takeaways from my first summer at the klingenstein center

(I wrote most of this post before the school year started but didn’t get around to publishing—using the end of winter break to finish it off!)

I’ve been teaching pretty much non-stop since I started in 2012. Every summer I’ve been an instructor at a couple different summer academic programs (a.k.a. nerd camps). But this summer, I took a break from teaching to work on a master’s program in educational leadership. This meant taking five classes in six weeks:

  1. Negotiations
  2. Ethics & Education
  3. Private School Leadership
  4. Instructional Leadership
  5. Research Methods in Education

Key learnings

Negotiations

Pearls of wisdom

  • You can build or lose trust with very, very few words. Pay careful attention to the messages you signal to others by the details you choose to share about yourself. What do you emphasize, and how is that received?
  • An excellent negotiator creates value for everyone while protecting their own interests.
  • Attend to process—every negotiation has one, regardless of whether it is stated explicitly.
  • Time is one of the most critical aspects of negotiation. Life is timed and open-book. Time pressure can be a major enemy to a successful negotiation, so a good first move is to pause and consider process.
  • Too often, we focus on our positions rather than our interests. Always look for ways to reframe the problem.
  • It is important to prepare for negotiations and difficult conversations.
  • There are many techniques for blocking questions, such as changing the subject, over- or under-answering, answering only the helpful part, deflecting with humor, responding with a probing question, or ruling the question out of bounds. To get an answer, ask four questions about the same thing, since most people can only block once or twice.

Tensions in a negotiation

  • Creating vs. distributing value
    • There is a pervasive cultural assumption that negotiations are win-lose, with an adversarial relationship between parties.
  • Empathy vs. assertiveness
    • Our culture frequently holds these as contradictory, but they are better thought of as orthogonal. We should strive to both advocate for our own interests while seeking to understand the other person’s interests. Making your needs known is key—this invites the other person to problem-solve with you.
  • Principals vs. agents
    • In a school setting, many different actors can try to claim to be the agent acting on behalf of the principal (the student).
  • Being yourself vs. playing an institutional role

Difficult conversations

  • You don’t have to agree with another person’s story, but you do have to understand it.
  • Attend to your own feelings and decide if you want to share them. Make sure to anticipate others’ feelings and which might be hard to hear. (For example, hearing disappointment can be very challenging.)
  • Intentions can be positive, negative, or neutral. When you’re only perceiving negative intentions, brainstorm neutral or positive intentions that could be underlying the other person’s actions.
  • It’s critical not to conflate your work identity with your personal identity.

Ethics & Education

Pearls of wisdom

  • Case studies are an immensely powerful tool for identifying our values and drawing out our hidden assumptions as educators.
  • Studying ethics helps us understand why there is conflict and misunderstanding around decision-making in schools.
  • There is a key difference between reading a text to recover meaning (the “hermeneutics of recovery”) versus comparing it against an existing theory (the “hermeneutics of suspicion”)
  • The more we understand the nature of social power, the more we can understand human practices and thus change our actions, and moral agency becomes increasingly possible.

Private School Leadership

Pearls of wisdom

  • There’s an inherent tension in the model of “leader as learner”: if I still have so much to learn, how could I possibly think I’m ready to lead? This is normal.
  • Personal transformation is part of change leadership. If we can’t change, our schools can’t change!
  • Culture eats strategy for breakfast.
  • A useful metaphor for learning and growth is thinking about “maps and compasses.” When we operate using a map, we have clearly defined goals and the process feels comfortable. With a compass, we can go off the beaten path but there is more discomfort with the process.
  • The shift to thinking about adults as learners is fairly new in schools (~last 15 years), and especially independent schools.
  • It’s important to explore what it means to be a listening leader. Consider there ways in which listening is both a situated (i.e., context matters) and a fully embodied practice.
  • In decision-making, schools seek coherence, not consensus.
  • A soundwalk can be a valuable tool for understanding the life of the school in a new way by allowing you to pick up on what often goes unnoticed. Learning walks are a related tool to keep the pulse on the day-to-day. You can ask yourself:
    • How did I hear the mission of the school? What did I hope to hear?
    • What is the heard culture? The hidden culture?
    • What affirms or contradicts school values?
  • When analyzing a text, ask yourself:
    • What assumptions is the author making?
    • What do I agree with?
    • What do I find myself wanting to argue with?
    • What could I aspire to?
  • An excellent question to ask of any group of people is, “What went well?”
  • When introducing a new tool (such as the creation of an asset map) and folks are asking, “Is this right?”, it can be useful to offer the reframe, “Was this useful to you?” to emphasize that it’s not about following a formula, but about identifying and helpful practices and personalizing them.
  • There’s a difference between task leadership and social-emotional leadership. Both are critical components of change leadership, because there’s a gap between wanting change and actually going through with it.
  • There’s value in unfinished work—see the body of scholarship studying unfinished artwork.
  • When building out new structures, you need to establish rapport, which comes from presence and consistency.
  • Projects and core organizational change are managed very differently. But both need a vision and an elevator pitch others can echo.
  • As a leader, every word you use matters. For example, start with “anti-bias” if folks can’t hear “anti-racist.” It’s not cowardice, it’s a plan for moving to the next level of good.
  • When walking into a difficult meeting, pause to consider your purpose and how you intend to conduct yourself.

