I Like Triangles

Last night, I asked if anyone could point me back to this fantastic animated factorization visualization. (h/t @calcdave)

Now, I'm kicking myself for not thinking to use this in the first weeks of the school year.  Talk about some Fake World math doing a number on pseudo engagement strategies.

I started the animation at the end of each period and walked out to greet students as they walked in.  Once everyone got settled, I walked back in the room and each time the dots would circle up, I'd yell, "PRIME!"

"Alright, I'm up 1-0. PRIME!, man I'm smoking you guys."

Kids caught on really quick and started looking for the circled numbers.  In fact, it took many students a while to realize that the applet literally said "prime."

We started out by looking at the patterns and how each number was represented visually.

But, next I said, "You know what, I really like triangles. What is the smallest number that will give us a triangle?"

This one's easy. 

"Alright, what's the next number that will give us nothing but triangles?  Write your guesses on your easel."

Guesses were about 50-50 between 6 and 9. 

"Alright, what about the next one?"  

Still guesses were a little sporadic.  But by the time we got to 81, most students thought they figured out a pattern.  From 243 on, we were at about 100%. 

After about 10 minutes of doing this and discussing our results, I put up this slide.

There was a nice discussion on clarifying our question.  Three different student offerings illustrated the idea of First Idea; Best Idea. 

Student 1:  At what stage is each triangle?

Student 2:  How many triangles are in the green circle?

Student 3:  How many dots are there in each circle?  *Boom*

Now get to it and be prepared to justify your answer.  

The first student said there were 9, 27, 81 and 243 dots.

"Ok, great. So how did you do that?"

"Well, the green circle has 9 dots, then I multiplied by 3 to get the red. Multiplied by 3 again to get the blue and then by 3 again to get the black."

"Alright, so let's press on this idea a little."

I know what question I want to ask, but I just bit my tongue until a student speaks up.

"How can you be sure that there are 9 dots in the green circle?"  *There it is*

"I estimated."

Here's where it gets good. 

 From across the room, I hear, "It looks like there are 9 dots in the green circle, but we have to look past that."

Wait, what?

"Yeah, we can't trust the picture because the dots are too small.  We know there are 6,561 dots on the whole page and there are three black circles of dots.  We have to start there."

So, Dave, keep this link handy, I'm sure I'll be asking for it again next year.  

First Idea; Best Idea...

...and the Worst Idea

Creating a Culture of Questions was, by far, the most popular post on this blog until someone somewhere starting linking to the post on Exponent Rules.

I think a natural follow up to the Culture piece would be with regards to establishing a classroom culture where feedback is given and accepted.


The First Idea is the Best Idea and the Worst Idea

The first time students hear this, I usually get, "Gosh, that's mean."

But we discuss how the first person who puts forth an idea holds the best idea as there is nothing to which we can compare it.  But using the same logic, this idea should be the worst.  This assumes the flow of ideas that should follow.

I think this encourages two important things:

1.  "If I go first, it doesn't matter that my idea isn't fully formed."  This student has established a floor on which each other student can stand and/or build.

2.  "I can take someone's idea and help them make it better."  The real work is done by the first follower.  This student chips away at any imperfections and helps the first student refine her idea.  Subsequent students then follow suit.


What's this look like?

Yesterday, we trying to determine the equation between the points below and students wanted the y-intercept.

Students were using what they knew about slope to find other points and had to wrestle with the fact this particular line doesn't have a lattice point for a y-intercept.  Once we were finished, I asked students to write down any questions they had.

Student 1:  "I have a comment."

"Ok, what is it?"

Student 1: "No matter which points we choose, the slope simplifies to the same thing."

"Can you turn your observation into a question?"

Student 1: "Will that happen all the time?"

Now here is where it happens.

"I can misunderstand [Student 1]'s question, can we make this more precise?"

Student 2: "Will the slopes always simplify to the same thing?"

