Reflections of a Techie

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.

Failed public policy---where's the teacher expertise that is need to save our schools?

To tell you the truth, my head's been spinning with ideas.  I've spent two days with some of the best teachers in my state...Kansas.  Award winners, new teachers and preservice teachers...we had them all.  The first day of the conference was pretty much...ho-hum.

The second day I called Fabulous Friday.  We had roundtable discussions of what public education should look like, what reforms we saw as critical, how we thought we might link today's schools to tomorrow and so on. 

Yes there was the sentimental videos with wide-eyed children and the typical encouragement to do more for the good of the kids.  But on the whole, there was much more focus to what it is that we need to do to improve learning....where it is the future of our profession lies...and what role teachers need to step into in order for things to change.

What was clear is that these expert teachers don't believe that policymakers have been successful with their vision of improvement ala NCLB.  It is more of a failed policy than success.  If nothing else NCLB helped teachers wake up to the fact (and probably begin to accept the fact) that accountability is here to stay, it can be a good thing and that if you let people who don't understand the complexities of the education business dictate how you will conduct your classroom....well, chaos will ensue. 

There's big problems in Kansas.  Teacher salaries are in the basement.  They need and want more.  It doesn't take an Einstein to see what impact low salaries are having....  A recent survey of Kansas school needs breaks down like this....we are going to need 375 math and 307 science teachers next year.  With the decrease in enrollment in the state colleges and universities, we will only have 115 math and 63 science teacher graduates to fill those vacancies.  There is an 86% decline in teaching licenses issued to biology teachers, a 50% decline in chemistry licenses and a 67% decline in physic licenses.  Students in these fields are going elsewhere and money is the big factor.

Coinciding with these new graduate numbers is the aging of our statewide faculty.  36% will be eligible to retire with in the next 5 years and about 42% leave the profession within 7 years of starting.  The ed schools have experienced a 25% decrease in enrollment.  Money and lack of support in the job is cited as the major reasons.

Teachers have lots of great ideas for ways to recruit more and retain more.  We've tried it the way the Kansas legislators think works and see the results.  No money.  No teachers.  That seems like another example of failed public policy.  Legislators can whine all they want to that teachers are overpaid and continue to cut off reasonable pay increases.  If they haven't figured it out yet, they will soon enough when there are severe shortages in schools and there aren't teachers for their children.  Does it have to hit them personally with a daughter, a son, a niece or neighbor before they "get it"?  I have always hoped that legislators have bigger vision than that...but it sure doesn't seem like.  Come on....suck it up and find a way to pay teachers for their work in a way that encourages people to go into the profession instead of running the other way. 

These conversations reinforced my belief that we need teacher leaders to step up and learn how to enter to policy arena.  They need to figure out to show that accountability doesn't scare us and that we can prove what we do makes a difference.

Student generated work---Properties of Water

One of my students took on the challenging of conveying what he'd learned about the properties of fresh and salt water using Voicethread.  What I really like about his work is that it is well written, has his voice and that he cited his sources.  He did a terrific job.  See what you think.

Blabberize.com! - Blabberizing Math and Science

Link: Blabberize.com! - Blabberize Your Pictures.   

Maybe this is the cool way to present converting fractions to decimals to percents.  AGAIN.  They'll be rolling in the isles and it may even top the Molecules In Motion Teacher Tube video.

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.

Tying History Inquiry to Earth Science---It's the Droughts that Did It!!!!

In the middle of my Earth's Water unit I knew I needed some more punch...something to perk student interests in the ever present water cycle (how many times does it appear in the curriculum???? a million????) and ocean chemistry.  Climate change current events is everywhere so I knew that something could pique the interests of my students.

Sure enough it was a book about Droughts.....sounds pretty exciting doesn't it.  Well, the topic is even if a whole book isn't.  I knew that they were especially interested in the Dust Bowl.  Heck many of their grandparents remember  the 1930s.  So I thought I utilize that as my hook.

I created another Voicethread that merged our science content with historical pieces from the Kansas Historical Society.  They went nuts reading the newspaper article accounts of dust storm encounters.  They were amazed at the photographs that are available.  In a little over a week they left over 100 comments about what they were seeing and what interested them.

Take a look at the work they created and the comments they left.

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.

Drawing experts into science class--f2f, Skype, Google Groups???? So many options, so little experience.

One of my students has offered to loan us an expert....her dad, of course.  He's a hydrologist and we're right in the middle of studying Earth's Waters.  How cool would that be if he could really do this or wanted to?

