Touching Calculus

     When he told me that a square could have wiggly lines and 72 degree angles, I felt a panic attack coming on.  Dizziness, increased heart rate, a gnawing sense of dread mixed with a strange giddiness.  I mean, if that were true, if a square could have wiggly lines and 72 degree angles, then what else might be possible?  There are certain unshakable pillars that hold the world in place, and one of them is that a square has straight sides and 90 degree angles.  But the guy holds a Ph.D. in Mathematics, with expertise in Geometry, Topology, Discrete Mathematics, and Combinatorics.  He is a professor at the University of Arkansas.  He's co-authored research like Intrinsic Defects, Fluctuations of the Local Shape, and the Photo-Oxidation of Black Phosphorous.  He's the Academic Director of the Epsilon Camp.  And, along with Alex Bellos, he's co-authored Patterns of the Universe.  So when he says that a square can have wiggly lines and angles of 72 degrees, who am I to argue?

Do you see the square?

     For my morning session at TMC '16, I chose Tessellation Nation.  There were several other interesting options, but the temptation of spending 6 hours over 3 days playing in the Triangleman's sandbox, in his Minneapolis backyard, was too hard to resist.  We would be given the freedom to explore whatever we wanted, for however long we wanted, with whomever we wanted. All Christopher asked was that we start with a question or goal, no matter how vague or obscure. Fortunately Max went before me, and I adopted his:

When does play become math?

I went right for the turtles.

Elizabeth was interested Escher drawings, and spent time exploring, tracing, and creating designs.

Bryan and Malke worked with these shapes...

...and had some questions.  Good thing John Golden was around to explain periodic tiling.

Jose got all 3-D on us.

Graham worked on hexagons...

...and so did Henri.  "Hexagons will humble you," he said.  And he was absolutely correct.

Max and Malke disappeared for a while.  They had an idea and needed more space.

    Edmund Harriss brought along some laser cut shapes for us to explore, and told us that if we put them together we could make a ball.  I had seen Edmund present a My Favorite at TMC '15, and knew him as the co-author of the brilliant book Patterns of the Universe.  But I hadn't had the chance to meet and get to know him, and the shapes looked cool, so I dropped the turtles and, along with Bryan Anderson, jumped in.

First we had to remove the shapes from the sheet.  They could be joined together by tabs.

That was fun!

      Then he gave us another project to work on.  He told us that if we put the shapes together in just the right way, we could create a cool tessellation that would ripple.  He explained that, around any vertex, we would need to alternate 2 triangles, a square, another triangle, and then another square.

And that's when I got dizzy.
Because I didn't see any triangles or squares.
And that's when my play became math.
Because I wanted to know what in the world he was talking about.

   He pointed out a shape that had 4 wiggly sides:

I tried to imagine the wiggly sides pulled taut to form straight lines that intersected at 90 degree angles.

Me: But Edmund, that's not a square!  A square has straight sides and angles of 90 degrees!  This shape has wiggly sides and angles of... ?

Edmund: 72 degrees.

Me: But that's not a square!

Edmund: Really?

Me: Well a square has 4 sides, all the same length, and this shape has 4 sides, all the same length.  And a square has 4 vertices and so does this shape.  And in a square all the angles are equal, and all the angles in this shape are equal.  So maybe it is a square?

Edmund: The most important thing I learned getting my Ph.D. was to be flexible.  I had to learn how to bend myself around the math.  In order for us to talk about what's happening here we are going to call this shape a square.

Me: Not for nothing, but it sounds like you're bending the math around yourself!  But I'm over it.  A square it is!

     I felt an exciting sense of liberation.  Math, which had always seemed to me to be a rigid, inflexible, no nonsense, no talking back, no coloring outside the lines set of arbitrary and infuriating rules and regulations, suddenly had become, in Edmund's hands, something to play with.   In my hands it was lifeless.  In Edmund's, it was art.  I was finally beginning to understand what all the fuss was about.
     Edmund explained that what we would be creating was called a uniform tiling with curvature.  He said that to understand what was happening mathematically we would need to use something called Differential Geometry, which is calculus in several dimensions.  He said the equations would be quite complex.

I touch calculus.  I've never been this close to it, nor ever wanted to be.  I'm going to be 55 years old in just a few weeks.  I'm ready to learn now.

