Reflection on Practice - Part I

Our second class of the day at PCMI was Reflections on Practice, where we focused on formative assessment. My favorite thing about this class is that it has left me with a lot of questions. Many of these questions don’t have quick and easy answers, but instead will shape the choices I make about how I structure my classroom. Going in to next school year, I will do my best to think through as many of these questions as I can and make instructional decisions based on this thinking. But I imagine that some of these questions I will revisit and revise my answers to for a long time yet to come. It’s the act of thinking through the answers, rather than the answers themselves, that will make me a better teacher.

Throughout reflections on practice, we looked at a lot of research surrounding formative assessment. Here are the ideas that stuck out most to me and the questions that they make me ask. I will do a whole separate post for my take-aways and questions based on our class's video conference with Dylan William.

1. Evaluative vs. Interpretive vs. Generative Listening

This was a framework for types of listening, and really types of teaching, that we analyzed various classroom videos with. With evaluative listening, the teacher comes into the class with a plan for the path that the class will take to reach the instructional goal. If the student responses are what the teacher expects, then the teacher can easily continue with the plan. If student responses are not what the teacher expects, the teacher will simply say the idea him/herself and continue on with the plan. Somewhat opposite to this is generative listening, where the teacher has a goal but uses student work and understanding in order to choose the path for how to get to the goal. In the middle is interpretive listening, which I am the least clear on. I believe with interpretive listening, the teacher comes in with a goal and a path to get to the goal. The teacher is interested in surfacing student thinking and using that to help students along the predetermined path.

Throughout the whole course, we did not explicitly place more value on some types of listening than others. However it was implied that generative listening was preferable to evaluative listening. On the surface, this certainly fits with my teacher training and own educational values. I strongly believe that students will be more invested and understand better if the math that they are doing is connected to and based on their own thinking. But I also think it is much more complicated than saying that teachers should only ever do generative listening. I think that there is a place for being evaluative or interpretative. For example, if I have a group doing a card matching, I might be evaluative and tell them that 6 out of 8 of their matches are correct and to identify and fix the mistakes. That is not always the instructional move that I would make in that situation, but it is one I have made and would make again if I could tell that the group understood the big idea and had either made a careless error or needed to focus in on nuance. 

Therefore, I am left with the following questions:

- What is the purpose of each type of listening?

- In what situation would I be trying to achieve that purpose?

- Based on the previous two questions, is there an ideal balance between the three types of listening?

- What are the common pitfalls in executing each of these type of listening? How do I make sure that I am actually achieving the purpose that I would like?

2. Hinge Questions (See Dylan William’s explanation here)

Hinge questions are in a category of moves where the teacher assesses the whole class and then makes an instructional decision about what to do next based on the assessment. However there are some particular characteristics of hinge questions that increase the efficacy of this process:

- kids should not be able to get the question right for the wrong reason

- the whole class answers the question in only a couple of minutes

- all responses are assessed in under a minute

- the teacher is ready and able to change the lesson based on the assessment

I make instructional decisions all the time based on a formal or informal assessment of the whole class. For example, I will use exit tickets in partnership with informal observation to make decisions about what I will do the next day in class. When I grade homework, I check one pre-selected question for accuracy. This is a question that I have written to be a mid-level question that addresses the main objective from the previous day. Based on responses to that question, I may then decide if we are going to go over the question, if I want to clarify something before we start the day’s lesson, if I am going to check in with particular students during the day, or many other responses. Here’s how I see hinge questions as slightly different from what I already do: it is done in the middle of class and the decision about what to do next is even faster. Due to this the teacher not only has to craft a really efficient question, but also plan in advance the possible directions the class could go and the threshold of understanding for each direction.

This leaves me with the following questions:

- How can I plan and structure lessons in order to have maximum flexibility within the lesson to react and change course depending on how it is going?

- If students are struggling to grasp an idea at a time when I expect them to have grasped it, in what situations is it necessary to react immediately and in what situations can I wait until the next day to address this? How does the percent of students who are struggling with the idea affect the required reaction time?

