Class Commences – an hour I won’t soon forget

Recently, Michael Pershan unearthed a Shell Centre gem straight from the 80’s (literally).  This collection of materials is fantastic, and hopefully demonstrates to both students and teachers that engaging in rich tasks and high-level thinking is timeless.

I decided to give the function unit a shot in my Algebra 2 class today.  Some background on this group of students:  there are 38 juniors and seniors, last hour of the day, in a class geared toward lower-level students.   So far though, the only thing that’s been “lower” in this class is the number of empty desks I have.   I handed out this task, gave minimal directions and let them go for a few minutes on their own:


from:  Shell Centre for Mathematical Education, University of Nottingham, 1985

from: Shell Centre for Mathematical Education, University of Nottingham, 1985

It was so interesting to watch the different ways each of them started.  Some began with 7, since that was the first you saw when reading the graph from left to right.  Others insisted to work from 1 to 7, identifying the corresponding people along the way.  A few worked the other way around, from the people to the graph.

I walked around to make sure each student was able to get started and that those who thought they had determined a solution also supported their claims.  Then, I wrote the numbers 1 – 7 on the dry-erase board, stepped back, and let these kids amaze me.
One student volunteered an answer, and then handed the marker off to another.  I intervened only briefly to make sure that every student had an opportunity to contribute if he or she wanted.  Once 7 names were completed, I knew a couple of them were out of place.  I sat and said nothing, and this entire class showed me what they are capable of.  Here was a class full of students labeled mathematical underachievers completely nailing SMP #3.  Their arguments were viable, their critiques constructive, their discussion productive.  It bothered a few of them that I wouldn’t let them know if/when they were correct.   But most of them are starting to understand that my main focus here is not the correct answer, but the incredibly rich and interesting process they used on their journey to finding it.  They came up with multiple ways to support their answers and noticed tiny details about the people that supported their findings.  For example, did you notice that Alice is wearing heels? According to my students, that is perhaps why she appears slightly taller than Errol.

I had a heart-to-heart with this group when we were done about how proud I was at how they conducted themselves throughout this task.  I’m really thoroughly looking forward to a fantastic trimester with this special group of kids.  Their work on this task gives both of us the confidence that they can tackle something more difficult next time, and they are capable of mastering high-level mathematics this trimester.

Notice and Wonder with Gusto

My daughter was very content on the airplane ride from Fort Meyers to Minneapolis watching Frozen for the 102nd time.  I took this opportunity to read the Noticing and Wondering chapter of Powerful Problem Solving, the superb new publication from Max Ray and the Math Forum crew.  I took so many notes on this chapter since this is a strategy that I think every teacher can implement, no matter their apprehension about new strategies.  It is such an easy set of questions to ask:  What do you notice?  What does that make you wonder? Those two questions can open up an entire class period of rich discussion and mathematical exploration.  No one explains this classroom strategy better than Annie Fetter of the Math Forum in her Ignite Talk.  (Seriously, if you have not seen this 5 minute, dynamite, game-changing video, stop reading and go there now. ) 

Last Thursday was day 1 of our high school’s third trimester.  The first day of the slide into the end of the year. Regardless, the first day of the trimester always seems like the first day of school: the anticipation of a scenario that’s been played over and over in the minds of teachers and students becomes reality.  For me, this day meant the last hour of the day I would be met with 38 (you read that right) “lower level” Algebra 2 students.  My class is most likely the last high school math class that these juniors and seniors will take, and many of them do not like math or are convinced they are not any good at it.

This class has been in the forefront of my mind most of the year for a lot of reasons.   One of those reasons being that after Jo Boaler’s class this summer, I know that a huge barrier to raising the achievement levels of students in this class is the students’ beliefs that they are capable of doing high level mathematics.  And I also know that a key component to getting these kids to perform better is to give them feedback that allows them to believe that they are capable of it in the first place.

