The Stupidity of Number Flexibility (#TMWYK)

I’d compare the struggle between teachers and learners at the end of the year to that of a parent trying to carry a limp child to their bed.  Eventually they will both get there, but the parent is frustrated and the child is attempting to make things as difficult as possible.   In the end, neither party is probably happy.

Rather than focus on that inevitable struggle, I want to detail a fun experience (for me) that I had with my daughter this past weekend. Her four-year-old rebellion has included a resistance to completing her math and language at school and a refusal to engage in those conversations at home.  Grandma and Grandpa were in town this weekend, which gave me an opportunity to exploit her desire to impress them.

At school, Maria is given “problems” similar to the ones on this sheet.  They seem randomly chosen, and the children are given beads to model the problem if needed.  0526150926-1


Given the opportunity to exploit the situation, I handed her 12 beads and wrote down 4 + 4, 5+3. 3+5, 2+6, 6+2, 1 + 7, and 7+1.

After protesting that 5 + 5 (her favorite after 4 + 4) wasn’t on there, she started sorting the beads into two piles.


I noticed:

  • She knew 4 + 4 by memory and did not use the beads. (Same with 5 + 5)
  • She sorted 5 + 3 into a pile of 5 beads and a pile of three beads.
  • She did not grab the beads to do 3 + 5 but rather recognized it was the same two numbers and therefore totaled 8.
  • 6 + 2 required her to count the beads but she did not grab new beads.  She simply rearranged the original piles.
  • At 2 + 6 she was onto me and simply filled in “8” for the remaining answers.

Maria:  Mommy, this math is stupid.

Me:  Why do you think it is stupid, my sweet little bucket of sunshine?

Maria:  All of these are the same answer.

Me:  And what makes that stupid to you?

Maria:  It is just stupid.  Math is stupid. I never want to do math again.


This reaction makes me very curious about where her feelings of “this is stupid” comes from.  She’s only 4.  She has much less experience with “answer getting” than, say, a teenager.  Yet, her evaluation of the task being “stupid” seems to stem from the idea that if the answer is always the same, why do the problem in the first place?  Is the mentality of “answer over process” more innate than we think?  Or is it simply so pervasive in our education system that even my 4 year old has picked up on it?

Number flexibility is something I’ve made routine in my classroom as of late.  Detailing different strategies to arriving at the same result gives students a stronger foundation on which to build algebraic thinking.

Sigh.  It might be a long summer, but she’ll learn I don’t give up so easily.

Bingo Lingo

This time of year, the standards used to measure the success of a lesson may look different than they do at other times of the year.  For example, some teachers might consider “Students not using worksheets to have paper airplane throwing contest” to consistute a lesson well executed.  To a certain extent, I am joking, but there’s a thread of reality there.   Think back to a time when your excitement for a future event prevented you from doing anything productive.  Now imagine leading a room full of 32 people with that same excitement and handing them a manual for their new scanner/copier.  You get the idea.

I can usually distinguish between my being pleased with a lesson based on lowered expectations and my being pleased with a lesson because of a high level of learning and collaboration.  Today was the latter, with my 9th grade probability and statistics class again.

I found this on Don Steward’s website.  If you have seen his blog and are not fascinated, or at least intrigued, we cannot be friends.  He comes up with some amazingly simple, yet elegant classroom problems.

Picture1  Picture2 Picture3


We started this yesterday.  They are in groups of 4; the oldest student in the group got to choose first and so on.  Then they played three “games” using a pair of dice and a whiteboard with their numbers on it.  Today, they worked on figuring out why the “6,7” card was the best and determining how to rearrange the numbers  on the cards to make them all equally likely to win.

I’ve had this glossy paper in my room forever, so I decided to have them make a mini-poster with their solution and some reasoning.  Here are my two favorites:

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Talking Pizza and Pennies

Today was a banner day in my ninth grade probability and statistics class.

First, our number talk was a bite out of the real-world and not the “you and 5 friends share 8 pizzas” kind of real-world.  When my daughter has a babysitter, as she did last night, I usually spring for pizza.  (Yes, our vegan lifestyle maintains a real iron-grip on nutrition when mom and dad are gone.)  I even splurge on the good stuff:  $5 Pizza.

