Reflections of a Techie

The Second most famous irrational number & golden ratio

Drawing the nautilus today was our culminating activity of our weeklong study of Fibanacci numbers and the golden ratio.  In days just before we had looked for the golden ratio in our bodies by measuring tapes, studied DaVinci's Vitruvian Man , beforpracticed golden problems.  We'd talked about how the ratio of short/long and long/short...they definitely were tuned into 1.618 and .618.  One student even downloaded the first 10,000 places of phi!!!!

I'd say they were pretty much in love with the second most famous irrational number.

Our read aloud, Wild Fiboncci, was delightful.  We had a great worksheet activity from Fibonacci Numbers and they did this to figure out the technique.  Since we are near the end of the school year, we'd moved to the commons so they could spread out on the floor.  Once they had perfected the technique, I had them move to the lunch tables...and use a scale factor to enlarge the worksheet rectangle onto BIG graph paper.   Some of our drawings were 3 feet across.

Now all this is a great activity, but what was so fun was to think about the math with them.  They first noticed that each square had a Fiboncci number for side length.  They were amazed to find out that it started at 21, then went to 13, then to 8 then to 5 to 3 and so on.  Lots of excitement actually filled the air as they uncovered this truth.  We "wondered" if we blew the drawing up by scale factors of 2, 3 or 5 if it would still work. It did!!!

Then that smarty kid wanted to know if we had a 0.5 scale factor if the pattern would still hold.  I gave them paper and they feverishly worked on it.

It was a great day and a fantastic way to end the school year.  They were excited about math and they were explorers.  Gosh I'll miss them next year.

Finding your voice in writing about math

Yesterday I spent most of the day working with colleagues on how we'd differentiate a new course offered next year.  A "Plus" math class to compliment the regular math class.  My opinions of this decision is a whole other post...and not relevant to thinking about how I'm going to deal with reality.

We went through each unit and pulled the most challenging problems...those that were extensions of the idea into new realms.  Or ones that had students access information from previous units...in the attempt to tie all things together.  I think that works out pretty well.

As we began to talk, a colleague really hammered home the need to "frontload" vocabulary.  If you are working with a math curriculum that is story problem based, it is essential for them to use words as precisely as possible.  That thought then got me considering other things...such as writing in their math journals.

I'm definitely going back to the word wall idea next year.  Don't care if my middle school colleagues laugh.  I think it helps to "read" the wall...use that flashlight to practice vocabulary.

Right now I'm pretty limited in knowing how to teach writing in math class.  I have a set of "signal words" from a LA teacher.  That has helped me require students to improve on their word choice depending on the types of text answer they are attempting.  It has worked pretty well, but it is still quite difficult.  At the beginning of the year when students would try and describe a process they'd used to solve the problem, you'd hear all sorts of artiifical langauage...they'd inject after, before, later, then, when, not long after, etc etc etc.  No matter how well it fit the situation, it was there...maybe all of them would be there!!!!!  Slowly they began to use the language better.

I still need to improve dramatically in how I teach them to write about what they know in math.  I want to incorporate more than word choice...I really want to expand into organization...and do you think there's voice in this type of writing.  I think so.  If I think about really great expository writing, there's voice in there.  Now I don't know if you consider math journals expository or technical writing.  Note to self:  Find that out.  It probably will help you determine if there's voice or not.

Here's what I really know.  There is a void of writing examples in math.  You try finding something that isn't an elementary school student.  I haven't been able to find a middle school example.  I am going to try and work with our school improvement specialist on this. 

Photos are licensed as Creative Commons images from www.origamitessellations.com/category/mentions/.

Dust off Donald, Let's see the Magic in the Golden Ratio

I'm old so I remember the glory days of Disney.  I have to admit that this little jewel had slipped my memory, but when we dusted it off and used it with our students....wow....the power of Disney to make something interesting came to life.

Donald in MathMagic Land does many great things.  We are in the process of studying ratios, specifically the golden ratio.  This is the perfect movie to show them.  I think one of the most amazing parts of the movie is where Donald shows how this ratio comes into play with musical instruments...ha!  get the pun?  (We have an old VCR copy of the movie, but you'll get the idea from this YouTube clip.)

