Being Wrong, and Learning

I was reminded of this story by a conversation I recently had about the purpose of teachers.  I think that one of a teacher’s most important roles is as a model of a good learner.  I often try to model learning for my students by asking them difficult questions that I don’t know the answer to, and working out the answer as a class, or by allowing them to pose questions and discussing their work with them.  Sometimes, though, it happens accidentally.  Sometimes I think or say something that’s just plain wrong – and these times can be the most educational for all.  Here’s a story of a time when I accidentally put myself in a situation of not knowing and being wrong, and having to learn.

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In one of my 6th grade classes, several months ago, we were investigating the sum of the measures of the angles of a polygon.  I’d discovered the previous week that they had firmly entrenched in their heads that the measures of all the angles in a triangle sum to 180 degrees, and weren’t super interested in knowing why.  They didn’t all seem to know that the angles of a quadrilateral sum to 360 degrees, though.  I really wanted them to come up with a way of determining the angle sum of any polygon.  So, I set up an investigation in which I hoped they would “discover” the angle-sum formula for polygons.

The investigation went something like this: Kids, (I asked them) for any polygon with any number of sides, what is the least number of triangles you can break that polygon into?  I drew some polygons on a sheet for them to break apart and encouraged them to create their own polygons, as motivated by their investigations.  This was a homework assignment.

As far as I knew, the least number of triangles you could break a polygon with n sides into was n-2.  And, because the angles of a triangle sum to 180 degrees and there are n-2 triangles in an n-gon, its angle-sum is 180*(n-2).  I hadn’t done a particularly large amount of research into proofs of and justifications for the angle-sum formula, but this made sense to me.  It also seemed like an accessible investigation for 6th graders, with a pretty clear ending-point.

The next day, my expectations appeared to be justified.  Several students reported the results I’d anticipated, and I started to go into why this showed interesting things about the angle-sum of a polygon… that is, until one kid brought up a monster.  This polygon had one non-convex angle.  And it had one less triangle than expected.  So, according to the formula I was developing, its angle-sum was going to be 180 degrees less than other polygons with the same number of angles.  Uh-oh.

No one was sure what to do, not even me.  That’s right – I didn’t know what to do.

I’d like (maybe?) to say that this hardly ever happens – not knowing the answer to a mathematical question that comes up in one of my classes – but it does happen periodically, especially when I’m teaching things for the first time (which, being relatively new to the business, is often).  This situation was slightly different (maybe even worse?) because not only did I not know what to do, but my original approach, on which I’d based the entire lesson, was clearly wrong.  I was claiming that all polygons with the same number of sides had the same angle-sum because their least-number-of-triangles-number was the same.  And here was a counter-example.  Yikes!

So, here’s what I did.  I said to my class of 6th graders, “Guys, I don’t know what’s going on here.  I know that the formula I shared with you is correct.  What about this polygon seems different from the other polygons we’ve been looking at, and how is that causing problems with this theory?”

So, we puzzled over the weird polygon together.  It was a student – not me – who finally found the root of our problems.  In all of the other polygons we’d studied, the vertices of the triangles all met at the vertices of the polygon.  But in this goofy polygon, the concave angle was constructed from the vertex of a triangle and the edge of another triangle.  So the sum of the angles of the triangles wasn’t contributing to a whole 180 degrees of the polygon angle-sum!  That was where we could find the missing 180 degrees!  And it explained the lack of a single triangle!

Armed with this information, we created a class version of my theory: The angle-sum of a polygon is 180*(least number of triangles you can break that polygon into), as long as the vertices of the triangles meet at the vertices of the polygon in question.  Nobody has come up with a counter-example to that, yet.

I’m not proud of the lack of preparation that I put into this investigation.  As an investigation in which students would uncover the angle-sum formula, it wasn’t so great.  Even if the students had found that the least number of triangles you can break a polygon into is always 2 less than the number of sides, making the connection between that result and the fact that 180*(number of sides – 2) = angle-sum of a polygon was going to be a stretch for them.  I was going to have to give them that result, which wasn’t my goal.  So, for the future, I’m going to have to come up with a better investigation for this lesson.