With tax, my vegan-less vice cost $5.36.  I gave the cashier $20.11.  How much change did I receive?

Lots of great strategies:  counting up, counting down, counting to the middle even.  It’s worth noting that the two students in each class that insisted on stacking the numbers and borrowing were not able to do so correctly.  I say this not to discount the standard algorithm.  Rather I wish to point out that in this case, when it’s necessary to borrow three times, the standard algorithm is blatantly inefficient.

The students had to know why on earth I would give the cashier $20.11 rather than just $20.  The answer: Quarters.  Because if you’re at the store with a 4 year old and you do not have a quarter for a gumball machine, god help you.

The main portion of the lesson was the real magic. This problem is from Strength in Numbers by Ilana Horn:

Imagine that you have two pockets and that each pocket contains a penny, a nickel and a dime.  You reach in and remove one coin from each pocket.  Assume that for each pocket, the penny, the nickel, and the dime are equally likely to be removed.  What is the probability that your two coins will total exactly two cents?

They sit in groups of three or four.  I gave each group a large piece of paper, had them put a circle in the middle for their final solution and then divide the paper into 4 sections for their individual work.  When looking through my pictures of student work, I noticed that I have a tendency to capture correct work (but differing methods), but I do not take photos very often of incorrect work.  Today, I changed that.

Here is a sample of their strategies for determining the number of outcomes:

0519151329a 0519150944b 0519150943c 0519150943 0519150943b0519151329a

The level of discussion was exquisite.    But what’s more important was that they were able to work together to organize their thinking and to make sense of their solution.  They built on what they knew an gained conceptual understanding as a result.  In addition, they were able to focus on understanding their path to the solution rather than simply being satisfied with the solution itself.  I’m very proud of them.

Making Math Talks a Habit

How many dots are there?  

One of the best experiences about being a teacher is the opportunity to bear witness to student sense-making.  I enjoy hearing learners help one another develop different ways of approaching problems because I know this is a skill that will transcend mathematics class into when-are-we-ever-going-to-use-this land.

I was first introduced to the idea of a Math Talk when I was taking Jo Boaler’s online course How to Learn Math.  This one is simple enough that anyone able to count can do it.  Seriously, take a second and give this one a go:

How many dots are on the card?  How did you determine your answer?  

How many dots are on the card?  How did you determine your answer?

The answer of ten is hopefully quite obvious to your students.  But it’s the incredible number of ways in which they determined that answer that blows me away.  Is it two rows of 3 and two rows of 2?  Or is it 4 diagonals of 1, 2, 3, and 4?  Maybe 5 in the top 2 rows and 5 in the bottom 2 rows?  Perhaps 5 pairs of vertical dots catches their eye?  THESE ARE JUST DOTS, PEOPLE!  All of this awesome thinking over dots arranged strategically on a piece of paper.  But these dots opened the door to my getting my students to explain their thinking to one another.

Fast forward to MCTM this past weekend.  I was reminded of the power of the Math Talk at a session hosted by Christy Pettis and Terry Wyberg.  I knew Fawn Nguyen had some wonderful examples on her website, so I jumped in.

The results have been lovely.

Monday:  Which is greater 79×25 or 75×29?

Tuesday: Visual Pattern #10

How would you have determined that there were 85 puppies in step 43?

How would you have determined that there were 85 puppies in step 43?

Wednesday:  Which is greater 12/17 or 5/8?

There were many lovely responses to all of these questions in each of my classes. But the one that stands out as my favorite was Caytlin in my 5th period Algebra 2 class.  For Wednesday’s problem, Caytlin says that it’s easier to compare the reciprocals of those fractions, so she flipped them over to compare 17/12 and 8/5.  When converted into a mixed number, 1 and 5/12 is smaller than 1 and 3/5.  The opposite would be true for the reciprocals of the numbers.  Therefore, 12/17 is larger than 5/8 since its reciprocal is smaller.

Honestly, isn’t that golden!?  What I love about math talks is that students are asked to make sense of the problem themselves.  They aren’t shown an example or taught a rule.  They develop their own method and then help their classmates by sharing it.  There have been a lot of good experiences in my classroom this year, and math talks rank up there near the top.

(For additional information on math talks, I recommend the book Making Number Talks Matter by Cathy Humphreys and Ruth Parker)