Instructional Leadership

Pearls of wisdom

  • It’s crucial to distinguish between “effort” and “effective effort.” We have to explicitly teach what the latter looks like inside and outside of class.
  • Most schools are not set up to promote long-term learning.
  • When asked to reflect, most people just describe or narrate what happened. We have to explicitly teach what meaningful reflection looks like. People learn from reflection on an experience, not the experience itself. This is because memory is the residue of thought (see Daniel Willingham’s writing).
  • Feedback given to students is only useful insofar as students engage with it to increase their level of understanding.
  • The reason looking over notes is ineffective is because you’re recognizing, not retrieving. We can think about forgetting as retrieval failure.
  • Struggle is key to deep, enduring learning. However, struggle brings up emotion, and we have to attend to it. First, we need to build up a culture of trust and intentionality.
  • Forgetting is essential to learning. We need to remind students that this is a normal part of the process. Re-learning goes much more quickly and reinforces those connections.
  • A great interview question is, “If you ran the new teachers program, what three principles of learning would you make sure everyone grasped?”
  • Put the big ideas of a course out early, then provide spaced practice and interleave them.
  • Expertise is not gained simply through hours spent—it’s about deliberate, effective effort.
  • When planning for class, practice taking each individual student’s perspective rather than the students’ perspective as a whole.
  • In terms of feedback, we should be giving baseball cards (with have multiple measures), not grades!
  • Design lessons to require students to construct meaning for themselves. (See the memory demonstration described here.)
  • Effective feedback doesn’t change the underlying product (external). It changes the learner’s understanding (internal).

Questions to ask students

  • What did we learn yesterday?
  • What are the most important ideas you will be tested on?
  • What for you is worth remembering?
  • What would I ask you if you were stuck in class?
  • What would I ask you if I were there with you? (reflective question for students to consider while working on homework)

Metacognitive questions

  • What do I do when I’m stuck?
  • How does this connect to prior knowledge or past experience?
  • What alternative approaches are available?
  • How do I know that I understand?
  • What am I noticing? What am I wondering?
  • If I were to do this again, what would I do differently?
  • How effective was my planning and my effort?
  • When did I pause to evaluate how I was doing?

Research Methods in Education

Pearls of wisdom

  • In general, we are not sufficiently critical of summaries of educational research.
  • (sorry, ran out of time!)

Readings to revisit

Beyond Winning. The chapters specific to lawyering can be skipped.

Source: Mnookin, Robert H. Beyond Winning Negotiating to Create Value in Deals and Disputes. Harvard: Harvard University Press, 2004.

Difficult Conversations. This is worth rereading a chapter or two every month.

Source: Stone, Douglas, Bruce Patton, and Sheila Heen. Difficult Conversations: How to Discuss What Matters Most. Yuan Liou Publishing, 2014.

Epistemic Injustice: Power and the Ethics of Knowing. The concept of “testimonial injustice” is useful for naming systems in which marginalized individuals’ voices are met with doubt and disbelief. The first chapter of Fricker’s book very carefully builds up definitions along with examples and non-examples. I’d like to read the rest of this book where she develops the idea of testimonial justice.

Source: Fricker, M. (2011). Epistemic injustice: Power and the ethics of knowing. Oxford: Oxford University Press.

