Online Math Teacher Community Survey

Bet you were surprised to see a (2) next to my blog in your Google Reader!

After NCTM, Kate and I have been talking about working on a "Welcome to the internet, math teachers" page where we could help new teachers get involved with the online math teacher community more easily. To start the process we've created this survey for active members of the community.(commenters, bloggers, tweeters, lurkers are all welcome to fill it out) If you feel the online math teaching community has helped you out, please take the survey so we can help get others excited and involved!

Geogebra Derivatives with Limits Activity

Yesterday morning I had an epiphany and started on a new idea for a lesson which was rolled out exactly 43 minutes later. I've been trying to use GeoGebra more in my classes and I thought of an idea that would not only let me do that, but also help out with the actual concept of limits which some of my students were struggling with.  While they can calculate limits pretty well and can throw back some of the things I've thrown at them(phrases like "what the function is approaching" and "what the point should be"), I've had this underlying feeling that there's still something missing in their understanding.  We're also moving into derivatives, and I wanted a nice visual way to show them what exactly we are going to do.  So I whipped up a few of these:


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)


I split the class into a few small groups, had each group use their laptops to manipulate the GeoGebra files and gave them this worksheet which has them find various slopes at points on the curves, then had them explain their process and discuss what exactly that has to do with limits. The graphs increased in difficulty so the first answer was 0, which is intuitive just by moving the point around.  The second answer was 1, which is also pretty intuitive, but a little more challenging.  For questions 3 and 4 they had to get a little deeper into how exactly how to calculate the slope.

It was important to me that I didn't tell them *how* exactly how to find the slope, so that they could figure it out on their own. I did mention that if they put the points directly on top of each other, they wouldn't have a line anymore. For most of the students, I felt this activity went over really well.(A few students did struggle with not being told exactly how to do it and how exactly to discuss it with their group to make forward progress) For the most part they were able to figure out how to get the different slopes. A couple had an issue with the question about how this dealt with limits until the following exchanges:

"Okay so what exactly did you do to find it"
-"We just moved the point as close as we could to... Ooooh!"

or

"Hey wait a minute, you were assuming that point was (4, 6) but it's actually not exactly"
-"So as I move the point it's getting closer and closer to (4, 6) so.. Ooooh!!!"

That "Ooooh" moment was awesome, and exactly what I was looking to see.  Through this activity, a lot of my students ended up with a better understanding of what limits are, and the stage was set much better this year for coming up with the limit definition of a derivative.  I'm hoping this will lead to my students to a much better understanding of the relationship between limits and derivatives this year, but we'll have to wait and see.

I didn't end up with a ton of time to plan this activity, and though I'm happy with what came of it I would welcome any comments you have for improving it.

NCTM thoughts aka We don't even exist!

aka You won't care about any of this if you aren't a math teacher.

I went to NCTM last week, where I met up with Kate, Jessica, Nick and Jackie.  I had a good time, and as I don't have a lot of real life high school math teacher friends, it was a lot of fun to have people that can completely relate to everything I do for my job everyday.  Fun random story:  In the first session I went to, this girl briskly walked out and said to the guy next to her "Pseu-do-con-text"  I laughed because of what she said and because she tripped over a bookbag immediately after. Later when I met up with other online math teachers, it turned out that girl was none other than @smallesttwine! Anyway...

If I had to choose one thing that stood out that I learned from NCTM it's that a ridiculous number of math teachers are completely unaware of the online math teacher community. I naively had it in my mind that when I walked around the conference center with *the* Kate Nowak that math teacher groupies would be startstruck left and right.  This was not the case.  In fact, I was given the impression that the 5 of us that met to hang out were some of the only few people in attendance who were aware of the amazing online math teacher community that we have here between blogging, comments and twitter.  I have learned quite a bit, found some awesome lessons, and gotten plenty of help from said community, yet there are so few math teachers out there seem to know about us.  I mean when you think about it, with 4000 teachers in attendance, and 5 of us... that's 0.125%  and I didn't forget to move the decimal point!  To a statistician, WE DON'T EVEN EXIST!  Granted, there could have been more in attendance that we were unaware of, but I doubt it could be that many.  Out of curiosity I started asking people near me at a few of the workshops if they knew of Dan Meyer and none of them did. (a sample size of like 5 is good enough, right?)

