Monday, May 25, 2015

Questioning Strategies from the #MTBoS

This past week, a fellow asked me about questioning strategies.
I'm not sure where your brain goes when someone asks about questioning, but there's at least a few places my brain goes:
  1. What questioning strategies can we use to unstick students when they're stuck?
  2. What questioning strategies can we use to explore student (mis)understanding?
  3. What questions can we ask when designing lessons/activities/tasks?
  4. What questions can we ask when launching lessons/activities/tasks? 
  5. What questions can we ask...
Thanks to Twitter and the amazing #MTBoS, here's the list you all gave me. Hope this is helpful. If I missed any resources, please add them in the comments.


P.S. Here's the list.

Monday, May 18, 2015

Ketchup (Guess vs. Estimate)

I had breakfast at a restaurant this weekend and noticed the ketchup bottle on the table. You know, the bottles that are red plastic? That are supposed to appear full? I always get a kick out of these bottles. Here's why..
I immediately wonder how much ketchup is in the bottle?

Take a second to think how almost any answer is pure GUESS.

WHAT information would you want to know here to make an estimate and not a guess?
HOW would you go about getting the information to make an estimate and not a guess?

The second I do this...
I know WAY more information. It's no longer a guess.

Think of other senses that could be used to make a better estimate. 
*One scenario would be something along the lines of me watching the customer(s) before me to see how they held the bottle. How did they shake the bottle? How many people at the table used ketchup and how much?
But that's just plain weird...

Sure, pick up the bottle. Formulate an answer and be ready to back it up with a reason. Don't skip this reasoning part. The bill depends on it!

Let's now move to the answer. Let's say we have more information now.
How would you describe your answer?
How might someone else describe their answer?

Could we say any of the following?
  • It's half full.
  • It's about 3 squirts.
  • It's two-thirds empty.
  • It's about a pound.
  • I could eat 5 french fries with that.
  • It's about 8 ounces.
  • [insert other]
Here's where specificity matters. How should we agree to quantify the amount of ketchup in the bottle? Should we agree at all?

This ketchup bottle context is one of the simplest contexts I've come across in awhile. Here's why:
  • The question is straightforward.
  • You demand more information to do anything better than a pure guess.
  • With one small piece of information, your guess should now be an estimate!
In case you're wondering about the time of day, I don't think it really matters here. The bottle was about a quarter full and this was at breakfast time. It's not like someone went around the night before and filled every ketchup bottle. Which begs the question: 
Is it more efficient for an employee to go around lifting all of the ketchup bottles to determine if it needs refilling or should they just wait until a customer says, "The ketchup bottle is empty, can we get a new one?"
Why haven't you seen more Estimation 180 challenges that deal with weight, density, etc? They're tricky to capture. I wish I could fix that, but I digress. I'll put that onus on you. 

My charge to you is:
No matter what grade level you teach, bring in an item like the ketchup bottle. Ask a simple question where the answer is pure guess and students demand more information to make an estimate. Literally, keep track of all the questions/demands students formulate. Report back.
Classroom (or lesson design) application:
  • Design lessons with less. (notice "less" is in "lessons")
  • Ask straightforward questions that demand more information.
  • Use information to move away from guesses and into estimates.
  • Is it more effecient to go around asking our students what they're stuck on and re-filling them with information or should we wait until they realize their stuck and we help them get unstuck?
Lots for me to think about. Feel free to chime in with some advice. Thanks.


P.S. This reminds me of one of my favorite jokes:
A momma tomato and baby tomato are walking down the street. The baby tomato falls behind because it's going slower. The momma tomato turns around and stomps on the baby tomato, yelling "Catch-up!"

