Today, my 7th graders were dealing with a problem that eventually they will solve by setting up a system of equations. Right now, they don't have that in the tool kit, so guess-and-check would be the strategy of choice. Here is the problem and our first step:

One student was able to come up with the equation: 1.50 a + 2.00 d = 360, but another was quick to point out that there would be many different solutions, so we couldn't use it...yet. So the first guess is 20 and we recognize that the total is too high. Usually that indicates that the first guess is too high, so we normally go with a lower second guess. However, this time, Aaron pointed out that it didn't make sense to decrease the guess on advanced tickets because they are cheaper. In fact we want to increase that one. This is right about the time I almost lost my poker face:

Aaron: "So if we add to the advanced tickets our total will actually go down."

"Why?"

"Because for every $1.50 we add, we lose $2.00. So every time we change the tickets by one, we drop $.50. Since we need to drop $30, we need to sell 60 more tickets in advance."

At this point, I almost blew it. Nearly jumped right in and said something stupid like, "That's right Aaron. You guys get it?" To which they would have all nodded "uh-huh" and we would have moved on. But I caught myself, gave him the "eh, I am not sure about that" and removed myself from the conversation. So Aaron had to try to re-explain to his classmates what he was saying. I could tell that a couple of them got it, but many were still perplexed. After a couple of minutes I asked Jose if he could explain what Aaron was saying. Jose nailed it and a bunch of kids have an "a-ha" moment. Good stuff.

Later I asked Jose why he didn't speak up a little sooner. He said that he didn't figure that he needed to ask any questions because he understood it. I asked for a show of hands on how many understood after Jose's explanation and that's when about 15 hands shoot up.

We have been talking a lot in my classes about how important it is to join the conversation. Some kids still think that it's just about them. They don't realize that if they offer something to the conversation, not only do they benefit from explaining something they already understand, but there is no telling how many other kids benefit too.

Today, I think Jose gets it.

## Thursday, October 29, 2009

## Monday, October 26, 2009

### Our Grading System

**Standards Based Instruction**

We decided a few years ago that the sequence of our texts don't really work for us. As a result, we began teaching one standard at a time. We have taken our state standards set up a sequence and pace in a way that makes sense to us. Some of the standards we have broken down into specific skills and these skills are what make up our gradebook.

**Common Formative Assessment**

Once we have done our initial classroom instruction, we give a common multiple choice assessment. This assessment is graded on a 1-4 rubric. A students next step is based on how well they do on the assessment which is a pretty fair cross-section of the different problems a student may be expect to do on the given standard. We determine the initial score based on the percentage of problems they get correct. 85-100% = 4, 70-84% = 3, 55-69% = 2 and below 55% = 1.

**Less Than Proficient**

The level of understand a student demonstrates determines what happens next. If a student scores a 1 or 2, he will do a series of activities that may include defining basic terms and demonstrating pre-requisite skills. Once finished with these activities, the student will then take the problems that were missed on the CFA and not only correct them, but explain what went wrong (verbally or in writing) and how to work the problem correctly. Students who score a 3 on the initial assessment are just required to make the necessary corrections with explanation. Once the corrections have been made, the student is then given a re-assessment and the new score replaces the old score in the gradebook.

**Proficient**

Although students may get 100% of the problems correct on the initial assessment, we still give them a 4. Our reasoning is that the multiple choice test doesn't allow our students to demonstrate understanding that goes beyond classroom instruction, but it does allow them to show proficient understanding. Students who demonstrate proficient understanding on the assessment are given the opportunity to turn the 4 into a 5 by choosing from two categories of activities. Examples of activities may be creating a mini lesson, peer tutoring or some other project agreed upon by teacher and student. The second activity is some sort of writing assignment that may require the student to explain the process or describe what skills a student may need in order to be successful with this standard. Once a student has completed these activities and has shown the ability to explain his work the score in the gradebook will be turned into a 5.

