Calling for a Compass Comeback

Have you ever handed a compass to a child? After they satisfy their initial curiosity that one of the ends is pointy, have you ever then seen what a child does with that compass? I recently attempted this experiment with my 7 year old daughter. Her immediate discovery: She could make a flower by strategically constructing 7 circles.

I wanted to have a conversation with her about the 7-circle arrangement and see what geometric understanding we could build on from what she noticed and wondered.

She identified shapes and “non-shapes,” talked about shapes that could fit inside other shapes, and developed methods of determining whether or not certain shapes were congruent. She was most interested in cutting out figures and seeing what the result was when she folded them in different ways. My favorite observation of hers was with respect to the image on the left. She described it as “the top of a rocket ship being built on a really strong stick.” What’s important here is she made sense of what she observed and then justified it based on what she knows to be true: The rocket needs to balance on a strong, vertical stick in order to be completed. She is developing her own understanding of geometric relationships. It is much less important that she can produce an output that matches one of the 1st grade Minnesota Academic Standards in Mathematics. We can build up to understanding specific standards, but only after we have a foundation based on what she has already made sense of herself. In this case, the sense-making began with my handing her a compass and then having a conversation with her as she explored with it.

If you walk into a typical secondary math department, often the compasses can be found collecting dust and developing rusty hinges in the back of the supply closet. And these ancient tools have been traded for more updated technology such as animations of classic geometric theorems such as the perpendicular bisector theorem. Here is an example from a popular geometry textbook:

(Source: Envision Geometry, Pearson 2017)

You can draw your own conclusions about the usefulness of this example and how it can help students develop an understanding of perpendicular bisectors. But what I notice, besides that the perpendicular bisector theorem has been dwindled down to a series of steps to understand, is that the reasoning behind the compass arcs is unclear and somewhat ambiguous. Additionally, the students are not given an opportunity to explore their own curiosities and play with the mathematics. They are left with a set of steps to follow without ever having connected this idea to a circle or a compass.

What would happen if instead we gave students a copy of this image (or better yet, let them draw it for themselves):

(Source: Don Steward)

What questions would they ask? What properties would they explore? What would they notice and wonder? And if they didn’t hypothesize about perpendicular bisectors (or regular hexagons or symmetry), how could we as secondary math teachers lead them down the path of making sense of this theorem?

Here is another image from Don Steward’s blog which could ignite similar curiosities:

(Source: Don Steward)

He begins with an initial question, but what else might your students ask about this particular design?

Geometry class is ripe and ready for these kinds of interesting, beautiful explorations, yet our textbooks insist that students need to be shown steps and procedures to understand these theorems well enough for a standardized test.

Sometimes we are fearful of using manipulatives in math class because we are afraid that kids will “play around” with them. But in order for them to do sense-making, to develop a lifelong appreciation for, or even love of mathematics, isn’t that exactly what we have to let them do? They need a chance to play and wonder, conjecture and experiment. We need to believe that children (and yes, high schoolers) can develop powerful mathematical ideas, and if these ideas do not perfectly align with our mathematics standards, we need to develop strategies to build from student sense-making.

Spiraling the Hundred Chart and Beyond

Now that I have a little more time on my hands, I decided to write up some of my spiral explorations so that others can more easily play along or use them in the classroom.

Spiral Instructions

Templates:

Spiral Templates – 100 1 sheet

Spiral Templates – 100-4sheet

Spiral Template 1 to 196

Spiral Template 1 to 1089

Google Document Files:  Instructions   Templates

Suggestions:

  • Because my school is not 1:1, I printed these out and had the kids use markers or colored pencils.
  • I have them start with the spiraled hundred chart (printed 4 to a page) to have them play around with patterns at first. This is also a good opportunity for them to test conjectures.
  • Connecting the numbers in the pattern adds another element of visual interest.
  • Students looking for an extra challenge can try a variety of quadratic patterns.  I usually have them put a function into Desmos or a graphing calculator and use the table.

Here are the pentagonal numbers done on the large spiral:Pentagonal.jpg

#NCTMRegionals Chicago Presentation

As promised, here are the slides and resources from our NCTM Regional session, “Stats Trumps Hate.”

I want to thank Carl Oliver for coming all the way from New York for the day and leading this presentation with me.  Carl, you are such an incredible human being, and you have so much awesome to offer the math educator community.  I am truly humbled that I have gotten to work with you so many times on this topic that is close to my heart.

https://docs.google.com/presentation/d/117234ae5AWL_2P3UODDqJGUO1CvOvs_VuoECkMoj3fQ/edit?usp=sharing

…But the End is Beautiful

Every term, a lot teachers (myself included) are disappointed that students didn’t learn exactly what we intended them to. Furthermore, they didn’t learn as much as we wanted or thought they could. I’ve been there before many times, and that result is the main reason I changed the structure of my class to focus on student discourse and making sense of problems.