Caring: A Feminine Approach to Ethics & Moral Education. In the first chapter of this book (“Why Care About Caring?”), Noddings develops the concept of “care ethics.” Whenever I get bogged down in the details of math pedagogy, this chapter will be an important resource for humanizing the process of education and our students’ needs as human beings.

Source: Noddings, N. (2003). Caring: A feminine approach to ethics & moral education. Berkeley: University of California Press.

Democracy and Education. In the chapter “Education as Growth,” Dewey names a mechanism through which much of the deficit thinking about children and adolescents arises. When we think about students as unfinished adults, we necessarily close ourselves off to the powers and strengths unique to young people. This chapter is worth rereading every year.

Source: Dewey, J., & Boydston, J. A. (2008). The middle works, 1899–1924. Carbondale: Southern Ill. Univ. Press.

Educator Competencies for Personalized, Learner-Centered Teaching. This resource enumerates and unpacks the skills, understandings, and dispositions teachers need to enact student-centered practices. I’ve previously focused on making sense of the instructional and cognitive competencies, and this report also dives into the interpersonal and intrapersonal dimensions of student-centered teaching. I could see myself using this to ground myself as I analyze the competencies that are present in myself and the educators I’m working with.

Source: Jobs for the Future & the Council of Chief State School Officers. 2015. Educator Competencies for Personalized, Learner-Centered Teaching. Boston, MA: Jobs for the Future.

Leadership for Increasingly Diverse Schools. An equity audit can be used to identify and address issues concerning proportional representation in various areas of the school. The suggested timeline for accomplishing goals is about three years, and it’s crucial to link these goals to student learning. This tool provides a clear structure for meeting a particular type of school-based inequity head-on.

Source: Theoharis, George, and Martin K. Scanlan. Leadership for Increasingly Diverse Schools. Routledge, 2015.

Organizational Culture and Leadership. The opening chapters from this resource pushed my understanding of organizational culture forward by laying out three levels of culture. I particularly appreciated the explanatory power of invisible but deeply influential underlying assumptions when we encounter structures and behaviors that don’t align with our publicly stated beliefs. I could see myself appealing to the lily pond metaphor to explain this idea to others who are invested in change. I’d like to read more out of this.

Source: Schein, Edgar H. Organizational Culture and Leadership. 5th ed., Wiley, 2017.

Dilemmas of Educational Ethics: Cases and Commentaries. Our ethics professor suggested we read this. Reading and thinking through case studies was one of my favorite activities of the class, so I’m looking forward to reading this.

Source: Levinson, Meira, and Jacob Fay. Dilemmas of Educational Ethics: Cases and Commentaries. Cambridge, MA: Harvard Education Press, 2016.

Practices to implement

Visible random groupings.

Retrieval practice.

Spaced practice.

Interleaving.

Connecting to prior knowledge.

Coaching. By offering help to my advisees in a non-directive way, I can honor their autonomy and ability to solve their own problems. There’s a toolkit I can use that focuses on the now and the future and spends more time on the values, wants, passions, and motivations of the person I’m working with. This is more effective than jumping right into the logical and logistical domains. I can also practice asking more “what” than “why” questions to keep the focus on coaching the person, not the problem.

Sharing hopes/fears anonymously.

Listening walks. Slow down in the hallways.

Teaching metacognitition explicitly.

Teaching executive functioning skills explicitly. For example, have everyone write on an index card the “movie script” of how they did their homework. Where were they physically? What time was it? Where was their phone? What resources did they use? Collect, shuffle, and redistribute the cards. In groups, compare the approaches. Share out the ones that sounded most effective.

why i “leaked” my logarithms test

In the somewhat rushed lead-up to our unit test on logarithms in my Advanced Algebra II class, I realized I was asking students to study from an enormous corpus of practice material I’d provided. I could tell that this was overwhelming, so I got the idea to give the test to my students in advance.

Not the full test—just a redacted version:

I had dropped a hard copy in one student’s open backpack during a break, and then I made a big show of asking him to pull out the paper marked “top secret.” Then I was all, “ohmygoodness what’s that?!” and “Did I accidentally post a copy of the test to Google Classroom?” My students were thrilled. (I did make it clear that this was a one-time thing!)

