Friday, December 16, 2011

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

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.

Monday, December 5, 2011

Iron Sharpens Iron

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.  

Saturday, December 3, 2011

GeoGebra: Leveled Applets

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[ehg, 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.  


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

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.  

Thursday, December 1, 2011

The Timeline of Awesome

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.

Monday, November 28, 2011

Mixed Up Mixture Problems

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.


 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.  

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.

Thursday, October 20, 2011

This is Gonna Hurt

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.

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.

Wednesday, October 5, 2011

The Un-Lecture


3x - 5y = 15

x intercept:  ______

y intercept:  ______


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 



And Finally

Hit the refresh icon and Repeat

Exit Slip

7x - 3y = 21

x intercept: ______

y intercept: ______


90% correct

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

Friday, September 23, 2011

Time to Stretch

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.

Wednesday, September 21, 2011

Time to Breathe

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.

Friday, August 12, 2011

Adventures in Pedagogy: Units Matter

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?"



- Posted using BlogPress from my iPad

Wednesday, July 27, 2011

Popping Popcorn in a Popcorn Popper

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

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.

Wednesday, July 6, 2011

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

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.

Wednesday, June 8, 2011

Adventures in Pedagogy: No Solution

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?"



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?"

"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

Friday, May 27, 2011

And Now a Word From Elisabeth

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.

Thursday, May 26, 2011

Something Different

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.

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

I think you physics folks would say something like this:

Tuesday, May 17, 2011

( )conceptions

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.


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)


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

Sunday, May 8, 2011

A Tale of Two Teachers

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.

Friday, May 6, 2011

Well, Since You Asked

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."

Tuesday, April 19, 2011

Point of View

I can just sit and stare at both of these.

What (if any) questions do these provoke?

Saturday, April 16, 2011

Adventures in Pedagogy: Shrinking Rocks

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

Thursday, April 14, 2011

Teachable Moments

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*


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?"


"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?"


"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

Wednesday, March 30, 2011

Quadratics Revisited: The Falling Object Model (part 2)

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?"


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."


"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?"


"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.

Monday, March 28, 2011

To LCD or Not To LCD

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

Saturday, March 26, 2011

I Made Me Do It

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?).


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 - it looks like you don't have Java installed, please go to