At the conference, a workshop might go through one or two good problems or lessons in an hour, and it was only possible to go to a few workshops a day.  There a tons and tons of great problems and lessons online right now that teachers could find if they knew where to look.  So I think it would be awesome if we could come together and work towards building our little corner of the Internet.  Some ideas I'm thinking about are working collaboratively on a "Welcome to the online math teacher community" introduction/roadmap/guide for newbs and working together to make a live presentation₁ that we could give in our own respective corners of the world. Watch this space, and let's get the word out and show off how awesome this community is.

_{1- We could use Google Wave! Oh wait...}

Pimp My Catapult

*Fffwwwooooooooooooooooooosssshh* That is the sound of me blowing the dust off of my blog. Welcome back to my little corner of the Interwebs. Between getting married, buying a house and my math programming side project I've had little time to share anything on here, but I'm back in action and ready to start updating more regularly, so here we go...

I've been tweaking my catapult design a bit each year, and think I've finally found the best design to fire accurately(even in student hands).  So if you tried the catapult project last year, but had trouble with student accuracy this post might help if you're willing to give it another shot. If you haven't tried the project yet but want to this year, I would highly suggest using the design and tips below.  Here is the new design:

Highlights:

1)   Mini clothespins give a more appropriate distance and height.

2)   I now make the basket out of a thin strip of paper, complete the circle with a small piece of scotch tape and use a bit of glue to hold the basket on.  Having a basket that is just barely bigger than an m&m laying flat helps accuracy.  Make sure students know to load the m&m laying flat each time. 

3)   Cut off the rounded edge of the popsicle stick behind the basket.  The flat edge helps fire more consistently.

4)   This new firing mechanism allows student to essentially just pressing a button to fire.  There is still room for some error here, so make sure they know to hit the button SLOWLY each time. The back clothespin is positioned so that it just barely overlaps the back of the catapult when it is all the way down.  Here's a close up with the catapult loaded:

 5)   The front of the catapult is tilted forward so that it goes further and not as high.

6)   Okay, so there's no 6 on the picture, but thinking about it now it would make more sense to move the guide rails on the main catapult much closer to the back.

So there you have it. I hope this helps.  Let me know if you have any other ideas for improvements in the comments.

Previous catapult posts:
http://sweeneymath.blogspot.com/2009/08/m-catapult-project-pt-1-catapult-plans.html
http://sweeneymath.blogspot.com/2009/09/m-catapult-project-pt-2-project.html


BONUS:  I let one of my classes build their own catapults last year, and after finishing the project one group decided to come in to my classroom during some free periods and make this ridiculously awesome trebuchet.

Yearbook Signatures

Posts have been lacking in the last 2 weeks because I've started developing a math note/workbook program for students to more easily take notes or do work on problems, which has been taking up all of my free time.

Pretty much every time I try to write words in a yearbook for a student it comes out completely and utterly cliche. (You're a great student! Have a good summer!  Good luck!)  So, I've decided to stop writing words altogether and usually draw a picture or do some sort of relevant-to-them thing.  Here's a solid work of genius I came up with today:

The answer is pretty worth it, IMHO, so give it a try if you've got some free time!  You can do the harder parts with wolframalpha if you want to cheat. (The integral from ___ of ___, the derivative of ___ at ___, the lim as x approaches ___ of ___)  Sam and Dave, this is a no calculator question for you!

PS- I triple checked the answer would come out right, but I still have a sinking feeling I messed something up.

Exam Preparation Reflection

Last year I tried something new with my seniors.   A lot of my students(and I'm sure yours too) tend to not realize that their preparation for a test has a lot to do with the outcome.  So many seem to think that you are just born with math ability and that directly determines how well you end up doing on tests.  I wanted them to be able to reflect on how they prepared for an exam, and how that led to their grade.  I started crafting a project for my students to help with this.  I had the following goals:

1) Students who give 100% in every class and don't have to study much shouldn't be punished
2) Students should have the option of not doing anything at all.  The fact is some kids don't study, and I want them to be able to face it without punishing them further.
3) Students who work hard should be rewarded.
4) Students would be forced to think about how they prepared while seeing their exam grade.

So, I decided that for the project they would hand in all of the materials they used to study and write up a one paragraph explanation detailing exactly what they did.  I made it clear that if they wanted to they could hand in a paper that said "I didn't do anything to prepare" and it wouldn't be detrimental to their grade so long as they did well on their exam.  The worst grade they could get on their project was their exam grade but they could do better on their project if they prepared well.  I gave them the previous year's exam as practice and gave them some general suggestions on what they could do to best prepare.

I thought this would be a good reflection for my students and that they would learn a lot and it was, but it turned out that I learned from it too.  I had been under the assumption that some students studied well, but struggled on my tests and that other students didn't prepare at all, but ended up doing well.  To my surprise, students' grades lined up almost exactly with the way they prepared.