Even though I had already provided a list of learning targets, I had two main goals by “leaking” the test itself:

  1. Reduce student anxiety by providing the parameters of the test (number of problems, layout, types of question prompts, etc.) very explicitly
  2. Focus students’ studying on the ideas I considered most important

This was a spur-of-the-moment idea, but definitely one I would use again under the right circumstances!

understanding pushback on inquiry-based learning

I teach in a progressive math department committed to inquiry-based learning, and I’ve been working on sharpening how I talk about why I believe in this approach.

A conversation will often start when I hear a student or a parent say one of these:

  • “I learn better with a traditional approach where we learn something and then practice it a bunch.”
  • “My kid needs a teacher who actually teaches.”
  • “I get that some students can figure out how to do math without being told how, but my kid just needs you to show them the steps to follow.”
  • “My teacher made us figure most of the material out on our own last year, which was frustrating. Instead of telling us the answer, they would just ask us more questions!”

It’s hard to imagine what an inquiry-based math class looks like if you’ve never been in one. I suspect that if you’re a parent and your child is experiencing math-related stress or frustration, there are a few reasons you might order if this way of learning is best for your child. These are some of the images you might conjure up:

  • Students working on a hard problem they’ve never seen before as the teacher watches silently from the side of the classroom
  • Students “learning” an incorrect approach and leaving the class with serious misconceptions
  • A teacher who de-emphasizes procedural fluency to the point that students while students can explain a concept or idea in broad strokes, they can’t actually solve any math problems
  • Students who are stuck, frustrated, and angry

When I think about an inquiry-based math class looks like, I see something very different:

  • Students engaged in problem-solving requiring them to notice patterns and wonder why they occur
  • Students having rich conversations with one another about math under the guidance of a thoughtful, attentive teacher
  • Students working hard to satisfy their own curiosity while being simultaneously “egged on” and supported by a caring teacher

Ultimately, every parent comes to these conversations about math education with the goal of seeing their child successful and happy. Keeping this in mind is critical for addressing their concerns.

Over at the IBL Blog, Stan Yoshinobu writes about the feelings can arise when students haven’t bought into the value of productive struggle:

Mindsets are at the core causes student buy-in issues. When students don’t buy it, it’s often because they don’t like being stuck or that being stuck implies there is something wrong with the problem, them, or the teacher or all of the above and more.

I was digging more into the IBL Blog and discovered this lovely metaphor of “I don’t learn this way” as the tip of the iceberg:

Iceberg 1

This imagery reminds us to look beneath the surface to find the source of resistance to inquiry-based learning. Only then can I speak to my beliefs and vision for my students.

I believe in inquiry-based learning because I think students learn math best when…

  • … they have a chance to explore ideas on their own before being told what the “best” strategy is for solving a problem.
  • … their mathematical ideas are affirmed and valued, even when they’re not fully clarified or correct yet.
  • … they are given opportunities to develop intuition before technical vocabulary and formalism are introduced.
  • … they are invited (and expected) to look for patterns and are regularly asked, “What do you notice?”
  • … teachers explicitly and implicitly communicate to students that mathematical knowledge is not isolated to select “experts” (like math teachers) who then dispense it to others, but rather that mathematical creativity is broadly accessible.
  • … they are invited (and expected) to pose questions of their own and are regularly asked, “What do you wonder?”
  • … teachers explicitly and implicitly communicate to students that the teacher’s mathematical questions are not the only interesting ones, but rather that the ability to ask a rich, thought-provoking question about math is broadly accessible.
  • … they see their peers employ a variety of successful strategies to solve a problem and are encouraged to understand multiple approaches.
  • … they are given opportunities to communicate their understanding to their classmates and receive guidance on how to improve their oral and written communication.
  • … there are structures to support collaboration with their peers.
  • … they spend most of their time in the sweet spot of productive struggle.
  • … they are given opportunities to apply the fruits of their intellectual labor during focused practice, building mastery and supporting long-term retention.

I’ll conclude with some prose from Joshua Bowman, who recently shared a preface he wrote to an IBL course. Here’s an excerpt:

… [T]he success of the class will depend on the pursuit of both individual excellence and collective achievement. Like a musician in an orchestra, you should bring your best work and be prepared to blend it with others’ contributions.

