Things Change, but Organization is Still Key

At the school I am now at, the students do not have a strong mathematics background.  When I say that, I do not speak of their basic computation or their ability to manipulated algebraically, I refer to the ability to think conceptually.

This is something that can be fixed; however, this causes a big problem when it comes to Calculus and Precalculus.  Precalculus has a ton of conceptual mathematics early on in the course, including parent graphs, transformation, limits, and end behavior.  AP Calculus starts off with a huge amount of limits, which is a very strange concept for students who are weak in conceptually.

Luckily, organization can be a miracle worker!  Currently, my precalculus students are organizing information and working with laboratory activities.  With Calculus, I decided to go back to the source.  The year began, instead of with numerical and graphical approximations of limits, with infinitesimal and algebraic methods of taking the derivative.  I know that infinitesimal are not part of the AP curriculum, but for my students, it seems to have worked well.  They can take the derivative of anything I through at them, and they do not even know what a limit is (as their precalculus teacher last year told me, before she retired at the end of last year, that she did not cover anything about limits).  I think this is an exciting find for my students, they understand how to take a derivative and that the derivative stands for the slope of a line and have gained a lot of confidence in their abilities.  After their first test tomorrow, I will enter into basic integration.

The point is this, I have organized my courses to be easy for students to gain confidence while still being rigorous in the amount of material covered and pacing of the subject.  I am also being strict on my students organization, which I will talk about in my next post.

Exam Design

It is almost spring break, which for me means designing my exams for the courses that do not have a state or national exam.  Over the years, my exam format has changed, but I now believe that I have a strong, concise, format for exams.

The first consideration is exam length.  Yes, my exam has to be long enough to take an amount of time close to that provided by the school.  That being said, I also need an exam length that I am willing to grade.  What I have established is the follow.

• 30 Multiple choice questions
• 5 Short answer questions
• 5 Free Response questions

Yes, this is only a 40 question exam, which is short; however, it isn't the amount of questions you have, it is the quality of the questions that counts.

The thirty multiple choice questions comprise 50% of the exam grade.  These questions are basic computational or recall problems.  Each question has 4 to 5 possible answer choices, and yes, answers that would be obtained if a student made a common error are some of the choices.  Though these are the most basic of questions, it takes real thought to answer them correctly.

The five short answer questions are conceptual in nature and comprise 20% of the exam grade.  Students have to explain, in two to three sentences, the concept being discussed.  My favorite type of short answer question is error analysis, where I give to sample sets of work and the student must identify which one is the correct method.  Another popular style of question involves applications to mathematics; for example, I show students a graph discussing a companies profit, then ask about the significance of the slope or the y-intercept of the graph.

The five free response questions consist of multiple parts and comprise 30% of the exam grade (for those keeping count, this should be 100%).  This is the part of the exam where students combine their computational and conceptual skills.  Each part of each question builds off of the previous part, but if a student gets part (a) incorrect, they can still earn the points in part (b) if they use their answer from part (a) correctly.  For these questions, a strong, but consistent rubric is designed that requires them to show their work in each question.

Prior to the exam, I tell students how many questions are on the exam, what percentage each group of questions count, and the topics that will be covered.  I do not, however, conduct a formal in class review.  I can get away with this "lack of compassion" (as one parent described it) because all of my test are cumulative, therefore I should not have to review extensively in class if every student has completed their corrections from previous test.  I do make myself available before and after school, meaning that the two weeks before the exam I arrive at 7 AM sharp and leave a 5-5:30 PM every day, regardless of if a student has come for a review.

One final note, the first year I formatted my exam like this, one student looked at the number of questions and said he would be done in twenty minutes.  One hour and forty-seven minutes after the exam started he turned it in and said "I have never had an exam that short that involved so much thinking."  Remember, quality is far more important that quantity.

Goals for Next Year

Here are my plans for this summer.

1. Write all exams for all courses
2. Write all test for all courses
3. Create/update pacing guides
4. Create/update unit plans
5. Fix nuclear physics and differential equations lessons

This list is daunting, but manageable.  Also, if I get through all of this in the summer, then next year will be so much easier on me.

I find it amazing how the last part of the year in teaching seems to always be focused on the next.

