Contexts

Unlikely Students in Unknown Places

I recently got back from Santa Fe.  I was attending a conference there for a few days last week and afterward, I drove to El Paso to visit my brother’s family (he’s currently stationed in the middle east so I didn’t get to see him – unless you count face time) for a day before flying home.

Let me preface this story by saying that we all probably have a story similar this, but how we handle it can be a possible game changer.

Somewhere on my long drive, I stopped at a fast food restaurant to grab a quick bite.  So, I went inside and got in line.  The following outlines the beginning of our interaction as I stepped to the counter:

Cashier:  May I take your order?

Me:  Yes, please.  I’d like a number 2.

Cashier:  Large or medium.

Me:  Medium, please.

Cashier:  (after pushing more buttons than is conceivably necessary to enter my choice of “medium”): Your total will be $6.05.

I dug through my wallet (receipts from the trip and everything) and found that all I had was a $10 bill, so I handed it to her.  She entered $10.00 correctly and the correct change of $3.95 showed up on the little screen.  At just about that point, I remembered that I had a bunch of change in my pocket and said quite enthusiastically, “Oh, wait, I think I have a nickel.”  Who wants to carry around $0.95 in change in their pocket.

The cashier didn’t miss a beat, and said, “So, your change will be $4.00 even.”  I kind of smiled as I continued to look through my change, proud that she had a mental strategy to adjust to the situation and that she seemed quite confident and comfortable using it in this situation.

Unfortunately, I didn’t have a nickel, but I still didn’t want change falling out of my pocket into the depths of the rental car, never to be seen by me again.  So, I told her, “Oh, I’m sorry, I don’t have a nickel, but I do have a dime.”

As I handed her the dime, I saw her face morph from a confident smirk to a confused, almost terrified look of despair .  I had just taken her from a mathematical point of “Yeah, I can do this math stuff.  I may not use the computer for the rest of my shift” to “Holy $#!+, what the #=|| just happened!”

I went into math teacher mode and waited patiently for her to begin breathing again.  And then I waited for her begin thinking.  She adjusted my change with my introduction of the idea of a nickel, why not a dime? After what seemed like 5 minutes (probably more to her), it was painfully obvious to all around that her anxiety in this situation was taking over her ability to tackle this problem. So, I tried to think of a “least helpful question” to ask.  Now I put myself on the spot.  If she only knew that we were both now feeling some of this pressure.

So, I finally asked her my question and she gave me the correct change within a few seconds.  She smiled as she gave me my change and my new “unknown” student and I parted ways.  I know I felt good about helping someone develop a strategy outside of the classroom.  I hope she had a similar feeling about learning to make sense (no pun here) of making change.

Being a math teacher is a 24-7 job sometimes and we can find our students anywhere – even in a fast food restaurant in New Mexico!

What you would have asked the cashier in this situation.  I’d love to hear what your “least helpful” question would have been. No pressure, take as long as you like.  No one is waiting in line behind you!

Feed the hungry!

Oh, here’s my question:  “If you could change the dime into some other coins, what would you change it for?”

Perplexing Donuts

A good friend and colleague, Krystal Shaw, tweeted this article about Krispy Kreme Donuts in the UK a while back and it immediately got me thinking. . . so I really liked it and wanted to use it with kids.  To plan for the lesson, I started to take myself through this problem as if it were a 3-act task (I wasn’t sure it would become one, but I wanted to see where this would lead).  I looked at the picture:

Top of Box

and jotted down what I noticed. Then I began wondering:

  • How many donuts are in that big box?
  • What are the dimensions of the box?
  • Is there more than one layer of donuts in the box?
  • How many rows of donuts are there?
  • How big is (What is the diameter of) a Krispy Kreme donut?
  • When I was finished (or thought I was finished) wondering, I began to seek the information needed to answer my questions.

I found some nice strategies for determining the number of donuts in the box.  Strategies accessible for 4th grade students.  I was happy, so I moved on to the next question: What are the dimensions of the box?

This is when it happened.

I was stuck.

Perfect.

Challenge accepted.