… Mistakes are inevitable, and they should not be an obstacle to further progress. It’s normal to struggle and be confused as you work through new material. Accepting that means you can keep working even while feeling stuck, until you overcome and reach even greater accomplishments.

interesting numbers v2

This is a follow-up post to my first day math problem from last year.

I’ve only posted once in a while over the past few years, but I’m very glad I spent the time last year to write about how my first day math problem went. After re-reading my post from last year, I was able to rapidly load the problem and how it went back into my memory, and I decided to make some of the changes I had written about.

Here’s the updated version of the problem, which adds scaffolding and will hopefully keep groups moving along who aren’t as comfortable with this type of open-ended exploration.

And here’s the checklist I’m going to use to keep myself organized as I walk around and monitor each group’s progress. (This idea comes from 5 Practices for Orchestrating Productive Mathematics Discussions by by Margaret S. Smith and  Mary Kay Stein.)

Feel free to pass along any feedback or take this a step further!

Resources:

math is like a pomegranate

This is my contribution to The Virtual Conference of Mathematical Flavors.

“Math is like a pomegranate—intimidating, and kinda scary looking at first, but also incredibly fascinating and vibrant.”

In order to figure out what flavor of math I’ve been serving up in my classrooms over the past six years, I’m going to take a stab at Sam Shah’s idea of working backwards from what students have written about their experiences in my math classes. (Spoiler alert: the answer is apparently pomegranate; who knew?!) Of course, not all of my students have had transformative experiences and others have straight up had a bad time. But right now, I’m going to focus on the students who have been positively impacted in order to articulate what the best implementation of my ideals has felt like.

But to be honest, it feels way scarier to share the positive things students have written about me over the years than anything critical. When I was younger, I used to brag and show off; I thought that if people knew about all the things I was good at, they would have to like me. Once I figured out that this is not how relationships work, the pendulum swung hard in the other direction for me. I grew increasingly uncomfortable accepting compliments and I minimized my achievements, working to avoid even the appearance of self-promotion. It’s an ongoing struggle to get right-sized, but lately I’ve begun to internalize the idea that being excessively diminutive is its own barrier to connection.

So with that confession out in the open, here are some of my favorite reflections students have written. (The title of this post comes from one of these!)

A few years ago, I started asking students to write advice to next year’s students. And when I remember, I make sure to share this advice once the new group arrives. Here are some examples of what my students have written.

Reading these, some of the core beliefs I bring to my teaching are apparent to students in different courses and at different schools. I hope that my students internalize them as well:

  • Math is something to get excited about. I want my math-skeptical students stay curious about why certain people openly love math, and I want them to find reasons of their own for loving math. I’m not shy about telling them when I get goosebumps when talking about math, and I don’t hesitate to make corny memes—and be super proud of them—to show how highly I think of a mathematical idea or how much their understanding has grown. (See the “extending the definition of sine and cosine” trigonometry meme I made this year.)
  • Math is a playground for creativity. You can ask and answer your own questions. There are games to make up and play, connections to establish, new approaches and representations to develop, and structures to create and explore. Some of my favorite moments come when a student (or even better, a group of students) comes up with a solution pathway I’ve never considered or notices a pattern I’ve never seen before.
  • Engaging with math is an opportunity to build confidence. No matter where you are in your mathematical journey, there are ideas to wrestle with in math that are hard, but not impossible. It’s like an infinite gym for your brain with an endless selection of workouts. Realizing that you can do something you’d previously found scary, intimidating, or intractable is incredibly empowering.
  • Expect and welcome obstacles. In math, there’s nothing wrong with being wrong, and getting stumped is an invitation to push your thinking deeper or to try something else. For this reason, I react like it’s the most normal thing in the world when a student tells me their approach didn’t work or when they don’t know what to do next. Sometimes they get a little peeved when I don’t rescue them right away, and that’s okay!
  • Math is especially enjoyable when shared with others in a caring and trusting community. Despite the cultural trope of the solitary mathematical genius, there is no rule saying that math has to be a solo sport. The process of guiding another person to a mathematical idea you’ve uncovered requires patience, clear thinking, and careful consideration of what the other person is comprehending. Similarly, the practice of asking for and receiving guidance requires humility, self-awareness, and careful articulation of what you’re understanding and where you’re feeling fuzzy. To help with this, I treat the word obvious like a cuss word in math class, and students usually buy in pretty quickly!

