Questions?

On Teaching By Learning

2012-02-01T20:56:00.000-08:00

One of the things that I have learned over the years is to let go of any preconceptions I have about how a problem should be solved. I have methods I prefer, but my students need to develop their own. Never has this been more obvious than it was today.

We are beginning to play around with non-linear functions and so I gave my class the following problem:

You're going to build a garden and need to build a fence around it. If you have 120' of fencing, how would you set it up in order to have the biggest garden?

...or something like that.

I didn't specify rectangle although I figured most kids would default to that. I didn't mention the barn in the back of the property that could be used as one of the sides. I just kept the prompt as simple as possible so I could see where they took it.

Many assumed a square right off the bat. And one group felt pretty ambitious and looked up some shapes in their school planner only to settle on the typical decagon. (Good luck calculating the area of that one, fellas.)

So, I'm walking around and seeing groups all proud of themselves by defining x as one side and y as the other and they're arguing about whether 2x + 2y = 120 or x + y = 60 is the better equation. I ask a few questions like, "so how will that equation help you determine the largest garden" to which they reply, "we'll find the maximum."

"Mmm-kay," I say as I mutter to myself, "I'll be back..."

Then I walk up to a group of three boys who usually push the envelope when it comes to creative problem solving and I see the equation x + y = 60. I think to myself, "oh, no not you too."

Then I look a little closer and I see this other thing they're working on.

xy > 900

I double take.

"Wait, what?"

Before any of them even address my incredulous look, one of them says, "let's go put it in GeoGebra."

"So how'd you come up with that?"

"Well we figured that x + y = 60 tells us how much fencing we have. And since a square would give us 900 ft², we want to know if there is an area out there that's greater than 900."

"Uh, yeah. That's, uh, yeah, that's exactly how I'd do it."

Not really. This is what I really said.

These kids used a system. Not in a million years would I have ever considered using a system to solve this problem and three 13 year-olds set me straight.

Man, I love this job.

Something's Wrong Here

2012-01-25T20:45:00.000-08:00

Search Institute _[1] has compiled a list of 40 Developmental Assets that may protect youth from high-risk behaviors while promoting positive attitudes and behaviors. I don't know how valid the research is, but I found the list to be interesting and the data to be fascinating.

Each of the 40 assets remain the same for each developmental stage while the application of assets change based on the age of the child. They're worth a look.

Early Childhood (ages 3 to 5)

Grades K-3 (ages 5 to 9)

Middle Childhood (ages 8 to 12)

Adolescents (ages 12 to 18)

I took a particular interest as I have a son in each of these developmental stages. Suffice it to say, I have some reading to do.

When I first saw this chart, I thought, "Man, I gotta get my kids 31 of these assets."

But then, when I saw this chart I thought, "Holy, crap. We're doing something really wrong."

Figure it out in the comments, will ya?

_[1] I have no idea if this is even a credible organization. I learned about this today at a meeting I attended and it was the one thing that didn't make me want to stab myself in the eye with a pencil.

All I Really Need to Know About Teaching I Learned From...

2011-12-16T13:30:00.000-08:00

Mike Krzyzewski:

“The truth is that many people set rules to keep from making decisions. Not me. I don’t want to be a manager or a dictator. I want to be a leader—and leadership is ongoing, adjustable, flexible, and dynamic. As such, leaders have to maintain a certain amount of discretion.”

Look, it doesn't matter if we are talking about lesson design, assessment (formative/summative or whateverative), reporting, feedback or any other thing you can get yourself riled up about. The bottom line is that leaders--decision makers-- will find a way to be successful. They'll find a way to be successful because they realize that anything worth doing is about relationships. And relationships are dynamic. Relationships are messy. They're frustrating and they sure as heck don't come in a box. If you're looking for a program, system or formula to guarantee success for yourself or your students, STOP! It doesn't exist!

That's all.

Iron Sharpens Iron

2011-12-05T19:51:00.001-08:00

Based on the feedback I've received in the comments and on Twitter, I have an updated version of this applet. Linda has also created a screencast that will probably prove to be way more cogent than my attempt at describing the process.

Thanks to @jk_herbert, @mrhodotnet, @MrPicc112, @mrautomatic and @mathhombre for the suggestions on Twitter.

Updated version addresses the following:

Writing equation given slope and y-intercept places less cognitive demand on a student than writing the equation based on the graph. Levels 3 and 4 have been switched.
Points and answers can't be changed after answer has been submitted.
A running total has been added so student and teacher can view overall performance.
Correct answer shown once answer is submitted giving immediate feedback to students on all levels.

There were also a couple of suggestions that were already embedded in the applet. There is a "reset" and "go to next level" button that can be accessed in the object properties. Double clicking on the applet should open it in a separate window which will allow you to save as well as make any changes you'd like.

Updated Applet

Update (12/6/11): Here is a version for student practice that includes both the "Reset" and "Next Level" buttons.

GeoGebra: Leveled Applets

2011-12-03T23:28:00.001-08:00

This stuff is crazy. We can actually make leveled applets that allow students to move on only after they've been successful with the previous level. I saw this applet the other day and was blown away. The applet itself is pretty simple, but the fact that it requires students to complete a specified number of exercises perfectly before moving on is the part that really interests me. The problem is that the thing is in German and there are a bunch of unnecessary steps. So, looking through the construction protocol proved to be fruitless. I'm pretty sure the guy who built it is way smarter than I am, so I'll try to simplify this the best I can.

Keeping track of student success pretty much requires three things.

True or False

Conditions must be set to determine whether the student's answer agrees with the target answer. This part made my head hurt. Having different levels made setting the conditions tough at first, but once I got a feel for what I was doing, the work started to flow.

Let's take a look at my level 1 problem.

In order for a level 1 problem to be considered correct, two conditions had to be met:

1. The line graphed by the student (h) had to be the same as the line generated by the applet (e).

2. The "Check Answer" button had to be clicked. The button was tied to boolean value g.

I entered the conditions for each problem type's correctness into the GGB spreadsheet and this what was entered into cell C2:

=If[e ≟ h ∧ g, true, false]

Each subsequent cell was used for the next level. (ie. C2 -> Level 1, C3-> Level 2, etc.)

Each individual condition for correctness was tied to a global correct boolean value named AnswerCorrect.

The condition for AnswerCorrect to be true is below.

If[C2 ≟ true ∧ ActualLevel ≟ 1 ∨ C3 ≟ true ∧ ActualLevel ≟ 2 ∨ C4 ≟ true ∧ ActualLevel ≟ 3 ∨ C5 ≟ true ∧ ActualLevel ≟ 4 ∨ C6 ≟ true ∧ ActualLevel ≟ 5 ∨ C7 ≟ true ∧ ActualLevel ≟ 6 ∨ C8 ≟ true ∧ ActualLevel ≟ 7, true, false]

The blue text represents the condition for a Level 1 problem.

Buttons

The AnswerCorrect and AnswerWrong booleans were tied to two buttons: ButAnswerCorrect and ButAnswerWrong. These show up with the basic condition under the advanced tab.

Scripts

This is where the magic happens. I'm still learning how to use the scripts, but this is where the levels advance, construction is reset and a new problem is generated. Both buttons have scripts, but the ButAnswerCorrect button is the most complex. These scripts can be used as a template for future applets. This is a good thing because there is no way I could create this on my own.

The applet I created is here. Double click the applet to open it in a GeoGebra window. You can then save it and play around with making your own.

I'd really appreciate feedback on this. If you have any questions, leave them in the comments and I'll do my best to answer them.

Big thanks to Linda for helping me weed through the junk on this.

The Timeline of Awesome

2011-12-01T10:44:00.000-08:00

Friday August 12, 2011

Kate poses a great problem.

Thursday, November 17, 2011

Dan asks a great question.

To which I responded something like, "yeah, prolly, but it'd take a bunch of brute force."

Saturday, November 19, 2011

I forward it to the GeoGebra Forum.

Sunday, November 20, 2011

Raymond responds.

This flow of information absolutely amazes me. I mean, I loved the question after Kate posted it. In fact, I immediately created an applet and had used the problem with my advanced class early in the first quarter. They struggled a bit with it, but then when Dan asked about highlighting the squares and doing some of the counting, things changed.

I consider myself to be a little better than average when using GeoGebra, but Raymond is a freaking Jedi. Take a look at his stuff. He takes an applet that I thought would require a number of tedious steps and bangs it out using 6 steps--and within 24 hours. That's ridiculous.

The applet is here.

Mixed Up Mixture Problems

2011-11-28T19:56:00.001-08:00

A former student of mine (and future math teacher) just posted this problem on Facebook:

Soybean meal is 16% protein and cornmeal is 8% protein. How many lbs. of each should be added to get a 320 lb mix that is 14% protein?

I've never been a big fan of 8th grade students having to work through mixture problems, but maybe that has to do with the way I've taught it.

Every year I would come up with a new way to encourage students to set up equations to solve these problems but we'd always end up with some variation of this:

Let x = lb of cornmeal

Let y = lb of soybean meal

x + y = 320

.08x + .16y = .14(320)

Solve for x and y.

And for my advanced classes, that was fine. They already knew how to solve systems of equations and it just became another jumpable (yeah, that's a new word. Deal with it.) hoop.

It never set well with me.

This year, it kinda pisses me off. I've got kids who are able to think, but this kind of abstraction just kills them. It may seem intuitive to freaks like us, but for most kids, it's just a ridiculous application of an already ridiculous skill.

Enter proportions.

We live in the agricultural capital of the world. People mix stuff here all the time--and they don't use systems of equations to do it. They use common-freakin'-sense.

Mira.

Step 1

Step 2

Step 3

Step 4

Step 5
Sit back and relax while the other ~~losers~~ diligent students are using systems of equations to solve this ~~disaster~~ challenging problem that includes not only rigor but relevance.

I find this not only easier for students to do, but it appeals to a skill (proportional reasoning) that they may actually use 10 years from now as opposed to a skill (systems of equations) that they will use for about however long it takes to pass the test.

Question
How do I introduce this so that it appeals to my students' intuition in a way that keeps it from being just another trick they learn?

Applet is available for download or online use here.

This is Gonna Hurt

2011-10-20T18:36:00.000-07:00

These kids aren't dumb. But they think they are. What's dumb is the way math has been done to them. I didn't think so before. I do now.

I made a commitment to do a few things differently this year.

Consistent Format
I've created a lesson template in SmartNotebook that is really quite simple. Each file contains lessons one week at a time (even though I may only lesson plan one day at a time). Files are named Week1, Week2, etc and slides are named Day1, Day2, etc. Each day contains a slide for the Opener, Number Talk, Lesson for the Day, Homework Assignment and Exit Slip. Any handouts, links or any other resources I need for the week are attached to the notebook file. I've linked the previous day's homework slide to the current day's opener slide--mostly because I have a tendency to forget to talk about homework. Another cool feature I've started to utilize is the ability to link to these resources from within the actual Smart slides. Now, I can access the web, applet or any other resource with just a click on the slide.

This has been really good for me because it's allowed me to really focus on the content of the activity which ends up paying dividends for the students as well. This makes the design a non issue as I'm just replacing old with new and everything has it's place. Next year it will be nice as well because my resources will all be aggregated.

Opener
I've always been a believer in having something for students to do at the start of the period but having things structured has helped me be more focused on my daily first impression. These problems usually contain a review problem, some sort of pattern recognition problem and a maybe random fact involving numbers that I re-word so students have to make an educated guess.

Number Talk
I read about these in What's Math Got To Do With It and it immediately clicked. I've been wanting to help kids with their basic numeracy and tried to deal with it as it came up. Having a daily Number Talk has embedded it into my lessons. This has paid off greatly in the first few weeks of the semester. Metacognition is one of the bonus words in the faculty meeting bingo game and it gets thrown around like LMNOP in a pre-school class. We think we know what it means, but really have no idea. Metacognition isn't something you can talk about, it's something you have to be about. (h/t @jybuell) A Number Talk is all about Metacognition. I'm sure there could me more done with it than what I'm doing, but so far, so good. It's a travesty that so many kids haven't learned to decompose and recompose numbers. It's almost like I've had to give them permission to do so. And there's a positive correlation between kids who struggle and kids who don't know how to break numbers apart and put them back together. These kids are what we've begun to call "Stackers." They want to mentally stack the numbers they are adding or multiplying and work the standard algorithms mentally which is the most difficulty way to do mental math.

This activity has "win" written all over it. Not only do kids have to think about their thinking, but they get a chance to discuss it with others in a pretty non-threatening way. I can already see the culture of the classroom changing. I still have kids who are resistant because they are painfully afraid to share their thinking, but more are jumping on board than not.

