Math is Personal - Week 1 Reflection

“The truth is this: You’re not born being a math smarty pants—it is something you learn. The real trick to being a math smarty pants is believing that math makes sense, or can make sense if you put your mind to it.” ---Marilyn Burns Math for Smarty Pants pg. 9

This past year, I was lucky enough to start working on a team that was explicitly teaching and reinforcing growth mindset. In short, this is the idea that you aren’t born smart, but instead, effective effort makes you smarter.  I think that this is such an important idea for students to embrace, particularly with mathematics, where so many people seem to think they are a “math person” or they aren’t. (Jo Boaler offers two interesting online courses—one for students and one for teachers that tackle this idea as well.) In reading the introduction to Math for Smarty Pants for my the Math is Personal smOOC, I found that growth mindset was really the foundation on which the rest of Marilyn Burns’ conceit was built.

Burns explains to her reader that there are different ways of being smart in math: with numbers, with shapes, with visualization, with strategy games, with puzzles, etc. This reminded me of the mathematical competencies that Ilana Horn presented in Strength in Numbers: quick and accurate calculation, posing interesting questions, making astute connections, representing ideas clearly, developing logical explanations, working systematically, and extending ideas (30).

This idea of different ways of being smart or different types of competencies in the math classroom is so important to me not only because I believe it is true, but because so many of my students do not. As Burns says, “some people think that being good in math is just getting the right answers to problems.” People think this because that is what many math classes (including mine) reinforce. As I continue this class, I want to think about how the choices I make about the structures in my own classroom reinforce the value of quickly getting the right answer. I want to think about how I can change the structure, change the way I speak, change the way I present problems to get at the heart of mathematics. Because the wide array of mathematical competencies gives more students an entry point to get smarter at math. It's also why math is interesting, beautiful, and ultimately a subject that I love.

Automathography

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.

Math Teachers' Circle: Mathematical Empiricism & Its Role in Education

I am spending the week at a math teachers' circle retreat and loving every moment of it. It's great to talk with other middle school math teachers about their experiences, resources, and what they are passionate about. I also love having the opportunity to spend a significant number of hours a day doing math with other people. Here's what I am still thinking about from Glenn Stevens' session: Mathematical Empiricism & Its Role in Education.

 

Knowing --> Explaining

The central idea of this session was that mathematics is first empirical and then deductive. That is, we come to realize that something is true based on experimentation and observation. It is only after we've experienced this truth that we are driven to prove or explain it. For example, after calcualating the squares of the sides of x number of right triangles, you might know with all your heart that a^2+b^2=c^2. But you would still wonder why. This curiosity would then drive you to prove the pattern. However, we often teach in the exact opposite way. First a teacher explains an idea and then students use or apply it. In doing this, students don't get to experience the curiosity that comes with the expectation of structure and the drive to explain why it works.

I find this to be an extremely convincing framework for the doing of mathematics. In the problems we've worked on this week, I have found that I am reasonably convinced of a conjecture (based on a combination of experimentation and intuition) and then want to figure out why and prove beyond a doubt that it is true. In thinking about the school year, I want to make sure that my students get the opportunity to have this type of experience in doing mathematics.

One of the other teachers posed follow-up question: If students can thoroughly observe and describe a pattern, do they then need to be able to explain why it happens? I think this is a tough question because for me, the answer is "it depends." It depends on the topic and the student's own need to know why. Because it is at the heart of mathematics, I would like to cultivate in all of my students this compulsion to explain. However, sometimes I may need students to trust their own powers of observation without explanation of why a phenomenon exists. If I go this route, I need to have a good reason for why, as a class, we might not pursue an explanation.

In talking about doing mathematics, Professor Stevens also shared the following ideas:
- math is natural
- math exists independent of us
- experience precedes formality
- math is the science of structure
- math is the art of figuring things out

When asked about structure, Professor Stevens said "I can't define it, but I know it when I see it." This pushed me to think about how I define structure, particularly in the context of Math Practice 7: Look for and Make Use of Structure. I think that structure is any type of organizing principle. And that it is these organizing principles that allow us to be convinced of the truth of a conjecture before the proof, but also to create the proof itself.

