Sep 162012

Where to start.

I have been thinking and working with the Exeter materials quite a bit in the last 3 months. I have come to see the value in the methods and the questions, and the way the questions cycle from lower levels to higher levels.

But I have to say I don’t see the Exeter curriculum as a magic bullet. It isn’t. There is no such thing as a magic bullet for math education. There is a lot of hard work. There are a lot of relationships to build with learners. There are many hours to put into lessons that engage learners to think deeper about the mathematical issues.

The Exeter Curriculum is a part of this process, not the end of this process. It is not something that will solve any problems. It is however, something that will help me, as a math teacher trying to improve my classroom, to engage learners, to develop deeper thinking, and to push the high standards of the Common Core into classrooms.

I am not confident of the efforts offered by the textbook publishers. Here are two examples of why:

If the CCSS is going to actually impact the classroom in a positive manner, we can’t take the same ol’ same ol’ materials and just slap on a new label. We need to structurally change and improve what we are doing.

That is where the Exeter Curriculum can come into play and help, and it creates the next problem I, as a public school teacher have. And this goes back to the first post I made, Exeter we have a problem. I had flashbacks of Apollo 13 as I wrote it because it is relevant. As the quote goes, “Houston, we have a problem” and the problem was absolutely centered in that little capsule. The experts who developed the program were on the ground and could go home safe and sound at the end of the day, but those astronauts needed to step out of their comfort zone and do something above and beyond.

As a public school teacher, I am in the same capsule. Our comfort zone has been stripped away and completely new standards pushed on us. We need to step up, or step out. It really does come down to that. The old guard who doesn’t want to change will be forced out through the new “evaluation” procedures that also have been forced down our throats by people who have no clue about education.

Okay, so the stage is set. Nothing I wrote above will change. Stop complaining.  What the heck am I going to do about it.

The Plan (or WCWDWT):

As part of our evaluation process I had to create a Professional Growth Plan. The plan I proposed and was approved was to take the Math 1 Exeter Curriculum and align it with the Common Core State Standards as well as simultaneously give the problems keywords and strands.

In addition, I have spoken with the two very nice and enthusiastic gentlemen from who have the technical background to take the entire project, upload it to their website, and host the problem sets, alignment, stranding, keywords, AND make it all searchable, selectable and downloadable for FREE (and that is free as in air).

What Can We Do With This? We can create a database of problems that are rich. We can create a database of problems aligned to the CCSS that are searchable, selectable and downloadable for use in the classroom by math teachers around the world.

What can we do with it then? That hasn’t been explored. We have to create the foundation before we can build the building. I have spoken with someone at Exeter and they are interested in the project. Of course, they can not help much. It isn’t their burden to take on, it is ours (and now mine!).

I have another teacher at my school who has agreed to take on this with me. She is absolutely crazy to do so, which means I am completely insane.

Feb 212012

I tried something very new in AP Stats this year. Okay, it may not be all that new, but it was new for me. Last year when teaching confidence intervals, I taught it as it shows in the book, first 1 prop z, then 2 prop z, then inference testing, then 1 sample t, then 2 sample t.

I ended up with a class that saw confidence intervals as 4 separate things, and never once (except for those few exceptional learners) connected the dots to see that all 4 intervals, 5 actually, because you can have 1 prop z and 1 sample z, were all the same, exact idea separated only by what kind of data you have.

This year, while working with a colleague in another state (thank you @druinok and your blog) I learned that while the curriculum to AP Stats is pretty set, the creativity to teach it better comes from me. So, I changed it up. Last year, my problem was that the learners did not see the intervals as the same thing.

This year, I decided to teach all 4 intervals at the same time. Slight exaggeration. I taught the 1 prop z interval, and the conditions for it, and how to interpret, and how to do them. Then, I offhandedly mentioned, “and you know, there are other types of intervals we will get to as well.” In the restaurant business, that is called planting the seed. English teachers call it foreshadowing. I call it darn good stuff.

The reaction from the learners was immediate. “What are they?” “Are they different?” “How are they different?” were some of the immediate questions. So I went into them. Next I covered the 1 sample t interval. Here are the conditions, here is where you use them, etc.

They were hooked. The class, as a whole immediately saw that the intervals really weren’t all that different. Next, I made a worksheet with 12 problems. Several from each type, but purposefully NOT 3 of each type. Actually, 3 of the 1 prop/sample z, 2 of the 2 prop z, 5 of the 1 sample t, and 2 of the 2 sample t. The learners cut out the 12 problems, and had to sort them. There were a couple of purposefully tricky examples, like:

Suppose the average height of randomly sampled 100 male students at University of Reno is 67.45 inches with a standard deviation of 2.93 inches. Find a 95% confidence interval estimating the height.

