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!

Wednesday, January 20, 2016

What I Think a Rate Is Right Now

Stating a few assumptions before I get into this:

I'm going to explain how I use the word "rate" and the phrase "unit rate" (and also throw around the word "ratio" somewhat recklessly) and it might not match what's in your textbook or how you use the words in your classroom. Some textbooks proclaim that ratios may only involve like units whereas rates use unlike units. In the physical sciences they typically use "rate" to refer to a measurement with respect to time, specifically. All names for things are conventions. I'm not trying to say that you or your textbook or the physics teacher are wrong. Here is a complete list of the arbiters of correctness when it comes to conventions: 
  1. mathematical consistency
  2. people in the act of communicating about the same situation understand each other
  3. you're not setting up a person for massive confusion later on 
The definition of a trapezoid is a good example. Is a trapezoid a quadrilateral with one and only one pair of parallel sides, or is it a quadrilateral with at least one pair of parallel sides? Said another way, is a parallelogram a special type of trapezoid, or is a parallelogram by definition never also a trapezoid? Answer: ¯\_(ツ)_/¯ It depends on a choice made by a person. Textbooks often present definitions like, "This is what the word means!" when they really mean something more like, "This is a choice we made in order to move forward." 

I'm working on a common core aligned math curriculum for sixth grade. So something to understand as a consequence of that: I'm thinking about how to make these ideas make sense to kids in middle school. So I don't want to write a post about mathematically ironclad definitions that would pass muster with research mathematicians; I want to write a post about stuff that it would be wonderful for kids age 11-13 to understand and is also flexible and useful to build on in later studies.

And one last preliminary: sometimes it's important for teachers to understand some nuances and it's not as important for students to understand them at the same level of detail. So, I'm not suggesting that any of this post is appropriate for instructional or assessment purposes with students. For example, an appropriate question for a student might be "In a fruit punch, the ratio of cups of grape juice to cups of soda water is 2:5. How many cups of grape juice for every cup of soda water?" But this question would not be appropriate for sixth graders: "In the ratio 2:5, what is the unit rate?" Because, ew.

Okay so here we go

A tortoise travels 10 inches in 3 minutes. A snail travels 8 inches in 3 minutes. Are they traveling at the same rate? (Assuming they're both traveling at a constant rate.)

No they are not traveling at the same rate, but I hope you didn't need to compute anything to know that. You can tell because they traveled different distances in the same amount of time. In this context, you have their distances traveled, you have the time it took, but then you have this third thing that means something concrete in the context -- how fast they are going. Their rates. We can express the tortoise's rate as 10 inches in 3 minutes or around 3.33 inches per minute or a foot-and-a-quarter every 270 seconds but the real live concrete in-context rate is the concept of how fast (or in this case, slow) he is moving.

(Note for curriculum nerds: at some point you have to make it explicit to students that "are these happening at the same rate?" is structurally the same question as "are these equivalent ratios?" Not super relevant to this discussion but it seems worth mentioning.)

At one store, 2 pounds of M&M's cost $14. At a different store, 2 pounds of M&M's cost $16.95. Which is a better deal? Did you have to compute anything to know that? No, you can compare the good-deal-ness without computing the cost of 1 pound or how many pounds you can get for $1. The rate is a third thing going on here capturing how-good-is-this-deal that could be expressed in different ways, one of which is a unit price.

Those examples were different types of quantities (distance and time, weight and cost) but we can talk about rates with same quantities like volume and volume. 

Kate mixes 2 oz of gin with 5 oz of tonic water. Ashli mixes 3 oz of gin with 7 oz of tonic water. Are they sipping the same beverage? This is not so easy because none of the quantities match up. So now we need to find how many ounces of gin for one ounce of tonic water for each beverage right? We could, for sure, but we don't have to. We just have to compare equivalent ratios for the same amounts in the different concoctions:


If I used 3 oz of gin to mix a beverage that tastes the same as my original drink, I would need 7.5 oz of tonic water. Ashli only mixed her 3 oz of gin with 7 oz of tonic water, so Ashli's was a bit stronger than mine. Here, that third hidden thing going on is the potency of the beverage, and I'm still asserting that it's a rate, and I still haven't figured out how many of anything per one of anything.
So, let's sum up what we have so far: in any set of equivalent ratios that represents a context, there is a third thing that characterizes something meaningful about those two things happening at the same time. It could be land speed, how much of a good deal you are getting, beverage strength, the tempo of a song (number of beats to number of minutes), how crowded my neighborhood feels (number of people to square miles)... This third thing hidden within a set of equivalent ratios is a concept I'm calling a rate.