**Strengths**

At this point, I can say that the strengths of this system are:

- Students grade is based on what they understand and not a mere accumulation of points.
- Students are allowed to re-assess and new understanding replaces old understanding in the gradebook.
- Students seem to understand where they are having trouble and what skills they need to remediate.
- Learning has become a conversation between teacher and student because in order to re-assess, student needs to articulate previous misunderstanding and current understanding.
- Students have a choice on when to re-assess. They can work at their own pace.
- Students also have some choice on which activities to do in order to demonstrate understanding.
- The dross has burned off the grade. Students' grades are based on what they understand rather than things like effort, homework or extra credit.

**Weaknesses**

- Students have to take more ownership of their learning which means they have to un-learn some bad habits. Not sure that it is a weakness in our system specifically or an indictment of the educational system in general.
- Teachers are having to re-think classroom management when students are working on different activities.
- We are having to decide if some of our standards actually lend themselves to "advanced" work or if later standards are the advanced version of some previous standards.
- Need to develop more advanced activities for students to do while working towards a 5. We allow for students to create their own activity as long as it has been agreed upon by the teacher, however many students don't know what to do with that kind of freedom.

**Questions**

- What's the best way to take a series of 1's, 2's, 3's, 4's and 5's that are based on levels of understanding and turn them into a letter grade? Do you use mean, median or mode?
- If we go with some sort of average from 1-5, what percentage do you use for an A, B, C, D or F? Currently we are going with average where > 4.5 = A, > 4.0 = B, > 3.0 = C, > 2.0 = D and <2.0 = F.

**Adaptation for the Advanced Class**

Because my classes are advanced, I have to adapt this system to suit my students' needs. Basically, I am using the standards as the "basic skills" for my class. I have a posted a series of mathcasts and study guides on each standard and the students are expected to view the online examples and do the problems in the study guide prior to taking the pretest. If a student scores above 90% on the initial assessment, there is no other work to be done on the standard--they receive a 5. Students who score below 90%, need to correct their errors, explain to me that they understand why they made their mistakes and how to fix them. Once I am convinced that they have truly corrected their errors, I give them a second assessment and the new score replaces the old one.

My reason for allowing students to earn a 5 right off the bat is that 90% of our classwork is problem solving that uses the standards as the jumping off point. Students in my 8th grade class receive two grades. They are all enrolled in an algebra class as well as a geometry class. We treat the algebra class as the "grade level" basic skills class and the geometry class is the "advanced" class. The geometry we are doing is analytical so students are having to use the algebra at a much higher level...so I'm ok with not making them jump through hoops in the algebra class.

**Note:**My department is awesome. I truly loved my math department at my previous school and it was really tough to leave them. However, I couldn't imagine working with a group of teachers more willing to try new things. We have pretty good discussions in our department meetings and there is plenty push back. But at the end of the day, we are all trying to find the best way to educate our students.

## Sunday, October 25, 2009

### We've Had It All Wrong

There have been quite a few blog posts lately discussing how one would explain to a "non-math" person why a negative times a negative is a positive. I think the reason we have such a hard time explaining it is because it's not true. We need look no further than California's own Nancy Pelosi for the explanation. (Check right around 3:50)

In an interview with CNBC's Maria Bartiromo, Pelosi was asked if the expiration of the Bush tax cuts would amount to a tax increase.

Her response:

"It's not a tax increase. It is eliminating a tax decrease that was there."

So there you have it--eliminating a decrease does

In an interview with CNBC's Maria Bartiromo, Pelosi was asked if the expiration of the Bush tax cuts would amount to a tax increase.

Her response:

"It's not a tax increase. It is eliminating a tax decrease that was there."

