# 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.

# Summer Acceptance and Finally Freedom

July 4th. Independence Day. Let freedom ring while we eat a variety of barbecued unmentionables and enjoy pyrotechnicians creating art in the sky.  I hesitated in writing about this because I don’t want my blog to be a venting space, but I realized the eclectic nature of my posts are what make it uniquely mine.

Backstory:  Getting through the spring trimester seemed insurmountable because on March 2nd, a gentlemen, presumably heading to work just as I was, failed to look in the direction he was driving and smashed into my car.  My car was totaled and the base of my thumb was crushed by the airbag.

What was hurting me most though, was the resentment I had over this injury and the recovery over which I had no control.  And when you are an alcoholic, resentment has the power to destroy, and I felt very powerless over letting it tear me apart.  I barely got myself out of bed on weekends and paid little attention to my daughter and husband.  I ignored emails from my mom and shut myself out from letting her help me. I lashed out at people on Twitter, both overtly and in subtle ways.  I pushed away friends and neglected relationships, some of which I may not be able to recover.

Step Ten of Alcoholics Anonymous states, “Continued to take personal inventory and when we were wrong, promptly admitted it.” I’m thankful that I am a teacher for a multitude of reasons, and the summer off is giving me the time to find clarity and strength to rebuild what I have broken down in my state of depression.  Since school has ended, I have gotten myself out to visit with the three dimensional people on weekends, and I’m working on interacting more positively on Twitter.  I’m trying to repair broken relationships with people I pushed out of my life especially my mom, who I know always loves me.  And I’ve spent quality time with my child and my spouse.  And I am happy again. Genuinely joyful and self-accepting.  And free form the burden of resentment.

I went to an AA meeting recently, and someone made a reference to a paragraph in the Big Book on acceptance.  I marked it, have read it many times in the last few months, and am going to end this post with it:

And acceptance is the answer to all my problems today.  When I am disturbed, it is because I find some person, place, thing or situation unacceptable to me, and I can find no serenity until I accept that person, place, thing or situation as being exactly the way it is supposed to be at this moment.  Until I could accept my alcoholism, I could not stay sober; unless I accept life completely on life’s terms, I cannot be happy.  I need to concentrate not so much on what needs to be changed in the world as on what needs to be changed in me and in my attitudes.

Alcoholics Anonymous Big Book, Chapter 16, page 417

# 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:

# Welding Math and Metal – Day 2

If I needed to choose the most productive portion of most students’ week, Monday morning, first hour would be pretty low on the hierarchy of engagement.  I was undeterred because making sure we had the correct solution was important.

We discussed that the radius of the spool would decrease every time a layer of wire was used.  They began calculating the resulting wire as layers were removed.  This served as an excellent opportunity to introduce summation notation and a great practical use for the mathematics behind it.  It seemed like a much better option than to add up dozens of calculations anyway.

When we arrived at our correct answer (with the desired units) of 1.98 miles, the questions and estimating didn’t end.  They wanted to know how far they could stretch such a wire.  Would it go to the edge of our campus and back?  Would it go from here to the middle school?  Could you go all the way to the grocery store?

They settled on taking the wire, running it out to the edge of the soccer practice fields and then running it all the way to the middle school sign.  It ended up being, to the hundredth, the exact amount of wire we had, provided that someone would stand and hold the wire at the edge of the soccer field.  I loved the attention to precision. I also loved that they were so savvy with Google Earth.

# 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.