Problems

Finally, two awesome problems that I spent a lot of time thinking about.

1. In how many ways can one spell out "ABRACADABRA" by traversing the following diamond, always going from one letter to an adjacent one? (from Polya)

2. With Pascal's triangle, how many odd numbers are in the 100th row?

Tackling Group Work - Part 1

Another thing that I want to incorporate into my classroom this year is more structured partner (and eventually 3- or 4-person group) work. This is because I strongly believe that when people talk about their ideas, they push each other and themselves to clarify, develop, question, integrate, and add to ideas in ways that no one person could do alone. Collaboration also reinforces the idea that I, as the teacher, am not the only source of knowledge in the room, instead it builds a belief in classmates and in self.

In the past, I have simply told students that they are welcome to work with a partner during most work time in class. As a result, some students would seek out other students for help when they got stuck and others would work independently. When we were working on a longer problem for the whole period, I would often be more direct in my instructions. I would have students take a couple minutes of PTT and then ask students to share with a partner what they had done so far. Students would then write brief notes on the similarities and differences in each other's work. After that, I would instruct them to work together to solve the problem. This was slightly more successful, but often ended up with one student doing the majority of the thinking and the other student copying down their partner's work.

To support my students to fully take advantage of everything that partner/group work has to offer, I've decided to do two things: structure partner/group work and explicitly teach it. To help me with this goal, I did some reading about exploratory talk, complex instruction, and accountable talk.

Before I get into the specifics of what group work will look like and how I will teach and support it, it was important to me to figure out what I valued and wanted to promote in group work. Maybe I will primarily re-inforce these values implicitly, maybe I will teach them explicitly, and/or maybe I will guide my students in creating their own norms around these values (see here or here or here or here for examples of what this might look like). To help me figure out what these values were, though, I pulled group work norms/ground rules from four sources, and categorized them.

Values of Group Work	Strength in Numbers (51)	Shell Centre PD Module	The Value of Exploratory Talk (Chapt. 4, pg. 68)	Thinking Together (University of Cambridge)
Participation	- everyone gets a turn	1) Give everyone in your group a chance to speak	- everyone participates	3 We will ask everyone to say what they think. 2 We will share what we know with each other. 6 We will pay attention and try to think of good ideas
Understanding	- give reasons for ideas	4) Try to understand what is said 6) Demand good explanations	- ideas are elaborated when necessary, so that everyone understands what is meant	5 We will give reasons for what we say.
Listening	- listen to different ideas	2) Listen to what people say 3) Check that everyone else listens		4 Everyone will listen carefully to others and consider what we hear.
Respect		8) Treat all opinions with respect	- tentative ideas are treated with respect
Interact with Ideas		5) Build on what others have said 7) Challenge what is said	- partners engage critically but constructively with each other’s ideas - ideas offered for joint consideration may be challenged - challenges are justified and alternative ideas or understandings are offered
Joint Responsibility		9) Share responsibility 10) Reach agreement	- opinions are sought and considered before decisions are jointly made - knowledge is made publicly accountable (and so reasoning is visible in the talk)	1 We will talk together to think about what to do.  7 We will decide what to do only when everyone has said all they want. 8 We will try to agree about what we think

Synthesizing the table above, I came up with the following values for myself as I move forward in thinking about group work:

• Everyone gets a turn to share their ideas and everyone listens
• Explain reasons or ask for explanation to build group understanding
• Respectfully challenge and build on each other’s ideas
• Make decisions together after everyone has shared

It's also extremely important for students to feel comfortable sharing half-formed ideas. This is something that I am hoping to normalize and value in my class overall, but I may emphasize this in group work.

Finally, the following norms for group work are also presented in Strength in Numbers (48). These were separated from their norms for group discussion, which are in the table above. I definitely plan to implement these guidelines for the mechanics of group work in my class.

• stay focused on your group’s work--no talking outside your group
• you have the right to ask for help
• you have the responsibility to give help to anyone who asks
• helping is not the same thing as telling