New Old Thoughts on Schools’ Use of Time

A guy I work with said a few years ago, “We don’t know how long it takes to learn Algebra 2. We know how long it takes to teach it – it takes 9 months – but we don’t know how long it takes to learn it.”

In a recent conversation about work/life balance, someone commented that our shift from an eight-period day to a seven-period day pushed more of her prep and grading to the evening because of there were fewer periods for the meetings that happen each week. My colleague’s earlier comment immediately ran through my head: we don’t really know how much time we need for our meetings. We know how long they take – they take 45 minutes because that’s how long our periods are – but we don’t know how time we actually need.

The decrease in free periods has also reduced our ability to watch each other teach, so our Faculty Council recently incentivized class visits: if we get a group of five people together and visit each other’s classes, the school will pay for us all to go out to dinner and talk about what we saw. This has been incredibly effective. I recently visited a World History class for the first time in my 19-year teaching career, and on Friday I added Engineering and Fiber Arts to that list.

Our Engineering class is self-paced and self-directed. Students work through a curriculum, and the teacher is in the room to help and troubleshoot, but the students drive how much gets done each day. When I visited, some groups were building a spreadsheet to determine gear ratios for a robot. It was pretty cool to watch, but I was struck by the fact that no matter what the students were doing, when the bell rang, the work was done. Groups in the middle of tracking down a formula mistake had to stop, save their work, and move on to their next class. They needed more time, but they couldn’t have it, because we don’t know how long it takes to debug a spreadsheet but we know how much time they have to do it: 45 minutes.

Fiber Arts is not unlike Engineering. There’s a little more instruction and there is time when the class stops to share or critique work, but ultimately students work on their own projects at their own pace. I was struck not by this similarity but by the use of time in that class. Several students had projects where they would do some work and then let it sit for at least a few minutes and in some cases overnight. Watching them, I got to thinking: what if…

  • …the students’ time was more flexible?
  • …they had the opportunity to work on their art projects, and when it needed time to dry they could turn their attention to their troublesome spreadsheets?
  • …they could work on their spreadsheets until they had resolved the issues and hit a good stopping point, return to their art project to complete the next step, and use the subsequent down time to review the teacher’s comments on the rough draft of their World History term papers?
  • …the students had a target they had to hit at the end of each day, or each week, but they had more control in how they went about it?
  • …we no longer measured learning in 45-minute increments but in increments of days, weeks, or months – or maybe not in terms of time at all, but in terms of projects completed?

J Coady E Ruane Collaboratory 2015 (5)It’s not as far-fetched as it sounds. For a few years, a colleague and I co-taught a course in physics and precalculus. Our class was double-blocked, meaning we met 90 minutes three times a week and 150 minutes once a week. Some days we divided the time evenly. Some days we focused mostly on one topic or idea. And some days, we made decisions on the fly to reallocate time based on where the students were. If it worked for two of us, could we make it work for six of us? Could we put 60 students and 6 teachers together and build a community where students learned in increments that were determined not by bells but by the needs of their work at any given time?

I think we could. I think we should. And in doing so, I think we could do school in a more fun, effective, and productive way. I’m not the first to say it, and I certainly won’t be the last. But maybe if enough of us say it, schools will actually move in this direction, so consider this my contribution to the cause.

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Why Do I Teach Synthetic Division?

On Twitter today, Drew Lewis posed the following question:

I was going to try to answer but realized that 140 characters wasn’t enough and the tweetstorm I’d have to resort to was going to be too long. Fortunately it’s summertime, so I can actually sit down at the blog and explain why I do it.

First things first: I teach it in our Honor Precalculus course. If I were to teach a different course, I’d either do it differently than I’m about to describe, or I wouldn’t do it at all.

The reason I teach it is because it is a small piece of a big puzzle that prepares students for calculus, not from a content perspective (OK, a little from a content perspective) but from a keeping-track-of-all-the-details-in-a-big-complicated-problem perspective.