Student 3: "Will the slopes between two points always simplify to the same thing?"

"Are we only using two points?"

Student 4: "Will the slopes between three points always simplify to the same thing?"

Student 5:  "Will the slopes between any two pairs of points always simplify to the same thing?"

Student 6: "Are the slopes between any two pairs of points always equal?"

"Are we really talking about any 4 points here?"

Student 7: "Are the slopes between any two pairs of points on a line always equal?"

( )conceptions

Preconceptions?  Misconceptions?  Heck, I don't care what we call them.  All I know is that I have kids coming to class and making decisions with their heart and not their head.  Intuition is great.  Inductive logic is great.  But it just isn't enough.  Back it up.  Verify it.  Embrace the conflict that arises when what you thought was true turns out to be, well, not so much.  

I've taken to putting these (   )conceptions front and center.  Put them out there for kids to wrestle with. Plug in some numbers.  Argue.  Get frustrated.  And then walk away with a little more understanding than they did before.

Today's episode centered on the equation:
(2/3) (3x + 14) = 7x + 6 and students were asked to multiply both sides by 3.

And, of course, they came up with 2(9x + 42) = 21x + 18.

Why?

Well because, naturally, a(bc) = (ab) (bc).

So what do you do?

The younger me would have said something profound like, "You don't distribute multiplication over multiplication.  I'll say it again slower for those of you taking notes.  You. Don't. Distribute. Multiplication. Over. Multiplication."

There ya go. My finest pedagogical moments summed up by slowly repeating a negative definition of a property they obviously don't fully understand.

The older, wiser, 5-kid-having self is a bit more patient.

Up on the board goes:

True or False

2(3 · 4) = 2(3) · 2(4)

and

2( 3 + 4 ) = 2(3) + 2(4)

Most kids said that their gut told them that both equations were true.  In fact, many said, "true" before the virtual ink had dried.

"But what does your head tell you?  Verify that both equations are true."

Oh, no they're not both true.  

"Ok, good.  So now you have a conflict.  What you think should be true is different than what you know is true.  Why?"

This is why I have been calling these things preconceptions.  Students bring something to the task.  Always.  They never come empty handed.  These responses that #needaredstamp are usually a right idea used at the wrong time.  It's like a kid who has never played sports before goes from learning basketball to soccer.  Coach says dribble and the kid picks up the ball and bounces it as he runs down the pitch.  Right rule; wrong application.

I've had kids tell me that they do certain operations on a problem because "it just felt right."  I'm not sure how to address that other than to put them in a position for their feelings to betray them and help them deal with the disappointment in a constructive way.

Next weeks episode:  Why Love Isn't an Emotion

Creating a Culture of Questions

Virtual Conference on Soft Skills _[1]

[Note: This post was pretty much written by all of you.  There are no unique ideas here (save for a few anecdotes) but I still found it productive to try to flesh out exactly what it is that I do to promote this culture of questions in my classroom. It's a huge part of what I do, but I'm afraid my ability to articulate the process may be lacking.  I asked for a bunch of feedback via facebook/email from former students for this; kids ranging from the classes of '99 to '14 and they really helped shape this post.]  So here it goes:

I learned how to learn when I was in college.  No one told me.  It just happened.  As a teacher I have tried to help this process along a bit for my students because it kinda pissed me off that I spent 14 years in school and no one actually told me, "Learning is about the questions you ask, not the answers."  So that pretty sums up my teaching philosophy.  It hasn't  changed much in 16 years.

Kids don't get that.  They think that as long as they get the right answer, who cares about the how and the why?  Questions and answers are on opposite teams.  Answers get my work put on the refrigerator.  Questions mean I don't know the answers.  Answers mean I know.  Questions mean I don't know. I can't let 'em know I don't know.

That's wrong. It's our job to change that.  It's not easy and you have to set the table for yourself.  The pillars to a questioning classroom involve: Truth, Trust, Togetherness and Transparency.

