# Tag Archives: fingers

## Math for the very young

It’s title promotes unnecessary parental mania, but once you get past that bit of self-marketing, I think this Atlantic article offers five great ideas for helping parents encourage numeracy (my word, not the article’s) in very young children.  Here are its key points.

1. Talk about numbers.  Don’t just count to ten, refer to physical objects to help young ones make connections.
2. Talk about spatial relations.  Learn names for different shapes and make comparisons.  [My addition: Continued conversations about comparisons are the seeds for phenomenal connections later.  One example is my ‘blog post here.]
3. Talk about math with your hands.  Point to objects as you talk about them.  Connections between multiple representations is huge for brain development.
4. Engage your child in spatial play.  The article talks about puzzles (more below), but I’d add rolling balls, stacking blocks, and more.
5. Engage your child in number play.   Play games like Chutes & Ladders to encourage counting.

These are all great points and not particularly revolutionary if you think about it.  So, I’d like to add a few topics to the list from my play with my own children.

• Play with blocks, Legos, etc.  Get down on the floor and stack those blocks yourself alongside your child.  Sometimes build your own tower and sometimes add to your child’s creations.  The connection time is great, and your child learns greater creativity by trying to imitate some of your more sophisticated constructions.  Talk about what you’re doing, but if your child doesn’t seem to care or follow, no problem!  Over time, the ideas will sink in, and you got quality time anyway.  As they get older, play Jenga or–even better–make your own game of Jenga using blocks.  Don’t forget to laugh and have fun when the tower falls and you get the chance to rebuild the original tower.  For young children, play with a much smaller stack of bigger blocks.  Check out this sophisticated Jenga tower.  What else can you and your child make with this game that can be good for years of play?
• Play with jigsaw and similar puzzles.  I’m quite fond of the great creativity of most of the Melissa & Doug and other similar puzzles, but they can be pricey.  I save lots of money on puzzles at local consignment sales.  Early puzzles for my kids have knobs on them to enable easier handling.

Long before they figure out how to put them back in place, you can use the pictures for conversations about names, colors, etc.  So much room for general creativity!

Before they can assemble them, we put puzzles like the one below in their play space.  Straight-edged puzzles eventually give way to more traditional jigsaw puzzles.  As they got more sophisticated in their thinking, we encouraged them to assemble connected puzzles outside their frames.  As always, other games with the puzzles are great:  How many pieces does the puzzles have?  What colors are there?  What is Pooh doing?  Tell me a story.

I was about to put away some very simple puzzles when my oldest daughter created a new game.  She knew the early puzzles were too simple, so she turned all the pieces upside down and tried to reassemble them without the aid of pictures.

We sometimes work more complicated, increasingly difficult jigsaw puzzles together.  Talking about shapes, colors, and searching for where individual pieces might fit into the big picture of the final puzzle are all great activities.  Pointing to part of the picture on a puzzle box and then pointing to the corresponding location in the puzzle as it is assembled is a tremendous lesson that helps children make connections between multiple representations of ideas.

We also have a big foam floor puzzle of the alphabet (Thanks, N!).  In the earliest days, it was a nice floor pad.  It comes apart and can be assembled in different ways.  Actually, being able to disassemble is a great early skill for children (good to remember when you’re annoyed the 100+ times you put it back together yourself).  Some pieces are easier to put back than others, but cheer every time they accomplish a new task. Sometimes we sit in another room and ask them to retrieve a specific letter.  We’ve encouraged early literacy by connecting every letter with something familiar: “D is for Daddy”, “Y is for yogurt”.  They remember these special relationships long before they’ve memorized the alphabet.  Literacy and numeracy are not isolated skills.

Don’t forget to try something unusual.  I’ve had great fun making blocks and fences out of this floor puzzle.  My girls giggle as we build interlocking walls around them, creating and filling in windows, etc.

