Making Sense of Arithmetic

Making-SenseHomeschoolers have an advantage in teaching math: As our students grow, our own understanding of math grows with them because we see how the ideas build on each other.

This is especially true for those of us with large families. We pass through the progression of concepts with each student, and every pass lays down another layer in our own minds.

If you’d like to short-cut that process, check out Graham Fletcher’s Making Sense of Elementary Math video series. He’ll walk you through the topics, showing how manipulatives help build early concepts and gradually give way to abstract calculations.

“Understanding the vertical progression of mathematics is really important in the conceptual development of everyone’s understanding. This whole Making Sense Series has truly forced me to be a better teacher.”

— Graham Fletcher

Continue reading Making Sense of Arithmetic

FAQ: Trouble Finding the Right Math Program

“I can’t find a home school math program my son likes. We’ve tried Singapore Math, Right Start, Saxon, and Math Mammoth. We subscribed to a month of IXL Math to keep him in practice, but he hates that, too. I know I shouldn’t have changed so many times, but this was our first year of homeschooling, and I was trying to please him. But I’m running out of things to try. Do you think Life of Fred might work?”


You’ve tried all those math programs in one year? Many people recommend that new homeschoolers take a few months off to “detox” from the classroom setting, to relax and enjoy the freedom of making their own choices. But your son might want a few months to detox from his homeschool experience.

I suggest you set aside all those books and focus on games and informal math. Try to avoid schoolish lessons until your son starts to enjoy learning for its own sake. The Internet offers an abundance of creative math ideas.

  • For example, download the Wuzzit Trouble or DragonBox apps to play with, but don’t make it a homework assignment.
  • Or let him choose one of the activities at Gordon Hamilton’s Math Pickle website and explore it for a day or a week or as long as it remains interesting.
  • Browse through the Primary Level 1 or Level 2 puzzles and games at the Nrich Mathematics website for more ideas.

Look for more playful math on my blog’s resource pages:

Explore Big Concepts: Infinity

Math that captures a child’s imagination can make the more tedious work seem bearable. For instance, in the 1920s, mathematician David Hilbert created a story about an imaginary grand hotel with an infinite number of rooms.

Explore Big Concepts: Fractals

Sierpinski-tortillasTake a mental trip to infinity by playing with fractals. Cynthia Lanius’s online Fractals Unit for Elementary and Middle School Students offers a child-friendly starting point.

Fractals are self-similar, which means that subsections of the object look like smaller versions of the whole thing.

Most children enjoy exploring the concept of infinity with hands-on fractal patterns, such as this Sierpinski triangle made of tortilla chips. Talk about what you notice and wonder: How does the triangle grow? How many chips will we need for the next stage?

The Daily Four

If you worry that your son needs to keep practicing traditional arithmetic during his break, try making him a series of Daily Four pages:

  • Fold a sheet of plain paper in half both ways, making four quarter sections.
  • Write one math problem in each part. Choose them from any of your math books.
  • Make sure each problem is different — one addition, one fractions, one multiplication, or whatever — and that none of them are hard enough to cause frustration.
  • Don’t worry about an answer sheet. Show him how to use a calculator to check his work.

You can make up a whole week’s worth of these problem sheets at once, with a balanced mix of problems for each day. Your son won’t feel overwhelmed, but you’ll know he’s reviewing his number skills.

Or download some of the Corbettmaths 5-a-Day practice sheets for him. Some problems may seem too easy while others require concepts he hasn’t studied yet. Easy review won’t hurt anything, but do let him skip the problems that feel too hard.


Credits: “Rock Surfer Boy” by Ken Bosma and “Boy” by Isengardt via Flickr. (CC BY 2.0) Hotel Infinity video by Tova Brown.

Click for details about Let's Play Math bookThis post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, as are many of the articles in my Let’s Play Math FAQ series.

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FAQ: He Won’t Stop Finger-Counting

“My oldest son has somehow developed the horrid habit of counting on his fingers. We worked on the math facts all summer. He knows the answers in simple form, such as 9 + 4, but if it’s in a bigger problem like 249 + 54, he counts up to add or counts down to subtract, all using fingers. My younger children have no problem with mental math, but he can’t seem to get it. Are there any tips or tricks to stop this?”

