Proofs (and writing) are difficult

The moment I started to have success helping student really learn how to write proof in geometry was the moment I realized that”The Proof” is nothing more than a persuasive essay converted to math class. It’s disciplinary literacy. And thinking of them as mathy-style essays can help us isolate some of the reasons the students struggle with proof in general. My experience leads me to think that many of the struggles are the same the ones the students experience with writing outside of math class. They don’t understand the structure, they don’t appreciate the value of honoring their target audience, and they don’t understand the content well enough.

Luckily for us, the ELA department has often hit those three points really hard. As math teachers, we just need to help the students bridge the gap.

What’s the structure to an essay? Thesis, supporting paragraphs, conclusion. Or, in math, “Given angle a is congruent to b, I’ll prove that segment a is congruent to segment b. Here’s the evidence I’m using to support my claim. And here’s what I just proved.”

Who’s the target audience? In math, it’s often either someone who doesn’t understand or someone who disagrees with you. That explains why you need to back up each statement with theorem, definition, or previously proven statement. Take nothing for granted or you’ll lose your reader.

And as for content? Well, have you ever read an essay from someone who plain ol’ doesn’t know what they are talking about? The best structure in the world isn’t going to save them if they can’t define the words they are trying to use.

So, when it comes to proof-writing, I think we math teachers need to appreciate that “writing” really is at the core of it, and the better we make that connection explicit to our proof-learning students, the more likely they are to be successful. And perhaps there’s a role for some meaning collaboration between high school math and ELA departments.

And with that, enjoy the latest “Instructional Tech in Under 3 Minutes” discussing, of all things, writing.

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New Podcast: Why I love this picture

So, I get the pleasure of supporting a different school community than I have been these past couple of years. As a part of that work, I have decided to start a short video podcast. (The Geogebra post was the first episode.)
Occasionally, that means I’ll be borrowing “thegeometryteacher” content and podcasting new life into it. This is one of those examples. I originally posted this last fall…

 

Here’s the picture…

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I took this picture this morning in Lansing, MI during some wonderful small group math talk. There is one device, an iPad, with an Osmo setup attached to it.

So, here’s why I love this picture.

There’s tech and…

… manipulatives and whiteboard markers and collaboration. Tech fits among the variety of tools available. It’s not the best tool unless it best supports the learning. And sometimes other tools work better. And in this case, the students were being led into learning with all the different tools.

The activity is built around the social nature of learning. 

The kids are clearly sharing their answers with each other and the teacher… there is a constant back-and-forth, sharing ideas and discussing them. They were seeing each other ideas, but…

Their strategies aren’t all the same. 

One girl is using an array. One girl is using groups of three. One girl wasn’t quite sure what to do (so it was a good thing she could see the other two girls’ work.)

The teacher’s hands are off. 

The students are doing the reaching, arranging, manipulating. Remember, the one that does the work will be the one that does the learning.

That’s why I love this picture. it captures so many wonderful things about the right kind of teaching and learning.

Rediscovering Modeling in Professional Learning

Teachers? How many of your PD presenters were willing to come into your classroom and demonstrate what you’re learning live with students in their natural element?

Principals? If a teacher you’re evaluating needs support in instructional or classroom management strategies, do you feel equipped to show them how it’s done?

PD Presenters? Do you ever get the opportunity to teach alongside someone who is learning from you?

These are the moves that make a difference. I’ve recently been reminded of this.

Since September, I’ve been involved in a new professional learning model that is built around job-embedded learning opportunities for one main reason.

To see if it works better. And it does.

Bad professional development is the worst-kept secret in education. I’ve attended them. Heck, I’ve given them. I’ve been called in to present some tech tool for a half-day to some captive staff and never heard from any of them again. Now, I’ve been told I put on a pretty good show. We laughed some. I used some fancy strategies.

But, I doubt they learned a thing. And what’s worse? Everyone seemed cool with that.

Well, our team stopped being cool with that. If it’s worth training, it’s worth putting a structure in place that will actually impact teacher and student experiences. And it required re-discovering modeling in the classroom.

