Plan, Teach, Grade. Rinse, Repeat.

The serendipitous timing of a three hour layover in Newark must be maximized with a reflective blog on my weekend in Pennsylvania.  Specifically, about my day spent observing Justin Aion in his classroom.  If you follow Justin on twitter, read his 180-blog, or have the privilege of knowing him personally, you’re no doubt familiar with his harsh assessment of himself as a teacher and his no-nonsense way of sharing his opinions.  I knew heading into a day of observing his class that I was in for a treat, whatever that meant.

Justin let me blog about the experience as his daily entry, so if you’re interesting in reading about the amazing time I had, head on over there and get reading.  In the meantime, I want to discuss some implications this day-long observation had on me in my role at my school.

As a department head, I have the chance to view the classes of multiple math teachers, once, sometimes twice a year.  I find these opportunities incredibly valuable and I always have a take-away from each observation I do to benefit my own teaching practice.  Usually, I have a 20-30 minute conference with the observed teacher where we can reflect on what is working and how the students responded to the lesson.  These “after parties” are, hands-down, the most important part of the observation process because it is in those post-conferences that enlightenment results from the insights an outside observer can provide. Unfortunately, because all of these interactions have to happen outside of the school day, many of the important issues that arise in these post-conference meetings cannot be dissected and fully examined and therefore result in little or no long-term changes. For example, in the post-conference, a question addressed is “What changes or modifications would you make in the lesson?”  A few lines are filled in on a form and off they go to teach the next group of students. The discussion simply doesn’t happen because there’s not enough time to have it.   And because these observations are shared only with a small group of people (an administrator and an instructional specialist), there is no discussion of what issues arise over and over again in multiple classrooms.  As a result, few can benefit from the individual observation.  And there are hundreds of them done in the district each year.  And it seems as though the math department as a whole isn’t able to reap the maximum benefits of having a trained observer visit classrooms.

What was different about observing Mr. Aion’s class was that the observation happened from 6:30am until 2:50pm, but the post-lesson conference went on for the rest of the weekend.  I’m obviously not suggesting that we all spend a weekend math retreat discussing pedagogy over Thai food and club soda, but the conversations we have as teachers with teachers are important.

Important and not valued.

In fact, any time that isn’t spent actively teaching or planning and grading is consistently shoved outside of the 7-3 school day.  And it seems as though the teachers who want to engage in those discussions and realize their value must sacrifice personal time in order to do so.

Here’s another example:  Just today, at lunch, one of my colleagues brought up his frustration with a student that we share.  His grievances aren’t unique to this particular student and highlight an ongoing struggle that all teachers in our department share.  We were able to have a very meaningful discussion and came to a workable solution for this kid.  The problem:  this conversation, while incredibly productive, had to happen during our already rushed lunch period.

I think this bears repeating:  Teachers need time during the school day to talk with other teachers.  And I don’t mean to lesson-plan for common preps.  I mean to really dig deep into the outcomes of our teaching and have meaningful discussions on how to better serve all of our students.

Are Teacher’s Safe in Admitting Mistakes?

Image from
Image from

Sometimes on the way home, I tune my Iphone to the TED Radio Hour podcast.  I love the way Guy Raz re-captivates his audience by highlighting numerous, already captivating TED talks.  Today, I was listening to Making Mistakes and the first segment was an interview with Dr. Brian Goldman.  His words here are striking:

The redefined physician is human, knows she’s human, accepts it, isn’t proud of making mistakes, but strives to learn one thing from what happened that she can teach to somebody else. She shares her experience with others. She’s supportive when other people talk about their mistakes. And she points out other people’s mistakes, not in a gotcha way, but in a loving, supportive way so that everybody can benefit. And she works in a culture of medicine that acknowledges that human beings run the system,and when human beings run the system, they will make mistakes from time to time. So the system is evolving to create backups that make it easier to detect those mistakes that humans inevitably make and also fosters in a loving, supportive way places where everybody who is observing in the health care system can actually point out things that could be potential mistakes and is rewarded for doing so, and especially people like me, when we do make mistakes, we’re rewarded for coming clean.

If we replace doctors with teachers and medicine with student learning, would we proclaim that our profession fosters still safe space for teachers to talk about their mistakes?  Do we have the opportunity to be vulnerable in our classroom approach in order to improve our practice and learn from others who have made similar mistakes in the past?  In an age of education the covers of news magazines equate bad teachers to rotten apples, are we really able to grow directly from what is not working in our classrooms?  Or are we lead to believe that mistakes lead to labels like “ineffective teacher”?

We preach constantly about creating safe mistake-making environments for our students.  This strategy needs to be applied to teaching as well. We, too, need this loving, supportive environment that all may benefit from.   How do we, as teachers, create this space for ourselves so that ultimately, we can learn from those mistakes and make our profession stronger?

