Sunday, August 10, 2014


Justin (@j_lanier) runs a smOOC (small open online course, as opposed to the massive ones) called Math is Personal.  The idea, as I understand it, is to help us reflect on our own relationship with math throughout our lives and to build on it in a way that makes it easier to connect with students in our classrooms.  Our first assignment is our math autobiography, or automathography.  Though there are many ways to tell a story, I didn't know the big themes until I finished writing, so it came out as a timeline.  I'm not going to pretend this was written for a public audience to quickly skim and learn something new, but if you have some time, I would love to hear how it connects to your own story with math or where you would push me to go deeper in my own reflection.

As far back as first grade, math was my favorite subject.  I was fast at it -- give me 5 minutes to do as many addition facts as I could on a sheet of 100 and I would hand it back to you done in 2 minutes.  We also did a math activity each day called Number DAP, essentially a solo activity to arrange manipulatives in a way that demonstrated understanding of a concept.  I would always finish one task quickly, but despite often finishing a second task during our work time, my teacher didn't make it back around to me since she was still checking off classmates.  This drove me crazy, since to me, math was a race, and I was super competitive.  At the end of first grade, my teacher moved me into the 2nd grade math workbook where I could work in the back of the room.  Two other students joined me shortly after, and by 2nd grade we were permanently moved a grade ahead in math.

In 2nd and 3rd grade, I worked in the back of the room during math time with my 3 other math buddies -- Sonja, Alex, and Kirsten.  Kirsten is now my wife, but since I didn't like girls at all in elementary school, there was no drama in the back of the room.  That said, we did have interesting dynamics.  Whenever I finished something before the others, they made me share my answers under threat of being poked with their pencils.  We never really got a math lesson -- we just got a brief concept presented to us and used the examples in the workbook and one another to figure it out.  It was nice to not have to slow down.

Moving up to 4th through 6th grade, we did our math in the classroom with older students.  This adjustment was fairly easy.  One of my favorite parts was having to walk to the middle school across the park each day with our growing group of advanced math students to attend class.  I was a shy kid, so socially this was actually pretty helpful for me to have a group with boys and girls that I talked to every day.  It was extra awesome when 6th grade was moved back to the elementary building so we had to walk over two years in a row!

My first deep struggle in math came from multiplying fractions.  Multiplying is a function used to make things bigger, but my teacher was up in front telling us lies about how 1/2 times 1/3 was 1/6, a number SMALLER than either starting value.  I just didn't get it and kept raising my hand to argue.  I felt bad since nobody else was raising their hand (which to me meant they all understood it), but it was so shaking to my entire understanding of math that I had to be heard.  I think in the end the word "of" turned the corner for me, such as taking half of 1/3 would make it smaller, but it still took a patient teacher and a lot of examples to come around.

Middle school math was not too bad either.  Most of the advanced math students took Algebra (not sure if it was a true Algebra 1 or more of a 8th grade type of math), and then in 8th grade learned Geometry.  That was my toughest class to date with all of the theorems and proofs.  Logic didn't feel like math to me, but was instead a challenge in organization and argument.  I still liked it, and my teacher was fantastic (though he didn't put up with any shenanigans from our crazy cohort).

My high school had a traditional schedule, but some of the math classes were offered in block.  This would give me the opportunity to go through math even faster after being slowed to a single rate for most of elementary and middle school.  Algebra 2 was a pretty easy start to high school -- the class moved pretty slow since most of my class was juniors who didn't understand math well.  Sophomore year trig was much more interesting and pushed me to think more, but the block format of the class gave me enough work time to avoid homework most nights.  Later that year, I had a brief stats class filled with a lot of graph sketching (which I found incredibly boring since I didn't like making neat, ruler-guided lines for everything).  Pre-calc seemed like a painfully detailed version of algebra, especially things like limit definitions of derivatives, and homework took a while.

Junior year rolled around with Calculus.  I hardly knew what it was until I was in the class, but I knew I wanted to get to it.  I didn't even know what came after it -- it was the highest math class offered in the building.  It was also the first time I had to work my butt off, every single day, to do well in math.  I'm not sure what about it was hard -- I didn't mind the algebra, I had a TI-89 (I was a bit of a calculator nerd at this stage of my life), and the visual part didn't scare me -- but every day was a struggle.  One side note about how I wrote out math problems in the past -- I showed as little work as possible and wrote as tiny as I could.  I prided myself in fitting 5+ complete assignments on a single sheet of paper.  Since math was about getting answers, that was all I needed to write.  This made a handful of teachers mad over the years, but most eventually caved since I could demonstrate that I knew how to do it.  Getting back to Calc class, this wasn't going to work.  However, I still couldn't bring myself to use a full sheet of paper or more for an assignment, so I turned to whiteboards.  Everyday I just stood up during work time and did my problems on the board.  Quickly I was joined by a few others who liked standing and writing big while doing math.  As a result of writing large and doing so next to peers, this was when I started to do a lot of peer teaching and learning.  I loved to explain my thinking to somebody who was struggling and I liked working through problems as a small group in class.  Before Calc, this would have slowed me down, but now it was actually saving me time and helping me understand.  At home, I spent 1-2 hours on my Calc homework most nights, learning almost everything by example as I worked through the solutions on a website that had a step by step guide to every problem in our textbook.  Some people criticize this method, but it worked really well for me to develop a solid understanding.  I still learn best by seeing lots of examples and drawing my own generalizations and conclusions.  Despite how much work it was, I must have enjoyed it since I always did it first (probably why I ended up using Sparknotes for a lot of my English novels instead of reading them).