And so, the former HS geometry teacher who’s last year in the classroom was 2014 with mostly 10th and 11th graders is going into early elementary classrooms and teaching math.

I promise, the students aren’t the only one learning something. Because I’ve discovered a paradox. In many ways, good teaching is good teaching. And in other ways, the early elementary classroom is a whole different world than the 10th grade classroom.

It isn’t always pretty. It is NEVER perfect. But it is almost always productive. And that is a massive step in the right direction from the standard remembrances of PD’s past.

Because here’s the reality. Can an elementary teacher learn from a HS teacher? Yes. But talk only goes so far. The PD presenter might say, “Your students need more opportunities to respond during your whole group time”. It is perfectly reasonable for the learner to say, “Can you show me what that looks like?” And instead of a cheeky demo on-the-spot, you make an appointment and a plan and go and teach that teacher’s students.

The feedback has been overwhelming. And the impact on teacher practice has followed suit.

And the stated difference in the feedback is the modeling. That has changed the game.

So, PD presenters: What options do you have to connect with folks you are presenting to? How might you get into classrooms to demonstrate?

Principals: How does your credibility spike when you can own a classroom for a half-hour to demonstrate good practice?

Teachers: If you have a trouble area in your practice, invite someone in.

It’s high time we start holding our professional learning to a higher standard.

Perplexity and how it appears…

Here’s a video (by Derek Alexander Muller) I think you should watch.

 

The critique of #flipclass aside, I’m intrigued by the way the narrator describes the value of “bringing up the misconception”. It’s almost like a thorn that creates some discomfort that only learning will relieve. This gets close to Dan Meyer’s use of the word “perplexity”.

From Dr. Meyer: “Perplexity comes along once in a while. What is it? It’s when a kid doesn’t know something, wants to know that thing, and believes that knowing that thing is within her power. That right there is some of the most powerful learning moments I’ve ever seen – so powerful that it’s really hard for me as a teacher to mess those up.”

There’s power in perplexity. I’ve seen this in my classroom on multiple occasions. It’s important to remember that there’s three distinct parts to creating what Dr. Meyer is describing. First, there needs to be something worth knowing. Second, you have to create the want. And finally, we need to empower the students so they feel enabled to know that thing. What Dr. Alexander suggests is that becoming aware of your misconception seems unsettling (leading to claims that videos were confusing), but also leads to more learning. The discomfort fed a drive to resolve the discomfort.

The tricky thing is that misconceptions are a tool you can use when they are available. Science provides a particularly fertile ground for misconceptions because so much of it is drawn from experiences many of us have regularly. Alexander uses the model of a ball flying through the air. This video uses the phases of the moon and the seasons.

The potential for misconceptions is necessarily lightened when there’s no misconceptions, so the quest for perplexity in math needs to take on a different look, proper planning and timing, and different strategies for when perplexity isn’t an available option. (Preconceived notions are just as good at times. After all, we ALL think we know something about squares!)

It’s like Dr. Meyer says, those wonderful perplexing moments only come along once in a while. We foster those moments when we have them, try to create as many as we can and we do our best every other time.

 

Math Talk by Necessity: a 4-year-old’s story

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My 4-year-old is chatty. And while he will occasionally talk to himself, he’d much rather talk to you. And to him nothing is more frustrating than not being understood.

So, he’s realized that he needs detail. Words like “lots” or “just a little” or “soon” are easily misunderstood. Like when Dad says, “Hang on, dinner will be ready soon.” He’d prefer to have a better understanding of whether you mean 30-seconds soon or 10-minutes soon.

The math teachers among us see this as adding units to your answer. We ask our students, “What was the final speed of the car?” We don’t expect them to say, “fast.” We’d prefer something like “54 m/s”. So would my 4-year-old. “Soon” is too loose a word.

Except, most of the typical units don’t mean anything to him because he is still developing his understanding of what “a mile” is. First, that it’s a unit of distance. Second, how far is one of those. These two things are essential to making meaning out of a statement like, “Well, the store is 6 miles from here.” Without those, statements like “This box is heavy. It’s, like, 6 miles!” are not uncommon around my house right now.