Female Feelings and Brash Boys??

It’s Friday, it’s Halloween, and our football team is playing tonight in the section finals for the first time in about 30 years.  To think that solidifying understanding of domain, range, increasing and decreasing functions was going to be a priority for my college algebra class was a farce and so I decided to make the class more productive.  We watched the Simpsons. And I ate my weight in Swedish Fish.

Image from Wikipedia

Image from Wikipedia

Girls Just Wanna Have Sums pokes fun at the stereotype that men do better in math and are inherently aggressive and women want to sit and talk about their feelings.  Consequently, Springfield Elementary is divided into a girls school and a boys school which embody those stereotypes.

(If you weren’t aware, every Simpson’s episode ever created is available on FX’s new website, Simpsons World.)

After the show, I asked the students to write about their feelings on this stereotype and how it plays out in the United States.  I then asked them to describe a satirical jab that they found particularly disturbing or upsetting.  As a math teacher, I sometimes lack the inclination to ask students how they feel on a particular controversial topic; but I’m always glad when I do ask.  Acknowledging differences that exist in the ways males and females approach math is important.  But this episode was more about educational access to high level mathematics.

The majority of what they found disturbing was that the girls weren’t given the same opportunities as the boys to experience challenging mathematics.  As I responded to their submissions, I took the opportunity to push their thinking further and ask “Are there instances, other than gender, where students are not given the same educational opportunities?”  I hope that my feedback will foster a dialog about the importance of all students having access to a high quality education, beyond inequalities based on gender.  Because these inequalities exist, maybe not based on gender, but definitely based on race and socioeconomic background.  I’d like to continue to help them think about what that means for those students.

Talking about Black and White rather than X and Y


I was out of my classroom yesterday, attending an AP Stats workshop with my coworker and friend, Dianna Hazelton.  Upon my return, per the usual, I learned that students struggled with the assignment I left for them.  Naturally, it would seem that the most important task to be completed today was to address their issues with yesterday’s work.  But something much more important came up:  a discussion on racism and sexism.

The natural opportunities to discuss race in a mathematics classroom in rural Minnesota are not numerous.   I usually need to carefully weave them into the topics and diligently ensure that equity is valued when a student brings up situations of racism.   The minority population in our district is not high but being the only student of color in a classroom is challenging for them.  These students can’t be expected to just blend with their white classmates when their needs aren’t being addressed.

Somehow today, instead of practicing line graphing on whiteboards, we discussed race and gender when a student expressed her discomfort while attending a concert in Minneapolis.  Her comments were respectful, but her concerns legitimate.  As a teenage girl, when at social events in the city, she and her friends feel vulnerable and sometimes threatened by the sexual advances of men.  She started off pointing out black men specifically, but the conversation progressed to a point where she acknowledged that  first, her isolated experience shouldn’t shape a stereotype about all black men and secondly, white men engage in this behavior as well causing the same discomfort for her.  She quickly realized that her assessment of dealing with harassment shouldn’t be examined through the lens of race.

Rafranz Davis, a woman whose fearless, relentless advocacy of kids I highly admire and respect, summed this up perfectly on her blog:

Students carry unique perspectives about their experiences and until these issues, along with the countless others unaddressed, are met head on through discussion and action, these tensions and perspectives will never change.

I was very proud of this student  for acknowledging her initial prejudice, and as a result, we were able to have an equally productive conversation about gender as well.  And something I didn’t expect happened:  the boys just listened.  They just listened to the girls talk about cat calls and being whistled at.  “Just come and say hello, my name is so-and-so,” one girl said, “that’s much more of a turn on than being harassed.”  And at the end of the class period, one of the boys went up to that girl and said, “hi my name is…”  Bingo.

These conversations are difficult, but when a student is willing to admit their prejudice, the teacher doesn’t only have an opportunity, but a duty to help foster positive change.  Graphing 3x + y = 10 can wait until tomorrow.  The real problem of the day, and every day, is that these kids come to our schools for 7, 8, 9 hours a day and we spend such a small percentage of that time listening to their voices and giving value to who they are inside.



A Conceptual Victory

College Algebra, reviewing the graphs of polynomial functions.  Each student has a whiteboard.  We started with y=(x+2)(x-3) for simplicity.

Me:  What do we know about this graph?

Student(s): It has x-intercepts at -2 and 3 (or something along those lines)

Me:  What else do we know?

Student(s):  It’s a parabola (or some version of that)

Me:  What else do we know?

Student(s): [Crickets] (or owls or frogs or some other creature that makes noises when all else is silent.)

Me:  Does this parabola open up or down?

Student(s): Up. Down. no Up. no Down.

At this point I’m shocked that they do not remember the one polynomial coefficient that they all nail down in algebra 2:  The a value.  But I shouldn’t have been.  Rather than asking “What will the sign on the x^2 term be?” I decided to approach it differently to see if I could garner some conceptual understanding.