3. Jo Boaler’s Norms:

We also looked at these positive classroom norms from Jo Boaler:

- everyone can learn math to the highest levels

- mistakes are valuable

- questions are really important

- math is about creativity and making sense

- math is about connections and communicating

- math class is about learning not performing

- depth is more important than speed

These norms align strongly with my own values and math education philosophy. But in the current popularity and push towards growth mindset, I feel like much of the focus is on the language that we use in talking about these ideas. While I think language is extremely important, I want to spend more time making sure that my actions actually match the language that I am adopting. While I believe all of these things, the ways that I structure my class do not always reflect my beliefs. And students pick up on that. For example, it doesn’t matter how many times a teacher says that mistakes make your brain grow if the student work or ideas that the teacher highlights are always mistake-less.

So here are my questions:

- Which of Jo Boaler's norms are most important to me? Are there other values that I have about math learning that are not represented here?

- What structures do I have in place that reflect my values?

- What structures do I have in place that are at odds with my values?

- What changes can I make in my classroom structures in order to better align with my values?

Morning Math - Part II

I have already mentioned that morning math was one of the highlights of my PCMI experience. Over the course of the three weeks, I realized that this was because the purpose of the class was to give us opportunity to deepen and connect our mathematical knowledge.  This is in sharp contrast to the point being to finish problems. This made me wonder what were the conditions of this class that fostered this goal of us more deeply understanding math rather than completing problems? And of course, though I won’t address it in this post, how can I make this happen in my math classroom?

First, a couple of caveats given by Darryl on the last day:

- this way of structuring a class does not lend itself to getting everyone to the same place

- given the time and effort that goes into the creation of the problem sets, it is impractical  to write them every day for a school year

With that in mind, how did they foster this type of mathematical inquiry?

The Problem Sets

The problem sets each day were about four pages of math problems split into four different sections—the opener, the important stuff, the neat stuff, and the tough stuff. To directly quote Bowen and Darryl “Check out the Opener and the Important Stuff first. All the mathematics that is central to the course can be found and developed there. That’s why it’s Important Stuff. Everything else is just neat or tough.” It’s hard to describe the brilliance of these problem sets. Better to just check them out here.

Here are some of the attributes that I think made them so engaging to me:

- They let you think and struggle: There were many, many times that an important stuff problem made me think of a question.  Then, over the next couple of days, there were problems in each of the sections that were related to the question, but didn’t force me to answer the question by giving the exact scaffolding I needed. Eventually my question did show up in the important stuff at which point I was ready to answer it. The work I had done on the previous days had built my insights and understanding up to the point that I was ready to tackle that question myself. I had developed all the tools I needed—I didn’t need to be given anything.

- All of the important stuff questions were enterable with a basic high school understanding of math. In the first week, all the content was centered at Algebra I or earlier. Later on, they added conics and complex numbers. But there was no expectation that I had to remember formulas or laws or other easily memorized, easily forgotten math knowledge. Anything I needed I was able to construct or figure out myself as I was working.

- We got to make connections between seemingly disparate topics. Think about the question “Find the length, width, and height of a rectangular prism whose surface area is equal to its volume.” Now think about the question “Determine values of a, b, and c to make the following equation true: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}.” In many ways, those are the same question. That’s just really cool.

- There was something for everyone. I spent the majority of the time living in the first 3/4 of each day’s important stuff. I skimmed through the neat stuff and the tough stuff, but rarely had the chance to spend very much time on those problems. I felt totally fine with that because I was able to follow the progression and think about the big ideas that were being developed throughout the course. At many times the important stuff was simultaneously important, neat, and tough. But for people who moved more quickly through the important stuff there was always something else to think and be excited about.

 - Finally, I can’t talk about the problem sets without at least mentioning the jokes. The problem sets didn’t take themselves too seriously, so neither did we. On the papers where I was doing the math, I put in exclamation marks when I found something particularly cool or satisfying. I’ve never done that before. I think part of it is because there was personality and voice in the problem sets. Therefore, I felt like there was a place for my personality and voice in my own math work.

