Thursday, June 28, 2007


I know it feels a little dead around here lately... I'm leaving the country for most of July and probably won't be posting. I'll be doing my best to not be thinking about math or school. But rest assured, once we hit the middle of August, I'll be back and posting regularly. There will be big changes in my Numeracy class that should be quite interesting to document, and I'm sure lots of other fun things to think about. See you then!

Friday, June 22, 2007

Common Denominator Division

I just finished helping facilitate a week-long workshop for most of the people teaching 5th - 9th grade math in two of our school districts. It was very interesting to be a part of, and to see what kinds of content and pedagogical knowledge teachers do (and don't) have. I'm not going to write much more about it, except I did want to post about common denominator division (CDD).

I brought this technique up to see if any teachers use it, and none had even heard of it. I've been teaching it to my students, as part of the scaffolding toward understanding the "multiply by the reciprocal" algorithm. Though not as efficient (typically), I think the CDD method can really help students see what is going on in fraction division, in terms of "the number of times the divisor fits into the dividend". It also fits nicely with multiplication, as "divide across" works just like "multiply across". The only problem is that "divide across" only comes out nicely when the denominator of the divisor is a factor of the denominator of the dividend, which is why (I assume) it is not the standard algorithm that is taught.

Once students know both methods, I give them a set of fraction division problems, and ask them to solve them with both methods. They then discuss which types of problems are better solved with which method, and why.

Here are some pictures that illustrate what is happening:

In this last one, you can see that the bottom shaded area fits into the top shaded area 1.5 times. This is the type of problem that I think really shows the benefit of CDD.

Here is a short proof that "divide across" works. Note: when the denominators are the same, dividing across yields 1 on the bottom, which I didn't write above.

Thursday, June 14, 2007

Last Day

Another year, come and gone. We had an end of the year assembly today that was super long - about 2.5 hours. There were lots of recognitions of student successes, good-byes to leaving teachers, the Numeracy Project played, I presented the winners of the Algebra 2 treasure hunt, and we saw the DCP mini-movie (once it gets published to the web, I'll link to it).

I gave back the finals to my Algebra 2 honors students - all of them passed the final but two, and everyone did end up passing the class. Here's how the course grades were distributed:

9|0 0 2 2 4
8|0 1 2 2 2 5 5 6 6 7
7|0 4 7 8 9

The student with the 106% is a math monster... She's gotten more than 100% on just about every single assessment I've given (she always gets the bonus question, and usually doesn't miss any regular credit). She is also one of the pair who found the treasure. When I gave out class awards, she won the "I'll know more math than Mr. Greene in a few years" award.

After the assembly, there was a pizza lunch (along with food tasting from our new food company for next year, Revolution Foods), music, dancing, soccer, yearbook signing, and so on. I did photo shoots with different students, informed kids that need to take summer school (I wanted to tell them first before they got a notice in the mail, and encourage them to look at it as a chance to improve their skills), and got a marriage proposal from the student I wrote about in my previous post! I declined, but told her I'd be happy to help her make a study plan next year instead. I got some notes of thanks, and the best student gift ever: some of my A2H boys called me over, clustered around me, and gave me a bird-cage shaped present wrapped in toilet paper and tape. They said it was in appreciation of all that I've done for them this year. It was a Transformer from the new movie coming out this summer (they remembered me talking about my love for Transformers all the way back to the functions unit, when we did transformations and translations). This year, like all years, has had its fair share of challenges and problems. But it was nice to end it like this - out in the sunshine, shaking hands and giving hugs, congratulating students, and wishing them a good summer.

Tomorrow, I'll finish cleaning out my classrooms and packing, and tomorrow evening is graduation. Then, the year will really be over. I can't believe that I've been here for 7 years now, but I don't know how I can ever tear myself away from these kids.

Next week, I'll be helping facilitate a math/pedagogy workshop for all the elementary and middle school teachers in a couple of local school districts. The education department at San Jose State got a large grant to put this together. I don't know how it will go, but it will be very interesting. I'll post what I can about it as it happens.

Tuesday, June 12, 2007

The Final

Tomorrow, my students are taking their Algebra 2 Honors final exam. It was hard, as always, to try to distill a year's worth of material into a test that is comprehensive yet fair, that can be completed in 2 hours.

I decided to give them 25 multiple choice questions that cover lots of the smaller topics, like dividing complex numbers, simplifying radicals, powers of i, associative/commutative properties, etc. These questions are worth 50 points, or 1/3 of the total. I expect students to finish them in 30 - 45 minutes.

The remaining time should be spent on the free-response, worth 100 points. To be as fair as possible, I gave students a sheet listing the content that would be tested in these 11 problems. Though it does not include all of the material we covered this year, I tried to pull out the topics we focused most heavily on.

