Looking Back & Looking Forward

Looking Back...

One of my main goals this past school year was to have my Geometry students see math from multiple perspectives.  This goal was a direct result of Jo Boaler's "How to Learn Math" MOOC.  The fact that very simple problems can be seen from multiple perspectives is one aspect of mathematics that I truly love and my hope was to instill that love, or at least an appreciation, with my students during the last school year. My second goal was to have my students see the value of mistakes - that mistakes are part of learning.  Not all student answers are perfect, but we can learn from all student answers.

So...how did I do?  I asked students at the beginning of the year to use 2 words to describe math and I had them do that again at the end of the year.  The first Wordle was based on the words students used from the beginning of the year and the second Wordle was based on the words students used from the end of the year. The larger the words, the more the students used that word. (Note: There are some interesting words that only a few students used.  I recommend zooming in on the pic to see them.)

Beginning of School Year	What are the "Top 5" words students used to describe math prior to the school year starting? 1) interesting 2) challenging 3) complex 4) fun 5) ubiquitous
End of School Year	What are the "Top 5" words students used at the end of the school year? 1) useful 2) challenging 3) interesting 4) fun 5) ubiquitous

It is interesting to note that not many words in the "Top 5" list changed.  Although some students still found math "confusing" or "intimidating" at the end of the school year, there are many new words in this list had including, rewarding, elegant, stimulating and exhilirating.  Not all of my students loved math like I did, but there was more of an appreciation of its power at the end of the school year.

But what about my orignial goals: math from multiple perspectives and students valuing mistakes for what can be learned from the mistakes.  I asked about this in a google form and although some students said simply "Yes" or "Sure", others gave more detailed responses.

One student said the following: I think that mistakes were also seen as ways to learn in the class, but not personally because I hate making mistakes. I do realize that I should learn from them more now, from this class.  Embracing mistakes is something that I still personally find challenging and I can relate to the struggles of this student.

Another student said: Viewing math from different perspectives not only showed us the many dimensions of mathematics, but also helped those who may learn differently than others. By looking at math in different ways we grow to appreciate the different ways of thinking of our classmates. And here I wanted my students to see math from multiple perspectives to help them understand the beauty and connections within mathematics.  But this student gets that different perspectives are important because (drumroll, please) not all students learn the same way!  I know this, but didn't think of this as a reason for pursuing the goal of multiple perspectives.

Looking Forward...

In the fall, I will continue to teach Honors level Geometry.  But I will also have a section of College Prep Geometry.  How do I get this group of students to undertand that there is value for them in seeing math from multiple perspectives?  How will I get them to embrace mistakes as a form of learning?  Of course I can do some of the same things I did last year, but these students haven't had as much success with math and are less enamored with the subject.

What will I try? We need something that is accessible to all students, but can challenge those students who are ready for higher order thinking. Enter Low Floor High Ceiling tasks, some of which can be found at youcubed.org and others at nrich.maths.org.  In the next week or two, I have the opportunity to try out a problem or two in a summer transitions program. Two tasks I am considering using are What's the Secret Code? and Beelines. Beelines seems quite challening, but I like the visual aspect of this problem and the ability to link it to GeoGebra or keep it simple with paper and pencil. Here is a video about the Beelines problem.  Looking forward to hearing the Buzzzzzing of voices as students tackle this one.

Around the World: Surprise! (Part 2)

Before you read part 2 of this blog, I suggest you read part 1.  The problem I gave my students was as follows:

Imagine that the Earth is a perfect sphere and that a metal wire is snugly wrapped around its equator. Now imagine that we cut this wire in one spot and splice in an additional 100 meters of wire.  We take up the slack by using posts to raise the wire an equal distance all the way around the Earth.  How high above the surface of the earth will the wire be?

Students worked together to find an answer of about 15.9 meters.  Groups had to explain their solution to me before they could continue on to the next phase - finding the solution for a second spherical object.  

On the front board I had written the following:

1 - baseball

2 - moon

3 - Jupiter

4 - beach ball

5 - basketball

Groups selected an index card at random and were assigned the object based on what they selected.  They needed to research the dimensions for their particular object and do the calculations again.  They submitted their results through a google form and when I displayed the results to the entire class, this is what they saw. (Note: These results are for all 3 of my classes combined.)

WHAT????  

No matter how big the object, the wire would always be 15.9 meters above its surface. 

Wait...WHAT????  But, why???

That was the reaction I was hoping for.  But these are the reactions that I got.

Period A - Huh. Interseting, I guess. (I think they were still on spring break.  It was Monday and 8:20 AM.)

Period B - That's cool.  Why does that work?

Period F - We are not surprised at all.  You do stuff like that with us all the time.  

So...now you may be asking Wait...WHAT Why does this work?  The derivation I did with one of the classes appears below.  The reason it works is because there is a direct linear relationship between the radius of an object and its circumference.  (Students know this direct linear relationship as C = 2*pi*r.) When the circumference increases by x, the radius increases by (x/2) divided by pi.

I love problems like this that are a SURPRISE to students and go against what student think should actually happen.  I would love to have more problems like this to share with my students.  If you have any problems like this, please send them my way.

Around the World: Surprise! (Part 1)

Back in the late 90's I worked with The Math Forum Problems of the Week. (POWs) Specifically I worked on the Trig/Calculus POW
and the Discrete Math POW.  Submissions came in from around the globe and I enjoyed seeing the variety of methods students used to solve the problems. There was even one time when a student used calculus to prove she found the minimum number of moves needed to solve the discrete math problem for that week!

This past week I was on spring break and I pulled out a wonderful GeoPOW problem from my files.  I plan to use it on Monday, because we often miss students on the first day back from spring break due to families extending their spring break.  Rather than having several students miss new material, I thought it would be fun to look at this surprising problem.  In the GeoPOW files, this problem was called "All Around the World".

Imagine that the Earth is a perfect sphere*, and that a metal wire is snugly wrapped around its equator. Now imagine that we cut this wire in one spot and splice in an additional 100 meters of wire.  We take up the slack by using posts to raise the wire an equal distance all the way around the Earth.  How high above the surface of the earth will the wire be?

*many students are surprised to learn that the earth is not a perfect sphere*

Collaboration is a big part of learning math in my classroom; students will work on this problem in groups.  After each group gives me an explanation that convinces me that they have the correct answer (Note: The answer is about 15.9 meters.), the group will reach into a hat and pull out a slip of paper that will have one of the following items written on it: basketball, moon, Jupiter, beach ball or Mars.  The groups will then repeat the problem that they just did with the Earth, but with their new spherical object.  After students have finished the problem, they will go to the tinyurl written at the front of the room to enter data into the following form:

After students click the Submit button, they will see the message "We will review the results from the class shortly."  The surprise will come when I turn off the AV Mute button on my projector and we see the results at the same time!  Come back for part 2 of this blog when I share how the lesson went and my students' reactions.  But before then, do the math and predict why my students will be surprised.