Norms

I learned the most math and enjoyed my work time the most when I was with a table that really embodied the norms. Here are the norms that Darryl and Bowen set:

- don’t worry about answering all the questions

- don’t worry about getting to a certain problem number

- stop and smell the roses

- be excellent to each other

- teach only if you have to

- each day has its stuff

On top of these norms, Darryl and Bowen reinforced a couple of things about morning math at the beginning of the second week. One of them was that we should not be working on these problem sets outside of class. This message was really for the people who were feeling bad that they hadn’t finished whatever they felt like they should have finished in the previous class. They mentioned this in order to try to rid people of guilt about “not having gotten far enough” and it made me feel so much better about how I was choosing to spend time in class. I think they would have been fine with people continuing to do work on the problem sets outside of class—but only if it was because people were so excited and interested in continuing their investigations. Not because of guilt or living up to someone else’s expectations.

These norms meant that I was really able to explore my preferences for how to work during morning math. I certainly fell into the group of participants who liked to explore “rabbit holes.” Often when an idea was introduced for the first time, it was through a problem that was pretty straightforward to answer, but the answer illuminated something that was interesting or surprising. For example, executing a transformation that ends up creating a non-similar figure, but the area of the transformed figure is always 10 times the area of the original figure (see Day 6, problem 5). The problem set moved on at this point, but I was left with the question why does this happen? What about this transformation algebraically and/or geometrically scales the area by a scale factor of 10? How can I prove that this always happens? (I am still left with all of these questions, by the way). At this point, instead of continuing on with a new problem, I really enjoyed trying to answer my why questions. Sometimes I was able to answer them that day and sometimes I wasn’t. They often came up in the neat/tough stuff or in the important stuff in the next couple days. But even when they didn’t, I still felt like it was a good use of my time.

And that gets to ultimately what I felt was so valuable about this experience for me. The problem sets, the norms, the whole environment was designed so that the focus was on deepening our mathematical understanding, not finishing problems. And while I have experienced this before, it has been in the context of deepening my mathematical understanding in order to make instructional decisions. I have been cut off in my thinking about a specific math problem in order to think about how what I am doing relates to my students. And that is certainly worthwhile. But this allowed me to deepen my mathematical understanding for myself, in the directions that I was interested in and excited about. While I certainly don’t think this will make me a worse teacher in any way, it was an extremely freeing experience to just be able to focus on myself and the joys of thinking mathematically.

Morning Math - Part I

I just spent three weeks at Park City Math Institute. Morning math, the two hours that we spent working problem sets, was one of my favorite parts of the day. The problem sets were written and facilitated by Bowen Kerrins and Darryl Yong centering on “Some Applications of Geometric Thinking”. I strongly recommend checking them out here.

At the end of each week, we took some time to record our insights/connections we had made during the week as well as some questions we still had. Here’s what I came up with. If you’re thinking about doing some or all of the problem sets, I would strongly recommend not reading any further.

Rectangles, Boxes, and their Relationships to Quadratics and Unit Fractions

- The length and width of a rectangle are the solutions to the quadratic equation \(x^2-\frac {p}{2}\ +a=0\) where p represents the perimeter and a represents the area of the rectangle. A way to think about these solutions is to think about what two numbers have a sum of \(\frac{p}{2}\) and a product of a. You can find these values using the following expression: \(\frac{p}{4}\pm\sqrt{(\frac{p}{4})^2-a}\). When we set p=a, the l and w solutions satisfy the equation \(\frac{1}{l}+\frac{1}{w}=\frac{1}{2}\). A rectangle exists only if \(a\leq(\frac{p}{4})^2\), which can be restated as \(p\geq4\sqrt{a}\). A square maximizes the area and minimizes the perimeter of a rectangle.

- Similarly, we can think about the relationship between surface area and volume for rectangular prisms. \(\frac{1}{l}+\frac{1}{w}+\frac{1}{h}=\frac{1}{2}\)when SA=V. This same relationship exists for polygons that “surround” a vertex. When 3 regular polygons surround a vertex, \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}\), where a, b, and c are the number of sides of each polygon. There are ten unique whole number solutions to this equation.