Because of the structure of some of our high school courses, most of these students have not had experience with higher degree graphs, equations, or functions.  They may have seen something similar in their science coursework, but quadratics have not formally been introduced.

I gave them the following graph along with the scenario and let the noticing and wondering begin:  Mrs. Bergman likes to golf and her golf shot can be modeled by the equation: y= -0.0015x(x-280).

A couple of them stuck to non-math related Noticings (the graph is in black and white), but almost all of them noted multiple key characteristics of the equation and/or the graph.  Some highlights:

  • The graph doesn’t have a title and it needs one.
  • Both heights are in yards
  • Horizontal distance goes up by 80.  Height by 5.
  • The peak is in the middle of the graph.
  • The graph is symmetrical
  • The maximum height is about 28 – 29 yards
  • The distance at the maximum height was about 120 yards
  • She hit the ball 280 yards.
  • The number in front of x is negative
  • The graph curves downward
  • It has an increase in height and then a decrease in height.
  • As the ball reaches the peak height, the rate the ball climbs slows.

The list of Wonderings was even more impressive to me. A lot of them wondered things like what kind of club she was using, if the wind was a factor, did she have a golf glove, how much power she used to hit the ball, the brand of her tees, clubs, glove, ball, etc.  Then one student laid out something so profound, it made the entire class stop and and acknowledge the excellent contribution:

“What distance would the ball have traveled if the maximum height were 20 yards rather than 28?” (audible ooo’s here)

After this student said that, the floodgates opened with great questions from others:

  • What was her average height for the shot?
  • What is the maximum height that she is capable of hitting the ball?
  • Is this a typical shot for this golfer?
  • If the maximum height was higher, like 35 yards, how far would she hit the ball?
  • What is the exact maximum height that she hit the ball and how far did she hit it when it reaches that maximum

There were still a few that couldn’t get passed what kind of glove she was wearing or tee she was using, but most of the students stepped up their Wonder Game when one single student demonstrated a rich example.

What I really love about this strategy is that it is so easy to implement into your classroom routine with the resources you already have.  For example, rather than starting with a procedure for solving quadratic equations, simply ask the students what they notice about the structure of the problem.  How is it the same or different from problems they have done recently?  Ask them to list attributes of the equation.  I have found most often, the noticing of one student triggers the noticings of others and the list becomes progressively more sophisticated.

I have heard from some teachers that they do not use try this strategy out of fear of students making a list of trivial noticings (like, the graph is black and white).  They will include those every time; expect it.  But by acknowledging those seemingly trivial items, that student, who would not have dreamt of entering the conversation before now has received validation of his or her contribution to the discussion.  And when students feel heard and their opinions valued, their contributions will start to become more profound.

I’m very proud of this class.  I’m really looking forward to the creative perspective that their noticing and wondering will bring.

She Defines Resilience – One Year Later

We are defined not only by what we do, day to day, but also by how we react and utilize our seemingly random hand of cards in life.  I’ve found over 33 years of life, the mark of character that differentiates those who excel and succeed from those who do not is resilience.  I can point to former students as examples:  The college graduate who grew up with an absent mother and a drug addicted father? Resilience.  The home-care nurse whose parents didn’t value an education past high school?  Resilience.  The successful plumber whose best friend committed suicide his sophomore year of high school?  Resilience.

It seems fitting to give a Webster’s definition of resilience here, however, I think that we all can picture individuals who personify our meaning of the word.   For me, above all, those people are my brother, Matthew, and my sister-in-law Danielle.   This story isn’t about me, or my reaction to this event.  It’s about them and what they have taught the world about resilience and the power of hope.  I hope my intentions come across as I recap their story.

