Back to work

It's been a lovely summer, but now it is time to get back to work. I will be teaching Algebra 2 this year (4 sections) and Algebra 1 (1 section). So, most of my lesson posts are going to be about Algebra 2.

I have been working with H. on creating a skills mastery assessment system. She has posted a lot about it, so I won't bother repeating it. You can catch up here.

I just found out about box.net and I will be posting my files there as I go this year. There is a handy widget on the blog now (if you are reading this by RSS), and you can even subscribe to that by RSS. Feel free to look at and use anything I post, but of course nothing may be published/sold/turned in to your ed school instructor without my permission.

I describe how my system will work in one of the files, but here it is if you don't want to bother downloading it:

Each skill test will be given at least twice – the scores are added together. The purpose of this is to promote retention of concepts. If students receive a perfect score on the second administration of the test, the first score will be raised to a perfect as well. Students are able to retake these tests as many times as they want before the end of the semester, though they can’t take the same test more than once per day. The questions on the skills tests are all single topic items that reflect typical STAR questions.

Homework will be graded. To make this possible, students will be called on to grade their own work. This will require trust and buy-in on the students’ part – but, the weight is limited to 10% to limit temptations for cheating. Each class, students will be shown the answers to the problems, and five to ten problems will be selected for grading. Each problem is either right or wrong, no partial credit. However, students may do corrections to the incorrect problems and turn them in the following class to earn their points back.

Thus, a full 60% of the points are “recoverable”, and students who put in sufficient effort should be able to earn the full amount of points. A student who earns all of the recoverable points needs only a 25% average on comprehensive tests and the final in order to earn a 70% in the class. This way, the class is passable by lower ability students who are willing to work hard, and these students will hopefully be better prepared to take the STAR test. Students who want to earn a B or an A must also do well on the comprehensive tests; this allows me to create tests that are more challenging and focus on analysis and synthesis problems.

In the Unit Skills Lists, skills that are marked CE will be tested only on a comprehensive exam. These are skills that are not required by the benchmarks/released questions, but are key to progressing to higher level math.

This system is still in its development phase, so any feedback you may have would be much appreciated.

Next Lesson: Operations on Complex Numbers

Tomorrow, the Do Now will introduce the idea of finding the distance to the origin on a coordinate plane by creating a right triangle and using the Pythagorean Theorem. Students must calculate the shortest distance from a given point back to the origin, where they left their sweeties, and therefore want to hurry back to.

Following this, I will do direct instruction on addition, subtraction, multiplication, and division. Multiplying by the complex conjugate for division will be a nice follow up to simplifying fractions like 2/(3 + root2) from the previous lessons. Then, we will recall what was learned during the Do Now to understand how to calculate the absolute value of a complex number. Students will plot complex numbers in the complex plane, and then draw a right triangle and calculate the length of the hypotenuse. I will ask students to figure out other complex numbers that have an equal absolute value, and to determine, given a set of complex numbers, which one is farthest from the origin.

Finally, students will have about 20 minutes to work on these types of problems in pairs and ask for help. Not a very exciting lesson, to be sure, but, for some reason, complex numbers and their operations seem to be hit pretty heavily on the STAR test, so it needs a lot of class time. I don't think students will be asked to find the absolute value of a complex number, but it is good scaffolding for the distance formula, which is an indispensable tool in any high school mathematician's bag of tricks.

The lesson will be posted on ILoveMath.

Standards Coverage

In a comment in the previous post, Darren says:

I find the Algebra 2 standards to be fairly "aggressive", like drinking water from a fire hose. How you find time to add in topics that are not in the standards, I don't know but would be most interested in learning.

(I think it is worth starting a new post to answer this... I'm interested, as always, to hear people's thoughts.)

Well, you're right. We are not able to adequately cover all the standards as they are written, especially since our students start so far behind the curve. What we've done is to strand out the standards over the 4 years. We leave some things out of Algebra 1, and push them into Algebra 2. Some of the Algebra 2 standards we then move into Pre-Calc. And some, we decide not to do at all.

When deciding what to cover (and in what depth) in Algebra 2 Honors, I look not just at the list of state standards, but at the blueprint for the standardized test (i.e. what are the key standards that comprise the bulk of the test), at what will be covered in Geometry and Pre-Calc, and at what will be most useful for students moving in a trajectory toward Calculus.

I know we can't do it all, so my goal is to balance the required coverage with a strong scaffolding for success in Calculus - all with an eye to who our target student is.

For example, though I think conic sections is a great topic, a thorough unit would take my students many lessons to master - yet there is only one single question on the test. So we leave it for Pre-Calc.

The same is true for combinations, permutations, probability and stats, mathematical induction, series - all together these topics comprise just about 20% of the test. I choose to focus on the rest, and go deeper by including relevant scaffolding, math analysis components, word problems, and so forth.

With our students, I believe this actually yields higher test scores than covering everything more shallowly would.

Update:
There is discussion of this going on at Darren's site if you are interested.

NCTM standards vs. California Algebra standards?

As a relatively new teacher, it's hard for me to really know what to believe. In my math methods class, I was taught the socio-constructivist philosophy. I accepted it not blindly, but because I had already been teaching for a couple of years (I took afternoon/evening classes over a period of 2 years while teaching on my intern credential), and the ideas really resonated with me, based on what I saw from my students. One of the units I developed in the class was the slope unit that I mentioned in an earlier post, and when I implemented that unit, it worked much better with my students than anything I had previously tried. We read a lot of research and several case studies, and there seemed to be a lot of evidence supporting the benefits of this approach. Plus, it just made a lot of sense to me.

This philosophy is different than what has been described as the purely constructivist, or inquiry-based learning that many people seem to abhor. And it is quite different than the behaviorist (teacher as source of all knowledge) philosohpy.

I have read many posts and articles from mathematicians who are opposed to the NCTM and its beliefs. But here is a quote from their standards document pertaining to high school level:

Because students' interests and aspirations may change during and after high school, their mathematics education should guarantee access to a broad spectrum of career and educational options. They should experience the interplay of algebra, geometry, statistics, probability, and discrete mathematics. They need to understand the fundamental mathematical concepts of function and relation, invariance, and transformation. They should be adept at visualizing, describing, and analyzing situations in mathematical terms. And they need to be able to justify and prove mathematically based ideas.

This seems reasonable to me, and I wonder if those of you who take issue with the NCTM's beliefs could comment specifically on what the concerns are. Is it the content of their standards, or the way they are implemented, or something else?

I am bound by the CA content standards and STAR testing. I feel that most of the standards are things that a student taking algebra, for example, should know how to do. But there seems to be very little emphasis on problem solving, critical thinking, and application. And the STAR tests themselves do not really assess these things, only the fundamental "tools" of algebra. As teachers know, if it is not assessed, it will not get done - especially when teachers and schools are under the gun to raise test scores. So is the plan that students will master the tools of math in high school, and then will somehow be able to become problem solvers in college? And what about the students who don't go to college?

What thoughts do people have on the CA standards (or, if you are familiar with another state's standards)? Are they a subset of a good math education? Do they mesh with the NCTM standards at all? Are the state standards counterproductive? (As for the STAR testing, that will probably be a good topic for a later posting).

And finally, why is there such hostility over these issues? What I've read seems more like political partisanship, and less like people trying to collaboratively build a consensus as to how best to teach our country's students.