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:0126161730-1.jpg

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.  0126161732-2.jpg

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.



Somewhere between Concrete Sequential and Abstract Random

It occurred to be relatively early in during the trimester this past fall that my college algebra students (generally) have no idea what I mean when I say “quadratic function.”  This isn’t because they have never heard it, learned it, or used it.  But that technical of a term simply has not stuck around in their long term memory.

So, similar to linear functions, we start with a pattern:


from and

I then had them make posters including how it is growing, what the 10th, 100th, 0th, and -1st cases would look like, table, graph, expression, and relationship to the pattern.  Instead of one large poster, they use 4 smaller pieces of paper and tape them together.  That way each group member can contribute simultaneously.

I noticed:

  • It is difficult for students to describe how an irregular shape is growing.
  • It is even more difficult for them to describe something abstract like the -1 case.
  • Many of them expressed the overall growth as “exponential.”
  • Most could easily see the two rectangles formed and determine the dimensions with respect to “n.”

I wondered:

  • If they could connect the work with patterns to other quadratics.
  • How to have a meaningful discussion around the “exponential growth” issue.

Their homework was to answer similar questions for this pattern from You Cubed’s Week of Inspirational Math:


Spoiler alert: the rule for the pattern is f(n) = (x + 1)^2 or f(n) = x^2 + 2x + 1

So where do we go from here, two days before Winter Break?  My goals are to review some specifics on quadratic functions and simultaneously help the students make connections between different representations.  I know what I must do.  I must channel my inner Triangleman.

[Backstory:  Christopher Danielson and I go way back. At least to 2014. Maybe even 2013.  Seriously though, I strive to organize my college algebra class the way Professor Danielson describes in his blog.  I have picked his brain on more than a few occasions and he is gracious enough to give me advice in certain curricular areas. In short, his philosophy titled “They’ll Need it for Calculus” is the foundation of my College Algebra course. ]

Ok, back to room C118.

Me: Write down everything you know about the function y = (x+1)^2

(Most write down the expanded form, some start to graph, but not many)

Me: What other ways can we represent this function?

Students: Tables! Graphs! Pictures! Words! Patterns! Licorice!

Me: Sweet!  Let’s do all of that, minus the Licorice.

(I give them a few minutes to create a table and a graph.)

Me: NOW, write down everything you know about this function.

I circulate and hand each group a half sheet of paper.

Me: Write down the most important thing on your groups list.

At first I wasn’t really concerned what exactly they wrote down, but how they defended their choice. Then I came across this in all of my classes:


We came up with a pleasing list of attributes of a positive parabola that included vertex placement, end behavior, leading coefficients, and rate of change.

Next up for discussion: Parabolas grow exponentially.

Me: Turn to your partner and tell them whether you agree or disagree with this statement and defend your choice.

Students: Yes, words, words, words.

Me: Ok, now the other partner, say how you know something grows exponentially.

Students: Multiplied every time, more words, blah blah blah.

So we agreed that 2, 4, 8, 16, 32, 64… is an example of something that grows exponentially.

Me: Numerically, how can we tell how something is growing?

Students: (eventually) Rate of Change!

We came up with this table and agreed that these two functions were definitely NOT growing in a similar way.


Now on to helping them understand what it means for something to grow quadratically…


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.


East Coast meets Minnesota Nice


You guys have something called the “Trap Team?”


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


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

Using Local Data to Teach Statistics (1)

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.

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.


Thanks, Jenks

When you build up a future experience in your mind, it is not often BETTER than how you envisioned it.  Twitter Math Camp was that experience for me.  It was so much better than it looked on a hashtag.

In 2008, I began my twitter journey.  I mostly followed celebrities and friends.  My brother swore that twitter’s true gold was in following real people that have similar interests and ideas.  As it turns out, he was right.  Since jumping head first into the Mathtwitterblogosphere, I’ve experienced nothing but a genuine willingness to help one another become better educators.  TMC solidified my understanding of this network of delightful people that make up the math-educator-online community.

Recently, twitter was abuzz over the thought that TMC should be more theory, less play.  Part of the beauty of this experience was the organic nature in which everyone gathered and collaborated.  At professional conferences, you never see groups of teachers still talking pedagogy at 6pm, still at 8pm, and at midnight, and still at 2am. This went on for FOUR solid days.  Can you imagine this happening at school:  students staying after school into the night to work on the math investigation that they can’t stop talking about?  It doesn’t happen.  But anyone who’s been a summer camp counselor knows that there’s always that group of kids that can’t get enough interaction with their peers and choose to forgo sleep to soak it all in.  That’s why the C in TMC stands for CAMP and not Conference.

