Tag Archives: number sense

Recreational math

A few weeks ago I saw an article in the New York Times, “How to Fix Our Math Education.” The authors suggest replacing the theoretical teaching of math with applicable math. “The truth is that different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact,” mathematicians Sol Garfunkel and David Mumford wrote in their op-ed piece.

Most people never need to solve the quadratic formula or use complex numbers, so why do we spend so much time teaching so many children math skills they will never use? Sounds rational.

The authors suggest “teaching relevant problems that will lead students to appreciate how a mathematical formula models and clarifies real-world situations.” Again, entirely rational and reasonable. Perhaps if people understood math better on a relevant, everyday basis, fewer people would have signed mortgages they could never pay. (I have no proof of this, but I can’t help fantasizing about a couple ridiculing a mortgage broker proposing ludicrous terms.)

Kari Kling teaching a Math Mania class in her home.

I started teaching in 1993 at Aztec Elementary School in Scottsdale. One of the lead teachers responsible for designing the school, Kari Kling, is an expert at making curriculum relevant to students based on how their brains develop. She understands that awareness of the brain development in children should be driving what is presented in the classroom.

In 1997 she published a book called It’s Not About Math, It’s About Life. It relates math to the real world, where it belongs. The authors of the Times article would probably approve of her approach.

Kling says, “We use the language of math to communicate about something in real life. Numbers aren’t just random, they stand for things.” She teaches a class called Math Mania “to teach kids about numbers as a foundational piece so they understand math when it gets more complicated. Kids need to have real experiences with numbers. If they hear something weighs 14 pounds they need to know what 14 pounds feels like.”

When I taught third and fourth graders, we were learning about the Titanic. They read that the doomed ship was 882 feet long. I asked if that was bigger than a football field. Some said yes, some said no. I told them a football field was 300 feet, goal line to goal line. Silence. One kid who understood numbers said, “The Titanic is almost three times the size of a football field?” He was stunned. I asked if they thought the Titanic would fit on our playground. Some said yes, some said no.

Dice for generating sums.

The next day I brought my 100-foot extension cord to measure the playground. By the time we had got to 882 feet, they had a clear sense of how big the Titanic was. Teachers call this number sense and it is what Kling focused her second and third graders on. What good is manipulating numbers in an algorithm if there is no understanding what those numbers mean?

I went to Kling’s house a few weeks ago to watch her teach a Math Mania class for second and third graders. She had the kids come up with as many ways as they could to add two numbers to make 10 as a short cut to adding large columns of numbers. She calls this “bridging to 10.” It yielded a pattern as pictured in the photo at left. Hopefully the student will see the numbers on the left go from 1 to 9 and the numbers in the middle are 9 to 1 and apply this pattern for any sum.

Then they rolled two or three dice and added the numbers to discover what combinations the dice yielded. They kept track of the sums on a chart. In a subsequent class they would analyze which sums came up the most often, which pairs of numbers came up and why.

The kids were all doing math but they were playing with numbers and playing with dice and writing about the activities in their journals. It wasn’t more “kill and drill” of worksheets or page after page of problem sets. They were playing with numbers in their heads, on paper and with other kids.

There are parents who think math doesn’t need to be fun, it has to be learned. But these are kids, and kids — and adults, really — remember things that are fun because so many parts of their brains is being stimulated all at once.

Plus, it’s better to think of math as fun than as a chore.

Story and photos by Daniel Friedman

Students keep track of what number combinations and sums they roll.



Numbers, sensibly

10 CheeriosTeaching new skills and concepts depends on understanding what the student already knows. This applies to adults as well as children. When kids enter school and learn to read they need to know their alphabet and have an inkling that letters correspond to sounds. Most kids learn the alphabet by singing the ABC song as well as seeing letters in books, on signs, on television, etc.

Math is the same way. Before kids learn 1+1=2, they need to know what numbers are and what they represent. They must develop number sense, which is the understanding that numbers relate to the real world. Young children have number sense when they know that three people can fit in their family car, but 30 could not.

A study by David Geary at the University of Missouri confirms scientifically what parents and teachers know: that number sense is as essential to learning math as the alphabet is to learning reading. The study followed 177 elementary school kids for five years to track the differences in progress amongst the group of elementary school aged kids.

press release about the study reports, “A long-term psychology study indicates that beginning first graders that understand numbers, the quantities those numbers represent and low-level arithmetic will have better success in learning mathematics through the end of fifth grade, and other studies suggest throughout the rest of their lives.”

Number sense is part of the standards as set by the Arizona Department of Education for first grade but number sense is part of a youngster’s life when parents count things and play games with quantities.

When I taught third, fourth and fifth graders, I still taught number sense, though with a more advanced perspective than one would for a preschool student. When teaching multiplication, I needed to make sure my students understood that multiplying two numbers in the hundreds would necessarily be in the tens of thousands or in the millions — not just thousands. Number sense bolsters estimation skills also.

From the National Council for the Teaching of Mathematics: “Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts and relating them in ways that are not limited by traditional algorithms (Howden, 1989).”broccoli5

Hmmm. That’s a complicated way of saying kids should know that the five pieces of broccoli they don’t want to eat is four pieces of broccoli after they manage to choke down one piece of broccoli. 5-1=4 means very little to children without number sense, even though they may be able to count on their fingers and get the answer right on a worksheet. Kids understand math when they have number sense.broccoli4

Before you rush out to find a cool number sense smartphone app, remember, number sense is a real-world, three dimension experience, not the flat screen of a smart phone or computer. Numbers count things. With your child, count the number of Cheerios that fit on a spoon or in a cereal bowl, or discover what 100 Cheerios looks like. Or count all the socks that have no mates. In the car count the number of orange cars compared to the number of white cars.spoon full of cheerios

“Cognitive Predictors of Achievement Growth in Mathematics: A Five Year Longitudinal Study,” will be published in the journal Developmental Psychology. But really, you don’t need to read it. — Dan Friedman