Developing Understanding in Algebra and Geometry

Last week, my district brought in a Pre-AP math consultant to work with us on Pre-AP Strategies. Initally, most of us were not pleased with having to miss a day of school to attend a PD day, mainly because we had missed 2 weeks of school in February due to a blizzard and we have end of course exams coming in mid April, so a lost instructional day was not high on our to-do list. However, we were also very grateful that the district was trying to bring in PD that would be applicable and ready to implement. In a nutshell, we had a great day :) Here are some tidbits of knowledge that I took from it...

General comments
"You (the student) will feel uncomfortable in this room! Your brain will hurt daily!"

The "Rule of 4" should be on every assessment and potentially every problem.

Make the kids communicate in complete mathematical sentences. Communication helps ALL kids - the answer alone does not. With just the answer, the struggling kid still doesn't know what to do or where to start.

Students should be asked to read, interpret, communicate, draw, label, compute, and explain/justify on a regular basis.

We as teachers should not have to put "Show all work" on every test - this should be an implied directive that they do no matter what.

More Specific Ideas
One of the key components of the workshop was the focus on how to communicate mathematically both your process and your product. One of the activities she used came from the Math Connections book by Dale Seymour. She used the "What are you likely to be asked" activity, but in the Amazon link, you can see an example of the "How do you know" activity. I can really see both of these as great warmup/journal prompts or even as exit slips.

Another idea she had was based on the game "Would you rather?" I think it would be neat to sit down and brainstorm a bunch of these for various courses. This again could be a neat journal activity.

She also refered us to the Algebra Lab website and we looked at the All Tied Up in Knots problem, which explores a system of equations using two ropes, tying knots in each of them and measuring their length. This could be a great pre-ap Algebra problem as you could go into what the y-intercept and slope really mean, which is a vital skill in AP Statistics. I'm eager to go through the other activities on that site to see what else I could use in my classes.

We also explored some neat geometry problems such as constructing a tangram using compass/straightedge, a fun manipulative proof the Pythagorean Theorem, 3D solid constructions using pipecleaner pieces and coffee straws, and peeling an orange to discover that the surface area of a sphere is 4(pi)r^2.

All in all, it was a fun and productive day that really allowed us to get some creative juices flowing.

Mastery of Objectives

So during a discussion on twitter today, @misscalcul8 and I were discussing the number of objectives taught per week. On average, in Algebra 2, I would say I cover 3-4 objectives. Her question, and a very valid one was this:

So how do you teach to mastery in one day of experience with a LT? I don't meant that critically.

Now that response is way over 140 characters, so I told her I would blog my answer :)

In short, the answer is "You don't." Mastery is not a one-day process - it is a long term process that differs for every child. You have to continue spiraling the curriculum so that over time, mastery is obtaining. Some of you may have seen the "forgetting curve"...

Here's how the forgetting curve works for me...
On the first day of an objective, I teach, they take notes, do problems, etc like normal. Then the next day, I give a "Checkpoint" - this is an ungraded formative assessment so I can provide written feedback and correct errors. On day 3, they get the checkpoint back, look over it, read my notes. (Note: during days 2 and 3, I am teaching new, but related objectives that I try to continue tying into each other). A few days later, there is some kind of cumulative activity to see how objectives work together, review multiple objectives, etc. There are also Jing videos that I have created and put on the class website that they are supposed to refer to if they need additional instruction. About once a week, there is a SBG quiz that covers that week's objectives. Again, lots of written feedback if needed. Over the next few weeks they are working on those objectives using self-check materials, the Jing videos, tutoring, etc so they can reassess. (In order to reassess, they must provide evidence that they have worked on mastering the objective)

In a nutshell -
From start to finish, the student has seen and worked with the objective many many many times, so that they are constantly bringing it to the forefront of their mind, spiraling the material so that they will achieve mastery.

Definitely longer than 140 characters :)

Spring Break :)

I have been meaning to post for a few days, but I'm struggling with forming coherent thoughts right now... This post will be somewhat of a rambling one, so bear with me...

At school, some of the teachers have decided to get together for lunch once a week in order to discuss vertical alignment. This has caused me to pull out old Pre-AP materials from workshops, do some research online, and even do a few calculus problems. What I have learned is that at the time I took calculus in high school and college, I really was clueless about what was going on. It makes so much more sense now that I've taught geometry and algebra 2 and have the conceptual maturity to get it!

As an extension of that, I've really been feeling a need to do a better job in Algebra 2, specifically with making connections. So I am going to steal borrow from the wisdom of my amazing blogger and twitter friends to rework the exponential/logarithm chapter with more active learning techniques. That was part of my goal today and I have 1 lesson done out of 8.. woot! :)

While I definitely enjoy having a break and being able to sleep in, I really am not a fan of this time of the school year. For some reason, each year in March and April, I look back over the year and thing, gosh, I did a cruddy job :( Then I start my to-do list of things to do better next year and the cycle repeats itself....

I need a nap...