The class put this in the “t interval” category at first, and that categorization would probably not be wrong on an AP test. It fits better in the “z” category though. Why? This was tricky because it doesn’t say we actually DID take a sample of 100. It says “Suppose ….” Yup, this is Mr. Waddell being a jerk and trying to trick the learners. But they got that. It was the only question worded that way on purpose.

At the end of class, the learners had 4 stacks of problems. I worked 1 problem all the way out using PANIC (Parameter of interest, Assumptions check, Name the interval do the math, Interval in correct notation, Conclusion in context). They had to pick 1 from each stack and do the problem.

They left class in good spirits with some very complex problems. They left feeling like they understood something important. I walked away chalking the last 3 days up as a success. Now it is just up to them to recall it.

Jan 212012

Really, I found a use for the boxes of old scantrons I have in storage! I didn’t think of it myself, though. It came from here.

Provide each student with a scantron sheet and ask them to guess which would be the correct answer to the first question, if you were unable to see the question.. Talk about the percentage of the students in the class which would have guessed correctly. Extend this to two questions and so on. You can then talk about the math behind probability.

In AP Statistics, as well as the Algebra 3 class I teach, we do binomial probability. It is very abstract, and difficult material. For the Alg 3 course, we incorporate Pascal’s Triangle, as well as the combinatorials, and it becomes a very interesting lesson. But there are still always learners who can never figure out what I mean when I ask:

Now suppose you are taking a 12-question multiple choice test with four possible answers to each question. You need to pass the test in order to get ‘un-grounded’ for that crap you pulled last weekend.

  1. If you are totally guessing on every question, what is the probability that you will get a passing grade of at least 60% on the test?
  2. Suppose all those minutes of studying pay off and you are able to eliminate one answer for each question. Now what is the probability that you will pass the test and get to go out on Friday?

Yes, this is a real question I ask in Alg 3. The fact that you need 8 out of 12 questions right really stumps the learner, because they try to use that instead of the (.25)(.75) probability required.

But if they actually had a scantron in front of them, would they do better? I don’t know, having taught the material in Alg3 several weeks ago (before Christmas break). But my AP Stats class will be doing the binomial distribution this week. I am going to try it as an introduction. We can then extend the formula to the normal model afterwards.

I will report back what I find out. [which means that one of my goals this year is more active blogging and sharing. I have said that before, but never give up!]

Aug 042011

I have done a very poor job of writing about advanced algebra, the course I helped co-author 4 years ago with 5 other teachers in my district. I would like to rectify that this year, and explain more about the course and honestly, get better ideas for the course.

The course is more project based, and it has four distinct, but overlapping sections. Quarter 1 is financial math, quarter 2 is math in art, quarter 3 is math in technology and quarter 4 is math in health / human body. If you go to the site you will see the basic structure and some of the lessons / sites I use to teach the course.

Starting off this year, I am going to do something different with the quarter 1. It always felt a little disjointed to me. We use the materials from NEFE to get us started, and then we jump in much much deeper than NEFE goes. We spend a lot of time on spreadsheets (which are nothing more than giant algebra problems using variables) polynomial equations and rational equations (another way to think of Pert and other compounding equations, ie. purchasing a car and annuities) as well as some basic ideas of personal finance.

This year, instead of just teaching each module as a standalone, I am going to tie all of the quarter 1 together with this outline.

At the end of this project, you will need to show the Banker your portfolio that demonstrates you have the financial knowledge and ability to purchase a house in the North Valleys.

Part 1: Setting S.M.A.R.T. Goals

1. Short term, medium term, & long term goals

2. Tracking your goals

3. Adjusting and rewriting your goals

Part 2: Savings & Compound Interest

1. Calculating compounding interest for n

2. Calculating compounding interest for continuously

3. Finding rate or time in both kinds of compounding (working backwards)

4. Knowing and demonstrating differences between APR and interest rate

Part 3: Career & Income

1. Identify 3 careers for yourself

2. Calculate lifetime earnings

3. Calculate $1,000,000 earnings timeline

4. Comparing 2 and 3 for your careers

Part 4: Purchasing a Car

1. Calculating your payment

2. Deciding on years of repayment

3. Comparing years to payment and making a good choice

Part 5: Investing & Credit

1. Risk vs. Reward

2. “Safe” vs. “Intermediate” vs. “Risky” investments

3. Annuities

4. Credit Cards

5. Credit Scores

Part 6: Budgeting your spending and savings

1. Creating a budget (will be working on all quarter)

2. Projecting income

3. Projecting expenses

Part 7: Buying your home

1. Put it all together for the Banker, and using the financial information gathered to justify to the banker that you are a good loan candidate

That’s right. The end goal and purpose of the portfolio will be to purchase a house. It is the biggest investment a person generally makes, but I also know of 4 learners who graduated within the last 3 years who are now homeowners. It is a reality they can achieve now, whereas 3 years ago it was out of reach.