As a final thought, it feels liberating to put this out there in a less formal way than I’ve articulated aspects of my educational philosophy in the past. For comparison, this is what I’ve used in previous job searches, and almost all of it was composed in 2014, before any of the student reflections above were written.

I still stand by everything in this document, but I really appreciate the type of unencumbered sharing Sam’s framing of the prompt for this virtual conference has facilitated. In other words, asking “What mathematical flavor are your serving up?” rather than “What’s your theory of mathematics education?” seems more likely to inspire folks to share a healthy multiplicity of approaches instead of competing formal philosophies. And, it gives us an opportunity to celebrate our wins instead of worrying about all of the things we’re not doing.

initiation problems for optimization unit

I’ve had the pleasure of teaching standard-level calculus with Sam Shah this school year, and recently we’ve been working with the students on optimization. Rather than starting with the canonical calculus optimization problems, we decided to jump in with maximizing the area of various shapes under curves:

Feb-01-2018 16-37-59.gif

I think I first saw a problem like this in a textbook called Advanced Mathematics by Richard G. Brown (page 167, #12):

Untitled3

I’ve always been fascinated by these types of problems because they’re easy to understand and make guesses about but often have unexpected solutions. I wanted to use Desmos to bring this problem to life, so I put together a Desmos activity and companion sheet (.docx version) to look at four of these problems.

Students worked in groups, and they while they all had their own screen, they were expected to move together and come to consensus on the best possible shape before moving on. This fostered lively debate among students as they tried different shapes and improved their guesses by manually calculating areas.

For the “isosceles triangle under a parabola” problem shown above, there were a variety of responses, but there was convergence around the optimal triangle (whose vertex in Quadrant I is (\sqrt{3}, 6):

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Students were also asked to identify the constraints of their shapes before moving on to the next challenge. After locking in their guesses for each challenge, students had to really dig in to the second challenge (the “rectangle under a parabola” problem). This time, we provided a slider that would calculate the area of the rectangle for them as they changed the x-coordinate of the vertex in Quadrant I:

Feb-01-2018 16-47-47

Student were expected to record these data points on the companion sheet to form a sketch of the area function:

challenge2.png

Many students initially assumed that this area function was going to form a parabola, but after plotting more points, the class decided that it couldn’t be because of the lack of symmetry. But this function has a peak—how could they find it? This is where the calculus kicked in!

After carefully taking the derivative of the area function, setting it to 0, solving, and determining the dimensions of the best possible rectangle, students were able to finally determine which group came the closest with their initial attempt. They were also ready to tackle the remaining challenges on the second part of the companion sheet (.docx).

Before jumping in, I was also able to recognize groups for getting the closest to the best possible shape while also pointing out that they could do even better!

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All in all, I thought this was a super fun way to kick off the optimization unit while keeping engagement high and providing valuable practice with the non-calculus algebra that can trip students up. Most importantly, calculus was positioned as the aspirin for the headache posted by the Desmos activity.

I would love to see these types of optimization problems become more popular!

Resources:

P.S. See Sam’s post on a lovely lesson he put together called POP! Popcorn Optimization Problem, which is a way more engaging way for students to tackle traditional optimization problems that usually look like this:

original-439030-1.jpg

first day math problem 2017–18: interesting numbers

I decided to re-sequence my start-of-the-year activities and to lead with a low-floor, high-ceiling problem in assigned random groups of three or four students.

Here is the problem, which comes from Phillips Exeter Academy’s Math 1 curriculum:

pea-1-92-3.png

I told the groups to figure out everything they could about this situation with prompts like, “What do you notice about interesting numbers? What do you wonder about them?”

As I watched twelve groups of students explore this problem over three classes, I began to see students latch onto different aspects of this problem. All of these questions and discoveries are inter-related, so I’m writing them down now so that I can map them out in the future.

Questions:

  1. Which numbers up through 20 (or so) are interesting?
  2. Why are powers of 2 interesting?
  3. Are powers of 2 the only interesting numbers?
  4. Are there any interesting odd numbers?
  5. What happens when I sum any two consecutive positive integers?
  6. What happens when I sum any three consecutive positive integers?
  7. If is odd, what happens when I sum any n consecutive positive integers?
  8. If is even, what happens when I sum any n consecutive positive integers?
  9. How can I decompose any odd number?
  10. How can I decompose any multiple of 3?
  11. If is odd, how can I decompose any multiple of n?
  12. How can I decompose any even number?
  13. Is there a general algorithm for decomposing any number?
  14. How many ways are there to decompose a given number?