But, as an investigation in which students and teacher were uncovering interesting and surprising mathematics together, it was fantastic – and was made all the better, I think, because I got stuck and needed the brain-power of my students to continue the lesson.  My bafflement and intellectual excitement at being presented with that monster polygon were genuine.  I was able to take part in the investigation, the thrill of asking new questions and uncovering new monsters, and the joy at having reached the result we wanted as a fellow learner, as another person who loves to do and learn mathematics.  Through my not knowing, I transferred my enthusiasm to my students and helped to make the classroom a more dynamic and supportive environment for discovery.  And I was able to model being a mathematician.  This may have been more valuable for my students than the tighter investigation with a clearer goal that I could have constructed, with more time and thought.

By the way, if you keep looking into non-convex polygons, another interesting pattern relating the number of sides of the polygon to the least number of triangles it can be broken into appears.  My 6th grade class eventually developed a formula for the least number of triangles for any polygon – convex or not – so long as the triangles don’t overlap.  But, if they do overlap…  Oh, the questions!  Do they ever end?

I hope not.

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Two Conversations with a Third-Grader

Happy Pi Day!

Before we begin, you really must watch this video if you haven’t seen it already (and, actually, you should watch it again if you have – it’s that awesome):

Now down to business.  There are some kids who just love everything you ask them to do.  Some of them love it because they love, love, love math in all its forms.  Some of them love it because they love you so very much.  (This is true in 3rd grade, at least – it’s less true after that, sadly.)  But some kids have a lot of trouble digging in.  The things you say, the activities you plan, the enthusiasm you bring – none of it grabs them and compels them to make a niche in the mathematics you’re studying, to make math their own.

The third grader whose conversations with me are the topic of this post is one of those hard-to-grab students.  She LOVES to draw, to make up characters, and to create cartoon stories.  She HATES math.  Or so she told me today.  Here’s conversation number one (clearly not verbatim – I don’t have that great a memory.  But it’s close enough.):

Note: I’m calling her Olive, but that isn’t her name.

OLIVE:  Can I draw my ballerinas instead of doing math today?

ME:  I think not.  We have lots of things to do!

OLIVE:  What are we doing?

ME: We’re going to learn about division!  (Exclamation points indicate me communicating enthusiasm!)

OLIVE:  I don’t want to do division.  I don’t like division.  I hate math.

ME:  What about math do you dislike?

OLIVE:  I don’t know.  I just don’t like it.

ME:  It’ll help me to understand what you’re thinking and what we can do about it if you can give me some specifics.  Do you think math is boring?  Do you think it’s too hard?  Do you not like certain parts of math?  What do you think?

OLIVE:  Math is too hard.  It’s too hard because I don’t know the basics.

ME:  (Taken aback a little by response.  Not what I expected to hear.)  Can you explain what you mean by that?  What basics do you feel like you don’t know?

OLIVE:  I can do some things, but I don’t know the basics.

ME:  Ok.  Here’s a basic – do you know how to add?

OLIVE:  I can add two-digit numbers.

ME:  Great!  You sure can!  I’ve seen you do it!  How do you add two-digit numbers?

OLIVE:  Well, you put one number here (gesturing) and the other number here (gesturing below the first number) and you line up the digits and you add them.

ME:  Awesome!  Can you add three-digit numbers?

OLIVE:  Yes, I can.  You do the same thing.  But I can’t add one-digit numbers.

ME: (Pause.)  No?

OLIVE:  Adding one-digit numbers is harder.

ME:  Huh.  Why don’t you show me how you’ll add two-digit numbers.  (Grabbing paper and pencil.)  Ok – what’s 99 plus 99?

OLIVE:  (Writes numbers stacked digit-wise.)  9 plus 9 is 18, put the 8 here, carry the 1.  1 plus 9 plus 9 is 19.  So it’s 198.

ME:  Super!  Ok, so what’s 999 plus 999.

OLIVE:  (Adds.)  It’s 1998.

ME: Alrighty.  9999 plus 9999?

OLIVE:  That’s easy, I’ve already done that.  Just stick another 9 on the inside.  Ooo, that’s an interesting pattern.

ME:  Yeah, it is.  Olive, look – (pointing to the nines in the ones column) – you just did one-digit addition here.

OLIVE:  (Pause.)  I know.  But it’s harder when it’s just one digit.

ME:  What about it is harder?