Some highlights for me: 

  • Justin Aion is the same ball-of-fun in person that he seems online.  I’m grateful for getting to spend time with him.
  • Max Ray is an artist at facilitating problem-solving.  His session was masterfully orchestrated.
  • Steve Leinwand is a humble communicator but an electrifying presenter.  I was moved by his keynote very much.
  • Malke and Christopher’s willingness to teach Math in Your Feet afterhours was generously spectacular. I was skeptical at first about my ability to engage, but I’m so thankful that I was pushed to do so.
  • Bob Lochel knows more stats activities than pages in a textbook.  I enjoyed working with him in the morning sessions very much.
  • Glenn Waddell is an amazing human being.  I’m humbled to have gotten to steal some of his attention this weekend.
  • Eli Luberoff is a humble genius and a class act.
  • I have the two greatest coworker friends, Teresa and Dianna, who came with me to Jenks and dove head first into the awesomeness of this community.


I had hundreds of interactions with some fantastic people.  This isn’t something that can be re-created online, despite the fact that the community began there.  Thank you, Jenks, for hosting such an incredible event.


Brain Sweat

I’ve talked about my Algebra 2 class at length on this blog over the last 2 months, and as the trimester comes to a close, I want to celebrate the positives in this class as much as possible.  They frustrate me sometimes, but the bottom line is I’m willing to fight and fight hard to make their experience with math more positive.  Ultimately, they’ve been dealt an unfair hand:  crammed into giant classes and labeled incapable of high-level mathematics.  They are capable of more than they give, but they also deserve much more than they’ve been given.

The perpetual optimist in me wants to continue to celebrate their achievements and play the hand they’ve been dealt as best we can.  Today we took on Robert Kaplinsky’s Cheeseburger Lesson.  I’m not sure why I’m constantly drawn to this lesson, since the picture of the 100×100 makes me a little ill.  Perhaps it’s the constant student engagement I get from it, time after time.  The intriguing thought that someone actually purchased this godzilla-burger hooks students every time.

What I liked most about my class’s efforts toward this task was the multiple revisions they had before arriving at the correct answer.  I had many students assume that a 3×3 cost the same as three cheeseburgers, only to find that their burger only needed one bun.

Below is a student’s work that I really appreciated.  At the end of the activity, he said,

Mrs. Schmidt, I’m sweating.  I thought so hard on this problem that I’m sweating.  But I believe I have the right answer.”

photo 5

If I’m being completely honest overall, this class has tested me, day in and day out.  I’ve worked very hard, but in the end, I’m not sure I taught them much of anything worthwhile.  I hope I have, but I’m not sure I did.  A class size of 36 seemed insurmountable, and perhaps in some ways, I never really overcame it.  Unfortunately, next year’s class size projections promise more of the same.  The silver lining, however, is that I get another crack at teaching this same course, and I’m 100% sure I can do it better the next time around.

Probability Ponderings

It’s been a great week in my probability and statistics classes.  I’m not sure why I’m pleasantly surprised.  This time of year it’s absolutely essential that we engage kids in meaningful mathematics and when we do, they respond well.

Monday, we did expected value and Dan Meyer’s Money Duck.  See Monday’s blog post for details.  Extra Credit if you can find my duck pun in there.

Tuesday, after assessing expected value, we moved to tree diagrams and conditional probability.

Wednesday, I used Nrich’s In a Box problem to create some discussion about dependent and independent events.  

I started with a bag with unifix cubes and had them do some experimenting to see if the game was fair.  What I love about this problem is that the initial answers that the kids come up with are usually completely wrong.  It really allows the teacher to identify the misconceptions.  Additionally, this problem is so easy to extend.  Simply have the students come up with a scenario of ribbons that creates a fair game.  Most will come up with something like 2 red and 2 blue. Have them test their theory, find out it’s wrong and then test another.  Even when they find the magic combination that creates a fair game, there is still the task of generalizing the results that’s challenging.     

Thursday, I totally stole Andrew Stadel’s 4! lesson.  What a great intro to the idea of factorial.  Last trimester I used IMP’s ice cream bowls and cones, which I still might refer to.  I felt like having a few students up in front at the beginning got everyone on the same page at the same time.  It was completely awesome to see the different methods for solving this.  I love the repeated reasoning here:


Plus, opportunities to use animal counters in HS math are scarce.

What’s the most pleasing about this week is that I think that this group’s conceptual foundation of these concepts is more solid than it has been in any previous year.  We still have practice to do, but I feel like they have made a good connection to what their answers represent.  In the past, my formula driven instruction didn’t bode well for retention of the concepts. I’m more hopeful this time around.