Realizations:

  1. All powers of 2 are interesting.
  2. Only powers of 2 are interesting.
  3. No odd numbers are interesting.
  4. The sum of two consecutive positive integers is odd.
  5. The sum of three consecutive positive integers is a multiple of 3.
  6. If n is odd, the sum of n consecutive positive integers is a multiple of n.
  7. If n is even, the sum of n consecutive positive integers is n/2 more than a multiple of n.
  8. There is an algorithm for decomposing even numbers.
  9. There is exactly one way to decompose a prime number greater than 2.
  10. The powers of 2 are exactly the whole numbers without odd factors.

There was a split between groups that started by trying to answer (the very natural) question #1 (and thus getting to realizations #1 and #2) and those that started by generating and then trying to answer questions #5 and 6 (and thus getting to realizations #4 and #5). There was also one group in one class that decided to explore the sum of the first n consecutive integers (i.e., they wanted to know about the triangular numbers).

I think I will definitely use this problem again, with perhaps a bit more structure and guided mini-explorations along the way as groups arrive at various questions and realizations. It would probably be worth making a checklist for each group to help keep me organized as I keep tabs on each group’s progress.

Related:

more on grading—a synthesis of some my favorite thinkers (part two)

[A continuation of part one.]

Cohen, Guskey, Schimmer, Wormeli

Many teachers worship at the church of the arithmetic mean.

In Fair Isn’t Always Equal (2006), Rick Wormeli writes:

… it’s easier to defend a grade to students and their parents when the numbers add up to what we proclaim. It’s when we seriously reflect on student mastery and make a professional decision that some teachers get nervous, doubt themselves, and worry about rationalizing a grade. These reflections are made against clear criteria, however, and they are based on our professional expertise, so they are often more accurate. Sterling Middle School assistant principal Tom Pollack agrees. He comments, “If teachers are just mathematically averaging grades, we’re in bad shape.” (p. 153)

The best case I’ve been able  to make for why the practice of averaging is so fraught is given by Thomas Guskey in On Your Mark (2014):

Can you imagine, for example, the karate teacher suggesting that a student who starts with a white belt but then progresses to achieve a black belt actually deserves a gray belt? (p. 89)

Tom Schimmer hammered this point home in a December 2013 webinar called “Accurate Grading with a Standards-based Mindset”:

Adults are rarely mean averaged and certainly, it is irrelevant to an adult that they used to not know how to do something. Yet for a student, these two factors are dominant in their school experience.

In his article published in the April 2016 issue of “Educational Leadership,” Guskey echoes Wormeli’s point that defensibility and the perception of objectivity are highly prized among many teachers:

In teachers’ minds, these dispassionate mathematical calculations make grades fairer and more objective. Explaining grades to students, parents, or school leaders involves simply “doing the math.” Doubting their own professional judgment, teachers often believe that grades calculated from statistical algorithms are more accurate and more reliable.

In this blog post, David B. Cohen makes the case for reforms many folks in the TTOG community have been pushing for for some time:

We need to relinquish our preconceptions about the meanings of specific numbers and percents. Giving up the idea of points altogether would help; points are a convenient fiction, as long as you don’t think too hard about what they supposedly represent.

Cohen recommends ditching the 100-point system:

Why do we need 100 points then? That’s a level of definition that has no meaning. It would be like having a weather report stating today’s high temperature was 58.3 degrees, or including cents in conversations about rents or mortgage payments.

All of these points and reforms encounter institutional resistance, however, because of how much they ask teachers to make major shifts in their practice.

For me, though, it’s worth it. I was so glad to see this article by Alex Carpenter and Alberto Oros in the August 2016 edition of “Educational Leadership,” which made the connection explicit between grading practices and enacting a social justice pedagogy. The authors implore us to “take a moment, right now, to think about how we can modify our gradebooks in the name of justice.”

I’ll reiterate my questions from a year ago, because they are still very fresh on my mind.