Truth

I talk to my students.  A lot. In just about every year I've taught, I've had some variation of the same discussion with my classes.  For some classes, this talk happens in the first week.  For others it doesn't happen until the second semester.  But inevitably, it happens. 

It goes something like this:

Suppose all the information in the world is contained in this circle.

Here's you.

Some of them get offended until I show them this:

We're kinda in the same boat.  Not much to brag about and not much to be embarrassed about.

What happens when we share?

And the more we learn, the sooner we realize that this circle is waaaay too small. 
If our goal is to learn everything there is to learn, then we are chasing a finish line that's moving away from us much faster than we're moving towards it. 

The only way we begin to know is when we realize that we don't know...and become OK with that.

So, how do we share our knowledge?

Questions.


Trust 

Once we've established that we're all really in the same boat, there's no excuse for false pride. It's time to build trust.  Do I really believe what I told them?  I make it very clear to my students that I'm not smarter than they are; I just have more experience.  I make it very clear that I don't tolerate anyone looking down on others for asking questions.  I'm not so much of a there's no such thing as a dumb question kinda guy as much as I am a who are you to look down on someone who doesn't know? You didn't know the first time you tried kinda guy. This always leads to a digression about the first time we all tried to walk or ride a bike (my kids turn out to be great fodder for these kinds of discussions) which furthers the trust factor as I let them into my life a bit. 

I don't let my students say, "That's easy" when someone asks a question because it deflates the questioner.  Quickly.   First time that comes out of someone's mouth I say, "Everything's easy once you get it."

I have to be honest, establishing trust was much easier for me when I taught high school.  It's been a bit more of a challenge with 7th and 8th graders because I have a Kate-ish thing for truth.  Sometimes I have to dial it back a bit without becoming falsely warm and fuzzy. 

Togetherness

It takes some students quite a while to adapt to my questioning style in class.  I've had kids want to drop my class (especially when I was at the high school) because "he doesn't give me the answers" "he never answers my questions."  It's tough sometimes because kids are resolute.  They'll try to corner you into taking the pencil out of their hand.  The key is consistency.  The more questions I ask, the more willing they are to ask.

One of the things I've done to help establish a willingness to ask questions is give the class lateral thinking puzzles.  I mean, c'mon,  there's no way you can guess the answer on some of these on the first try.  Asking questions and being wrong are part of the process because they help us eliminate potential possibilities.  It takes kids time to figure this out because they want to be specific with their questions at first which is very counterproductive.  They have to learn to start with very general questions and the yes/no answer tells them whether or not to continue down that path.  Once they get the hang of this, they start to realize that there are common themes in many of these puzzles which can be applied to later puzzles.  This works nicely with problem solving strategies down the road.

At some point I throw out the challenge for them to find a lateral thinking puzzle that will stump me, which can't be done (or so I say).  The trash talk ensues and I model the heck out of how one goes from the very general question to more specific questions.  

The important thing here is that we do these as a class, together. 

Transparency  

Transparency has two meanings here: The I'm-not-here-to-trick-you-here's-really-what-I expect kind of transparency and the physical-posture-I'll-take-in-class-so-become-invisible kind of transparency.

Expectations
Expectations/standards/topics...whatever, have to be absolutely clear.  The What? can't change.  It's the How? that is up for discussion.  I try things. I show my students first hand that I don't have all the answers and I am constantly trying to find better ways to help them learn.  I fail. A lot.  That's because I don't believe I have to have all the kinks worked out before I do something with my students.  Our entire class is one big question and that question is this:

What do we need to do in order for us to learn?

