Wednesday, July 20, 2011

Thoughts on Algebra 1

After a five week mad sprint to teach my students everything they'll really need to know about Algebra 1 going forward, I have a few thoughts on the subject.

First, as far as I can tell, there are two "Big Ideas" of Algebra 1:
  • Everything students already know about arithmetical expressions (adding, subtracting, multiplying, exponents, radicals, fractions, simplifying, word problems) can be extended in a consistent way to situations involving variables.
  • Equations with variables give you the power to describe situations more generally than a single set of numbers allows.
If you've taught Algebra 1, do you agree with those? Am I missing some? If I ever find myself teaching this class again, I'd like to try to present those up front and highlight them repeatedly as they come up, rather than gradually discovering them for myself and trying to make them clear to my students over the course of the class.

I think one of the most problematic concepts of Algebra 1 is "Simplifying". It's so very easy to slip into "here are the rules for what constitutes simplified in this case, and this case, and this case..." (polynomials, exponents, rationals, radicals, ...) and so very difficult to access a big idea like, "We're trying to make our lives easier when we move on to the next step of solving or using a complicated equation that describes some situation in real life like particle physics or the stock market or the operation of an engine or the forces on a bridge." I can say that to students, but I have a hard time showing them or getting them to experience it...

Second, I find that I have some startlingly clear memories of my own Algebra 1 class in eighth grade in 1990, probably because some of the subject matter I learned never got used again. It seems like there's a lot of manipulation and variations on a theme that we teach students but aren't actually critical to their ongoing life as mathematical practicioners.

For example:
  • I've used a lot of math in my life and graphed a lot of linear functions and data. All you really need is a deep and thorough understanding of y = mx + b. You do not need the point-slope form of a line. You might need the standard form when you get around to doing linear functions at a deeper level in Linear Algebra (and all of its applications in physics and economics and ...) in college, but could it wait until then?
  • I've used a lot of math in my life, and while being able to factor out the greatest common factor of a polynomial is a deeply practical skill for all the math it can make easier, and factoring x^2 + bx + c offers a sort of aesthetic pleasure and can be used to ensure that students understand FOIL at a deeper, backwards operation level, no one ever tries to factor ax^2 + bx + c . You pull out the quadratic formula, because it's so much more general and so much less guess and check. Why do we then waste time teaching an essentially guess-and-check approach to solving it?
What's your perception of which parts of the standard Algebra 1 curriculum are actually essential and useful to practicing users of math in this modern, computational age, and which parts are historical cruft?

Standards-Based Grading, Take One (The Reflection)

As I procrastinate take a break from making my final for the Algebra 1 summer school course I've been teaching for the past five weeks, I find myself reflecting on my first try at Standards Based Grading this summer.

Things that are excellent:
  1. I had a constant read on what my students did and didn't get, individually and as a whole class, which allowed a lot of course corrections, both major and minor, and would have allowed some awesome differentiation if I had more time for that kind of planning.
  2. My students and their parents had a constant read on what they did and didn't get, which let motivated students target the areas on which they needed help and reassessment and let motivated parents push their less-motivated students into the same.
  3. When I had a co-teacher for the last week of classes, I could send off the kids struggling to meet my goal of mastering at least half the concepts on my concept list, and she had instant access to where each student needed targeted help.
Things that need work:
  1. I stole my starting concept list pretty whole-sale from Dan Meyers (here). While it was great to have a starting place (thanks, Dan!), it needs some tweaking to serve the constraints of my class.
    • First: I only had 5 weeks (of 4 hours per day, 4 days / week) instead of a full year. While I had basically 80 instructional hours (so, theoretically almost a semester of time) and some pretty awesome kids, I still found myself slashing out some pretty critical concepts (solving systems of equations is the major one, but some other pretty important things also fell by the wayside). I ended up leaving out some core topics that we'll cover in the math class these students will take next year (solving systems, quadratic functions and graphs thereof, exponential growth and decay) and keeping some less central concepts that are at the back of most algebra books and many of many students hadn't seen yet (rational and radical expressions).
    • Second: I wanted to work a lot on my students' mathematical reasoning abilities with open-ended problems, and their comfort with dissecting and translating word problems into math. Yet I have a single concept on my final list for word problems, none for mathematical reasoning, and the word problem standard I have is by far the one the students still have the most trouble with. If I teach this class again, i want to think more about what I want students to be able to do beyond check off the key procedural and conceptual skills asked by a typical Algebra 1 textbook.
  2. Reassessment, Part 1- I tried stealing Sam Shah's reassesment request form but in a five-week class with limited other use of the class website and no paper copy available in my classroom, basically one student used it once. It was too cumbersome and / or too much overhead for students to use and for me to enforce. This may be somewhat different in a full year course, but at the very least I need to make sure I get a paper copy available in person the next time I do this.
  3. Reassessment, Part 2 - I need a way to get the less motivated and more struggling students in for help and reassessment. Probably the most fundamental thing I need to do is to make my expectations about this crystal clear up front to both students and parents. Also, I should think about more creative ways of making the right choice the easy choice. This could constitute a mandatory 1 hour after-school session this week that you have to attend if you have less than a 4 on any concepts that I'm not assessing any more. Or it could constitute some part of a regular class period when the non-struggling students do a fairly independent activity (could use suggestions of these) and I focus on the struggling students. This summer I did it by setting my co-teacher loose on the most struggling students for the last three days of the course, but that's probably not a luxury I'll have in the regular year (but it may be worth brainstorming ways I could make that work).
  4. Work load - grading 40 quizzes with 6-8 questions per day on top of planning 4 hours of new instruction a day almost killed me. This is probably partially because I'm a newbie teacher who started from scratch in terms of... everything... this summer, and it might be better in the future just because I'd be tweaking this year rather than making everything from scratch. But I wonder if I could get 80% of the results from either fewer daily quiz questions (3 instead of 6?) or every-other-day quizzes? Realistically, I could only do a good job of introducing two new concepts per day, but the need to give students multiple chances to assess past concepts drove up the number of questions. At the least, I think I need to give one "what do you already know?" assessment of every single standard at the beginning of the summer and then do my darndest not to waste the students' or my time by assessing a standard before I've had time to teach it, which happened multiple times this summer.
More thoughts on Algebra 1 as a class in the next post...

Wednesday, July 13, 2011

Lessons of the day

Things I learned in summer school today:

0) Homemade Algebra Tiles are a lot cheaper than the $70 sets from Amazon, and work fine. (But if you have the funds, a more substantial set might be nice.)

1) Algebra Tiles are pretty cool, and have a lot of potential, especially for making kids really consider what the meaning of integer addition, subtraction, and mutliplication are. I definitely saw some lightbulbs go off when we worked through "4*3 means four sets of 3".

2) Trying to introduce Algebra Tiles from scratch and accelerate up to multiplying and factoring polynomials in 60 minutes is a recipe for disaster. (I should have known better.)

If I were to teach this class next year, I would introduce the Algebra tiles the first week, and embed them as we reviewed integer arithmetic, solved simple equations for x, and worked our way up to polynomials. I think they'd be a useful addition to the boardwork / notes / practice / fun outside problems mix I've developed this summer.

Live and learn.

Useful algebra tiles resources on the web:

  • Good power point overview and a solid series of worksheets escalating from integer addition through polynomial division are here.

  • Good overview of a lesson flow for introducing the tiles is here.