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.