RAFT Activity Kit: Static Merry-go-Round

Monday, September 28, 2015

It’s Okay to Make Mistakes!



By Jeanne Lazzarini, Math Master Educator/R&D Specialist, RAFT

I have often shared with my students that I make mistakes, and I have learned so much because of them! Sometimes I even purposefully made a mistake in a math lesson to see if students take notice! Let your students know it is okay to make mistakes, and when you do, your brain is developing new insights, new ways of thinking, and bursts of conceptual understanding!  





From an early age many of us are taught that it’s bad to make mistakes, to fear failures, and to avoid them all costs. However, the truth is that failure and making mistakes are a necessary part of growing up and of being successful and should never be avoided! 

So, you might ask, how do I encourage students to feel okay about making mistakes?  Talk with them about mistakes and failures, including:
·         Have students investigate “famous” people who have made mistakes, then share them with the class!  They’ll be very surprised at these stories of success from failures!  (see:  http://www.onlinecollege.org/2010/02/16/50-famously-successful-people-who-failed-at-first/ )
·         Encourage alternate ways of expressing thoughts; verbally, written, artistically, acted out, or whatever. Even if that thought is off-target, it often leads to other ideas that may not have otherwise been discovered!
·         Failure and mistakes teach us an approach may not be right for a particular solution, but opens the door to investigating alternate approaches.
·         Inspire stepping out of a “comfort zone” and trying something new! This leads to new insights and self-realization!  And each time you fail, your fear of failure becomes smaller, allowing you to take on bigger challenges!
·         Each failure brings you closer to your goals and makes you stronger and better.  This brings to mind the saying “Nothing ventured, nothing gained”….
·         Learn from your mistakes by thinking about where you can go beyond them to get better.  You will never fail as long as you
never give up! 
·         All “successful” people have failed and understand the value of not giving up! 
·         Research shows when students make mistakes, brains grow!


So, it is good to make mistakes, and it is very important to talk about this with your students! Share examples, encourage alternate ways of thinking through a problem, and you’ll see students blossom with a new enthusiasm for learning!

Tuesday, September 15, 2015

How does math relate to real life?


By Jeanne Lazzarini, Math Master Educator/R&D Specialist, RAFT

How does math relate to real life?  One way is to take a look at the shape of a cloud, a mountain, a coastline, or a tree!  You might be surprised to find that many patterns in nature, called fractals, including growth patterns, have very peculiar mathematical properties ---  even though these natural shapes are not perfect spheres, circles, cones, triangles, or even straight lines! 

3D Fractals For Inspiration

 
So, what is a fractal?  Benoit Mandelbrot (November 20, 1924 – October 14, 2010) is commonly called the father of fractals. He created the term “fractal” to describe curves, surfaces and objects that have some very peculiar properties. A fractal is a geometric shape which is both self-similar and has fractional dimension.  

Daydreaming fractals


Ok, so what does that mean?  Well, “self-similar” means that when you magnify an object, each of its smaller parts still look much the same as the larger whole part. And, “fractal dimension” is different from what we use to describe shapes such as lines, flat objects, and geometric solids.  Simple curves, such as lines, have one dimension.  Squares, rectangles, circles, polygons, etc. have two dimensions, while solid objects such as cubes and polyhedra, have three dimensions.  Some say time is the fourth dimension.  In all these cases, dimension, based on Euclidean Geometry, is described as an integer: 1, 2, 3, 4, … 
 
But a fractal curve could have a dimensionality of 1.4332, for example, rather than 1!  A fractal’s dimension indicates its degree of detail, or crinkliness and how much space it occupies between the Euclidean Geometric dimensions.  Most objects in nature aren’t formed of squares or triangles, but of more complex fractal shapes, such as ferns, flowers, coastlines, clouds, leaves, trees, mountains, blood vessels, broccoli, weather, lightening, fluid flow, river estuaries, circulatory systems, geologic activity, fault patterns, planetary orbits, animal group behavior, music, and so forth. 

Whew! By understanding fractal dimension, mathematicians can now measure forms that once were thought to be immeasurable!   


Romanesco broccoli fractals

Have fun discovering “fractals” with RAFT’s “Freaky Fractals” activity kit!  Use the kit to create a fractal shape resembling “arteries”, “coral”, “a heart”, “a brain”, “tree branches”, etc. Then go to the store, buy some broccoli or cauliflower, then take a  close look! Break off a branch and what do you see?  The smaller branch looks just like a miniature copy of the whole vegetable!  Now look around you and you’ll notice thousands of living examples of self-similarity in ferns, coastlines, clouds, leaf veins, trees, and the formation of shells, mountains, blood vessels, lightening, river estuaries, circulatory systems, fault patterns, galaxies, musical compositions, and so forth!  By understanding fractal dimension, mathematicians can now measure shapes, such as coastlines and so forth that once were thought to be immeasurable! Fractals are AWESOME!  Math really is all around you when you stop to look!