We spent the whole class on the Monday after exams reflecting on exam prep. I opened by having each student write down their answers to the questions below.

In preparing for your exam...

What did you do that worked?

What did you do that didn't work?

What could you have done better?

I handed back their graded exams and we went through the following stats one at a time(red results are from this year):

Class Average: 83% (83.9%)
Highest grade : 93% (99%)
Average of Students who didn't turn anything in: 72%
Average of Students who "looked over their notes": 74%
Average of students who didn't do any of the practice exam: 75%
Average of students who did the easy problems for them on the practice exam: 79% (76.8%)
Average of the students who did all of the hard parts of the practice exam: 87.4% (88.2%)
Average of students who identified and targeted what was difficult for them: 90% (97.25%)

Then we discussed the results and talked about effective studying.  We talked about how "looking over notes" didn't do much of anything, and how even doing problems that are easy for them didn't really do much either.  We talked about how the largest jump came from the people who made sure to figure out how to do the problems that were harder for them and practiced those problems.  We went over some highlights of the Sweeney Study Method and their answers to the questions from the beginning of class, then used the remaining time to start on test corrections.

Overall, I was really happy with the results of this project.  It forced my students to see how their study habits directly influence their exam grades, and was an interesting learning experience for me as well.  I'd really be interested in seeing results from other classes, so if you try something like this, make sure to let me know how it works out!

The Banana Rule

Another concrete rule I use to help my students is the Banana Rule.  I noticed that my students often struggle with simplifying things like this(when we aren't directly learning about it):

Sometimes they don't realize that you can add them, but often times they realize they are like terms but end up messing with the exponents.  To me, it's really just a matter of how they are looking at it.  My best attempt at getting my students to see this problem the way that I see it is by telling them that if everything in a term is the same except for the coefficient(1) is the same you can think of it as a banana. Therefore, the problem above is simply:

Now, they see the answer is 4, but don't totally get it right away, so the banana explanation is quickly followed by:

"So, when you add a banana and 3 bananas, does the fact that you're adding bananas change?"  

"No!"  

"So the answer is 4 bananas, and what did we say was a banana?", etc

During that exchange the lightbulb goes off and they get it.

Just like the rainbow rule, I'll go through this with a class the first time time it comes up naturally in some problem, and then refresh it as it comes up throughout the year.  This method seems to reinforce that it's a tool, and not just "this is how you do this kind of problem".  If I have time, I might go off on a tangent about how it works with anything even if they haven't seen it before and go through some quick examples with trig, logs, integrals or crazy fractions and roots.  Also like rainbow rule, it helps kids put a name to something they struggle with and attaches an intuitive process to it.

(1)Okay, I usually say "number in front" instead of coefficient at this juncture, but we're trying to make it simple right? Please let me keep my math teacher card.

The Rainbow Rule

A couple weeks ago Sam urged people to talk about little tips and tricks done in their classrooms.  Since I found THE CLAW so helpful, I thought it would be only fair if I shared a trick of mine.

The Rainbow rule, a Sweeney original, is my favorite as it's useful for all of my classes from Algebra 1 to Honors Calculus.

Whether my algebra students come across this...

or my calculus students come across this...

...they either don't seem to know how to do it, or are confident about doing it the wrong way.

 Enter the rainbow rule:

Since we don't all have the luxury of excess time and image editing software when working out problems, the actual version looks and works like this:

This rule has been very helpful for my students because it gives a name to this situation which is both easy to remember and helps to avoid the common mistake of crossing the two lines (rainbows don't do that, after all).  While not every students gets it totally right, it still has really improved my students' ability to deal with fractions, especially those who I've had for more than a year.

Student centered learning using WolframAlpha

Recently I had a really successful lesson on exponent rules.  Every year I go through exponent rules with my algebra 1 class, and though it's likely they've seen much of them before, their retention is such that it feels like they are seeing everything for the first time.  This year I tried to change it up a bit in order to do two things that I've been trying to get right this year:  Increasing my student centered learning and using wolfram alpha as an effective learning tool in class.





At the beginning of class, students were given what was last year's quiz I gave after teaching and practicing some exponent rules.  The instructions:

1)       Answer every question with your best guess.  We haven't learned this stuff yet, so I don't expect you to get many of the questions right, but I do want you to try to make some sort of guess for each.

2)      Use your laptop to go to wolfram alpha to check each of your answers.  Write the correct answer separate from your original answer and try to figure out what each exponent rule is.

