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.

Reflections on a Device-Free Day, Part 2

As I mentioned in my last post, our school went device-free for twenty four hours earlier this week. In addition to experiencing life without technology and reflecting on the role it plays in my teaching (and other areas of my professional life), I spent some time considering the implementation of the day itself. In thinking about what worked, what didn’t, and how we might do it better next time, it occurred to me that our collective experience of this day was a microcosm of how to, and how not to, implement change in schools.

I will say right up front that my thinking about this topic of school changed is heavily influenced by Grant Lichtman, Julie Wilson, and Jonathan Martin, among others. These are people who are thinking about what conditions must be present for effective change on both the macro and micro scales, and their work has given me a lens for thinking about change that I am involved in, whether it’s as a leader or a participant.

While I had a great time going device free, my experience does not seem to have been shared by the majority of faculty and students. Responses to the day ran the gamut from complete abstinence to not even acknowledging the day. It would be easy to say that students were on their devices because the faculty weren’t setting a good example, or that the faculty were on their devices because they didn’t think the students wouldn’t be able to stay off them anyway. Both of these conclusions are too simplistic and miss some much bigger lessons that we can learn from the day. Here, then, is my assessment of the critical conditions of effective change and how we fared under each of these criteria.

  1. Know what problem you’re trying to solve – change for the sake of change won’t get much traction; without a problem to be solved through change, there’s little reason for people to embrace it. In this case, the “problem” was clear: we have all integrated technology into our lives to such an extent that we aren’t even aware of it at times. This can have both positive and negative consequences, and a day free from electronic devices is a good way to identify those consequences.
  2. Make a compelling case for change – even with the problem identified, one has to convince others that a change is needed to address the problem. The device-free day was first mentioned in a talk given by the headmaster, and was followed up with e-mails to the faculty and the students, but this didn’t go far enough in making the case for why we needed to do this. People didn’t see a compelling reason for being device free, so they saw no reason to participate in this event.
  3. Obtain stakeholder buy-in – even when the case for change is compelling, people will only commit to it to the extent they feel ownership of it. The school’s Academic Council was consulted about the timing and calendar, but there was no broader effort to get input from faculty or students into the device-free day. The best chance for success depended on people being committed to this, but with no say in the mechanics of the day there was no reason to feel that sense of commitment.
  4. Don’t offer a way out – when change is presented as optional, those opposed to it will opt not to participate. The device-free day was presented as entirely voluntary. Students were told there would be no policing of this, no steps taken to address those found using devices when they shouldn’t have been. Given the option to avoid the challenge presented, many people took it.

This analysis might lead one to believe that our Device-Free Day was a waste of time. Beyond the tremendous personal value I found in the day, I actually think it was institutionally significant. On a small scale, it showed us how change will be received, so we’ll know better next time how to proceed in implementing change. As long as we can learn something from the day, and I think it’s clear that we can, this event can significantly influence how we proceed as we continue down the path of growth as a school.