Invisibility
I don't have just one (physical) focal point in my room.  I have a SmartBoard on one wall, a dry erase board on the adjacent wall, multiple dry erase easels spread out throughout the room and I roam around using a wireless tablet that lets me annotate my slides.  The focal point is the content, not a person.  Students need to learn that the teacher is just one of many resources and doesn't necessarily have to be the primary resource.  This means that I have to make myself invisible at times.  I usually walk to the opposite side of the room as the person who is talking or maybe I actually take a seat during discussion and stare at the floor.  I've found that kids start to depend on each other if I'm not there (so to speak).  And when they depend on each other, they tend to start asking questions. 

So, I guess at the end of the day, I try to be as real with my students as I can.  This all comes down to relationships founded on truth; a truth that we can only catch glimpses of.  We often times beat ourselves up because we don't see the fruit of our labor.  These "soft skills" (who coined that term, anyway?) are really the reason we do what we do.  We spend a copious number hours finding ways to offer immediate feedback to our students but our feedback is much more slow cookin'.  We won't know if the time we spend with our kids will pay them dividends down the road, especially when it comes to these "soft skills."  That comes when we see our students after they have finished college (or maybe they didn't  go to college and went straight to work) and started their own families.  That's when we see the fruit.  So be patient, the harvest is comin'.


_[1] Huge thanks to Riley for putting this together. The posts that have been assembled have been phenomenal and everyone who took the time to put something together should be commended. 

Update (as per Dave's request) 
My class layout.  I teach in a an "art room"  but since we don't offer classes like that anymore, I'm in it.  Students sit in pods (equipped with outlets making computer time nice), 4 or 5 to a group depending on class size. I'm hardly in the same place for more than 5 minutes.  

My Apologies

...but I can't get this one out of my head:

"From Wojtyla's perspective, the moral relativism that has resulted from the modern "turn to the subject" is only the product of an anthropological stagnation in the absence of faith.  In other words, man turns to himself and "stays" there, failing to see that his own humanity points him beyond himself, failing to see that anthropology points to theology

(Christopher West, Theology of the Body Explained)

Not trying to get preachy here, promise.  But here's the thing:

We spend all kinds of time discussing how to better educate our students.  For what?  Aren't we using our content areas as vehicles to produce better citizens?  (or at least that's what we say.)  It's not about math, English, science, social studies or P. freakin' E.  It's about human beings. It's about the dignity of the individual, right? But if we never help students get past the fact the world doesn't revolve around them, what good is it?  Why bother?

So if we are trying to help students get past self and become part of a greater good, what is this greater good?  Where does faith enter into this equation?  Or does it?  Should it?

We have problems in our schools.  Serious problems.  Most of our kids could give a rip about the Pythagorean Theorem, slope-intercept form or whether or not they put their name in the right place on the paper. Our kids are trying to figure out where they fit into this world and they're thinking "if it's not about me, then who's it about, you? Why should I care about you?"

Most of us don't want to discuss our politics let alone our theology.  It's too personal.  But there's no way it doesn't affect our pedagogy.

Are we really the answer? Or as we become a little more self-aware should that point us to something greater us?  If so, where's that in the curriculum?  If it's about more than us, yet us is the only thing we can talk about in schools, we're trying to build a fire without oxygen.

Standard Deviation

For those of you who have been reading a while, you know that I'm not the best lesson planner in the world.  My SmartBoard slides aren't always the best designed, although that's something I'd like to improve upon.  And I'm not that creative with activities.  But today, I realized that I'm pretty good at getting out of the way when some real inquiry shows up. 

My 7th graders just finished state testing yesterday and I wanted to help set the stage for what I'm going to expect from them next year as 8th graders.  The 7th grade curriculum is not much different than grades 2-6; it's very skill based.  My kids are very good duplicators, but I want them to become good applicators and eventually creators.  I explicitly said that to them today.  I want them to focus on representing their ideas as many ways as they can. 

Tom Henderson:
@mathpunk

Anyway, the theory goes that you don’t understand a mathematical concept until you understand it in TWO modalities. I do very well with visual knowledge, so my notes of understanding are full of color and pictures and mindmaps and arrows linking concepts, and I highlight the holy hell out of math books. However, I don’t believe I KNOW a concept until I can explain it verbally, because I can barely understand anything if someone just talks it at me.