Context, Context, Context
I've made a conscious effort to introduce everything via some sort of problem solving context. These contexts are sometimes application problems but, more often than not, they have been in the form of some sort of problem that gives students the opportunity to identify a pattern and use multiple representations to describe the pattern. I started this very early because I want them to realize that words, pictures, tables, equations and graphs are just different ways to tell the same story. I'm finding that kids who have scored very low on standardized tests have been very successful in taking a pattern and generalizing as well as graphing with not much trouble at all (which just further demonstrates that it's not the kids failing math, it's the math failing the kids). I've also found that it's much easier to differentiate when you have students engaged in an activity than it is when you're in the middle of a lecture--news flash, right? I'll go into more detail about the actual activities I've had students do in a future post. But, suffice it to say, I'm convinced that these kids can do math.

Things That Need Work
I've been spoiled over the past five years. The kids I've had come through my class have been extremely mature and hard working. I would still have kids who would lose focus and say and do things they shouldn't, but the overall direction of the class has always been pretty easy to maintain. But again, that mostly has to do with the fact that the advanced kids usually have parents up in their grill about anything less than an A.

First quarter just ended and, while we are still having our ups and downs, I can see some adjustments I need to make.

I have to be more mindful of the timing of things. I'm still having trouble determining how long an activity should take. As a result, I'm either running out of time or expecting things to go longer than they actually do. It's a different type of planning for sure.

I can't expect all these kids to be mathematicians. Yet. Yesterday, I had one of my advanced kids discover the point-slope form of a linear equation. All I did was give the class a point and a slope and asked them to figure out a way to show me the equation. It was beautiful. And messy. And it was all his. In fact, instead of using (x₁,y₁), we used (c, a) for the given point for no other reason than those were the kid's initials. I can't just drop a problem on my other two classes and get out of the way. I can however, offer opportunities for them to engage in math in real ways.

It's not their fault. I'm all about kids taking responsibility, but I have to remember that they have ~~been forced~~ ~~been trained~~ learned to wait for directions. I have to help them unlearn this learned helplessness. But I also have to be gentle when pointing it out. I'm not very subtle at times and I think that's done more to hurt the progress of some kids than help.

I have to find a balance between context and abstraction. Even my struggling kids can take a context and have a decent discussion on how the graph helps tell the story. But if I give them the exact same equation to graph without any context, I get a bunch of I've-never-seen-this-before looks. This isn't a matter of not being able to do it; it's just a matter of recognizing it when there isn't a context.

I'm probably not going to fix all these things before the year's out, but that's ok...there's always tomorrow.

The Un-Lecture

2011-10-05T15:13:00.000-07:00

Pretest

3x - 5y = 15

x intercept: ______

y intercept: ______

Results

0% correct

The Non-Khan Academy Un-Lecture Prompt

Me: Tell me what to do.

Them: Put the red dot on ____ and the blue dot on _____.

Gentle Feedback for Misconceptions

Encouragement

Scaffolding

And Finally

Hit the refresh icon and Repeat

Exit Slip

7x - 3y = 21

x intercept: ______

y intercept: ______

Results

90% correct

The applet. (includes original .ggb file as well as jar files.)

Time to Stretch

2011-09-23T18:36:00.000-07:00

The thing that convinced me to leave the high school classroom was the chance to work with a bunch of precocious pre-teens and follow them through middle school, hopefully sending them off to the big bad world of high school a little better off than they were when we met. This opportunity, coupled with meeting y'all has created a professional development explosion. It was a perfect storm of honestly asking the question: "how can I make this matter?", a group of ~~math~~ educators willing to push back on ideas while simultaneously offering unconditional support, an administrator willing to let go of the leash and a group of kids who constantly push me to be better.

The tough part has been the assumed disqualification when it comes to conversations about pedagogy because the stuff I do only works for "my kids." Let's just say, I feel Shawn on this front. This year, the gloves have come off. I have two completely heterogeneous classes with skills ranging from Advanced to Far Below Basic (see Jason for explanation) and 2/3 of the kids are less than proficient based on previous years' scores. So, basically, this year I have to put up or shut up. And some things have to change.

Let me be clear: I'm not changing my expectations; I still believe that all kids can do math. But my planning has to change.

In my experience, "advanced" kids fall into one of two categories: Advanced duplicators and advanced thinkers. Advanced duplicators are the kids who take copious notes, ask if "this is going in the grade book", want to retake tests minutes after turning the original test in and will do absolutely anything the teacher asks. They are compliant. Advanced thinkers will often be the kid who gets labeled as lazy and distracted because, well, they are lazy and have distracted. Problem is, they aren't lazy, the lesson just sucked. We ride the backs of these advanced duplicators because they are good for test scores. They make up for everything that a teacher lacks because they don't necessarily need a teacher to learn skill duplication (see: Khan Academy) and they are willing to do it because, well, that's what good students do, right?

I've spend the better part of that past five years trying to find ways to have students explore, invent and discover while maintaining fidelity to our state standards. My focus has always been on pushing kids beyond--but I've ended up learning as much from them as they have from me. My goals have been singular in that my planning has been framed by the question "how far can we go with this?" I've really learned a lot about exploring the right side of the bell shaped curve. Keep going until maybe only three kids get it. Forget Madeline Hunter, I'm using the Daniel Tosh lesson planning model.

Now it's time to look at the other side of the curve.

Time to Breathe

2011-09-21T19:47:00.000-07:00

So, I planned on writing a really nice reflective post on the past school year which turned into a based-on-last-year's-experience-here's-what-Imma-do-next-year piece which became a Start 'o the School Year Here's What's Happening that morphed into a Dude, Do You Still Have a Blog? kind of thing.

I missed the deadlines. But I have a note. Ask Alex.

Last year, the district bought my afternoons so I could get out there and work with other teachers in the most organic way possible. It really looked like an opportunity to be part of something pretty cool in that there really wasn't a detailed description other than "let's try to get teachers talking about best practices." The flexibility was what drew me to this but it also scared the heck out of me. The problem was that I had "technology" attached to my name and began fielding questions like "how do I turn this thing on?" Not what I signed up for. But, all in all, the conversations I had with teachers and administrators were very good. Based on the feedback they received, the district office wanted to keep it going this year, but my site took a beating in the afternoons. Class sizes were high and test scores were low due to the fact that the sections I'd normally teach were just absorbed by the other teachers.

I learned a lot about myself and the profession last year. I learned that I love to talk about teaching and really trying to figure out the best way to help kids understand this thing we call math. I also learned that many other teachers don't like to talk about it quite so much. But I can't really blame them. The system we are in has made many teachers feel like Big Brother is looking over their shoulder. This creates quite a dilemma in that innovation is not a chance many are willing to take. We've created a system where teachers want to be told what to do so that they are covered if it doesn't work. Tough to grow in that soil, let me tell you.

I learned that I'm not really interested in being a "tech guy" although I do believe that technology has it's place and if it can be a conversation starter, then I'm for it. Some of the things I saw in the classrooms I visited showed me that elementary teachers work their butts off. I don't see how those people do it. But it was also apparent that we need vertical articulation. We have to have a common vision K-12 with respect to how we have our kids do math. We have pacing guides, benchmarks, common formative assessments and even a standards based report card, but there's really not much continuity between how kids approach mathematics from elementary to middle to high school. I think that needs to change. We can spend all the time we want on unpacking standards and developing assessments, but somehow, it has to affect what happens in the classroom. We have to go beyond using the ancillary materials that come with a textbook adoption and start teaching our kids to do math. Now, if that job opens up, I'm interested.

Even though there is a lot of potential with having a teacher have some flexibility in the schedule, my site needed a full time teacher and the district wasn't ready to make what I did a full time position. So I'm back in the classroom full time this year.

More to follow; gotta get the kids to bed.

Adventures in Pedagogy: Units Matter

2011-08-12T17:19:00.001-07:00

Nevan is measuring the length of an aquarium when he exclaims:

"Dad, this thing is 3 feet long!"

"No it's not, its 36 inches."

"Hmm." *looks at the tape measure* "Oh, 36 inches and 3 feet are the same."

"No they're not. One is 3 and the other is 36. 36 is waaay bigger than 3."

"Dad. Look. Inches are smaller than feet. There's an amount of inches in a foot."

"Oh. How many?"

"12."

"Ok."

- Posted using BlogPress from my iPad

Popping Popcorn in a Popcorn Popper

2011-07-27T10:06:00.000-07:00

Popcorn Question from David Cox on Vimeo.

Most of my previous attempts with these story problems have resulted in me slapping on a timecode and cutting the video. This one had me thinking a bit. I'm not sure I got it.

Your assignment:

1. What question does this provoke?

2. If you have a tough time answering #1, what question do you think I was after? And what can I do to help that question along?

Act 2's a Killer

2011-07-27T09:37:00.000-07:00

I need a little help. I think I've nailed the question:

Barbecue Q2 from David Cox on Vimeo.

But I can't figure out what to give students to help them through Act 2.

Here's the raw footage and the current conversation and Greg's run at the data. (Thanks to @maxmathforum for archiving)

Any help would be appreciated.

Virtual Conference on Core Values: Treat 'Em Like They're My Own

2011-07-06T07:20:00.000-07:00

Conference is here.

What's at the center of my classroom?

It's the same thing that drives my parenting: I'm raising them to leave.

My home is differentiated. My wife and I have boys ages 12, 9, 6, 4 and 1.5. We don't treat them according to their age; we treat them according to how ready they are to be independent. It's based on this unwritten authority/influence continuum that seems to be as dynamic as anything I've ever encountered. At one end, we have authority which is determined by the decisions we make for our children. At the other, we have influence which eventually becomes the decisions they makes for themselves. And in between? We have all kinds of decisions we make together.

We start out by feeding, burping, bathing and changing. It's a no-brainer, because an infant can't do these things for himself. Trust me; I've tried.

As the child grows, he begins to do more for himself. The part we have to embrace is the messiness that ensues as the spoon leaves our hand and moves into the hand of the child. Let go of the spoon too soon and you'll be cleaning the ceiling for months; let go too late and the child may never learn to feed himself. But as you begin to let go, make no mistake, it's going to be messy. That part is hard because sometimes it's just easier to feed the kid yourself. Parents get this all kinds of messed up and the child eventually pays for it. We've all seen it: helicopter parents who make decisions for their kids, resolve their conflicts and clean up all their messes.

There may be times when the child may look like he's taking responsibility, but all he is doing is following the lead of his parents.

The way this works into my classroom is simple: Never do for them what they can do for themselves.

Our initial placement on the continuum is critical. The only way to assess that is to provide activities that can be differentiated in terms of our involvement: How much do we show them? How much do we explicitly tell them? What questions do we ask and how helpful should they be? We can't turn them loose too soon, but the goal has to be to let go.

We are preparing them to leave.

I think once we wrap our mind around that concept, the continuum begins to look more like this:

See, most of the teaching I've been around assumes the teacher to be the source of all knowledge, like a breathing encyclopedia. I realize we are all dealing with mandated curriculum and most students probably aren't quite ready to choose their own adventure anyway. I'm not trying to discuss what we teach (that's a topic for another post); I'm talking about how. But sometimes, the spoon never leaves the hand of the teacher because it's assumed the student can't feed himself. The complexity of the material may increase, but the cognitive demand of the delivery never leaves modelling and explicit directions in the form of statements--or maybe, if we're lucky, closed questions. The student may look like he's functioning at a high level, but all he's doing is following the lead of his teacher.

I have to constantly take a look at how much of what I say ends with a period and how much ends with a question mark. I also have to be aware that not all questions are created equal--are my questions pointed and closed or are they open, allowing for multiple entry and exit points? If the best I can do is offer closed questions to my students, then the best they'll do is depend on me to be the one asking the questions. The really interesting part is how a student and I can move all over the continuum during a single conversation. Sometimes we may start with the open questions (which is always best, in my opinion) and based on the student's response, I may have to just get out of the way and let him go--or we keep moving towards modelling until we find a spot where the student is comfortable. It's really up to him. The key is to let the student lead. This may be problematic at first because, just like learning the tendencies of a new dance partner (sorry for changing metaphors there, but it had to happen. Besides, I do dance with my kids, so it kinda fits.), many students are conditioned to follow. They'll sit and stare for a while until you let them know, "the music's playin', kid, time to bust a move."

So, whether we are talking about feeding a child or dancing, the point remains the same: the student determines my level of involvement and it's important to never underestimate how independent he can truly be. It's a tough call sometimes because the little boogers'll sandbag, for sure.