One year ago, January 30th, 2013, Danielle, while finishing up a nursing clinical suffered a massive hemorrhage resulting from a burst aneurysm on the right side of her brain.  She was rushed to the local hospital where she was taken into surgery and given a very bleak prognosis.  The sobbing ER doctor explained to my brother that his wife was probably going to die.  My younger brother, who I’d always joked as being “30 going on 19” now was faced with an incomprehensible, life-altering situation.  He captures his emotion poignantly on a Caring Bridge post about the account of the moment when he told that doctor, as well as the hospital chaplain to F-ing get his wife to Iowa City!  I think those words have defined his attitude on the situation that it does not matter what has plagued us in the past.  He knew she had much more to give this world, so let’s get out of her way so she can fight to give it.

Reflecting during anniversaries of events seems to be a cultural norm and a time to remind ourselves of where we came from and how much further we have to go. A year ago today, we watched in udder horror and shock as Danielle lay motionless, lifeless, with small tubes ushering blood from her brain.  Furthermore, we observed silently as every half an hour, a nurse would shine a light in her eyes and ask for a reaction that never came.  “No change,” became the most chilling words I’ve ever heard.  I didn’t say it at the time, but I went to bed that night believing our precious Danielle was most likely gone.

The next day brought new light, and a miracle.  The overnight nurse said she had never seen anything like it.  When prompted to wiggle her toes, Danielle obliged.  “Thumbs up if you hear me, Danielle?”  And it was the most beautiful thumb I’ve ever seen.  She began her recovery that day and has not stopped since.  In one year, Danielle has gone from “probably going to die” to thriving and living.  Her personality, again, lights up the room as it always had.  She walks with less and less assistance each day and remains poised and confident that she will walk in the Bix 7 this summer.  Every day my brother is there by her side, emotionally and physically.  From the hospital ICU to a rehabilitation center in Ankeny, Iowa.  And now back home, where he’d turn their house upside down if he had to in order to ensure her comfort.

One of Danielle’s doctors said, “When you’ve seen one brain injury, you’ve seen…one brain injury.”  I believe these words are not necessarily a testament to the brain alone but the person in control of it.  Danielle proved that her fate was not finalized and her husband stood by her side believing the same.  These two incredible people inspire me every day to be a better person and to remember that all people fight a battle, in their bodies and their minds.  And I am so thankful for their presence in my life, and the opportunity to learn from them.

Danielle with my daughter, Maria this Christmas.

Danielle with my daughter, Maria this Christmas.

Pattern Power

If you have little kids and you’ve been privy to an episode of Team Umizoomi, then perhaps the title of this post evoked a little jingle in your head. You’re welcome; I’m here all day.

My daughter, although she doesn’t choose Umizoomi over Mickey Mouse as often as I’d like, picked up on patterns relatively quickly after watching this show a couple of times.  She’s 3 years old, and she finds patterns all over the place.  Mostly color and shape patterns, but a string of alternating letters can usually get her attention as well.  These observations of hers made me realize that pattern seeking is something that is innate and our built-in desire for order seeks it out.

High school students search patterns out as well.  For example, I put the numbers 4, 4, 5, 5, 5, 6, 4 so that the custodian knew how many desks should be in each row after it was swept.  It drove students absolutely CRAZY trying to figure out what these numbers meant.  I almost didn’t want to tell them what it really was as I knew they’d be disappointed that it lacked any real mathematical structure.

I’m not as familiar with the elementary and middle school math standards as perhaps I should be, but I’m confident that patterns are almost completely absent from most high school curriculum.  Why are most high school math classes completely devoid of something that is so natural for us?

Dan Meyer tossed out some quotes from David Pimm’s Speaking Mathematically for us to ponder.  This one in particular sheds light on this absence of pattern working in high school mathematics:

Premature symbolization is a common feature of mathematics in schools, and has as much to do with questions of status as with those of need or advantage. (pg. 128)

In other words, we jump to an abstract version of mathematical ideas and see patterns as lacking the “sophistication” that higher-level math is known for.  To be completely honest, this mathematical snobbery is one of the reasons I discounted Visual Patterns at first.  Maybe it was Fawn Nguyen’s charisma that drew me back there, but those patterns have allowed for some pretty powerful interactions in my classroom.   I’ve used them in every class I teach, from remedial mathematics up to college algebra because they are so easy to  differentiate.

