Photo courtesy of Steve Snodgrass
The more I talk to people about math, the more I hear this refrain: “I don’t like math, because math problems have only one answer.”
Peshaw!
Okay, so it’s not such a crazy idea. Most math problems do have one answer (as long as we agree with some basic premises, like that we’re working in base ten). But math can be a very creative pursuit — and I’m not talking about knot theory or fractals or any of those other advanced math concepts.
I have a friend who is crazy good at doing mental math. She can split the bill at a table of 15 — even when each person had a completely different meal and everyone shared four appetizers – without a calculator, smart phone or pencil and paper! This amazed me, so I asked her how she does it. And what I discovered was pretty surprising. She approaches these simple arithmetic problems in ways that I never would have thought of. She subtracts to solve addition problems, divides to multiply. And estimation? Boy howdy, does the girl estimate. In other words, she gets creative.
(She also has a pretty darned good understanding of how numbers work together, which is probably the biggest reason she can accomplish these feats of restaurant arithmetic.)
While there may be one absolutely, without-a-doubt, perfectly correct answer to “How much do I owe the waiter?” there are dozens of ways to get to that answer. Problem is, your fourth grade math teacher probably didn’t want to hear about your creative approach.
See, when we learn math as kids, we’re focused on computation through algorithms. (In case you’re not familiar with the word, algorithms are step-by-step procedures designed to get you to the answer.) You did drill after drill of multiplication, long division, finding the LCM (Least Common Multiple) and converting percents to fractions. But nobody ever asked you, “How would you do it in your head?”
The good news is that now you’re all grown up. There’s not a single teacher who is looking over your shoulder to see if you lined up your decimal points and carried the 2. You can chart your own path! And when people are given this freedom, they often find really interesting ways to solve problems.
Don’t believe me? Try this out: Add 73 and 38 in your head. How did you do it? Now pose the question to someone else. Did they do something different? If not, ask someone else. I will guarantee that among your friends and family, you’ll find at least three different ways of approaching this addition problem.
So, let’s do this experiment here. In the comments section, post how you solved 73 + 38 without a calculator or paper and pencil. Then come back later to see if someone else had a different approach. If you’re feeling really bold, post this question as your Facebook status, then report the results in the comments section.
And while you’re at Facebook, be sure to visit and like the Math For Grownups Facebook fan page!
Old-fashiond way: I added the 3 and the 8, and that gave me a 1 for the ones column. Then I added 7 + 3 + 1 = 111. A more creative way would be to find out how far 73 was from 100. (27) and subtract that from 38, leaving 11.
I think your first approach is really common, Not Evelyn. (Not that you are!) This is the algorithm that we’re taught in school. My problem is that I can’t do that in my head. *smile*
Your second approach would never occur to me in a million years. *smile*
Laura
It’s only fair that I play too! The first thing I notice is that 7 + 3 = 10. (Or 70 + 30 = 100.) So the answer will be larger than 100. Then I add 3 and 8 to get 11. Finally 100 + 11 = 111.
I added 70+30 for 100 and 3+8 for 11, so 111. On paper I use algorithms. In my head I do whatever seems easiest.
My first reaction for 73+38 was to rename these numbers as 70+30+3+8 which is 100+11 for a sum of 111. Since that was already mentioned I thought of a different one: 73+30+8 which is 103+8 for a sum of 111.
I agree with your point that many of us (adults) were taught to do the algorithm and I am still ‘programmed’ to do that but I push myself to do mental math. The algorithm will always work for large or not-so-friendly numbers but mental math strategies have more meaning and are usually more efficient in our daily lives.
I am very happy that this is being considered and encouraged in our current math classrooms. Students make meaning for themselves and use strategies that they have connected with. This kind of math will ‘stick’ with a person rather than students forgetting what they did last year or even last month
I couldn’t agree with you more, Cathy. Whether we solve these problems with algorithms or geometric series, the point is that we can get to the answer without killing too many brain cells. Thanks for chiming in!
At first I remembered how much I like the number 9, so I multiplied both of the numbers by 9. 73 times 9 is 73 times 10 (730) minus 73. 730 minus 73 is 700 minus 43, is 660 minus 3, is 657. Similarly I found 38 times 9 is 342. Then I looked at the numbers:
657
342
And noticed each column adds up to 9 (a number I like), which pleased me. So these add up to 999 which is 10^3-1.
Then I looked at the question, which was to find out 73+38.
I new it was (10^3-1)/9 which is (10^3-1)/(10-1)- which reminded me of the formula for a geometric series. Yeah, that’s equal to 1 + 10 + 100.
So the answer is 111.
Kudos to anyone who solves addition by way of a geometric series!
It is impressive, yes? I had to read it aloud to a couple of my math buddies.
Thanks!
Of course I was just joking about the technique. I wanted to demonstrate your point that there are so many different paths you can take to get the answer of even such a simple maths question. What’s always amazing (although I guess it shouldn’t be) is that no matter what silly route you take, everything always works out by the end.
M
I figured out a long time ago that it’s easier to do mental math from left to right. So I added 70+30 and then 3+8.
Alex figured out on her own how to do two-digit addition a little while back. She also starts from the left. Something about it apparently just makes sense.
Left-to-right addition is becoming a more common algorithm for elementary students to use. The key is understanding place value — whether you’re adding the tens place or the ones place, etc. I believe some of the newer math curricula call this “chunking” or something similar. Unfortunately, some kids have the tendency to “carry” backwards when they add this way.
But that’s a testament to the value of conceptual teaching. If we can find out how a kid thinks, we can find out how to teach him. Fortunately for us adults, we can teach ourselves! (As long as we can let go of our previous ideas about math!)
73 + 38
We need two tens and a seven to get 73 to 100.
Take two tens and a seven from 38.
That leaves 11.
111.
I had wondered where this #1 was (it isn’t tagged or in the category of secrets). I’m glad you did the round-up of these!
Regarding “one right answer” – boy! Nothing makes me madder than hearing that. Okay, there are a couple of things, but not many MORE make me madder than hearing that.
Kids continue to deal with these horrid teachers who force this “only one method” crap down their tiny throats. And parents feel forced to let them.
It’s time for all this to change. And it starts with the grownups.
So listen up, grownups! Be creative. Follow Laura’s suggestions. And PASS THAT ONTO YOUR KIDS!
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