Here’s a pretty typical instruction from a Calculus textbook (Calculus, Anton, Bivens, and Davis, 10th edition, pg. 264): “Give a graph of the function and identify the locations of all critical and inflection points.” Here’s a pretty typical sequence of steps a student would take to answer this question:

  1. Take the derivative of the function.
  2. Set the derivative equal to zero and solve the resulting equation (keep in mind this could be a polynomial, rational, exponential, logarithmic, or transcendental function).
  3. Check the derivative for any points at which it is undefined.
  4. Take the second derivative of the function.
  5. Set the second derivative equal to zero and solve the resulting equation.
  6. Set up the appropriate tables or structures to identify intervals on which the function is increasing and decreasing, and justify the answer.
  7. Set up the appropriate tables or structures to identify intervals on which the function is concave up and concave down, and justify the answer.
  8. Find the y-intercept of the function.
  9. (Probably) Find the x-intercepts of the function.
  10. Sketch the graph of the function using the information gained from doing all this work.

Compare this to the following pretty typical instruction a student would encounter in a precalculus textbook: “Analyze the given polynomial function and sketch its resulting graph.” Here’s a pretty typical sequence of steps a student would take to answer this question:

  1. Factor the function if possible.
  2. If there are factors that are not linear or quadratic, ensure the function has integer coefficients or rewrite it so that it does.
  3. State all factors of the leading coefficient and constant terms, and use these factors to list all possible rational roots of the function.
  4. Test the possible rational roots until you find one that is a zero of the function.
  5. Divide the function by the appropriate factor to depress the degree of the polynomial.
  6. Continue steps 4 and 5 until you are left with linear and quadratic factors.
  7. Use the quadratic formula to obtain all linear factors over the set of real numbers.
  8. State all x-intercepts based on the factors you found.
  9. Find the y-intercept of the function.
  10. Determine the end behaviour of the function using the leading coefficient test.
  11. Sketch the graph of the function using the information gained from doing all this work.

As you can see, the process to answer each of these questions is quite long, requiring the use of definitions and theorems; recall of algebraic facts and algorithms; and the ability to keep the question asked in mind when the work involved is long and complicated. Are they conceptually equivalent? A little. Are they challenging for the students taking the respective courses? Yes. Does answering the precalculus question serve as a good foundation for answering the calculus question? I believe it does.

So why do I teach synthetic division? I teach it because it is a quick and easy algorithm to use in performing step 5 of the precalculus question, which is to say I’m giving them both one more tool in their bags and one more thing to understand and be able to apply. I think this helps prepare them for calculus. Otherwise, I can’t think of any reason why I would teach it (or most of the other stuff in that list).

Launching Our Integrated Math Curriculum

A look through my blog posts of the past year shows a lot of references to our developing integrated math curriculum, and more than one mention of diving into the details in a future post. Well, that post is finally here.

In just less than two weeks, our first group of new freshmen students (we call them third form students) will embark on a journey that has been two years in the making. It is only through countless hours spent with  committed colleagues, and the trust and confidence of our department chair and the school administration, that we have arrived at the point of rolling out a new curriculum, one we essentially developed from scratch. Here, then, is an overview of the year, followed by some detail on what we’ll do and why we’ll do it.

The Oveview

In the first year of integrated mathematics, most of our third form students will explore the following topics, in this order:

  • One-variable statistics
  • Two-variable statistics
  • Constant Rates of Change
  • Systems of Linear Equations
  • Geometry of Lines, Rays, and Angles
  • One- and Two-variable Inequalities
  • Constant Percentage Rates of Change
  • Sequences
  • Consumer Finance

The Details

As you can see, this is a non-traditional sequence blending some topics typically covered in Algebra 1, Geometry, and Algebra 2 courses, plus some topics that are covered only briefly or not at all. Here’s why we’re doing what we’re doing:

  • One-variable statistics
    • Students come into the school with a variety of math backgrounds. This will start them all at the same place, with something familiar that we can quickly expand on.
    • We’re starting the year with a concrete topic: data and how we make sense of it.
    • Exploring single-variable data allows us to dig into some common mathematical approaches and mindsets, including multiple representations.
    • Students will use their TI-84 calculators and Mircosoft Excel spreadsheets in the first three weeks of the year, but they’ll have to know the math before they learn the technology.
    • Students will start writing about mathematics early in their high school careers, working on clear communication in math at the same time they’re working on it in English and history.
  • Two-variable statistics
    • It’s a logical jump to go from one variable to two and see what happens – a mathematical mindset we’ll explore repeatedly over the integrated math sequence.
    • Students will consider what approaches from single-variable statistics will work for them and what they’ll have to change, all while continuing to work with multiple representations.
    • We will develop notions of association, correlation, and causation early in the year, supporting the future work these students will do in their science classes.
    • We’ll do lines of best fit by hand, then get regression equations on the calculator and in Excel.
    • The topic is still concrete, allowing us to explore lines in an applied context and introduce the language of rates of change.
  • Constant Rates of Change
    • This is a natural next step, motivated by the an exploration of data that is perfectly linearly related.
    • This is a fairly typical topic, covering slope, intercepts, and forms of lines.
    • Because students have a lot of concrete examples and experience describing things in context from the first two topics, we’ll build from the concrete to the abstract – a natural progression for these third form students and a nice way for them to view mathematics at this stage.
  • Systems of Linear Equations
    • Again, a natural next step: what if we have two data sets that share a variable? What can we learn by exploring them both at the same time?
  • Geometry of Lines, Rays, and Angles
    • This is the second time we’ll raise the level of abstraction and challenge the students to respond.
    • The basic idea is to take the systems of linear equations we just explored, remove the Cartesian plane, and then explore what’s left. We’ll look for patterns, consider multiple representations, and make use of other mathematical mindsets we’ve been developing.
    • We won’t introduce formal proofs, but we will introduce the notion of logical structure, an axiomatic system, and building an argument. This is an extension of our work with statistics, where students took data, analyzed it, and built an argument based on their understanding.
  • One- and Two-variable Inequalities
    • This topic doesn’t flow as nicely from the previous one as other topics did, but it’s an important and accessible one to work in.
    • Students will already have explored comparisons in their other work, like which rate of change or which angle is bigger, so this is a chance to formalize the idea of comparison.
    • After seeing single-variable inequalities, we will work on our notion of extending our understanding and seeing what works by looking at inequalities involving lines and systems.
  • Constant Percentage Rates of Change
    • This is a big jump, but one that opens a whole new world of mathematics for the students.
    • We will once again begin with concrete data, look for patterns, apply what we know to figure out what works and what needs to be modified, and use multiple representations in our work.
    • Students will learn about the rules for exponents and radicals, exponential operations and equations, and logarithmic operations and equations.
    • The formality of inverse functions will be left alone for now; instead, we’ll opt to explore logarithms as “the things that undo exponentials” in the same way we’ll explore radicals as “the things that undo powers.”
  • Sequences
    • With linear and exponential behaviour under our belts, as well as an understanding of the difference between discrete data and continuous equations, sequences are a natural topic to introduce here.
    • The build-up will be along familiar lines: move from concrete to abstract, use what we already know, and represent and communicate this mathematics.
  • Consumer Finance
    • This is the culminating topic of the year, and rightly so. It pulls together virtually all the math we’ve done this year: discrete vs. continuous, concrete and abstract, and constant and constant percentage change.
    • It also pull together our mindsets – looking for patterns, using multiple representations, and figuring out how to use what we have – and our technology, as we’ll return to Excel and the calculator to see their power and limitations.
    • This gets at a concern we see identified  regularly: students who do well in school but don’t understand some of the basics they need to get along on their own in college and beyond. By the time students finish exploring savings, consumer credit, loans, and the time value of money, they should have a firm understanding of what they’ll soon by facing and what the math they’ve been learning has to do with it.

So, that’s the plan for the year, and that’s why we’re doing it. Like I said, it’s been a long time coming, and I’m both excited and nervous to put our money where our mouths are and see how things go. Whatever the outcome, I cannot stress enough what an amazing experience it’s been to question everything we do; to be empowered to act on the responses to those questions; and to collaborate with colleagues who inspire me to do more and go further than I ever thought I could. As my collaborator and co-conspirator Bill likes to say, we have created the course we’d love to teach. Now it’s time to find out if our students will love to learn.