So now what? How do you move the spoon from your hand to the hand of your student?

Sorry, it's your spoon.

Adventures in Pedagogy: No Solution

2011-06-08T15:10:00.001-07:00

Dawson (12) is ready to begin 7th grade and I'm taking over the curriculum duties. We are starting off with some of James Tanton's Math Without Words

One of the early puzzles looks like this:

Dawson jumps in and devours these things. Right up until he encounters a puzzle like this:

"Dad, I don't get it."

"Get what?"

"This puzzle. I don't understand it."

"What's the rule?"

"I have to connect the two dots by going through each box."

"Is that the whole rule?"

"Yeah. No. I can only go through each box once and I can't go diagonally."

"Does that rule work for all the others you've done?"

"Yeah."

"Hmm."

I go back to doing the dishes as Dawson and Nevan (9) discuss what's "wrong" with this particular puzzle. Once I'm finished, I chime back in.

"So, have you figured it out?"

"No. I just don't understand?"

"Have you considered that maybe this particular puzzle doesn't have a solution?"

*perplexed*
"You mean, that's allowed?"

"Yeah. Sometimes problems don't have answers."

*points to a different puzzle on the page*
"Oh, then this one doesn't have an answer either."

- Posted using BlogPress from my iPad

And Now a Word From Elisabeth

2011-05-27T08:45:00.000-07:00

Elisabeth decided to define her own project. She became interested in the Golden Ratio. She came to me with a boatload of questions she wanted to tackle ranging from: 1) How can I create the Golden Spiral? to 2) Why is the Golden Rectangle aesthetically pleasing?

I said, "yeah, that's some good stuff there. I trust you. Do it."

Two days later, I sit down with her as she's working and watch her think. She has this perplexed look on her face as she is shuffling through her notes. She has about a half-dozen different rectangles sketched with different dimensions along with more questions than what she started with.

She looks up and says, "The more I look through all this, I'm wondering, 'what's my question?'"

That's right, Elisabeth. Don't ever forget that.

Something Different

2011-05-26T11:50:00.000-07:00

This year, I decided to take a much more hands-off approach when it came to student projects. There were some homeruns, but there were too many swings-and-misses. Some students opted not to even step to the plate. I suppose that's what happens when students are offered more autonomy. But, I didn't do enough to prepare them to make decisions in such an open ended environment. I think I was too hands-off.

For the final project, I gave my 8th graders seven choices; one of which was to determine the angle that would maximize the distance traveled by a projectile.

What they knew:

Linear motion model.
Vertical motion model.

What they didn't know:

Vertical and horizontal motion do not affect one another.
How vertical and horizontal motion work together to determine the path of a projectile.
Trig ratios

Last year I had students do an investigation on trig ratios prior to working with projectile motion. But due to a shortened school year and the fact that all of my students will be taking geometry next year, I had to cut something.

It took a few short conversations for the group to get the fact that horizontal and vertical work together to determine the path and that they needed to use the vertical motion model to determine how long the ball would be in the air. From there, they could figure out how far it would go.

But there was one problem: they didn't know how fast the ball was travelling which made it impossible to determine the vertical and horizontal components.

The Process

Q: How fast is the ball travelling when it is hit?
A: I didn't specify, did I?

This led to a nice conversation on how we need to eliminate as many variables as we can.

Solution: Pick a velocity and work with it. They chose 100 ft/sec.

Q: So how fast is the ball travelling vertically and how fast is it travelling horizontally?
A: That depends.
Q: On what?

So we took turns pushing Joey around the room from behind and the side simultaneously. Each time one person pushed harder than the other.

Conclusion: If the person from the back pushes harder, Joey goes forward more. If the person from the side pushes harder, Joey moves to his left more.

Then we talked about how the velocities can be modeled using vectors and we can use what we know about triangles. Since the forces are perpendicular, we have a right triangle.

Q: If all we know is the hypotenuse of the right triangle, how do we find the other lengths?
A: Is that really all you know?

Solution: They settled on using a 45-45-90 since that is the only way they could figure out the other two sides.

Q: But what do we do for other angles?
A: Yeah, that's kinda tough, huh? Why don't you use a protractor to draw the angle you want, build the triangle you want and measure.
Q: Can we use GeoGebra?
A: Or that.

They used an applet with a fixed hypotenuse of 100 and gathered data on the other two sides.

Q: Is there an easier way?
A: Yeah. It's called sine and cosine. See how these ratios don't change as long as the angle remains constant? (it took a little longer than that, but you get the point)

They were off and running.

Conclusions

45 degrees maximizes distance.
Complementary angles yield the same distance.
Oh, and this:

I think you physics folks would say something like this:

( )conceptions

2011-05-17T10:30:00.000-07:00

Preconceptions? Misconceptions? Heck, I don't care what we call them. All I know is that I have kids coming to class and making decisions with their heart and not their head. Intuition is great. Inductive logic is great. But it just isn't enough. Back it up. Verify it. Embrace the conflict that arises when what you thought was true turns out to be, well, not so much.

I've taken to putting these ( )conceptions front and center. Put them out there for kids to wrestle with. Plug in some numbers. Argue. Get frustrated. And then walk away with a little more understanding than they did before.

Today's episode centered on the equation:
(2/3) (3x + 14) = 7x + 6 and students were asked to multiply both sides by 3.

And, of course, they came up with 2(9x + 42) = 21x + 18.

Why?

Well because, naturally, a(bc) = (ab) (bc).

So what do you do?

The younger me would have said something profound like, "You don't distribute multiplication over multiplication. I'll say it again slower for those of you taking notes. You. Don't. Distribute. Multiplication. Over. Multiplication."

There ya go. My finest pedagogical moments summed up by slowly repeating a negative definition of a property they obviously don't fully understand.

The older, wiser, 5-kid-having self is a bit more patient.

Up on the board goes:

True or False

2(3 · 4) = 2(3) · 2(4)

and

2( 3 + 4 ) = 2(3) + 2(4)

Most kids said that their gut told them that both equations were true. In fact, many said, "true" before the virtual ink had dried.

"But what does your head tell you? Verify that both equations are true."

Oh, no they're not both true.

"Ok, good. So now you have a conflict. What you think should be true is different than what you know is true. Why?"

This is why I have been calling these things preconceptions. Students bring something to the task. Always. They never come empty handed. These responses that #needaredstamp are usually a right idea used at the wrong time. It's like a kid who has never played sports before goes from learning basketball to soccer. Coach says dribble and the kid picks up the ball and bounces it as he runs down the pitch. Right rule; wrong application.

I've had kids tell me that they do certain operations on a problem because "it just felt right." I'm not sure how to address that other than to put them in a position for their feelings to betray them and help them deal with the disappointment in a constructive way.

Next weeks episode: Why Love Isn't an Emotion

A Tale of Two Teachers

2011-05-08T07:00:00.000-07:00

My day begins at the door where I greet my students with handshakes and fist bumps.
Her day begins with five little boys climbing into bed and dog-piling her.

My lesson planning is done sitting at my desk.
Her lesson planning is often done at the bottom of that pile.

My lessons are informed by pacing guides, practice tests and proficiency levels.
Her lessons are informed by bugs in the backyard, bicycle tires and brotherly conflict.

My students use manipulatives to learn counting techniques.
Her students count out baby carrots as they make Dad's lunch.

I use rabbits to teach about exponential growth.
She uses a persuasive essay so her students can decide if they really want a rabbit.

My students solve fraction problems about baking and measuring.
Her students cut recipes in half, measure and bake.

My students eat lunch and go out to a treeless field for recess.
Her students eat lunch in the tree.

My classes end when the bell rings.
Her classes end at bedtime.

I use formative assessment to shape lessons.
She uses formative assessment to shape lives.

My students call me "Mister."
Her students call her "Mommy."

One of us teaches. The other pseudo-teaches.

Well, Since You Asked

2011-05-06T11:42:00.000-07:00

I had a copy of John Van de Walle's book on my desk the other day when a student asks, "Mr. Cox, what are you reading?"

"Oh, this book on how to teach math. It's pretty interesting."

"Oh, yeah? What's so interesting about it?"

"Well, I like this chapter on Teaching Through Problem Solving. It mentions three different ways problem solving can be taught: Teaching for problem solving, Teaching about problem solving and Teaching through problem solving."

"What's the difference?"

"Well, teaching for problem solving is when students will learn a certain set of skills and then later be asked to solve a problem using those skills. Teaching about problem solving is when students learn about particular strategies for solving problems. And teaching through problem solving is when students are given the problem first and then they figure out how they want to solve it. It's kind of the opposite of teaching for problem solving."

"Hmm. So teaching for problem solving means the teacher shows us how to do stuff first?"

"Yeah, pretty much."

"They don't think we can think for ourselves? That's kind of offensive."

Point of View

2011-04-19T15:29:00.000-07:00

I can just sit and stare at both of these.

What (if any) questions do these provoke?

Adventures in Pedagogy: Shrinking Rocks

2011-04-16T09:50:00.001-07:00

Inquiry can get messy.

For those of you following along at home:

Rocks = Dirt Clods

YouTube Video

Our family has also verified that charcoal doesn't swim, either.

- Posted from my iPhone

Teachable Moments

2011-04-14T20:45:00.001-07:00

or How a Conversation About Tall Mountains Turned Into a Lesson on Derivatives

I usually only share the good stuff and I think this qualifies.

I have an interesting group 3rd period. It's a geometry/exploratory math class and the kids in there are very curious. About everything. Today they were arguing about the world's tallest mountain and what considerations are used to determine height.

One student went to Wikipedia and pulled up this page and the highlighted graphic got us talking about relative maximum and minimum.

So I sketch a run-of-the-mill cubic function on the board and I ask what the maximum and minimum are.

"It goes on forever in both directions, Mr. Cox."

"Is it increasing or decreasing?"

"It goes up, then down, them up again."

"Ok, so what do these two points have in common?"
*pointing to relative max and min*

"Hmmm."

We talked about why the points were considered relative in terms of being a maximum or minimum and discussed how right before the relative max, the function is increasing. Right after the max it's decreasing.

They quickly connected their understanding of slope to the increasing or decreasing nature of the function. Then I pointed to the relative maximum and minimum and asked:

"So what's the slope here?"

"Zero."

"And, so what do they have in common?"

"Zero! The slope is zero at both points."

"How do you think we can figure the slope at different points on the graph?"

This time I sketched y = x² and we talked about how we need two points to determine a slope.

"Would we be more accurate if the points are far apart or close together?"

"Close."

"How close is close enough?"

It wasn't long before we had it.

4th quarter projects should be interesting.

- Posted using BlogPress from my iPad

Quadratics Revisited: The Falling Object Model (part 2)

2011-03-30T13:05:00.000-07:00

I worked on this applet for a while, but Frank offered some good pushback which got me to rethink how I wanted to go about this lesson.

Day 1+

Falling Object (Question) from David Cox on Vimeo.

Show the first part of the video and discuss what we notice. I

"9.8 meters per second per second."

I ask, "Alright, so what does that mean?"

"9.8 meters per second per second!"

"So what's the per second per second all about?"

"Hmm."

Show part 2.

Obvious question: "When is it going to hit the ground?"

Pass out the laminated picture and some Vis-a-vis markers (I knew those would come back in style).

"So what do we need to know?"

"The amount of time between each ball."

"The video was exported at 6 frames per second."

"Huh?"

"6 frames per second."

After a brief discussion, we were pretty clear that 6 fps meant that 1/6 second passed between strobes.

Let's go to work

Most kids realize that the strobes' rate of change has a rate of change and they're able to come within a few hundredths of a second from the answer.

Falling Object (Answer) from David Cox on Vimeo.

Day 2+

Project the same still and ask:

"Ok, so how fast is the ball travelling?"

"Huh?"

"How fast is the ball travelling? You guys were talking about 9.8 m/s² yesterday. What's that all about?"

"It's gravity."

"What does that mean?"

We discuss how gravity is an acceleration but we change from metric to ft ('cause that's what's gonna be on THE test) so the 9.8m becomes 32ft.

So how fast is the ball travelling after 1 second? 2 seconds? 3 seconds?

led to

What is the average rate after 1 second? 2 seconds? 3 seconds?

which led to

Then how far has the ball traveled after 1 second? 2 seconds? 3 seconds?

Once we got comfortable with 16t being a rate, it wasn't too far of a stretch to be able to say that 16t² is a distance.

Can we generalize the height with a given starting height (s)?