I looked at the pictures, found the information in the article, then began to question that information (and myself) as well as some critical friends.  This problem was getting better and better as I walked myself through it.  Fantastic!  SMP 3: Construct viable arguments and critique the reasoning of others, such as a Krispy Kreme representative from the UK or a USA Today reporter.  Maybe this question won’t have a third act, but the estimation and reasoning used to solve this could be extremely empowering for kids.

I challenge you to solve this problem with your class as well and share your results.     Challenge yourself and your students to construct a viable argument and critique the reasoning of others.  Does your math challenge the information in the article or support it.  Either way, integrate writing into math class in a meaningful way:

write to the reporter, Bruce Horovitz or Krispy Kreme UK: helpdesk@krispykreme.co.uk and tell them what you  discovered

Time for me to give this a try!  More in about a week.

By the way:  Krystal Shaw gave her amazing Mathletes after school club the task of writing a 3-act math lesson for their teachers to teach.  I think she should post it on her blog to share with the MTBoS!

Re-viewed: Children’s Mathematics. . . It’s a Beautiful Thing

About a month ago, I was asked to preview the new edition of Children’s Mathematics and write about it on this blog.  I was more than happy to oblige!  Children’s Mathematics is one of a select few books that I’ve read in the past decade that have really had an impact on how I teach mathematics.

Let me begin by saying that no matter which edition you read, it’s worth it.  If you’re a teacher (or parent) and have an unread edition of Children’s Mathematics sitting on your shelf (for whatever reason), do yourself and your students (or children) a huge favor and read it.  Then do exactly what it says to do!

The research-based approaches to teaching mathematics you’ll learn from the contents in this book are invaluable.  Cognitively Guided Instruction.  That’s what it’s called.  And it’s a beautiful thing to see in action.  And it’s even easier to see in action as you read the book (more on that later).  Empowering students to think, make sense of, and solve problems based on their own understanding.  Why don’t we all teach this way?  It makes so much sense.

To be clear, Cognitively Guided Instruction (CGI) is not a program or a curriculum.  It’s an approach to teaching and it’s based on research on children’s mathematical thinking and how it develops.  The idea that’s most intimidating to teachers (and parents) is that no direct instruction is used before giving students a problem.  Many would argue that students won’t know how to solve the problem unless they are shown how first.  This is so NOT TRUE.  The contexts of the problems give the students all they need to jump in to the problem.  Their pathways to solutions are defined by their own understandings.  For example, students may be given a problem such as:

Luke had 7 toy cars.  His friend gave him some more cars for his birthday.  Now Luke has 12 cars.  How many cars did his friend give him?

Students given this problem may solve it by counting down from 12 or up from seven,  They may begin by choosing 7 objects to represent the cars, then counting some more to get to 12.  They may even make two sets (one set of 12 and one set of 7). There are multiple ways students can represent the problem.  All of them valid.  Some are more efficient than others, but regardless of the strategies used, it’s a beautiful thing.  It’s especially beautiful when students share their strategies and learn from each other.  When we listen to students’ thinking we best know how to work with them in order to move them along their own mathematical journey.

Now this is all great, but you can get this and more from any of the earlier editions of this book.

Here’s some of the new goodness you get from the latest edition (out later this month):

  • A chapter dedicated to Base-Ten number concepts – this was nice to see, since base-10 understanding is a huge part of elementary mathematics.
  • Quotes from real teachers using CGI in the classroom.  These can be found at the beginning of each chapter.  Its a small part of the new edition, but really it’s one of the things I really enjoyed!
  • Video clips that you can watch as you read! No more CDs to have to load, or lose or break.  When you’re reading and want to see the accompanying video, just scan the QR code in the book with your phone.  It just pops right up!

Overall, this new edition has some updated content and makes it easier (thanks to technology) to see in action.  As one teacher from the book put it:

The better I get at listening to children, the clearer I hear them tell me how to teach them.

Have I said this already?  Beautiful.  Absolutely beautiful!

What are you waiting for?  Go out and get Children’s Mathematics and read it.

And then, go out and get the “sequels”:

Extending Children’s Mathematics: Fractions and Decimals

and

Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School

Check out the new edition here:  http://heinemann.com/ChildrensMath