math forum

Regional Reflection – Releasing my Grip

As humans, our complex brains are able to create such detailed visions of the future.  We build things up (or down) in our minds that reality can’t possibly compete with.  Until we let go of what we believe should happen, we are unable to fully experience the beauty of what actually is.

Proposals for the NCTM Regional Conference here in Minneapolis were due in September of 2014.  This means I have had over a year to continue to wind the anxiety yarn into one giant ball of stress.  But sweet relief occurred when I released my iron grip on my expectations and began to appreciate the phenomenal power of educators coming together.

First off, thank you, from the bottom of my heart, NCTM, for  your support of the MathTwitterBlogosphere at the NCTM conferences. I spent much of my time at the #MTBOS booth in the exhibit hall.  Sharing this wonderful, supportive, organic community with other math educators has been as fulfilling as it has been fun.

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East Coast meets Minnesota Nice

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You guys have something called the “Trap Team?”

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Woman: You didn’t say there would be math. Christopher: Actually, I said there would be nothing BUT math.

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When Nicole Bridge gets fired up, the magic happens.

When asking people in the Exhibit Hall “are you on Twitter?” the most common response was “yes, but I don’t tweet.  Think of the student in your class that thinks very deeply, submits very thoughtful work,but doesn’t raise his/her hand in class to volunteer his/her thinking.  I’d hope that most teachers would agree that these students are still valuable members of the classroom community.  It works the same with the online edu-community.  Plus, I’d venture to guess that many people who actively tweet with other math educators started by diving down the rabbit hole of math blogs.

Max Ray-Riek led a panel where we discussed this problem and blog post of mine.  Next week we venture into rational functions in college algebra and I anticipate good times to be had once again.

An hour later, Carl Oliver and I spoke on statistics, social justice, and how to have safe, productive conversations with students around the issue of race and equity. Here is the link to the slides.  The discussion centered around these data sets:

Using Local Data to Teach Statistics

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I really enjoyed giving our presentation and a lot of great discussion ensued.  But ultimately, I’m thankful to the MathTwitterBlogosphere for being the catalyst of the great discussion we get to take part in, day in and day out.  I had never met Carl Oliver in person before Wednesday.  But the powerful connections we (all of us) have made with one another, make it possible for an algebra teacher in New York and a stats teacher in Minnesota to get together and share their passions with fellow educators. It allowed a teacher in Massachusetts to spread the fire she started in Boston on to Atlantic City, Minneapolis, and Nashville.  And that fire is continually kindled as we welcome, share, engage, and support over and over and over again.  Thank you, #MTBoS for being the genuine, authentic community that has naturally produced so much awesome for so many teachers.

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Math is Messy. So Are Gender Roles.

I have been absolutely humbled by all of the positive feedback I have received from my previous post.  Thank you to infinity for taking the time to read, write, and share.  I believe that it is our common humanity that makes it possible for us to learn from one another, not necessarily our knowledge of content.  There is so much of my sobriety that goes into my teaching.  It is an incredibly freeing feeling to be able to be honest about that part of my life as I blog.

Rose Eveleth wrote a great piece about the roles that girls find themselves taking on in group work.  In short, Eveleth focuses on acknowledging that girls often self-assign the “recording” role, absolving (and downright excluding) themselves from a problem solving opportunity.  The end result, career-wise, may lead women away from high-profiled positions.   As teachers, it’s easy for us to overlook this discrepancy because girls, generally speaking, are neater and more organized, and may seem like the best fit for the job.  In a related article, Dale Baker does a great job of asking teachers to examine gender preferences that exist in our classrooms in order to help encourage all students to step into the “lime light.”

On Friday, I tried a simple version of this.  First, students were presented this scenario (taken from the Math Forum POW section):

The Student Council at Rahkenrole High School is planning a concert.  They’ve hired the Knox Mountain Boys, a popular local band, for $340.  A poll among the students has shown that if tickets cost $5, 140 people will come to the concert.  For every dollar the ticket price goes up, 10 fewer people will come, and for every dollar it goes down, 10 more people will come.  