OLIVE:  There’s less there.  There’s just one digit.  (Doing some gesturing indicating lining numbers up.)

ME:  (Really trying to understand what’s going on.)  Because there’s nothing to line up?  Because there’s more riding on that little addition, because it’s the whole answer?

OLIVE:  No, kinda, no.  It’s just harder.

ME:  Ok.  Well, what’s 4 plus 4?

OLIVE: 8!

And the conversation continues in this way for several more minutes, until class begins.

Now for conversation number two:

Olive is sitting on the rug with another girl.  Second girl is working on division sheet.  Olive is doodling.

ME:  Hey ladies.  What’s going on?

OLIVE:  Can I draw my ballerinas instead of doing the sheet?

ME:  That’s a great ballerina.  But how about you try this problem?

OLIVE:  No, I don’t want to!  I don’t like this.  I don’t like math.

ME:  I remember you said earlier that you didn’t like math because it was too hard, because you thought you didn’t know the basics.  Well, this kind of division is one of the basics.  It’s a good thing to learn if you want to know more basics.

OLIVE: No, that isn’t why I don’t like math.  I don’t know why I don’t like math.  I just don’t like it.

ME:  Well, what are some things that you like that math doesn’t have?

OLIVE:  Huh?

ME:  What are some things that you like to do that you don’t do in math?

OLIVE:  I like things involving pictures.

ME:  I have an idea – why don’t you make a picture for this division problem?  (Not really expecting her to go for this.)

OLIVE:  (Pause.)  That’s a really good idea!  8 divided by 2…

Starts drawing an 8 spliced into two pieces, top circle and bottom circle.  8 has arms and legs.  It is being menaced by a 2 – fanged, with a giant sword.

ME: I really like the teeth on the 2.  But what goes in the eight-halves?

OLIVE:  What?

ME: Well, what is each eight-half?  What goes in each circle?

OLIVE:  Ummmm.  4?

ME:  I think so!  What can you do with that?

OLIVE:  They’re boogie-ing!

Draws a little dancing 4 in each half of an eight.  Moves onto next problem – 14 divided by 7.

OLIVE:  How many pieces should I break the 14 into?

ME:  Well, what’s it being divided by?

OLIVE:  7?  That’s going to be hard to draw.

ME:  And what’s boogie-ing?

OLIVE:  No!  That’s too hard!  Can you tell me, pleeeeeeeease?

ME:  Olive!  It’s your artwork!  I can’t make part of it for you!

OLIVE:  Ok, ok!  (Really intensely focused expression crosses face.)  2, 4, 6, 8, 10, 12, 14.  The twos are boogie-ing.

Olive and I spend the next 45 minutes doing seven very basic division problems this way.

Pros:  She dug in.  She was so very into her artwork that we worked almost a half-hour beyond math class on this.  This is one of the only times this year that she’s chosen to do math over other things.

Cons:  We didn’t do much dividing.  We did a lot of personifying numbers and the act of division, which is important for her understanding of division.  But we didn’t get much hands-on experience with numbers and division.

She loves to draw, and I like that we found a way for her to incorporate drawing into math.  But I’m not sure we did enough math.  She wants to learn math – in her own words, she recognizes that her skills are weak, and she wants them to be stronger.  But she isn’t compelled to work on those skills.  This doodling division was great for today, but I need better ways to get her invested in doing mathematics – for the long haul.

What Can You Do With What Can You Do With This?

I’ve been on a Dan Meyer “What Can You Do With This?” kick for the past week in my 6th grade classes.  They had so much fun doing the Apple Countdown (some of them stayed up to see if they could get it – one kid claims that he watched and waited for hours, got bored, started watching videos on YouTube, and then missed it by mere minutes), that I thought I’d see what else we could do with What Can You Do With This?  Here’s what we did (and what I learned while doing it).

On Monday, we watched the Boat In The River video.  I particularly like this video because, like the Apple Countdown, it poses a question that is really compelling.  Everyone (I hope it’s not just me) has wondered, at one point in their life, how hard it would be to walk up the down escalator.  It’s a somewhat subversive thought – you aren’t supposed to go up the down escalator.  But here’s a video of Dan doing just that, giving you the chance to answer that burning question!  It’s also one of those top-notch What Can You Do With This? videos that contains all of the information you need to answer your question.  No external research or measurements necessary.  (Though requiring research has its pluses, as I learned later)

We watched it twice before I took suggestions for questions.  I got three different questions:

  1. How long with it take him to climb the down escalator?
  2. What song is he listening to?
  3. How did he make the movie, with himself in it four times?