  1. Place numbers in the correct locations on a Venn Diagram of the complex number system.
  2. Given 3 points on a parabola, find the function of the parabola in standard form. You must be able to write and solve a 3 x 3 system of equations to do this.
  3. Graph a piecewise function.
  4. Answer graphical analysis questions (given a graph, determine domain, range, values of x and f(x), find when f(x) <0, etc.)
  5. Solve a polynomial inequality with a number line model.
  6. Given a verbal situation, set up a model of an exponential function and then solve with logarithms. For example: the current value of my car is $12,000 and it is decreasing by 9% each year; how long will it take for my car to be worth only $9000?
  7. Solve a logarithmic equation using properties of logs, and eliminating extraneous solutions.
  8. Translate/transform a graph. For example, given the graph of f(x), draw the graph of y = 2f(x + 3) – 7.
  9. Simplify a rational function, indicating values excluded from the domain. Determine intercepts, holes, and asymptotes, and make a graph.
  10. Solve a quadratic equation (with the quadratic formula) that has imaginary solutions.
  11. Divide with polynomial (or synthetic) division.
Extra credit:
  1. Given a rational function, determine its inverse, and the domain and range of both.
  2. Prove the quadratic formula by completing the square.
This afternoon is our final review session, which I expect to be very well attended. Looking over that list of topics reminds me how proud I am of what these students have been able to accomplish this year - especially those students I was teaching fractions to last year.

Friday, June 08, 2007

Software assisted differentiation

Is there anyone out there with experience using software to help differentiate instruction for students in math?

A couple of programs I've looked into: Aleks and Agile Mind.

I am interested in seeing if I can use something like this to help support my Numeracy students. Of course the websites are filled with anecdotal success stories and even data to support their claims of success. Has anyone reading this tried to use something like this? I'd really like to hear your experiences if you have.

Brainslug asks a clarifying question in the comments, and the answer is long, so I'm posting it here...

Short answer:
"differentiating" is edu-babble for providing different students with different instruction and/or assessment, as opposed to teaching the same thing/same way to all students in the class. Software might help with this greatly in my Numeracy class.

Long answer:
We have all of our students who test below 7th grade take our Numeracy class concurrently with Algebra 1. The problem is that skills range from around 2nd to 7th grade levels. Some kids need to work on place value and subtraction, while others are ready to tackle fractions. We decided not to split the class up into two or more levels to avoid the pitfalls of tracking in a small school.

So, to help students most efficiently, they need to be provided with instruction where they are ready to learn. One solution can be to split kids into flexible groups within the class, where the different groups are working on different skills. However, aside from this being an exorbitant amount of planning time, our freshmen generally do not have the student skills needed to work independently for long periods of time, or the ability to learn from static worksheets without direct instruction and good coaching.

The software that I mentioned above assesses students' "knowledge space" and then only lets them work on the skills they are ready for. The software provides explanations, examples, feedback, and so on. It also allows you to easily provide individualized homework and assessments. In my ideal scenario, I'd set up the class as follows:

Each week (or so), a new skill in math is taught. At the beginning of the week, all students take a quick diagnostic. If they pass, they don't participate in the lesson: they use the time to instead work on the software, on whatever skills they are currently building - plus, maybe some other problem solving curriculum. If they don't pass, they spend half the class (which is an 80 minute block) working with me on the lesson as normal: direct instruction of conceptual and procedural understanding, manipulatives when appropriate, and pair/individual practice. They would then spend the second half of the class working on their individual objectives via the software.

If this works, it would allow class time to be much more efficient, as students would only ever be working on material that was needed, and at the appropriate level. Of course, this all hinges on the software being able to make good on its promises. I'm hoping that the software is both understandable enough and engaging enough that my students can actually learn from it. Computer based learning could be just like a glowing worksheet, or it could make good use of video, animation, interactive demonstrations, and so forth to really move students forward. So my question: has anyone tried this with students?

Thursday, June 07, 2007

The end is near...

Like many other people, I've felt the frustration that comes along with the run-up to finals. Students who know they don't have a chance at passing have checked out; lots of students are still on the fence, but you can't manage to get them all on the greener side. Teaching new material is fun and exciting; reviewing old material again and again is, well, not. Especially to those students who seem to think you are still teaching them new topics. But I know that this is all part of the game, and I am not too discouraged. Some students will fail, but they'll have another shot over the summer or next year. Ultimately, most will be fine. And, there have been lots of success stories this year too - those students who I wasn't sure had it in them yet, and yet they found it.

My algebra 2 honors class, though, has been and continues to be the best class I've ever taught. (I haven't told them that yet - when I do, I know I'm going to get lots of "awwwwwws!") They have been working hard this past week getting ready for the final, and I haven't even had to prepare anything for them. I decided to see what would happen if I just handed out blank copies of the old unit tests and midterms and let them work and ask questions. It's worked amazingly well, and I'm pretty sure that every single kid is going to not only pass the final, but actually pass the class. If every student passes the class, it may be a first in DCP history!

To celebrate their hard work, tomorrow I will stop class a bit early so I can thank each student, say a few words about them, and present them with an award that I whipped up in photoshop: