Listening

If you love board games I can't recommend Qwixx enough.  It's got some chance, some strategy and lots of math!

Without getting too much into game rules or strategies, it will be enough to know that a 7 on your first role (summing a white and any other die) is bad.

K: "What are the chances that you get a 7 on the first role?"

I'm super excited that she's even asking a probability related question, though I immediately recognize this would be a really tough one to answer.

Me: "Well first we have to know how many different outcomes are possible and then decide how many of them have 7s in them.  I think there are going to be some pretty big numbers.  Can we figure out how many outcomes there are first?"

K: "Sure"

Me: "Any thoughts on how many outcomes there are for six dice?"

K: "No."

Me: "How about 1 die?"

K: "6"

Me: "How about 2 dice?"

K: "11"

Me: "How did you get that?"

K: "Is it right?"

Quick aside here.  For years of our mathematical conversations, I haven't been telling K if her answers are right or wrong until after she answers "How did you get that?".  Lately I've dropped the question for arithmetic that I'm confident that she's mastered.  I almost said, no, but thankfully, this time I told her that I wanted her to explain where she got 11 from.

K: "Well you could roll a 2, a 3, a 4 and so on up to a 12. So there are 11 possibilities"

I'm so glad that I took a moment to ask where she got 11 from.  Based on her understanding of the word outcome and the types of questions she was asking 11, makes total sense. 

We continued to talk about how many outcomes there were for 2 dice, which had me acknowledging her answer as correct and offering another, different definition related to the actual dice as opposed to just their sum.  Then we did lots of thinking and listing, eventually working up to a general formula and an answer to how many outcomes exist for 6 dice.  We did not however answer the original question, but that's OK too!

Responsive Instruction in Math Class

Last week, I facilitated a session on responsive instruction in the math classroom.  The structure of the session was to start with a Which One Doesn't Belong, because we always have to start with some math. 

A little twist on this classic, I asked my participants to explain their choice via a creative response in goformative.com without using any words.  Here are some of the responses:

In these two responses, we made the connection between opening up and down and the coefficient of the x^2 term.

These two examples gave us an opportunity to talk about x-intercepts, as well as some representational limitations, with the quadrant picture highlighting that even though we can't see it in the picture, that the top right graph will eventually be in quadrant 2 as well as quadrant 1.  

After we discussed our WODB answers, we got in groups to get to the heart of the session.  Groups were asked to highlight areas that are consistently troublesome for students.  Part of my responsive instruction was to anticipate participant responses, so I guessed that fractions and signed arithmetic would be two areas that were common problems.  Sure enough, my participants agreed, adding also some trouble with creating and solving equations.

Our goal was to highlight trouble areas and come up with ways to help students before, during and after instruction.  The desired outcome was to go back to GoFormative and draw a representation of how to teach the topics with responsive instruction in mind.  To accomplish this, I provided participants with some resources to research their anticipated trouble spots.  We used the Progressions Documents and Nix The Tricks. 

The discussions that ensued with participants started to really blow me away.  Participants went back and forth between the Progressions and Nix The Tricks.  Questions surrounding signed arithmetic started popping up.  Teachers found a lot of usefulness in representing the number line vertically, thinking it would be more intuitive for their students.  Then came a struggle with how to use the number line to represent subtracting a negative.  Here I pivoted the conversation to the order of operations and how students make sense of that.  I referred participants to the PEMDAS section of Nix the Tricks.  We talked about the advantages of switching to GEMA and getting students to understand the definition of subtraction as adding opposites.  I was truly impressed by my participants openness and willingness to engage with these new thoughts on classic topics.  As we debated the merits of the two approaches, we also incorporated the other aspects of GEMA: the lack of division, the idea that we don't need to go left to right (and by extension the commutative and associative properties), the broader aspect of Grouping symbols versus Parentheses, and the connection to inputs in the calculator.  Here are some responses to this activity:

 Working with my colleagues, seeing their excitement about some new approaches, has really invigorated me at the beginning of this summer, and I can't wait to check in with them during the school year to hear how these approaches are going!

Permutations Of Minions

We’re getting ready for my daughter’s Minion themed birthday party.  She’s turning 8.  Where does the time go?  Since we had a snow day today, we were all home during the day and making preparations.  My wife made the poster below.

Kayla couldn’t figure out which minion was which.  She kept saying their names in different orders.

“Kevin, Stuart, Bob”

“Bob, Kevin, Stuart”

Not expecting much conversation I asked how many ways the three could be arranged.  Without shying away from the challenge, she sat and thought for a minute.  Deciding she needed to write down her thinking, she got some paper and made the lists below before confidently announcing her answer of 6.

I praised her for both showing her thinking and the organization of her method.  She had been stating aloud, “well there are 2 ways that they could stand with Kevin in front, then there are two ways that they could stand with Bob in front, and then there are 2 ways that they can stand with Kevin in front”.

Curious how far I could I push, I asked her about including another minion, Dave.  She struggled more with this one, initially listing 17 possibilities.  When I told her that she missed some possibilities, instead of going back to her list, she surprised me by trying to go into some arithmetic.  She started with a “4 times 3 times 4….48”.  I asked her how she got that and she started explaining that there are 4 minions and then you multiply by 3 because there are three left, but she couldn’t explain where the other 4 was from.

She revised her thinking to just the 4 times 3, but then realized that she had already created more than 12 possibilities.  I asked her what she knew about arranging 3 minions.  She said she knew there were 6 ways to do it.  I asked what happens when we add another minion.  She connected the dots and was able to realize that there would be 4 times 6 possibilities for arranging 4 minions, where the 4 was “the number of minions that could be first and 6 was the number of ways that the other 3 could be lined up”. 

To build on the momentum, I asked about adding a fifth minion, Jerry.  Here she struggled again, so my wife asked her what patterns she noticed.  We started listing the ways to line up 2 minions, then 3, then 4.  Kayla recognized the recursive nature of the pattern that a 5^th minion would mean 5 times as many possibilities as 4 minions, so she set to work on multiplying 5 by 24, which was immediately followed by 6 times 120, at which point her desire to do multiplication stopped.

My wife then gave Kayla a brief explanation about what a factorial is, how to calculate it and how to represent it.

 

Besides being ridiculously excited that I could bring Kayla into this conversation, I reflected that this is exactly the type of You do, We do, I do model of teaching that I espoused in my Ignite talk. 

You Do – Kayla had a prompt about lining up minions and wrestled with the task with no input from me.

We Do – In the extension problems of 4 and 5 minions, my wife and I asked Kayla questions to understand her thinking and refine her thought process.  By asking about the origins of the values in her arithmetic, she was able to independently identify flaws in her reasoning.  Some direct guidance was also provided.  I told her 17 was not correct, but didn’t offer any additional strategies.

I Do – Once a pattern was identified, Kayla was in a place where she could attach some new vocabulary (factorial) to an experience.