I am participating in Justin Lanier's smOOC and our first assignment is to write up some of our "automathography," which is an account of our mathematical experiences. Here is what I have come up with:

Math has always been my favorite subject. I realize that
this is not exactly true, as I was a music major and I preferred my music
classes to my math classes in college. But being a “math person” has been my
identity for as long as I can remember. In elementary, middle, and high school,
I excelled in mathematics. I loved the adrenaline rushes of perfectly
completing timed tests on number facts, and understood procedures from examples
and could effortlessly apply them. I liked that most problems had one right
answer and it was one that I could check. I was drawn to the order of pure
mathematics and uninterested in contrived real-world applications. My parents
gave me math-related books and math puzzles. I tutored in math classes during
my study halls in high school. I had math teachers who I idolized and wanted to
be. While I excelled in all classes, math was my favorite.

So I went into college thinking that maybe I would want to be
a math major. My first semester, I took linear algebra. Linear algebra was fine.
It didn’t rock my world and I certainly didn’t rock its world. My favorite
class that semester was my Chinese class because it was immensely challenging
and required daily, sustained effort. My linear algebra class was not similarly
consuming. I went on to take multivariable calculus and differential equations
and felt similarly towards them. I found the math moderately interesting and
doable, but nothing more. Instead, I was most engaged in classes that pushed me
to my limits in terms of thinking. I did passably well in math classes, but I
knew I could absolutely do better if I devoted more time. But I just found
other classes more interesting. I ended up deciding to minor in math and major
in music because I loved my music classes.

For me, the most interesting part of my math classes was the
interaction that I had with other students when doing problem sets. I made sure
that each semester I found at least one other person to work with, even when I
was in a class when I didn’t know anybody else. While I might have been able to
do the problem sets alone, I certainly didn’t want to. I knew that I was more
efficient, more interested, and more motivated, when working collaboratively. I
think this is part of the reason that I feel so strongly about incorporating group
work in my class.

The exception to my luke-warm feelings about math was the
last math class that I took in college—graph theory. I loved graph theory and
spent time explaining what it was and how awesome it was to all of my friends. I
was so into the class that I wanted my friends to find me a Bridges of Konigsburg mug like the one my professor had (sadly he got it at a math
conference and they are not widely available). I’m not exactly sure why the
class was so amazing for me.

Basically, class time was spent with our extremely engaging professor
explaining definitions and concepts to us (the ones that corresponded to the
section in the textbook we were working on). Then we had two problem sets a
week, which mostly consisted of proving or disproving all sorts of graph-theory
related statements. This was the first class that I took that required proofs
and I was initially terrified. I didn’t know how to write a proof and only had
vague memories of the nightmare that was the two-column proofs in my 9

^{th}-grade geometry class. But I quickly learned that I didn’t have to write anything particularly formally—just clearly. I took the class with one of my best friends and we would just sit and work through proofs. We wrote them up separately (as mandated by our professor), but we thought through every single one of them together. About half the time I had the strike of inspiration that helped us figure out the proof and about half of the time he did. And it was the talking with each other that helped us get there. It was the perfect math-working relationship.
So why did I like this class? At the time, I remember saying
that I liked it because the thinking was complex, but we could represent
basically everything with pictures. Even when trying to prove theorems
involving infinite numbers or graphs that can’t be represented
two-dimensionally, it was always possible to draw a smaller problem. Looking
back on the class, I think it was also because writing proofs pushed me to
think in a way that the applying procedures of my previous math classes had
not. Perhaps I should have taken more upper-level math classes like graph
theory.

There’s one particular statement that my graph theory
professor made, which has stuck with me ever since. Professor Schmitt claimed that you became a
mathematician when you made your first original proof. He talked about how for
most math students, it’s not until grad school (or even after?) that people do
this. I was enamored with the idea and at the time considered myself to
absolutely not be a mathematician. Jumping forward a little bit, when I brought
this up with another student in my grad program (who had left a math PhD
program), she basically just rolled her eyes and told me that was nonsense from
elitists. And while it is semantics, of a sort, this is a question that still
plagues me. On one hand, the elitism of my professor’s definition of
mathematician has some pull for me. There is something exciting and
distinguishing about proving something no one has ever proved before. On the
other hand, I believe that all people authentically engage in mathematics. Not
just ones who do so in a formalized or academic way. And I don’t think there
should be a great divide between the two. I want my students to realize that
they are legitimately doing math. It’s not just something that you spend 17+
years practicing before you get the chance to actually do.

Then in my teacher-training program, my math methods class
challenged all of my pre-conceived ideas about mathematics, and made me love math
in a way that I never had felt before. I think this transformation was
primarily based on two inter-related concepts. First, we did a lot of high
cognitive demand problems. As defined by QUASAR, “doing mathematics” tasks
require complex and non-algorithmic thinking, explore the nature of math
concepts/processes/relationships, require analysis of constraints and
connections, and may involve some level of anxiety due to unpredictability. Second,
one of my professors repeatedly told us, “I don’t care about the solution to
the problem. You are never going to see this problem again.” Instead, she, and
by extension we, only cared about how we thought about the problems. We
practiced notating our thinking rather than notating a finished product. We
used the lenses of the math practices—particularly 2, 7, and 8, to think about
more broadly applicable math concepts such as structure and patterns. I
absolutely thrived on this type of process. I got to think and experiment and
represent and apply my mathematical brain to all sorts of challenges. As
student teachers, we tried to bring type of mathematics into our classrooms. I
wanted students to be doing the type of math that was so fun for me.