Day 3+

Now what if I threw the ball down at a rate of 10 ft/s?

The major preconception here was that many students wanted the initial velocity of 10 ft/s to be an acceleration so they just added it to the 16 ft/s². I tried to stay out of it as much as I could and cooler heads prevailed. We eventually decided that the height of a ball after t seconds could be modeled with:

h(t) = -16t² - vt + s

The key to unlocking this function was for each term to be able to stand alone on its own. Once everyone realized that 16t², vt and s are all distances and 16t² has to be negative since the distance has to be subtracted from s in order to get the height, we were good to go.

Now let's have some fun with rational expressions.

To LCD or Not To LCD

2011-03-28T12:22:00.000-07:00

Is finding the LCD necessary? I know that it makes the numbers smaller when adding fractions, but does it help conceptually?

Would teaching fraction addition like this: $\frac{a}{b}+ \frac{c}{d} = \frac{ad+bc}{bd}$

and then reducing later make adding polynomial fractions like $\frac{x}{x+2}+ \frac{3x}{x-3}$ a bit more intuitive?

Does teaching LCD make an already difficult concept more difficult? Or does it help?

Here are a couple of applets that illustrate the point. I'm wondering which one would be more helpful for 5th - 7th grade kids.

With LCD and without LCD.

If you want the original GeoGebra files: LCD, NO LCD

I Made Me Do It

2011-03-26T08:00:00.000-07:00

or Dave Made Dave Do It

or @calcdave and @dcox21 are pretty much the same person so this is really a post about a guy talking about himself to himself (or would that be to himself about himself?).

Whatever.

So apparently Dave was eaves dropping on one of my conversations (the nerve of some people) and started asking some interesting questions. I particularly liked the triangle image and couldn't help myself especially since Jason encouraged me via GoogleChat last night (or was it GoogleTalk?) to keep doing this GeoGebra stuff.

Oh, and Jason and I live in California (which makes us pretty much the same person too, except that he lives in the Bay Area and I live in the Central Valley so we are the same person kinda like Massachusetts and Texas are the same state in that they are both part of the contiguous United States) and he lives, like right next door to Dan who's been to Canada.

So I guess y'all have to thank Canada for this applet, eh.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Adventures in Pedagogy: Check for Understanding

2011-03-17T19:31:00.001-07:00

or Context Clues (part 2)

Nevan's helping me build planter boxes by holding the 2x6s in place while I screw them together when I ask:

"Nevan, are those two pieces flush?"

"Yeah, Dad."

"Do you know what 'flush' means?"

"Mmm. No."

"What do you think it means?"

"I assumed it meant 'on track' or something."

So when I'm asked, "how do you know that they know?"

That's how.

The challenge is to care enough to ask.

- Posted using BlogPress from my iPad

Adventures in Pedagogy: Percentages

2011-03-17T12:41:00.001-07:00

Sitting here in the car with the boys as my wife runs into Target to pick something up "really quick." (I believed her. She can be quick—in Target, right?)

Anyway, I find a spot that gives us a clear view of the entrance and Dawson says:

"Hey Dad, see that white car in front of us? Why don't you park in front of it? It's closer."

"Can't. See that other car? It's like halfway in the parking space."

"Halfway? That's a bit of an exaggeration."

"Alright, then what percentage of the space is it taking up?"

"About 1/11."

"I'll give you that, but what percentage?"

"About 9 point something percent."

"Not 10%?"

"No, a little less."

Right about then, my wife returned. She was pretty quick.

And she brought me a Drumstick.

- Posted (on location) from my mobile

Adventures in Pedagogy: Problem Solving

2011-03-14T10:30:00.001-07:00

Dawson is playing a game on his DS and I plop down next to him and ask:

"Whatcha doin'?"

"Oh man, Dad, I'm on this level that took me forever to figure out."

"Really? So what do you do when you get to a level and you don't know what to do?"

"I experiment."

"What do you mean?"

"I try stuff and keep trying until something works."

"You mean you don't just sit there, throw your hands up and say, 'I don't know what to do, so I quit?'"

"Nope."

- Posted from my iPhone

Similar Triangles and Algebra

2011-03-07T19:21:00.001-08:00

Dr. Hung-Hsi Wu pg. 58:

The reason for the critical need of a definition of similarity is that a working knowledge of similar triangles is absolutely essential for students to achieve algebra. Without this knowledge, they would have no hope of understanding the interplay between a linear equation of two variables and its graph, which is a major topic in beginning algebra.

So, why do you think that is?

- Posted using BlogPress from my iPad

Fraction Multiplication

2011-03-03T15:59:00.000-08:00

I've been spending some time lately thinking about how to help elementary school teachers teach math conceptually even if they don't consider themselves strong in math. I keep coming back to interactive applets (read: GeoGebra) because it's kind of a one-stop shop. Yesterday, I tried a few out with a 4th grade class and they responded very favorably. So, here are the latest two I've done on fraction multiplication.

Click on "File" and "Save" if you'd like a copy of either applet.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Please forgive the crowding, I had to minimize everything in order to fit them into the post.

Another First

2011-03-02T19:41:00.000-08:00

Usually kids raise their hands to show what they know.

Today, M.C. politely asked me to get out of the way, walked up to the document camera, put her work under it and said, "I need your help, guys. Really. I want your feedback. I don't know how to do this."

The class responded. No one ridiculed her. She asked them questions. They asked her questions.

All I could do was sit down and watch.

Today, I caught a glimpse of what school should be.

Let 'Em Drive

2011-03-01T13:04:00.000-08:00

The Problem:

Find the x-intercepts of y = x² + 10x + 16.

Me: How many different ways could we do this?

Class: Quadratic Formula, completing the square and factoring.

Student 1: Mr. Cox, why does y = 0? Isn't this a function where y can equal a bunch of things?

Student 2: Yeah, but we want y = 0 because that's when it crosses the x-axis.

Student 1: Ok, so we are just focusing on one possible value of y.

Student 3: So then, if we let y = 0, then we are finding the x values that make y = 0, right? But what if we want to know when y = 1? Can we do that too?

Student 4: I guess we should just let y = 1, but we'd have to subtract it from both sides so we get:
0 = x² + 10x + 15. So now we are finding the x-intercept again.

Me: Is this an x-intercept? What does y equal?

Student 4: Oh, no, y = 1.

Student 5: Mr. Cox, can we see what this looks like in GeoGebra or something?

We graphed it and then moved a point around on the parabola to verify our results and looked at how the points where y =1 maintained the same symmetry with the x-intercepts. Then, a kid pipes up:

Are there graphs that have more than two x-intercepts?

Me: Hmm. Maybe. I'll tell you what...you guys give me a function that has x intercepts at -4, 0 and 3. You have 15 minutes. Go!

About 5 minutes later, J.V. walks up with this:

y = (x + 4)(x - 0)(x - 3)

y = x³ + x² - 12x

And I ask him, "How did you come up with this?"

J.V.: Remember the other day when we had to come up with parabolas with x-intercepts? I figured I could just work backwards like we did then.

I'm amazed at how often these kids will take the lesson into places I wouldn't have thought to go. It's just a matter of letting go a little. And the more I look for open ended opportunities, the return is exponential.

No Title Necessary

2011-02-26T07:39:00.001-08:00

- Posted using BlogPress from my iPad

Some GeoGebra Love

2011-02-14T09:28:00.000-08:00

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Inspired by Greg Hitt

I'm Holding Somebody Responsible For This!

2011-02-10T16:48:00.000-08:00

Step 1: Choose values for 'a' and 'c'.
Step 2: Move the 'b' slider from -10 to 10.
Step 3: Select all the points step 2 puts in the spreadsheet. Right click and select "Create list of points."
Step 4: Click on the "Conic through 5 points" tool.
Step 5: Click on 5 of your points generated in step 3.
Step 6: Right click on the resulting function in the algebra window and change standard form
Step 7: Compare this function to the Starting Function
Step 8: Explain to me why I never knew this before.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Quadratics Revisited: The Falling Object Model

2011-02-02T20:32:00.000-08:00

I keep hoping that I can use this to help kids derive the falling object model. I'm getting close. I exported the video at 6fps so 1/6 second elapses between strobes. I'd like some feedback on this before I roll it out to my students. How intuitive is it to use both applets together?

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Once you have plotted your points, use the FitPoly function. Simply enter "fitpoly[A,B,C,D,E,F,G,H,I,2]" to plot a quadratic function.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

If this proves to be useful, I'll dial it all in and post the still, video and applets for download.

Mixture Problems

2011-01-21T12:21:00.000-08:00

You know this one too:

Given:

Solution A contains 20% acid

Solution B contains 60% acid

Find: The number of mL of each necessary to obtain 450mL a solution that contains 30% acid.

I usually teach this the same way I do work and rate problems. Find what we call a "guiding equation" that is based on the units available in the problem and set up some sort of table to organize the information. Depending on the given information, we may use two variables and solve a system of equations. Maybe not. But at the end of the day, this is going to be some complex variation of a rate problem.

I have a colleague who sees these problems in terms of proportions and our recent conversations have inspired this GeoGebra applet.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

There may be some rounding error, but I think this gives an interesting conceptual spin to a problem that can very easily become an algorithmic nightmare.

We can make some quick adjustments for the good ol' price per pound problems as well.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Download % Mixture applet
Download Price per Pound applet

Work Problems

2011-01-10T13:00:00.000-08:00

Y'all know the drill.

Person A does a job in A hours, Person B does the same job in B hours. How long does it take them to do the job together?

Teaching the work problem can turn into a debacle fast. Really fast. Most teachers boil these down to an algorithm they themselves might not understand.

"Here's the formula, kids. Learn it. Know it. Use it. Now let's move on to something else."

That's never set well and a couple of years ago, it dawned on me that a work problem is nothing more than a complex rate problem.

Person A can do 1/A of the job per hour and person person B can do 1/B of the job per hour. How much of the job do they complete per hour when working together?

This has been a game changer. Find the common denominator and add 'em up. Now we have a common rate. From there it's pretty easy to determine how long it takes to complete the job. We give no formulas and kids understand how to find an answer and why it works.

We had kids derive two different algorithms for work problems.

x = (AB)/(A+B)

and

1/A + 1/B = 1/X

Sure, kids. Use it. G'ahead.

But then we added the dreaded twist.

Person A can do the job in A hours. If person B helps, they can do the job in X hours. How long would it take person B to do the job when working alone?

I give you, Clinton's Theorem:

I kid you not, this kid came up with this all himself. By "integer portion of X" he means, everything to the left of the decimal. He's truncated the number to make things "simpler."

As he was presenting this to us before we left for break, I was baffled. It works, but why? So today, I sat down next to him and had him explain the formula to me again, I took notes and then simplified the expression. I won't deprive you the joy of doing so yourself.

Seriously, have at it.

The Monkey Hunter

2010-12-30T15:50:00.000-08:00

Yesterday, Dan threw out this tweet:

So, naturally, I had to try this:

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

If I were using this in class, I'd probably animate it so students had to choose values for velocity, start height and distance and hit "GO."

Download original .ggb file.

Adventures in Pedagogy: Number Sense

2010-12-21T14:55:00.001-08:00

Nevan: Dad when do you have to go back to work?

Me: I'm off for three weeks.

Nevan: Ok. Well, today's the 20th. Plus 7, that's the 27th. 4 more to the 31st. The other 3 for January 3rd. Plus 7 more...the 10th. You go back to work on the 10th.

- Posted from my iPhone

Volume

2010-12-17T08:00:00.000-08:00

We had a few presentations to watch due to absences and one in particular sparked a pretty good discussion. One, in particular, focused on how to find the volume of a cylinder. The presenter told the class how to find the volume and then proceeded to explain why we find the volume of a cylinder by using the formula:

V = πr²h

But she didn't stop there. She asked herself and the class:

Why is that the formula?

She went on to describe how it's like taking a bunch of CDs and stacking them. We know we can find the area of a CD.

That's when things got interesting. I heard comments like these:

We use the height to determine how many circles it takes to make up the cylinder.
But it's just like a line [it has no height]. How do you get volume with that?
If you have a thing of CDs and stack them...you're going from 2D to 3D.
But area has no height. We're stacking things that have no height.

So we talked a little about finding the volume of a rectangular prism is just counting the number of cubic units it takes to fill the prism and how doing that with a cylinder is tough because its round. I liked the direction they were going with this by trying to talk about something larger by defining it by its smaller parts. So small, in fact, that the height is practically 0.

I ask, So how can you find the volume of a sheet of paper?

