Alert!

Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!

Friday, June 9, 2017

FAQ: So When Do I Teach?

We are putting the finishing touches on the IM middle school math curriculum published by Open Up Resources (For early access to sample units in the pilot, you'll have to share your contact info with us here, but we're looking at mid-July for the release of version 1.)

We're often in the position of talking to teachers who have heard about the materials and are evaluating them, or whose district has adopted them and they are just learning about them. I'm putting together a FAQ for people in our organization so they are prepared for questions we know they will get. I am thinking to hash some of the Q's out in blog form, first. So theoretically this one in the first in a series. If you want to fight with me on anything I have to say, please speak up!

Imagine this scenario: you demonstrate a problem-based activity with a group of teachers. You let them know that this is a grade 6 task where students have already learned to use double number lines and tables to represent a set of equivalent ratios. By this point, students are also familiar with recipe contexts; they know that an equivalent ratio of a recipe tastes the same. Here is the task:

Lin and Noah each have their own recipe for making sparkling orange juice.
  • Lin mixes 3 liters of orange juice with 4 liters of soda water.
  • Noah mixes 4 liters of orange juice with 5 liters of soda water.
How do the two mixtures compare in taste? Explain your reasoning.

The task is launched with a notice and wonder, they start happily working away, and you monitor what they are doing. You invite a few of them to make their reasoning visible to everyone, deliberately selecting them to share in a way that highlights a particular nuance you want to make sure everyone will understand, making mathematical connections between their approaches. (If you're savvy, you'll recognize this structure as Smith and Stein's 5 Practices, though my short description here isn't really doing it justice.) After conducting this discussion, many voices have contributed. Earlier in the day, you did another activity that loosely followed this same structure. You think, hey, I've done a pretty good job demonstrating the basics of how a problem-based classroom is meant to operate.

Then you get the question, maybe timid but very curious, "So, when do I teach?"

So here is a response that I'm turning over.
Can you say a little more about what it looks like when you teach, as it looks in your mind, here? Okay, it sounds like synonyms for what you are describing might be telling or explaining. Is that fair? Okay. It's expected that you'll do some telling and explaining when using our stuff as it's meant to be used. The difference is in the timing. Let's think about what we did in the sparkling orange juice activity. You had a chance to work on a task, a few people shared their approaches, and then we made some observations about their approaches. What do you think the mathematical learning goal of that activity was? 
"Well, I remember seeing two sets of equivalent ratios represented with a double number line and with a table, and then so-and-so explained how she computed how much orange juice for 1 liter of soda water for both mixtures. It seemed like the point was that when you want to know which mixture tastes stronger, you need to create equivalent ratios so that one of the quantities is the same for each mixture. For example if orange juice to soda water is expressed as $15:20$ and $16:20$, you know that the second recipe tastes stronger." 
Okay cool. Do you think you got out of that activity what was intended? Does that mean you learned something? Does that mean teaching happened? 
There's still telling and explaining. Mathematical playtime is awesome, but a problem-based classroom is not just about mathematical playtime. We have clear learning goals for the course, each instructional unit, each lesson, and each activity. 
The way it's different than you might be used to is when the explaining happens. Perhaps you are used to first explaining something, and then kids do some work on the thing you just explained. In problem-based instruction, this is reversed. Kids have a chance to try and figure some stuff out first, you see what they come up with, and then after they've had a chance to get good and familiar with the context, the question being asked, the constraints, and they at least make some progress. . . then you take steps to make sure the relevant learning goals are made visible. Sometimes this part looks like explaining or telling.
I'd suggest that teaching is a really broad and complex set of skills and behaviors, and telling or explaining is just one of them, and that telling or explaining isn't the only way to help kids understand something. In fact, does that approach work well for every student? How much do your students remember of what you explained the next day, or the next week? If you're completely satisfied with how things are going, awesome, but I bet you're here because either you or someone in your school endeavored to look for ways of conducting a math class that might work better for more kids, so that things made sense to them and the learning stuck around. 

Did I miss anything to address this particular question? (Please note that this is one vignette from two days of learning, and we spend time on a whole bunch of other things as well.) Does any of that come across badly? I want to acknowledge the person's completely understandable discomfort but also not shy away from asserting that teaching and learning happen in a problem-based classroom, and that we did it this way because we think better teaching and learning happen.

Friday, March 3, 2017

Anyone Want to Classroom Test Something? (grade 7)

Hi! We are field testing all of our new materials in pilot schools, but I have one activity where the first draft was unworkable, and we have to come up with something totally new, and since the pilot schools are past this point I can't throw another version back to them. So...Internet... want to try something out for me? This is working toward the CCSS standard 7.EE.B.4a, so it's for seventh graders or students working on grade 7 material. The assumption is that they already have some strategies for reasoning about and solving equations of the form p(x+q)=r and px+q=r but that throwing negative numbers into the mix is relatively new.

Mainly what I am worried about here is that question 2 will go awry and students will go overboard and way far away from equation types they know about. And I also don't know whether that would be a good thing that students and teachers can just roll with, or if it's going to present challenges that are too much for too many people.

So, if (and only if) this fits in with your plans, please try it out and let me know how it goes! Thanks in advance!

Okay here's the task:


1. Here are some equations that all have the same solution. Explain how you know that each equation has the same solution as the previous equation. Pause for discussion before moving to the next question.


x = -2
x - 3 = -5
-5 = x - 3
500 = -100(x - 3)
500 = (x - 3) ᐧ -100
500 = -100x + 300

2. Keep your work secret from your partner. Start with the equation -5 = x. Do the same thing to each side at least three times to create an equation that has the same solution as the starting equation.

3. Write the equation you ended up with on a slip of paper, and trade equations with your partner. See if you can figure out what steps they used to transform -5 = x into their equation. When you think you know, check with them to see if you are right.

Saturday, February 25, 2017

Is This Thing On?

Hello, Blogoworld! I'm not sure if anyone is still listening, but if you are, I have a short assignment for you. I'm preparing a talk where I'll show different people's sample work to the same problem. So I'd like to collect a bunch of different responses. Here is the problem:

A sloth can go 50 feet in 7 and a half minutes. How far can it go in an hour and a half?

If you'd like to participate, I need a good photo of your hand-written work. Upload it wherever, and share in the comments of this post. Bonus points for use of representations with more structure than dividing and multiplying. If you have access to a young person, it would be cool to have some samples that are in little kid handwriting. Thank you!