features hundred charts, coordinate grids, assorted graph paper, and all the game boards for the Math You Can Play: Prealgebra & Geometry book.

Each player or team needs a separate gameboard.

If you’re not using printed gameboards, each player or team makes a chart with two columns labeled Mine and Theirs. List the letters A–J in each column. Next to each letter in your own column, write one of the whole numbers 0–9 in any mixed-up order. Fold your paper to keep that column hidden from your opponent. Slip a paperclip over the fold for greater security.

How to Play

On your turn, ask for the result of a basic arithmetic calculation involving two different letters. For example, you might ask:

• A + B = ?
• C − D = ?
• E × F = ?
• G ÷ H = ?

But you may not ask for something like “I − I” or “J ÷ J.”

The other player/team figures out the answer to your question, using their values for each letter, but they do not tell you the number. If the answer is on their chart, they tell you the letter for that value. Or they tell you that the result is not one of their letters.

Players never say a number in response to a question. They only say a letter or “Not a letter.”

Write notes to help you remember each clue. You’ll need these to deduce your opponent’s number code.

Then it is their turn to ask you for a clue about your secret code.

When one player/team claims to have broken the opponent’s number code, the game is over. They read aloud the number values they have figured out. If they’re entirely correct, they win the game. But if any of the answers are wrong, the opponent wins.

Variations

Don’t end the game after the first code is guessed. Whether the guessing team won or lost, the other team keeps asking questions until they also break the code. If both teams fail, the game is a tie.

For Beginners: Start with only four unknown numbers, and allow players to give number answers in response to each question.

Keeper of the Code: (Mastermind-style) One player makes the code, and the other players try to crack it. Players take turns asking for clues, but they may discuss the answers and plan a strategy together. This variation is good for younger players, if you let them be the Keeper while older students or adults are guessing.

History

Math lecturer David Butler created this game and writes:

“I was struck by how quickly the logic got complicated, and then how quickly it all cascaded into finding everything when I finally got a few different numbers.”

Butler also shared a printable gameboard that can fold to stand on the table Battleship-style, hiding your secret code.

Writing to Learn Math: What did the author mean? Put the thought in your own words. Do you agree or disagree? Why?

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Quotation Prompt

Let students choose how they want to react to the quotation. Or offer one of the following questions:

• What did the author mean? Put the thought in your own words.
• Do you agree or disagree? If you agree, can you think of someone who would disagree? Why?
• Is this quote a general principle, or only for specific situations? Describe a time when it might apply, or when it might not.
• Tell a time in your life when you lived up to the quotation — or when you wish you had.
• How does the quote relate to math, science, history, or another subject?

Short exercises are great writing practice. But occasionally you’ll want to assign deeper essay topics, such as:

• Look up the author’s name online. Who are/were they, and why do people care what they said?
• Quotes are often misattributed. Did the author really say this?
• What have others said about the same topic? Search out a variety of quotes related to this one. How are they similar? How are they different?
• Does thinking about the quotation make you want to change anything, in yourself or in the world? How could you put that idea into action?

Quotation from John von Neumann

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

—John von Neumann

It began with some 4th-8th grade friends who met in my dining room to work math puzzles and play games. At first, a few of the kids wondered how anyone could have fun with math. But we did enjoy ourselves, and Math Club grew until we couldn’t fit anyone else around the table.

When one girl had to move away, I thought, “Why not send Math Club with her?” Thus was born Math Club by Mail newsletter (later renamed Mathematical Adventures), a 4-year snail-mail newsletter that brought puzzles, strategy games, art projects, historical tidbits, and more to readers as far away as Hawaii.

Never very many readers, I admit—certainly never enough to make a profit—but it was a blast while it lasted.

So I thought you might like to read Alex’s story from the beginning, as my original math club kids did. The stories will appear sporadically, scattered between my more typical math-activity and teaching tip posts.

I hope you will have as much fun reading these adventures as I had writing them.