But yesterday, something new happened. Together, he and I made Alton Brown’s Hot Cocoa Mix . To go from ingredients to a drink, you have four units of measure. (cups, teaspoons, tablespoons, and ounces)

So, midway through as we were chatting about this and that… together making the distinction between teaspoons and tablespoons… why two cups of one ingredient looks so much different than two teaspoons of another… stuff like that. Then, we came to the part where we needed to scoop the mix into the mug and I handed him a “scooper” and he says, “I need two… two… of THESE, but I don’t know what it’s NAME is? Dad… what’s this one’s name?”

Did you catch that? There was a step forward in that.

He knew the number wasn’t enough. “Put two in there” is really easily misunderstood. That would be a problem. Two of what?

He also knew that it totally mattered what “name” he gave the scooper. If he called it “a cup”, he risked calling it the wrong thing. Two cups of mix makes a totally different amount of hot cocoa that two tablespoons. And both could be called “a scoop”. (Two scoops from the coffee can is different that the two scoops of raisins on the cereal box.)

Finally, he figured out that this thing had a name. That I knew it. That he needed to know it and that it wouldn’t work to make it up. If he is going to communicate this amount to others, he needed to know the ACTUAL name of this scooper. Not his preferred name.

There you have it. Math talk by necessity. Lends credence to the notion that if you put students in rich enough environments, you won’t have to mandate good math talk. They simply won’t be able to communicate effectively without it.

Now, we just have to deal with the fact that when Chef Alton says “two tablespoons”, Dad tries to measure accurately while the 4-year-old would prefer that mean “the maximum possible amount of hot cocoa mix that the tablespoon will transfer to the mug.”

Growth Mindset – It’s not that simple

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Photo Credit:  Flickr user “WoodleyWonderWorks”

I am often not on the intellectual cutting edge. Pretty much any idea that I’ve had, someone else has had before. Usually lots of people, actually.

This discussion of growth mindset is no different. The Atlantic (specifically Christine Gross-Loh) and the Harvard Business Review (written by Growth Mindset author Carol Dweck) beat me to this one. And that’s okay. They have folks whose job it is to write. I do this as a hobby. I’m just grateful for their shoulders to stand on.

Growth Mindset is becoming a fad. Of course, Dr. Carol Dweck (who is responsible for coining the term and the supporting research) understands this better than anyone. What will happen to the term “growth mindset” when it gets unleashed on the public and Twitter gets a hold of it?

Well, certainly the nuance is gone. The public arena (and it’s right arm: social media) tends to treat nuanced arguments like they belong on a dollar-store clearance rack. So, at the very least, you can count on growth mindset being over-simplified (probably quite significantly at that).

With over simplification comes the inevitable misunderstanding. If all you know about this idea a series of talking points laid out 140-characters at a time, the chances of you missing an essential component are quite good. Of course accidental misunderstanding is independent to one’s confidence to propagate their (mis)understandings on the topic.

And of course, researchers have a love-hate relationship with this phenomena. On one hand, they are getting often getting criticized for statements made by people who don’t really understand the actual point they are making. Not fair, but common. (Ask our Lord Jesus Christ, Charles Darwin, Martin Luther…).

On the other hand, your idea becoming a household-familiar is evidence of success. The constant (though flawed) propagation of your idea means you’ve created something that’s resonated with the masses. (Or, as Chamillionare said, “Weird Al isn’t going to do a parody of your song if you’re not doing it big.”)

And of course, this explains why growth mindset is talked about with such a wide range of implications. Sometimes, it takes more than Yoda and Darth Vader posters to explain complex topics… particularly topics wrapped around the way an individual sees the world around them.

Says Dweck (in The Atlantic article), “Many people understood growth mindset deeply and implemented it in a very sophisticated and effective way. However, there were many others who understood it in a way that wasn’t quite accurate, or distilled it down to something that wasn’t quite effective, or assimilated it into something they already knew.”

As with most things, in order to experience the full benefit, there needs to be fidelity with the implementation. Otherwise, what gets left out might be the thing that makes it effective.