A couple questions on my mind

  1. What practices do you, your department, and/or your institution have in place to facilitate difficult conversations about grading, reporting, and assessment?
  2. To what extent would it be a useful exercise for each department within a school to produce its own purpose statement for grading? (“The purpose of grades within the ___ department at ____ School is …”)

more on grading—a synthesis of some my favorite thinkers (part one)

This is part one of a series I’ll be writing on grading.

Guskey, Kashtan, and Reeves

On his blog, Douglas Reeves writes:

I know of few educational issues that are more fraught with emotion than grading. Disputes about grading are rarely polite professional disagreements. Superintendents have been fired, teachers have held candle-light vigils, board seats have been contested, and state legislatures have been angrily engaged over such issues as the use of standards-based grading systems, the elimination of the zero on a 100-point scale, and the opportunities for students to re-submit late or inadequate work.

Miki Kashtan, co-founder of Bay Area Nonviolent Communication, succinctly and insightfully explain  what’s needed to ground intense conversations in cooperation and goodwill:

Focusing on a shared purpose and on solutions that work for everyone brings attention to what a group has in common and what brings them together. This builds trust in the group, and consequently the urge to protect and defend a particular position diminishes.

In On Your Mark (Solution Tree, 2014), Thomas Guskey backs up Kashtan and calls upon the work of Jay McTighe and Grant Wiggins on backward design when he writes, “Method follows purpose.” (p. 15)

Guskey continues to emphasize the importance of beginning with the end in mind when we come together to discuss our craft with other educators:

Reform initiatives that set out to improve grading and reporting procedures must begin with comprehensive discussions about the purpose of grades … (p. 21)

Summary

  • Discussing grading can quickly become prohibitively emotional. (Reeves)
  • Focusing on a shared purpose helps those of us who have already put a stake in the ground to be willing, eager and able to move it. (Kashtan)
  • Before considering the “how” of grading, deeply consider the “why.” (Guskey)

A couple questions on my mind

  1. What practices do you, your department, and/or your institution have in place to facilitate difficult conversations about grading, reporting, and assessment?
  2. To what extent would it be a useful exercise for each department within a school to produce its own purpose statement for grading? (“The purpose of grades within the ___ department at ____ School is …”)

More to come.

thoughts on resilience & grading

I spent the last three days helping to facilitate a leadership retreat for some of our rising 10th, 11th, and 12th graders. This year’s theme was resilience, which we linked closely to one’s relationship with failure.

In several different ways, we asked students to reflect on the extent to which the school provides opportunities for them to fail, process what happened, make adjustments, and persevere through a difficult situation.

As we concluded the retreat this morning, we invited the students to consider how they and the adults at our school could facilitate the development of resilience during the upcoming school year. I was overjoyed with the first comment a boy put forward, which he intended for both students and adults:

Too often we get so focused on grades that we lose sight of the learning. Let’s keep the conversations about the learning rather than the grade.

I was blown away because I had hoped a student would bring this up, and this boy came right out with it. I’d like to make some strategic changes in my messaging around grading, reporting, and assessment this school year, and making the connection to resilience explicit could help keep these shifts rooted in a value to which the community has expressed a commitment.

My guiding question is this: What grading, reporting, and assessment practices (and policies) most effectively promote resilience in students?

There are many broad categories of issues come to mind, but in my current context I’d like to focus on redos and retakes.

I would like to try to assemble the most concise, convincing evidence that allowing multiple attempts at demonstrations of mastery facilitates the development of resilience. (I would go further and say that the practice of averaging in the scores of unsuccessful attempts impedes the development of resilience.)

Here’s a selection of articles I’ve read that support this view.

As Thomas Guskey writes in On Your Mark, we won’t get very far if we don’t agree on the purpose of grades, so the goal here is to convince someone who believes that the primary purpose of grades (in math class especially) is to summarize performance on one-time tests (via the arithmetic mean).

What do you think?

  1. What grading, reporting, and assessment practices (and policies) most effectively promote resilience in students?
  2. What is the most concise, convincing evidence you know of that allowing multiple attempts at demonstrations of mastery facilitates the development of resilience?

P.S. The value of mastery-based (competency-based) learning has begun to make its way to the independent school world as well: in this article from 2014, David Cutler writes about his expectation that traditional grades will be obsolete by 2034.