Introducing Alexandria Jones

My newsletter stories followed the adventures of homeschooler Alexandria Jones, her archaeologist father, and the rest of their family as they discovered a triangular treasure from Ancient Egypt, puzzled over patterns in Pythagorean pebbles, or explored the applications of math in daily life.

We learned about hieroglyphic math, fractals and chaos theory, logarithms, geometric algebra, and how to multiply any numbers using only the times-two table.

As I attempt to resurrect Alexandria Jones and her family here, the stories will appear sporadically, along with snippets from history and a side dish of math puzzles.

The first story, down below, actually appeared later in the original newsletters, but it makes a nice introduction.

Alexandria’s Dog is Now a Teacher

From the beginning, Alex’s dog supported her adventures with eager energy. In the original stories, Rammy was named after a famous Indian mathematician, but a reader of Indian descent was kind enough to write to me about a difference in our cultures.

In the US, or at least in the parts with which I am familiar, it is common to name one’s pet after a famous person. In India, however, to name a dog after a human is a very deep insult.

Therefore, I have re-named Alexandria Jones‘s dog after a Westerner. I would like to keep the tradition of naming the characters in my Alex stories after people in math history, so I am hoping this one will not offend anyone.

Peter Ramus: Homeschooler, Teacher, Magnet for Trouble

Peter Ramus is also known as Petrus Ramus or as Pierre de la Ramée. Born in France in 1515, he was educated at home until the age of twelve, when he entered the Collège de Navarre in Paris. He worked his way through college, graduated with a master’s degree at the age of 21, and began teaching philosophy—until his attacks on the dominant Aristotelian curriculum of his day got him banned.

Ramus switched to the study of mathematics, was reinstated, and rose to become the head of the the Collège de Presles. Always involved in controversy, however, he converted to Calvinism and spent several years in exile due to religious wars.

In addition to being a popular lecturer and author, Ramus was determined to reform educational philosophy. He had no patience for mnemonic tricks or rote learning. Instead, he thought it was the teacher’s job to break each of the liberal arts down into its most basic principles and organize these into a systematic and practical method of study. Above all else, education should be useful.

Due to his growing conviction about the fundamental importance of mathematics, he proposed a chair of mathematics at the University of Paris. Putting his money where is mouth was, Ramus endowed the chair in his will.

In 1572, he was killed in a massacre of French Huguenots.

A New Style of Teaching

Peter Ramus promoted many educational reforms, including:

• the simplification of knowledge, using his “laws of method”
• attention to pedagogy, the science of effective teaching
• increased attention to teaching physics and other sciences
• an emphasis on experimentation and practical application
• a de-emphasis of theoretical speculation (including abstract geometry proofs)

Ramus had great respect for Aristotle and other ancient philosophers, but fought repeatedly with the Aristotelians of his day. For at least a century after he died, Ramists and Aristotelians continued to argue about what logic really was and how we should teach students.

Enough Background! Here’s a Story: The Mysterious Temporal Freeze

Alexandria Jones stepped into the huge tent that protected her father’s excavation site from the desert winds. She laughed to herself. It was like walking into a circus.

She knelt down to whisper in the ear of her faithful dog Ramus.

“In this ring, grad students carefully brush away another layer of sand. In the next ring, the artist sketches every piece as it is found.” She waved her arm. “And over there, our flashiest attraction—drum roll, please—the photographers shoot each shard of pottery from every possible angle. But where is the Master of Ceremonies?”

Alex and Rammy found Professor Jones near the back of the tent, talking to another student. While she waited for her dad, she looked through an assortment of numbered artifacts that were ready to be packed and sent to the museum.

Fibonacci Jones had discovered the tomb of a scribe, an ancient Egyptian secretary and bookkeeper. It didn’t look like much—merely a hole in the ground, way different from a pharaoh’s hoard—but Alex knew her father. To him, a roll of papyrus was a treasure more precious than jewels.