A student takes a sheet of paper puts a ruler up to it as if he's going to measure the height of one sheet of paper and one of his group members says:

Dude, why don't we just make a stack that's 1" tall and divide by the number of sheets it took to make it?

They all agreed that would work and I thought we were ready to move on when a boy asks, "So what's the volume for a sphere?"

They wrestle with this for a minute or two when one of them turns to me and says, "Mr. Cox, you know, huh?"

Yep.

But you're not going to tell us are you?

Nope.

A student opens his planner to the Math Formulas page and starts to tell us what the formula is when one says,

I don't care what the formula is. I want to know why that's the formula.

Next semester's gonna be fun.

Exterior Angles

2010-12-16T12:36:00.000-08:00

"Hey, Mr. Cox. Is there a theorem that says this?"

I'm not sure. Why do you think that's true?

"Well if a triangle has 180^o and I know two of them, then the third one has to be whatever's left over from 180^o. But that third angle and x make a straight line so they have to add up to 180^o too."

If it's true, what would you call it?

"I remember seeing something about 'exterior angle' in the index one time, so that's taken. Maybe I'll look it up and see what that one means."

*goes and looks up Exterior Angle Theorem*

"Ah, man! Someone already discovered it."

Download Time

2010-12-15T09:44:00.000-08:00

I kept reading about all the fun everyone was having with the ProjectEuler thing, so I thought I'd give it a try. I thought it was pretty easy until I got to that one problem where they ask you to add up all the multiples of 3 or 5 that are less than 1000. I hear that people are using Python and letting the computer do the number crunching.

So, I think to myself, "Self, you took Pascal ~~about 20 years ago~~ ~~back when RAM was measured in single digits and you said 'bless you' if someone said, 'gigahertz.'~~ in college and ~~absolutely hated that freaking class~~ came out a better person because of it. This may be ~~something you don't have time for, I mean you have 5 kids. Wake the heck up~~ an enriching experience. And besides...Pascal...Python, they both start with Ps ~~yeah, that's probably the only thing they have in common~~. You got this, man. G'head, give it a try."

I go to the Python site and read every thing I can to figure out which version would best suit me. (There's a lot to choose from, you know.) I can't find anything anywhere. It's all so confusing. Please. Someone. Give me a nudge in the right direction.

So I take a shot in the dark and go with v.2.7.

Anyway, this isn't really about Python. It's about the video I grabbed while downloading the program. I'm going to have my kids wrestle with it after break.

Download Time (Question) from David Cox on Vimeo.

I may not ever find the solution to that really difficult problem about multiples of 3 and 5, but at least I got a lesson out of it.

Rates of Change

2010-12-15T08:38:00.000-08:00

Sometimes we just stumble onto stuff. I've taught slope intercept form for a lot of years, but the more I try to stay true to the concept, the more my students demonstrate innovative ways of seeing things. We just finished up linear equations and their graphs. The equations part of things usually culminates with a student taking two points and writing the standard form of the equation by way of this process:

Step 1: Find slope
Step 2: Write in point slope form
Step 3: Solve for y to get slope-intercept form
Step 4: Rewrite in Standard Form

I've usually encouraged this process as it seemed to be the efficient way to cover the majority of these skills. But this year, I kept it much more loose. We all agreed that if the two points given define a line with an integer for the y-intercept, then it's pretty easy to write in slope intercept form.

I mean, c'mon, we've got the slope and we see the y-intercept plain as day.

But what happens here?

We know the slope is 1/7 and we can estimate the y-intercept, but how can we know the y-intercept?

Rate of Change

Most of the time when we discuss Slope we talk about the path from one lattice point to another. For example, the slope between A and B is 1/7 because we go up 1 and over 7 to get from A to B.

But we have been doing more with rate of change as a unit rate which means that for every 1 unit of horizontal change, we increase 1/7. Yeah, I know that sounds elementary, but it makes a difference in how your kids see these things.

If each step increases 1/7, we increase a total of 4/7 to get to the y axis. Therefore, y-intercept is 4 4/7. We still talk about point-slope form, but this beats the heck out of the plug and chug that usually goes along with it.

Semester Projects

2010-12-14T11:42:00.000-08:00

I'm still a believer in the taxonomy I wrote about a while back. I can teach duplication like no body's business and finding ways to get students to apply what we are learning isn't a problem. It's the creation part that I get hung up on. I want my students to find ways to make their own connections with the math.

Our attempt at creation.

Assignment: Pick something that interests you; something that you can connect to one of the concepts we've covered this semester. Learn as much as you can about it. Tell us what you've learned.

I only had one requirement: Don't ask yourself, "What is Mr. Cox looking for?"

Keep in mind that I get the kids who taught Clever Hans how to read people. They are fantastic kids, but they also have learned how to game the system. Find a way to jump through the fewest hoops (or at least the most efficient way to jump through them all) while gathering as many points as possible on the way to be-all-end-all of educational existence: The A.

This assignment scared the crap out of some of them. This was the most unhelpful I've ever been. I allowed them to work on their own or in groups (up to 3). We discussed the benefits and pitfalls to working alone vs. working with a group.

We discussed what type of media would best serve their presentation. I learned that when I say, "presentation," kids hear "PowerPoint." I told them that it didn't matter what they used to convey their learning, but if they were going to use visuals, they needed to work with their content, not against it. Does that mean I talked to them about design? I guess I've come a long way since this comment. (Full comment here.)

As we approached the due date, we started to discuss how this assignment should be graded. Again, I took a back seat on this. We just recently opened up the sequoiahawks.org domain for our students to gain access to Google Apps, so this was a great opportunity to have them work collaboratively on a rubric. My 8th graders had some experience with this as we used G-Docs last year, but this was new to all of my 7th graders. The only input I had in this process was to create the document and ask a few questions. The rest was all them.

I found that some of the conversations that took place regarding what to include, what to exclude and what categories were the same (ie. "Aren't self interest and effort the same thing?") were very refreshing. These kids were really thinking about what was important.

The Presentations

It was nice to see students take things that we have done in class and either find a context for them or extend them beyond what we've done in class. There were a couple of presentations that stood out for their creativity and some for their thought provoking qualities. We had a few nice class conversations that came from questions posed in the presentations. Both classes had someone ask the question: is .999...=1? which provided a great opportunity for discussion. The 8th graders were pretty curious and this led to a conversation on ∑ notation, ∞ and limits.

We did have a few groups that mailed in the projects, though. I had a tough time not airing them out publicly (in one case I did) as they showed complete disrespect for their classmates' time. This demonstrates that the hoop jumping is not quite out of their system, although I'd say we are off to a decent start.

Build Your Own

2010-12-03T19:30:00.000-08:00

What can you build with this?

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Download GGB File

Ticket Prices

2010-12-02T12:36:00.000-08:00

The other night, Dawson and I were doing some math together and we ran across a problem that asked to interpret a scatterplot for the average movie ticket prices for the past 10 years. We poked around online until we found this image:

http://www.the-numbers.com/market/

We used the information to come up with an average price and dropped it into Excel and came up with a best fit line and all. Personally, I find Excel clunky, but since we could use it to quickly calculate average price, we went with it. But it just so happens that I'm doing linear relationships with my 7th graders, so I get a two-fer with this one.

Step 1: Calculate average ticket prices and created a scatter plot.

This went pretty quickly, but one group mistook the raw number for the average ticket price. For example: from 2003 to 2004 the revenue went down, but so did the number of tickets sold. The raw numbers led this group to believe that the prices went down. After a quick conversation, they realized that if the number of tickets sold decreases as well, the price can actually increase.

Step 2: Predict the price of a ticket in 2020 and justify answer.

Most groups used the trend of the graph to predict, but one group actually calculated the average rate of change from year to year and came up with $.24/year. They used this to extrapolate a price in 2020.

Step 3: Decide what type of curve best fits the data.

Two camps on this one: Those who thought it was linear and those who thought it would be like "the graph we got when we did compound interest." Nice work kids.

Note: We will explore that exponential thingy but we ran out of time today.

Step 4: What line would best fit the data?

They were creating a group graph so they took a meter stick and just plopped it down where they thought the line should go.

Step 5: Estimate the equation of that line.

We played with this applet yesterday, so students had a pretty good idea how to use the graph to predict what the equation should look like. I was pleasantly surprised at the fact that all groups decided the rate of change was somewhere between $.22 and $.25 and all said the initial condition was $4.34.

Step 6: Let GeoGebra work her magic.

Add the ordered pairs and use the "Best Fit Line" tool to calculate the regression.

Step 7: How good was our guess?

All Present and Accounted For

2010-11-30T10:35:00.000-08:00

Finally, everything from the old blog is here. The process was actually pretty simple:

1. Export posts from Wordpress to XML. I had to do this in two increments because the conversion site only works for files smaller than 1MB.

2. Convert the XML using Wordpress2Blogger.

3. Import the converted XML to Blogger.

Done!

Now What?

2010-11-30T09:27:00.000-08:00

Paul Lockhart:

In particular, you can’t teach teaching. Schools of education are a complete crock. Oh, you can take classes in early childhood development and whatnot, and you can be trained to use a blackboard “effectively” and to prepare an organized “lesson plan” (which, by the way, insures that your lesson will be planned, and therefore false), but you will never be a real teacher if you are unwilling to be a real person. Teaching means openness and honesty, an ability to share excitement, and a love of learning. Without these, all the education degrees in the world won’t help you, and with them they are completely unnecessary.

Don't be too quick to write this off as impossible given our current system.

Insomnia

2010-11-27T00:54:00.000-08:00

I went to bed with this problem swimming around in my mind. I kept trying to figure out how to construct in GeoGebra. I told it to be quiet so I could sleep. It wouldn't.

Problem:

Given two squares, deconstruct and reconstruct as one square.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Double click to open in a new window.

It's almost 1am. I'm going to bed now.

I Love This Problem

2010-11-26T19:49:00.000-08:00

I love simple problems with simple proofs. And I love GeoGebra even more for making it possible for my students to use their induction to help the proof along.

Problem:

Find the shortest path from point C to point D that touches line AB.₁

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

¹ This problem is one of the many reasons I'm glad I finally got around to reading this.

Video Assessment

2010-11-08T16:32:00.000-08:00

I mentioned a while back that I thought a Flip camera may be a pretty decent assessment tool. Today, I had one group having a great discussion about distilling the process for finding a perpendicular bisector into a few steps. I grabbed the Flip and sat down with them. Forgive me if you can't read their steps very well, I'm still working out the kinks.

Perpendicular Bisector from David Cox on Vimeo.

I think I see some real potential here. Kids can record themselves and then we can go to After Effects and add some stills of their work. I don't know, maybe I'm getting ahead of myself. Feedback ~~welcome~~ necessary, please.

WCYDWT: Projectiles

2010-11-04T13:42:00.000-07:00

The camera angle messes with the perspective a bit, but I still find these images interesting. Does it mess with the perspective so much that it ruins the problem or does it just lead to another discussion on how things aren't always as they seem. I'm thinking that I'll show the video, but make digital and hard copies of the photos available to students.

Question #1

Projectile Question (YG) from David Cox on Vimeo.

Question #2

Projectile Question (PG) from David Cox on Vimeo.

Answer #1

Projectile Answer (YG) from David Cox on Vimeo.

Answer #2

Projectile Answer (PG) from David Cox on Vimeo.

Next up:
Create something that helps kids see vertical and horizontal motion independence.

Which One's Best?

2010-11-03T09:48:00.000-07:00

Two questions:

1. Is the question clear?
2. Which image asks it best?

#1

Toaster Remix

2010-10-29T14:13:00.000-07:00

Dan's getting more mileage out of this problem than I am. Either way, there are some good questions being asked re: pseudocontext, relevancy and so on. So I'm curious, which one would you use and why?

(Note: don't check the answer until you've done the regression. It's much more fun that way.)

The Original (setting 1,2 5 and 7)

Toaster Question from David Cox on Vimeo.

Or this?

Toaster Question (1-4) from David Cox on Vimeo.

Or this?

Toaster Question 2 (1-4) from David Cox on Vimeo.

The Answer

Toaster Answer from David Cox on Vimeo.

WCYDWT: Toast

2010-10-27T21:58:00.000-07:00

Funny what you come up with when making breakfast for your kids.

Toaster Question from David Cox on Vimeo.

Crushing

2010-10-20T20:41:00.001-07:00

Between 2nd and 3rd periods I usually step outside class and wander the plaza as kids are heading to their next classes. Today I happened upon a group of students who were waiting for their teacher—many of whom are in my 4th/5th block.

