# Rational Function Fan Fair

Sometimes when planning a unit, I browse through the Desmos Activity Builder.  When searching for Rational Functions, I came across Dylan Kane’s Building Rational Functions Activity.  Excellent.  I now had a muse.  Here is what I came up with for college algebra:

I like to gush over my students when they do awesome stuff, and this was no exception.  I love it when my classroom is abuzz with sense-making conversations.  I feel like this activity helped students become more comfortable with the structure of rational functions and how that equation structure is reflected in the graphs. Thanks, Dylan, for inspiring some awesome thinking in my class today.

# Watching Solitaire in Silence

Remember Windows Solitaire? I have fond memories playing this fantastic digital distracter with my high school beau on his brand new Gateway computer.  We would take turns striving for success in this card-clicking frenzy, the other watching and waiting patiently for the deck to empty.

But have you ever watched someone play solitaire on the computer?  It is so…what word comes to mind?  Frustrating?  Infuriating?  Aggravating, perhaps?  And why is that?

Check out this screenshot:

What if the player was about to click on that blue, flowery deck of cards…would you be fighting the urge to save them from their potentially game-ending error of failing to move the sequence beginning with the six of spades to its rightful place atop the seven of hearts?  Or would you idly sit by and let them to figure out that solitaire is won by carefully searching for card moves before drawing from the deck?  Would you make any suggestions for improving their game once failure was inevitable?

I think this solitaire analogy is a lot like teaching.   I realized fully today why the “productive struggle” is so hard to sustain and perhaps why teachers so often fall back on traditional methods of delivering information to students:  Watching people struggle without intervening is difficult. Just as it’s natural to want to smooth out the path for our children, it’s also tempting to do the same for our students.  It’s just easier (and so much faster) to zip Maria’s (my daughter) coat or buckle her seat belt or pick up her toys.

As a simple, mathematical example, imagine one of your students is attempting to solve a quadratic equation. They start off like this:

Being the savvy algebra teacher you are, you can anticipate the error that the student is most likely going to make.  You’ve seen it hundreds, if not thousands of times.  Your inner teacher voice might be thinking, “For the love of humanity, Herbie (not your real name), set the dang thing equal to zero!  Quadratic formula!  IT’S GOT A SONG, FOR GOODNESS SAKE!”

Instead, you do not impede their solving and let them continue on their merry, algebraic way.

Re-enter teacher voice in your head, “Now look what you did, Herbie.  You’ve gone and…wait…one of those answers is right.  Great.  Now we’ve really got issues.”

So what do we do about this?  Clearly the student needs some redirection and the teacher’s role is to guide the learning.  But had we intervened during earlier steps, we rob this student of a golden opportunity for brain growth.  Plus, we deprive the rest of the class the chance to learn from the misconception.  Even more, what a fantastic extension we have here:  why did the student get part of the problem correct and part incorrect?

In summary, we deny students the opportunity to learn from mistakes if we  prevent them from making mistakes in the first place.

Related Side Note:  I’m currently reading The Gift of Failure by Jessica Lahey.  Her introduction about her son’s shoelace-tying trials seems strikingly familiar.  And I can use this antedote as a reminder when encountering the zippers, and the seat belts, in addition to quadratic equations.

# Bingo Lingo

This time of year, the standards used to measure the success of a lesson may look different than they do at other times of the year.  For example, some teachers might consider “Students not using worksheets to have paper airplane throwing contest” to consistute a lesson well executed.  To a certain extent, I am joking, but there’s a thread of reality there.   Think back to a time when your excitement for a future event prevented you from doing anything productive.  Now imagine leading a room full of 32 people with that same excitement and handing them a manual for their new scanner/copier.  You get the idea.

I can usually distinguish between my being pleased with a lesson based on lowered expectations and my being pleased with a lesson because of a high level of learning and collaboration.  Today was the latter, with my 9th grade probability and statistics class again.

I found this on Don Steward’s website.  If you have seen his blog and are not fascinated, or at least intrigued, we cannot be friends.  He comes up with some amazingly simple, yet elegant classroom problems.