I think high school kids can gain a more conceptual understanding of algebraic functions with the use of patterns.  For example, this Nrich task asks students to maximize the area of a pen with a given perimeter.   The students were able to use their pattern-seeking skills to generalize the area of the pen much  more easily than if they had jumped right from the problem context to the abstract formula.  

I also notice that the great high school math textbooks include patterns as a foundation for their algebra curriculum.  For example, Discovering Advanced Algebra begins with recursively defined sequences.  IMP also starts with a unit titled Patterns.   I think these programs highlight what a lot of traditional math curriculums too quickly dismiss:  patterns need to be not only elementary noticings of young math learners but  also valued as an integral part of a rich high school classroom.

Engaging with Engagement

High school students are inherently unpredictable. I’ve been told it’s the condition of their pre-frontal cortex and they can’t help it. I’m sometimes baffled and confused by what intrigues and engages them. If you’ve seen their obsessions with Snapchat, you know what I mean.
Something that always gets teenagers riled up, however, is a statement that challenges their peer group. In fact, I found today, that they’ll engage at a much higher level when presented with data that questions their level of engagement.

After a little guessing and estimating, I revealed this graph resulting from a recent Gallup poll on student engagement during my 9th grade statistics class today:

Gallup Graph

The kids were fired up right away.  Even if students agreed with the representation, it seemed as though every kid wanted to share his or her interpretation of how student engagement changes over time.  They shared their experiences from their formative years of education and respectfully expressed their frustrations for how much more difficult school gets each year.  Surprisingly, the students seemed to place blame for the overall decline in curriculum immersion on themselves.

Until one boy opened up the floodgates with the proclamation, “In elementary school we get to learn by messing around with stuff.  In high school, all we ever do is listen to the teacher talk and do boring worksheets.”  Expecting me to dismiss this kid’s comment for daring to suggest that the burden of student engagement also lies on the teacher, the class was relieved when I asked this student to expand on his thoughts. Almost simultaneously, multiple hands shot up in the air agreeing with this sad truth many of them were thinking and this young man had the courage to say out loud.  A rich, important, respectful discussion ensued about the difference between being busy in class copying, listening, and doing and being engrossed in activities that facilitate learning.

We continued the conversation by critiquing the methodology used to collect the data for this poll and the misleading representation in the graph.  Sorry, Gallup, my 9th graders spotted the flaw in the using in a self-selected study to represent all students right away.  They also debated the validity of broad categories such as “Elementary School” represented only by 5th graders rather than K – 5.

We discovered that the actual Gallup Student Poll is available online.  The students agreed that Friday was probably not a good day to do a survey about school engagement, but we’re really looking forward to collect and analyze the data on their classmates.

Curiosity Driven Mathematics

In my very first years of teaching, I used to have students ask me, in that age-old, cliche teenage fashion, “When are we ever going to use this?”  I vividly remember my response being, “Maybe never.  But there are plenty of other things we do in life, like play video games, that have no real-world application. That doesn’t seem to bother us too much.”

In fact, if every moment of our lives needed to apply to the bigger picture, the REAL-world, when would we do anything for pure enjoyment? or challenge?  or even spite?  I know kids are capable of this because some of them spend hours upon hours a day engaging not only with a video game but also collaborating with other people through their game system.

And furthermore, where do we think this resentment for learning math really comes from?  I have a guess…probably adults who have realized that through the course of their lives, being able to solve a polynomial equation algebraically is not all that useful! News flash, math teachers:  Our secret is out! 

There are many kids across all levels of achievement that will not engage in the learning process simply because the state mandates it or the teacher swears by its real-world relevance.  Students (and arguably people in general) are motivated by immediate consequences and results and cannot easily connect that the algebra they are learning today will be the key to success in the future.  They do not care that if they don’t nail down lines, they’ll never have a prayer understanding quadratics.  If they are bored to death by linear functions, I can’t imagine that they have even an inkling of desire to comprehend the inner workings of a parabola.  

