Friday, June 30, 2006

The end of education schools

Check out this post on Robert Talbert's Casting Out Nines.

He is referring to an essay by Peter Wood that predicts that education schools will be gone by 2036. It seems like there isn't a lot of support out there for the stats quo. I wonder if there will be much change in the near future?

I gave my own comments on the issue on Joanne Jacob's blog.

If you are interested in elementary math education, I recommend the book Knowing and Teaching Elementary Mathematics, by Liping Ma. It shows some very scary stuff about elementary school teachers' lack of mathematical knowledge. Most of the American elementary teachers they studied couldn't correctly divide fractions (let alone come up with appropriate story problems to illustrate the problems).

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.

Wednesday, June 28, 2006

Wanted: A Good Conceptual Structure for Algebra 2

In my note-taking post, I talked about structuring my summer Geometry class conceptually.

I am interested in doing the same thing for my Algebra 2 Honors class this fall. I want to figure out an overarching structure for the Algebra 2 standards. Here is my very initial idea for categories:

- Manipulating Expressions
(simplifying, factoring, polynomial operations, etc.)
- Solving Equations
(solving by factoring, quadratic formula, completing the square, finding roots of a graph, using logarithms, etc.)
- Working with Inequalities
(absolute value inequalities, polynomial inequalities, graphing inequalities, linear programming)
- Graphing Functions
(quadratics, higher order polynomials, rationals, logs and exponentials, etc.)
- The Number System
(sets of numbers, properties)

Any thoughts? If you have experience with Algebra 2 concepts, how would you categorize them? This definitely needs revision and I'm sure I've forgotten things. Also, there is the problem of topics that cross over sections. I am thinking about including some sort of cross-referencing strategy (but maybe that's too much). Good thing I have all summer to mull this one over.

So far, I've had one student tell me (unsolicited) that she really likes the use of the 3-column note taking system. That's not bad for only 3 days of use!

Tuesday, June 27, 2006

5-hour classes...

...didn't seem that tough at the time... I thought I was going to have a nice, easy summer. I don't know what I was thinking. Planning 5 hours of lecture, handouts, selected readings, sketchpad labs, quizzes, and so on is pretty demanding. So far, it's taken me about 4 - 5 hours of planning a day. So, instead of leaving school at 5 like I was hoping, I'm there till 7:30 or 8. I'm thinking that I will get more efficient at it as we get farther into the summer - well, here's hoping anyway. But it's going pretty well - the students have actually commented on the fact that the day goes by relatively quickly.

The note taking system seems to be working pretty well so far. As the daily quizzes start happening (we've only had one so far), I'll have more of a sense as to whether they are able to process and retain the vast amount of material I am trying to teach them.

We finally got a computer lab at school, and it's pretty cool. I am able to present on the projector while they work, and with Remote Desktop, I can take control of their machines and even send them to the projector. Today, I sent a student's screen to the projector so he could show the class how he solved a problem (figuring out how to construct a pair of complementary angles). He talked it through from his seat, using his mouse as a pointer - then, as I walked around, I saw other students begin to copy his technique and run with the problem. Technology is pretty nice when you have it, and it works!

The nice thing about having a 5 hour class is that we can work on the same content in multiple ways during the same day - from guided exploration on sketchpad to lecture / reading to practice problems, followed up with a formative assessment the following morning. I'm hoping that this reinforcement from multiple ways of presenting information will help them absorb it.

Monday, June 26, 2006

Yours is not to reason why...

...just invert and multiply!

This seems to be one of the fundamental philosophical questions in math education. Do you teach tricks and rules, or do you let students explore and construct their own knowledge? Does it matter what type of students you are working with? Behaviorist, Constructivist, Socio-Constructivist?

I, personally, believe in Socio-Constructivism. The idea is that the teacher creates a well-structured pathway for learning that takes students through the levels of understanding. Students start off being given basic facts or information that they will need, and then are given time to work (individually or collaboratively) on expoloration or inquiry based activities. Through this process, they begin to generate conjectures and create a quasi-mathematical understanding. Then, through class discussion and direct instruction, the teacher helps correct misunderstandings and formalize the knowledge (i.e. algorithms, processes, etc.) The drawback to this, of course, is that it takes a lot more time - both to plan the materials, and actual class time. Is it worth it? That's the real question. I believe it is, but I think there are those who disagree.