From The Atlantic article:

But Dweck recently noticed a trend: a widespread embrace of what she refers to as “false growth mindset”—a misunderstanding of the idea’s core message. Growth mindset’s popularity was leading some educators to believe that it was simpler than it was, that it was only about putting forth effort or that a teacher could foster growth mindset merely by telling kids to try hard. A teacher might applaud a child for making an effort on a science test even if he’d failed it, for instance, believing that doing so would promote growth mindset in that student regardless of the outcome.

Or as Dweck says later in the piece:

another misunderstanding [of growth mindset] that might apply to lower-achieving children is the oversimplification of growth mindset into just [being about] effort. Teachers were just praising effort that was not effective, saying “Wow, you tried really hard!” But students know that if they didn’t make progress and you’re praising them, it’s a consolation prize. They also know you think they can’t do any better.

So, there’s a little more to fostering Growth Mindset than “praise the effort, not the outcome.” A lot more, it turns out. From Dweck’s article in Harvard Business Review:

“Even if we correct these misconceptions, it’s still not easy to attain a growth mindset. One reason why is we all have our own fixed-mindset triggers. When we face challenges, receive criticism, or fare poorly compared with others, we can easily fall into insecurity or defensiveness, a response that inhibits growth. Our work environments, too, can be full of fixed-mindset triggers.”

It involves really understanding our own inner balance of fixed and growth mindsets. Growth mindset isn’t the easiest way. It’s the most valuable way. Are you sure you really want a growth mindset? Remember, thinking you’ve maxed out your ability provides a neat little excuse to only “do your best” as opposed to what WMU football coach PJ Fleck regularly calls “changing your best.” And even if you are convinced in the value of growth mindset, you still have to convince your students.

Dweck beat me to that one, too. From the Atlantic article:

Finally we talked about why someone would want a growth mindset. We realized that some kids would be overjoyed to hear you can develop your intellectual abilities, but others might not think it was the most exciting thing. So we then had a whole section on why you might want to develop your mind. Teenagers are really excited about the idea that they can do something to make the world a better place. So we asked them what they want to make their contribution to in the future—family, community, or societal problems—and then talked about how having a strong mind could help them make their future contribution.

Nestled deep within growth mindset is “growth”. Growth toward what? Well, certainly away from what we currently are (because it’s not good enough yet) toward something else (presumably something better.) This mindset is fundamentally at odds with a philosophy that says, “Don’t worry. You’re perfect just the way you are.” If that were true, why would we need growth? Why are we trying like crazy to help our students “grow”? Why should we be trying like crazy to model that in ourselves?

Certainly because we’re not all “perfect just the way we are.” In fact, in a time of struggle, it might be more helpful to be told, “perfect or not, you’re not stuck the way you are. You can be more. I’ll help you. And if you try for it, your effort will be rewarded.” That’s what growth  mindset brings to the table.

So, how do you operationalize that? We’ll have to save that for another post. (Although, as usual, I’d love to hear your thoughts.)

 

Why I love this picture…

I want to tell you why I love this picture.

2016-11-15-09-06-34

I took this picture this morning in Lansing, MI during some wonderful small group math talk. There is one device, an iPad, with an Osmo setup attached to it.

So, here’s why I love this picture.

There’s tech and…

… manipulatives and whiteboard markers and collaboration. Tech fits among the variety of tools available. It’s not the best tool unless it best supports the learning. And sometimes other tools work better. And in this case, the students were being led into learning with all the different tools.

The activity is built around the social nature of learning. 

The kids are clearly sharing their answers with each other and the teacher… there is a constant back-and-forth, sharing ideas and discussing them. They were seeing each other ideas, but…

Their strategies aren’t all the same. 

One girl is using an array. One girl is using groups of three. One girl wasn’t quite sure what to do (so it was a good thing she could see the other two girls’ work.)

The teacher’s hands are off. 

The students are doing the reaching, arranging, manipulating. Remember, the one that does the work will be the one that does the learning.

That’s why I love this picture. it captures so many wonderful things about the right kind of teaching and learning.