In my senior year, I signed up to take Differential Equations at the liberal arts college in town.  This was when I stopped liking math.  Not coincidentally, it is the first place where I couldn't use effort to overcome a lack of understanding.  There was a hard-to-follow lecture with no answer key and homework problems that assumed prior knowledge of physics (which I had not yet taken).  Eventually, I managed to wedge my way into a homework study group on campus at night.  I don't know how my parents felt about me being out past midnight on a school night on a college campus, but that was my only path to survival.  I contributed almost nothing to the group but was so thankful they let the little high school kid tag along.

A couple things that were going on during high school that shaped my experience were my deteriorating vision and my participation (and evangelization) of math team.  When I was 21, I was diagnosed with a rare eye condition called keratoconus.  Before then, I just had my mom and my friends all telling me I was blind, but when I went in to get glasses, they never improved my vision (I see okay now thanks to a hard contact layered on a soft one in each eye).  As a result, I was in denial about my eye problems yet could not see the board in class without squinting, so I never watched the teacher's examples that closely and often developed my own similar ways of solving problems.  As for math team, I joined my freshman year and loved an event that was challenging, competitive, team-based, and something I was good at.  By the time I was a junior, we had new advisors that worked hard to get lots of students to attend the monthly meets (mainly through extra credit), so it was extra exciting to be on the "varsity" group whose score counted for our school point total (I treated it like a real sport, to my friends' amusement).  I was so enthusiastic about it, the advisors asked me and another energetic senior to walk around to the math classes to promote how awesome math team was.  In the meet hosted at our school, we managed to get almost 70 students to come, so we were all pretty excited.  Despite the many interesting, non-traditional problems I was exposed to through my math team participation, it was the competition and the team aspects that motivated me the most.

College was a complete pivot in a lot of things for me, including my involvement with math.  I went to Olin College of Engineering, a new design and project-based school of ~350 students near Boston.  Math class was fully integrated freshman year with programming and physics, and all students were in the same course together.  We were asked to do a lot of open-ended tasks, mostly around modeling and simulating physical scenarios.  My goal of getting into a good college and racing to do as much math as possible were no longer relevant, so I lost my drive to push for higher and higher levels of math.  I took a greater interest in circuits and programming, both of which used some math, but the math was simple and used in a completely different way.  I actually avoided pure math as much as possible.  The low point was taking Diff Eq (the same one I took in high school) with a visiting professor -- I once again had almost no idea what was going on and once again got a C.  My favorites were Discrete Math with a professor who worked for the NSA for a while and Statistics with one of my programming professors (read his free and amazing textbook).

College was also where I developed my obsession with education.  In high school, I always enjoyed tutoring others and once tried to write a Calculus textbook that explained things in a much simpler way than our class textbook (that project didn't get too far), but I never truly considered teaching as something I would actually do as a career.  Near the end of my freshman year of college, however, I suddenly became on fire not for what I was learning, but how I was learning.  I loved the hands-on, fully integrated way of learning that I never got in high school.  I wanted to design schools that would teach this way.  I took one class my sophomore year, Improving Schools, that convinced me that there were already a few awesome models out there that could be built upon.  I was so into education-related things that I was running out of time for my engineering coursework, pushing me to take a year off of college to dedicate myself fully to education.  I spent about 15 hours / week preparing for and teaching a Saturday STEM class for Black and Hispanic high school students west of Boston, but most of my time was spent working on a ed-tech startup with a group of five other friends who also took a year off.  In the end, the business didn't pan out, but I was determined to stay in education to try to figure out the real problems and hopefully some solutions.

Three weeks after graduation, I got married, so I knew I would be moving to Minnesota where my wife was finishing a physician assistant's program.  I found a one-year teacher certification program through Winona Stats University in Rochester where I could take a summer of classes but spend the majority of the year learning by doing in the classroom.  I wanted to be a science teacher, probably physics or chemistry, but my college coursework didn't show the right prerequisites for the program.  Only by being a math teacher could I get certified and in the classroom quickly, and since that was my main goal, I went for it.  I figured I would spend three years teaching and move into administration as soon as possible so I could get closer to my bigger goal at the time, starting a new school.  I'm so thankful that I decided to actually put myself in a situation where I could be in the classroom as a teacher every day for the past three years, because now, I appreciate how much more complicated things really are.  Before I would even consider trying to start a school, I want to see what kind of innovation, especially in math education, can be done in our traditional system.  Though biased from working in a forward-thinking district, I'm finding that many of the limitations are not from "the system", but from our lack of imagination on how things could be.  That's the problem I've been struggling with for three years and counting.  One big piece of the solution is a much deeper, richer, more nuanced understanding of math and how we can think about it and use it.  That's where I hope my journey as a math learner takes me next.


