Real World, Unreal School

When I walked into Masha Albrecht’s geometry class at Berkeley High School last week, her students were holding hands. It wasn’t budding romance. It was math. Before I explain, I have to tell you about Masha.

I met her when I was a senior at Cornell in the 1970s. I decided to do an independent study that involved working with students in an elementary school classroom and she was the most eager, bright-eyed fourth grader in the class. Masha’s brother Bobby was in an adjacent room and the two classes were team-taught. I hit it off with both siblings, and before the end of the semester I had been to their house several times, met their parents, and spent some enjoyable after-school hours together—doing math. The three of us had delved into Harold Jacobs’s masterful Mathematics: A Human Endeavor, then in its first edition, and we tackled the mind-stretching, often funny, problem sets with gusto. “This is how math should be taught in school,” exclaimed Mrs. Albrecht. [Read more…]

Trick or Treat… and Teach!

As a child, I was filled with awe as I contemplated ordinary things. With their enormous numbers, sizes and distances, the stars amazed me. One day, visiting a museum with my parents, I focused on much closer, much smaller objects, looking through a microscope at hairs from my head and tiny creatures in a drop of pond water. I wondered how many hundreds or thousands of them would be as big as a flea! My mother and father were not scientists or mathematicians, but they encouraged me to ask questions like these, to find math everywhere.

Now, as a math author and educator, I have traveled the world speaking to children, teachers and parents. I’ve noticed that the kids who are successful in math are often those whose parents have helped them find mathematics at home and in family activities. These families know that it only takes a moment to make math meaningful.
– David Schwartz

Costumes … pumpkins … candy … math! All can be part of Halloween fun. With their hard-earned loot spread on the kitchen table, children see caloric pleasure, but parents can look at the same piles of sweets and see Math Moments.

The brightly wrapped candies can become “manipulatives,” or math props that parents can serve to their children as calorie-free math learning treats.

Children usually need no prompting for the math to begin. The first thing they usually do, after admiring their pillage, is to sort the jumble of candies into those they like and those they could live without. Parents can encourage other kinds of sorting by asking them to group candy by various characteristics. Sorting is an important early math skill that involves seeing similarities and differences, and classifying things according to characteristics – a mode of thinking used in higher-level math.

At age 3, Chaney Grover can find the square and round candies in his heap and sort them into two piles. By next year’s Halloween, he may be able to add a few more shapes to his sorting vocabulary – rectangles, ovals and cones, perhaps – and a year or two later, he’ll start to sort by multiple characteristics: “Do we have any candy that is round and brown?” his parents might ask. Or, “Can you find something cone-shaped and silver?” Or, even harder,“What is yellow, but not round?” The many characteristics of packaged candy can lead to impromptu games. “How are these two candies similar? Different?” Parents can invent a 20
Questions-type game: Parent: “I’m thinking of a candy. Find out
which one.”
Child: “Is it a square?”
Parent:“No.”
Child: “Is it orange?”
Parent:“Yes.”
And so on.Then switch roles.
Of course, children will probably
be more concerned with quantity
than qualities, which can lead
to counting and computation.
Chaney can count to 10 but,
like many 3-year-olds, when he
counts objects, he doesn’t always
start at one. Using candies, his
mom Meliss helps him get the idea.
She puts down three candies and
Chaney tells her how many. She
adds two more and Chaney starts
the count over again from one.
With practice, he will see short
cuts, like “counting on” (starting
from the previous total). Later, he will
simply add: three candies plus two candies
make five candies.
Children do not learn these math
basics by memorizing rules. Practice gives
them the chance to discover for themselves
how numbers work, and discovery
means understanding.
When more than one sibling is counting
his or her own sweet treasure, comparisons
(and boasts) are inevitable: “I
have nine Butterfingers™, and you have
only four!” Don’t let such an opportunity
get away without a Math Moment:“So
how many more do you have?”
If that’s too easy, try phrasing it another
way:“How many more would Sabrina
need to have as many as you?” It may seem obvious that the two questions are
equivalent, but children can draw blanks
when a simple question is posed in an
unfamiliar way.
Munchy Manipulatives
Daniel Pascal, 5, is very good at
adding small sums, and he has decided to
view his haul by lining up each kind of
candy in a separate column, like a bar
graph. He thinks this will make it easier
to compare quantities, but he soon sees
that a taller column doesn’t necessarily
mean more candy.The height of the column
depends on the size of the package,
not just the number of pieces, so he
pairs the candies side-by-side across the
columns. Now it’s easy to see that there
are more Dove™ chocolates than
Snickers™ bars, even though the line of
Snickers bars is taller.
Though Daniel can count to 100, he’s
a little shaky on some of the larger number
names. His dad, Steven, sees a chance
to give him some practice:“How many
candies do you think you have?” Daniel’s
guess is 33.
“OK, let’s count.” With a little help on
the harder transitions (49 to 50, 59 to
60) he completes the count.There are 70.
His father takes it one step further:“If
you can eat two a day, how long will it take
to eat all of these?” To answer the question,
the candies can be used as manipulatives,
taking two at a time to represent a
day’s ration. It’s a challenging problem for a
5-year-old, but Steven Pascal has always
challenged his kids with math.
Five years earlier, when Daniel’s older
sister Emily was 6, Steven took her to
inspect a new wing of their house, then
under construction. Referring to the
architect’s plans, they used a measuring
tape to find the future location of doors
and windows. Steven remembers telling
Emily that the doorknob would be
halfway up the door. He asked her to
mark the exact location of the doorknob.
Having been nurtured on a steady diet
of real-life math, Emily has excelled in the
subject at school. Last summer, she participated
in an intensive four-week math and
science program for middle school girls.
Math with Jack
You can find Halloween Math
Moments even before the first treat falls
into your child’s sack. Pumpkin-carving
gives children a chance to guess (and then
measure) their pumpkin’s weight and
dimensions. One mother asked her three
kids to cut a piece of string just long
enough to go around the pumpkin at its
widest place. All three kids cut strings
that were way too short. (Try it with any
spherical or cylindrical object.) This led to
a discussion of circumference,
the distance around a circle or
sphere.
She also had the children
guess how many seeds were in
their pumpkins, and once the
seeds had been scooped out,
they refined their guesses.They
divided the seeds scooped
from each pumpkin into four
equal piles, counted the number
of seeds in one- fourth of
a pumpkin, and used that
amount to estimate the total.
Then the 10-year-old wondered
aloud, “Does a heavier
pumpkin always have more
seeds than a lighter one?”
They got out the scale and weighed
their pumpkins and try to answer the
question with data they’d collected.They
were not only thinking mathematically, but
acting scientifically. And it all began with a
few simple Math Moments.

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