Now, I really wanted the kids to answer the first question.  I tried to steer them towards it.  And we watched the video several more times, gathering information and using it in various ways to figure out the answer to our question – as a class.  Pretty productive, no?

But it didn’t feel great.  If you’re a teacher, you know the feeling in a class when things really are great.  And this wasn’t it.  Things were good – but not totally awesome.

For homework, I gave them a sheet with the picture of the gumball machine with a kid standing next to it.  I didn’t give them the one with the measurements – just the kid and the machine.  I asked them to come up with an interesting mathematical question to ask about the picture, and to make a list of the other questions they’d need to answer, formulas they’d need to research, and measurements they’d need to take to answer their question.

We were going to return to the assignment on Wednesday, because Tuesdays in our class are Skills Challenge/Free-Choice days (basically, a quiz and math activity free-for-all).  But, to my surprise, two of the students were so excited about their gumball machine questions that they wanted to share them with me and work on them during free-choice time.  And the things they shared with me really changed my perspective on this activity.

The first student had posed the “intended” question: how many gumballs are in the machine?  Not surprising.  What was surprising was that he had already answered the question – without me giving him the diameter of the gumball or container!  He did this by measuring the teeny, tiny gumballs in the photograph, Googling the size of an average gumball, finding the ratio of the two sizes, and using that to scale-up the radius of the container that he measured from the picture.  He then looked up the formula for the surface area of a sphere (an unfortunate but understandable mistake, because we hadn’t talked about volume yet in class at all) and calculated… I think it was something along the lines of the number of gumballs needed to wall-paper the container.  But he’d done all of this by himself!  I gave him a bit of re-direction during class (distinguishing between volume and surface area), but this was an example of student-powered investigation at its best.

From this student, I learned that I really didn’t need to – and probably shouldn’t – give the students much guidance in measurements or methods.  This student didn’t just learn about volume and scaling – he also learned how to do research, use the tools you’re given resourcefully, plan an investigation, and troubleshoot during an investigation.  In the grand scheme of things, what do I care more that my students learn?  Volume, or those other things?  I think the latter win.

The second student didn’t pose the “intended” question.  She asked: how tall is the kid?  My first reaction when she shared this question with me was, “Ok, great.  But is there anything else you want to know?”  I had intended for this to be a finding-volumes-of-spheres activity.  But there wasn’t anything else she wanted to know.  And when she started showing me the work she’d done, I saw how genuinely excited she was about her question and how much great thinking she was doing to answer it.

From this student, I learned that if I was going to ask them what they wanted to do with a situation, I had to let them do what they wanted to do.  One of my goals was for them to develop as mathematical question-posers.  I wanted them to use this activity to exercise their curiosity, and I wanted them to know that I valued their ideas.  Again, what was more important to me that they learn?  Volume, or curiosity?  Curiosity takes the cake any day.

Armed with this knowledge, here’s what we did in class on Wednesday.  I said, Kids!  Your choice!  You can work on your gumball question (whatever it is), work on your escalator question (whatever it is, so long as it’s mathematical), or pose a question about these pictures of Russian stacking dolls (here and here).  Rulers, calculators, the internet, your classmates, your teachers – all are tools at your disposal.  Go for it!

And they sure did go for it!  The questions in the room ranged from the classics – how many gumballs, and what’s the size of the next-smallest doll? – to the very personal – such as, how many of my favorite kind of frog will fit in the gumball machine?  And, how many average-sized Russian stacking dolls would fit in my friend?  Because some of them were watching the escalator video on computers (we have a class set of Chromebooks), they started browsing Dan’s blog for more pictures and videos, and got more and more inspired.

This made for a rather hectic class for me.  Everyone wanted to share what they were doing with me, and every time someone needed my help, I had to learn about their unique question.  But the classroom was bursting with energy, creativity, whimsy, resourcefulness, curiosity, and excitement.  It genuinely felt great.

My question for you, readers, is where do you think I should go from here?  How do I build on this momentum?

