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.

Thursday, June 30, 2011

Summer School Midterm

I'm two mind-blowing weeks into the five weeks of my first summer school teaching experience. I gave the students a Teacher Evaluation as the exit ticket today, and I'm in a more reflective mood myself, so it seems like a good time to jot down some notes.

Things that worked well

Curriculum-wise (huge thanks to Dan Meyer for many of the items on this list):
* Algebra cups
* Graphical fractions (week 2)
* Chewing gum measuring stick
* Origins of the word "Algebra" (modified for ninth-grade reading level from this)
* Get the Math
* Stacking cups
* Russian dolls (inspired by the comments on this post, especially this image)
* Graphing stories
* "Leveling-up" linear functions worksheets (25LevelOne.pdf from Dan's algebra handouts - linked at the bottom of this page)
* Daily warm-ups - the kids really do need to warm up their brains at 8 a.m. Trying to sneak in higher order thinking and group work will not work well at this point in the day.
* Judicious use of the powerpoints from the textbook to define the key concepts and walk through scaffolded examples

* Concept quizzes (once I scaled them to the topics I was really covering and a grading load I could handle)
* Concept checklist

Classroom management / Relationships:
* Remembering that the students have reasons for acting in ways I find challenging, and I'll do better to find out what those root causes are and address them than to just try to force them into acting a different way.
* Asking students their opinion on the class atmosphere and environment and things they like and don't like about the class, incorporating that feedback where possible, explain why not when it's not possible, and being flexible and creative.
* My best example of being flexible and creative: some students want music when doing independent work and some want silence --> you can listen to your mp3 player during that time, as long as no one else is disturbed by it.

Lessons Learned
* Still need to follow up Algebra cups with formal instruction on steps of solving equations

* Don't pay for and pour hours into a new-ish online gradebook system when they just changed their whole infrastructure a few weeks ago, aren't expected to support summer school, and you're their unknowing beta tester

* Find out on Day 1 what the students' self-assessment of their ability on your course material is. If they think they already took and mastered Algebra 1, and that's all you're planning to teach them, give them a version of your final ASAP so you can kick them up to a harder class, send them home for the summer, or show them (and even more critically, their parents) that they do need your class after all.

* If it all possible, solidify the curriculum, schedule, and course resources before the second week of the class. (I did try, it's just that before trying to teach this and working out some of the kinks the first week, it wasn't clear to me what they should be. I got a lot more clarity this week, and I'm feeling good about this now, and like I'll be in a much better place if I do this again next year.)

* Sometimes I really just need to give clear structured notes, and clear, basic opportunities to practice (even though it seems boring to me).

Things to work on
* Helping kids see how to follow up on non-mastery-level concepts
* More efficient planning

Sunday, March 20, 2011

Red Cross Pedagogy

Yesterday I had the privilege of spending 8 hours in the hands of the Red Cross, freshening up my adult and child CPR and first aid and meeting AEDs for the first time.

Here are the pedagogical approaches they used that I learned from in a way I might remember if an emergency actually hit:
  • hands on practice in moving a choking friend back and forth between back blows and abdominal thrusts
  • hands on using the plastic torsos to practice CPR for a decent chunk of time - half an hour maybe? (I could also feel some muscle memory from the last time I did this in approximately 1998)
  • hands on using the plastic torsos to simulate using the AED
  • hands on tying a bandage on my friend and having him tie a sling on me.
Here are the pedagogical approaches they used that I learned from for just long enough to pass their test end of they day multiple choice test:
  • lecture
  • reading
  • demonstration of one student rolling another student
  • videos
  • last minute pre-test cramming of needed information into our head ("teaching to the test" in the purest form I've ever seen)
It's always good to be forced to spend a day in a situation I'm not particularly intrinsically motivated for, just to see how most of my students feel most days.

I think what surprises me the most is how little the demonstration did for me -- it was a purely visual flow of information that wasn't really much more engaging than a video. I tend to try to substitute demos for hands-on experiences in my lesson planning because it's so much more efficient in limited time, and I guess this says I need to try to squeeze my efficiencies out some other way -- like cutting down the lecture even more.

Something that was surprisingly effective for me was the continual stream of verbal directions and reinforcement of critical information as I was doing the hands-on work. Maybe that should be my main method of information transfer rather than lecture.