Well, we tried it.  Some of the strings students went and dragged their instruments into our classroom.  At first we couldn't stretch the string tight enough to just do as in the movie.  But eventually we were able to replicate Donald's experience.

Students at this time of the year are darn hard to engage nevermind impress.  This old, old movie did it.  Now the kids are totally pumped to learn about the 2nd most famous irrational number.  In our class opinion, pi takes 1st place and phi takes 2nd place.

It will be fun to see what kinds of drawings they are able to make of the nautilus shell.  The best part of all this is that learning is fun and it is setting them up for their next unit at the start of 7th grade.

Math Wars Aside....It Takes a Great Teacher

Rarely do I read something that urges me to say "amen" but in reading Ramblings of a Math Mom today, I wanted to cheer.  The author spoke to the issue that teaching math is hard and it requires a masterful teacher. 

I have other friends whose school does an awesome job with Everyday Math, though I realize that that appears to be the exception and not the rule with that curriculum. What is key is great teachers, in either case. But that is all I am saying about particular curricula.....

While many teachers can and do follow the script provided to them by their adopted instructional text, it takes someone with vision and knowledge to make that conversation come alive.  It is SO not about which text you adopt....and it is SO about the expertise that the teacher brings to the table.

I can tell you that this is true of Connected Math Program.  I use this curriculum in my room and it has taken me years to get really good at it....to make it seem fun and easy....to instill the commitment to work hard even though you don't start off well.....to know the connections to help students make...to know the patterns that should be uncovered and how to lead them to finding those....and on and on and on.  It is the years of experience and thought that has created this environment and the capacity I now have to be good at teaching math.

Teaching isn't for anyone who doesn't want to work hard....who doesn't want to believe that all can do it....that empowering children to believe in their inate ability to do well in math.  And I'm here to tell you that these kinds of teachers don't come along everyday. 

Our country needs to wake up and figure out that we need to be more rigorous in the preparation of teahcers.  Then we need to figure out how to strengthen them once they are in the classroom....and I'm not talking about the typical mentoring programs.  yes, you do need to figure out classroom managment but it is so much bigger than that.  It needs to be about falling in love with mathematics.  It's about sharing that love of the content with children in powerful, compelling ways that make them interested and excited to work hard and to uncover the secrets/patterns that await them.

First of several Pi or Pie Days

With only 3 days left before spring break, we thought it perfect time to celebrate math.  Heck we've been toiling away in test prep and they've gone about as "fer" as they can go. 

Our 6th graders teamed up today with 8th grade partners to explore that mysterious ratio of pi.  We started off by reading aloud from Sir Circumference....moaning at all the puns and enjoying the language.   Then we saranaded them (yes, lots of rolling eyes but when we realized we weren't quitting we were encouraged to sing verse 2) with Oh Number Pi . 

Next we looked at the phenonmenon by having them use 5 different sizes of cylinders and lids...cutting bits of string for the circumference and the diameter.  Comparing how many diameters it took to make one circumference.  Here was a place the 8th graders could help get things taped into their math journals.  Download circumference_activity.doc

Then we worked with Sticky Pi.  It's a very cool activity.  You get rolls of 1centimeter graph paper...so the paper is about 3 feet wide.  They create the independent axis by tracing the diameters of many containers.  Then the dependent variable, circumference, goes on the y axis.  Students showed circumference by taking masking tape and cutting a piece of tape that fit around the container...then they aligned that tape with the end of their diameter.  Amazingly, they showed a linear relationship...lining up.  This activity was fabulous because the 6th graders could benefit from seeing how it worked everytime and the 8th graders appreciated the nuance of the fact that the rate of change was about 3+ everytime.

The 43 minutes was far too short.

I liked collaborative 8th grade partners.  They really pitched in and helped our 6th graders...and I wasn't sure how that would go with almost 150 kids in the cafeteria all working on the same activity.  It was great.

Tomorrow is PIE day.  That will be great, too....much less waist friendly...but fun nevertheless.  We saved the rest of the read aloud to eat with our pie.

World Maths Day

Maths...not math.  Be clear about that.  It was the smallest of things that hooked my students into wanting to know more.

