Dimensions of Intelligence

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My children are smarter than me.

Allow me to define “smart” for my purposes. I am certainly more knowledgeable and experienced than my 9-year-olds. I am better read than they are and more able to find practical solutions to problems, whether academic or everyday. I am far better at explaining complex concepts to people than Misses Giggles and Distractable. My ability to critically examine an argument is currently, at age 36, quite a bit better than J and M’s at age 9.

M and J, however, have always been better at absorbing new concepts than I was at the same age. Their minds work faster and burrow deeper. They see connections and parallels that would have never occurred to me. I have no reason to believe that this general trend won’t continue. As far as I can predict, when they are 36 years old, their brains will process ideas more effectively and deeply than mine does today.

The only milestone I beat them to was reading. According to my mother, I read at age 2. J and M were 3 before they were reading independently.

The fact that my daughters are smarter than me makes me proud. Perhaps if I had fewer academic successes under my belt, I would feel diminished by being outshone by my children. Perhaps if I were less egotistical, I wouldn’t be confident that I am just as smart as I need to be. I’m not in competition with my children. My task is give them the tools, skills, and support to be the best M and the best J they can be. I certainly aim to be the best Sadia I can be.

I am not a trained teacher, but I’m a proud nerd and I love getting others excited about knowledge. When my daughters learn a new concept at school, I often expand on it with them at home. It was while doing this that I confessed to them, for the first time, that they’re both smarter than me.

The children were studying 3D shapes in their regular 3rd grade math class. They told me all they knew about rectangular prisms, pyramids and cylinders. I asked if they knew why they were called 3D shapes.

They didn’t.

A mom explains the third and fourth dimensions to her kids, and is at peace knowing that they learn more easily than she did at their age.

The “D”, I told them, stood for “dimensional”. They could think of a dimension as a direction that exists in a shape.

  • A dot has no dimensions because you can’t move around inside it.
  • A line has one dimension because there’s no room to turn around.
  • A plane, I told them using a piece of paper to illustrate, has two dimensions. You can go back and forward or side to side. By combining those two motions, you can get anywhere on the sheet of paper.
  • If you jump off the sheet of paper, you’re in three dimensions. That’s the world we inhabit. Back and forward. Side to side. Up and down. Ocean creatures experience the three dimensions more fully than we do, being able to move vertically with ease.
  • The fourth dimension, I told my girls, was time. That took a little more convincing.

I still had the 2D piece of paper in hand, so I rolled it up to illustrate.

Sadia uses a rolled up sheet of paper to explain to her daughters why time is the fourth dimension.

Imagine, I told them, that there was an ant walking around on my sheet of paper. His world is two-dimensional. He’s not aware of what’s off the paper. Whether the sheet is flat or curved until opposite edges touch, he’s moving around in two dimensions. Even if I wave the paper through the air, the ant probably doesn’t know that it’s being moved. His entire universe is that 2D sheet of paper.

We are similarly unaware of moving through time. Right now, we’re in the dining room, playing with paper. Count to three, and we’re in the same place in the three dimensions we can navigate, but in a new second in the fourth dimension of time.

How to visualize time as the fourth dimension.

J and M said that made sense. “I’m in a new time now!” exclaimed M. “And now… and now. And I hardly wiggled!”

J took the next logical step. “Is there a fifth dimension, mommy?”

“Yes,” I told her. “I’ve read about theories of physics that argue that there must be a fifth dimension.”

“Show me, mommy!” J demanded. “Explain me the fifth dimension.”

“Little J, I recognize the concept, but I can’t see it in my mind. Without a picture, I have to use words. My best explanation is to say it’s the next logical step in the ant analogy.”

“So the fifth dimension is of the parallel universes, mom!” J realized. “Why didn’t you just say that?”

“I didn’t say it because I didn’t understand it. I can’t see it clearly the way you can right now. I’ll do my best to create a metaphor and picture in my mind, but it’s going to take me some time.”

“Mom! It’s obvious,” J told me, more than slightly irritated.

“Sweetheart, you’re going to run into a lot of people who have a harder time understanding ideas than you. Please be patient.”

“But mom,” J pointed out, “you’re mom.”

“I know sweet girl, but as you get older, you’re going to know and understand more and more things that you’ll have to explain to me instead of the other way around. There’s a lot I don’t know, and a lot it’ll take hard work for me to understand. Some of those things will come really easily to you, and that makes me happy.”