But then, it's often convenient to refer to the special equivalent ratio that is something-paired-with-a-one: "how many of these for every one of those?" It is convenient for at least two reasons (and probably more). First, it helps you solve equivalent ratio problems pretty quickly. For example, I know that I get a certain lovely shade of orange acrylic paint if I mix 3 teaspoons of yellow paint with 2 teaspoons of red paint, but I want to make the same shade and I need alot of it so I want to use up the 9 teaspoons of red paint I have on hand. How much yellow paint should I mix it with? I might approach that problem in any number of ways, but a good way is to reason that 2:3 is equivalent to 1:1.5, so to solve 9:? I just need to multiply 1.5 by 9. (This explanation would be clearer if I drew you a ratio table or another double number line but I am getting tired and it's almost cocktail hour.) It's convenient to use a word to name the 1.5, and "unit rate" is as good a name as any. I like how the "unit" part reinforces that it has something to do with 1. A question kids should be able to answer as part of their process is, "What does the 1.5 mean in this context?" and they should be able to say "there are 1.5 teaspoons of yellow paint for every 1 teaspoon of red paint."

Second, it's a way to express that third thing in a set of equivalent ratios with just a single value which can be algorithmetized (like if you want to tell a computer how to do it.) In the gin-and-tonic example above, we could have computed that Kate's drink had 2/5 oz gin for every ounce of tonic water, and Ashli's drink had 3/7 ounce of gin per ounce of tonic water, and since 3/7 is greater than 2/5, Ashli's was stronger.

Then actually later in seventh grade I could write an equation for the relationship that is my recipe, g = 2/5 t, where t is volume of tonic water and g is volume of gin, and 2/5 is re-named the constant of proportionality for the set of all gins and tonics of that particular strength, and I could graph this equation and an equation representing Ashli's recipe and note that the line representing her recipe is steeper, but we're really getting ahead of ourselves here.

Okay, this was a long post, but we're almost done. I believe that my interpretation is supported by the CCSS standards and the RP progression document, although I also believe that those documents also allow you to conclude that rate only means "how many of these for every one of those" (because the only examples they give for "rate" are quantities per 1). But if you're going to use rate to mean how much of this for every one of that, I think you need to come up with another word for that third-thing physical quantity that I am calling rate.

Alright. Comments are on. Come at me, nerds.

Monday, January 4, 2016

In Defense of Unsexy

At IM we're writing a sixth grade curriculum, and much of my time is spent writing, reviewing, and begging other people to write and review new grade 6 tasks that really just hit the fundamental stuff. After five-ish years of accepting task submissions, we have some holes. Because nobody wants to write easy questions.

Do you know what teachers have the hardest time finding?

Quality, basic stuff.

It is getting relatively easy to find the rather-complicated application problems, the projects, the whole new grading systems, the elegant warmups, the pinterest-worthy graphic organizers. Many people have invented and shared some very sexy, awesome stuff, and they are changing many teachers' and kids' experience of math. Hooray for that!

But so many teachers aren't helped by sexy stuff. I think one reason is that they don't think the payoff is worth the time investment. Or maybe that changing the whole way they run class is too intimidating. I'd be happy to entertain alternative theories.

Here's a cooking show analogy: in the early-mid 2000's people enjoyed Mario Batali's homemade gnocchi and Bobby Flay's 90-ingredient curries and Alton Brown's coconut cake that takes THREE DAYS (I'm not kidding. Three days.) But you know who I watched every day at 4:30? Rachel Ray. I suffered through her saying "yummo" and "EVOO-that-means-extra-virgin-olive-oil" approximately 19 times per episode, and she taught me how to get a reasonable meal on a plate in 30 minutes and how to chop a damn onion.

This blog grew in popularity (and stays relatively popular even though I neglect it so) not because I invented something big and sexy but because it offered relatively easy swaps for practice worksheets and ugly, fresh-off-the-smartboard rewrites of high school lessons that made the kids do a tiny bit more thinking than usual.

So I'm starting to hear my low-level angst echoed elsewhere and it's bubbling over. I have a request for you if you are a math teacher and you have a blog.

Share your kinda-borderline boring stuff. Your small tweaks that unloaded the right amount of cognitive lifting onto the kiddos. Your rather-basic task or set of tasks that don't seem that exciting, but your kids always seem to readily grasp that topic. Your snippets of classroom dialog where everybody ended up going OHHHH. Your artful arrangement of pieces of instructional units you found lying around. How you took that cool instructional idea you read in that book and figured out how to do it in a congruent triangles lesson.

We need you. Your kinda-lame-but-seems-to-do-the-trick exponent rule investigation is going to make you somebody's superhero. If you share them with me (add a comment on this post, tweet them at me, whatever), I'll re-share them and compile them in new posts. (And probably they will get added to some of those wonderful virtual filing cabinets and wikis.)

Update: Some gems in the comments. And some shared on Twitter. 


Saturday, November 21, 2015

NCTM Nashville Presentation

I had the pleasure of attending the NCTM regional meeting in Nashville this week. I learned some cool stuff that I'm still processing, and I got to do a presentation. In the presentation I tried to explore whether the way I would rewrite and rework lessons when I was a high school teacher can be generalized and communicated to other people. I was, I think, marginally successful.

NCTM is trying this cool pilot where participants can engage with presenters after the conference. So instead of sharing stuff about my presentation here, I'm going to send you over to the presentation page on their site.

