King's X From the Math Monster
"What about times tables?"
The fascination some parents have with the magical power of knowing times tables amazes me each time I hear that question.
I stopped counting up around the fiftieth time.
"Do your kids do times tables?"
What does "do" mean in relation to a table? Discover it? Too late. Write one out? This "table" of which they speak is a block of data, a pattern of numbers, involving relationship and function. What's to do? It's done.
My kids can "do it."
Kate Stavisky wrote:
True confession: I was hung up on the "times tables" issue a couple of years ago, and I've been learning to make my peace with it, when this story happened.
We decided to start giving our 6 year old a $5 a week allowance last month. I showed him where I would put an "x" on the calendar every Friday, so we'd know if I skipped a week by accident. He asked me what would happen, then? I told him that I'd make up the difference -- that is, if I skipped 1 week, I'd give him $10 the next week, or if I skipped two weeks, I'd give him $15. I might have said, you know, because 5 + 5 is 10, and 5, 3 times, is 15, but I certainly didn't harp on it. He thought for a minute and then told me that if I skipped 4 weeeks, I'd have to give him $20, and if I skipped 5 weeks, $25 . . . and so on up to 35 or 40. I was astonished . . . told him that was exactly right, and how did he know that? He said he learned counting by 5s from the "halftime quiz show" section of the PBS Kids show "Fetch with Ruff Ruffman." (So actually I guess this is a story about learning being everywhere, too . . .)
That came by e-mail, and I wrote Good story! Do you want me to add it to the page? I can... —Sandra
Or how about this one that happened moments ago? I said it was time to
leave in 10 minutes. He said, is that 600 seconds? I blinked at him. Yes
. . . I couldn't help myself from asking him how he knows that. 5 minutes
is 300 seconds, he said, and 3 and 3 is 6, so 300 and 300 is 600 . . . I'm
just amazed! I love unschooling!!
Pam Sorooshian wrote:
I would like
to talk about the idea of unschoolers fostering interest in math by
utilizing commercial math programs.
I also love math and have at least one child with a lot of aptitude
for it who has enjoyed it, too, since she was little. I teach college-
level economics (which includes a fair amount of math) as well as
statistics and I've been tutoring high school-age kids in math. I am
very very aware of what these older teens can do and what they
understand in math - after many years of taking math classes. They
can "do" math if it is presented to them in a textbook format. They
have very minimal understand of what they are doing, why it works,
why they'd want to do it. They can't generalize the techniques to
utilize them when an appropriate situation presents itself and they
can't combine techniques to solve more complex problems.
As an unschooler, I would seriously warn people away from supporting
their young child's interest by providing math programs or textbooks.
I very honestly feel it is not a good idea for kids to use any kind
of program to memorize multiplication facts, for example, and I mean
that especially for those kids who have a natural interest in
mathematics! YES, support their interest, but do it through
conversation, games, experiences, responding to their questions as
they arise. Relax and let them develop conceptual understanding
slowly, over time. Don't encourage them to memorize anything - the
problem is that once people memorize a technique or a "fact," they
have the feeling that they "know it" and they stop questioning it or
wondering about it. Learning is stunted. I've talked to a lot of
people who say they didn't understand until they were adults that
multiplication was just a short-cut way to do addition or division
was a short-cut way to do subtraction. They thought of multiplication
and division as "facts" - just something to memorize and use. The
biggest danger is that thinking of math like that, as a series of
things to memorize and use, means that kids/people don't expect it to
all make sense, they don't expect to understand why and how things
work. And it soon begins to seem normal to them that they don't fully
grasp math, that they feel it is beyond them to truly understand it.
"Doing math" without understanding what they are doing is very
discouraging and people frequently report that they spent years like
they were going to be "found out," or it would "catch up to them,"
eventually. Even students who got good grades will later admit that
they felt like fakes and it was very unpleasant. This is one of the
most common sources of the epidemic of math anxiety we have in this
I'm beginning to think that some people's hang ups with learning things like multiplication tables is part of a much bigger picture than just worrying that our kids will "get behind". I think a lot of it is fear of change. They did things a certain way, learned things a certain way, so that "seems" like the best way. (Even though they had no choice to do it any other way, and were probably punished if they tried.)
