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:

Multiplication Square C thumb (1) thumb

 

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.

 

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.

The Anti-Answer-Getter

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:Untitled

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:

Make 37 1885 C

 

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:  IMG_6508

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.  IMG_6505 IMG_6506

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.

image

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:

IMG_4814

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:

IMG_4816IMG_4815IMG_4817

 

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?

IMG_4817

 

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.

 

Pair Products – An Nrich Favorite

In a few short weeks, I will be making a presentation at Twitter Math Camp on my favorite Nrich Tasks.  I know a lot of teachers have reservations about integrating rich mathematical tasks into their regular routines so I want to focus on problems that have that “traditional” feel while still allowing students to explore mathematical relationships more deeply.

Pair Products is an amazing offering by Nrich and its low barrier to entry makes it accessible for all students.  After working through the problem myself, Nrich offers additional questions to raise the ceiling.

Pair Products C

Additional Questions to Ask:

  1. What happens when you use 4 consecutive even or odd numbers? 5? 6? n?
  2. What happens when you use 4, 5, 6, n consecutive multiples of 3? Multiples of 4? 5? 6?
  3.  (My Favorite) What happens when you use n consecutive multiples of w?
  4. Does your generalization from #4 hold for numbers that increase by .5?  (For example: 3, 3.5, 4, 4.5)

My favorite Nrich pair, Charlie and Alison, offer two different approaches.  Charlie explains a clear algebraic manipulation to arrive at two expressions with a numerical difference.  Alison, on the other hand, represents the product of numbers with an area model.

Alison

An interesting challenge might be to ask students to show the area model that Alison employs for some of the additional questions.

 

Twiddle dee Twiddla

Yesterday was our first official day of SUMMER.  So after a thunderstorm curtailed my gardening plans, I thought I’d check out some apps that have been on my to-do list for a while.   First up:  Twiddla, an online collaborative whiteboard.  Why a collaborative whiteboard?  Our school district uses Google Apps and there are many beneficial collaborative options through Google docs, sheets, etc. The problem:  Mathematics just doesn’t translate very well when typed or through a computer medium.  If I’d like kids to collaborate in real-time via the web, Twiddla might be a viable option for students to collaborate in real time online, with a blank canvas.

What I like:

  • No login required.  Just post the web address and kids are good to go.
  • PDF’s and images are insertable into the background.
  • There is a grid background as well.
  • Students can “chat” or audio conference while working.
  • A variety of colors, shapes, and line thicknesses can be utilized.
  • The Pro version (usually $14/month) is free for educators and students.
  • The writing is very smooth without a stylus.

What I did not like as much:

  • Annotations are added when writer “pauses” rather than as they are writing.
  • An “undo” button would be helpful.

Some screenshots from my twiddla-created session:

 IMG_4068 IMG_4069 IMG_4071 IMG_4072

Now, I’ll have to wait until Fall to test this app out with students, but I’m optimistic about it’s potential.  It could just be one of those things that’s “cool” but in reality, pencil and paper will do.

Yesterday:  As much as I get frustrated by the attitudes and actions of my 5th hour, much of my resentment stems from the fact that I believe the situation in my class is my fault.  I feel like I’ve conditioned them by accepting disrespectful behavior in order to keep kids in the classroom.  As a result, the entire learning environment has suffered.

Today: So that was the beginning of yesterday’s post. I was concerned going into today’s class. Last Friday of the year and the fact that the school has been a circus compounds the issue. I was expecting chaos, but what I got was mathematical success. The difference was I demanded their attention in a more respectful way. I was firm, but polite, and it payed it’s dividends in student engagement.
We began with a simple math talk that I modeled from Fawn Nguyen’s March 21st math talk:

Today is the 30th day of the month.  Write as many equations you can that equal 30.

I gave them about 5 silent minutes. Then I let them use their calculators to come up with more gems.  At the end, I had them share their favorite or most complicated equation on the whiteboard.

IMG_5388 IMG_5389 IMG_5390

 

Here’s where the real magic happened.

Me:  Look up here and see if there are any equations you disagree with

Lots of discussions ensued about order of operations, square roots, rounding, parentheses, etc.  Overall, the activity lasted 30 minutes, which was about 29 more minutes of math than we did yesterday.

But the fun doesn’t stop there.  To boot, I introduced the Mathalicious Decoder Ring Lesson.  We watched the Christmas Story clip and talked about what a decoder ring does.  What I liked is that most of them were trying to figure out how the decoding worked, rather than just “get the worksheet done.”

Justin Aion stated poetically on his blog today the exact way I feel about this class:

We teach students long-term strategies to accomplish short-term goals and often don’t see any progress.  If we are very lucky, we’ll see the kind of growth we want by the end of the school year, but the growing season on students isn’t as regular as it is for other crops.  Each seed needs its own time to grow.We desperately need to get away from the notion that if it hasn’t sprouted by the beginning of June, then it must be a defective seed.