Wednesday, September 14, 2016

It's Never Too Late to Learn

Some teachers wonder then what their role in the classroom could be if it is not focused on demonstrating and explaining strategies to students and monitoring students' progress in using these strategies.

Susan B. Empson and Linda Levi

      I'm late to the math party.  I came to my current position as a K-5 specialist from a general elementary education background.  I taught all subjects during the 23 years I spent in my grades 2 and 3 classrooms, and math didn't stand out in any significant way.  I knew who Marilyn Burns was, but if I had ever met the luminaries that my colleagues in the MTBoS speak of so highly, people like Van de Walle and Fosnot and Kamii and Richardson, for example, it was long ago in teacher school, and I'd certainly forgotten who they were.  So I play catch up.
    Cognitively Guided Instruction (CGI) is something that, upon hitting the MTBoS, I heard referred to again and again.  Glowingly.  Reverently.  And so this summer I decided to find out what the fuss was all about.  I wanted to explore Extending Children's Mathematics: Fractions and Decimals, and put together a small, very informal book study PLC composed of our two grade 4 teachers and my former partner and fellow specialist Theresa.  We met twice in July and once in August to discuss our reading assignments and share the work we had done.

Kudos to my colleagues who gave up some of their summer evenings to participate in this project.

    It was an incredible experience, and we learned so much from the book and from each other.  We increased our own content knowledge, and are now armed with some tools that I hope we can employ to better analyze student work.  
     A few weeks ago I came across a set of photos I had taken late last school year of responses to an item on a grade 5 unit assessment.  The problem seemed pretty simple, and I was struck by the many different ways students had attempted a solution.  At the time I was ignorant of CGI, but looking at them now through a CGI lens I can better understand and describe what's happening.

The student multiplied 4 x 2 and 4 x 5/8 and added the two products together.  This was a good illustration of the distributive property of multiplication over addition.  This student was also able to use relational thinking to express 20/8 as 2 1/2.  He is able to decompose 20/8 into 8/8 + 8/8 + 4/8.  He understands that 8/8 equals 1 whole, and that 4/8 is equal to 1/2.

    The students who got it right were almost uniform in implementing the above strategy, although there were some outliers:

Convoluted, but it worked.  This student is also able to use relational thinking to express 2 5/8 as 21/8 and 4 as 32/8.   Why did he feel it necessary to have common denominators?

   But what had really caused me to stop and take some pictures was the many and varied ways that students had managed to be wrong.

This student converted 2 5/8 into an improper fraction.   Did he really understand why 2 5/8 was equal to 21/8 or was he just following a procedure: multiply the denominator by the whole number and then add the numerator?   The multiplication across the numerators is incorrect.  Careless mistake or confusion about the identity property of multiplication?

No question here.  Issues with the identity property of multiplication.

This was a common mistake: multiplying 4 x 2 then just adding the 5/8.  This student has used an equal sign to separate the expressions, rendering an incorrect equation. 4 x 2 does not equal 8 + 5/8.

This is diagrammed as if the student were multiplying 42 x 5/8.  When multiplying 5/8 x 4 the student flipped the 4 into 1/4.  Perhaps he had heard something about "invert and multiply"?  However he did not do this when multiplying 5/8 x 2.  In the top left corner he is adding his two products, but how he got 10/628 is a mystery.

With no actual work, I deduce that this student multiplied 4 x 2 to get 8, then, in the mixed number, multiplied 2 x 5 and 2 x 8 to get 10/16.  He finished by adding 8 and 10/16.

At first look it appears as if this student multiplied 4 by every digit he saw in the mixed number.  How did he get 160?  By multiplying 8 x 20?

     These fifth graders are now off to middle school.  But their work remains, and looking at it through the lens of what I've learned this summer is going to inform what I do, not only with the new batch of fifth graders, but with all math learners in our school.  If I read my CGI right, these kids are victims of an over-reliance on procedural memorization.  They have bits and pieces of algorithms that they can't put together because they have weak conceptual underpinnings.  They're easy to spot. However it's possible that many of the kids who did get the correct answer also have shaky conceptual foundations, but are just better at memorizing a procedure.  How will we find them?
     CGI argues for a decrease in the amount of time teachers spend, "demonstrating and explaining computation and problem-solving strategies to students."  Instead, teachers are encouraged to allow their students' intuitive strategies to emerge first.  "Children have some conceptually sound understanding of fractions, even before instruction," Empson and Levi write, "(but they) can learn to ignore this understanding in favor of models introduced in school that portray fractions in narrow ways."
     This is a tremendous shift away from the traditional "I do, we do, you do" model, and entails teachers taking on a different role.  "This new role," the authors of CGI explain, "Centers on helping students communicate strategies to other students, directing questions to specific students to help them draw connections between these strategies and more basic strategies, introducing equations to represent students' strategies, and highlighting the fundamental properties of operations and equality that that underpin these strategies."
     That's a lot to ask.  But I've read the book and I'm a believer.  Let's roll up our sleeves and get to work.