Leadership as Defined by Michelle Obama

A couple weeks ago, I was working on a summer assignment for the students in the leadership seminar class I’ll be teaching this year. Wanting them to think about what leadership means to them without explicitly asking them that question, I was writing questions about people and characters in popular culture and in their summer reading. I was looking for a question or topic related to politics when I remembered Michelle Obama’s speech at the Democratic National Convention.

I watched the speech again, anticipating asking my students if (and how) she was acting as a leader at that moment, when something else about it struck me. If you listen closely, you can hear Mrs. Obama not just be a leader but offer her personal definition of leadership. Consider the following:

“Remember how I told you about his character and conviction, his decency and his grace, the traits that we’ve seen every day that he’s served our country in the White House.”

“…doing the relentless, thankless work to actually make a difference in their lives.”

“She never buckles under pressure. She never takes the easy way out. And Hillary Clinton has never quit on anything in her life.”

“You can’t make snap decisions. You can’t have a thin skin or a tendency to lash out. You need to be steady and measured and well informed.”

“Someone who’s life’s work shows our children that we don’t chase fame and fortune for ourselves, we fight to give everyone a chance to succeed. And we give back even when we’re struggling ourselves because we know that there is always someone worse off.”

In her own words, Mrs. Obama is defining what a leader is to her. Sure, she could have just used the buzz words we so often use in talking about leadership – “servant leader,” “ethical leader,” “clear sense of purpose” – but she didn’t. She stuck to her own words, but the picture she painted is clear, simple, and unmistakeable.

Two things occurred to me about this. First, what a great definition this is. As teachers, coaches, and dorm parents, who among us wouldn’t want a team captain, dormitory prefect, or student in class who “never buckles under pressure,” doesn’t “chase fame and fortune” for himself or herself, and does “the relentless, thankless work to actually make a difference” in the lives of his or her peers?

Second, I wonder how it would go if we asked our students to write definitions of leadership that are just this simple, clear, and direct? What if we did it before it was time to choose team captains or student government representatives? How would school look if students first decided what leadership was to them, and then looked for those who most closely met those ideals?

And, to extend Mrs. Obama’s argument to its logical conclusion, what if students held their peers to these self-defined standards and let them know when they had let us down, just like people let their political representatives know every day? How would student leadership, and how would school, look then?

Resuming an Old Road Trip

Twenty-four years ago, I embarked on an adventure. It started when the wilderness program director at my boarding school invited me to assemble a group of peers to spend the day doing a low and high ropes course at the Santa Fe Mountain Center. It was my first foray into formal leadership and group dynamics work, and it set me down a path that I have pursued off and on ever since – learning about leadership, intentionally developing my own leadership repretoire, and helping others develop as leaders.

I’ve had a lot of great experiences on my travels through the world of leadership. Some of them were planned, like the summer I spent facilitating a low ropes and my participation in the Gardner Carney Leadership Institute. Others were much more spur-of-the-moment, like my volunteering to coordinate prefect training without really thinking about how much work would be involved. Like any good road trip, I’ve had a general plan but have always looked for scenic detours, and I’ve rarely been disappointed with where I’ve been or what I’ve done.

A couple weeks ago, I was offered the opportunity to start a new leg of my trip through the world of leadership. In addition my other responsibilities, I will now be the Student Leadership Programs Coordinator at The Hill School. In this newly-created position, I’ll have the opportunity to work with students and colleagues to advance our current leadership development work, create new programs and activities, and blend it all into a comprehensive leadership program that will define our students’ experiences as we pursue the School’s mission to help them “be prepared to lead as citizens of the world, uniquely guided by our motto, ‘Whatsoever Things Are True.'”

So, here I go again. I’m in the planning stages, looking at possible routes to our destination and things I want to see along the way. I’ll do my best to share my plans here as they develop, and I look forward to the on-going feedback and support of those who are traveling with me or just following from afar. 

The City Slickers Model of Leadership Development

(A word of warning: most of my posts over the next few weeks will start with “A few weeks ago…” because, well, when you hit the spring term as a teacher it gets hard to keep up with day-to-day life, much less find time to blog about the important things that are happening around you. My apologies in advance. I’ll start blogging my current ideas soon.)