• When they’re older, draw pictures of their room or maps of your home.  It doesn’t matter if the scales are right or the pictures are accurate.  Show them on maps of your neighborhood, city, state, country, or world where they live and where special family and friends live.  Revisit this when you travel or walk with your child.  Connections between the real world and 2D maps can be tough, but are phenomenal skills for later mathematical abilities.
• Older games we’ve used that also happen to be great for visual-spatial development Connect 4 and the absolutely glorious Blokus.  I’ll post more on this game another time.

WARNING:  I’ve said this before, but I worry about parents who might succumb to the mania suggested by the Atlantic article’s title.  Education of children should not be about competition or creating math whizzes.  Play, model creative play yourself to encourage their creativity, cheer for your kids when they do something new to encourage out-of-the-box thinking, don’t worry if your kids don’t get it right away, be patient, be 100% willing to move to a different task/game if your child isn’t interested, and make connections.

Growth will happen if you keep them surrounded by challenges and point out how much fun it is to think, to create, and to solve.

## “Digit”al Multiplication

So here’s another musing I had on a beach visit. I don’t recall where I learned this trick, but I’ve had it for decades.  I suspect most of you already know how you can multiply by 9 using your fingers, but I’ll briefly explain just in case.  An extension follows.

Start by laying out both hands with all fingers outstretched.  Number your fingers from 1 to 10 from left to right.

To compute $9*n$ for integer values of $n$ between 1 and 10, fold down the $n^{th}$ finger and count the number of still-extended fingers before and after the folded finger. Thinking of those two numbers as a two-digit number gives your answer. For example, to compute $9*9$, fold down the $9^{th}$ finger as shown below.

Because there are 8 fingers before the fold and 1 after, $9*9=81$. Simple.

On the beach, I recalled this little trick and first extended it to two other simple multiples of 9. Computing $9*1$ is simple, but folding down the $1^{st}$ finger confirms the answer by showing 0 before and 9 fingers after the fold, so $9*1=09=9$.

Computing $9*10$ works the same way.  Folding down the $10^{th}$ finger shows 9 before and 0 fingers after the fold, so $9*10=90$.

As I lay on the sand, I marveled at how nice this worked, but was saddened that such a cute approach had such limited applicability.  Thinking about what was going on in this problem, the obvious observation was that all were products of 9 and worked from 10 initial fingers.  So what would happen if you used a different number of fingers?

Starting with 7 fingers, numbered as before, I thought I might be able to multiply by 6.  As a first attempt, I tried $6*6$, but the next image shows that my approach gives $6*6=51$–obviously not the correct result.

Then inspiration struck.  Perhaps the answer was good, but my interpretation was off.  If multiplication by 9s (using 10 fingers) gave an answer in base-10, perhaps multiplication by 6s (using 7 fingers) needed to be interpreted in base-7.  That is, $6*6=51_7=(5*7^1+1*7^0)_{10}=(35+1)_{10}=36_{10}$.  Eureka!  But does it always work?

Testing once more, I tried using 5 fingers meaning I would be multiplying by one less (4) and getting an answer in the base of the number of fingers I was using.  The next image shows $4*2=13_5=(1*5^1+3*5^0)_{10}=(5+3)_{10}=8_{10}$.

Generalizing, imagine that you could hold out any number of fingers.  The image below suggests that you extended $n$ “digits”, so you could use this to multiply by $(n-1)$.

The next image supposes that you hold down the $k^{th}$ finger leaving $k-1$ fingers before and $n-k$ fingers after.

That suggests $(n-1)*k=(k-1)(n-k)_n$ where $(k-1)(n-k)$ is a 2-digit number whose left digit is $(k-l)$ and whose right digit is $(n-k)$.  Expanding, $(k-1)(n-k)_n=((k-1)*n^1+(n-k)*n^0)_{10}$ and $(n-1)*k=(k-1)(n-k)_n=(k*n-k)_{10}$.  QED

So, if you just had enough “digits” (pun intended) and didn’t mind working in different number bases, the result of every single multiplication could be known with one simple finger fold!