Monday, August 29, 2016

Talking Math With Your Kid: End of Summer Edition

    Having aged out of camp, and unable to subsist on the cash received from her intermittent babysitting gigs, my daughter faced the same prospect her brother had faced two summers before: get a job. That's how, on a warm, late June afternoon, with her dad in tow for moral support and a folder full of very skimpy resumes, she found herself pounding the pavement in downtown Princeton, New Jersey.

Nassau St.

Throughout the spring, she had deflected every conversation on the subject of summer employment.  There was school, and dance, and a busy social calendar.  She expressed ambivalence about working at a day camp, and anxiety about working retail.  Never a good math student (and that's putting it kindly), she was afraid, she explained, of working the cash register.
   "I don't really know how to make change!  And I'm afraid that there will be a long line of customers and they'll all get mad at me!"
   "You don't have to make change," I answered.  "You just punch in the numbers, and the machine figures out the change for you."
   Variations of this conversation, and others concerning the urgency of looking for work while there were still positions available for teens, continued up through the final day of her high school sophomore year.   Finally, she ran out of time and excuses.   School ended on a Thursday; on Friday we were on the hunt.  I could sense her unease, and knew that the entire project could end up in tears, raised voices, slammed doors, and recrimination.  This was going to be a delicate operation; with a ticking time bomb of a soon-to-be 16 year old, it would test all my fatherly skills and powers.
    The plan was to walk around town, looking for businesses with Help Wanted signs in the window.  But she had her own, difficult-to-verbalize litmus test: regardless of whether or not they were hiring, some stores were simply off-limits.  We fell into a rhythm: JaZams toy store: No. Thomas Sweet: Absolutely not.  Nina's Waffles: A half-hearted yes.  Hulit's Shoes: Heck no! As frustrated as I was by her refusal to jump on every opportunity, I kept my cool as we trudged on.

Here was one that passed the test.

    We stopped for lunch at PJ's Pancake House (they were looking for help; another defiant no.)  She had left her resume at four stores.  It was a start.  We were emotionally exhausted.  We would go home and regroup, but not before making one more stop on our way out of town.

Bon Appetit.  A gourmet food store in the Princeton Shopping Center.  Her older cousin had worked there when he was in high school.

  She protested, but I put my foot down.  Sensing I wouldn't budge, she walked in reluctantly, asking for the manager as I browsed the shelves.  Several minutes later she found me.
   Panic and barely concealed tears.  "Dad!  They want to interview me!  I don't want to work here!  What do I do!  You made me come in here!"
  "Calm down.  Let them interview you.  It will be good practice.  Why don't you want to work here?  Ben worked here and he liked it."
   "I just don't don't want to work here!"
  "You might not even get the job.  Sit down with the manager.  It will be a good experience either way.  Ask him what he might want you to do."
   "But I don't want to work here!"
   Back and forth, our voices rising, then lowering so as not to make a scene, until she gave in.
   I waited outside, peering in the window, pacing up and down the outdoor patio.  I knew she was mad at me for having forced her into the store, and I hadn't thought things would proceed so far so fast.  But I  hoped that all would be forgotten in the afterglow of a successful interview.  Boy, was I wrong.
   When she came out to meet me, all the pent-up emotion came flooding out.
  "Dad!" she cried, fear and anger mixing together.  "They want to hire me!  Right now!!  I don't want to work here!  What do I do?"
  "That's great, honey!" I was proud she had made such a good impression.  "Why don't you try it out?  Did you ask what they wanted you to do?"
  She unfolded a piece of paper.  On it were the words: cash register.  Of course.
  "We've been over this a million times!"  I was exasperated.  She had a job offer in hand.  She could start tomorrow.   "You don't have to make change.  If someone gives you cash, the register will tell you how much change to give the customer."
  "I know that," she quavered, voice rising through the tears.  "It's that I don't know what coins to give back!"
   There it was.
   It had been there all along, but I didn't hear it.  Her fear wasn't about figuring out how much change was due.  True, she would have a lot of trouble doing that, but she knew the register acted as a calculator.  It was about finding the right combination of coins.  Not only that, she knew that a customer would expect the combination be comprised of the least number of coins possible.  She was unsure she would be able to do that, especially under pressure, with a line of impatient customers backing up out the door and onto the street.   Her litmus test, which had seemed so arbitrary, finally made sense: the store had to have as little foot traffic as possible, and at least the potential of non-register related work.
    I told this story one evening at Twitter Math Camp, along with what I had taken away:
  • It was possible for a student to go through elementary school, middle school, and Algebra 1 and Geometry in high school without mastering the ability to make change with the least number of coins;
  • That this was less a reflection on her than it was on a system of math education that, from early on, left her feeling inadequate, disconnected, and disaffected;
  • That math anxiety is real, that it interferes with performance, and that bad things happen when we put undue pressure and time constraints on kids.
And bless Andy Gael for his response: "I love how your daughter advocated for herself."