I've been thinking about ways beyond the traditional have him come in to talk to us...and while that would be totally cool...what would be the advantages of setting up a group conversation or maybe having him visit several times via Skype.

To my way of thinking, it might be very much more engaging for students to have a chance to mull over questions, do some research/thinking on their own and then write emails back and forth.  I guess the danger of that is the flood of emails that might occur....and the strain of providing thoughtful responses to 60+ kids!!!!  I've never setup a Google Group, but it can't be too hard.  I wonder about the security (beyond the need not to reveal who they actually are)...does it pose any dangers that I can't think of right now????

I really like the idea of seeing if he might virtually visit us for a half hour a couple of times.  We could still prepare our questions...maybe even send them to him in advance...and then he could guest speak without having to take off from his work and come to school.  They are just getting ready to start their independent research activities and expert insights would be very welcome...

Since today is a snow day, we weren't at school.  So I'm hoping we go back tomorrow, my student will have talked with her dad and I might know if he's up for something.  Keep your fingers crossed.

Super Tuesday--caucus was inspiring and nuts

I have strong political views but don't think this is my forum for talking about them.  Until last night. I still won't express specifics but what I witnessed last night was a transformation of our citizenry...from apathetic, I don't want to even talk about it....to this is my country too and I want to participate.

Oh, my gosh.

I was told that I could go to the Kansas caucuses last night...swing by about 6:10pm, stand up for my candidate and be gone in about 30-45 minutes.  Boy was that wrong, wrong, wrong.

I drove over the church where the caucus was being held and traffic at 6 was already piling up.  So I parked my car at the first place I could find and just walked.  I got in this long, long line that wound around from the back of the church, across the front lawn and into the parking lot.  It was there I stood for almost 1 1/2 hours waiting in the wind, snow flurries and cold just to get it.  Finally the police showed up and were overwhelmed with what to do...barricades to control the traffic, flares so people didn't get hit crossing the street and directing traffic.  They parked every where for a mile diameter around this church.

The line was a blast.  Of course I was there to support my candidate, but everyone was in a jovial mood....talking about why their choice was THE choice.  One lady standing near me was talking about how she had been to every caucus since the 1950s.  The amazing thing she said was that typically about 200-300 people show up. 

Not last night...she said that she hadn't seen this many people since the crowds came out to support JFK!!!!   JFK?????   JFK???????  That's a long time ago.  But there they were.  No one was leaving despite the traffic jams, the hours long wait, the disorganization or anything.  Everyone was talking about how much hope they had that something COULD be different and that they wanted to participate in our government.  That sentiment was expressed by every candidate's supporters.

All walks of life were there...young moms holding little kids' hands, grandmas, young couples, kids with high school letter jackets, grey hair, punk hair, no hair....they just seemed to want to be a part of repairing our country.  Infusing it with new energy.  Restoring what most thought we had lost over the past years.

After I had been inside for about another 1 1/2 hours, they announced the fire marshal had closed this building and they still had more people.  Apparently someone called the rabbi of the synagogue across the street and asked if they would open it up to people.  Now the line was wrapped around 2 places...the church and the synagogue. Even though it was so long, all the people around me kept mentioning how great it was to live in a place where you didn't have to worry about getting killed for voting or your house/business blown up for having a political view.  Thankfulness trumped impatience.   About 9:45 (more than 3 hours after I arrived) we were able vote for our candidate.

I got home in enough time to see the 10:00 news.  The caucus experience I had was repeated again and again around the whole Kansas City metropolitan area....some caucuses ran out of inside room so they move their crowds outside IN THE SNOW and COLD.  People just didn't leave. 

I don't think I've been so anything as politically inspirational as this since they decked out Boston for the Bi-Centennial 1976.   1976 is a long time ago.  Overwhelmingly the people in crowd were there to support Obama. By huge margins....4 to 1 easily. The spirit of hopefulness and pride in our country was pervasive in that long, cold line last night.

As I listened to the national news roundup, there was no mention of this phenomenon.  Hey...I know we're from Kansas and no one cares. We're not a big electoral state...we're not California, New York or Florida. Too bad because what I witnessed was huge. I'll bet the networks don't report this kind of thing (because you know it isn't limited to just here) because it isn't what the "experts" predicted.  Instead it people taking control of their future...and how do you analyze that????

It is the heart of what makes an American citizen special and I do believe it's more universal to our country.  I hope it was repeated (and not reported) again and again in all the Super Tuesday states.  I hope Kansas wasn't the only place in our country that our spirit is resurfacing.  Trust.  Hope.  Civic participation.

Very cool.

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!!!!!