Friday, October 2, 2015

Friday Favorites 7

Happy Friday! (It's really Saturday but I'm going to backdate this post and pretend it's Friday. Ha! Technology!) My reading and favoriting has slowed down because I have made the decision to limit my Twitter time, which is exceptionally mature of me, I think. (Using Stay Focusd, which is a chrome plugin that yells at you for not working. It's brilliant.) What I'm mostly doing these days is a zillion math problems, which is pretty fun, actually... You know how when professional chefs see a bag of onions, they get excited because they get to chop a bag of onions? That's how I feel about doing a bunch of math problems. It's a little bit drudgery, but satisfying. Still and all, when something gets a little mentally difficult it can't be too easy to distract myself. Twitter needs to not be an option in those moments.

This is not a favorite because I made it myself, but it's public, so I might as well share it. It's a place to stash mathematically interesting artifacts that I might turn into tasks or assessment questions or lessons. There's nothing worse than needing to write a question in a context and googling for hours. You're welcome, future Kate.

Now here are real favorites:

Capture Recapture with Goldfish

I did this lab in an Algebra 1 class ages ago. It reminds me of that illustration of statistics vs probability: If you know what's in the bag, reach in and grab a handful, and want to predict what's in your hand, that's probability. If you don't know what's in the bag, reach in and grab a handful, and use the handful to predict what's in the bag, that's statistics. It's a good activity, but my first or second year teacher self probably didn't do such a great job with it. Because, obviously, I didn't have Elizabeth and Julie's helpful writeups. I like the way Elizabeth frames how it fits into a bigger Algebra 1 picture. I could also see using it in a stats lab in a way that emphasizes sampling and sample proportions just as easily as a 7th grade-ish solving proportions lab.

Problematizing Geometry Constructions

I love everything about this. Using a popsicle stick as a straight edge: pro move.

How Parents and Students and Teachers Can Work Better Together

...is a better headline than the clickbaitey one they gave this article. Which is empathetic and treats everyone involved as a professional and a human. Forward anonymously to those parents whose first move is calling the Principal.

Michigan's Teaching and Learning Exploratory

Don't let the boring name fool you - Michigan has done an awesome thing here by posting hours and hours of unedited classroom footage. I learned in the last chapter of Why Don't Students Like School? that looking at video of yourself or someone you know is too scary a place to start, and it's easier to watch and practice constructively critiquing someone you don't know. This resource makes that a whole lot easier.

Thursday, October 1, 2015

Every Bit of This

Link
High schools focus on elementary applications of advanced mathematics whereas most people really make more use of sophisticated applications of elementary mathematics. … Many who master high school mathematics cannot think clearly about percentages or ratios.

Wednesday, September 30, 2015

Exponential Functions and also Area of a Triangle

That title is confusing, right? I know! I just wanted to alert y'all to some tasks that recently went up on Illustrative Mathematics that might address some of your needs, if you are teaching these things.

Exponential Functions: These tasks involve negative exponents in a functional relationship in a context and are aligned with F-LE.
  • Decaying Dice (It's like the penny lab for modeling half-life that kids often do in Earth Science... except with dice.)
  • Predicting the Past (Making sense of negative integers in the domain of a simple exponential growth function.)
  • All Your Base are Belong to Us (Exponential decay and negative exponents, together at last. Bonus points if you get the reference.)
  • DDT-Cay (Interpreting the exponent in a half-life equation.)

Area: These are meant to be used to build understanding as you're working toward a formula for area of a triangle in sixth grade (6-G.1). But they could be useful to reactivate knowledge at the beginning of a study of area in a later Geometry course.
And, hey, it is non-trivial for me to test stuff out with kids these days, so if YOU try them out and you notice stuff or have suggestions, you can comment here or better yet, right on the task on the IM site. (Please let me know if you do that - I don't think I get a notification. And thanks!)

I did draft the initial versions but I can't take credit for these. Tasks published on IM are very much a team effort. Many thanks to Ashli Black who is an ace reviewer and helped me make these a ton better.


Friday, August 28, 2015

Friday Favorites 6

Happy Friday! I am elbows deep in Trello, of all things, but the cat is good company. Here we go...

I took a stab at Activity Builder with an activity that deals with discovering pi and thinking of circumference vs diameter as a proportional relationship. And wow, it's so much better because of their Twitter interaction. I don't know if my favorite part is setting a table to make points draggable only vertically, or their suggestion to share Teacher Notes in a linked google document.

In case you haven't heard, there's also a repository of user-created Desmos activities here. Mileage may vary.

All the Math Talking Points

Are in this shared google folder. If you haven't grokked the magic of Talking Points yet, go read you some cheesemonkey wonders.

OER Curricula and Curricular Outlines

In case I haven't talked your ear off about it yet, I'm of the strong opinion that a school's math department should Decide on a Coherent Curriculum and riff off of that, rather than expecting their teachers to create a curriculum on the fly using random resources they find on the Internet. Some textbook series are good, and there are also decent OER (Open Educational Resource) ones are already out there, and too many people don't know about them.



This Coaching Model

Where your team gets a Teacher Partner - someone who teaches a few classes but also coordinates your collaborative teacher learning. I love this.

I might be a little obsessed with other people's planning documents.

Icebreakers That Won't Make you Cringe

You know what I'm talking about. h/t Lani.

John's Exhaustive Tour of the Good Stuff 

Where do I start? Here.