I hope that this confession, made with confidence and without apology, showed J and M that it’s okay to be smart without being smartEST. That was a lesson that I struggled with. It was quite the blow to my ego to realize that I wasn’t the top undergrad at my college. I was “only” in the top 10% based on the very narrow measure of GPA. I’ve since learned that being seen as the smartest person in the room is no measure of success.

Doing my best — that’s how I now measure success, even if that fifth dimension escapes me. And for the moment, I’m doing my best to raise two little girls who are officially smarter than me.

The Dad Network
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Math Fun: Pi for Elementary Students

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Pi Day is coming up on March 14. Get it? π = 3.14. March 14 =3/14. This year, 2015, makes Pi Day (3/14/15) all the cooler, because the first 5 digits of π are 3.1415. Next year gets its glory too, since π = 3.1416 if you obey rounding rules. It’s the little things that bring us joy in my family.

In the run-up to Pi Day, my 8-year-old twin daughters have been assigned π-related projects of their choosing in their Gifted and Talented class. M, ever the perfectionist, is still pondering her choices, but J has decided to calculate the volume of the sun. Along the way, J will learn how to calculate the volume of a sphere to teach her classmates.

It warmed my heart when, as J was excitedly telling a family friend all about her project, she said, “I already knew about pi, because Mom helped us discover it with coins and stuff. It’s the relationship between diameter and circumference of every circle.” I was especially happy to hear this 3 months after we did that exercise. Since it made such an impression on my girls, I thought I’d share the activity with the parents of mathematically minded children everywhere.

Teach children about pi by letting them discover it for themselves. Have them measure the diameter and circumferences of objects around the house and show them that d/c is always approximately 3.14.

In December, we spent a day with dear friends, both physicists by training and IT professionals by vocation, who are expecting their second child and first daughter on Pi Day. My 8-year-olds wanted in on the joke, so I promised to explain it to them when we got home.

We measured all sorts of round things: coins, pot lids, coffee mugs, you name it. We used a piece of string around the edges to capture the circumferences and another piece of string across the middle to find the diameter. We then compared the scraps of string, finding that the circumferences were always just over three times as long as the diameters.

We then took it a step further, using a ruler to get a more precise measurement of each piece of string. Once we had our list of numbers, we punched them into the calculator, dividing each circumference by its diameter. We kept arriving at something close to 3.14.

I told my daughters that they had discovered a universal constant. Pi is a special, almost magical, number that just is. I told them that scientists used it to design rocket ships. I told them that builders used it to estimate their supply needs. I told them that they could even use it to calculate how much air is needed to fill a soccer ball.

To ice the cake, I had J and M put the word “pi” in the all-knowing Google search field. When even Google confirmed their calculations, they were so excited that they began to dance and all our lengths of string went flying.

Is pi for elementary students? I think kids are capable of understanding most concepts, given the chance. Let’s just keep the idea that math might be boring or hard to ourselves, shall we?

Please note that my daughters’ mathematical interests are atypical for their age. This activity is appropriate only for children who are comfortable with the basics of division. They certainly don’t need to know how to do long division, but they should understand that division is the breaking of things into equal parts, and that those parts need not be whole numbers.

Thinking about trying this activity with your children? Please let us know how it goes!

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Party Trick: Mental Multiplication

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“What did you learn today?” I asked M, my nearly-8-year-old, as part of our bedtime snuggle-and-connect.
“What did I learn today? Did I learn anything today?” M mused. “Oh! Riley asked me what is 169 times 28. It’s 4,732!”
“How did you figure that out?”
“Well, I know that anything times 10 just puts a 0 at the end. So 169 x 10 = 1,690. And that two times is 3,380. Plus another 1,690 is 5,070. Then I did 169 x 2, which is, um…”
“You kind of did it already with 1,690.”
“Right! 338! And 5070 – 338 is 4,732. Obviously.”
“Obviously.”
“Riley didn’t even know the answer! But I do know it now.”

And this is what bedtime looks like around here.

Sadia (rhymes with Nadia) has been coordinating How Do You Do It? since late 2012. She is the divorced mother of 7-year-old monozygotic twins, M and J. She lives with them and their 3 cats in the Austin, TX suburbs and works full time as a business analyst. She retired her personal blog, Double the Fun, but now also blogs at Adoption.com and Multicultural Mothering.