We started this yesterday.  They are in groups of 4; the oldest student in the group got to choose first and so on.  Then they played three “games” using a pair of dice and a whiteboard with their numbers on it.  Today, they worked on figuring out why the “6,7” card was the best and determining how to rearrange the numbers  on the cards to make them all equally likely to win.

I’ve had this glossy paper in my room forever, so I decided to have them make a mini-poster with their solution and some reasoning.  Here are my two favorites:

# Sitting in a Circle, Talking about Numbers

“I feel like all we do is sit in a circle and talk about numbers.   It doesn’t even feel like work.”

“This class is more exhausting than my PE class!”

“It’s nice to be confused and then un-confuse ourselves.”

These are words I’ve overheard from my college algebra students this year.  I couldn’t be more pleased with the strides they are making with my problem-solving framework.  I learned the hard way last year that you cannot just throw a problem solving scenario at a student and expect them to immediately persevere, even if they understand the underlying mathematics involved.  Having learned from my mistake, I sequenced the problems this year in a way that has worked to build on their Algebra problem-solving skills.  Furthermore, I’ve put them in groups of 3-4, which has helped tremendously in getting them to talk about their approaches.  Last year, while in pairs, the conversations didn’t occur as naturally as I had hoped.    Here are a few of the problems we’ve tried:

Additionally, we’ve used other Nrich problems such as Odds, Evens, and More Evens.

And to add some non-dairy whipped topping to this algebra awesomeness, my students are breezing through visual patterns and having some great conversations about them.  Credit here is due to their fabulous algebra 2 teachers who began visual patterns with them last year and let them struggle with them.  The result has been deeper connections and a more thorough understanding.

I must start off today saying that I have never experienced such a fantastic start to the school year than I have this year.  The energy within our department is almost palpable, and I know that the students are catching on as well.  Here’s an email I got from one of my co-workers this morning:

I want to give credit to Teresa and Dianna because they were more of the driving force behind encouraging the use of Plickers.  I’m thrilled with the result nonetheless.

The group that impressed me the most today was my first hour, math recovery.  These are kids who have previously failed a math class and are recovering credit.  You can imagine the lack of math love in the room.  Here was their prompt:

SPOILER ALERT:  I’m going to reveal the answer so if you’d like to try it for yourself, stop reading.

I had them come up with ways they could make 37 using different amounts of numbers.  It seemed that we could get 36 using 10 numbers or 38 using 10 numbers but couldn’t quite get 37.  Then we tried getting 37 using 9 numbers or 7 numbers.  We had some good discussion about which strategy seemed the most useful.

One student in particular mentioned that he wanted to add some and subtract some but he felt he would always be short without a 2.  I had them share their results on the board and I was very satisfied with the effort I’d seen.

I was nervous about the answer reveal because as it turns out, it’s impossible to make 37 with 10 numbers.  What we were able to do is focus our attention on what we DID discover, rather than the fact that there was no answer.  We discovered that Odd + Odd = Even, Even + Even = Even, and Even + Odd = Odd.  Because there is an even number of odd numbers, an odd sum is not possible.  I was more pleased with this result than any single answer they could have given me.  I expected a backlash from a group of students used to answer-getting but found that they were able to embrace a learning activity that didn’t one final answer.  I’ll mark that class period in the win category.

# Talky, Talky, Talky. No More Talky.

Because I’m hyper-interested in helping to create a space where kids feel comfortable sharing ideas and making mistakes, I began my classes today with the Talking Points activity that Elizabeth Statmore (@cheesemonkeysf) shared at Twitter Math Camp this past summer.  Learning that a tight rule of No Comment was a cornerstone of the activity intrigued me to try it in my classroom.  Productive conversations in math class don’t happen automatically very often.  I’m hoping that using this process helps students to use exploratory talk around mathematics.

The No Comment was difficult for students, but I realized quickly, it was difficult for me as well.  For example, when debriefing with the whole class, I was tempted to comment…after each group presented.  I had to tell myself each time a group gave a summary that there wasn’t a need for my comment.  I was tempted to clarify thinking or give a follow up explanation.   I needed to let the groups own their experience.

