Creative Craziness

I teach a lot of 9th graders this trimester. We offer a class called probability and statistics 9 and it is open to 9th grade students who also will have had the quadratic portion of algebra 1 this year. I really enjoy this class for multiple reasons. First, it lends itself very well to applying math to real-world scenarios.  Secondly, the hands-on opportunities are endless.
One of the issues I have been committed to improving with my own professional demeanor is the way I deal with 9th grade boys. Nothing brings out my sarcastic, short-tempered, disagreeable side like the antics of freshman boys. There’s something about the decision to play soccer with a recycling bin that just invokes the my inpatient side. Regardless, I need to develop more patience with this demographic. Boys are unique, both in the way that they act and the way that they perceive acceptable behavior. I’m not talking about “I’m bored” acting out. I’m talking about the “I really need to see if this eraser will fit in this kids ear” kind of acting out. I think that my short fuse has more to do with my failure on my part to  fully understand them rather than gross misbehavior on their part. What I’m really trying to grasp here is not “why can’t these kids sit still?” But more “when they can’t sit still, what makes them want to kick a recycle bin around the room or toss magnets at the learning target?” I think if I had a better understanding of what drives those behaviors, I could deal with them more productively. Suggestions?

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.

Block Talk With My Kid

I often wonder if my daughter will view the occupational status of her parents through eyes of appreciation or resentment.  Will she loathe the fact that teacher parents are more aware of the goings-on of their child’s academic life? Or perhaps she will more often appreciate that math homework help will be easy to come by at home?  Both of these scenarios have the potential to have a positive impact on her achievement.  However, I had a profound realization today that what actually will help my daughter be successful is maybe neither of those things.  While she may not be able to get away with teenage class antics as easily as her peers whose parents are not teachers, I do not think that what my husband and I provide for our daughter’s intellectual growth is something that only teachers can give.  ANY parent can engage in rich conversations with their children and see a growth mindset at work.

I want to give proper credit to Christopher Danielson for his Talking Math with Your Kids website and book that drew my attention to how I was interacting with my child mathematically.  As a secondary teacher, I am grateful to have gotten a better understanding of how number sense develops in young children and how I can help foster that development.  Danielson’s website, Talking Math with Kids has been invaluable in recognizing the everyday math conversation opportunities to have with my daughter.

Her preschool days have been spent at a Montessori school.  Although the noodle necklaces, punch cards, and snips of paper are an everyday treat, I was particularly excited when Maria came home with this gem the other day:


Of course my immediate excitement was over the fact that the correct number of squares were colored for each number.  But then I realized that there might be a rich math conversation potential involving that worksheet.  But, I had no idea what to do with it.   I threw it out on Twitter and got many great ideas, including:

IMG_3226 Tweet Response 2 Tweet Response

Friday was Parent Day at Montessori Central, so I took this as an opportunity to talk some math with my kid.  Luckily, I talked her into this same worksheet.  We explored all kinds of great number-driven curiosities:  which ones make squares, which ones make rectangles, which ones make neither?  How many squares are left over after it’s colored?  How are the rectangles for 2, 4, 6, and 8 alike?

I realized after that experience the limitations of the number worksheet.  For instance, the numbers at the top, while providing an opportunity for “tracing,” don’t allow the learner to explore a particular number any further.  I wanted Maria to explore more ways to color 6 boxes, but her 3.5 year old brain saw the “7” above the next 10-block and would not allow it.  Some worksheet surgery might be in my future.

I do not think that being a teacher makes these conversations natural.  I saw something was mathematically correct, but I didn’t have much experience with turning that into a real mathematically rich conversation with my daughter.  I’m thankful for both the math teacher and parent twitter community for throwing ideas my way.   Any parent is capable of taking something they like and turning it into a teachable moment.   When interacting with our kids, we need to do less showing and more asking; less telling and more listening.  She seems happy about it, doesn’t she?