Tag: kids

Learning Multiplication

I know a little bit about the differences between teaching with “blocking” or with “interleaving.” If you were teaching multiplication with blocking, you’d teach the “4 times table”, then the 5x, then the 6x and so on. With interleaving, you’d mix them up so students had to first identify what kind of question this is (“Oh, this is a 4x problem”) and then answer it. There is some nice work in cognitive psychology that shows interleaving leads to better retention.

I use the example of multiplication because that’s just what my 8-year-old was doing yesterday. He had a set of 5x flashcards that he asked me do with him. He didn’t have too much trouble, except for 5×9. As I was running through the cards, I thought of the multiplication table board I’d made a few years earlier when my other kid was learning multiplication. I remember being fascinated by a board like this when I was in Grade 3.

My home-made multiplication board. The green 1-10 along the top and left side are glued down. You put the 24-tile, for example, where the 4 and 6 intersect. Hmm, or the 6 and 4. Or 8 and 3 or 3 and 8!

The tiles are from a square hemlock spindle I got at the local hardware store, sliced on my mitre saw, sanded smooth, and then numbered. Green 1-10 tiles are glued across the top and down the left side. You put the blue 24-tile, for example, in the space where the 4 and 6 intersect.

My son and I got out this board, dumped the bag of tiles on the carpet, and he started to fill it in. The first row is easy: 1, 2, 3,…,10. Then he started on the second row. “2 times 1 is 2. 2 times 2 is 4. 2 time 3 is 6…” And about here, he stopped doing the multiplication and started counting by 2’s. I asked him how much 2 x 7 is, and he had to stop and think, despite the fact that he’d just placed the 10, 12, 14, and 16 tiles.

Uh-oh. My goal is to help him learn his times tables, I don’t want to reinforce repeated addition. What’ll he do with 104 x 56 next month?

So I turned the table on him. I started handing him tiles. “Here, where does this one go? How about this one?” That was pretty hard. For me, that is, because I had to quickly find the next tile I wanted in the big pile on the floor and hand it to him by the time he’d placed the current one.

I realize I was asking a different question: “What numbers multiply to give you 24?” is a lot different than, “What is 4 x 6?”. But this new version of the “game” worked nicely. He did some repeated addition in his head, searching for sequence that hit, say, 24. And he occasionally put a tile in the wrong place but I didn’t correct it. He discovered his mistakes as he lay down neighbouring tiles and the patterns were messed up.  It also forced him to estimate where in the empty board to place tiles any without neighbours.

He soon discovered repeated addition works vertically, too, so he could hop down the 4-column to find the 4x tables.

After a few minutes of this, I turned it up another notch (which was possible because there were fewer tiles left in the pile and I could find them faster.) I started handing him stacks of the same tile, for example, 4 40-tiles. “Here, put these down.” I only prompted him once or twice (“Well, if 4 x 10 is 40, what about 10 x 4?”) I was astonished how quickly he picked up the symmetry. “32 goes here at 4 times 8…Oh! And over here at 8 times 4!”

Older sister dropped by to help him with the high tiles – I don’t think my son’s had much practice with the 7x, 8x, 9x tables yet. The two of them finished off the board, fighting (in that friendly brother-sister way, of course) for who gets to put in the last tile.

There are so many patterns to explore on the completed board. We discovered where on the board to find the same tile (symmetric across the diagonal) :

Me: Here’s 7 x 3 and here’s 3 x 7. Here’s 4 x 6 and here’s 6 x 4. Hey, what’s up with 5 x 5? Where’s its match?
Him: It doesn’t have one, Dad, cause 5 x 5 is the same as 5 x 5. D’uh!

A friend dropped by with a 3-year-old and she asked the toddler to his age. My son helped, helping him find all the 3-tiles. And his 8-tiles. Then my daughter (who’s 11) dropped a little nugget that confirmed this “game” was worthwhile:

Well, I don’t have a tile because my age is a prime number.
[W00t! FTW, Dad!]

I’m sure the math ed people and elementary school teachers can tell me the history of multiplication boards and best practices for using them. But it was so much fun watching my son discover the patterns for himself. And to reinforce that math is something you can play with and — Zoinks! — even have fun with!

If you’ve got the tools and some patience (100 tiles is a LOT of tiles!) I highly recommend you make a set for your kid(s). Do you have any ingenious suggestions for what to use instead of cutting wooden tiles?

