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.

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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?