Fast-forward one year, and I have just now completed my
first year of teaching. I am struggling
with the balance of doing math through the lens that I was taught in grad
school with a more traditional method of skills building. I’ve been told that
it’s possible to teach and build skills exclusively through problem-solving, but
I don’t know how and I haven’t seen anyone do it. My math content methods
professors always said that students should have a “steady diet” of high
cognitive demand tasks, but I also found that maddeningly vague. I think, at
this point, that we do need to spend some days/time building skills through
more traditional practice (though hopefully in ways that aren’t completely
boring). I do have to teach my kids some procedures and have them develop
fluency with those procedures. But the point of that isn’t just to know
procedures. I spend time on procedures so students develop comfort and
flexibility that enables them to do exciting thinking. But this is a dangerous
way to think because often, the fluency threshold is much lower than I think it
is to engage in high-level thinking. “Math skills” shouldn’t be a prerequisite
for interesting, exciting problem-solving and thinking. But certain skills are
prerequisite or at least requisite for certain problems. So where is the
balance?

I’m going to end my authomathography, at least for now, by
talking about my most recent experience of doing mathematics. Last week, I went
to a week-long math teachers’ circle retreat. With 35 other middle school math
teachers, we devoted 95% of our time to doing math and about 5% of our time to
thinking about pedagogy. I had so much fun working on “good problems.” Many of
these problems involved proof, particularly after noticing a pattern (see this blog post). Many of these problems involved trying out “what ifs” and
hitting dead ends. Many of the problems involved multiple solution strategies
or even solutions. And they all involved talking to each other—justifying
solutions, comparing answers, and explaining our thinking in ways that helped
both our partners and ourselves. I’m really excited about continuing to meet in
this circle because I think it’s important that I continue to do math while
teaching math. It keeps my mathematical thinking skills sharp and reminds me of
the type of math experiences that I need to make sure that my students get.

You have had quite an extensive journey with mathematics so far. I like that you are coming from a pure mathematics perspective into teaching. I would love to talk more about the balance between teaching rote skills and high cognitive demand tasks. This is a huge question for me as a special ed math teacher.

ReplyDeleteAndrew, it would be great to talk more about balancing teaching skills and high cognitive demand tasks. I taught a special ed class last year and will be teaching two inclusion sections this year, and I am sure that your perspective will be valuable.

DeleteI see a common thread for many of us that we desire to do mathematics with others again as we did in college or at another point. I also love to think about that balance of and connection between skills and problem solving. It sounds like your summer course was very moving. I would like to take part in something like that at some point.

ReplyDeleteJasmine, I think that you are exactly right that we desire to do math with other people. I don't know where you are located, but there are math teachers' circles across the country. You can check out this site: http://www.mathteacherscircle.org/. I also think the desire to do math with others is something that I need to keep in mind for my students. I want to make sure that I foster and teach that skill.

DeleteI agree with Andrew and Jasmine. One thing that seems obvious again after reading this is how important it is to learn about pedagogy through quality experiences instead of reading them in a [text]book or hearing about them in a [lecture] presentation. The pedagogical insights come from reflecting on why you had such a powerful experience. Please keep reflecting on this in your blog!

ReplyDeleteHi Nicole,

ReplyDeleteThanks for sharing your math story!

When you were younger, who else around you was a “math person”? Who of your family? Who of your friends? What sorts of social interactions did you have around math that reinforced your math identity, or that challenged it?

How does/will your inclination toward pure math over real-world math impact your classroom? What are ways that you can help students with different tastes from yours connect with mathematics?

It’s awesome that your parents gave you mathy presents. Have you ever asked them about this? Were any of these books or puzzles particularly memorable for you?

It’s awesome that you had math teachers whom you wanted to be. I feel lucky to have had great high school math teachers myself, who inspired me to want to be a math teacher, too. Did it ever cross your mind to become a mathematician, rather than a math teacher? This was not at all on my radar when I was in high school. I didn’t know any mathematicians.

What math class before college most rocked your world? (In contrast with your lackluster Linear Algebra experience in college.)

“Professor Schmitt claimed that you became a mathematician when you made your first original proof. He talked about how for most math students, it’s not until grad school (or even after?) that people do this.” I think this is interesting. It’s not clear to me whether he meant proving something on your own, or proving something (perhaps already proven) in a way that no one has before, or proving something that no one ever has before. What other reasonable thresholds for becoming a mathematician might there be? One is: solving a problem that you’ve posed to yourself, unasked. Another is: doing any mathematics when it’s not required of you. Another: When someone who is a mathematician tells you that you are one, too. What others can you think of?

Do you consider yourself a mathematician now? Why or why not? If not, do you think you will be one eventually?

Also, I would claim many of the middle schoolers I’ve taught are mathematicians—in my opinion—by the “proving something on your own” criterion. And I wish every student had this feeling fairly early in their mathematical educations.

What was the biggest shift in your thinking about what “doing mathematics” is from when you were taking graph theory to when you were in your teacher-training program? What you’ve said of them seem similar to me—but maybe you were doing mathematics in graph theory but weren’t particularly thinking about the process of doing mathematics, and your teacher-training program helped to crystalize this for you?

“because often, the fluency threshold is much lower than I think it is to engage in high-level thinking.” I think you’re absolutely right.

I love the thoughtful questions you pose about skills and rich tasks, and how you’re grappling with it in your practice. And I’m excited to see how you’re continuing to engage in doing mathematics yourself with such passion as you begin your teaching career. I’m sure this passion for doing math carries over to your classroom.

Thanks again for sharing!

Justin