Friday, August 21, 2009

What to do...

I have a bit of a dilemma.  Three years ago I was brought to a GATE magnet school to teach a bunch of advanced kids. 

The thought was, "Rather than ship the really smart kids to high school to take the geometry class, we'll just bring the high school teacher to them." 

I didn't mind this idea.  In fact, it quickly began to grow on me.  I was teaching precalc, algebra 2 and algebra 1 at the high school and figured the change might do me some good.  The tough part was that I had no idea what to expect.  I was going from 55 minute periods to 94 minute daily blocks. 

What the heck am I going to do with a group of middle schoolers for 94 minutes? 

So after many discussions with my principal, we decided that since the 94 minute block was intended to allow for grade level instruction as well as time for intervention,  I'd just use my extra time for enrichment.  But what do you do to enrich them?  Do you accelerate the students so that they can be ready for algebra 2 as freshmen?  Do you spend the extra time doing all the cool stuff that no one else has time to do because they are worried about pacing guides and benchmarks?  We've decided to go ahead and accelerate them.  I'm still a bit unsettled about it, though.    Is it really that important for an 8th grader to complete geometry so she can take algebra 2 in 9th grade?  The kid is on pace to take calculus as a junior.  Is that better?  At this point, I don't know.  In the era of "the TEST", it seems that as long as I have the data to back up what I am doing, it doesn't matter what we choose.  My kids have the test scores, but I am not convinced that what I am doing is the right thing. 

I am definitely in unchartered waters.  As a high school teacher, I was simply one of many.  My last year there, we had a pacing guide and all common assessments.  It was pretty lock step and I was miserable--loved the staff and the students--but hated the system.  Now I have a bunch of autonomy because I can keep the powers that be off my back with some good scores, but I am not sure what direction to go.

Last year, the decision was a bit easier because I got to hand pick the students who would move into the geometry class. I taught both advanced 7th grade classes so I knew they were ready.   This year--not so much.  I have two classes made up of kids from two different teachers.  Some are ready to be accelerated and others look like they are going to need to spend some serious time with the algebra.  My question for you all is this:  how do you plan for these classes?  Half of my students have already been through algebra, a quarter of them have been exposed to it and the rest look like they may not know what a variable is.  I've never been here before, so I don't know what to expect.  If you've ever considered commenting on a post here, now's the time.  Hit me up.


Kate Nowak said...

Oof - I was with you until you said you decided to accelerate them. I'd love to hear/read about a public school who decided to DE-celerate, as in give kids/teachers the freedom to play with fun problems and discover math as they need it. Oh well. I'm in the same position you were at a high school, so I can't talk.

I'm a little confused about your question, though. I think you are saying you have 8th graders who are expected to take HS-level geometry, but you feel some of them aren't ready. Is that accurate?

Jessica said...

Most everything I do is a situation like the one you describe. I am in a very small school, so my classes are whoever needs the class or has nowhere else to go that block. Sometimes the ability levels are in the basement and the roof. Something that my colleague did with her Geometry class last year was to allow students to "Choose their own Adventure" of sorts. Any time there was going to be independent or paired practice, she started students with a low or medium challenge paper. If it was completed with minimal errors, they continued onto more challenging problems. I know, not always the best to be doing worksheets, but sometimes you need it.

Plan remedial mini-lessons for the kids who need it at the beginning of a topic while the other students are working on a related project or problem?

I'm only just starting my 2nd year, so I'm new at this. Thanks for starting the conversation.

David Cox said...

My question is two fold, I suppose. By acceleration, I mean kids have an opportunity to complete three levels of math in two years (ie. pre alg, algebra 1 and geometry). Of the 60 kids who come to us as GATE students, I would say 25 or so are really solid math students and the rest have just learned to play school very well. The primary question is should the program be geared towards moving students through the math that quickly...if they are ready for it? Or should we just try to have kids go deeper into the grade level curriculum?

Moving kids through geometry isn't very difficult to do when you get a class of kids who have all succeeded in algebra. For example, my 8th graders last year crushed the state test (4 proficient, 21 advanced) and about 12 of them successfully completed geometry. My standards are that they must score advanced on the state test and earn at least a B in my geometry class before I will recommend them to take algebra 2. Otherwise, they can just take geometry again as a freshman.
But what happens when the classes get mixed and some of them are going to need real instruction in algebra? Do I allow the kids who are geometry ready prove their proficiency in algebra then let them move through the geometry at their own pace? What do those lesson plans look like?

The problem is that this isn't simply a matter of differentiating instruction. I'm having to figure out how do differentiate the curriculum.

I like the idea of "choose your own adventure" lessons. Did this teacher create the curriculum for this? I don't mind trying to reach kids on their level when we are all covering the same topic. But I am struggling with how to have kids in two different curricula and at different levels within the curricula.