A few weeks ago, I was interviewing students who had applied to be prefects for next year (prefects are student leaders who help us run the dorms). I always ask the students if they have any questions, and a popular one is, “What’s the most important quality you look for in a prefect?” It’s a good question, one I usually turn around on them so I can learn more about what they think is important for the position.

One student asked me the question, but just a little differently than it’s ever been asked before: “What’s the one thing that’s most important for a prefect?” For reasons only a neuroscientist can explain, the phrase “one thing” immediately reminded me of this scene (caution: profanity approaching):

Instead of turning the question back on the student, I thought about this clip, and I answered it. I talked a little about the many qualities that are important for a prefect to be successful, from being organized to setting aside one’s own needs to help another. I said that all of these things are important, and that most of our candidates have most of these qualities. The one thing that’s most important, then, is the one thing the student doesn’t currently have. It’s different for every student, but when we work together we can find that one thing and then help the student develop in that area.

In a sense, this is how we often think about, and teach, leadership. We do things like Strengths Finder to figure out what our strengths are and how to leverage those, but then we also look at what’s not on the list and think about how we can grow in those areas. This is fine, but it’s also overwhelming, especially for students. It seems unfair, and unhealthy, to ask a student to work on her communication, organization, and public speaking while simultaneously asking her to work on her physics, tennis serve, and college essay.

I think Curly actually has it right: whether you are a prefect, team captain, or student government representative, find that one important thing and focus on it. For Curly, it’s the secret of life. For our students, it’s the secret to developing as leaders.

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Inspired by (Blatantly Stealing from?) Dan Meyer

Yesterday a colleague and I were discussing interval and set notation. Should we introduce it in the first year of our integrated math curriculum was our initial question, but we quickly moved into more general questions about these notations, what they’re good for, how much we expose students to them, and how much we expect students to use them.

And then, inspiration struck, and I could hear Dan Meyer in my head saying, “If set notation is the aspirin, what’s the headache?” (If you’ve never heard Dan ask this question before, then start here and read his collection of posts on questions like this.)

So, why do we need set notation in, say, an Algebra 1 or Algebra 2 class? My first response was, “Hey, I’ve seen it come up in some SAT II practice tests.” So we looked, and we found these:

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This problem has issues because it states that x is “equal to” the set instead of being “an element of” the set.

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This problem uses set notation in a way that’s fine but is, in the words of my colleague, just gift wrapping, i.e., completely unnecessary. I’m not sure what the purpose of including the notation in the question is. Are they trying to throw weaker students off? 
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This example uses the set notation just fine, but again, would anything be lost if we didn’t include it? So, “if you want to do well in the SAT” is probably not a sufficient headache to motivate the distribution of this aspirin.

My next idea was that set notation somehow simplifies communication of what we’re dealing with. After thinking about this for a few minutes, though, I don’t actually think that’s true. Whether a student writes “x = 3” or “{3}” makes no difference in a student’s attempt to clearly communicate her solution to an equation, and at the high school level we’re talking about, their ability to communicate this information is far more important to me than their ability to correctly apply a complex idea (sets) to a simple situation (the solution is 3).

Don’t get me wrong – there are ways in which ideas related to sets are very important to understand at this level. We want our students to know that two line segments cannot be called equal, why this is true, and why it matters. I’m just not seeing a reason to go beyond this set-based idea to actually introduce set notation and insist on its use in ways that don’t reinforce the important concepts that underlie the notation.

So what’s the right way to extend Dan Meyer’s headache-and-aspirin analogy? I think it’s to say that we have developed the bad habit of mathematically over-medicating. We all know the stories of people who demand antibiotics to fight off viral infections and what kind of problems this can lead to. I would argue that we do the same thing in high school math. We have taken problems that can be handled with a cup of soup and a good night’s sleep, and we’ve chosen to address them with IV medications (which is sometimes watered down, making it even worse). So I’ll keep looking for the headaches that need to be cured, but I’m going to start being a lot more aware of of the ones that can be taken care of with the remedies we already have available to us.