Creative Commons License Multiplication table  photo Peter Newbury is licensed under a Creative Commons Attribution 3.0 Unported License.

Galileoscope eyepieces

Galileoscope co-designer Stephen Pompea peers through his creation. (Dean Coppola / Contra Costa Times from Cosmic Log by Alan Boyle)

“I put my Galileoscope together. How do I use all these eyepieces?”

That’s a question I get all the time. There are three different eyepieces depending on how you assemble the components:

There are three eyepieces for the Galileoscope depending on how you assemble the components.

Creative Commons License Galileoscope eyepieces photo-illustration by Peter Newbury is licensed under a Creative Commons Attribution 3.0 Unported License.

The easiest way to use your Galileoscope is with eyepiece A. It gives a fairly widest field-of-view (you can see the largest region of the sky) with a 25x magnification. This is the combination I recommend to new users, parents and kids, and school groups. With this eyepiece, you can easily see the craters and shadows on the Moon and the moons of Jupiter.

The combination A+B+D gives an eyepiece with 50x magnification because B+D create a Barlow lens that doubles the magnification. The increase in magnification comes at a cost: a much smaller field-of-view and fainter image. It is almost impossible to use this 50x combination without a tripod (which the designers anticipated by building a nut into the bottom of the Galileoscope that fits any standard camera tripod.) If you have a tripod and a clear, dark skies, you can see the rings of Saturn. Yes, the rings of Saturn! And that’s magical.

Finally, there is a special lens combination included for historical (and educational) reasons. You see, the Galileoscope was designed as a cornerstone project of the 2009 International Year of Astronomy (IYA2009). That celebration marked the 400th anniversary of Galileo using his telescope to observe the Moon, Venus and, in 1610, the moons of Jupiter. The special “Galileo eyepiece” C+D mimics the view Galileo had, with a meager 17x magnification over a tiny field-of-view. The image appears right-side-up, though, unlike the 25x and 50x combinations which invert the image as most refracting telescopes do.

With all these eyepieces and magnifications, I still recommend the simplest one, just the 25x. In fact, when I’m doing “sidewalk astronomy” I keep the Barlow lenses in my pocket and pull them out only with the more advanced telescope users. Going from naked-eye to 25x already opens up a Universe of wonders.

Parents, teachers, sidewalk astronomers: The Galileoscope design team has put together a great collection of resources. You can order Galileoscopes directly from them, from Learning Encounters or check your local telescope store.

I’m really interested in learning to take pictures through my Galileoscope. If you’ve taken some good ones and have any tips, I hope you’ll share them below.

Why do we teach astronomy?

I just spent a week in Seattle at the 217th Meeting of the American Astronomical Society. If you’re here via my Twitter feed, you’ve been bombarded with my #aas217 tweets. I’ll be sharing some thoughts and experiences in future posts. There was one experience that really sticks in my memory, though.

Ed Prather from the Center for Astronomy Education led a workshop that I attended. I’ve been to a number CAE workshops with Ed. He’s intense. You don’t have “thin” conversations with Ed.

Ed and his colleagues are dedicated to teaching (and teaching  teachers to teach) “Astro 101”, the general education course that 100,000’s of non-Science undergraduates take each year. It’s likely their first, last and only science course. As Ed proclaims, and with which I wholeheartedly agree, we need to teach these people science. Not because they’re on their way to becoming scientists – that audience isn’t taking “Astro 101”. Rather, these people are the next generation of teachers, lawyers, politicians, journalists, parents.  In this age of technology, medical advances and global warming, it’s vital that the next generation of voters be scientifically literate.

Yes, YES! Just the pep talk that gets my heart pounding! And then Ed continued…

Why it is so critical? Because high-tech, science-related jobs in the United States are not being filled by Americans.


Don’t get me wrong — there is nothing wrong with patriotism. In fact, I admire how strong his convictions are. And if I dig deep enough in my brain and heart, I’ll probably say the same thing about Canadian kids. But I haven’t thought about it that way. I’m still at the “let’s do this for our kids because they’re inheriting our mess.” Maybe that’s naive of me. Or maybe it’s a Canadian/American thing. Either way, we all agree that scientifically literate citizens are critical to our — all of our — future.