Me:  If x is a really big positive number, like a million, what kind of number will we get for y?

Student(s):  A really big number.

Me:  Similarly, if x is a really big negative number, like negative a million, what kind of number will we get for y?

Student(s): [After much thought and group deliberation] A really big positive number…OH, then it opens up.

This wasn’t a huge victory, but it was satisfying.  Because not a single student mentioned an a value even if they were thinking it.  Additionally, when we moved to cubic functions like y=(x-2)(x-3)(x+4), they used the idea of substituting really big negative and positive numbers for x to determine which way the graph was trending in each direction.  We were then able to have a nice discussion about why a graph like y=(8.5-2x)(11-2x)(x) looks similar to y=(x-2)(x-3)(x+4) when the equations have so many differences.

When students learn a procedure, it’s very difficult for them to deviate from the steps in order to solidify their conceptual knowledge.  I’m very glad that on this Friday, their forgetfulness of the “steps” allowed us to have a nice discussion.

The Corn Sandbox

For an entire year, I’ve been anticipating our family’s return to Stade’s Shades of Autumn Festival.  My excitement has been building for one reason:  To estimate the amount of corn contained in their corn play area (or the Corn Sandbox as my 4 year-old has named it).

We went last Friday, armed with a measuring tape and a measuring cup.  The sign seemed to give away the answer of 800 bushels, but I wasn’t satisfied given that their was no mathematics to back up their claim.  We needed to attend to precision. Here are some of the photos I took:


IMG_20141017_111537.360  IMG_20141017_111603.675 IMG_20141017_111712.631

My favorite is the bottom photo, my parents counting kernels of corn in 1 cup. (There are 579, by the way)

I love when I present a problem to my class, and it takes longer than I anticipate for them to solve. There was supposed to be time for solving inequalities for the group that worked with this, but that will just have to wait until tomorrow.  I’m sure they were crushed.

The essential question we wanted to answer was:  How many kernels of corn are there in this corn sandbox?

Initial estimates were very low.  I let them revise after I revealed that 1 cup contained 579 kernels.


One approach:  Use the number of bushels to calculate kernels.

After looking up on Wolfram Alpha that 1 bushel = 9.31 gal, we determined that a reasonable calculation of the number of kernels, based on our 1 cup count, was 69,000,000.

Second Approach:  Use the volume of the enclosure to calculate kernels.

This was a little trickier given the irregular shape of the sandbox.  Numerous calculations and conversions later, we arrived at 81,667,931 kernels of corn.

We were uncomfortable with the over 12 million kernel discrepancy between our two methods.  It remains unclear which is more accurate given the fact that one includes actual measurements and assumptions and the other is provided by the farm.  Perhaps Stade’s Farm should expect a call from Mrs. Schmidt’s 2nd hour Algebra 2 class in the near future to clear this up.

Spiders Everywhere!

Steven Leinwand has a huge influence on how I approach a math lesson.  In my experience, one of the easiest ways math can be extrapolated from almost any task is by asking the questions:  How big? How far? and How Much?

This weekend, I came across this picture on social media, posted by David Roberts:


Of course the question I asked first was “How big is that tarp!?

Luckily, David was willing to make an estimate and allowed me to share his reasoning:

Screenshot_2014-10-11-18-20-20 (1) Screenshot_2014-10-11-18-20-27

I thought it might be interesting to present the photo to my first hour and see what questions they would ask about the photo.  Of course, the surface area of the tarp was on their list.

The original article including the photo added more depth to their questions.  As it turns out, the house is being fumigated after a spider infestation.  It seems as though their curiosity surrounded more the spiders than the tarp and legitimately so.  (The article estimates the house was infested with approximately 5,000 spiders)

I was pleased that my students used visual cues in the photo to make their estimates including the average height of a story of a house, the approximate height of the man in the photo, and the size of the window.  Luckily, we found another photo that gave us a better understanding of how much tarp was needed for the other side of the house.


I’m glad this man’s golf game was not disturbed by a spider problem in the distance.  (sarcasm)

Anyway, after making some calculations, my industrial-minded first hour realized that this type of tarp must have a somewhat standard size.  After doing some Google searching and actually calling one of the companies that manufactures these behemoth plastic coverings (authentic!  Yes!), they decided that there were four 84′ x 25′ tarps covering the house and the space surrounding it. (We were able to have a nice conversation about how multiplying each dimension by 4 was different that multiplying the area of the tarp by 4.)

In the end, their estimate was 8,500 square feet, approximately double of what David had estimated.  We then critiqued David’s argument and decided that based on the picture only, his calculations were reasonable.  Because we were able to dig for more information, my class believes that their estimate may be a tad more accurate.  Thanks, David for sparking our curiosity this morning.