So there you have it--eliminating a decrease does

*not*amount to an increase. I had better go fix those lesson plans. Thanks for clearing things up, Ms. Pelosi.## Wednesday, October 14, 2009

### Calculator : Arithmetic :: GeoGebra: ?

To tell you the truth, I don't really have a problem with my state's math standards (here and here). I do, however, have a serious problem when the standards become the target rather than the scope through which we aim at the target. So what's the target? What should be the point of math education today? It has become very clear to me that it has never been easier to find correct answers to anything rooted in computation. With WolframAlpha, GeoGebra and all the other resources available, there probably isn't a question we could ask a student where they couldn't quickly look up an answer. When I first started teaching, the big question was whether or not we should let our algebra students use a calculator. A lot of the "veteran" teachers were dead set against it because they "gotta add, subtract, multiply and divide, for cryin' out loud." But the calculator would allow a student to speed up all the calculations (read arithmetic) and get to the "math." Well can't the same be said for WolframAlpha or GeoGebra? Don't these tools give students access to certain problems where previously they would have been bogged down by calculations they couldn't complete?

I remember when basic skills were being able to perform the four operations over the Real Number system. What's a basic skill look like today? Is algebra the arithmetic of the 21st century?

I remember when basic skills were being able to perform the four operations over the Real Number system. What's a basic skill look like today? Is algebra the arithmetic of the 21st century?

## Monday, October 5, 2009

### Who Gets It?

Here's the problem:

An airplane is flying at 36,000 feet directly above Lincoln, Nebraska. A little later the plane is flying at 28,000 feet directly above Des Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of descent, predict how far from Des Moines the airplane will be when it lands. [1]

[1] Problem courtesy of Phillips Exeter Academy. Hat Tip: Alison Blank

An airplane is flying at 36,000 feet directly above Lincoln, Nebraska. A little later the plane is flying at 28,000 feet directly above Des Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of descent, predict how far from Des Moines the airplane will be when it lands. [1]

**Question:**Which student demonstrates better understanding? Why?## Student A

## Student B

## Student C

## Student D

[1] Problem courtesy of Phillips Exeter Academy. Hat Tip: Alison Blank

## Saturday, October 3, 2009

### Intro To Problem Solving

A while back, Justin Tolentino had a post asking how others might go about teaching problem solving strategies. Great question. I have started and scrapped a couple of responses to his post as well as a posts of my own on this topic. I think the easiest way for me to describe it is in the question I gave my 7th graders this morning.

A man lives on the 10th floor of a building. Each time he leaves his building, he will take the elevator from the 10th floor to the 1st floor. However, when he returns, he will take the elevator to the 7th floor and walk the remaining three flights to his apartment. Why does he do this?

I am sure that many of you have heard this lateral thinking puzzle before and may wonder how it belongs in a math class. I realize that this problem has nothing to do with math, but in my opinion, neither does problem solving. We use math as a vehicle to teach students how to take information they are given and then discern that which they can use and that which they must refuse. From there they can ask questions to gain any new information they need in order to solve the problem.

The only requirements I give students the first time I ask them one of these puzzles is that they ask "yes or no" questions. I also tell them that my answer to their questions will either be "yes, no or irrelevant." Then I turn them loose.

They start firing random questions like crazy.

"Is he afraid of heights?"

"Does the elevator work?"

"Does he need the exercise?"

These questions come in all shapes and sizes and many of them are very specific. After about 10 questions, I tell them I'm only going to give them 21 questions and I make a mark on the board after each question. At first, it doesn't deter them. They keep at it, often times repeating a question that was already asked. Then they get to about 15 and someone suggests that they slow down a bit and start thinking about what to ask next. Today, we got to 21 with no resolution and I was about to walk away from it when one of the kids who knew the answer asked if he could ask a question.

Sure.

"Does he

"Yes he does. And if you guys would have asked this question in the beginning, it would have kept you from having to waste some of your other questions."

This brings us to a great discussion on how we can look at a problem and ask general questions that eliminate the need to ask other more specific questions. As we carve out large chunks of potential questions, we begin to narrow our focus and become more specific.