I wish we'd done more writing in this class -- I'm wondering whether that would have fallen into the "doing something" category for me (resulting in some effective learning because I'd actually mentally had to process something and use it) or the "seeing / hearing something" category. Maybe it depends a lot on the kind of writing.

Overall, I think I need to learn to be more zen about teaching. I know that "doing" is almost always better than "seeing" and "hearing", but I also know that giving my students time to "do" always (a) takes more time than I expect and (b) is the fastest way I know to spread out my fast learners from my slow learners (maybe because they're actually learning!). I need to learn to accept a slower overall pace and the need for basic / crticial and extension levels of any meaningful assignment. And more "doing".

Thursday, March 17, 2011

Things I Learned at MACUL11

So, today I got to go to the MACUL Conference, courtesy of our Teaching with Technology class. Here are some things I learned:

  1. You can do really individualized, project-based, interdisciplinary learning that is driven by each student's interests, meets the state standards, and reaches kids who would otherwise drop out physically and / or mentally. But you may have to blow up the traditional school structure to do it.
  2. Using the annotate feature in YouTube lets you and/or your students put commentary on videos and make interactive "choose your own adventure" video sequences. Also, youtube is fun, and making youtube videos can make school more fun. And sometimes it can be educational too.
  3. Universal design for learning (UDL) is about giving kids access to the topics in multiple ways, giving them multiple ways to demonstrate their learning, and giving them choices so they can use the ways to learn that work best for them. There are lots of resources on the CAST site.
  4. Project-based learning is cool, and there are lots of resources for it here and here.
  5. You can use a combination of Google Sites and Google Docs to enable your students to collect their work over time in an e-portfolio and to enable you and your colleagues to share and collaborate on curriculum resources. There are resources and how-to's here, and it looks reasonably straightforward.
If there was a theme to my day, it was how to use technology to engage students in individualized, self-motivated, creative, project-based learning, and thinking about what constraints need to be shifted within the traditional model of school to make it work.

It seems like the most critical element is team-based teaching or some other way of solving the problem of "I want to study the Gulf of Mexico oil spill from a scientific, mathematical, political, literary, and foreign language points of view but my students all have five different teachers for each of those subjects, and that's a lot of cats to herd teachers to coordinate."

Also helpful: One-to-one computers. So your students can access your UDL-based electronic texts and interactive assignments, collaborate with each other even when they're at home, and get to their project and e-portfolio sites whenever they need to, regardless of their family's economic resources.

Saturday, March 12, 2011

Epsilon Greater-Than Vi Hart

This, and everything else on her site, are among the most wonderful math things I have encountered:

Sunday, January 30, 2011


How can I make the work of learning science and math this much fun?

Wednesday, January 19, 2011

Keepin' it real

I share this with you because it made me laugh out loud:

Since I'll probably never teach complex analysis, quantum mechanics, or in an environment where using the word s**t is appropriate, I can probably never use it in teaching.


Sunday, January 16, 2011

Note to self - Phone Home

When you think you've tried everything to reach a student, and it hasn't worked... ask yourself whether you've phoned home.

Other strategies that may come in handy:
  • If you've tried asking about alternate interests, and the only one offered is sports, don't despair just because I'm not personally into sports. Delve anyway: Do they play on a team? What position? When did they start? Why did they start? What do they like about it?
  • Potentially useful short-term, when-all-else-fails strategy: bribery.
From this you may deduce that I was trying really hard to help a student who didn't want to be helped on Monday. On Wednesday, I left a message for the student's mom. On Friday, they came in with a little bit of work done and a much better attitude, parroting something mom must have said about taking advantage of the help you're offered. And I gave that student a lollipop and got a solid hour of good focus and thinking in return.

I don't know how this coming week will go, but on Friday I did a little, intermediate victory dance in my head.

For reference, my awesome mentor teacher's recipe for a call home:
(1) Establish that you're on the same team -- you both want the student to succeed in your class.
(2) Tell the parents two things they need to know.
(3) Listen.
(4) Tell the parents two things you need from them.
(5) Thank the parents for their support.

Another good xkcd for the math teachers out there: Critical Content

Recommended by a friend who's teaching math out in Oregon, and was planning to use this to loosen her students up before a quiz:

Critical content