Edmund Harriss, right, with Henri Piccioto.  Twitter Math Camp, Minneapolis, MN, July 17, 2016.

My Problem With Word Problems

From Mark Anderson via a tweet from John Golden:

     I've spent a lot of time, and I know I'm not alone, helping struggling students find effective strategies to solve word problems.  Because so many of these students also read below grade level, much of that help is centered around improving comprehension.  After all, word problems are pieces of text, right?  Don't we also call them number stories?  It makes sense to locate the problem in an inability to fully understand what's happening in that text.
    Well yes, but there's more to it than that.  A comment thread on a post I published on this topic, along with an e-mail exchange with a colleague, has me reflecting on my own experience as a student.  I was, and remain to this day, an avid reader.  Reading (and writing) came easy to me.  My struggle was always with math.  So what was my excuse?  How come I had so much trouble with word problems?

Here's a typical example of a word problem, taken from our grade 4 curriculum:
   

     This piece of text wasn't written to inform, entertain, provoke, or enlighten.  It exists for one reason: to get kids to do math.  It was written to meet a standard, which makes it, by definition, contrived.  There's a subtle gesture towards kids; all kids like animals, right?  It does tell a story, but as a story it's neither compelling nor engaging. As a reader, I have no interest in what's happening.  Perhaps if I had been helped to feel the devastation of the storm, the thunder and the lightning, the sheets of rain, the flooding, the downed trees and wires; perhaps if I had been made to sympathize with the heroic attempts of the shelter staff to find homes for the poor animals, sweating in the heat with no air conditioning (or are they freezing in the cold with no warmth?), barking and meowing with fear in their dark cages, food spoiling in the refrigerators...perhaps then I might have cared about how many dogs and cats the Wyn and May shelters took.  I'm exaggerating of course, but couldn't the author of this problem made at least a small attempt to reach out and touch me?  As it is, I've given up long before I've reached the question.

     Robert J. Tierney and Jill LaZansky have written about the existence of, "An implicit contract between author and reader--a contract which defines what is allowable vis a vis the role of each in relation to the text."  They write:

     

"It seems reasonable to suggest that authors have a responsibility to their audience--a responsibility which necessitates that written communication be relevant, sincere, and worthwhile.  An author must predict the intentions and background of his or her audience.  If the author's predictions do not mesh with the reader, then the text the author has written may be deemed by its readers irrelevant, insincere, uninformative, ambiguous, or obscure; and it is likely the contract will be voided.  Symptomatic of this void, readers will likely criticize the author for being ambiguous, express an unwillingness to address the author's message, or misinterpret what has been written.  It is as if the reader would posit that the author has violated the terms of their contractual transaction." 

     Do the authors who write word problems for textbooks and standardized tests have a responsibility to their audience?  If they expect students to make good faith efforts to engage with their texts, is it reasonable to ask them to hold up their end of the bargain?  I wouldn't have been able to articulate this as a student, but as an adult reflecting on my own experience as a math learner, and as a teacher who has spent countless hours trying to help students navigate their way through number stories, I feel a rising sense of anger and frustration when I read problems like the one about the River Forest Pet Shelter.  Most students do not have the voice to criticize it on its own terms as a failed piece of writing, but many will react, as Tierney and LaZansky observe, by being unwilling to address its message (shutting down), or by misinterpreting what has been written.  As a student, I did both.  

    The MTBoS project endeavors to repair this contractual void.  What is a 3-Act if not an attempt by its author (teacher) to create a compelling narrative?  More than that, it offers an opportunity for the reader (student) to participate in the telling, making the author and reader partners in the construction of meaning.  In a good 3-Act, students may not always arrive at the correct solution, but they are rarely at a loss to understand what it is they're being asked to find, and, more often then not, they are eager to engage in the task. 

     The problem I had with word problems had nothing to do with literacy; I was a good reader with good comprehension skills.  I'm not saying that if they had been written with more craft and style that I would have been able to solve them, or even know what to do.  But at least I would have gotten to the question.  

     

    

     

  

7 x 4 x 7

One game that's gotten a lot of play this year is Salute!

The third graders play with multiplication and the second graders with addition.