One of the classs that I have developed is called Numeracy, in which we put freshmen who test below 7th grade level when they arrive (this is typically 70 - 80% of the class). We start off at basic operations and place value concepts - this takes the entire first semester. The second semester is all fractions. We spend a few weeks working with fraction circles, drawing fraction bars, using reasoning, etc. to compare, order, and evaluate fractions. We work on determining if a fraction is closer to 0, 1/2, or 1 whole. We figure out how you can compare 7/8 and 8/9 by reasoning. Then we spend 6 weeks adding and subtracting, with manipulatives, with pictures, and finally, with the algorithm. Then there are 6 weeks dedicated to multiplication and a few more for division. Yet at the end of all this, I still have a large number of students who haven't learned to work with fractions fluently. Sometimes, there is the temptation to teach the rules and then practice them to death, but in my heart I believe this won't work. Plus, if you blindly memorize the "flip and multiply" rule, you haven't really learned anything about division and you won't be able to apply that knowledge to other situations (i.e. Algebra). Also, if you develop no context for your algorithm, you have no way of knowing if what you are doing makes any sense.

I think students have been trained to want algorithms in math. They resist exploration. A noisy class will quiet down and get to work when a worksheet is put in front of them - why? Even if it is not being graded, they will rush to try and get answers down, regardless of whether they are learning or not. Like lemmings, they seem compelled to "finish the worksheet". It boggles the mind! "Mr. Greene, just tell us the easy way! Stop asking us questions!" How many times a day do I hear that in Numeracy? Students know that drawing fraction circles, for example, will help them solve a problem, but they would rather ask me for help, or just skip the question. When I force them to draw a picture, they begrudgingly do so, then look at their picture and say, "Oh, that's all you want us to do? That's easy!" and proceed to answer the question with little difficulty. Yet, on the next question, the process will repeat itself. Patience is definitely a learned skill!

I am happy to say that, at the end of Numeracy, most of the students have learned that, when adding or subtracting fractions, you don't add across!

Any thoughts on the matter?

Sunday, June 25, 2006

Which Geometry texts do you like?

In trying to get up to speed on Geometry teaching, I have been reviewing quite a few different textbooks. Each one, of course, has its good and its bad, and this is relative to the audience of the book (I believe). One reader emailed me to say that, in the course of supplementing her children's math education, came to detest the book that was being used (Key Curriculum Press: Discovering Geometry) and recommends Geometry, 2nd Edition by Harold Jacobs for its logical development, clear definitions, and foundations built on postulates. These are actually two of the books that I have been looking at, and I see some really good things in both of them.

We actually have a couple of shelves filled with the Harold Jacobs book which we purchased early on, and then found out that our students were unable to have much success with it. We are going to be reevaluating our Geometry curriculum for the following year, including testing out some different texts. Keep in mind that we are beholden to the state standards and STAR testing just like any public school... In all our decisions, we have to struggle between what is best for our students' mathematical development and access to college level work, and the spectre of the STAR test and API rankings. believe it or not, these do not always coincide!

If you teach (or have taught) Geometry, what texts are you using? Why do you like/dislike them? What type of student population are you working with?

Summer Geometry

Tomorrow starts our very first summer Geometry course.

All of our incoming students take Algebra 1 as freshmen (even though many have "passed" already in 8th grade, the majority don't know anything about Algebra). For our students who are better or more interested in math, this poses a problem, as it does not give them enough time to get to Calculus by their senior year.

Our original sequence was the standard Algebra 1, Geometry, Algebra 2, Precalculus. To get to Calculus as seniors, motivated students were able to take Algebra 2 as an intensive, 5-week course after their sophomore year, and Precalculus as juniors. I taught the Calculus class for the first two years, and it was extremely difficult, because the students were definitely underprepared in their algebra skills, and much of our time was spent relearning Algebra concepts.