- Somewhat surprisingly, if we think about four regular polygons surrounding a vertex, rather than three, we need \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1\), not \(\frac{1}{2}\).

- Consider finding all possible unique whole number solutions to the equation \(\frac{1}{a}+\frac{1}{b}=\frac{1}{n}\)for any given n.  b is related to a as follows: \(b=\frac{an}{a-n}\). You only need to check the set of solutions for whole number solutions when \(n+1\leq a\leq2n\).

Conic Sections

Parabola:

- All points are equidistant from a point (a, b) and a line \(x=n\).  Then the equation for this parabola can be written as follows: \(\sqrt{(x-a)^2+(y-b)^2}=x-n\). This is related to the equation below because when simplifying the coefficient for x² will be 0.
- \(Ax^2+Bx^2=C\), where A is equal to 0.

- Pour salt on circle that has a small circle cut out of it

- Slice a cone parallel to the side of the cone

Ellipse:

- All the points have the same sum of the distances to two points

- All points for \(0<e<1\), where e represents the ratio of the distance to the point (a, b) to the distance to the line x=n. The equation for this ellipse can be written as follows: \(\sqrt{(x-a)^2+(y-b)^2}=e(x-n)\). This is related to the equation below because when simplifying this equation, you would get coefficients of the same sign for both x² and y².

- \(Ax^2+Bx^2=C\), where A and B have the same sign.

- This ratio, e, is known as the eccentricity. It shows up in two places. As already mentioned, distance from focus to any point on the ellipse: distance from directrix to any point on the ellipse. Also, distance from the center to the focus: distance from the center to the vertex.

- Pour salt on a large piece of cardboard with a circle cut out of it
- Slice a cone from side to side

Circle:

- All the points for e = 0, where e represents to the ratio of the distance to the point (a, b) to the distance to the line x=n. The equation for this circle can be written as follows: \(\sqrt{(x-a)^2+(y-b)^2}=e(x-n)\).

- \(Ax^2+Bx^2=C\), where A and B are equal.

- Slice a cone parallel to the base

Hyperbola:

- All the points that have the same difference between the distances to two points

- All the points for some value e>1, where e represents the ratio of the distance to the point (a, b) to the distance to the line x=n. The equation for this hyperbola can be written as follows: \(\sqrt{(x-a)^2+(y-b)^2}=e(x-n)\). This is related to the equation below because when simplifying this equation, you would get a coefficients with opposite signs for x² and y².

- \(Ax^2+Bx^2=C\), where A and B have opposite signs.
- Pour salt on a large piece of cardboard with two circles cut out of it (I am still skeptical about this one)
- Slice two cones from base to base

Other Geometric Insights

- The shortest distance between any two points is a straight line. How can you use congruence to turn something into a straight line?

- Some transformations that are represented algebraically involve rotation and scaling, others just rotation. These transformations can be represented in the real plane or the complex plane

- When you square a complex number, its magnitude will also be squared (this seems trivial now that I write it, but was essential for figuring out the idea below)

- A way to create Pythagorean triples (… but not all Pythagorean triples can be created this way): Take any two number m and n. One leg is equal to \(m^2-n^2\), another leg is equal to \(2mn\), the hypotenuse is equal to \(m^2+n^2\).  If m and n share a factor, the triple will not be a primitive Pythagorean triple.

- A triangle with an incircle of radius 2 has an area that is equal to its perimeter. When you pour salt on a triangle, the center point of the incircle is the highest point. This is also the point of intersection for the three angle bisectors.

Some questions that I still have:

- Given a focus, directrix, and eccentricity, how do I accurately draw the conic section it describes?

- How can I predict what the geometric transformation will look like when given an algebraic transformation (with either real or complex numbers)?

- Why is the center point of the incircle at the intersection points of the angle bisectors in a triangle?

- How do you know there are only 10 possible boxes where the surface area and volume are the same? (I answered this on the last day, but I included it because it was the question that was most present in my mind from Day 5 on).