She picked up the scribe’s palette, a carved wooden block about the size of a box of watercolors. This palette had three bowl-shaped pits for ink. Two of these held dried-out remains of the scribe’s black ink, and in the last pit she could see a faint stain of red. Four thin reeds lay in a groove, ready for the scribe to dip in the ink and write.

(Styluses, she thought they were called. Or styli? Have to ask Dad.)

Alex looked up to see her father walking toward her. Strangely, he seemed to be moving in slow motion. He froze in mid-step and stood statue-still. When she opened her mouth to speak to him, no sound came out.

At her feet, Rammy whimpered, then fell silent. Alex tried to turn her head, but she couldn’t move.

What could be happening to them?

What Really Happened

The Alexandria Jones stories were originally written for my math newsletter. Since it was a one-person operation, when I got the flu, all production ground to a halt.

I recovered just before the next issue was due out, in time to send a one-page apology to my subscribers, promising them that they would still get their full number of issues. I wrote the vignette above for the apology letter.

My newsletter also included puzzles and games unrelated to the Alexandria Jones stories. Here is a word problem from the May/June 1998 issue:

Can You Solve This Story Problem Challenge?

You may estimate the pan of lasagna to be one square foot. An 11″×13″ pan is actually one square inch shy of that, but it won’t change the solution enough to matter—simply round your final answer to two significant figures.

(As any science teacher will tell you, you should never copy down all the numbers your calculator spews out.)

If you get stumped, my recent post on conversion factors might be helpful.

And in case you were wondering, InfoPlease Almanac lists the total area of Illinois as 57,915 square miles.

A Not-Quite Answer

I won’t give the solution away, but to put the answer in perspective:

• Columbus landed in the New World 5 centuries ago.
• Jesus of Nazareth lived 20 centuries ago.
• And about 46 centuries ago, the ancient Egyptians were building pyramids.

To eat one pan of lasagna in 3 seconds sounds like a speed-eater to me, but it still would take an unimaginably long time to eat a pasta the size of just one mid-sized state.

Isn’t the world an amazing place?

This game takes strategic thinking and planning ahead, and it makes a great journaling prompt, too.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Walter Joris’s Sequencium

Math Concepts: strategic thinking.

Players: two players or two teams.

Equipment: blank paper, colored pencils or markers.

How to Play

Draw a 6 × 6 grid of squares as a gameboard for players to share. Each player needs a different color pencil or pen.

Players begin by each writing the number 1 in opposite corner squares. Then take turns writing numbers until the grid is full.

On your turn, choose a square that touches any of your own previously numbered squares at a side or corner. Write the number that is one greater than the square you are touching. Draw a short line to connect those two numbers.

Your number sequence may branch and may cross the opponent’s line. Each branch continues the counting sequence from whichever square it touches.

If your new square is touching more than one of your numbers, continue counting from the higher number. Whoever reaches the greatest number wins the game.

Variations

Use the game as a journaling prompt. Here are some sample questions:

• What is your strategy for winning?

• Do you think this is a fair game, or does one player have an advantage?

• How would you count score, so you could compare your performance from one game to the next?

• How would you modify the game rules? Is your version easier or harder than the original game?

• Do you prefer logical strategy games or games of chance? Or do the best games have a bit of both? Explain.

Writing to Learn Math: When students create their own math, they forge a personal connection to mathematical concepts and relationships. And it’s fun!

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Create-Your-Own-Math Prompt

When students create their own math, they forge a personal connection to mathematical concepts and relationships. And it’s fun!

Children might make up a math game, write a story or poem, draw a comic, or pose a problem. Create math art, think up a challenging question, or write a puzzle. Since earlier chapters focused on writing and math art, most of these prompts involve creating puzzles or problems.

The “Story Problem Challenge” is one of my favorite math club activities. My students invent their own word problems in any style they like. They don’t have to know how to solve the problems they create. We read the stories aloud, and everyone works together to find the solutions.