What does resonate with learners is the satisfaction of completing a difficult task, puzzling through a complicated scenario, or engaging in something for pure enjoyment.  Kids are naturally problem-solving balls of curiosity.   There are ways to provoke curiosity and interest while simultaneously engaging in rich mathematics.  I think many teachers assume that in mathematics, especially Algebra, curiosity and deep understanding need to be mutually exclusive, and I’m positive that mindset is dead wrong.  For example, show this card trick to any group of kids, and you’d be hard-pressed to find a group who isn’t trying to figure out how it works.  I also think you’d be hard-pressed to find the real-world relevance to a card trick.  It’s still no less amazing, as well as algebraic.  



Pushy vs. Persistent

“Sharing is caring” does have a nice rhymey ring to it. Although lately, I’ve felt a little bit like my version comes off as ‘sharing is pushing and over-feeding’.  I’ve had teachers in my department inquire about problem solving and desire to get kids to invest and engage.  I like sharing what I’ve discovered and what I have found that works, but sometimes I get so excited about sharing resources that I end up like Tommy Boy and his pretty new pet.  I sometimes fail to realize that trying new approaches can be uncomfortable, unpredictable and downright scary and not all teachers want to dive into the change head first as I did.

Here’s a great example: we had final exams in 2-hour blocks right before Thanksgiving break.  To say that the kids get “restless” by the middle of the second day is sugar-coating it.  A new teacher in our department, (let’s call her Sheryl) sent this picture with the caption, “My algebra kids were bored after their final and built this with their textbooks.” booktower

Of course, my brain couldn’t just let that one go and say, “Nice book tower, Sheryl.” Dan Meyer calls this perplexity and modeling this behavior is a key to getting students curious. Instead, my eyes lit up and I thought, “what a great math problem!”  As we looked at this photo, I said, “what do you think kids will notice and wonder about this photo? Do you think you could get them to come up with how many books are in the 10th row or the nth row?”  Of course the question that’s raised, legitimately, is “what do you do when students say ‘there are green books and red books’ or ‘some are faced forward and some are faced backward’?”  This is the part that I believe is scary for a lot of teachers is relinquishing control of the immediate direction of the lesson and not being so certain about how students will respond.  At least when we give them a quadratic to factor, we have a pretty good idea of the limited number of directions they can move to arrive at a singular correct answer.

But what I believe is imperative here is validating and acknowledging those seemingly math-less observations and creating a math opportunity with it.  With the instance of “some are red and some are green,” we can now extend that declaration of color to ideas like percentages, ratios, and so forth.  But by first validating this red/green response, we’ve invited this student to the conversation and made them part of the creation of the problem we are about to solve.  Now they are empowered by the process and more motivated to step in to the problem-solving ring.  Whereas before, this same student might have disengaged completely.

A recent example of this from my own classroom:  We were beginning Dan Meyer’s 3-act task using the Penny Pyramid.  When collecting wonderings, one student asked how many 1996 pennies were in the pyramid. He was born in 1996, and was probably just fascinated by that year, but I didn’t want to dismiss that from the discussion.  I have a bucket of pennies in my room that I use occasionally for probability experiments, and I hoped that this kid could draw from his knowledge about samples to make a reasonable estimation of how many 1996 pennies were in that pyramid.  As it turns out, that students off-hand question turned into a great math discussion about random sampling.

But back to this book tower:  After I’m sure that I’ve thoroughly freaked out this new teacher with my enthusiasm over a book tower, something awesome happens.  This new teacher, races into my room after 1st period on Monday and says, “I did it!  I did the book tower, and it was AWESOME!”  I’ve had some great moments with other teachers, but that one is going to rank pretty high on my list for a long time. At lunch, she was STILL raving about it. She even said that the students were so engaged, that they ran out of time talking about it during class. Maybe there’s hope for Tommy Boy after all.