     A few weeks ago I found myself sitting on a carpet in the back of a third grade classroom, in a school not my own, with a group of four students getting ready to play.

Me: Salute! needs 3 people to play.  Two to hold the cards with the factors up against their foreheads and one to deal the cards, call out the product, and judge who got the missing factor first.   How do you play with four?

Student: The teacher said that two of us act as the dealer, product caller, and judge.

Me: OK.  Let me see how that works.

     As I watched them play, something occurred to me.  What if 3 kids held factors up against their foreheads?  Would the student acting as the dealer be able to multiply all three?  Would any of the kids be able to find the factor facing out from their forehead?  I didn't know these kids, or their skill levels, other than a word from their teacher that they were a "middle group."   How would they react? The kids in my school no longer flinch when I volunteer them as subjects in my little experiments, but these kids are not used to having some crazy math guy come in and disrupt their lives.  Should I wait until I got back to my school?

Me: You guys want to try something different?

Them (suspiciously):  Um, yes?

     I explained my idea.  One student agreed to be the dealer for our trial run.  Here were the three cards she saw facing out from her classmates' foreheads:

Before reading on, multiply these numbers together in your head.  How did you do it?

   After the cards were dealt, it dawned on me that I was going to have to multiply the three numbers together!!   After all, assuming she could do the multiplication, how would I know whether or not she was correct?   I looked over at her, and saw her eyes get a little wider.  We were both going to have to do a little thinking.
   I took a breath and focused back on the cards.  First I multiplied 7 x 7 and got 49.  That's really close to 50, I thought, and 50 x 4 was 200.  Well 49 + 49 + 49 + 49 had to be 4 less than 200.  200 - 4 was 196.  So the product was 196.  I checked and re-checked the math in my mind, and waited.
   I don't know how much time went by, maybe a minute more, maybe two or three.  Then, in a small voice, full of question and uncertainty, she spoke.
   "196?"
   "Yes!"  I said, relieved that we had agreed.  I turned to the three students, still sitting there with the cards held up against their foreheads.  It was their turn to sweat.  "OK!  The product is 196.  Can any of you figure out what number you've got?"
    Something akin to panic set in on two of the faces.  The other belonged to a boy whose eyes began to roll up to the top of his head, ever-so-slightly, as if he was looking into his brain.  He was holding one of the 7s.  I felt myself willing him to get it.  Finally, again after several minutes:

Student: I have a 7.

Me: How did you know?

Student: I saw a 7 and a 4 and I knew that was 28.  I thought how many 28s do I need to make 196?  28 is close to 30, and 7 x 30 is 210.  That's close to 196.  So I tried 7 x 28 in my mind.  I know 7 x 8 is 56, and 7 x 20 is 140.  140 + 56 = 196.  That's how I figured it out.

Me: Nice!  (Turning to the dealer)  How did you get 196?

Student: I knew 7 x 7 was 49.  To do 49 x 4 I first multiplied 49 x 2.  That's 98.  Then I added 98 +98.  I know 100 + 100 is 200, so 98 + 98 is 2 less than 200.  That's 196.

Me (to myself): Did we just have a number talk?

     It was time for the students to switch centers.  As the next group moved in (another group of 4 that the teacher had informed me was the "low" group),  I realized that I had been fortunate.  What if the factors had been different?  Say a 7, 8, and a 6?  That would have increased the difficulty, both for me and the kids!  Just to be on the safe side, I quickly rigged a deck with the numbers 1-5, a few 6s, and some 10s.

It was right in their wheelhouse.  The dealer multiplied 2 x 3 and got 6, then multiplied 6 and 6 to get 36.  The student with the 3 on his forehead responded first.  He said he knew 6 x 2 was 12, and that two 12s was 24, and another 12 made 36.  And that's how 4-Way Salute! was born.

I could hardly wait to get back to school and tell Theresa.  I showed her the 7, 7, and 4 and asked her how she might multiply the numbers together.  She thought for a moment.
  "I'd multiply 7 x 7 first and get 49.  Then 49 x 4.  40 x 4 is 160, 9 x 4 is 36.  160 + 36 = 196."
  A different, but no less effective strategy.

  Would you like to give it a try?

The product is 126.  What's the missing factor?  How did you figure it out?