More.  That's a mild understatement.  World Maths Day was March 5, 2008 and we participated in a world wide celebration where students did math problems.  Not terribly fancy, but highly engaging.  Simple concept.  Students from around the world answered 182,455, 169 questions correctly on 3/5/08.  That's a lota math questions and it gives you a scope of what this was all about.

Student logs on with user id I created for them and is matched with 1 or 2 other students.  These students could be from the UK or India or Malaysia or ANYWHERE.  A big world map scans for other student at your level and then zooms in to match you up.  Feels like a roulette wheel and adds to the fun of the drill and practice.  When we were practicing, there was about 11,000 Mathletes online at any given time....as I've gone back and checked the website I still find anywhere from 2,000 to 3,000 of them online practicing.

Then it's off....answering questions around the four biggies.  I mean addition, subtraction, multiplication and division of course.

My students answered 12,000+ during the warm up phase.  I had students from other teachers' classes asking me to create a log in for them so they could participate.  So it was definitely infectious.  Then when the calendar turned March 5th somewhere in the world the contest was on.  And stayed on until it was no longer march 5th anywhere in the world.

The class results page says that they answered 2, 173 questions correctly on March5th and that there was a 67.9% improvement in # of correct answers over the course of the day.  Honestly we would have answered more questions but we had internet connection issues and couldn't get online for a big chunk of the day.  One student improved 225%!!!!  boy that's a jump....makes you wonder how bad they were when they started!!!!

The big ah-ha for me was their enthusiasm to do rote math drills.  They loved it and they loved finding out who was their competition.  I heard them talking in halls, at lunch and after school about that "one kid from Australia" or "UB from the UK".  They had gotten to know some of the names and looked forward to matching up with them. 

Here's the simple power of the internet.  Take something we already know is important to do, something that is low interest for students on worksheets and turn it into something that is high interest and self-motivating.  I loved this.  Wish there was another maths day coming soon.

Would you dare to draw these shapes?

We're in the process of finishing up our algebra unit and, while I will continuously revisit these ideas with warm up problems, I am thinking about the scale factor and geometry unit that follows.  I have always wondered how to engage that artistic side with the math.  I found this "How To" article and think it may have some promise for my 6th graders.

What if they used this as their model and then created their own "How To"?

How to Draw an Impossible Triangle

from wikiHow - The How to Manual That You Can Edit

The "rule of three", where arrangements of triplets have a pleasing effect on the eye, makes this triangle an intriguing shape to ponder and to create. It appears frequently in the art of MC Escher. It is also known as a Penrose triangle or tribar.

Steps

1. Sketch an equilateral triangle. This will be the center of your triangle.
2. Lightly sketch two parallel lines outside one side of the triangle. The lines should be equally spaced. Take caution that your lines are drawn straight.
3. Do this for each of the other two sides. Your sketch should look like three triangles nested together.
4. Choose one side of the "center" triangle. Extend one end of that straight line until it reaches the "middle" triangle.
5. Find the same side of the "middle" triangle. Extend one end of that straight line, in the same direction as before, until it reaches the "outside" triangle.
6. Repeat steps for the other two sides of the triangle.
7. Erase short segments so that the triangle begins to look three-dimensional rather than flat. Each edge of this "3-D" shape should look like a reverse "L".
8. Add short segments at an angle in the corners. These short segments will finish off the outside points.
9. Cleanup your drawing by erasing the points outside of the short segments drawn in the previous step.
10. Add shading if desired.

Tips

• After you learn this basic optical illusion, you can experiment with more complex arrangements.

Related wikiHows

• How to Draw an Impossible Cube
• How to Draw an Impossible Cube with a Highlighter
• How to Draw an Impossible Cube Using a Pencil
• How to Draw the Triforce

Article provided by wikiHow, a collaborative writing project to build the world's largest, highest quality how-to manual. Please edit this article and find author credits at the original wikiHow article on How to Draw an Impossible Triangle. All content on wikiHow can be shared under a Creative Commons license.

Dynamic feedback loops OR Formative Assessment in Math & Test Prep

Having spent weeks on reviewing Number Sense, it was time to see how much students knew in a simulated testing situation.  I created a series of 3 - 10 question formatives which covered rational numbers.  I needed that dynamic situation for independent practice...something where I could watch all of them do their problem set but without interacting with them.