So I take a seat next one of my students as he's waiting for the door to be unlocked and we discuss how his faux-hawk could have been used to cut the stress joints in the concrete slab beneath our feet, you know, because that's just me.

So he turns to me and says, "You know, Mr. Cox, you're my only teacher who I could see talking to me outside of class."

Wait! What? Why?

"I dunno, other teachers just don't talk to me outside of class."

Don't they know who you are? They're missing out.

Now We're Cookin'

2010-10-06T06:38:00.001-07:00

We were going over our quizzes yesterday and I had a student make an interesting observation about the rubric we use to determine the overall score on a concept/skill.

5: Strong concept; No errors
4: Strong concept; Errors present
3: Flawed concept; No errors
2: Flawed concept; Errors present
1: Little or no attempt

She says, "This reminds me of the first time my brother and I made waffles without my dad helping us."

Oh yeah, what happened?

"We used baking soda instead of baking powder and they turned out really bad."

How's that like our quiz?

"We can know what we are doing but make a simple error and it messes the whole problem up."

So what would you rather do: know how to cook the waffles and use the wrong ingredients or use the right ingredients but not know how to cook?

"I'd rather make the mistake with the ingredients."

Why?

"Because all I have to do is use the right stuff and the waffles would've been good."

Is that easier to correct than if you had all the right stuff but you didn't know how to cook?

"Yeah. A lot easier."

Adventures in Pedagogy: Context Clues

2010-10-04T09:00:00.000-07:00

Dawson and I are watching an episode of Star Wars: The Clone Wars and the narrator opens with the usual recap of previous happenings. Towards the end, he says something like:

"...and Anakin Skywalker thwarted the nefarious plot."

So I ask Dawson, "What does nefarious mean?"

"What do you mean, Dad?"

What's nefarious mean? The guy just said that Anakin thwarted a nefarious plot.

*he thinks for a second*

"Well, it must be really bad because he was talking about the evil scientist."

It never ceases to amaze me how these teachable moments just fall in my lap. I just have to be paying enough attention to them and take the time to ask a couple of questions.

I wonder how many of them I actually miss, though.

Adventures in Pedagogy: Order of Operations

2010-10-03T19:07:00.000-07:00

Nevan (8) goes to the counter to grab a pack of gum from a box that contains four different flavors. He says, "Hey Dad, look. There's four different kinds of gum in here."

Oh, yeah? How many of each kind are there?

"Four."

Then how many packs of gum are in the box?

"Ten."

Really? How do you get ten?

"Yeah, we've eaten two."

Adventures in Pedagogy: Syllogism and Detachment

2010-10-02T21:00:00.000-07:00

On our way home Aidan (5) asks, "Dad does it hurt Jesus' hands when we drive?"

Wha?

"Does it hurt Jesus' hands when we drive?"

What do you mean?

"Our truck drives on tires."

Okay.

"And the tires have pokey things." (read: tread)

Right.

"And the pokey things hurt the road."

Yeah, kind of, I guess.

"And Jesus' hands are under the road holding it."

Why do you say that?

Dawson (12) says, "Dad, I think he thinking of 'He's got the whole world...in His hands...'"

Is that what you mean, Aidan?

"Yeah. So then, when we drive, it hurts Jesus' hands, right?"

A Conversation Worth Having

2010-09-30T14:57:00.000-07:00

I have a student who is very creative with her math. She just tries stuff, learns from mistakes and constructs her own understanding. It's pretty cool to watch. Today, she was describing to the class how she came up with the standard form of a linear equation we were playing with yesterday by using the slope of the line.

M.M. pipes up, "I don't get how she did that."

What he meant was, "Yeah, I see how that equation works, but I have no idea how she thought of doing that."

She follows her intuition and sees where it takes her.

I told M.M., "I'm not so sure that you really need to know how she came up with it any more than you need to know how an artist paints or how an author writes. What you need to appreciate is the fact that she created some understanding on her own."

I launched into a quick discussion on Duplication, Application and Creation and how many "advanced" kids actually live in the duplication stage, but they just do it faster than everyone else. I can teach duplication and provide opportunity for application, but creation is all them.

Here's what went down as a result.

I.N. : Mr. Cox, your class is the only one I do my homework in.

Why?

I.N. : Because you don't make us do it. Like at home, if my mom makes me go out for a sport, I don't really try because I feel like I'm forced to do it, but I'll go play sports on my own with friends or even with random people. I want to do my homework in here.

J.T.: Actually, Mr. Cox, you don't really force anything on us.

Try not to, anyway.

The conversation went to grades and grading in general and I said I don't really care for grades and wouldn't give them if I didn't have to.

S.N. : Then you need to start your own school, Mr. Cox.

Yeah, that'd be nice but I don't have that kind of money.

L.F.: I'll go work at McDonald's and give you my paycheck.

One kid said, "you're gonna change the system, Mr. Cox" and I kinda let the comment go. But after a few minutes I came back around to it.

You guys really think I'm gonna change the system?

*silence*

"Nah, we're gonna change the system."

Equations Three Ways

2010-09-28T15:37:00.000-07:00

I've never really been satisfied with how I teach students to solve equations. No matter what, it ends up being one big algorithm and kids have no idea why one side of the equation is equal to the other. Here's what I'm doing to try to fix that.

Modeling

Strength: It's not math. It's a puzzle.
Weakness: Dealing with negatives is a real pain in the butt.

Guess and Check

I actually really like this method. Guess and check is probably my most under-used problem solving strategy, but using it to solve equations has been really helpful. I've noticed a greater understanding of rate(s) of change, using information from wrong answers to help find right ones and checking answers--something most kids don't want to do--is embedded in the process.

We've gotten to the point where we can nail the answer on the third guess by using the information gained in the first two--even for equations with non-integer solutions.

Strength: Students understand that simplified expressions on each side of the equal sign end up looking the same every time (ax + b = cx + d). Rate of change is very useful. Being wrong helps you to become right. Did I mention they are checking their answers?
Weakness: Leave 'em in the comments.

From Construction to Deconstruction

We spend a lot of time teaching kids how to break things down whether it's reducing fractions, simplifying radicals or solving equations but we rarely (read: I rarely) have taught them how to construct things that may eventually need to be deconstructed.

Constructing a more complicated equation from a simple equation has helped my students understand that, no kidding, the two expressions on opposite sides of the equal sign are equivalent. I've used the just-unwrap-the-present illustration many times, but we really need to teach the students to wrap one up first. Having them list their steps for construction makes the process for actually solving the equation seem much more natural. When I say, "just use the inverse order of operations" --or whatever completely abstract thing I've been known to throw out there in order to make myself feel better when they keep screwing it up--it makes no sense to them. This helps.

Strength: Kids get a grasp of which operation to tackle first while solving for x.

Weakness: Very complicated equations with variables on both sides don't seem so natural when you begin with x = 2.

I've heard rumors that there are some teachers who actually teach solving equations by graphing. Never seen it in the wild, though.

Hijinx

2010-09-16T17:51:00.000-07:00

Teaching can be stressful. So, I find it important to loosen things up a little.

Like, maybe, accidentally leaving the guidance office with things that happen to belong on the secretaries' desks only to return them in the form of gifts on Secretary Appreciation Day in the Spring. We all get a good laugh out of it--except that one time a fight almost broke out because Mrs. P thought Ms. A had taken her tape dispenser.

Or, rearranging the furniture in a teacher's room while they're out on school business. (Note: don't do that one again.)

Or dressing up like a colleague on Nerd Day. (Well, I thought it was funny.)

Most of these antics were the result of me being part of a cohort of younger teachers who were now teaching at the same high school we had attended.

Things changed when I moved to the new middle school. Most of the "new" staff had come from the same school, so I had to take some time adjusting to the culture that many had brought with them. Seemed like the right thing to do. As a result, I quickly became good friends with Mr. G, who had come to us by way of a neighboring district. We'd usually end up in one of our rooms by the end of the lunch period. But one day, I walked into his room and he had a visitor. I felt like I had kind of intruded so the next time I walked up there, I knocked as I entered to let him know I was coming. He made the mistake of letting me know ~~I was now welcome to bang on his door every time I enter~~ had scared the crap out of him. Slowly, more staff have ~~put themselves in my crosshairs by thinking it'd be funny to get me first~~ decided to join in the fun. I have one rule: Administrators are off limits...except Mrs. H. But, she started it. She got me good.

Problem is, I can't scare the woman. She has this mother-grandma-teacher-administrator-eyes-in-the-back-of-my-head-I-feel-a-disturbance-in-the-force kinda Jedi thing going. I can't sneak up on her. This morning, I spotted her about 100' away. Her back was turned. Kids were walking around everywhere. Conditions were perfect. There was no chance she saw me. None. I get 10' away and she turns and says, "Good morning, David."

"Dangit! How do you do that?" (I think I actually jumped up and down while I said that.)

I'm not giving up. I'll get her someday. But in the meantime, she better keep a close eye on her stapler.

If Only It Was Always Like This

2010-09-15T10:47:00.000-07:00

Instead of working through the problems in the geometry book, we decided it would be fun to try to prove all the theorems as we come to them. A couple of weeks ago he proved the midsegment theorem. Now we are on to trying to prove that the centroid is 2/3 of the way down the median. So we sit around the white board easel and discuss how we might go about this. At this point, I don't even know how we're gonna prove this thing. We learned from the midsegment theorem that defining the vertices using (x₁, y₁), (x₂, y₂), (x₃, y₃) helped out greatly. So, what the heck, let's try it again.

We know we can define the midpoints generally as well, so that's what we do. Then it hits! We can define the vector from the vertex to the midtpoint of the opposite side, use a scalar of 2/3 to determine the vector from the vertex to what's supposed to be the centroid and then translate the vertex to said point.

Ta-freakin'-da

Turns out that the centroid is 1/3(x₁+ x₂+x₃, y₁+ y₂+y₃).

I didn't know that. If this is what it's like to become a co-learner with your students, then sign me the heck up.

WCYDWT: Space Shuttle Discovery

2010-09-14T09:48:00.000-07:00

I've tipped my hand here a bit because simply showing a shuttle launch would be way too open ended. I have the original lesson plan as designed by NASA and I'm pretty sure how I'd go about presenting this lesson, but I'm curious as to how y'all'd do it.

STS 121 Launch (Initial/Final values) from David Cox on Vimeo.

How Much is TOO Much?

2010-09-02T10:33:00.000-07:00

What else can I ask him to do?

He's 13.

I mean, c'mon. I can ask for a more rigorous proof, but the kid can manipulate these expressions and then say things like, "I think this looks really cool with all x's and y's all over the place. "

Exponent Rules

2010-08-26T08:00:00.000-07:00

I've grown tired of kids blindly following rules. Mine have the tendency to be the worst because they have always been really good at playing school. So we went for a different approach to exponent rules this week.

Prerequisites: Basic understanding of exponents

Instructions: Choose a rule that you would like to prove (read: demonstrate why it is a rule). As you are able to demonstrate an exponent rule, move downstream to the next group and help with that rule finally working on rule #8.

Here's the list...

...that culminates with the question:

8. What do you think x^1/2 represents?

I'll let them tell you._[1]

_[1]This conversation took about 12 minutes with the camera being shut off a couple of times. I limited the editing so as to try to capture the classroom vibe as naturally as possible. I asked a few questions that I'd like to take back, but... you live and learn.

Proper-tays

2010-08-25T11:21:00.000-07:00

I don't usually enjoy teaching properties because they seem so math-y. I like asking my kids to justify what they do, but for many, the properties just seem to be vocabulary that is forced upon them. Necessary evil, I guess? They are great for doing mental math tricks and kids use them without thinking of them, so I suppose there is no harm in giving a name to the stuff they already do.

Raise your hand if your kids mix up associative and commutative properties?

No more.

The Process

1. Give example of property with respect to addition.

ex: Associative: (2 + 3) + 4 = 2 + (3 + 4)

2. Ask students to write another example of their own.

3. Ask for a rule using a, b and c.

4. Ask students to write down the key characteristics of the associative property in Tweet form. (very few words)

Now here's the kicker:

5. Can you guess what the property for multiplication is going to look like?

This worked great for the associative, commutative and identity properties. A great discussion on the inverse property ensued and I ended up telling them that we want a multiplication problem that equals 1.

Done.

None of these properties are worth anything if we don't apply them. Next step is getting them to put words to all that stuff they "just do in my head."

6.

Let's synthesize this a bit more.

7. Now write a similar problem using the multiplicative properties.

I told the class to keep an eye out for times when we will use the inverse and identity properties--which will happen daily once we start solving equations.