Again, the product is 126.  But now you're able to see the 7 and 6.  Is finding the missing factor harder or easier?  Why?Did this player have an advantage?

What about now?

     I haven't had a chance to explore the many possibilities that this variation of traditional Salute! has to offer.  If you've got some time left in your school year, give it a try!

A Plea

     Can we stop giving kids worksheets like this?

Can we stop telling them how to solve problems?

     Recently observed:
     The class was in the middle of a series of lessons centered around developing strategies for multiplying 2-digit numbers by 1-digit numbers.  On this particular afternoon, a group of students was working independently on the above worksheet.  One student, after reading the first question, wrote 72 + 72 + 72 + 72 in a column in the work space.  As he began to add up the numbers, an instructional aide working in the room stopped him and said, "You have to use multiplication to solve this problem.  You're learning about multiplication, and it says in the directions to use multiplication."
    Obediently, but with an under-his-breath groan, the student erased all his work.
    I was not in a position to stop this from happening.
    Things like this happen all the time.
    If we're going to use worksheets like this, can we at least cut the top off first?

Let the kids work out the problems in ways that make sense to them, not to us.

We Noticed, We Wondered. Now What?

     Is there a corner of the known math education world that doesn't know about noticing and wondering?  Introduced by Annie Fetter, developed at The Math Forum, and popularized in Max Ray-Reik's book Powerful Problem Solving, this versatile prompt delivers tremendous value for minimal investment.
     I've seen it happening all over my school, and have been pleased with the results.  But it's time to take it up a notch.  Noticing and wondering is a means to an end, not an end in itself.  It's a problem solving strategy.  After all, it appears in a book called Powerful Problem Solving, in a chapter with the focal practice of SMP 1, Make sense of problems and persevere in solving them.  Here's Max:

    Noticing and wondering activities are very open-ended, and at first can lead to noticings and wonderings that are off-topic and even silly.  The initial process of writing noticing and wondering lists can take a long time, and students will notice details that they won't end up using as they solve the problem. (pg. 49-50)
 
     In my experience this is certainly the case:

Here's an example of some wondering a grade 3 class did recently about Graham Fletcher's 3-Act Share the Love.  By this time of year I would've hoped not to see wonderings like Why am I showing the video? What's the dad's name? 

     Accepting all student responses without judgment is an essential principle of noticing and wondering.  However this can become a source of frustration for teachers, who would like to gently nudge their students into more mathematical waters.  So what can we do?  Max again:
   
       Noticing and wondering is something that students get better at over time:  more focused, more relevant, more efficient, and more automatic.  Once students have become prolific noticers and wonderers, one simple prompt we've found to be helpful in focusing students is simply asking: 
     Which of these noticings have to do with math?
     Which of these wonderings could we use math to help us answer/prove? (pg. 50)

 Here are two adventures in trying to get better at noticing and wondering.

Grade 2:

     Last month I took a slightly altered version of Andrew Gael's 3-Act task Trail Mix on a tour of four second grade classrooms. 




  
As part of the Act 1 protocol the kids noticed and wondered:

   

After the lessons were over, I sorted through all the noticings and wonderings, choosing some that were overtly mathematical, some that had nothing to do with math, and some I felt might start an argument:

I came back to all four classes with the following task:

First the kids worked individually, then I put them in groups to discuss:

 They agreed, they disagreed, they defended their causes.  Some kids changed their minds, others stuck to their guns.

     We met back as a class to debrief.  I wanted to keep things moving, so rather than talk about each one, I asked if there were any that we could all agree on.  What do the Chex taste like?  and I wonder if that's Mr Schwartz? were unanimous no's.  How many pieces are in the bag? (which in fact is the focus question of the 3-Act) was a unanimous yes.  There was an unexpected controversy over I noticed the guy had a watch on.  I had included it as a no, but one student argued that a watch is for telling time, and telling time is math.  Nice!  All except one agreed that I noticed that there were 3 boxes was mathematical.  The lone dissenter argued that, since it couldn't be attached to a number model or equation, it was a no.  One student pointed out that you could add 1 box + 1 box + 1 box and get 3 boxes, and that made a number model.  Another said, "It's counting, and counting is math." She nodded, and changed the N to a Y.   