First swipe is through my best modality, second swipe is through my worst modality. The whole “learning style” thing may be overstated, but it remains true that getting students to understand things in a variety of modalities seems like the way to go.

So today the plan was to:

• Have a little chat re: expectations
• Work through some pattern recognition via sequences
• See if I can get them to describe the patterns at least two ways
• Do an activity with algebra tiles

 Chat went well.  And I showed this slide:

Note: I'm pretty sure you get sequences 1 and 2, but just in case, sequence 3 has to do with words. The first term has one one (11), the second term as two ones (21) and so on.  So n₅ = 111221.

Kids are really good at finding the next numbers but pretty lazy about writing a good description of the rule.  The had to add, "for example" to many of their written rules.  The challenge was to have them write a rule that was short but stood alone.  The got it. 

Challenge:  Can we write a rule in Mathanese?
I explained subscripts and how they can be used to name different values for the same variable.  For example: n₅ represents the 5th term in the sequence. It didn't take long for them to recognize that:
n₅ = n₄ + n₃

"So, if n₅ = n₄ + n₃, what does n_k equal?"

"n_k = n_j + n_i."

This led to a great discussion on how the alphabet doesn't nec-ess-a-ri-ly have one to one correspondence with the whole numbers.  Good talk nonetheless. 

We ultimately settle on n_k = n_k-1 + n_k-2 and for the most part the class was good with that.

Sequence three provoked something I hadn't considered before.  Luke asked if the fifth term could be 1231 or even 3112.  He explained:

"Each of the previous terms are describing what comes before it, but does it have to be as you read it left to right? Could it just be describing the amount of times the digit shows up in the term."

"So if I would have given you the fifth term, would your rule work?"

"Nope."

Then Joe comes out of nowhere.

"Are there more possible right answers?"

This is just too good to be true.  We now have two questions we need answered:

I let them choose which question interested them the most and split the class into two groups to investigate.  Group one came awful close to saying that n_k = n_k-1 + k - 1. They're not quite there so I'll let that one marinate for the night. Group two came to the conclusion that there were no other possible 5th terms. They were a bit disappointed because they thought that they got the wrong answer.

"No, no, no. You had a question. You investigated it. And you decided that the answer was no. Can you explain why the answer is no?"

"Yeah."

"Then that's a good day's work."



Gonna have to finish the algebra tiles tomorrow.

So What Does That Look Like?

I love all these great conversations via the blogs and Twitter. I really do. But someone's gotta help me out a bit. I keep reading about things like:

• have students take ownership of their learning
• let them decide what they want to learn
• let 'em "play with math"
• make sure you differentiate your instruction

But I've yet to see a concrete example of what it all looks like. Are these just the phrases we should use to demonstrate we're in the club like some sort of secret handshake?

Or was I absent that day it was all discussed?

If so, can I borrow your notes?

Nice Try

I feel like I've had the same conversation many times today as we review linear equations. Students have been given an equation in Standard Form and are asked for the x and/or y intercepts. It has gone something like this:

"Mr. Cox, I don't know how to do this."
"Do what?"

"Number 12."
"What's it asking you for?"

"I am supposed to find the x-intercept."
"What's the equation?"

"2x + 3y = 6."
"So what do you know about all x intercepts?"

"They're on the x-axis."
"Alright, then give me an example of an x-intercept."

"5."
"That's a number, give me an x-intercept."

"(4,0)."
"Give me another."

"(-10,0)."
"And one more."

"(7,0)."
"Ok, now what can you tell me about all of those x-intercepts? What do they have in common?"

"y = 0."
"So what do..."

"Oh, that's right, I let y = 0 and solve for x."

It's funny how the default is always "I don't get it." Don't let 'em fool ya. They know more than they let on.