I really like what I see when I present these puzzles to my students. Kids who won't normally offer much in a class discussion, will often times ask really good thoughtful questions. They feel safe to do so because the given information is so limited, there is no way to feel "stupid" for not knowing the answer. In fact, the entire process assumes that no one knows the answer.

This leads me to ask: How can we get students comfortable with what they

A man lives on the 10th floor of a building. Each time he leaves his building, he will take the elevator from the 10th floor to the 1st floor. However, when he returns, he will take the elevator to the 7th floor and walk the remaining three flights to his apartment. Why does he do this?

I am sure that many of you have heard this lateral thinking puzzle before and may wonder how it belongs in a math class. I realize that this problem has nothing to do with math, but in my opinion, neither does problem solving. We use math as a vehicle to teach students how to take information they are given and then discern that which they can use and that which they must refuse. From there they can ask questions to gain any new information they need in order to solve the problem.

The only requirements I give students the first time I ask them one of these puzzles is that they ask "yes or no" questions. I also tell them that my answer to their questions will either be "yes, no or irrelevant." Then I turn them loose.

They start firing random questions like crazy.

"Is he afraid of heights?"

"Does the elevator work?"

"Does he need the exercise?"

These questions come in all shapes and sizes and many of them are very specific. After about 10 questions, I tell them I'm only going to give them 21 questions and I make a mark on the board after each question. At first, it doesn't deter them. They keep at it, often times repeating a question that was already asked. Then they get to about 15 and someone suggests that they slow down a bit and start thinking about what to ask next. Today, we got to 21 with no resolution and I was about to walk away from it when one of the kids who knew the answer asked if he could ask a question.

Sure.

"Does he

*have*to walk the remaining three flights?""Yes he does. And if you guys would have asked this question in the beginning, it would have kept you from having to waste some of your other questions."

This brings us to a great discussion on how we can look at a problem and ask general questions that eliminate the need to ask other more specific questions. As we carve out large chunks of potential questions, we begin to narrow our focus and become more specific.

I really like what I see when I present these puzzles to my students. Kids who won't normally offer much in a class discussion, will often times ask really good thoughtful questions. They feel safe to do so because the given information is so limited, there is no way to feel "stupid" for not knowing the answer. In fact, the entire process assumes that no one knows the answer.

This leads me to ask: How can we get students comfortable with what they

*don't*know? How do we convince them that being educated isn't about knowing all the answers; it's about asking the right questions?## Thursday, October 1, 2009

### Thoughts I Have While Brushing My Teeth

### What Are You Looking At?

Today I gave my classes a survey as a way to gain some feedback on how the first quarter has gone. One of the questions was "What would make you more comfortable asking questions in class?"

Here is the response that really pushed back:

WOW! I had never really thought of that. Yeah, I guess if I am burning a whole through a kid with my gaze while I am answering a question, it may just make them think twice about asking another one. I don't think I do that, but perception is reality to these kids. So if she says I do it, I guess I do. Need to keep a watch out for that one.

Where do you look when you are answering a question from a student?

Here is the response that really pushed back:

Well, this may seem silly and childish, but you want the truth, right?

Well, when a student asks a question, you seem to direct your answer to the person who asked it, which makes me feel uncomfortabe. I mean, if other people don't understand, then why only talk to one person, instead of the whole class? It makes me feel weird, like I'm the only one who doesn't understand, and the teacher looking at one single student seems to cause everyone to look, making the student even MORE uncomfortable. As I read over this, I feel I want to delete it, because it seems so silly and unnecessary of mentioning. I won't delete it, I guess, because I suppose you want to know this, no matter how silly it (mine) is.

WOW! I had never really thought of that. Yeah, I guess if I am burning a whole through a kid with my gaze while I am answering a question, it may just make them think twice about asking another one. I don't think I do that, but perception is reality to these kids. So if she says I do it, I guess I do. Need to keep a watch out for that one.

Where do you look when you are answering a question from a student?

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