For puzzles where the child already knows the answers (for example, “Two Truths and a Lie”), let them trade with a friend. Can they each solve the other’s puzzle? Can they stump each other? Or save the child’s work and let them come back to it another day, after they’ve forgotten the answers.

And when students create something they’re proud of, let them share it with the world. Visit the Student Math Makers Gallery at tabletopacademy.net/math-makers to share your children’s math creations.

Journaling Prompt 205: Counting Puzzles

If I count by twos, I land on both 100 and 1,000. If I count by threes, I don’t hit either 100 or 1,000.

Is there any number I can count by to hit only one of them?

Make your own counting-by puzzles.

“I cannot teach anybody anything, I can only make them think.”

“There is no education but self-education and only as the young student works with his own mind is anything effected.”

—Charlotte Mason

Two great teachers of the past agree that a child’s mind is no bucket. We cannot shovel in whatever information we please and call that “learning.”

So how can we get children to think, to work with their own minds, to build their own relationships with knowledge?

I try to answer that question, at least for math, in my new book Charlotte Mason’s Living Math, which is now available exclusively on my Playful Math Store.

As I was finishing up the final bits of indexing and formatting the book for publication, I continued to tweak the manuscript. This ability to tinker endlessly is both a benefit and curse of indie publishing. For example, I added four new pages to explain my quibble with Socrates.

You see, Socratic questions are very much a case of “Do what I say, not what I do.”

Not Telling, but Asking Questions

Traditional math lessons often sound like a lecture. A living math lesson sounds more like a conversation, and children often do most of the talking. We pose a conundrum, a puzzle, a problem for our students to investigate. Then they do the work of thinking it through, comparing it with what they already know, making sense of the math.

There’s very little of what we might call teaching—that is, telling our children what to do. Instead, we want to draw out what’s in their minds, the knowledge they already have, and help them see it in new ways or apply it to new situations.

In his classic book, How to Solve It, mathematician George Pólya writes:

“A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations, he kills their interest, hampers their intellectual development, and misuses his opportunity.

“But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.”

—George Pólya

Don’t Follow Socrates

One of the best ways to get children reasoning about math, and to help them when they get stuck, is to ask questions.

But we need to be careful about the questions we ask. Don’t focus on procedures with questions like, “What kind of problem is this? What is the first step?”

Almost everyone has heard of the value of Socratic questions. But when I read Plato, I find it irritating that so many of those questions seem to be statements in disguise, where the answer must be a simple Yes or No, and the “right” response is clearly whatever the teacher wants to hear.

For example, when Socrates purports to ask Meno’s young servant about math: standardebooks.org/ebooks/plato/dialogues/benjamin-jowett/text/meno.

An Excerpt from Meno

Socrates, Meno, and a slave boy are talking about the problem of doubling the area of a square. At this point, Socrates has demonstrated that the intuitive answer—“make each side twice as long”—fails because the new square is four times as large as the original.

Now he begins leading the boy to the true answer, scratching a diagram in the dirt that begins as a single square, expands it into a square of area 4, then doubles that larger square by using its diagonal as the side of a new one that has area 8.

Socrates (to Meno): Mark now the farther development. I shall only ask him, and not teach him, and he shall share the enquiry with me: and do you watch and see if you find me telling or explaining anything to him, instead of eliciting his opinion.

Socrates: Tell me, boy, is not this a square of four feet which I have drawn?
[This is the original small square in the image above, so perhaps by “four feet” he means the side lengths. Later, he refers to the medium square above as four (square) feet.]

Boy: Yes.

Socrates: And now I add another square equal to the former one?

Boy: Yes.

Socrates: And a third, which is equal to either of them?

Boy: Yes.

Socrates: Suppose that we fill up the vacant corner?

Boy: Very good.

Socrates: Here, then, there are four equal spaces?