Aliens!

How do you get kids to tackle the most difficult problems with courage, grace, and an open mind?  How do you get them to look at a problem they have no idea how to solve, and NOT say, “This is TOO hard – I can’t do this!” but, instead, say, “Here I go!  I can marshal the resources, cleverness, and energy to do this problem!  I’m gonna keep on truckin’!”

This is a conundrum I (and you other teachers, too, I think) face with all of my students, but it often hits me most poignantly with my 3rd graders.  Maybe it’s because they’re so expressive.  Maybe it’s because they’re so cute – and I hate to see them sad.  Maybe it’s simply because my instincts for how 3rd graders’ minds work are less developed than my instincts for older kids are, so I more regularly give them activities that are over their heads.  I’ve fumbled around more than I’d like with giving my 3rd graders problems that challenge them but don’t make them want to give up.  On Friday and today, however, I gave them a problem that was VERY hard – and they are loving it!  I’m very proud of this success, so I thought I’d analyze how it worked (and bask in my pride by sharing it).

It was my co-teacher’s idea to give them this assignment.  If it had just been me, I wouldn’t have done it – because I honestly thought it was too hard.  This puzzle was very nearly too hard for me to solve when I first saw it (or, at least, I had a lot of trouble convincing myself that I could actually do it before I truly started working).  Many (most?) people have the same reaction that I did when I first saw this puzzle – serious anxiety, extreme confusion, and the thought, “This is impossible!”

What is it?  It’s what we at Saint Ann’s call The Alien Code.  I honestly don’t know much about its history and use, but I think it’s a message that we sent into space years ago with the hopes of communicating with sentient aliens.

If you want a very difficult, but eventually very satisfying, puzzle-solving experience, try your hand at cracking The Alien Code here.  (This isn’t the exact version that we gave our 3rd graders, but I believe it has the same content, just with simpler symbols.  I unfortunately can’t find that code at the moment.)  (UPDATE: I can’t find a full copy of any version of the code
!  The link I originally used no longer exists!  Here’s a picture of the first page.  If anyone can find the whole code, please let me know!)  But before you try, you might want to read what we told our 3rd graders, and what I think sold them on the project and helped them believe that they could crack it.

The point we strongly emphasized before we gave them the code was that, in working to crack this code, they were on an equal playing-field with any aliens who could potentially crack it, too.  None of the symbols are remotely familiar.  But they deal with ideas that (we hope, especially as mathematicians) are universal.  Hint, hint.

This statement rang bells with our 3rd graders at least in part because our curriculum emphasizes the development of ways of communicating quantity in human history.  We began the year by teaching them the “Ba-Na-Na” counting-system, used by entirely fictional cavemen in ancient times, in which a single Na signifies 1, Na-Na is 2, Na-Na-Na is 3, Ba is 4, Ba-Na is 5, etc.  As we grappled with communicating large numbers in this system, we discussed how and why our own base-10 system came into being.  We also learned about the ancient Egyptian number system, using this as an example of a system in which addition “feels” like addition (a wonderful phrase my co-teacher uses frequently), but the beautiful, simple addition “dance” (again, his term) we do with place-value isn’t possible.  (I take no credit for this wonderfully humanistic 3rd grade mathematics curriculum, but I am a very strong advocate of it.  All of our students – from the arithmetic whizzes to the book-worms to the dreamy doodlers – found it engaging, and I think everyone came away with a real appreciation for our numbers.)

By emphasizing that in working on this code, the students were aliens, we conveyed to them that cracking this code was going to be ridiculously hard – imagine you’re an alien receiving this message! – but completely do-able, because it conveys an idea common to all intelligent beings.  This made the kids feel challenged, but empowered.

We also told them about our own experiences with the code.  I said, with complete honesty, that I was convinced I couldn’t crack the code the first time I saw it.  My co-teacher echoed the sentiment.  But, we said, we did it!  And it felt great!

Armed with the knowledge that they were in for a crazy ride, but that two people they respected and related to had made it through and that they had the power to crack it simply by being human, our 3rd graders took to the code with awesome enthusiasm.  They’ve made it through about half of the first page.  After two days.  And they’re still pumped.  Not a single student has given up yet.

Hey, you’re human, too.  Want to give it a try?