I’ve been a huge fan of the Math Forum, long before I joined Twitter (and got to fangirl Max Ray at TMC14).  The reasons might not seem obvious from this scenario, but kids noticed right away that there was no question asked at the end.  What’s brilliant here is that there is literally an infinite number of questions that we could ask here.  Granted, some questions are more important than others, but I framed the task in a way that elicited what I needed.

I handed out a big white piece of paper to each group of 4 and had them divide the paper up into sections.  This way each person in the group was both the recorder and the problem solver.   I asked them to write down 2 questions they think that I would ask about the scenario and one question (anything) that they would ask.  They identified their group’s most important question and put it up on the whiteboards on the wall.

Low and Behold!  They READ MY MIND! They asked about maximizing profit, income, and people, and also requested modeling equations for each.  The excellence in this scenario (and the Math Forum in general) is that it can be applied to so many levels of math for so many reasons.  For example, most high school kids can make a table and figure out a reasonable answer for the maximization questions, and kids with more know-how can develop mathematical models.

Some great things happened:

  1. They knew they needed ONE set of answers in the center of their paper.  This meant they had to communicate the work in their section. The traditional group roles dissipated, and they all had equal stake in solving the problem.
  2. They solved the problem in so many different ways.  (Do you remember these types of questions from Algebra 1/2?  I’m sure they have a trendy textbook label that alludes me at the moment. But they are solved by making the variable “number of price increases. Interestingly, very few students solved it that way successfully.)
  3. They were messy. And I loved it.  In fact, I made the second class use markers exclusively so that they could not erase.
  4. They were uncomfortable leaving some of the questions unanswered.  When I didn’t label certain questions as “bonus” or “extension” they felt that all were necessary to be successful.  My goal was for them to collaborate with ownership in their individual contribution.  I may have gotten more joy out of this part than I should have 🙂

Here are some fun photos of their work:

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Do You Let Yourself Fail?

I sat down this weekend to do some recreational mathematics with a friend.  Maybe you know him; his name is Justin Aion.  He writes a pretty cool blog over at Re-Learning to Teach.

I made it a goal of mine this year to work on some geometry for a few reasons.  First, I’m not that great at it.  Second, the students at our school historically struggle with it as well.  Two of the problems we chose were from the Five Triangles blog.  And to be completely honest, I sucked.  I sucked a lot.  I sat there for much of the Google Hangout drawing and drawing the figures and then writing down what Justin had eloquently discovered.  And then nodding in agreement. Here are pictures of Justin’s and my respective work:

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Then we decided to work on something I thought was more my cup-o-mathematics tea.  Turning to the Math Forum, we tried this weeks scenario for Trig/Calculus.  How silly of me to assume that since this is just the beginning of the school year, perhaps the task could be solved using Algebra.  Of course Justin busted out the calculus seamlessly and like a pig in numerical-feces, excitedly worked his way to viable solution. (It turns out that applying algebra to this problem was not as straight forward as it might have seemed.) Again, I felt defeated by the mathematics.

The point here is that doing math that’s unfamiliar is hard.   Thinking deeply about problems is hard work.  Applying previous knowledge to a new situation is also taxing.  What I really took away from hours of difficult mathematics was an empathy for the anxiety of many of my students when I ask them to do the same.  It is disingenuous of me to expect my students to persevere through problems if I’m not willing to do the same.  So, I’m committing to being uncomfortable, mathematically, and I will get better.  My geometry skills will improve, and perhaps I’ll be able to revisit my long lost calculus pals, derivative and integral.  The important thing is that I’m willing to try and willing to fail.   In the long run, I think my students will benefit, and I know that I will as a teacher.