3)     For each answer you got wrong originally, make up 3 similar problems that use the same rule(s), answer them and check them using Wolfram Alpha.

4)    Once you are confident that you could get every problem right without help, I will give you your quiz.

The result?  Students did better on the quiz than in previous years where I taught the material.  Granted, there are some other variables involved in that, but it was clear from moving around and talking to them that figuring out the rules themselves gave them a better understanding of what was going on. The whole thing took two 45 minute periods, part of which was getting them used to typing the expressions and a discussion about why they shouldn't use wolfram alpha to do all of their homework for them.

You may be saying to yourself "Hey wait a minute, you could've just given them an answer sheet and done the same thing."  Well yes, and no.  Answer sheets would take away the flexibility of being able to check answers to the problems that they make up.  The making up step also helps them to understand what they are really looking for and is a good strategy they can use to help study for tests.  Also, part of the point(which I discussed with them) was to get them used to using Wolfram alpha to check their work any time, like for homework or studying for instant feedback.

Now as pleased as I am with how this went I'm certainly not ready to give up explaining things at the board ever.  Exponent rules lend themselves well to this sort of activity as they are reasonable to figure out with the answers and I'm definitely going to look for more topics that would work well with this kind of activity in the future.  Can you think of any?

Failure is not an option

I created this blog to provide practical examples of things that teachers can do inside of their classrooms, and wanted to try to avoid posts that were just my opinions on general theory of teaching.  So much for that.  I've been kicking this idea around in my head over the last couple weeks and I feel it's just too important not to share.

"We need to let kids fail so that they can learn important lessons about working hard and resilience.  People are going to have failures in life and they need to prepare for that."  I hear ideas like this a lot throughout blogs, twitter, in professional development sessions and within my school.  Whenever I hear it, I immediately agree.  It makes a lot of sense, and I feel like nowadays kids are given things too easily and protected from setbacks and failure too much.

However, in math especially it simply can't be an option to let kids fail.  When I say "failing" I mean failing a test, or even just failing on an important concept within a test.  I'm not by any means suggesting giving students grades they don't deserve, what I am suggesting is forcing students to deserve good grades in math.  Now this is easier for me because I have fewer students than most teachers, but I think the concept in general still applies to math courses across the board and we can build our classes in a way that helps students to succeed without sweeping problems under the rug.

So why am I presenting such conflicting statements?  I agree that kids should be able to fail, but I don't "let" them fail in my class.  The reason is because of how math is different than other subjects.  Math always builds.  That's not to say there's no building in other courses, but it's more gradual, more encapsulated and there's more chance to catch up.

In math courses, students need much of what they learn to be able to succeed in future math(and science) classes.  Let's say, for instance, a kid fails my test on solving equations in algebra 1 and it's totally his fault.  He didn't pay attention in class, didn't do his homework, didn't study and didn't ask for help.  Now if you ignore what the topic is and just look at what he did, does he deserve to fail? Sure.  Here's where the problem sets in though.  If I just let it go here, I'm not just letting him fail.  I'm setting him up for failure.  Unless this kid is explicitly taught how to solve equations somewhere else(and he very likely won't be), then I'm setting him up for failure in every following high school and college math class as well as Chemistry and Physics.  Am I taking too much responsibility here?  I don't think so.  It's my job to teach him the concepts of Algebra 1, and his future teachers will expect that he knows it if he passes the course.  If I don't intervene it's not as simple as letting him learn his lesson, it's dooming him to future failure as well.

Therefore, it is essential that in math courses students have option to (or better yet, they must) improve upon past grades for topics that they didn't master.  Dan seems to have a pretty good system which I'll get around to implementing at some point, but for now I just have mandatory retests in some cases and optional retests for anything.  I'll also find ways to revisit topics that students generally understood but didn't master (like warm up activities where they have to get every question perfect for some sort of motivator).

Are there topics in math that are less important that I'll let go if a student doesn't totally understand them?  Sure.  In general though, I get freedom over my curriculum so there aren't a whole ton of topics that aren't important later. There certainly aren't enough side topics for a student to be able to fail my class despite mastering the important stuff.

So how are my students going to be taught that important lesson on failure? I say, let their history teacher do it.   Seriously.  Not necessarily history, but anywhere where the loss of one unit isn't possibly the loss of that entire subject.  Living in a world where people advertise if not brag about how bad they are at math when I tell them my profession just solidifies my feelings on this topic.  Students that miss out on really learning how to solve an equation, or other core building topics will grow up to be those people;  There are already too many of them in the world, and I refuse to take part in creating more.



What are your thoughts?