Boy: Yes.

Socrates: And how many times larger is this space than this other?

Boy: Four times.

Socrates: But it ought to have been twice only, as you will remember.

Boy: True.

Socrates: And does not this line, reaching from corner to corner, bisect each of these spaces?
[The diagonal of the medium square.]

Boy: Yes.

Socrates: And are there not here four equal lines which contain this space?
[The largest square.]

Boy: There are.

Socrates: Look and see how much this space is.
[How big is the largest square?]

Boy: I do not understand.

Socrates: Has not each interior line cut off half of the four spaces?
[Each triangular quarter of the largest square is half of the medium square.]

Boy: Yes.

Socrates: And how many spaces are there in this section?
[The medium square.]

Boy: Four.

Socrates: And how many in this?
[The triangular half of the medium square.]

Boy: Two.

Socrates: And four is how many times two?

Boy: Twice.

Socrates: And this space is of how many feet?
[The whole large square is how many square feet?]

Boy: Of eight feet.

Socrates: And from what line do you get this figure?
[What is the side of this largest square?]

Boy: From this.

Socrates: That is, from the line which extends from corner to corner of the figure of four feet?

Boy: Yes.

Socrates: And that is the line which the learned call the diagonal. And if this is the proper name, then you, Meno’s slave, are prepared to affirm that the double space is the square of the diagonal?

Boy: Certainly, Socrates.

Socrates: What do you say of him, Meno? Were not all these answers given out of his own head?

Meno: Yes, they were all his own.

How to Teach with Questions

If we try to teach like this, we commit the pedagogical sin of funneling, as described in David Butler’s article “Twelve matchsticks: focus or funnel.”

Socratic questions are a valid literary device because the other characters in Plato’s dialogues are mere foils, existing only to show off the wisdom of the teacher.

Our students, however, are not foils but persons with valid ideas of their own. Therefore, try never to ask a question to which you already know the answer. Instead, ask open-ended questions that probe your child’s thinking, like:

• What do you notice?
• What do you wonder?
• Does the problem remind you of anything else you’ve learned?
• Can you describe what you’re trying to find?
• Is there anything you might try, even if you don’t think it will work?

And after you find a solution, that’s never the final end of the story. You can always look back and discover something more.

• Can you think of a way to check your answer?
• Could you do it differently? How many ways might we find to solve this problem?
• What if we changed something in the problem? How would that affect the solution?

“A great discovery solves a great problem but there is a grain of discovery in the solution of any problem.

“Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.

“Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.”

—George Pólya

“This book gives parents permission to believe that mathematics is something children can reason about, wonder about, discuss, investigate, and enjoy.

“It invites parents to treat children as people who have mathematical ideas worth listening to.

“For many parents, that shift in orientation will be far more valuable than another collection of lesson plans.

—Sonya Post

Review of Charlotte Mason a Living Math

This game builds visual thinking skills.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Symmetry Challenge

Math Concepts: symmetry, geometry.

Players: only two.

Equipment: graph paper or blank hundred chart, colored pencils or markers.

Set-Up

Each player will need a printed sheet of graph paper or a blank hundred chart. Draw a line that divides the page in half. Your line may be vertical, horizontal, or diagonal.

Or draw two lines that split the chart into fourths. With two lines of symmetry, the pattern will match in both directions: top reflects bottom, and left reflects right.

How to Play

With felt-tip markers or colored pencils, draw lines or fill in squares to make a pattern in one section (half or fourth) of your chart.

Trade papers with a friend. Can you complete each other’s puzzle? Whatever shapes you see on one side of a mirror line, draw their reflections on the other side.

Variation

Advanced Puzzle: Color shapes in different sections of the chart. Your shapes should not match each other symmetrically, but when all of them have been reflected across the line(s) of symmetry, they will create a finished design.

Writing to Learn Math: Writing helps students stretch their thinking and make sense of new ideas.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Writing Prompt

Writing helps students stretch their thinking and make sense of new ideas.