The last question on the final today was “Tell me something you learned about yourself this trimester. (It does not need to be math related).” Of course, it’s a math test so most of them tell me something math related anyway which doesn’t bother me one bit.

I originally was going to just share some highlights, but reading these answers brought me so much joy, I am just going to share the whole list.

I learned:

  • I understand math better when I’m given problems to try rather than just talking notes.

  • How to help myself learn on my own by asking more questions to deepen my understanding.

  • I can learn math with a group of people I don’t know at all and we can have fun as well.

  • I can develop my own solutions to problems and make sense of ideas myself.

  • I could be independent and do my work with less and less help.

  • It’s ok to ask questions but I learned how to believe that I can learn math myself and figure things out.

  • I don’t give myself enough credit for what I am able to do.

  • I learn things a lot better when working with others.

  • I am capable of being independent but still learn what I need to in order to be successful.

  • I am able to figure things out myself even if it’s not explained to me first.

  • Making sense of WHY I am getting an answer is much more important than the answer itself.

  • Understanding why something happens is much more useful than understanding just how to do it.

  • If I work through problems and bounce ideas off my teammates I CAN figure things out instead of being shown first. I enjoyed this, which surprised me.

  • Applying yourself to learn the material instead of memorizing it is much easier.

  • I CAN in fact understand the abstract concepts of math.

  • If I can’t explain WHY something works, then I don’t fully understand the concept.

  • I am a very hard worker and determined to reach my goals. I’m unstoppable when starting a problem.

  • I am able to graph functions without a calculator because I understand how they work.

  • The graphs of equations are really satisfying once you understand why they work.

  • I enjoy being able to figure things out myself first.

  • I really like to work with people to figure things out and I didn’t think I would.

  • I am able to make sense of my answers and not just get the answer, which is much more important.

  • Once the connections between concepts were more clear, math becomes much easier to understand.

  • While I’m not the best at understanding things while I’m being taught, when I teach other people, I retain the information.

  • How to explain myself better by teaching math to others.

  • When I need to explain something to someone else, I understand it better.

  • If someone doesn’t understand something, I am able to explain it in a different way that they can understand.

  • The more I am engaged with the math, the better I do.

  • Ask when you get stuck. The teacher will help you.

  • I’m glad I took another year of math even though I don’t like it.

  • I am able to teach other students when they don’t understand.

  • How to work through things better with a group of people.

  • The struggle is part of the process and you have to go through that barrier to learn new things.

  • I can handle more stress than I thought I could.

  • I am a visual, hands-on learner rather than just being shown how to do it.

  • I can be a mean person and that can turn people off from friendships.

  • I can better understand if I apply my mind.

  • Applying what I learn to everyday activities, I understand them better.

  • I should cherish my time with my friends and family before it’s too late.

  • I need to stop slacking on homework because it only gets harder from here.

  • There’s only so much in my control and not everyone will care about some things like I do.

  • If I actually take time to study and make sure I understand, I will do better.

  • I should probably study more in college.

  • It’s ok to not understand and ask questions.

  • When left to my own devices, I don’t do anything productive. I need to plan.

  • I do better in a classroom where there aren’t strict rules and things are more free flowing.

  • Don’t judge a person because you don’t really know what they are going through.

  • Not to overwhelm myself with school work and if I’m tired, I should go to bed.

  • If I study for my college algebra tests, I usually do better on them.

  • I love my job and my coworkers.

  • I don’t need to stress so much about the future.

  • If I want to get good grades in a college class, I need to study more.

    I am better at math when I have a therapy dog.

A Giant Leap into the Unknown

It was homecoming Friday.  The week had been absolutely crazy (in a good way) with activities and celebrations.  The week had also been crazy in a lot of not-so-good ways as my ability to handle the normalized chaos of my job had reached a breaking point.  I arrived at school, set down my bag, and started to cry.  And then I was sobbing, and I couldn’t stop.  And couldn’t breathe.  My angel-of-a-coworker pulled me out of my room and into the math office and got me calmed down eventually, but having a panic attack at work was not something I was quite prepared for.

Over the next few weeks, I maintained my composure while at school, but at home, I existed as an empty shell, literally unable to communicate with my husband and daughter, let alone anyone else in my life.  I spent most weekends in bed, hoping that something miraculous would come along and pull me out of this mental cage I was locked in.