I think some people think we unschoolers are trying to change the shape of the box, the box that has been a part of their lives for so long and is so comfortable that it scares them when they think someone is trying to change it. (Even though they were forced to get into the box.)
I think they fear that THEY will be the one's that are left behind. What we do and the way we live are confusing to them, and goes against everything they've ever been told. They fear it because they don't understand it. It's not part of the box.
A relative told me once that we are just trying to "make waves." I replied that we're not trying to make waves, we just want to stay on top of the ones that are already there.
We don't make waves, we just ride them for all they're worth.
I just had to share what happened today! My ds is 7, so…you know, the “multiplication table” age lol, and we unschool. Sometimes I worry about learning math since we don’t use curriculum or do seat-work and don’t really have structure. This morning we had such a great unschooling math moment that my fears have temporarily diminished!
He was watching the movie Matilda that we have on video and at the end came to me asking “What’s ‘times’ mean?” I didn’t know what he was talking about at first, I was like times? What? He kept asking “What’s ‘times’? You know, Mrs. Honeys says ‘537 times 821…what’s ‘times’?” I just made those numbers up but I realized he was talking about multiplication. I said it basically meant having 537 groups of 821 things, so basically you add 537+537+537… 821 times. He thought that was really cool! I said, since there are 12 bagels in a dozen and you, me, and daddy each had our own dozen, we could say “12 times 3 people” and have 36 bagels. Or you could say 3 times 8, 17 times 2345, or 9 times 5, or any other numbers you wanted! He thought about what I said for a second then said, out of the blue “45”! Huh??? “9 times 5”, he said, “that’d be 45”. I was floored and asked how he knew that, and so quickly (probably should have let it be but oh well). He shrugged. “Half of 9 is 4.5 so move the dot and that’s 45”. Calmly, confidently, nonchalant and acting like it was no big deal. I didn’t really understand how he got that because I wasn’t thinking that hard I was just surprised, and he couldn't even explain exactly how his method worked. He "just knew."
In a few minutes when I could really think about it, I realized how cool and creative insightful and “unschoolish” his answer was! He didn’t just had 9 and 9 and 9 and 9 and 9 like I said multiplication was. First he divided nine by 2, then recognized that moving a decimal place to the right is the same as multiplying by 10. I wasn’t even aware he knew about place value or decimals! He also must have made the connections in DIVISION, beyond “multiplication tables”, that 10 divided by 2 is the equivalent of multiplying by 5. These moments so restore my confidence in our decision to homeschool! I love how he can learn math from watching a movie or know division without being taught! I love it! Unschooling is awesome! Just thought I’d share. ;)
What my kids can't do is to answer speed quizzes like "What is six times nine?"
Meanwhile, they can calculate 6% (local sales tax) and 30% (the discount Kirby gets on gaming supplies) in their heads.
Once a much younger Kirby wanted to walk to the corner store for a Dr Pepper. he asked what he could do to earn money. I whipped up a blank table with 1-9 across and 1-9 down, showed him what the multiplicatory deal was, and said I'd give him a penny a square.
"How much money will that be?"
"Same as the answer in this square," and I indicated the 9x9 square.
Marty was only four or five, but he wanted in on this money-making puzzle, so I made him one too.
They helped each other, filled theirs in, figured who was right when there were discrepancies, LOVED finding the patterns, and nearly forgot about the sodas.
When they got back from the store (with change, as it hadn't occurred to them to stop when they had reached the price of the soda), they were still excited about that chart.
We had a big new concrete patio, and one night the whole family and some friends were playing outside. When a spinning, jump-roping, free-for-all game got old and they were tired, I chalked out a chart about 8 feet square, and the kids started filling in the easy parts—2's, 5's… soon there were only six or seven squares left, like 7x8 and other intuition-avoiding combos.
I have never before or since seen such total –doing- of times tables.
Jody Hegener once told me a tale of having been challenged by kids from a Waldorf school who waited until they had an eleven or twelve year old Jody headed out on horseback at a ranch with them to ask the big question: "Do you know your times tables?"
Jody told them she didn't know if she did or not, since she didn't know what "times tables" meant.