     

Grade 3

     Shannon's third graders got a number story to notice and wonder about:

No, not this!

This!

I collected and sorted their responses, and came back a few days later with this:

     The most interesting discussion arose from the very first notice, the one about Delilah's bus stop being far away.  All but one student had classified it as non-mathematical, primarily due to the fact that, "It had no numbers."  But one student argued that the notice had something to do with distance, and since distance can be measured, it should be classified as a yes.  That one student managed to convince all of his classmates to change their answers!

Some observations and reflections:

•  At some point we need to call out the elephant in the room.  It's math class, and our noticings and wonderings need to be mathematical. However...
• ...When it comes to number stories, especially ones with a lot of text, there are non-mathematical things kids may notice and wonder about that may help them understand the narrative of the story (such as it is.)  These observations may aid them as they make attempts to work towards a solution.
• Kids have definite notions about what makes something mathematical.  Seeing numbers is a tip-off, but with enough experience I believe they can expand their ability to mathematize.  Conversely, students can help teachers see math in ways they never thought of before.
• Having students classify the noticings and wonderings, and then have to defend their decisions, is a great way to start a fight in math class.
• Having our students notice and wonder about pictures, videos, and number stories is a wonderful way to lower the barrier to entry and engage all learners in math class.  But if we want to leverage that engagement into improving problem solving skills, we need to up our game.  After our initial forays into the practice, we need to carefully guide our students into becoming better mathematizers, and then show them how to apply that habit of mind to problems they encounter in class.  Max, one last time:

     Adding especially mathematical noticing and wondering skills (noticing quantities and relationships, wondering strategically) to students' repertoire increases the usefulness of noticing and wondering.  As students get better at targeted, mathematical noticing and wondering, and as they begin to notice and wonder automatically (as mathematicians do), they may find that all of the other problem-solving strategies become easier to learn as well. (pg. 55)

  

     

Second Grade for the Win

  One morning, a few weeks after publishing this post detailing the struggles some of our third and fourth graders were experiencing with the traditional algorithm, I found myself in Maggie's second grade classroom.  A unit progress check was looming, and the kids were hard at work on a review page.  As I walked around observing their efforts, I found myself admiring the varied ways they were solving this addition problem:

I asked the three other teachers on the team to save the work their students had done.  Here's a representative sample of what I saw:

Number lines

Starting with 12 + 8 was common.

This student decomposed the 50 into 5 jumps of 10...

...while this one took the entire 50 in one jump.

Although it came last in the sequence of addends, this student started at 54.  A great question to ask him would be, "Why did you do that?"  He also broke the 12 up into a 10 and a 2 with the 8 in between.

Partial Sums

This student added the tens, then the ones, then put them together.

This student also started by adding the tens, then put the 8 and the 2 from 12 together to make another 10.  She added that ten to get 70, then added the 4 left from 54.

This student started by putting the 12 and 8 together to get 20.  Then added 20 and 50, and tacked the 4 on at the end. 

Same as above, but he used words to describe what he did.

Can the traditional algorithm be far behind?  It's close, and when it comes it will have a conceptual underpinning.

Base 10 Block Representations

This student is transitioning from the representational to the abstract...

...while this one is not quite ready yet.

This example would provide a nice opportunity to discuss equations.  

Another student in transition.

    One after another, these beautiful examples of student thinking passed before my eyes.  And out of about 70, I saw only one traditional algorithm.
     I have not been shy about calling out our curriculum when I feel it's been lacking (see here and here.)  But I have to give credit where credit is due.  The work I saw was a result of the focus that the new, Everyday Mathematics 4 has placed on computation strategies that include finding number bonds for 10, decomposing multi-digit numbers into tens and ones, and using open number lines and base 10 block representations.  The teachers have maintained fidelity to the program, and in addition have started to explore number talk routines.  The results are plain to see.  I've been teaching elementary math for 30 years, and I've never seen anything like this.  The second grade teachers haven't been at it that long, but neither have they.  Does it represent a sea change, or is this an isolated phenomenon?  What will the third grade teachers make of these students?  Will they notice a difference?  How will they bring them forward?  It's questions like this that keep me excited to come to work every day.  Stay tuned and we'll find out together.