Today we dragged the mobile carts into the pod and got started.  As they took their quizzes, I would watch their questions stream into my grid.

"OK, Bill...be sure to check #2 again"....squint a bit more at the screen and realize that everyone was missing #9.  Buggers that means I really need to do more with teaching estimating with percents.  New numbers would post to my screen and I could tell the 2 kids in the back of the room where messing around, clicking through and not showing me what they knew/didn't know.  I walked back there and set them straight...by the time I returned to monitoring my screen they were in "review" mode and changing their answers.

Here is proof to me that dynamic test results can be powerful.  All the groundwork I had invested in teaching the concepts, having them drill on the concepts, allow a little time to go by and let it sink into long term memory (or escape) and blam.  See what they remembered and what they could still do.  The capability to watch their scores pour in, question by question, is remarkable.

I thought they were getting messed up on estimation problems.  Turns out I was all wrong.  They were missing the multi-step problems.  Without instant results, I would have operated from the wrong assumption and addressed a need they didn't have.  This is powerful stuff.  It is a feedback loop that can change how kids feel about math...

If I know what they "get" and "don't get", I have a much better chance at helping them learn the tough stuff.

It will be interesting to compare tomorrow's results with today's.  I will be watching to see if they soar and falter in the same places they did today.  Whatever the outcome, it will inform my decision about what to teach the next day.  That's just cool and humbles me to think about the potential we have together because of the leveraged knowledge I have with technology.

Header/footer titles for math graphs in SmartBoard--my new discovery

We are working our way through understanding when to use a graph or a table.  Which kinds of questions are better answered with tables and which are better for a graph.  That is tough.

Today we took data presented from one source as a table and another source as a graph.  Students tried to make comparisons and found out how hard this was to do.  Impossible many of them would probably say.   To remedy that problem, we added another row to the table and interpreted the graph's points to fill in comparable data points in the table...and then added the table data to the graph.

OK...so none of this is unusual and is the textbook activity.  Today we used the SmartBoard (I didn't have this tool last year when we did this unit).  So it was terrific watching kids come up and do all this...they worked so collaboratively to help the person at the board get everything just right.

I have been saving the class work as .pdf files for a bit.  But today I learned that you can actually add a title and footer to each of those pages.  Well, that's pretty darn cool.  I was able to annotate which lesson it went with in their book and then posted it to our class website.  I think this kind of description will help them find it .

So it's not the most revolutionary discovery I've ever made.  Usually I think it's the little things that make big differences for your kids.  This is one of those kinds...most people will just roll their eyes at my hurray...but it's a big deal.

Math Procedures---breaking it down and lots of practice

Today I worked with students on ordering fractions.  We studied fractions, decimals and percents all fall semester.  Amazingly they still seem to remember some of the things we learned back then.

Our state standards call for them to order fractions (with different denominators).  First there is the hurdle of paying attention to whether problem wants you to order from least to greatest OR greatest to least.  READING.  Not math determines part of the score here.  So we practice reading!!!!!

OK...now that we know which way we'll go, you have to find that pesky common denominator.  Sure there are some easy combinations of 4 or 5 fraction denominators.  We've tried many methods...and I still think showing multiple ways is good since not everyone takes to the same method.  Whatever approach, it's many steps.

Finally, you have the ability to find equivalent fractions with common denominators and order them.  Phew....

Lots of steps.  Lots of opportunity for mistakes.  Lots of things to remember.

It makes me reflect on the fact that it just takes time.  With liberal doses of patience.  I think I'm finally getting better at teaching this with the realization that you have to break it down into small enough pieces that they can grasp it.  Then practice those pieces over and over again until they build some automaticness to doing the process.  It really helps, too, if you teach them to "debug" their erros along the way so they know how to spot mistakes in the process.  That just takes experience and lots of dialog.

I guess no one became a basketball star without tons of hard work.  It is the same for mathematicians...lots of repetition, talking about it, practicing multiple ways to arrive at the same place, and repetition.  Oh, I said that.  But it takes repetition!!!!!