This realization made me cognizant of the other times a comment by me is unnecessary following a student response.   How many times have I insisted on having the last word in the class?  How many times have I summarized a student’s thinking for him or her?  Hopefully, as students move toward being more exploratory with their discussions, I can move toward being less dominant in the conversation.

# Facing Fear

It’s always fascinating to me to watch students step into a new classroom and immediately search for their social comfort zone.  Students aren’t unique in this phenomenon; they are just the group of humans in which I interact the most.  Today being the first day of school, the visible and invisible social boundaries that students draw between one another were clear as I silently observed.

As someone who struggled fitting into a unique social group growing up, I’m most interested in encouraging kids to break away from their cliques. After reading much of what Ilana Horn has written on the subject, I also began to see links between being socially extroverted and status in the mathematics classroom.  For example, kids who are quiet and mostly keep to themselves don’t often have opportunities to display their “smartness,” whereas an outgoing kid willing to contribute voluntarily to class discussion would have their “smartness showcased regularly.  Interestingly enough, when doing the “personality coordinates” activity with my college algebra class today, one group created this graph:

They defined social achievements as number of friends and academic achievements as GPA.  It allowed us to have a nice discussion about grades and overall intelligence as well as some lovely talk regarding different definitions of social achievement. I look forward to continuing these conversations over the course of the trimester and challenging them to let their popularity guards down.

On a similar note, I tried the Blanket Challenge in my Algebra 2 class.  If you have not read this chapter in Powerful Problem Solving, I’m not sure why you are still sitting here.  Go read it! What impressed me with this group of kids, was they were willing to step out of physical comfort in order to achieve the result they wanted.

On the first day of school, in a class that’s tough to adjust to, I can’t begin to express how proud I am of this group of kids for their willingness to work together respectfully and successfully.  I’m hoping to build on the results from this activity in the days to come.

# Torch Relays

Two 12-hr work days down, 5 days until school officially starts. (Cliche about how there’s never enough time). I’m optimistic about this year, but I can’t remember a school year that I didn’t have a positive outlook. (Incurable, I’m told).
Yes, this summer, I attended Twitter Math Camp, and there’s a lot of residual glow that transfers easily to energy toward my classroom. But what’s really got me charged this year is watching my two co-workers, who joined me at TMC, prepare for the school year by igniting the rest of our department with the torch they’ve had burning since we got back from Jenks. These two awesome women (@tootalltrees and @d_Hazelton) have courageously engaged the other math teachers at the highschool in important conversations about how students learn mathematics best. And it’s catching on. Hopefully like wildfire.
I put my desks in groups of 4 today and took a neat panoramic picture with my new phone. I’m excited to see if it’s a successful, productive room arrangement.

# 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:

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:

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?

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.

# Confession: I’ve never really been good at math

Here’s a confession of mine:  I’ve never really thought of myself as ‘good at math.’  Yep, I’m a high school math teacher proclaiming my discontent with my mathematical abilities.  Ironic?  Sad?  Make you want to hide your children?  Read on, it’s not as bad as you think.

Being a math teacher was a second career for me, as my undergraduate degree is in accounting.  I dabbled in a minor in mathematics while at the University of Iowa but let a ‘C’ in Linear Algebra from a cold professor change my trajectory for the next 4 years.   When I went to graduate school to earn my masters in Mathematics Education, I was always intimidated by the math undergrads who were much more polished and current on mathematical theory.  Recently I came across this article which shed some light onto what often happens with girls in areas like mathematics. In short, women tend to give up on themselves more quickly because of their strong inner voice.   I know that I was never discouraged from pursuing difficult challenges by my parents, especially academically.  I came from a family that was very supportive of my education.  It was my own inner-voice telling me that I wasn’t as good at pure mathematics, which was the lingering after effect of that C grade.

Recently, Rafranz Davis wrote a blog post about the transformation of twitter admiration into palatable inspiration.   This post was timely for me since as summer conference season reaches its peak, I’ll be attending Twitter Math Camp starting on Thursday with dozens of other math tweeps with whom I’ve admired and been inspired by.  These positive interactions have projected me to a place where I’m comfortable with my mathematical abilities and completely humbled by my ability to participate with such a wonderful group of educators across social media.