Ooh, they had her good. But they told her what it was that she knew so little about, and they demonstrated, zipping on up to the times TWELVE they knew (way better than the nine or ten limit of public schools).
So Jody was up with them and getting the pattern on the elevens and twelves, and said "What about 13?"
They didn't have to know 13. It wasn't so fun anymore. Jody told me she was multiplying by 14 when they finally got her to stop.
I imagine she was thinking of how very easy fifteens would be.
But "times fifteen" won't be on the test.
This summer Kirby was overheard explaining to some other teens at the gaming shop how to multiply by 18. Do it by 20, and subtract two for each one you have. No pencil, no paper, and the school-labeled "learning disadvantaged" friend totally understood.
The adults who overheard this expressed amazement.
The other homeschoolers who heard about it were amazed that adults had been amazed.
Perhaps knowing the times tables, "doing" the times tables, is a magic safety from further math trauma. "I know my times tables" is like "King's X" from being tagged by the math monster. My kids think math is a tool and a toy and a game. Why would they want to be saved from it?
"We don't have to know that" isn't anything I have ever heard my children say. Because there is nothing they –do- "have to learn," there is nothing that is off their learning list either. In artistic terms, without the object there is no field. In math-lingo, they have the infinite universal set. In a philosophical light, they avoid the dualism of learning and not-learning.
If you are new to math in the wild, I have some recommendations of things that worked well for us. First, don't be teacherly about any of this. Don't use the talking-to-a-poodle voice (and if you don't have one or don't know what I'm talking about, GOOD!). Then, gradually gather some subset of this sort of math stuff, plus other things it might remind you of:
two or more matching geoboards and colored rubber bands (you could get just one, but then you can't copy each other's designs)
pattern blocks, even if your kids are older (we have the wooden ones with the yellow hexagons being the biggest), because there are angles to mess with and because it's very soothing
Cuisenaire rods if you can get them for less than full price (or if you're rich), but don't worry about the "real" exercises
Yahtzee, Bazaar, Master Mind for Kids (cuter, friendlier, simpler version than adult Master Mind), playing cards, poker chips (not necessarily for playing poker, just for messing around with, making patterns), and Clue (Kirby says there is a D&D Clue game out. We have the Simpsons version.)
real money to spend
computer and video games like Zoombinis, KidPix, Harvest Moon, Mario, Link & Zelda
don't discourage Magic, Pokemon or other "CCG's" (collectible card games); D&D and other dice games involve much math and strategy
The other day in my kitchen I said, "Hey, Kirby, I might make $100 for an article on you teaching someone to multiply by 18, and you didn't even know I overheard you talking about it, so I'll give you $10."
"Cool. 10%. I'm like an agent or something."
His 23-yr-old first-time visitor got big eyes and said, "You were teaching someone to multiply by 18?!"
The article above is a corrected version of what appeared in the January-February 2002 Issue of Home Education Magazine.
Lisa Wendell on the Always Learning list:
I do know it is hard to let go of the fear - but this list has been the best for
me/us. I joined a year ago (last summer) because the doubts were creeping in as
my son turned 13 and he was now a teen and it was much easier when he was
younger ....... This list helped me over that hump and I've relaxed again into
the belief that he will learn it when he needs to/wants to.
For example our son being 14 makes me worry sometimes (more like a fleeting
thought) when he asks things like "mom, is 2+6, 8?" To which I answered "yes"
THEN the very next day we were at Costco - which is a large wholesale place
that sells in multi-packs - Hubby and I were looking at something that was 4
packs of 30 - Zac walked up and said there are 120 in the box. He did
multiplication in his head instantly - didn't even stop to think about it. I
could have sworn he does not know multiplication ;)
My daughter when she was five had memorized the six times tables up
to 15, but ONLY the sixes...that was done all on her own and done
because the tax on a dollar here in Ohio was 6% at that time...so she
needed to know how much money she really needed for things at the
dollar store so then gramma and grampa started sending gift checks
for $10.60...(tax included) marked on the check...she never bothered
to sit and memorize the rest of the tables but seems to get along
fine now and is almost 15
(elwazani, on AlwaysLearning)
mindful parenting, on
ideas for parents.