The purpose of this structure is to create a buy-in. Now they see the end goal. This goes hand in hand with backwards design and Understanding by Design principles. I have emailed out a draft to the other Advanced Algebra teachers, and once I have it more fleshed out I will email out another copy. I also hope to get some feedback from them to see what they would add as well.

So, what do you think? Is this a viable way to put together a quarter on personal finance? Let me know in the comments.

Jun 222011

When I last left this topic, I had a rather different arrangement of the Essential Understandings based on a theme of graphing, algebraic arithmetic and solving. We took this to our department head, and had a discussion with her about these. Her very valid concern with that arrangement was that the learners may not see the connections between quadratics and graphing, factoring, completing the square, etc. and polynomials and graphing, etc.

So, we have re-arranged our essential understandings by “theme” of type of function.

Below you will find our new arrangement for the Alg 2 first semester. The “Keys to Success” is a series of brochures and self check mastery, and the parenthetical numbers are the book section numbers. The book is not the driving force on this though, the district’s blueprint is. We won’t get the final version of that until July sometime.

The goal is to have the learners connect the graph, the algebra, and everything else we do inside the them together. We will be doing essentially the same mathematics 4 times, and then I need to make the connections clear and consistent through the classwork and assignments.

Now that we have an list of essential understandings, we need to go through and decide what demonstration and transfer of skills will be considered as evidence for the learner understanding the understanding. That is the next step.

Keys to Success (chap 1 & 2: reviews of alg 1)

  1. Integers
  2. Expressions
  3. Evaluate
  4. Solve
  5. Slope
  6. Graph Lines
  7. Equations – Lines
  8. Exponents
  9. Factoring
  10. Parent Functions

theme: Linear functions

  • Can you graph more than one equation on the same graph? (3.1)
  • Can you explain where the multiple equations intersect or not intersect, and what that means? (3.1)
  • Can you solve a system of equations with substitution? (3.2)
  • Can you solve a system of equations with elimination? (3.2)
  • Can you graph two or more inequalities on the same graph? (3.30
  • Can you shade correctly the two or more regions indicated by the inequality? (3.3)
  • Can you add and subtract matrices? (3.5)
  • Can you do scalar multiplication with matrices? (3.5)
  • Can you find the determinant of a 2×2 matrix or larger matrix with and without technology? (3.6, 3.7)
  • Can you multiply two matrices together with and without technology? (3.6, 3.7)
  • Can you use inverse matrices to solve linear systems with more than 2 equations? (3.4 & 3.8)

theme: Quadratic Functions

  • Can you graph a quadratic function, labeling the values of the vertex, axis of symmetry, and the minimum or maximum & solutions or zeros? (4.1 & 4.2)
  • Can you solve quadratic equations by factoring where a = 0?  (4.3 & 4.4)
  • Can you solve quadratic equations by factoring where a  0? (4.3 & 4.4)
  • Can you solve quadratic equations using square roots? (4.5)
  • Can you compare and contrast the different methods of solving quadratics? (4.5)
  • Can you complete the square for a = 1? (4.7)
  • Can you complete the square for a  1? (4.7)
  • Can you add and subtract complex numbers? (4.6)
  • Can you multiply and divide complex numbers? (4.6)
  • Can you graph complex numbers? (4.6)
  • Can you use the quadratic formula to solve quadratic equations? (4.8)
  • Can you use the discriminant to determine if the roots are real or complex? (4.8)
  • (rethink in July)
  • Can you use quadratics in real world situations? (4.10)

theme: Polynomial Functions

  • Can you describe the end state, rise and fall, max and min, and zeros of a polynomial function? (5.2)
  • Can you evaluate a polynomial function given a value? (5.2)
  • Can you add and subtract polynomial functions? (5.3)
  • Can you multiply polynomial functions? (5.3)
  • Can you factor perfect cube trinomials? (5.4)
  • Can you factor non perfect cubes by grouping? (5.4)
  • Can you divide polynomials through standard long division? (5.5)
  • Can you divide polynomials through synthetic division? (5.5)