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Algebraic Anguish

The following prompt presented at Twitter Math Camp by the Mighty  Max Math Forum (aka Max Ray) has been rattling around in my brain for the last few weeks.  Here a grid representing streets in Ursala’s town:

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The problem-solving session, masterfully orchestrated by Max, allowed each group of teachers to develop their own representation of the situation and think about what questions could be asked. For example, if Ursala is at point 1 and needs to get to point 19 along the line segments, without backtracking, how many ways are there for her to travel?  Lots of discussion ensued at our table including the definition of backtracking.

I’ve been at school the last few days and anyone who has sat near me at a meeting in the last few weeks has seen me doodle this scenario, I’m sure wondering what my nerdy math-brain was concocting:

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Simplifying the grid and turning it into a pattern expanded the questions that I wanted to ask.  For instance, how many line segments (or streets) in Ursala’s case) are used in step n?

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What I’m still grappling with is how to expand my wonder about this scenario past the algebraic representations.  In talking with other teachers recently, it seems as though many of us have been programmed to solve these, and many other problems algebraically.  I recognize that many students won’t reach for the algebraic aid.  So my next step is to try to see this situation in other ways, sans algebra to better understand how my students are likely to see it.

 

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Authentic, Value-Added Algebra

About a month ago, my algebra class was working on the Math Forum’s Free Scenario called Val’s Values.  

There was a lot to question here (which they did) and a lot to wonder (which they did as well).  Something that was unsettling, however, was that they did not know the age of Val or Amir which they felt was pertinent to answering a major question:  Who has spent more on jackets in his/her lifetime?

We made some age estimates and answered our own question as best we could, but it felt less authentic than it could have been.  So, we submitted a comment on Valerie’s blog and today our attention to precision was answered with a response.

I gave my class another go at figuring out who spent more on jackets.  Here are a couple of their responses:

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What impressed me overall with their approach to this problem was not necessarily the mathematics itself.  The magic was in their careful identification of important variables and analysis of what mattered and what didn’t.  Additionally, they were able to look past the “right” answer and truly own THEIR answer from THEIR assumptions.  I had very few students ask Is this right?  Instead, they were communicating their methods with one another and challenging the reasonableness of their results.  The spark:  the flexibility of the scenario and the real response from a Math Forum Team Member willing to help add some authenticity to a classroom task.  Thank you, Valerie Klein.  We appreciate it.

 

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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.

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Full Circle Reflection

It’s almost the end of the trimester already which made today my last official “teaching” day with my Algebra class.  I’ve used a lot of the Math Forum’s Problems of the Week in this class.  Since this is a college algebra class, I use the POWS more as problems of the day rather than the week.  As a member, I have access to the library of problems, which I scour quite frequently to find just the right problem to fit the topic at hand.

Today’s adventure:  Rational Functions

I used a POW in which the first four terms of a patterned sequence of A’s and B’s are shown.  The students are asked to create an expression to represent the number of B’s in the nth term and then create an expression to represent the ratio of B’s to the total number of letters in the nth term.  What I like about this task in particular is that it isn’t a completely obvious fraction-ladened, asymptote-wielding, makes-a-student-want-to-cry rational function.  The students are able to work through most of the problem forgetting that this is in fact THAT type of function.  In fact, since they weren’t immediately scared off with a 1/x or the like, it seemed easier for them to make connections from their solutions to the graph and equation of the function.

What was particularly fantastic about this problem was that the growth of these students in the problem solving process was so evident.  It was clear as I circulated the room that over the course of a trimester, these students’ goals as mathematicians were evolving:  from “fast and correct” to “patient and curious.”

For example, when asked to find the term that results in 35% B’s, I had many students make a table with # of A’s, B’s, Total letters, and ratio of B’s to Total letters.  At the beginning of the trimester, these kids would accept their correct answer, but then reject their method of arriving at the answer because it was not as quick as those able to recall an equation or procedural method.  Now, after 13 weeks, these same kids are able to look at their table and appreciate the extra questions they can now address about this pattern scenario.  Additionally, some students were willing to attempt multiple methods in arriving at the answer.    It was a pretty profound moment for them as problem solvers and me as an algebra teacher.  I don’t know who was more proud, them or me.

Here are some samples of their work:

 

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