  1. Thanks for sharing your story, Andy.
    I had a similar experience as you—math was something that helped me to forge social connections with my peers that was difficult for me otherwise. A big example of this was as a high school freshman, when I did a lot of peer tutoring after school in my teacher's classroom.
    Looking back on your acceleration in math at an early age, what was good about it and were there any downsides? How do you feel about acceleration for your own students?
    Something that comes up several times in your story is you figuring things out yourself or with a group of peers—rather than being taught or guided through math by a teacher. That's really interesting. Does that ring true to you? Is it the same or different from others' experiences? How does your past in this respect affect your own classroom and your attitude towards learning math?
    I’m glad you had a patient teacher when you had all those thoughts about multiplying fractions to resolve!
    “Logic didn't feel like math to me, but was instead a challenge in organization and argument.” I’ve encountered this thought pretty often from kids, and I don’t fully understand it. I’d be interested in teasing apart the difference in feeling of “math” and “logic”.
    Graph sketching is the worst. :)
    “Calculus. I hardly knew what it was until I was in the class, but I knew I wanted to get to it.” On the one hand, I suppose this is perfectly natural, but I also think this pretty standard feeling is also a little bonkers. Do you remember Calculus living up to your expectations? How would you talk with a kid about Calculus (as a subject, as the “top of the ladder”) to a younger high school student who was in a similar position as you were?
    Your experience with whiteboards is really cool. I appreciate your perspective on collaborative work, both here and in your elementary years and in what you say later about the year you took off from college.
    “I still learn best by seeing lots of examples and drawing my own generalizations and conclusions.” What are some other ways that people learn math best?
    Your DiffEq experience is a bummer to read about. It sounds like the class was poorly set up. I’m interested, though, in the arc you’ve traced. It seems like math was easy for a long time for you. And then Calculus was a challenge, and then DiffEq was a wall that shut you down. I feel like the basic shape of this experience is a pretty common one, though for some people the wall is fractions and for others it’s Algebra 1 or Linear Algebra. Which is a bummer and, on some level, so tragic—because where’s the support? It’s almost as though this is the way math education is *supposed* to work.
    Do you still feel competitive about math? How does this—or the change in this—shape your experience of the subject as an adult?
    I had a similar experience in figuring out ways to get into the classroom as a teacher as quickly as possible after undergrad. It’s what led me to independent schools, where I didn’t need a credential.
    “I'm finding that many of the limitations are not from "the system", but from our lack of imagination on how things could be.” Love it. Who we are as teachers is the biggest ceiling on what can happen in our classrooms. It’s a huge challenge and undertaking, but we’ve got to find great resources and colleagues and collectively and individually pull ourselves up by our own bootstraps.
    What’s an example of a “deeper, richer, more nuanced understanding of math” that you’ve gained so far?
    I’m glad for the happy accident of you becoming a math teacher! Thanks again for sharing, Andy.

    1. Wow -- thanks for all of the thoughts and questions! Here are some of my responses:

      The good and bad of early-age acceleration: I actually didn't see any major downsides to accelerating. If I was instead allowed to go at the pace I wanted to instead of being constantly slowed down, I think I would have hit the "challenge" stage much sooner than Calc and developed a healthier attitude of math as a place to explore, not a place to "win". I wish more schools were setup in a way that would encourage more students to work ahead and explore at a younger age.

      How does my peer/self learning affect my teaching?: In stats, where most of my curriculum development efforts have been, I try to design experiences that make it easier to discover big ideas and make sense of them with explicit instruction from the teacher. That said, I've created hundreds of videos that are almost all direct instruction in the past 3 years. With these, students choose the pace and whether or not they need them, which I also value, but they don't require an inquiry approach to learning. Though I'm too far on the pendulum towards direct instruction, a balance is needed.

      Calc -- the top of the ladder?: Perhaps I'm too into stats these days, but I see high school math diverging down two paths -- algebra/calculus and statistics/modeling/simulation. In college, as an engineering student, I didn't use almost any analytical techniques to solve calculus-type problems, instead using computer simulation. Most careers, including many techy careers, use tons of stats and little algebra/calculus. Thus, I wish we would stop treating it like it is so special and push more kids towards stats. So that's how I personally treat it.

      Do I still compete in math? What is an example of a nuanced understanding I now have?: I do not thankfully feel like I'm competing in math anymore. Part of that is that I am not chasing new frontiers / topics, but instead working within to re-think and build new connections between topics. In a competition, you want to check off boxes fast. In an exploration, you want to see the details and look closely at things in new ways. One example of this is my realization that addition is repeated counting, multiplication is repeated addition, and powers are repeated multiplication, and that forms the basis of most of our operations. I think I will try to writeup a blog post that expands more on this soon.

  2. I have definitely had students who were like you and tried to write down the minimum necessary to answer the problem in teeny tiny handwriting. I think it's interesting that you ended up preferring doing math on the whiteboard. I'll have to keep that in mind with for my students. I like your comment at the end about using our imaginations. I'll remember that this year as I'm teaching.