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Spiral Learning: Permutations for Elementary Students

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Categories Development, Education, From the Mouths of Multiples, Higher-Order Multiples, Identical, Parenting, School-AgeTags , 8 Comments

Permutations for Elementary Students

When I was browsing the lovely photos on MathiasQuads.org yesterday for this morning’s post, my daughter M took great care to read the names in each photo caption. She wanted to be sure to match each face to the right name. As an identical multiple herself, she understood how important it was to see Mary Claire, Anna, Grace and Emily as individuals.

M, aged 7, observed that they were rarely in the same order between photos.

M: There’s 16 ways for them to be lined up.
Me: How did you figure that out?
M: Because there’s 4 sisters and 4 spots and 4 times 4 is 16.
Me: That’s a very good deduction, my mathematician girl, but it’s actually 24. Can I show you how?

Is 7 a little young for combinatorics? Sure, but M showed an interest in it, so I dug back into my 8th grade math memories. I drew her a picture to show her how to think of permutations. She picked the colours for each sister.

Explaining permutations for elementary students. Showing them the first quarter of the pattern allows them to derive the pattern themselves. From hdydi.com

Me: There are 4 sisters who can go in the first spot. I’m just going to draw one of them. Once she’s in her place, there are only 3 sisters left to go second.
M: Then 2, then 1!
Me: Exactly. So there are 6 orders available for each sister who goes in the first spot.
M: And 6 times 4 is 12 and 12 is 24.
Me: Which is also 4 times 3 times 2 times 1.
M: Well, that was easy.

We’ll probably chat about combinations tonight during bath time.

Spiral Learning

I’ve always taken this approach to educating my daughters. If one or both of them is interested in something that illustrates a larger pattern or important skill, I explain it to them at a level that is pertinent, interesting, and within their abilities. Later on, when they’re more intellectually mature, I’ll come back to it. In a couple of years, I’ll show M how to use factorial notation.

My teacher friend Kaylan tells me that the eduspeak term for this is “spiral learning.”

Spiral learning is the practice of returning to a topic over time to build an increasingly sophisticated understanding

What sparks your child’s interest? What’s your approach to teaching?

Sadia (rhymes with Nadia) has been coordinating How Do You Do It? since late 2012. She is the divorced mother of 7-year-old monozygotic twins, M and J. She lives with them and their 3 cats in the Austin, TX suburbs and works full time as a business analyst. She retired her personal blog, Double the Fun, when the girls entered elementary school and also blogs at Adoption.com and Multicultural Mothering.

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Two of Me

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Categories Attitude, Difference, From the Mouths of Multiples, Mommy Issues, Perspective, RelationshipsTags , , , , , 4 Comments

I needed to assemble some new furniture recently. I put the first bookshelf together while my 6-year-old daughters were sleeping and presented it to them proudly when they awoke. J was unimpressed.

J: You did that by yourself.
Me: Yes, honey. Do you like it?
J: How did you do it by yourself?
Me: The same way I did the dining table. I just followed the instructions.
J: It’s supposed to take two people.
Me: I could see it being easier with two, but I was fine by myself.
J: Last time you had someone else.
Me: I don’t think so. Do you want to help me with the others? I’d love some help putting your book bag cubbies together!
J: You need two people. Two of me is one you. M is another me because we’re sisters and twins. Sometimes she has some different thoughts, but really, she’s another me. So me and M together is one you and we’ll help.

They did end up helping me assemble the cubbies we’re now using to house their schoolbags, dance bags, and piano books. M’s contribution was minimal, since she spent so long washing her hands that we were nearly done by the time she showed up.

When the girls were first born, I would have bristled at anyone saying that M was “another” J. Over the years, though, I’ve learned to embrace the similarities and closeness between my girls, while also celebrating their individuality and differences. Both my girls are well-adjusted, independent, and happy. Most of the time, they love being together, but sometimes they need time apart and they argue often.

I don’t think J’s conception of M as her other self was imposed on her from outside. It’s just one more aspect of the relationship that M and J share, one that might have existed even if they weren’t identical, even if they weren’t twins, perhaps even if they weren’t sisters. I kind of like the idea of my daughters adding up to “another me” when it comes to physical labour, too.

How do your multiples perceive their siblings in relation to themselves?

Sadia is a divorced mother of 6-year-old twin girls, living in the Austin, TX area.

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