Toy Cars

2010-08-21T20:46:00.000-07:00

Prompt: How far would you have to pull the car back in order to get it to go 100'?

Materials: Toy cars, meter sticks.

Hand them the cars, ask the question and get out of the way.

Question #1
But Mr. Cox, the farthest we can get the car to go is around 10'. We can only pull it back so far until it starts clicking.

Right. So if you could build a car that could be pulled back farther, how far would you need in order for the car to go 100'?

Question #2
Mr Cox, what do we do if our car keeps turning?

Yeah, I'm ~~a cheapskate~~ ~~a father of 5 on a single teacher's income. Give a guy a break will ya?~~ very thrifty. So what can we do to estimate the distance the car travels?

The two groups that had problems with the car came up with two different solutions. One group decided to tie a piece of string to the spoiler and measured the amount of string the car pulled past the starting line and the other group simply estimated by breaking the curve down into short line segments. (Oh man, do you guys just realized you set me up for a lesson plan in May? Can you say calculus?)

Question #3

Mr. Cox, if I pull the car back 3", it goes 30", but if she pulls it back 3", it goes 36". Why?

Turns out that one kid pushed down on the car harder than the other which kept the tires from sliding.

Our Findings
I'm not sure if it is supposed to be or not, but the data was pretty linear. One student wondered why it would be linear since the car takes time to speed up and slow down and the shorter distance it travels, the more energy it is using to simply get up to speed.

Reason #421 Twitter is awesome

I tweet some pics of the activity and Frank asks me if I'm going to have a contest to see which group can get their car closest to the line.

*ahem* Of course I'm going to have a contest at the end to see who can best predict the distance their car travels.

Three groups were able to get within 1.5". (Two of them were the groups whose cars turned). The best was this:

Mixed Meta Four

2010-08-03T08:20:00.000-07:00

If you haven't read about the Wolverine_[1] (here and here) or Sam's Exasperating Problem, ~~you need to get your priorities straight~~ go read 'em now. Sam's wondering if the problem can be scaled down so a precalculus class can handle it. I see the problem and think, "GEOGEBRA! I CAN USE GEOGEBRA!" Mr. H beat me to it. (If you haven't seen his applet, ~~I question your dedication to the cause~~ go ahead, we'll wait.)

I love this! We are always looking for ways to iterate problems and extend them, but there's nothing to extend with this problem. It's all ready for the wolverine wrangler to do his stuff. I'm looking for the guy who can make this wolverine sit and quit bearing its teeth so my 8th graders can pet it for a second. GeoGebra does this. Mr. H's applet makes this problem accessible to an 8 year old. In fact, my son was so mesmerized by the animation that I swear I heard him muttering, "Heffalumps and Woozles. Heffalumps and Woozles." Heck, I found a strange urge to put on some Pink Floyd myself.

Can you imagine starting a problem in middle school and finishing it with calculus? That's how beautiful (that's right I said it!) this problem is. Why can't we let these younger kids see the beauty of the wolverine without actually having to be the one to handle it? I can see posing the problem, setting the kids up with GeoGebra (with minimal prerequisites) and turning them loose. They'll see the pattern, make a conjecture and inductively decide the answer. Show the applet which demonstrates the first 360 cases and inevitably, the question will be:

Why?!

Now, talk about storytelling. The table's been set for the sequel that the kid's gonna have to wait a couple of years to see. precalculus kids can actually calculate the answer and the trilogy will be complete once they have the tools to actually prove that for n chords, the product is n+1. This problem can span four years. At least.

_[1] Apologies if I misused the metaphor.

Creating a Culture of Questions

2010-07-27T21:58:00.000-07:00

Virtual Conference on Soft Skills _[1]

[Note: This post was pretty much written by all of you. There are no unique ideas here (save for a few anecdotes) but I still found it productive to try to flesh out exactly what it is that I do to promote this culture of questions in my classroom. It's a huge part of what I do, but I'm afraid my ability to articulate the process may be lacking. I asked for a bunch of feedback via facebook/email from former students for this; kids ranging from the classes of '99 to '14 and they really helped shape this post.] So here it goes:

I learned how to learn when I was in college. No one told me. It just happened. As a teacher I have tried to help this process along a bit for my students because it kinda pissed me off that I spent 14 years in school and no one actually told me, "Learning is about the questions you ask, not the answers." So that pretty sums up my teaching philosophy. It hasn't changed much in 16 years.

Kids don't get that. They think that as long as they get the right answer, who cares about the how and the why? Questions and answers are on opposite teams. Answers get my work put on the refrigerator. Questions mean I don't know the answers. Answers mean I know. Questions mean I don't know. I can't let 'em know I don't know.

That's wrong. It's our job to change that. It's not easy and you have to set the table for yourself. The pillars to a questioning classroom involve: Truth, Trust, Togetherness and Transparency.

Truth

I talk to my students. A lot. In just about every year I've taught, I've had some variation of the same discussion with my classes. For some classes, this talk happens in the first week. For others it doesn't happen until the second semester. But inevitably, it happens.

It goes something like this:

Suppose all the information in the world is contained in this circle.

Here's you.

Some of them get offended until I show them this:

We're kinda in the same boat. Not much to brag about and not much to be embarrassed about.

What happens when we share?

And the more we learn, the sooner we realize that this circle is waaaay too small.
If our goal is to learn everything there is to learn, then we are chasing a finish line that's moving away from us much faster than we're moving towards it.

The only way we begin to know is when we realize that we don't know...and become OK with that.

So, how do we share our knowledge?

Questions.

Trust

Once we've established that we're all really in the same boat, there's no excuse for false pride. It's time to build trust. Do I really believe what I told them? I make it very clear to my students that I'm not smarter than they are; I just have more experience. I make it very clear that I don't tolerate anyone looking down on others for asking questions. I'm not so much of a there's no such thing as a dumb question kinda guy as much as I am a who are you to look down on someone who doesn't know? You didn't know the first time you tried kinda guy. This always leads to a digression about the first time we all tried to walk or ride a bike (my kids turn out to be great fodder for these kinds of discussions) which furthers the trust factor as I let them into my life a bit.

I don't let my students say, "That's easy" when someone asks a question because it deflates the questioner. Quickly. First time that comes out of someone's mouth I say, "Everything's easy once you get it."

I have to be honest, establishing trust was much easier for me when I taught high school. It's been a bit more of a challenge with 7th and 8th graders because I have a Kate-ish thing for truth. Sometimes I have to dial it back a bit without becoming falsely warm and fuzzy.

Togetherness

It takes some students quite a while to adapt to my questioning style in class. I've had kids want to drop my class (especially when I was at the high school) because ~~"he doesn't give me the answers"~~ "he never answers my questions." It's tough sometimes because kids are resolute. They'll try to corner you into taking the pencil out of their hand. The key is consistency. The more questions I ask, the more willing they are to ask.

One of the things I've done to help establish a willingness to ask questions is give the class lateral thinking puzzles. I mean, c'mon, there's no way you can guess the answer on some of these on the first try. Asking questions and being wrong are part of the process because they help us eliminate potential possibilities. It takes kids time to figure this out because they want to be specific with their questions at first which is very counterproductive. They have to learn to start with very general questions and the yes/no answer tells them whether or not to continue down that path. Once they get the hang of this, they start to realize that there are common themes in many of these puzzles which can be applied to later puzzles. This works nicely with problem solving strategies down the road.

At some point I throw out the challenge for them to find a lateral thinking puzzle that will stump me, which can't be done (or so I say). The trash talk ensues and I model the heck out of how one goes from the very general question to more specific questions.

The important thing here is that we do these as a class, together.

Transparency

Transparency has two meanings here: The I'm-not-here-to-trick-you-here's-really-what-I expect kind of transparency and the physical-posture-I'll-take-in-class-so-become-invisible kind of transparency.

Expectations
Expectations/standards/topics...whatever, have to be absolutely clear. The What? can't change. It's the How? that is up for discussion. I try things. I show my students first hand that I don't have all the answers and I am constantly trying to find better ways to help them learn. I fail. A lot. That's because I don't believe I have to have all the kinks worked out before I do something with my students. Our entire class is one big question and that question is this:

What do we need to do in order for us to learn?

Invisibility
I don't have just one (physical) focal point in my room. I have a SmartBoard on one wall, a dry erase board on the adjacent wall, multiple dry erase easels spread out throughout the room and I roam around using a wireless tablet that lets me annotate my slides. The focal point is the content, not a person. Students need to learn that the teacher is just one of many resources and doesn't necessarily have to be the primary resource. This means that I have to make myself invisible at times. I usually walk to the opposite side of the room as the person who is talking or maybe I actually take a seat during discussion and stare at the floor. I've found that kids start to depend on each other if I'm not there (so to speak). And when they depend on each other, they tend to start asking questions.

So, I guess at the end of the day, I try to be as real with my students as I can. This all comes down to relationships founded on truth; a truth that we can only catch glimpses of. We often times beat ourselves up because we don't see the fruit of our labor. These "soft skills" (who coined that term, anyway?) are really the reason we do what we do. We spend a copious number hours finding ways to offer immediate feedback to our students but our feedback is much more slow cookin'. We won't know if the time we spend with our kids will pay them dividends down the road, especially when it comes to these "soft skills." That comes when we see our students after they have finished college (or maybe they didn't go to college and went straight to work) and started their own families. That's when we see the fruit. So be patient, the harvest is comin'.

_[1] Huge thanks to Riley for putting this together. The posts that have been assembled have been phenomenal and everyone who took the time to put something together should be commended.

Update (as per Dave's request)
My class layout. I teach in a an "art room" but since we don't offer classes like that anymore, I'm in it. Students sit in pods (equipped with outlets making computer time nice), 4 or 5 to a group depending on class size. I'm hardly in the same place for more than 5 minutes.

What We Have Here...

2010-07-26T16:52:00.001-07:00

Last month I sent out this tweet as a result of a conversation I was having with my principal. He was telling me about a workshop he had attended about leadership and the facilitator said that the way one leads must depend on the generation of those being led.
For example, if a baby boomer is called into the office, he assumes that it must be bad news since he's been pretty much left alone his whole life unless he's screwed up. Whereas a worker from Generation Y would think that the same call to te office would signify some sort of attaboy since he's been raised by helicopter parents who made sure juice boxes, snacks and trophies came after every game win or lose "just for being you." (I guess the Gen Xers never got out of bed to get the note.)

I called BS and sent out the tweet knowing full well that I was sending it out to a skewed sample on account of, as Dan so aptly put it, y'all are freaks.

Your responses were overwhelmingly negative mixed in with a couple of "they probably want me to take on some new reaponsibility" and Matt's "I'm getting a promotion" (which he already got, so that's kinda like cheating). I was surprised to know that you are all so scared of your admin until I remembered a few years back when I got an email from my principal asking I "come see her at brunch", how I freaked out and responded with, "is there anything I can do to prepare?" only to find out that I was being surprised with the Wal*Mart Teacher of the Year award (still have the vest to prove it).

This makes me wonder:

WHY?

Why do we assume the negative? And more importantly, why do our students? Again I'll say, they see us the same way we see our administrators. It's no wonder they don't want to ask questions in class.

#sbar to #fubar

2010-07-18T15:23:00.001-07:00

Careful there, folks. Standards based grading isn't gonna fix everything. So before you go and throw yourself in front of the train, there's a few things you should know.

1. The ideas behind SBAR aren't new. Teachers were caring about student learning like 10, 20—3000 years ago. Back then, they called it "doing their job."

2. You're still going to have to distill all this learning into some sort of a letter grade thereby making it a shadow of the real thing.

3. All perfect assessments aren't. (see #3 here)

4. Don't overdefine this stuff. This will keep a good idea from becoming just another term in a long list of edujargon that forces people to earn a Master's Degree just so they can learn how to metacognate their pedagogical assessment practice and use it in a sentence.

5. I typed this on my phone with my right thumb during nap time.

Other than that, I like standards based grading. Just trying to keep it real.

What Experience Gives You

2010-07-16T09:34:00.001-07:00

1. You're not going to save the world so quit beating yourself up over the one(s) that got away.

2. You're not going to destroy the world so shake off the bad lesson/day/week/month—whatever. I've had to shake off a bad year—or three.

3. No matter how much time you put into developing the perfect assessment system, you'll rarely be surprised at what your kids can do. They show you every day—if you pay attention.

4. It is OK to smile before Christmas. In fact, the teachers who told you that are probably the reason your kids hate school.