Sunday, May 3, 2015

The Ultimate Task for Vertical Planning: Stacking Cups

This past week, I submitted a speaker proposal for NCTM 2016 in San Fransisco. The proposal is for a Grade 6-8 Burst (30 minutes) with the exact same title as this blog post: The Ultimate Task for Vertical Planning: Stacking Cups. I figure if I don't get accepted, at least I can share my thoughts here and you all can help spread the word about my idea if you think it has potential. If it does get accepted, I look forward to giving an update a year from now at NCTM. Here's my session description:
Who says you can't use the same task each year? Come see why Stacking Cups might be the single best secondary math task to get teachers at your school, district, or state to see the importance and necessity of vertical planning. Use tasks that utilize connections from the previous year and extend the mathematics each year. Work smarter, not harder. 
Let's first back up a bit. I attended Alex Overwijk's session at NCTM Boston a few weeks back. I had already read his awesome blog post "Open Strategy Cup Stacking" and knew there are multiple teaching moments with Stacking Cups. I remember teaching Math 8 a few years ago and getting a lot of use out of Stacking Cups as you can see a couple times here and here. I was preparing for a training with math teachers from grades 6-12 and THAT's when it hit me: I could have a room full of math teachers from grades six through twelve and they all could:
  • be working on this task
  • see the different skills and tools necessary for solving
  • know the expectation of each grade level
I've heard comments from teachers numerous times like, 
"Well, if they do File Cabinet in 6th grade, I can't do it in 7th grade with my students."
"If they've done Stacking Cups in Math 8, then I can't do it in Algebra."
"If the 5th grade teachers use Estimation 180 with students, then I can't." 
YES! YOU CAN! It's called vertical planning.

YES, YOU CAN! Instead, let's ask different questions like, "How can we use the same task to extend the mathematics each year?" and  "How can we make connections to prior learning from the previous grade level?"

Let's work smarter, not harder.

I will spend the rest of this blog post highlighting each grade level and suggested uses for Stacking Cups. It won't be complete or the final version as this is through the lens of one person. I'm confident, with your help and critique, we can make it even better.
Math 6
Question: How many cups do we need to stack (alternating) to reach someone's height?
We talk about rate. We organize our information on a number line, in a table, using a tape diagram, etc. We explore the rates using various models.

Math 7
Question 1: How many cups do we need to stack (alternating) to reach someone's height?
We continue the conversation started in Math 6 revolving around rates, using constant of proportionality. All of this can be represented in a table, as an equation, and in a coordinate plane.

Question 2: How many cups do we need to stack (consecutively) to reach someone's height?
We now shift our thinking a bit where there is still a constant increase with each cup, but there is an initial amount (the cup handle). Students explore how to write an equation to represent this situation and solve it.

Question 3What would be possible dimensions of a box that would contain the cups to stack to someone's heightWhich dimensions would be the most cost effective?
Imagine students understanding surface area and volume and how they're related to each other, especially if we model with mathematics, by identifying variables such as:
  • cardboard cost
  • delivery truck capacity 
  • store storage sizes
  • consumer trends with buying cups
  • more

Math 8
Question 1: How many cups do we need to stack (consecutively) to reach someone's height?
Similar to question 2 in Math 7. However, we extend the mathematical understanding as we explore constant rate of change (slope), input and output, linear, and how our situation can be represented in the form y = mx + b.

Question 2: When will two stacks of different sized cups be equal in height and have the same number of cups in each stack?
We introduce students to linear systems using this task. Students can organize the information about each cup in a table. We can extend prior knowledge to represent the situation using graphs, equations, and functions.
*By the end of Math 8, it might be helpful to mention (at least informally) to students the significance of discrete functions.

We tighten up the math (both questions) previously learned in Math 8. How can we extend the mathematics. Add more challenging situations like the stacks start on different objects like desks, boxes, etc.
Question 3: How many cups would we need to stack in a triangular formation to someone's height?
This questions really extends the mathematics for students, but we can still use the tools they've learned from previous grades. Maybe students start by organizing the data in a table. Maybe they graph the data and notice it isn't linear. Maybe we can use desmos with sliders or a line of regression to explore quadratics.

Beyond Algebra and Geometry:
I'll admit this is where I'm a little rusty and would need you high school pros to jump in and contribute. I think with the triangle stacking, it can be taken from quadratic to a divergent series. I've also seen high school teachers come up with the following representations:

Al Overwijk also stacked cups in a triangular pyramid which is awesome.

Let's keep this vertical planning going. If you would like a couple charges, here you go:
Go to your site and/or district and push for Stacking Cups to be a signature task at all sites and secondary grade levels. Help support your colleagues with vertical planning. Report back.
Look for other tasks out there like Robert Kaplinsky's Hot Dogs or Dan Meyer's Penny Circle or Mathalicious' Wheel of Fortune or Graham Fletcher's Water Boy that can be used with vertical planning. Report back.