Darren Kuropatwa said...

I agree with Kate although I would describe her approach as acceleration; by which I mean acceleration of learning as opposed to acceleration of "covering content."

I would look for ways to have the students create content that educates. i.e. design their own problems, based on what they've learned, then have them solve, annotate and share what they've learned.

The challenge is coming up with problems that are rich enough to be explored at multiple levels. That's where I would spend my prep time.

An example:

Phase One
How many square are there on a chess board?
Hint: the answer isn't 64.

This is an excellent problem to teach strategies such as:
solve a simpler related problem (a 2x2 chess board)
draw a diagram (pictures almost always help)
look for a pattern
... I'm sure others will arise in conversation with the students; I've often been blown away by their ingenuity.

Phase Two
What about a 7x7 chess board? 6x6? 5x5? Can we generalize a solution? Predict the solution for a 9x9, 10x10, or 20x20 board.

Phase Three
How many rectangles are there on a chess board?
Interesting aside: What's a rectangle? Does limiting the "kind" (dimensions) of the rectangles under consideration lead to different solution spaces? (What the heck is a solution space?!?)

Can you generalize each of these solutions?

Phase Four
Based on what you've learned, can you suggest/create a new problem? Maybe one in three dimensions? (Rubik's cubes come to mind.) Can we play with other geometric shapes in 2D? 3d? Do you know what a tesseract is? (Let them research that last one.)

The nice thing about all this is you'll be able to show how the algebra and geometry are connected and each helps us to understand the other.

Introduce them to Pascal's Triangle and, without going into the binomial theorem, have them do a few binomial expansions (a + b)^2, (a + b)^3, and so on, then remind them to explore the limiting cases (a + b)^1, (a + b)^0 ... can they find any related patterns?

Explore Pascal's Triangle for: powers of 2, powers of 11, counting numbers, triangular numbers, hexagonal numbers, other figurate numbers (what's a figurate number?), Fibonacci numbers.

Fibonacci opens a door into exploring math in nature, leaves on plants, they way trees branch and grow, the way bees and rabbits reproduce, ... FIELD TRIP! Send them out with digital cameras to find math in nature, publish the pics to flickr and annotate them (here's how I did that).

You might also explore some sort of long term projects with them, I called this Developing Expert Voices in my classes; you can see some student work here or there, and these are from this year.

I hope this long answer doesn't come across as "too much." I couldn't help, it's seems to me you're walking into such a golden opportunity to do some real teaching with your kids.

I tend to think about it like this: the more teaching they do the more they'll learn.

Kate Nowak said...

OK so then I vote for "go deeper into the curriculum." Rushing any of them through algebra is not going to serve them well. Why not let the group discover/invent algebraic reasoning together collaboratively by working on interesting and fun problems. The quicker students will learn the material way better if they assume some responsibility for bringing the rest of the class along on the journey.

Darren lays out a nice progression. I might back up one step with the initial question. Imagine a triangle where you drop a bunch of segments from one vertex to the opposite side. How many total triangles?

David Cox said...

Hi Darren
Thanks. The long answer is exactly what I needed. The real beauty of my situations is that as far as the geometry is concerned, I am not confined by pacing guides, standards and/or state tests. My kids will all take the algebra test. This fact isn't lost on me. I don't want acceleration to simply mean "cover content" either. However, that seems to be the way things are structured. So the real question may be: What does the geometry curriculum need to look like? But is there a better course to allow kids to explore than geometry?

David Cox said...

I agree that rushing them isn't going to help. That was a big concern. The thing is this: my 7th graders get nearly a full year of algebra and then we do things like the Farming with Google project when they are in 8th grade. So my first priority is definitely making sure they have a solid algebra foundation. Thanks for the input. I knew I could count on you. :)

Alison Blank said...

Hi David,
I recently acquired a list of meaty geometry questions that might be appropriate in terms of going deeper into the curriculum. They've been culled from the Phillips Exeter Academy collection. If you'd like to check them out, you can email me at alison.blank at

Darren Kuropatwa said...

I found this. I very much like some of the problems in the problem sets. Thanks for the tip Alison!

Jessica said...


The teacher in question did create the "choose your own adventure" curriculum at the different levels. She used many of the textbook resources that we have, but she did the majority of the work (and it took a lot of time). Our school is such that we often get kids that are not developmentally ready for the classes, but end up sitting in a state standards level course, and sometimes two courses happen in the same room at the same time. I'm looking for suggestions on how to remediate Algebra with a student that isn't very independent (1:1 class) while at the same time teach AP Calc to a very independent student, but one that deserves some quality instruction, too. Any suggestions for two different classes at the same time are wonderful and welcome. :-)