When students wrestle their thoughts into shape and create explanations, they do the same sort of work that mathematicians do every day. It’s difficult for children (or anyone) to capture a thought and cage it in words. But it’s great practice for life.

Students may supplement their writing with illustrations. Sketch drawings can be a wonderful aid to mathematical thinking.

For the poetry prompts, students should aim for evocative descriptions, vivid verbs, and playful words. If your child can’t think of where to start a poem, try brainstorming a list of sensory details.

You may reuse writing prompts as often as you like. Change the question, if you wish — but even when the prompt remains the same, the students have changed since the last time they wrote about it. Today is a new day, so they are seeing with fresh eyes and thinking different thoughts.

Journaling Prompt 277: Words Help Us Think

Make a math “word wall” on your journaling page. Write all the math vocabulary words you know.

Decorate them with frames, or write them crossword-style or in different directions. Or make it fancy any way you like.

Optional: Include definitions or pictures explaining the hardest-to-remember words.

It’s a great topic because the calculations are relatively easy, mostly multiplication and division. But there’s a lot of thinking that goes into figuring out just what calculations we need to do.

Counting is not only for little kids! It’s part of discrete mathematics, which means the study of things that are not continuous—things that come in discrete chunks—and it’s important in game design, computer programing, and cryptography.

This post is a section of my Prealgebra & Geometry Games book where my daughter and I explore the fundamental counting principle, the key to solving many counting and probability puzzles.

The Science of Counting

We spent a bit of time counting basic numbers and counting items in sets. Then we moved to the more interesting type of mathematical counting. Combinatorics is the art of counting possibilities: all the choices we can make, the things we may do, the ways we might combine stuff.

For example, imagine you are choosing your avatar for a new online game. You need to pick an outfit, and you have the following options:

• Leggings: brown or black, tucked into high leather boots.
• Top: green, purple, or blue.
• Cloak or no cloak.
• Head piece: hat, scarf, eye patch, or none.

Focus on the first two choices, the leggings and top. Imagine these choices as paths through a mental landscape, branching off each other to form a tree diagram. Your first option creates two paths: brown leggings, or black leggings. Each of these paths leads to a new question: Which top? At that point, you must choose between three branching paths.

The total number of branches in a tree diagram is the product of the choices at each branch point. In this case, 2 × 3 = 6 possible combinations.

The Fundamental Counting Principle

As we add more choices, the paths continue to branch out. Our tree diagram grows unwieldy, and we desperately need a new counting tool.

The fundamental counting principle comes to our rescue. We can count the possibilities of a situation by multiplying our options.

If you have A options for your first choice, B options for your second choice, C options for your third choice, and so on, then you can find the total number of potential outcomes by multiplying:

A × B × C ×…

To choose our game avatar, we have to make four choices—leggings, top, cloak (or not), and head piece—with two, three, two, and four options respectively. So we calculate the total number of possible outfits:

2 × 3 × 2 × 4 = 48

For More Information

We spent a whole year in middle school playing with the science of counting. Here are two of my favorite resources:

• Keone Hon’s lesson on counting and probability
• Jason Batterson’s puzzle sets

Batterson later published his puzzles in the book Competition Math for Middle School (affiliate link), which includes solutions.

This game builds understanding of decimals, and also of the inverse relationship between multiplication and division.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Decimal Product Game

Math Concepts: multiplying decimals, decimal place value.

Players: two players or two teams.

Equipment: printed game board, colored markers or a set of matching tokens for each player, two paperclips or other small tokens to mark the factors.

Set-Up

The free 44-page PDF Multiplication & Fraction Printables file features two decks of mathematical model playing cards, plus hundred charts and all the game boards for the Math You Can Play: Multiplication & Fractions book.

How to Play

The first player places a paperclip on any one of the factors at the bottom of the board. The second player places the other paperclip on a factor — the same or different — and then marks the product of those two numbers by coloring the square or placing a token.