This year, we have switched the sequence to (what we think is a more logical) Algebra 1, Algebra 2, Geometry, Precalculus. The summer class then becomes Geometry. This gives students an entire year to work on their Algebra 2 skills instead of a ridiculously short 5 weeks. Of course, the same thing will happen in Geometry, but our feeling is that there are significantly fewer skills in Geometry that are needed to be successful at higher level math. It's hard to make these sacrifices, but we have to do so all the time.

So, that being said, tomorrow I will start teaching the summer Geometry class. I've never taught Geometry at all, and I have to figure out how to boil down the essentials into 5 weeks of daily 5-hour classes. (Whatever else happens, I must say I'm impressed with the students who have elected to spend their vacation taking this class! We talk about ganas a lot at DCP, and these kids really exemplify that ideal!)

What successes / failures have you had (or heard of) in trying to bring low skilled, underserved, or underachieving students to higher levels of math?

How do you have students take and use notes?

This is a huge problem at our school! Our students have never learned how to effectively take notes in class, and then how to actually use them for completing homework and studying.

When I was in high school (and even college), all I did was write down what was on the board and important things said by the teacher, and it was clear to me how to use these notes to study from. I don't know how or when I learned to do this - and, since I don't remember learning how to make sense of this, I don't have a good idea of how to teach it.

The "just write it down and study it" method does not work at all for most of my students. I can get them to write things down pretty well, but I have had no success with getting them to actually keep and use these notes effectively. My students end up having binders stuffed with notes and handouts, but whenever they need to find something, they start at the beginning and flip randomly through it until they either find it (rare) or give up (frequent).

This coming year, a colleague of mine and I have decided to have students organize their binders conceptually instead of chronologically. He will do this for Geometry and I will do this for Algebra 2. Our idea is that, at the beginning of the year, we will come up with a conceptual structure for the entire year, and students will be required to keep their binders structured that way. For example, Geometry may have the following categories: Points, Lines, Planes, and Angles; Triangles; Quadrilaterals; General Polygons; Circles and Spheres; Logic and Proof; Synthesis. These will be the sections in their binders. Additionally, there will be a summary sheet at the beginning divided into these same categories. At the end of each lesson, students will be asked to file their notes in the appropriate section (this will be scaffolded away as the year progresses), and to make an entry on their summary sheets. Homework and other handouts will also be filed in the appropriate section, next to any relevant notes.

We think that this will help students access their notes much more effectively - which will encourage them to actually use them for studying! For example, if they see a problem with a diagram of a right triangle and a missing side length, they may not know what information they need, or when that information was taught (do we even remember when we taught what?) but they will know to look in the "triangle" section. Then, if their notes are complete, it should be a lot easier to find something useful.

Any thoughts on this? Has anyone tried this? How do you get students to effectively keep and use their notes?

Saturday, June 24, 2006


Hi! I have been teaching high school math for the last 6 years at Downtown College Prep, a charter school in San Jose, CA. Our students are primarily Latino, are far below grade level in their math and reading skills, and will be the first in their families to go to college. We refer to our students as being on an exponential learning curve: the average level in math of our incoming freshmen is 5th grade, and we need to get them to a 12th grade level in 4 short years.

Everything that I know about teaching math comes from what I have learned over the years by reading, experimenting, and collaborating with colleagues. I search the web a lot for ideas, but it's hard to find good, consolidated material that targets this population. Every so often, I find a good article in the NCTM magazine or a book that I can adapt, which provides a springboard for a great new unit - but I know there are a ton of successful and innovative ideas and strategies out there. My hope in starting this blog is to try to start a forum in which math teachers can collaborate and share their ideas for creatively and effectively teaching specific concepts and structuring their courses.

I plan on posting strategies that I am trying; but more importantly, I plan on posting questions I have, and am looking forward to the comments and discussion that will follow. If you have a question or topic that you want feedback on, just email me and I'll post it for comments. Also, please forward the address of this blog to any math teachers that you know so the ideas can multiply.