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?

The Sweeney Study Method

To be honest, I'm not entirely sure of the origin of my studying method.  My best guess is that I came up with it my freshman year of college and developed it as the years went by.  I probably used ideas that my teachers had been telling me all along in high school, but weren't necessary at the time since I found if I paid attention in high school, studying wasn't really necessary.  Since I came up with it to be as efficient as possible, I can pitch it to my students as a plan that helps you study quickly and effectively.

I usually don't have time to share the entire method with my classes, but I pick and choose parts to share with them to help throughout.  I do try to share as much as possible with my seniors as they will definitely need to figure out how to study in college if they've just been getting by without doing so in high school.

I wrote this as if talking to a student to make it easier to write.  Keep in mind this strategy works well for any subject.  It was actually more useful for my other classes as I clearly excelled in math.  So, on with the show...

Preparation:  Find somewhere quiet to work with no distractions.  No TV, no instant messages, no texts.  There are educational reasons for this, but the most important reason for me was that if I did that stuff, I would actually have to study for LONGER.  I could get more done in 10 minutes of good studying than I could get done in an hour of studying with distractions.  Why not just get the studying done quickly and then go have fun without having to worry about studying for so long?

Step 1:  Gather your materials.  In theory this should take almost no time to do.  Grab your book, notes and any old tests or quizzes that might be relevant to the test you're going to take.  In math especially, old tests and quizzes are incredibly important because not only do they provide you with a good prediction of what your teacher is going to put on the test/exam, they also tell you where you messed up the first time around. This is most likely where you'll mess up again!

Step 2:  Start with whatever you think is most important.  For instance, old tests would be most important when taking an exam.  Your book might be most important if your teacher takes a lot of questions right from the book.  Now, get a couple blank sheets of paper and a pencil.

Step 3:   Here's the important part.  You're not going to be studying or even reading everything.  It's time to separate the things you know from the things you don't know.  There's no point wasting time with the stuff that you're already sure of, you just want to focus now on the info that you don't remember already.  It's also important to be very honest with yourself.  Scan the book for bold words, information in boxes, diagrams and anything that looks important. Before you read the whole thing, ask yourself if you know it already.  For instance, you see "A quadratic equation..." If you can finish the sentence in your own words "an equation with an x squared in it" then you know it and you don't need to write it down.  It's also worth while to think about the way your teacher tests before writing things down.  You might forget some obscure detail that appears in your book.  Will your teacher test on it?  Maybe not.  If you're sure she won't, then skip it.

If you can't remember something, write it down in your own words. Never ever write down a definition directly from the book.  Math definitions in your book are oftentimes overly complicated and take a lot of decoding.  Instead of decoding the mystery of what the math book is saying every time you study it, translate it to yourself once and write that down.  You might use a picture to help.  Here's an example:  "A parabola is... shoot I don't remember.   The vertex of a quadratic function is... ugh i don't remember that either."  What you write only has to make sense to you.  Here's what I came up with:

So, let's say you read something and once you read it you suddenly "remember" it from class.  It's important to still write this down.  Here you are just familiar with the topic. You might remember it, but you shouldn't assume you're going to after reading it. After all, you did forget it once already didn't you?  So, write down the information you're just familiar with as well as the stuff you don't know at all.

With example problems, it's important to test yourself.  Try each example in your head or on paper if necessary BEFORE looking at the answer in the book.  This will help you decide what you don't know and need to write down, and help you focus on what you don't know.  If there are problems you don't understand and can't figure out: ask your teacher, ask a friend or ask Google.  In math you should really strive to understand every topic with help, and the test should just boil down to testing what you remember.

With old tests and quizzes focus on the problems that you got wrong, but take a look at some of the more difficult problems you got right as well just in case.

Writing down all the information you don't know won't take nearly as long as you think it might. With a focused goal and no distractions, you'll fly through the information. The best part is you're actually studying by putting things into your own words, testing yourself and writing things down.  Once you've written everything down, you're mostly done.

If your teacher allows a sheet of paper on an exam and it only took one piece of paper to write everything down, you're done studying!  Everything you don't know is on a sheet of paper and you can use that sheet for your final.  Sweet!

Step 4:  This step is key.  Now that you've used all of your resources to compile a list of everything you didn't know, get rid of all the resources(tests, books, etc).  Everything you need to know is on this sheet. Get a new blank sheet of paper and start from the top of your note paper. Now, repeat step 3 with your note sheet replacing your book/tests and a new sheet of blank paper.

As you work you're going to realize that you remember a lot of what you didn't know the first time around.  In fact, on your second sheet you'll probably have far less that half of what you had on your first sheet.  There will still be some things you just don't get, and that's okay, just write them down on your NEW note sheet.

You can repeat this part a third or even fourth time if you think it will be useful to you.  If your teacher lets you use an index card, you can stop whenever you think you can fit the information onto an index card, put it on and then you're done!

At some point, there will just be a few lingering things that you can't seem to get in your head.  If that's the case, move onto step 5.

Step 5:  Now that you have a short list of things you just can't seem to get into your brain, it's time to intensely focus on those topics. Use flash cards, make up mnemonics (PEMDAS for example), make up a song, do a bunch of practice problems or just do whatever you can to help you remember those last couple details.  I would usually just write the information over and over again because that seemed to work for me, but what's best for me not be what's best for you.  Try to do something here that suits your learning style.

Once you feel confident with everything, go have some fun!

----

So that's it.  I hope this is helpful to you or your students.  Feel free to share this with your students or other teachers.  If you're feeling extra ambitious you could have students read this post, do the study method for a test or quiz and have them hand in their notes sheet(s) for a project.

I'll have a more studying tips in the future as well as studying project I gave a class last year with some really interesting results.  Stay tuned.

Questions, comments or suggestions? Let me know!

Sweeney Math has been nominated for an award!

The annual Edublog nominations are out, and Sweeney Math has been nominated for the "Best New Blog" category!  I'd like to give a big thanks to those who nominated me.


If you enjoy my blog, I'd be honored to have your vote.


If you haven't perused the Edublog Awards already, it's a great place to find some new and exciting blogs to follow.  Check it out!