On each succeeding turn, a player moves just one paperclip to a new number and then marks the product of those two factors. If both players agree that all possible moves have already been colored in, the player whose turn it is may make a fresh start by moving both paperclips.

Whichever player marks four (or more) squares in a row — horizontally, vertically, or diagonally — wins the game. The squares must touch each other at edges or corners, with no gaps. If neither player can make four in a row, then the player who has the most sets of three in a row wins.

History

This variation on The Product Game comes from John Golden’s blog. He writes about his conversation with fifth-grade students:

“We discussed how to get hundredths. I think they knew on some level it was tenths times tenths, but had a bit of the ‘we haven’t been taught this yet’ syndrome. I used the analogy of a dime being a tenth of a dollar, so what’s a tenth of a dime?

“‘A penny!’

“And what part of a dollar is a penny? How many does it take to make one dollar? It is terrific that they are used to seeing one cent written as .01.”

Writing to Learn Math: At its heart, geometry is all about seeing connections and relationships.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Geometry Prompt

At its heart, geometry is all about seeing connections and relationships. How can students break shapes apart, put them together, move them around the page, turn them, or distort them? Which properties change, and which stay the same?

Every activity has the potential to spawn hundreds of variations. Alter something in the prompt to make a fresh investigation. Tweak the size, shape, or other properties of interest. What new things can your children see in the math? What questions can they ask?

For older students, use algebra to put some teeth in the relationships they see. Give the points names. Identify the line segments. Can your students write any equations about them? Which distances are equal to other distances, or areas equal to other areas? How can they know for sure? When they add new points, lines, or circles to the diagram, what new connections do they find?

Journaling Prompt 217: Pentagram

Draw a pentagon (five-sided shape). Draw diagonal lines inside the pentagon so that each corner is connected with a line to every other corner.

The diagonals make a shape called a pentagram.

What do you notice? What patterns do you see?

Where might you add new lines to the shape?

How many questions can you ask about this shape? Can you figure out the answers to any of your questions?

Hooray!

If you’d like to know more about the book, check out my interview with Cindy Rollins on The New Mason Jar podcast:

• S11E144: Charlotte Mason’s Living Math with Denise Gaskins

…and here’s the book description:

Apply the Principles of Education to Help Children Develop Mathematical Reasoning

“The Principality of Mathematics is a mountainous land, but the air is very fine and health-giving. People who seek their work or play in this principality find themselves braced by effort and satisfied with truth.”

—Charlotte Mason

Are you a parent or teacher? Here’s how to:

• Feed your children’s minds with big ideas.
• Give them confidence they can learn and make sense of things.
• Develop intuition and creative thinking about math.

Explore the world of mathematics with your children. Notice connections. Wonder about relationships. Delight in understanding.

Author Denise Gaskins shows you how to introduce your child to the living ideas of mathematics. Because when the ideas of math come to life for our children, their minds find joy in seeing numbers and shapes connect to each other and exploring these relations.

From early learning to algebra, Charlotte Mason’s Living Math offers a practical, activity-filled exploration of what it means to learn math as a family.

Chapters include:

• Practice Your Principles: Discover the fundamental principles of math education, the essentials of understanding who children are and how they learn.
• How Shall We Teach?: Get practical tips for guiding your kids, drawing out their curiosity with puzzles, and helping them master math topics.
• When Students Get Stuck: Help children find their own way around any barrier with a classic problem-solving approach they can use throughout their lives.
• Supplement Your Lessons: Find out how books, games, and math art contribute to a living education.
• And much more.

You’ll love how Charlotte Mason’s Living Math transforms your children’s experience of math, awakening their imagination about numbers, shapes, and patterns. Because the Principality of Mathematics is a wonderful place to explore with your kids!

So grab your copy of Charlotte Mason’s Living Math today, and launch your family’s adventure of learning.