Here’s a one-minute video that introduces the problem from The 1-to-10 Card Investigation I wrote about in a blog about a year and a half ago.
I used to have the order written down and I haven’t used this for years. I can’t find the card order and have been working on it for about 20 minutes. Can someone give me the card order? Thanks.
I don’t have the card order written down either, but I got a deck of cards and fished out ten cards, Ace through 10. I’ve solved this problem many times over many years so it really isn’t a problem for me any longer. I was able to figure out the card order fairly quickly.
Then I thought about how to respond to the request for the solution. I began a conversation with myself. Providing the answer would be quick and easy now that I’ve solved it again, but I wondered about what additional information I might also include. Actually, I had quite a long conversation with myself about how to respond.
The conversation in my head went something like this:
— Hmm, in the classroom, I resist giving students answers. Should I do it here?
— Well, why not? She obviously really wants to know and why should I withhold information that I have?
— But, then again, I worry that giving the answer betrays my educational belief that productive struggle has value. And what I especially love about this problem is that you know that you’ve solved it when you can deal the cards the way I showed on the video. No answer book is needed.
— But if a problem is frustrating, then struggle may not be productive. Maybe she doesn’t want to spend any more time on trying to figure it out. Also, if someone really wants the answer, maybe that will help spark some thinking.
— A confession: There are times, most recently when I was tutoring my granddaughter in high school math, when I check the answer in the answer key at the back of the textbook and then work backward from that to figuring out a solution strategy.
— So if the answer can be useful, why didn’t I just put the answer above in the original blog?
— Well, I worry that the solution would either be a spoiler for people who were curious or a so-what for people who weren’t interested anyway. In this instance, I think the answer would most likely close down thinking.
— About the problem itself: Why did I select it for the blog and make that hokey video at my kitchen table? The problem really isn’t a biggie, not up there on the top ten problems I love. (Well, maybe it’s close.) But if you never came across this problem, or never figured out a solution, it’s no big deal. You’ve gotten this far in life without knowing how to arrange ten cards so they come up in order in the very specific way I modeled of dealing and putting cards underneath. How important is that?
— Well, then, that supports my not giving the solution. I don’t think that knowing the answer will be a huge enhancement to anyone’s life, mathematical or otherwise. And not knowing probably won’t be much of a detriment.
— Back to the classroom: When teaching, I want to give students challenges that ask them to make sense of problems, encourage them to persevere when solving them, push them to express their solution strategies with precision, have them communicate their ideas to their classmates, talk with them about patterns and structure that can help them solve related problems, and more. All of these ideas directly relate to the Standards for Mathematical Practice, which describe important ideas for doing mathematics.
— Well, where’s the mathematics payoff in this problem? The numbered cards seem to imply that the problem has something to do with math, but I could just have easily labeled the cards with letters from the alphabet―a-b-c-d-e-f-g-h-i-j―and presented the same challenge. Would that be a problem I would use in math class?
— Yes, because I think it has the potential to engage students with the Standards for Mathematical Practice, as I mentioned above, in a way that’s playful, low stakes, and hopefully engaging.
— OK, should I give the answer and be done with it?
— I wouldn’t do that in class.
— But I’m not in class. I’m at home in front of my computer reading a comment from a person who is interested and stuck. She worked on the problem for 20 minutes and hasn’t felt successful. She made the effort to reach out on my blog site. I want to encourage her to continue to engage with the ideas I express in my blog posts. I want to be helpful.
— Hmm, what’s this about working on a problem for 20 minutes? How much time should someone spend trying to solve a problem. I googled “how long to work on a math problem.” My search returned over 92 million hits. (This conversation with myself was now becoming a time suck.) One site caught my attention and seemed promising. It’s from the Mathematics Stack Exchange, which describes itself as a question and answer site for people studying math at any level and professionals in related fields. It had a post titled How much time is too much (to put into a single problem)?. See what you think.
Now this is really turning into way too much of a time suck. It’s time to get back to life.
Thanks, Jo, for your comment. It got me thinking and for that I’m appreciative.
ALERT: THE SOLUTION JO REQUESTED IS AT THE END OF THIS POST. IF IT WILL BE A SPOILER FOR YOU, HERE’S MY ADVICE: STOP READING, CLOSE YOUR BROWSER, AND GET SOME CARDS.
A Last Comment
I’ve posted the answer below, Jo. For a challenge, here’s an extension: Solve the problem again, but this time after turning over a card, instead of dealing just one card to the bottom of the deck, deal two cards. Can you arrange the ten cards so they’ll come up in the correct order?
I’ve solved that problem before but I don’t have the solution. I’d have to work it out again. Let me know if you get stuck.
Oh, and the solution: 1-6-2-10-3-7-4-9-5-8.