Geometry... well this is Different

For the first time since my student teaching, I will be teaching geometry.  Now I am very knowledgeable about geometry, I understand the subject, and I am very competent when it comes to working with geometric principles; however, I do not enjoy geometry.  I am a strong believer in the principle that if you do not enjoy a topic, then you will not be as good of a teacher of that subject.  That being said, I do not have a choice, I will be teaching geometry because one of my colleagues will no longer be with the school.

So now I need to get back into the mindset of geometry.  For me, this means spending time this summer focusing on planning my geometry course.  I say my geometry course because, as I have stated before, I do not teach kids to the test, I teach the kids mathematics.  Yes, I will cover all of the topics and standards covered in state standards, but I will cover it my way.

I believe I will start the year off with constructions.  If the students can construct any geometric object, I think this will give them a better understanding of the information covered in the course.  This would mean a lot of up front work and a hard road as far as pacing is concerned, but if they can build the situation, then they can work with the situation more affectively.

The Importance of Foundation

When I went into teaching seven years ago, I did not know (even with the education degree) exactly what it meant to be a teacher.  I am hoping that I will never truly know what it is to be a teacher, but I do like having a better idea about it at this time.

As a math teacher, I am really an architect, building manager, construction worker, and psychotherapist.  As an architect, I design my lessons, assessments, tutoring sessions, and the like with a clear goal in mind; that goal is NOT having students pass an End of Course Exam.  Instead, that goal is to get the students to the point that they can perform well in a mathematical environment.  If they can do this, then any test should be a piece of cake to pass.

I am a building manager because I oversee everything that is done throughout the mathematics department (and science department for that matter).  At my current school, the person who teaches the highest level of the subject is the "department head;" since I teach AP Calculus AB and AP Physics B, I am technically the head of both the math and science department.  As such, I oversee all of the other math and science teachers, making sure that the overall design is kept in mind as teachers do their jobs.

I am a construction worker because I sit in the trenches everyday and slowly build the knowledge of students and help them hone there skills.  The key to this, in my experience, is providing students with a solid foundation to build themselves up from.  For example, I refuse to teach the ever popular FOIL method for multiplying binomial expressions because I view it as a crutch that makes it more difficult for students to have success when they reach multiplying true polynomials.  The stronger the foundation, the less chance that the house will crumble in a storm.

Finally, I am a psychotherapist because I continually council students out of their fear of mathematics.  As the title of this blog indicates, mathematics is art.  It is a wonderfully beautiful collage of logic, problem solving, poetry (yes, I said poetry), and images.  I am happy to say that many of my students have seen the beauty of mathematics.

In short, I am a teacher... and I am always looking to improve.

Test-Correct-Recall.... for the mathematics teacher.

First off, I will not take credit for creating this method.  I learned this method from Greg Jacobs, a Physics teacher and all around great person from Woodberry Forrest School.

The first thing you have to make sure you have for this method is a strong rubric set up for all questions.  Most of my rubrics are based off of AP rubrics for Calculus AB and Physics B; meaning they are measuring specific skills, including the ability to explain mathematics and to set up the appropriate equations and situations.

After the test is created and all rubrics written, the students take the test.  The test is graded strictly on the rubric.  The important thing about the rubric is students can only loose a point once, meaning if they make a mistake early on, but do all of the mathematics correctly from their on out, they only are penalized the one point they missed (even if the final answer does not match up).

After the test, all of the students to corrections.  I give them a simplified rubric so they can see where they missed the points, then I give them some short questions to answer based on the original test questions.  In order to earn credit for their corrections, then they have to correct the original question and answer the shorter questions given to them.

Finally, a few days after the corrections are completed, I give a brief recall quiz based on the test.  This quiz focuses on the fundamentals and less on the calculations, and acts basically as a solidifier to commonly missed concepts.

Curve Sketching in AP Calculus

In the past, I have taught strictly the curriculum when it comes to curve sketching.  The AP exam has deemed that students should be able to look at a graph and identify aspects about the derivative or the original function based off of the given graph.

This year, I decided that instead of just looking at graphs the entire time and identifying information from the graph that I would go through the entire process of curve sketching.  So far, the students seem to be reacting well to this set up.  We have spent the past few days creating tables to allow us to graph the original function by identifying aspect of that function from the first and second derivatives.  Tomorrow I get to see the payoff, as we will be looking only at graphs and identifying (and graphing) straight from the given graph.

Lets see if this works!!