New Crutches

Counting on fingers is not a horrid habit, it is a crutch. Please think for a moment about the purpose of crutches. The blasted things are an uncomfortable nuisance, but there are times when you can’t get anywhere without them. And if you need them, it does you no good for a friend to insist you should crawl along on your own.

That is how your son feels right now about his fingers. He is struggling with something his younger siblings find easy, and he can tell that you are frustrated. His confidence is broken, in a cast, and needs time for healing. So he falls back on what he knows he can do, counting up the answer.

Think positive: this means he still believes that math ought to make sense — that to understand what he is doing is more important than to guess at an answer. You want him to value sense-making, because otherwise he will try to memorize his way through middle school and high school math. That is the road to disaster.

“Schools spend a lot of time working with young children to get these facts memorized, but many children aren’t ready for that task yet. They’ll count on their fingers, and may be reprimanded for it.
“What happens when a person becomes embarrassed about counting on their fingers? If they still want to think, they’ll hide it. That’s the better option. The worse option that way too many students choose? They start guessing. When math becomes too incomprehensible, or not living up to someone else’s expectations becomes too painful, many students give up on math, and then they just guess.
“We count on our fingers as part of a thinking process. Perhaps the thing I want to figure can be memorized. But if I haven’t memorized it yet myself, the most efficient way to figure it will likely involve fingers.

—Sue VanHattum

The Problem of Transfer

What you describe is called the problem of transfer, and it is one of the huge, unsolved problems of education.

We can train someone to do a simple, limited task such as answering flash cards. But how do we get that knowledge to sink in, to become part of the mind, so they can use it in all sorts of different situations?

No one has figured that out.

There is no easy solution. It requires patience, and providing a variety of experiences, and patience, and pointing out connections, and asking the student to think of connections, and lots more patience.

Some Things to Try

It might help to do fewer math problems in a day, so you can take time to work more deeply on each one. Talk together about the different ways you might solve it. Make it a challenge: “Can we think of three different ways to do it?”

In math, there is never just one way to get a solution. Thinking about alternatives will help your son develop that transfer of skills.

Or pick up some workbooks that target mental math methods. The Mental Math workbook series by Jack Hope and Barbara and Robert Reys will help him master the techniques your younger kids learned without effort. It may still take him longer to do a calculation than what you are used to with the other children, but these books will give him a boost in recognizing the types of mental tools he can use.

Here are a few of my previous blog posts that include mental math tips:

Or perhaps encourage him to keep using his fingers, but to switch to a more efficient system, such as Chisenbop. According to math education expert Jo Boaler, research shows that finger-counting supports mathematical understanding.

Mental Math: A Battle Worth Fighting

Jumping into mental math is hard for an older child who wasn’t taught that way. I believe it’s a battle worth fighting, because those mental math techniques build understanding of the fundamental properties of numbers.

But the main goal is for him to recognize his options and build flexibility, not to do each calculation as fast as possible.

And be sure he no longer needs those crutches before you try to take them away.


Photo credits: “Stryde Walking To School on his New Crutches” by Jim Larrison and “Silhouette of a boy” by TimOve via Flickr. (CC BY 2.0)

Click for details about Let's Play Math bookThis post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, as are many of the articles in my Let’s Play Math FAQ series.

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Playing with Math Shapes

Playing-with-shapesI love it when a plan — or rather, a series of math thoughts — comes together.

On Monday, Emily Grosvenor (author of the Tessalation! picture book) asked me how parents who are insecure in math could help their children learn through play, and I responded with this quote from my Let’s Play Math book:

If you are intimidated by numbers, you can look for patterns of shape and color. Pay attention to how they grow. Talk about what your children notice.

But I wasn’t entirely satisfied with that answer. So many adults have come away from their own school experience thinking math is only numbers. Even with shapes, isn’t it the numbers about them — how many sides, what size of angles, calculate the the area or perimeter — that are important? That’s what school math tends to focus on.

Those of us who are comfortable with math know that there are many more things to notice and think about than just numbers. We know that it’s this noticing, thinking, and wondering that is at the heart of math. And that just playing with shapes can build a powerful foundation for future math learning.