     She went back in and told the manager she'd get back to him.
     She eventually found a job:

JerZJump: A family entertainment center.  Birthday parties, open play, camp field trips.  No cash register.

Her first paycheck.

Here's how the story ends:
     Summer is over, and the novelty of the job has worn off.  The smile you see above hasn't been seen in quite a while.  There's no air-conditioning, the hours are irregular, the pay stinks, and the kids are uncooperative and unruly.  Lately she's taken to spending time with her babysitting loot and the contents of the house change collection:

"Make 38 cents with the fewest possible coins."

Nothing like a little intellectual need.


Monday, August 15, 2016

Real Mathematicians

   First on my to-do list when I got back from TMC '16 was get a new notebook, because in Minneapolis I learned that real mathematicians go graph ruled.

My old notebook (left). My new notebook (right).

     Day 1 of Tessellation Nation. Michelle Niemi  is hard at work with a folded piece of paper and a scissors.  "I wonder if I could get a snowflake to tesselate?"
     I'm sitting next to her, deep into Christopher Danielson's turtles.  "I don't know," I tell her.  "But someone here probably does."
     Enter Dr. Edmund Harriss.  Co-author of Patterns of the Universe, researcher, professor at the University of Arkansas, and all-around great guy, Edmund would later blow my mind when he explained to me why a square can have wiggly sides and 72 degree angles.  But now he's got a problem to work out.
     I watch him sit down next to Michelle.  After listening to her for a few moments, he takes out a pad of graph ruled paper and a pencil.

     Putting the turtles aside, I turn my attention towards Edmund as he begins to explain to Michelle how he is going to accomplish this task, getting a snowflake to tessellate.  He starts drawing squares.  He's visualizing the folds.  He's marking where to cut.  He sees symmetrical designs in his mind's eye.  Mid-summer, and it's snowing in his head.   Michelle has questions.  She wants to know what he's thinking.   So do I.  Edmund is patient.  What's clear to him isn't clear to us.  He's got to go over it several times.
     And it occurs to me: I am watching a real mathematician solve a problem.  And it's thrilling!   Because I've never been this close before!  I'm not sure I really understand what he's talking about, and I don't care!  I'm just caught up in the excitement of watching him work.  And I think to myself, "This is what mathematicians do.  They solve problems.  On graph paper.  I need a graph ruled notebook.  And I need to stop thinking of myself as only a math teacher.  I need to be more of a math doer."
     Edmund finishes.  He takes a piece of paper, folds it into squares, and makes the cuts.  He's left with the pieces of a snowflake.
     "You could have every child in your class cut one of these out, and put all the pieces together to make a tessellation."    

     That evening: 

Once I started looking, I saw graph-ruled notebooks everywhere.

Henri Piccioto explained how to create a graph ruled notebook.

Megan Schmidt's spiral obsession, inspired by Edmund Harriss, began innocently during a school meeting as she kept herself a graph ruled notebook.

Of course sometimes graph ruled notebooks aren't available, and mathematicians need to improvise:

Jonathan Claydon did calculus on a napkin at dinner one night.

I love my new notebook:

I'm using it to solve problems for an informal summer book club...

...and it really came in handy for helping me understand the tessellation in my uniform tiling with curvature problem.

Up until last month, the closest I'd ever come to watching real mathematicians at work was at the movies.  Both real...

He helped defeat the Nazis!

...and otherwise.

Nice use of a vertical non-permanent surface, Will Hunting!

One of the great thrills of TMC is getting to watch real mathematicians do their thing live and unplugged.

John Golden (left) and Henri Piccioto (right).

So, nearly a month after it ended, I've finally found my #TMC1thing.  It's to put as much effort into doing math as I do into teaching math.  On a vertical non-permanent surface when I can, on a napkin if I have to, but mostly in a graph ruled notebook.  Like real mathematicians do.

Monday, July 25, 2016

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.

Friday, July 8, 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.  



Tuesday, June 14, 2016

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.
   "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!