5. Let 'em go to the bathroom.

6. You really don't have to spend 60-80 hours per week to be effective. It's tough to keep that pace up for a career.

7. Teachers:Admin::Students:Teachers. You see staff meetings the same way students see your class. How will you deal with that?

8. When dealing with staff, make few statements but ask lots of questions. Asking questions always works because they're fair. Sometimes you have to get people to actually verbalize their bad ideas before they see the fault in them.

9. Lighten up a little.

10. Not everything you try is going to work. So what? Keep trying; it sets a good example for your students.

Adventures in Pedagogy: Lessons From Limes

2010-07-09T22:14:00.000-07:00

I'm cutting some limes in half then quartering the halves when Nevan (my 8 year old) walks up, watches for a moment and says:

8 slices per lime.
Oh yeah? I just cut 4 limes, so how many slices are there?

36.
36, huh? How'd you come up with that?

8 and 8 is 16 and 16 times two is...no wait, 34. There's 34 slices.
Alright, explain that one then.

*Begins using finger and countertop as if it's chalk and slate*
No, no, no. It's 32. 32 slices.
Why?

Well it has to be even.
*teachable moment antennae go up*
Oh? Why even?

16 plus 16 has to be even because an even plus an even is an even.
You sure?

Yeah. Even plus even is even and odd plus odd is even too. For example, 8 plus 8 is 16. Even. And 7 plus 7 is 14. Even.
So how do we get an odd?

We get those by adding an even plus an odd.
Nuh uh.

Yeah, lemme show you.
*And he does*

You know, this is just too easy. Math is all over the place around here. I'm a big fan of WCYDWT and all, but I've always struggled with running into a great problem that fits something I have to teach. But since we've decided to take over the education of our children, I don't care what fits. If it's interesting, we're talking about it.

The best part of this is that I don't even have to bring in pictures of hot dogs and buns in order to teach LCM. I can actually have that conversation with my son in the grocery store on July 3 because we really don't want to have food go to waste.

That's it! I'm homeschooling my students next year.

Adventures in Pedagogy: Conversations

2010-07-07T10:58:00.000-07:00

My 11 year old was asked to describe what he could about the relationships between 1/a and 1/b if a < b (a, b are Natural numbers) and a/b and b/a.

He explained to me that 1/a > 1/b since it had a smaller denominator.

Why?

Well because if the denominator is larger, it's broken up into smaller pieces.

OK, so what about a/b and b/a?

a/b is a proper fraction and b/a is an improper fraction.

Then which one's bigger?

b/a.

Why?

*thinks for a moment*

Because b/a will have a whole number and a/b won't.

Then what can you tell me about all improper fractions?

*looks confused*

I get up and motion for Dawson to do the same.

If I'm 0 and you're 1, where would the proper fractions be?

*points between us*
Here.

Then where are the improper fractions?

*points away from me*
There.

Then what can you tell me about all improper fractions?

They're all bigger than 1.

Then what can you tell me about all proper...

They're all between 0 and 1.

There you go.

So do I have to do this problem?

Didn't you just do it?

Well yeah, but I didn't write it down.

Will writing it down help you understand it better?

Not really.

*Looking my son in the eyes*

*Smiles*
Cool.

Adventures in Pedagogy: Questioning

2010-06-29T17:00:00.000-07:00

Questioning is a huge part of my own classroom but working with my own kids has been a great mirror for me. I missed a few opportunities with my son here. See if you can pick them out.

YouTube Video

Adventures in Pedagogy: Motivation

2010-06-28T16:01:00.001-07:00

I'm not exactly sure what to make of it, but my children's motivation towards learning can be modeled by y = k/x where y = motivation and x = age. We've got Aidan (5)...

YouTube Video

And Nevan (8)...

...who was dialed-in to this really simple applet I built for Dawson (11).

That's not to say Dawson has no motivation, but it's clear that he's been affected by the point- gathering-culture that is school.

Nevan spent a lot of time simply investigating integer addition and it isn't even on his radar right now as far as planned curriculum goes. He played with (+)+(+) and moved on his own to (-)+(-).

During lunch he asked, "Dad, can you teach me to add a positive plus a negative?"

Guess what my answer was. Go ahead, guess.

I built a couple of assessments in ExamView for them and Dawson was all jacked at his instant-feedback-Dad-I-got-em-all-right performance.

Now they're asking for assessments.

We're not paying them, either. With points or anything.

Problem Solving: Rubric

2010-06-26T20:54:00.000-07:00

This one's a follow up to a previous post on which I'd still appreciate some push-back. (So if you haven't read that one, go there first) I'm still convinced that problem-solving/synthesis/application or anything upstream from skill duplication needs to be handled separately from the skill itself. That's not to say that a student can't validate a score on a particular skill(s) based on how they do on a richer assessment, but these types of problems are difficult to reassess for reasons stated in the Taxonomy post. I did some tinkering with a rubric and tried it out with my 7th graders. I liked what I saw.

I'm trying to make my students aware of four things:

There's often times more than one way to solve a problem.
There are usually four ways to represent a problem/solution: verbally, numerically, symbolically and graphically.
Multiple skills can be put together to create new understanding.
Strategies can often be generalized into a rule.

I gave my students a pretty open ended problem...

Tell me everything you can about the relationship described by the points: (2, 5) (-1, -1) and (5, 11).

...and gave them a page with two copies of this rubric. One for them to fill out when they finished the problem and one for me to fill out after they turned it in.

I'm not completely pleased with this rubric because there will be times when there may not be multiple points of entry to a problem or all four ways to represent a problem may not be applicable. Giving them the rubric was huge. From that day on, student's showing work ceased being a problem. They all realized that everything in class became an opportunity for assessment, I was clear about my expectations and they rose to meet them.

How does this work with SBG?
Each of the four columns will have it's own standard. I may weigh them as I think this is the stuff we're really after. I will also allow a student to call on skills to be validated. For example, if a student can identify finding slope as a skill needed to solve this problem and performs it correctly, the score on the slope standard becomes a 5. I do think that the student needs to identify the skill in order for it to be validated, though.

I know these ideas need some sharpening and I may be missing something, but I'm throwing them out to you anyway. So in all your free time...

Adventures in Pedagogy: Experiments

2010-06-21T08:50:00.000-07:00

Homeschooling has been interesting. Suffice it to say, my kids have developed some bad habits very quickly. It's hard to admit that I've let my own children become answer chasers right under my nose. I ask questions and we have a lot of conversations, but in a typical school day, my kids are gone from 8:00am-3:30pm and then have ~~busywork~~ ~~waste-of-time-packet-work~~, er, homework to do. By the time they were done, there wasn't really any desire to talk about the ~~hoops they were jumping through~~ stuff they were learning.

This morning was fantastic. Nevan, my 8-year-old, was stoked that he had a science experiment to do. He was looking through his text and saw the experiments and just lit up when my wife told him we were actually going to do them.

"Mom, my book at school had all kinds of experiments, but we never got to do them."

Wait, what?

Now, I'm no science teacher, but I couldn't agree more with Shawn's assessment of the situation.

I think the look on this kid's face speaks for itself.

My Apologies

2010-06-18T22:11:00.000-07:00

...but I can't get this one out of my head:

"From Wojtyla's perspective, the moral relativism that has resulted from the modern "turn to the subject" is only the product of an anthropological stagnation in the absence of faith. In other words, man turns to himself and "stays" there, failing to see that his own humanity points him beyond himself, failing to see that anthropology points to theology

(Christopher West, Theology of the Body Explained)

Not trying to get preachy here, promise. But here's the thing:

We spend all kinds of time discussing how to better educate our students. For what? Aren't we using our content areas as vehicles to produce better citizens? (or at least that's what we say.) It's not about math, English, science, social studies or P. freakin' E. It's about human beings. It's about the dignity of the individual, right? But if we never help students get past the fact the world doesn't revolve around them, what good is it? Why bother?

So if we are trying to help students get past self and become part of a greater good, what is this greater good? Where does faith enter into this equation? Or does it? Should it?

We have problems in our schools. Serious problems. Most of our kids could give a rip about the Pythagorean Theorem, slope-intercept form or whether or not they put their name in the right place on the paper. Our kids are trying to figure out where they fit into this world and they're thinking "if it's not about me, then who's it about, you? Why should I care about you?"

Most of us don't want to discuss our politics let alone our theology. It's too personal. But there's no way it doesn't affect our pedagogy.

Are we really the answer? Or as we become a little more self-aware should that point us to something greater us? If so, where's that in the curriculum? If it's about more than us, yet us is the only thing we can talk about in schools, we're trying to build a fire without oxygen.

Ground Floor

2010-06-18T13:22:00.000-07:00

About three weeks before the end of the school year, I had a meeting with our district's IT director. We had a really good discussion about how our we have committed many resources towards keeping the tech running but nothing in the way of developing a plan for implementation. This conversation quickly turned into the district deciding to free up my afternoons to help with this.

Things I know

Technology isn't going to make a bad teacher a good teacher.
Using tech to do the exact same things we've done without tech isn't productive.
Teachers don't like change.
Anything that comes from the district office is treated like a mandate and resisted.
"How do I use this?" is the wrong question.
"Why should I use this?" is a better question.
Professional development in the way of workshops isn't very effective.
Conversations are.

Questions I have

Where does assessment/grading fit into this conversation?
Are we looking to transform what we do in the classroom?
Why should anyone listen to what I have to say?
Where do I start?

I'm not really interested in throwing a bunch of tools into the laps of teachers. However, I'm very interested in engaging in conversations with teachers about why we do what we do and questioning whether or not there exists a tool that will help us do things better. I threw a bunch of these questions in the lap of my IT director and found myself sitting before our superintendent of curriculum and instruction.

I can't imagine the meeting going better. It was great to see someone with 40+ years in education saying that we need to look at what we do and why we do it. Many of the questions I had, she had as well. One thing we knew for sure was that this vision is fluid and we are in the process of creating it. Questions are encouraged. Push back is needed. Transparency is critical.

Lots of rope. Hope I don't hang myself with it.

What Else Ya Got?

2010-06-12T07:07:00.000-07:00

I know that there are many who are questioning whether or not to make the jump to standards based grading for whatever reason. But the more I think about my own children's education, the more I realize that anything else is crap. My wife and I are probably going to homeschool our boys next year and you know what our main question is? What do we want them to learn?

That's it.

Grades? Nope.

Due dates? Nope.

Packets? Heck no!

Homework? That'd be kinda redundant.

Finish when you finish. Learn when you learn.

Sure, in our classrooms, we have to teach things because we have deadlines imposed upon us and we have to work within them. Kids' learning doesn't give a rip about our deadlines.

So when you decide how you're going to do it next year ask yourself this question:

How would you want your own kids to be taught?

Deal With It

2010-06-11T16:33:00.000-07:00

So what do you do?

My 8 year old wanted to stay behind with my wife and two of his brothers to do some more shopping while I took the rest of the clan home. As we were leaving the parking lot of our local Costco, Nevan changed his mind. He wanted to go with Dad. Alright, so we made the quick switch and we went our separate ways. Unfortunately, Nevan thought twice about his decision but it was too late. I had frozen foods and a 35 minute drive ahead, so, tough luck son.

Then came the meltdown. And what a meltdown it was.

I've seen so many parents use the redirect with a kid having a meltdown. Make 'em think about something else and the behavior changes. Seems simple enough, but the problem with that is that the kid learns nothing. It's no more than behavior modification via manipulation. Yeah, I don't usually take that route.

I think it's important for my son to learn how his thinking is selfish. Can he be disappointed? Sure. But does he need to make everyone else pay for it? No way.

So we have a conversation about how whining doesn't do anything to change things. We talk about how he was allowed to make a decision and now he has to deal with the consequences of his decision--for better or worse. We discuss how it isn't fair to the others around him for him to carry on as he was. Sure you're disappointed. Sure you're big brown eyes well up with those crocodile tears which make me, for a moment, reconsider the importance of getting the bag of frozen berries home before they melt. But the meltdown has to stop.

Would it have been easier to redirect? Yeah, but it would have been a temporary fix. I want my kids to grow up being able to take their thoughts captive and keep them from becoming behaviors. I don't want to paint over the cracks in my kids character; I want to fix the foundation. That takes time. Lots of time.

So does teaching. When you see a kid doing a procedure incorrectly, do you just show them the right way or even worse (and if you do this, please stop) take the pencil out of their hands and do it for them? Or if you are discussing a concept and a student demonstrates a misconception, do you simply correct it or do you ask questions to get them to see the error of their thinking?

Good teaching takes time. Patience. Relationships.

Everything else is just painting cracks.