Then came that Sunday in late October. I was sitting in my car, in the driveway, motor running, heat on full blast, chair reclined, and letting the hum of the engine calm me trying to resist a complete meltdown.  My spouse came outside, opened the car door and said, “Maybe it’s time we consider what this job is doing to your mental health.”

Teaching is a second career for me after a failed attempt at being an accountant in a Wells Fargo cubicle farm.  From the moment I stepped into the classroom, I knew I was called to be a teacher.  I’ve grown as an educator, persevered through difficult transitions, and made pedagogical changes that were both scary and energizing.  And year after year, I felt it was worth the personal and emotional sacrifice that my job demanded because what I did made a difference to many.  And I also knew that no 6-figure, office high-rise job could replace that feeling.  But after a life-long battle with anxiety and depression, plus 5.5 years of sobriety under my belt, I also know that the status of my mental health is not something I can afford to gamble with.

So after 12 years as a high school math teacher, I am taking a medical leave of absence for the remainder of the school year.  I’m not taking on any other projects, and I’m not starting anything new.  I simply need time to let my brain settle down from the chaos that has taken over.  I’ll work to restore relationships with my spouse, my daughter, my family, and my friends that I simply have not had the mental energy to attend to.  My daughter will get to ride the bus to school and will get a couple of extra hours of sleep each night.  My weekends will become a time of family and togetherness and interaction again, rather than a time where I pull the sheets over my head and will the pain to go away.  After the holidays are through and 2018 has ushered in the ice and snow, I’ll be able to re-evaluate my ability to manage the pace at which I need to move in order to be the educator I want to be.

A Thank You Note for my College Algebra Students

Today was the last teaching day for my college algebra class for the trimester. I’ve had to clear a lot of hurdles this year, both personally and professionally. Yet I’m prouder than ever at what these young people have accomplished in the last 12 weeks, and so today, I read them a letter of gratitude:

I’ve only cried in front of a class once, and that was first hour, the day after the 2016 election. So I’m going to try to not do it again today, but I make no promises.
This trimester, my college algebra classes have done some amazing, special, and unique things. When I started teaching this course, I was told to lecture from bell to bell, the kids need to memorize the formulas, odd answers are in the back so assign the evens. Every trimester, I’ve tried to make some improvements on that model and, although not anywhere near perfect, I’m really proud of where this class is now. That has less to do with what I do to modify the course and more to do with how you wonderful young people engage in it. The marble slides activity on Wednesday was a defining moment for me with this course. Lecturing from bell to bell wouldn’t have made students as successful with that activity as you were. Not even close. The creativity and curiosity you bring to school with you every day is such a treat to get to experience for me as an educator. It has been my absolute privilege to get to learn from you this trimester. Thank you from the bottom of my heart for not only letting me experience you grow as mathematicians, but for the energy you brought to collaborate and push each other to engage in making sense of the mathematics we have sworn to you is important for your future. You’re going to encounter math that will be more difficult than the math you encountered here and that’s ok. What you’ve done here is shown me that what is asked of you in math class should make sense, and by working together to provide one another with your ideas and your thinking,you’ve built a foundation for understanding the math you’ll encounter down the road. I believe in you as you go out there and make the world a better place.

Rebuilding the Wall

On the first day of school, I was delighted to hear from multiple students in multiple classes, “Mrs. Schmidt, where did the dog wall go?!”  I moved into a new classroom this year.  At the end of last school year, I had my wonderful student aid carefully dismantle the dog wall and carefully box up all of the pictures so that they could be placed in their new location.

And today, a student aid began to put the pictures back up.  And immediately the atmosphere in the room changed.  Students pulled out their phones and were so eager for me to see their furry ball of joy that provides them with that unconditional love.  I saw videos and pictures, and heard memories and felt that powerful bond between my students and their pets.  Is there anything else that students are willing to share with so much happiness and passion?  Perhaps you have examples, but in my experience, nothing has collectively drawn out a students’ willingness to take a vulnerable, emotional risk than sharing a picture or story about their pets.

There is a love that a dog can give you that humans are just not capable of. Whether it’s that tail-wagging excitement when you get home for the day, or the head-on-your-chest affection when you need it the most.   Here are my two beagles, Herbie and Stella, and they’ve saved my life more times than I can count, especially on the days where the pain seems to overtake my life.   The slow deep breath and soft love of that creature beside me is often enough to calm the raging anxiety and clear the irrationality from my head.

IMG_20170723_142117465

And here is the progress thus far on the dog wall.  I’m looking forward to some cute new additions this year.

img_20170912_093700469.jpg