And then yesterday, Malke Rosenfeld posted a beautiful article about a paper manipulative created by Paula Krieg. Which included this video:

The ability to create, and maintain, and manipulate shapes mentally — that’s the goal. Just like kids who can put numbers together in their heads, kids who can rotate, flip, and think of how shapes fit together in their heads have a powerful tool to analyze not only simple shape puzzles, but dividing up an area that’s a more complex room shape … to look at a piece of artwork … or look at a building … For these kids, all the world around becomes a playground to use mathematical ideas.

— Doug Clements
Problem Solving Development: Composing Shapes

Of course, pattern blocks are good for much more than just filling in worksheet pictures. But I love this peek into how a child’s understanding grows, in bits and spurts — without any numbers at all — until the world itself becomes a playground for mathematical ideas.

Want more?

You know what? Children like mathematics. Children see the world mathematically … When we do a puzzle, when we count things, when we see who’s got more, or who’s taller … Play and mathematics are not on opposite sides of the stage.

— Doug Clements
Why Early Childhood is the Right Time to Start Learning Math

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Memorizing the Math Facts

Central City Times Tables[Photo by dsb nola via flickr.]

The most effective and powerful way I’ve found to commit math facts to memory is to try to understand why they’re true in as many ways as possible. It’s a very slow process, but the fact becomes permanently lodged, and I usually learn a lot of surrounding information as well that helps me use it more effectively.

Actually, a close friend of mine describes this same experience: he couldn’t learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can’t remember 9×5, but I know 8×5 is half of 8×10, which is 80, so 8×5 must be 40, and 9×5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college and is now a cancer researcher who deals with a lot of statistics.

Ben Blum-Smith
Comment on Math Mama’s post What must be memorized?

The entire discussion (article and comments) is well worth reading:

You may also enjoy:

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March 2016 Math Calendars

Once again, a few of my favorite bloggers have come through with math calendars for our students to puzzle over. Check them out:

algebra calendar

Things to Do with a Math Calendar

At home:
Post the calendar on your refrigerator. Use each math puzzle as a daily review “mini-quiz” for your children (or yourself).

In the classroom:
Post today’s calculation on the board as a warm-up puzzle. Encourage your students to make up “Today is…” puzzles of their own.

As a puzzle:
Cut the calendar squares apart and trim off the dates. Then challenge your students to arrange them in ascending (or descending) order.

Make up problems to fill a new calendar for next month.
And if you do, please share!

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Active Math Game: Rock

Gordon Hamilton of Math Pickle posted Rock, a new active math game for grades K–2. If you have a set of kids and a few minutes to spare, give it a try!

How to Play Rock

  • Everyone makes a rock shape with eyes closed.
  • Everyone chooses a number: 0, 1, 2, 3, 4, 5, 6, 7, 8 …
  • Teacher calls out numbers consecutively, starting at 0.
  • When a student hears their number being called they immediately raise a hand. When the teacher tags the hand, they stand up.
  • If more than one hand was raised, those students lose. They become your helpers, tagging raised hands.
  • If only one hand was raised, that child wins the round.


“Each game takes about 45 seconds,” Hamilton says. “This is part of the key to its success. Children who have not learned the art of losing are quickly thrown into another game before they have a chance to get sad.”

The experience of mathematics should be profound and beautiful. Too much of the regular K-12 mathematics experience is trite and true. Children deserve tough, beautiful puzzles.

Gordon Hamilton

What Happens When Grownups Play Rock

What are the best numbers to pick? Patrick Vennebush hosted on online version of the game at his Math Jokes 4 Mathy Folks blog a few years back, though we didn’t have to bend over into rocks‌—‌which is a good thing for some of us older folks.

Vennebush also posted a finger-game version suitable for small groups of all ages, called Low-Sham-Bo:

  • On the count of 1-2-3, each person “throws” out a hand showing any number of fingers from zero to five.
  • The winner is the person who throws the smallest unique number.

You may want to count “Ready, set, go!” for throwing out fingers, so the numbers in the count don’t influence the play.

The official name for this sort of game is Lowest Unique Bid Auction.

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